The Human A Priori: Essays on How We Make Sense in Philosophy, Ethics, and Mathematics 0192871412, 9780192871411

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The Human A Priori

The Human A Priori Essays on How We Make Sense in Philosophy, Ethics, and Mathematics A. W. MOORE

Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © A. W. Moore 2023 The moral rights of the author have been asserted All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2023930736 ISBN 978–0–19–287141–1 DOI: 10.1093/oso/9780192871411.001.0001 Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work.

For Andrew

Contents Preface Publisher’s Acknowledgements

ix xi

Introduction

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PART I. THE NATURE, SCOPE, AND LIMITS OF A PRIORI SENSE-MAKING 1. Armchair Knowledge: Some Kantian Reflections (2023)

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2. On the Necessity of the Categories (written jointly with Anil Gomes and Andrew Stephenson, 2022)

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3. What Descartes Ought to have Thought about Modality (2019) and Postscript

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4. Varieties of Sense-Making (2013)

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P A R T I I . H O W WE M A K E S E N S E I N P H I L O S O P H Y 5. Sense-Making from a Human Point of View (2017)

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6. Not to be Taken at Face Value (2009)

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7. Carving at the Joints (2012)

127

8. The Concern with Truth, Sense, et al.—Androcentric or Anthropocentric? (2020)

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PART III. HOW WE M AKE S ENSE IN ETHICS 9. A Kantian View of Moral Luck (1990)

149

10. On There Being Nothing Else to Think, or Want, or Do (1996)

171

11. Conative Transcendental Arguments and the Question Whether There Can Be External Reasons (1999)

189

12. Maxims and Thick Ethical Concepts (2006)

210

13. Quasi-Realism and Relativism (2002)

226

14. From a Point of View (2012)

233

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15. Williams, Nietzsche, and the Meaninglessness of Immortality (2006)

241

P A RT I V . HO W W E M A K E S E N S E I N M A T HEM A T I C S 16. On the Right Track (2003)

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17. Wittgenstein and Infinity (2011)

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18. Wittgenstein’s Later Philosophy of Mathematics (2017)

291

19. A Problem for Intuitionism: The Apparent Possibility of Performing Infinitely Many Tasks in a Finite Time (1989–90)

306

20. More on ‘The Philosophical Significance of Gödel’s Theorem’ (1999)

320

Bibliography Index

337 353

Preface These essays are reprinted with relatively minor amendments. Many of the amendments are purely cosmetic. Some, such as the addition of some crossreferences and the introduction of some standardization, are for the sake of the volume. In a few cases I have corrected what I now see as simple philosophical or exegetical mistakes. I have made no attempt to eliminate repetition from one essay to another: this is partly to accentuate interconnections between the essays, partly to ensure that each essay remains self-contained. As far as the interconnections are concerned, I shall try to elucidate these in the Introduction. Three cases deserve special comment. Essay 2 is co-authored. It arose from an exegetical disagreement about Kant that I found I had with two friends and former students, Anil Gomes and Andrew Stephenson. The disagreement came to light during a course on Kant that Anil and I gave at the Oxford University Department for Continuing Education. Not only had each of us previously been unaware of this disagreement; each of us would previously have been inclined to regard the matter as uncontentious. Anil discussed our disagreement with Andrew, whose position was the same as his, and before long the three of us became embroiled in a fascinating trialogue in which we came to appreciate that the matter was both exegetically and philosophically much less straightforward than any of us had previously thought. Anil and Andrew were prompted to write a joint essay in defence of their position. After I had read their essay, and after we had engaged in further discussion of the issues, it evolved into what appears here, which is to say an essay by all three of us in which we moot an intermediate position that had not originally been on any of our radars—a real case of thesis, antithesis, and synthesis. I am very grateful to Anil and Andrew both for the stimulation provided by working on this essay together and for their permission to reproduce it in this volume. For reasons that I shall try to clarify in the Introduction, it very nicely captures one of the main threads that links the whole volume together. Essay 3 has a new postscript. This essay was originally written for a conference to mark the twentieth anniversary of the publication of Jim Conant’s wonderful essay ‘The Search for Logically Alien Thought: Descartes, Kant, Frege, and the Tractatus’. It engages with Jim’s discussion in that essay of Descartes. In the volume that grew out of the conference, which is where my own essay first appeared, there is a response by Jim. The purpose of my postscript is to correct a basic misunderstanding of my position on Jim’s part (albeit a misunderstanding that occurs within the context of yet further wonderful work in which he both develops and disrupts some of the main contentions of his own original essay).

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As it happens, for reasons that I shall again try to clarify in the Introduction, this material too very nicely captures the thread that I mentioned above that links this volume together. Finally, Essay 10 is the essay that has undergone the greatest revision. When I returned to it to consider it for inclusion in this volume I found much to dissatisfy me. Its imperfections were due partly to the fact that it ended with a statement of ideas which, though I had already defended them and would go on to develop them elsewhere, I did no more than state in this context. The result was hurried and—I now realize—bemusing. The revised version is a little less hurried, and I hope a little less bemusing. This was not however the only source of Essay 10’s imperfections. Another was that, so far from capturing the thread to which I have referred, it cut across it. This too is something that I shall try to clarify in the Introduction. But I have made no attempt to remove this imperfection since, in its own way, it helps to draw attention to that thread. In fact I shall use some of what is at issue here to structure the Introduction and to indicate what the thread is. I thank Peter Momtchiloff, philosophy editor at Oxford University Press, for his advice, encouragement, and support. I also thank the editors and publishers of the volumes in which these essays first appeared for permission to reprint them.

Publisher’s Acknowledgements Essay 2, ‘The Necessity of the Categories’, written jointly with Anil Gomes and Andrew Stephenson, was originally published in The Philosophical Review, 131 (2022): 129–68. Essay 3, ‘What Descartes Ought to Have Thought About Modality’, was originally published in Sofia Miguens (ed.), The Logical Alien: Conant and His Critics (Harvard UP 2019). Copyright © 2020 by the President and Fellows of Harvard College. Used by permission. All rights reserved. Essay 4, ‘Varieties of Sense-Making’, was originally published in Midwest Studies in Philosophy, 37 (2013): 1–10. Essay 5, ‘Sense-Making from a Human Point of View’, was originally published in Giuseppina d’Oro and Søren Overgaard (eds), The Cambridge Companion to Philosophical Methodology (Cambridge UP 2017): 44–55. Essay 6, ‘Not to be Taken at Face Value’, was originally published in Analysis, 69/1 (2009): 116–125. Essay 7, ‘Carving at The Joints’, was originally published in the London Review of Books, 34/16 (30 August 2012): 21–23. Essay 8, ‘The Concern With Truth, Sense, et al.—Androcentric or Anthropocentric?’, was originally published in Angelaki 25/1–2 (2020). Essay 9, ‘A Kantian View of Moral Luck’, was originally published in Philosophy, Vol. 65, no. 253 (1990): 297–321. Essay 10, ‘On There Being Nothing Else to Think, or Want, or Do’, was originally published in Sabina Lovibond and S. G. Williams (eds), Essays for David Wiggins: Identity, Truth and Value (Blackwell 1996): 165–84. Essay 11, ‘Conative Transcendental Arguments and the Question Whether There Can Be External Reasons’, was originally published in Robert Stern (ed.), Transcendental Arguments: Problems and Prospects (OUP 1999): 271–92. Essay 12, ‘Maxims and Thick Ethical Concepts’, was originally published in Ratio, 19 (2006): 129–147. Essay 13, ‘Quasi-Realism and Relativism’, was originally published in Philosophy and Phenomenological Research, Vol. 65, No. 1 (2002): 150–56. Essay 14, ‘From a Point of View’, was originally published in Philosophical Quarterly, Vol. 62, No. 247 (April 2012): 392–8.

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Essay 15, ‘Williams, Nietzsche, and the Meaninglessness of Immortality’, was originally published in Mind, Volume 115, Issue 458 (2006): 311–30. Essay 16, ‘On the Right Track’, was originally published in Mind, Volume 112, Issue 446 (2003): 307–22. Essay 17, ‘Wittgenstein and Infinity’, was originally published in Oskari Kuusela and Marie McGinn (eds), The Oxford Handbook of Wittgenstein (Oxford UP 2011): 105–21. Essay 18, ‘Wittgenstein’s Later Philosophy of Mathematics’, was originally published in Hans-Johann Glock and John Hyman (eds), A Companion to Wittgenstein (Blackwell 2017): 319–31. Essay 19, ‘A Problem for Intuitionism: The Apparent Possibility of Performing Infinitely Many Tasks in a Finite Time’, was originally published in Proceedings of the Aristotelian Society 90/1 (1989‒90): 17–34. Essay 20, ‘More on “The Philosophical Significance of Gödel’s Theorem”’, was originally published in Grazer Philosophische Studien, 55/1 (1999): 103–126. Permissions to republish are gratefully acknowledged.

Introduction Part of the rationale for collecting these essays together is that they are all concerned, in one way or another, with the a priori. But there is a more fundamental and more distinctive unifying theme: the essays all reckon, again in one way or another, with what I see as something ineliminably anthropocentric in our systematic pursuit of a priori sense-making. I shall not try to provide a precise definition of the a priori. Given the range of these essays, and given the extent to which their concern with the a priori is a matter of unspoken background presupposition rather than direct engagement, it suits my purposes to allow as much latitude as possible in how the term is to be understood. This includes latitude in how its very domain is to be understood: does the term apply to truths? to states of knowledge? to concepts? to modes of investigation? to justifications for what is believed? possibly even to features of reality? It is largely to accommodate this latitude that I have elected, in this Introduction, to use the blanket term ‘sense-making’ as the complement of ‘a priori’. For ‘sense-making’ can itself be understood in a suitably wide variety of ways. And even if it does not capture all of what has been classified by philosophers as ‘a priori’, its own classification as ‘a priori’ allows for extension to other cases: for instance, a truth may be said to be a priori if it can be known as a result of a priori sense-making. All that really matters, for current purposes, is that if something can be classified as ‘sense-making’, and if it manages to do whatever it is intended to do independently of experience, then it can also be classified as ‘a priori’. Just as I shall refrain from trying to provide a precise definition of the a priori, so too I shall refrain from trying to provide a precise definition of the anthropocentric. Again all that really matters, for current purposes, is that the term indicates what is from a human point of view, and that ‘human’ in turn is to be understood in relation to Homo sapiens. This reference to Homo sapiens might have been thought to go without saying. But it deserves to be made explicit, if only because of a non-biological understanding of the term ‘human’ that we find, at least arguably, and at least sometimes, in Kant. On that understanding the term denotes finite rationality.¹ Interestingly, this makes the concept of the human itself a priori—though, more interestingly still, there is an argument due to Michael

¹ See e.g. Kant (1996a), 4: 428 ff. and Kant (1996d), 6: 26 ff.

The Human A Priori: Essays on How We Make Sense in Philosophy, Ethics, and Mathematics. A. W. Moore, Oxford University Press. © A. W. Moore 2023. DOI: 10.1093/oso/9780192871411.003.0001

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Thompson that the concept of the human is a priori anyway, even when it is understood biologically.² This raises some fascinating issues that are clearly pertinent to what I have identified as the unifying theme of these essays. Even so, I mention them principally to set them aside. For the question whether or not the concept of the human is itself a priori is strictly orthogonal to the question whether or not what is a priori is bound up with the human in the way I am claiming. Either answer to the first question is compatible with either answer to the second. In order to give an initial indication of why I see the connection that I do between the a priori and the human, I am going to present something that I will call ‘the Basic Model’. In the Basic Model, there is some subject S who is in possession of some concept c which is integral to some a priori sense-making that S achieves, but there is also something radically parochial about S’s possession of c. A simple example would be a subject who, by virtue of possessing the concept of a wife, deduces a priori that there are at least as many women and girls as there are wives. The a priority of S’s deduction is in no way compromised by the fact that there is a complicated network of highly contingent social structures and values that support the institution of marriage and that serve as a precondition of any subject’s possessing any such concept in the first place. The Basic Model is therefore already enough to indicate how the a priori can be grounded in the parochial. It is not a huge leap from there to the thought that the a priori can be grounded in peculiarities of an entire species; nor from there to the thought that there can be a priori sense-making that may appropriately be said to be from the point of view of that species; nor from there to the thought that we humans and what accrues from our systematic pursuit of a priori sense-making are a case in point. I mentioned in the Preface that Essay 10 has what I now see as an important imperfection whereby it cuts across one of the main threads that links together this volume as a whole. In the bulk of what follows in this Introduction I shall say a little about each of the essays in the order in which they occur; but first I want to amplify on what I had in mind when I made that comment about Essay 10, and to draw on some related material in Essay 12, since this will help to clarify the Basic Model. Essay 10 is concerned with an idea that occurs in David Wiggins’s work: the idea of there being nothing else to think.³ In that essay I explore a way of construing this idea whereby the claim that there is nothing else to think but that p is equivalent to the claim that it is true that p. This in turn involves the following subsidiary idea: if it is true that p, then anyone who does not think that p pays a price. But what is it not to think that p? It is easy to assume, and in the essay I in effect did assume, that not thinking that p must take one of three forms: ² Thompson (2004). ³ For a fascinating discussion of this idea, and of other related ideas, see Diamond (2019).

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thinking the opposite; being self-consciously agnostic about the matter; or not even considering the matter, possibly not even being in a position to consider it. But even at the time of writing the essay I was aware of what many people, Bernard Williams in particular, would regard as an important fourth possibility. I gesture towards this possibility in footnote 29 of the essay, albeit only to register my disagreement with Williams. However, as I indicate in a parenthesis within that footnote added for the reprint, I have subsequently arrived at a more sympathetic view of what Williams has in mind. To understand what Williams does have in mind we can exploit some of the material that I present in Essay 12. I there offer a distinction between what I call a ‘disengaged’ grasp of a concept and an ‘engaged’ grasp of it—a distinction which I fudge in footnote 29 of Essay 10 when I talk of ‘having’ a thought, and which for that matter I fudge in Essay 10 as a whole when I talk of ‘thinking’ that something is the case. This distinction applies when a concept is what Williams would call a ‘thick’ concept, that is a concept with both a factual aspect and an evaluative aspect. An example is the concept of infidelity: if I claim that you have been unfaithful, then I say something straightforwardly false if you have not in fact gone back on any relevant agreement; but I also thereby censure you. Another example, albeit one in which the evaluative aspect is somewhat subtler, is that which I used to illustrate the Basic Model: the concept of a wife. To grasp a thick concept in the disengaged way is to be able to recognize when the concept would correctly be applied, to be able to understand others when they apply it, and so forth. To grasp such a concept in the engaged way is not only to be able to do these things but also to be prepared to apply it oneself and hence to share whatever beliefs, concerns, and values give application of the concept its point. Talk of ‘having’ a thought, or even of ‘thinking’ something, and other related talk, can then be understood in two corresponding ways: in the engaged way whereby it requires having an engaged grasp of all the relevant concepts; and in the disengaged way whereby it does not. And if ‘thinking’ that p is understood in the engaged way, then there is indeed a fourth form that not thinking that p can take: namely, ‘considering’ the matter, where this is understood in the disengaged way, and possibly even ‘recognizing’ that it is true that p, where this too is understood in the disengaged way, but not oneself being prepared to apply one of the relevant concepts and thus not oneself thinking that p. Moreover, all of this may be completely self-conscious. One may not think that p because one repudiates the concept in question as somehow pernicious. The reason why this poses a particular threat to my project in Essay 10 is that, if the concept is somehow pernicious, then the idea that one pays a price for not thinking that p when it is true that p is clearly compromised: the very perniciousness of the concept may mean that one is better off not thinking that p, because one is better off not thinking in such terms at all. The relevance of all of this to the Basic Model should be clear. I couched the Basic Model in terms of ‘sense-making’, a term whose versatility I have already

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heralded, and in terms of ‘possession’ of a concept, a term on which I did not expand. Importantly, the distinction between the engaged way of understanding a term and the disengaged way of doing so apply to both of these. In particular, both can be understood in the engaged way. And if they are, then the instance of the Basic Model that I gave concerning wives turns out to be just one of a whole family of instances that involve thick concepts. For there is certainly something parochial about anyone’s possession of a thick concept, so understood; and such possession can certainly be integral to a priori sense-making, so understood. In what follows I shall frequently return to the Basic Model. Now there are often thought to be three great exemplars of the systematic pursuit of a priori sense-making: philosophy, ethics, and mathematics. The essays in Parts II, III, and IV deal respectively with each of these. The essays that precede them in Part I deal with the very nature of a priori sense-making and introduce the anthropocentrism. Much of the attention throughout is devoted to the work of other philosophers. But, even when it is, I take it to be of more than exegetical interest. One of the lessons that I take to emerge, either in opposition to the views of these other philosophers or by invocation of their views, is that we humans achieve nothing of real significance in philosophy, ethics, or mathematics except from a human point of view. In itself this does not force us to conclude that there is anything ineliminably anthropocentric about our systematic pursuit of a priori sense-making. After all, it may be that none of these three disciplines is the systematic pursuit of a priori sense-making that it is taken to be. This is not in fact my own conclusion, although it would be striking enough if it were the only alternative. My own conclusion is that philosophy, ethics, and mathematics each betoken what may reasonably be called ‘the human a priori’.

1. Part I: The Nature, Scope, and Limits of A Priori Sense-Making Given what I have said so far, Kant might be expected to figure in these essays as a hero. Is he not the great champion of the human a priori? One of his primary metaphysical projects is, after all, to account for a certain kind of a priori sensemaking; and the way in which he does so is by appeal to experience-independent cognitive resources which we humans have and which, for all we know, only we humans have. Not only are these integral to the a priori sense-making in question, they are integral to it in a way that makes it entirely appropriate to say that such sense-making is from a human point of view—possibly even from a peculiarly human point of view.⁴

⁴ Cf. Kant (1998), A26/B42.

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In many ways Kant does figure in these essays as a hero. He is the focus of each of the first two essays. Nevertheless, the principal lesson of Essay 1 is that there is something badly wrong with Kant’s own vision of the human a priori. This vision has three critical components: (Necessity) When we make a priori sense of things from our human point of view, we make sense of them as necessarily being a certain way.⁵ (Dependence) (Inescapability)

Things’ being that way is dependent on that point of view. We cannot make sense of things except from that point of view.

But there is an incoherence in supposing that we can acknowledge all three of these. For to acknowledge Dependence is to acknowledge a contingency in things’ being that way. And, given Necessity, this is to make sense of things from other than our human point of view (which presumably means, in this context, from no point of view at all). But this is what Inescapability says we cannot do. In the penultimate section of Essay 1 I argue that a significant part of Kant’s problem is the nature of the experience-independent cognitive resources that he invokes to explain our a priori sense-making. He includes aspects of how we think. But he also includes aspects of how we receive material to think about. And he does the latter in such a way that he also includes aspects of that very material, specifically its spatiality and temporality. By the time he has done all of this he is committed to Dependence. Had he only included aspects of how we think, any anthropocentrism that this involves would not have infected the subject matter of our thoughts and would not have compromised the necessity in how we make sense of things as being. We can appreciate this by reconsidering the Basic Model. However parochial the fact that a given subject thinks in terms of wives, to revert to that example, this subject is in a position to see that there must be fewer of them than there are women and girls.⁶ I said that, had Kant only included aspects of how we think in the resources that he invokes to explain our a priori sense-making, ‘any anthropocentrism that this involves’ would not have infected the subject matter of our thoughts. But what anthropocentrism does this involve? Is it akin to the anthropocentrism involved in the other resources that he invokes, that is to say in the spatiality and the temporality that are operative in how we receive material to think about? In their case, although Kant thinks they are part of our human point of view, and although he thinks we can know this, this is the limit of what he thinks we can

⁵ This is not to be confused with the thesis labelled ‘Necessity’ in Essay 2. ⁶ In the final section of Essay 1 I moot another way in which the problem could be averted, albeit a way that would take us even further from Kant’s own position. We could develop a conception of a priori sense-making that allows for contingency in how things are thereby made sense of as being. But I shall not now dwell on the many further issues that this raises.

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know. We cannot, in Kant’s view, know whether they are part of the point of view of other finite sense-makers (if such there be) nor for that matter whether they are part of the point of view of all possible finite sense-makers.⁷ The issue is whether he adopts an analogous circumspection concerning the resources that are operative in how we think, or whether, in their case, he reckons that we can know more: specifically, that they are part of the point of view of all possible finite sensemakers. This is the issue that Anil Gomes, Andrew Stephenson, and I address in Essay 2.⁸ We end up mooting a second-order circumspection on Kant’s part whereby the answer is neither—although there are reasons of principle why Kant had better not explicitly endorse this position.⁹ In Essay 3 attention shifts to Kant’s predecessor Descartes. Descartes likewise sees an anthropocentrism in our a priori sense-making. And he likewise embraces a version of Necessity. Both of these are manifest when, in making a priori sense of things, we at the same time make sense of them as necessarily being a certain way. For, on Descartes’s conception, for things necessarily to be a certain way is for the denial that things are that way to ‘conflict with our human concepts’.¹⁰ Not only is there a version of Necessity at work here, though. There is also what appears to be a Kantian predicament in the offing, as we see when Descartes pits his conception of necessity against his conception of God. For he is reluctant to say that any necessity in how things are is necessity even for God. This is in large part because he believes that ‘every basis of truth . . . depends on [God’s] omnipotence’,¹¹ from which it follows that even those things that are necessarily a certain way are ultimately that way only because God decrees that they are. From this in turn it follows, or rather it seems to follow—I shall return to the significance of this qualification shortly—that the necessity in question is at most a necessity for us, a necessity resting on a deeper contingency about our human point of view and the play of our concepts there. This is not the contingency of Dependence: the link here is between our human point of view and the necessity itself, not between our human point of view and what the necessity attaches to. But it makes for similar trouble. And it does mean that, if an analogue of Inescapability is at work in Descartes, as it plausibly is, then the apparent Kantian predicament to which I have referred is a real one. In fact, however, it is the burden of Essay 3 to argue that it is merely apparent. Descartes is at perfect liberty to deny that what I said seems to follow does follow; and he is at perfect liberty to insist that the necessity in question is indeed ⁷ Kant (1998), A27/B43 and B72. ⁸ For those who have read my Preface and are curious to know what our disagreement was, I can add that I originally thought that I could defend the former of these exegetical alternatives, while Gomes and Stephenson originally thought that they could defend the latter. ⁹ Some readers familiar with other work of mine, on inexpressibility, may see the stamp of that work on this conclusion. They would be wrong to do so. The silence required of Kant here has nothing to do with inexpressibility. It is silence on an issue that I take myself to have just expressed. ¹⁰ Descartes (1984b), ‘Second Set of Replies’, p. 107. ¹¹ Descartes (1991), p. 359.

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necessity even for God. He can do these things by doing what his a priori reflections on these issues mean that he should only ever have been doing— albeit, for reasons that I have indicated, he is sometimes diffident about doing— namely, heeding the analogue of Inescapability and resting content with making sense of things from our human point of view. This makes any claim that the necessity in question is necessity even for God harmlessly anthropocentric. It does nothing to gainsay the fact that even those things that are necessarily a certain way are ultimately that way only because God decrees that they are. In saying that they are necessarily that way we are saying only that it would conflict with our human concepts to deny that that is how they are. (We are also alluding to our means of coming to know that that is how they are.) In its own way, then, Essay 3 clearly develops the theme of the human a priori. In a brief postscript to the essay I correct a misunderstanding of the essay due to James Conant that precisely fails to recognize this. Of the four essays in Part I, Essay 4 is the one that is least obviously about the a priori. It is targeted at what is commonly dubbed ‘the new atheism’. I use the essay to explore a conception of theistic sense-making for which the new atheism makes no allowance. As it happens I believe that this conception significantly overlaps with my broad conception of a priori sense-making; I also believe that, where it does, there is something fundamentally anthropocentric about it. So, although none of this is explicit in Essay 4, it does mean that the essay is not the incongruity which it may appear to be. Even so, the significance of the essay for the volume as a whole lies elsewhere. I have included it because of the way in which it draws attention to kinds of sense-making that are not characteristic of the natural sciences. My hope is that it thereby serves as a helpful prelude to Parts II, III, and IV. For I do not believe that we can properly grasp the anthropocentric element in philosophy, ethics, or mathematics until we have come to appreciate how deeply the sense-making involved in each of these differs from that involved in the natural sciences (whose systematic pursuit can reasonably include the aspiration to abandon the human point of view¹²). Part of the force of what is to come in the remaining essays, therefore—as of Essay 4 itself—is an antiscientism.

2. Part II: How We Make Sense in Philosophy Such anti-scientism is to the fore in Part II. Of the three disciplines around which the essays in this volume are structured—philosophy, ethics, and mathematics—it is philosophy that is in greatest danger of falling prey to scientism. In Essay 5

¹² This is something that I argue in A. W. Moore (1997), esp. ch. 4.

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I take as my starting point the view famously defended by Bernard Williams as a safeguard against this danger: that philosophy is a humanistic discipline. I consider some of the implications of this view. However, to the extent that Essay 5 is written in opposition to anyone, it is written in opposition, not to those philosophers, typically within the analytic tradition, who think on scientistic grounds that we can abandon the human point of view when practising philosophy, but to those philosophers, typically not within the analytic tradition, who think that we can do so on the very different grounds that ‘we’ do not have to understand ‘ourselves’ as human beings at all and should embrace what is sometimes called ‘the post-human’. If philosophers of the former kind are in too much thrall of the natural sciences, then philosophers of the latter kind are in too little thrall, it seems to me, of their own humanity. The sheer fact that they adopt such a stance indicates that they have a greater aversion to philosophical conservatism than they have to philosophical loss of identity. And while I am certainly conscious of the dangers of philosophical conservatism—to the extent that I agree that we should be ready to embrace the post-human—nevertheless there is something so important about our humanity that the dangers of philosophical loss of identity strike me as being altogether graver. It is in Essays 6 and 7 that my opposition to philosophers of the former kind— those who think on scientistic grounds that we can abandon the human point of view when practising philosophy—is most evident. Each of these essays is targeted at a book by a philosopher in the analytic tradition. The target in Essay 6 is The Philosophy of Philosophy by Timothy Williamson; the target in Essay 7 is Writing the Book of the World by Theodore Sider.¹³ And each of these books is a defence of what I see as just such a scientistic conception of philosophy—or rather, in the case of Sider’s book, of metaphysics, although I take metaphysics to be a subdiscipline of philosophy that is in relevant respects typical of the discipline as a whole. My opposition takes a somewhat different form in each of these two essays. In Essay 6 it takes a more piecemeal form. I there focus on a few characteristic examples of how Williamson’s scientistic conception of philosophy manifests itself, and I try to indicate in each case why I see things differently. It is worth noting that one of the clearest ways in which it manifests itself is in the doubt that Williamson casts, not on the view that philosophical sense-making is fundamentally anthropocentric, but rather on the view that it is fundamentally a priori. (Williamson is in general suspicious of the significance that philosophers attach to the a priori. He is even suspicious of the significance that they attach to it in connection with mathematics. Some of what I say in Essay 6 is a foretaste of some of what is to come in Part IV.) In trying to counteract Williamson’s conception of philosophy I thus have my work doubly cut out.

¹³ Williamson (2007) and Sider (2011), respectively.

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In Essay 7 my opposition takes a more systematic form. Sider sees metaphysics as continuous with physics; and he devotes some of his book to practising metaphysics, some of it to the meta-metaphysical task of reflecting on what he is thereby doing. Significantly, however, he acknowledges that some such reflection is already part of metaphysics itself. For he sees metaphysicians as standing in a similar relation to physicists as meta-metaphysicians stand to them (metaphysicians). More specifically, he sees metaphysicians as reckoning with the propriety and worth of what physicists are doing. The significance of this, as I urge in Essay 7, is that there can be no reckoning with the propriety and worth of anything except from a point of view that allows for due evaluation. In particular, there can be no reckoning with the propriety and worth of what physicists are doing except from a point of view that allows for due evaluation of various human endeavours, which is to say a human point of view. Sider’s acknowledgement that such reflection is part of metaphysics therefore constitutes a crucial concession to anyone who shares my conviction that there is something ineliminably anthropocentric about metaphysics in particular, and about philosophy in general. Not that Sider would agree. He would deny that the evaluation in question is linked to a point of view in the way I claim. He has to deny this: the alternative poses far too much of a threat, if indeed it does not deal a fatal blow, to his vision of metaphysics as continuous with physics. But then so much the worse, I say, for that vision. The target in Essay 8 is the work of another philosopher, although this time a philosopher less easily classified either as an analytic philosopher or as a nonanalytic philosopher: Pamela Sue Anderson. I believe that she errs in the opposite direction. Much of my opposition in the three previous essays has been to the view that philosophy can escape the human. Anderson advances reasons for opposing the view that philosophy can escape the gendered human. In particular she argues that some of my own philosophical work betrays my masculinity. I disagree, although I acknowledge that she thereby raises some very important issues about the relation between philosophy and the masculine, between philosophy and the feminine, and between philosophy and the human. Towards the end of Essay 8 I try to reinforce a recurring theme of all the essays in Part II by insisting that it is the third of these—the relation between philosophy and the human—that is overwhelmingly the most significant.

3. Part III: How We Make Sense in Ethics The essays in Part III are concerned with ethics. Of the three disciplines— philosophy, ethics, and mathematics—this is the one that is liable to provoke least resistance to the thought that it is fundamentally anthropocentric. On the other hand, it is also the one that is liable to provoke most resistance to the

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thought that it is fundamentally a priori. There is accordingly a kind of shift of onus in these essays compared with those in Part II. The first four essays of Part III direct us once again to Kant. One of the many reasons why Kant is significant in this context is that he is, in the respect to which I have just adverted, an outlier. On Kant’s view, ethics is not fundamentally anthropocentric, but it is fundamentally a priori. In so far as Kant is a champion of the human a priori, this is because of his views about a priori sense-making of the theoretical kind that is characteristic of philosophy and mathematics, not because of his views about a priori sense-making of the practical kind that is characteristic of ethics. The latter, for Kant, is at root neither more nor less than an exercise of pure reason. As such it can be implemented by any being whose faculties include reason, be the rest of that being’s constitution as it may. This means that it is not only a priori in a way that does not involve its being anthropocentric; it is a priori in a way that precludes its being anthropocentric. One of my aims in these four essays is to consider some of what makes the opposed idea that ethics is fundamentally anthropocentric so attractive, and to explore how much of Kant’s commitment to the a priority of ethics could survive its assimilation. Given what I have said so far, the answer is obviously not all of it. But it is not obviously not any of it. The upshot of these four essays is neither a simple defence of that commitment nor a simple attack on it, but rather, in keeping with the volume as a whole, a non-Kantian reconsideration of a priori sense-making as itself, even in its practical form, inextricably bound up with the human. In Essay 9 I consider some of the consequences of Kant’s view that, even though ethics is not fundamentally anthropocentric, there is something fundamentally anthropocentric, possibly even peculiarly anthropocentric, about the way in which its demands appear to us as obligations. The fact that we are not just rational beings, but rational animals—with all the needs, desires, and drives that this entails—means that we are not always inclined to do what it would be purely rational to do. Hence, as Kant himself points out, what we would willingly do if we were purely rational appears in the guise of what we ought to do.¹⁴ And when we do not do it, there are issues that arise, and that create a certain awkwardness for Kant, about what kind of a failing this is, about what kind of control we have over what we do instead, and about what kind of relationship there is between such control—or lack of control—and our blameworthiness. St Paul takes an extreme view in his letter to the Romans: ‘[W]hat I do is the wrong which is against my will; and if what I do is against my will, clearly it is no longer I who am the agent, but sin that has its lodging in me.’¹⁵ Kant is under pressure to say something similar. But it is pressure that he resists. (This is the primary reason for the

¹⁴ E.g. Kant (1996a), 4: 449.

¹⁵ Romans 7: 19–20.

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awkwardness to which I referred.) On Kant’s view, if I incur blame for doing something other than what I ought to do, then there is no question but that I myself am the agent. In exploring all of this, I am playing out a curious variation on the theme of Essay 3 concerning Descartes, in as much as I am directing attention, not at the content of any of our a priori insights, but at the kind of necessity that attaches to them from our human point of view. There is even a hint of something else from Essay 3: the idea that we can do greatest justice to the resultant position by adhering resolutely to the human point of view and not trying to make sense of any of this except from there. No matter how much justice we do to the resultant position, however, it will still contain elements that are mysterious, counterintuitive, or both, for instance the idea, which is related to the pressure that Kant is under malgré lui to align himself with St Paul, that there is no such thing as a totally free act of wrong-doing. It will also contain elements, as I finish the essay by briefly expounding, that reflect further discomfort on Kant’s part with what St Paul says. In particular, Kant will not want to join St Paul in saying that a person’s blameworthiness for doing something other than what they ought to do can be annulled by divine grace. For just as it goes against the Kantian grain to say that someone can incur blame as a result of something that sin does, so too it goes against the Kantian grain to say that someone can forego blame as a result of something that God does. For non-Christians this may seem a relatively arcane matter. But for Christians and non-Christians alike it serves as a reminder of the purity that Kant sees in our practical sense-making—while some of the other elements in the resultant position serve as a reminder of the messiness that he encounters in his attempt to make sense of that purity from our human point of view. But what about the question that I flagged above, about how much of what Kant sees in our practical sense-making can survive if the purity is removed, that is if such sense-making is itself reckoned to be from our human point of view? One thing seems clear. Whatever survives, we shall encounter a similar messiness, if not a much greater messiness, in trying to make sense of it from our human point of view. What is not clear is whether this matters. Once we have relinquished the view that our practical or ethical sense-making is an exercise of pure reason, we shall be less beholden to the particular notions of freedom, control, wrong-doing, and such like that made for mystery and counterintuitiveness in the messiness that Kant himself encountered. The messiness that we encounter is liable to strike us as simply the messiness of life. In fact there are some important lessons to be learned here about how the combination of the a priori with the human is always vulnerable to the interference with the a priori by the human. That is, there are some important lessons to be learned about how the attempt to make a priori sense of things from a human point of view has its own distinctive ways of meeting with failure.

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Essays 10 and 11 are likewise concerned with the kind of necessity that attaches to the ethical from our human point of view. Neither essay defends the view that ethics is an arena for the human a priori. Neither, come to that, defends the view that ethics is an arena for any kind of a priori. But both portray the necessity in question as having some affinity with the kind of necessity that characterizes our a priori sense-making, and both combine that with an indication of how such necessity has the special force that it has from a human point of view. I have already discussed the flaw in Essay 10, and the way in which this flaw, once perceived as such, puts us in mind of the Basic Model and thereby prepares us for the possibility of the human a priori. The significant point here, however, is that Essay 10 manages to prepare us for that possibility anyway. The crucial work is done by something that I refer to in Essay 10 as ‘the Basic Idea’—where that label, incidentally, does not betoken any special connection with what I have been calling ‘the Basic Model’. The Basic Idea is that human beings are finite, but have an aspiration to be infinite.¹⁶ It is this that allows the necessity to have the special force that it has from our human point of view; for the necessity is precisely to be explained in terms of certain marks of the infinitude to which, according to the Basic Idea, we aspire. Not that the details of the account (which are in any case very sketchy in Essay 10) are what really matter in this context. Much more important and much more fundamental than the Basic Idea itself—be the truth of the Basic Idea as it may, and indeed be the interpretation of the Basic Idea as it may—is the broader idea of some shared conative state among human beings that influences our sense of necessity. If there is any such state, then there is scope for it likewise to influence our a priori sense-making and to prepare the way once again for a kind of human a priori. That same broader idea plays a similarly crucial role, and a similarly relevant role, in Essay 11—where the necessity has a new guise, as the necessity that animates a kind of transcendental argument. More specifically, I argue in Essay 11 that, just as there may be transcendental arguments of a Kantian kind for the conclusion that things are thus and so, proceeding via the intermediate conclusion that it is necessary for us to believe that things are thus and so, so too there may be ‘conative’ variants of these transcendental arguments for the desirability that things are thus and so, proceeding via the intermediate conclusion that it is necessary for us to desire that things are thus and so (in some suitably broad sense of ‘desire’). And it is the necessity of our desiring that things are thus and so that exemplifies the broader idea: there is a conative state which, on the one hand, we all have because we cannot help having it, and which, on the other hand, influences our sense of necessity, including the very necessity of our having it. Much of Essay 11 is concerned with tracing these elaborate interconnections.

¹⁶ Cf. Cavell (1979), p. 109; and Conant (1991b), p. 634.

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Neither Essay 10 nor Essay 11 is primarily about Kant. In fact Essay 10, as I have already indicated, is primarily about David Wiggins, whose related idea of there being only one thing that one can think about a given issue is an idea that he would distance from anything peculiarly Kantian, or indeed from anything to do with the a priori.¹⁷ Nevertheless, both essays have clear Kantian resonances. And, in so far as they depart from Kant in what room they leave for an understanding of the ethical as a priori, they do so precisely by allowing aspects of our humanity back in. This they do by taking seriously the idea that we humans all share a conative state that shapes all our ethical deliberations, a priori and empirical alike. The fourth of the essays in this quartet is like the first in being more straightforwardly about Kant. And it is pivotal to the volume as a whole. For it is here, in Essay 12, that we find the most graphic illustration of the Basic Model. (This is why I had occasion to refer to Essay 12 earlier in the Introduction.) It is here too that we most directly confront the question of how much of Kant’s own commitment to the a priority of ethics could survive assimilation of the idea that ethics is fundamentally anthropocentric. Ethics, for Kant, is an exercise of pure reason. But even Kant acknowledges that ethics is applicable to issues that can be framed only in terms of concepts whose possession depends on highly contingent social structures. (It had better be applicable to such issues, if the exercise of pure reason in question is to be suitably practical.) Kant has no qualms, for example, about drawing ethical conclusions about the marriage contract.¹⁸ And such applicability is already an illustration of the Basic Model. For precisely what it involves is a priori sense-making that is achieved through the implementation of concepts whose possession is radically parochial. But now comes the twist. It would be possible to maintain a broadly Kantian view of ethics, while nevertheless departing from Kant himself and embracing the view that ethics is fundamentally anthropocentric, by conceiving of ethics as concerned not only with issues about how to respect whatever concepts we possess but also with issues about what concepts to possess in the first place—and, in particular, about what thick concepts to possess in the engaged way. On this extended conception of ethics—I say some more about the conception and about its rationale in the final section of Essay 12—ethics would involve negotiating certain basic facts of human nature that determine what concepts we are so much as capable of possessing. (This is not unrelated to Kant’s own concession that the exercise of pure reason that constitutes ethics sometimes involves negotiating certain basic facts of human nature that determine what we are capable of willing.¹⁹) Ethical sense-making could then reasonably be viewed as a prime example of sense-making that is both fundamentally a priori and fundamentally anthropocentric.

¹⁷ See Wiggins (1996).

¹⁸ Kant (1996c), 6: 279–80.

¹⁹ E.g. Kant (1996a), 4: 423–4.

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The remaining three essays in Part III are more concerned with the anthropocentric than with the a priori. But they are concerned with the anthropocentric in a way that connects importantly with what has gone before and continues to have implications for the human a priori. Essay 13 is a critique of Simon Blackburn’s meta-ethical ‘quasi-realism’, whereby ethical claims, though they are expressive of our conative states, also admit of truth or falsity. In this essay I argue that, despite Blackburn’s insistence to the contrary, he is committed to a relativism akin to what we have just witnessed in Essay 12. Moreover, I do so in a way that directly relates back to the discussion of Descartes in Essay 3. For Descartes’s conception of necessity and Blackburn’s conception of desirability are variations on a single theme: each adverts to what we are implicitly saying about ourselves when we make some claim about the notion in question. On Descartes’s conception of necessity, when we make some claim about how it is necessary for things to be, we are implicitly saying that it would conflict with certain concepts that we human beings possess for things not to be that way. On Blackburn’s conception of desirability, when we make some claim about how it is desirable for things to be, we are implicitly saying that it would conflict with certain conative states that we human beings have for things not to be that way. The reason why I take Blackburn to be committed to a kind of relativism is that I take it to follow from this that, had our conative states been relevantly different, which I believe his own quasi-realism compels him to say they could have been, then we would, quite rightly, have counted different things desirable. Interestingly, however, there is no reason to think that Descartes is committed to an analogous relativism. For, as I argue in Essay 3, there is not the same compulsion for Descartes to say that our concepts could have been relevantly different. Be that as it may, the label ‘anthropocentric’ looks entirely appropriate in both cases. (Not that Blackburn need demur. The relativism that he eschews, as the Cartesian case shows, is a separate matter.) In Essay 14 I turn to Derek Parfit’s very different meta-ethical views and reproach him for precisely failing to advert to, in fact for failing to respect, some of what we are implicitly saying about ourselves when we make ethical claims—or, in his extended discussion of these issues, when we make normative claims more generally. These claims, I urge, are irreducibly from some point of view: in making them we are implicitly saying something about our occupancy of that point of view. (This places me closer to Blackburn than to Parfit.) And, although there is nothing in Essay 14 to suggest that ‘we’ here means ‘we human beings’—the reach of the pronoun in any specific case may be either wider or, more probably, narrower—there is still something fundamental about the human at stake, if only because making ethical sense of things is itself an essential part of being human. For that matter, there is something fundamental about the human at stake in the very idea that making ethical sense of things involves making sense of things from some point of view, because we—however wide or narrow the reach of that

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pronoun—cannot make sense of things from any point of view that human beings are incapable of occupying. This is reminiscent of the way in which Essay 12 allowed for a conception of ethics as anthropocentric. What mattered in that case were the constraints imposed on our ethical sense-making by the fact that our ethical sense-making needs to involve concepts that human beings are capable of possessing. The final essay in Part III, Essay 15, is concerned with one very radical version of the idea that making ethical sense of things involves making sense of them from some point of view, a version that can be found in Nietzsche. This is the idea that making ethical sense of things involves making sense of them from ever different points of view. Not that this is the main focus of Essay 15. The main focus of Essay 15 is something quite different: Bernard Williams’s argument for the meaninglessness of immortality. Nietzsche’s relevance to this lies in an argument that I give to the effect that he (Nietzsche) can be seen as an unexpected ally of Williams, in as much as even to acknowledge our immortality, let alone to rejoice in it, would, on a Nietzschean conception, and contrary perhaps to appearances, thwart this continual making of new ethical sense of things from new points of view. This indicates one of many ways in which the human a priori needs to reckon with our very finitude (a reckoning that assumes even greater significance if we accept what I called in Essay 10 ‘the Basic Idea’: that human beings are finite, but have an aspiration to be infinite). This in turn is a good cue for the next section, where attention shifts from ethical sense-making to mathematical sense-making. For if the latter is an example of the human a priori, then it too must indicate how the human a priori needs to reckon with our finitude. This is because one of the most elemental tasks that we confront, when we engage in mathematical sensemaking, is to make sense, in particular, of the infinite; and this requires that we leverage our finite resources to achieve a grasp of that which precisely cannot be grasped by any straightforward use of any finite resources. Each essay in Part IV is concerned, to a greater or lesser extent, with what it takes for us to do this.

4. Part IV: How We Make Sense in Mathematics I said above that, of the three disciplines—philosophy, ethics, and mathematics— ethics is the one that is liable to provoke least resistance to the thought that it is fundamentally anthropocentric and most to the thought that it is fundamentally a priori. Mathematics is its polar opposite in this respect. Certainly there is liable to be great resistance to the thought that mathematics is fundamentally anthropocentric. The essays in Part IV go some way towards motivating the view that even so, in some sense, it is. This is not the crude relativist view that, when we humans claim that, say, twice two is four, what we mean is that twice two is four for us although it may have

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some other value for other beings. It is rather the view that, when we humans claim that twice two is four, even if what we claim holds both universally and necessarily, there are nevertheless some facts of human nature that not only enable us to make such sense of things but that enable such sense to be made of things at all. One exponent of this view is Kant. Kant would certainly deny that twice two may have some value other than four for other beings. But he does hold that, when we humans claim that twice two is four, we are making use of concepts that have been formed ‘through successive addition of units in time’;²⁰ and he also holds, as we noted earlier, that time is a feature of our human point of view. So any being that did not share this point of view and that knew nothing of it would not be able to make such sense of things. What this means is that the question of what twice two is would not so much as arise for such a being. It does not mean that the question would arise and somehow receive an alien answer. Nor does it mean that the sense-making involved in answering the question is anything less than a priori: it really just casts mathematical sense-making as an instance of the Basic Model. Call the view that mathematics is anthropocentric in this way the Anthropocentric View. I tried to indicate earlier why I think that Kant himself, by assigning time the role that he does in his own version of the Anthropocentric View, lapses into incoherence. But ‘his own version of the Anthropocentric View’ is the key phrase. The structure of the Anthropocentric View, and in particular the casting of mathematical sense-making as an instance of the Basic Model, is not in itself problematical. Kant’s critical error is to include in the experience-independent cognitive resources that he invokes to explain our mathematical sense-making aspects of what we think about, not just aspects of how we think. Had he done only the latter, he would have avoided any such incoherence. To the extent, therefore, that we can construe the experience-independent resources that equip us to engage in mathematics as a matter of how we think, not a matter of what we think about, we too shall avoid any such incoherence. This gives us scope to adopt an acceptable alternative to Kant’s version of the Anthropocentric View. On what I take to be the most attractive version, mathematics consists in developing, refining, consolidating, and implementing the very experience-independent cognitive resources that equip us to engage in it. Mathematics is a formation of mathematical concepts. But the concepts, once formed, exhibit a rigid interrelatedness that is made not a whit less rigid by whatever peculiarly human sensibilities and faculties were integral to their formation. Not only is twice two four: twice two must be four, always, everywhere, and for everyone.²¹ This view is essentially Wittgensteinian. Wittgenstein regards mathematics as a formation of mathematical concepts.²² He also has an acute sense of how the

²⁰ Kant (2002a), 4: 283. ²¹ This once again calls to mind the discussion of Descartes in Essay 3. ²² E.g. Wittgenstein (1978), pt IV, §§29–33.

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human is at work in sustaining our use of mathematical language.²³ Wittgenstein accordingly features prominently in Part IV. The first three essays in this part consist of more or less direct exegesis of his work—albeit, in the case of Essay 16, through the lens of Crispin Wright’s discussion and appropriation of it. Essay 16 includes a brief discussion of (Wright’s tentative defence of ) Wittgenstein’s argument against the possibility of a private language; but its primary focus is Wittgenstein’s later philosophy of mathematics, which is also the subject matter of Essay 18. In between, Essay 17 provides an overview of Wittgenstein’s treatment of the infinite. In all three essays I try, however subliminally, to motivate Wittgenstein’s version of the Anthropocentric View. One of the issues that I address in Essay 16 is that of how to avoid allowing self-conscious awareness of the anthropocentrism at stake to instil in us needless sceptical worries about whether our mathematical sense-making is as robust as it should be. In Essay 17 I address the more specific issue, adumbrated in my remarks at the end of the previous section, of how to avoid allowing that same self-consciousness to instil in us needless sceptical worries about whether our mathematical treatment of the infinite is as robust as it should be. I suggest that Wittgenstein himself does not always succeed in this respect. For he is led by these reflections into what I see as unacceptable fussing about standard mathematical accounts of the infinite. In fact he is led into outright disdain of them. This theme is pursued in Essay 18, where I place Wittgenstein’s scepticism about these accounts in a broader context. In particular, one thing that I do in Essay 18 is to highlight a tension that there appears to be between Wittgenstein’s philosophy of mathematics and his philosophy of philosophy. Wittgenstein’s philosophy of philosophy, familiarly, casts philosophy as a therapeutic exercise aimed at combating various confusions to which our mishandling of our own ways of making sense of things exposes us. There is no need, on this conception, for philosophers to reform how we make sense of things: indeed they had better not do so, for they simply run the risk of generating more such confusion if they do. It is in this connection that Wittgenstein says that philosophy ‘leaves everything as it is’²⁴—to which he immediately adds, in amplification, ‘It also leaves mathematics as it is.’ The apparent tension lies in the fact that his philosophy of mathematics seems not to respect this precept. Not only in his reflections on the infinite but elsewhere in his philosophy of mathematics, we find Wittgenstein taking continual philosophical exception to actual mathematical practice, and thereby to actual mathematical sense-making. (Another example will occur in Essay 20, on Gödel’s theorem.) To be sure, there is an obvious get-out clause for Wittgenstein: when he insists that philosophers had better not interfere with our ways of making sense of things, there is a tacit restriction to our legitimate ways of making sense of things. It is ²³ E.g. Wittgenstein (1967a), pt I, §§240–2, and pt II, pp. 226–7; and Wittgenstein (1978), pt I, §142. ²⁴ Wittgenstein (1967a), pt I, §124.

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entirely possible that the ways that mathematicians have of making sense of things are not always legitimate, but are sometimes corrupted, say by mathematicians’ view of the nature of their own discipline.²⁵ Whenever this is the case, mathematical sense-making is precisely ripe for philosophical interference on a Wittgensteinian conception. But, for reasons that I try to make clear in Essay 18, I do not myself believe that this get-out clause ultimately prevents the apparent tension between Wittgenstein’s philosophy of philosophy and his philosophy of mathematics from being real. And I end the essay by saying (albeit without elaborating) that I take the principal fault to lie with his philosophy of philosophy. Given what I have said so far in this Introduction, this may come as a surprise. Have I not been suggesting the very opposite: that the principal fault lies rather with his philosophy of mathematics? Not exactly. The point is this. Even when some mathematical way of making sense of things is legitimate, there may be an alternative that has certain practical advantages. Suppose there is. The fact remains that the sheer legitimacy of the original way of making sense of things means that Wittgenstein’s philosophy of philosophy requires philosophers simply to accede to it. And that seems to me unduly conservative. True, there is a risk that adopting the alternative will generate new philosophical confusion. But there is a risk that retaining the original will do that too. Indeed precisely one of the practical advantages of the alternative may be that it is less susceptible than the original to being mishandled in a way that throws us into confusion. And if that is the case, then not only is it entirely reasonable to advocate for the alternative, it is entirely reasonable to do so on philosophical grounds.²⁶ And this returns us to the main theme of this volume. For deciding which of the two ways of making sense of things is less susceptible to being mishandled in that way will require sensitivity to the various human sensibilities and faculties that are involved in our implementing each of them. The final two essays in Part IV are concerned with specific applications of the Wittgensteinian version of the Anthropocentric View. But the starting point of Essay 19 is provided by a non-Wittgensteinian version of the view, closer in many ways to what we find in Kant: namely, the view endorsed by intuitionists whereby the facts of human nature that enable mathematical issues to arise in the way in which they do are facts about our experience of the pure structure of time. Quite how closely or distantly this is related to Kant’s view is an issue that I touch on very briefly in §4 of the essay: perhaps distantly enough for intuitionists to avoid some of problems that afflict Kant himself. For, rather than casting time as the subject matter of our mathematical sense-making, they can arguably be seen as doing something more innocuous: casting temporally informed concepts as ²⁵ Cf. Wittgenstein (1967a), pt I, §254. ²⁶ For further discussion see A. W. Moore (forthcoming), esp. §5.

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among the primary tools of our mathematical sense-making. Nevertheless, intuitionists do confront a problem, which it is the main burden of Essay 19 to address. The problem is that their view completely loses its force if we allow ourselves to apply the concept of the infinite in a certain way. To solve this problem, and thus to prevent ourselves from applying the concept of the infinite in that way, what we need to do, I argue, is to pit such applications of the concept against the Wittgensteinian version of the Anthropocentric View and, more specifically, to pit them against some of the lessons of Essay 17. Finally, in Essay 20, I apply the Wittgensteinian version of the Anthropocentric View to consideration of the philosophical significance of Gödel’s theorem, in the context of a discussion of Michael Dummett’s essay on the same topic.²⁷ Not that Essay 20 is unreservedly Wittgensteinian. In the final section of the essay I discuss some of Wittgenstein’s own remarks on Gödel’s theorem, in the course of which he recoils from one standard way of stating the theorem. This way of stating the theorem is as follows: given any sound axiomatization of arithmetic, there are arithmetical truths that the axiomatization cannot be used to prove. Wittgenstein advocates a relativization of mathematical truth to some axiomatization which makes this way of stating the theorem inappropriate.²⁸ While I am more sympathetic to what Wittgenstein is doing here than most commentators, I nevertheless urge that this is another example of unacceptable meddling with actual mathematical sense-making. The bulk of Essay 20, however, is devoted to arguing that we can make best philosophical sense of Gödel’s theorem in broadly Wittgensteinian terms. In particular, it is in broadly Wittgensteinian terms that we can make best philosophical sense of what is involved in our advancing from acceptance of some axiomatization A of arithmetic to acceptance of the consistency of A, and thereby to acceptance of certain arithmetical truths that A cannot be used to prove. The forces that are at work here are the forces that are at work in the very formation of all the relevant mathematical concepts, such as the concept of a natural number, the concept of addition, and the concept of consistency. And they depend, as I try to indicate in Essay 20, on shared human reactions. Gödel’s theorem can thus be seen as a further illustration of the Anthropocentric View— as it can, therewith, of the human a priori.

²⁷ Dummett (1978c).

²⁸ Wittgenstein (1978), pt I, app. III, esp. §8.

PART I

T H E NA T U R E , S C O P E , A N D L I M I T S OF A PRIORI SENSE-MAKING

1 Armchair Knowledge Some Kantian Reflections

Abstract This essay considers a puzzle associated with ‘armchair knowledge’, that is to say, knowledge that is not warranted by experience. The puzzle is that each of the following claims seems true although they also seem mutually incompatible: there is armchair knowledge; some armchair knowledge, if such there be, concerns what is beyond the subject; and armchair knowledge does not involve any appeal to any particular encounter with anything beyond the subject. The Kantian solution to this puzzle, namely transcendental idealism, is a view whereby some of what the subject has knowledge of has a form that depends on the subject. After discussion of the scope and limits of this solution, it is argued both that it is the only available solution when the armchair knowledge in question is synthetic and that it is incoherent, from which it is concluded that there is no such thing as synthetic armchair knowledge. But this is all on the assumption that the armchair knowledge in question is knowledge of what is necessary. In the final section of the essay consideration is given to other solutions to the puzzle that may be available if the knowledge in question is knowledge of what is contingent.

1. A Kantian View of Armchair Knowledge One of the oldest of philosophical puzzles is to account for what I shall call ‘armchair knowledge’. By ‘armchair knowledge’ I mean knowledge that is independent of experience, in the sense that it is not warranted by experience. The rationale for the label is clear enough: a subject¹ who has such knowledge could have had it while remaining seated in an armchair.² I might just as well have used ¹ There will be frequent references in this essay to the ‘subject’. For remarks that are very pertinent to my use of this term see Kant (2000), 5: 401. ² That is, the subject could have had it while remaining seated in an armchair granted possession of the concepts involved: it is not precluded that the subject had to leave the armchair to acquire those concepts in the first place (cf. Kant (1998), B3). And it is important that the armchair should be nothing more than an inessential prop: one thing that a subject could know while remaining seated in an armchair is how comfortable the armchair is, but this, I hardly need say, is not an example of what I have in mind. (In the first and the most famous discussion of armchair knowledge in Western philosophy—a discussion that predates armchairs—the only significant prop involved is some sand: see Plato (1961d), 82b–85b.)

The Human A Priori: Essays on How We Make Sense in Philosophy, Ethics, and Mathematics. A. W. Moore, Oxford University Press. © A. W. Moore 2023. DOI: 10.1093/oso/9780192871411.003.0002

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the more familiar label ‘a priori knowledge’. But ‘a priori knowledge’ is sometimes applied more broadly—not just to knowledge that is independent of experience, but to knowledge that could have been independent of experience.³ I hope that my use of the less familiar label, along with my stipulative definition of it, helps to avoid confusion on that score.⁴ I also hope that it helps to highlight a crucial feature of such knowledge, or at least what appears to be a crucial feature of such knowledge: it does not involve any appeal to any particular encounter with anything beyond the subject. The puzzle to which I have referred arises from the fact that not only does there appear to be such knowledge, but some of it appears to concern what is beyond the subject. For how can it?—given that what is beyond the subject is irrelevant to it in that way. The puzzle is exacerbated by the fact that some of the knowledge in question appears to concern, not just some of what is beyond the subject, but all of what is beyond the subject; indeed, not just all of what is beyond the subject, but all of what could possibly be beyond the subject.⁵ To repeat, the puzzle arises because each of the following appears to be the case: (i)

there is armchair knowledge;

(ii)

some armchair knowledge, if such there be, concerns what is beyond the subject;

and (iii)

armchair knowledge does not involve any appeal to any particular encounter with anything beyond the subject.

Some philosophers think that the puzzle can be solved by denying the appearances. Thus certain empiricists simply deny (i). Other empiricists accept (i), but deny (ii): they hold that all armchair knowledge concerns the subject’s command of language, or the subject’s conceptual repertoire, or something of the sort. Certain Platonists accept (i) and (ii), but deny (iii): they hold that armchair

³ Thus my own knowledge that every natural number is the sum of four squares is based on an appeal to authority. So it is not included in what I am calling ‘armchair knowledge’. But it is included in what, on this broad usage, would be called ‘a priori knowledge’, since it is knowledge of a mathematical truth that could in principle have been independent of experience. A further complication is that the term ‘a priori’ is also sometimes applied, not to knowledge, but to truths: those that, in my terms, are potential items of armchair knowledge, in other words those that are knowable independently of experience (cf. Kant (1996b), 5: 31, and Frege (1980), §3). A yet further complication, which will prove to be pertinent in §6, is that the term ‘a priori’ is also sometimes applied to non-propositional entities, such as concepts. ⁴ For a second possible advantage of my use of the less familiar label—pertaining this time to the fact that ‘a priori knowledge’ is sometimes applied, not more broadly, but more narrowly—see n. 57 below, together with the accompanying text. ⁵ Cf. Kant (1998), B3–4.

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knowledge is acquired through, and involves appeal to, acquaintance with one or more Platonic idea or universal.⁶ But there are also philosophers who think that the puzzle can be solved without denying the appearances. They accept (i), (ii), and (iii). The way in which they solve the puzzle is by espousing some version of idealism, whereby some of what the subject has knowledge of, including, in this version, some of what is beyond the subject, has a form—a range of essential features—that depends on the subject. The armchair knowledge in question pertains to this form. Thus some of it is knowledge to the effect that whatever has the form is of such and such a kind; some of it is knowledge to the effect that whatever is of such and such a kind has the form. This means that it does indeed concern what is beyond the subject; for the form is the form of what is beyond the subject. But it also means that the knowledge does not involve any appeal to any particular encounter with anything beyond the subject, precisely because the form to which the knowledge pertains depends on the subject. This is Kant’s view.⁷ Its attractions are not confined to the fact that it can be used to account for armchair knowledge of what is beyond the subject. It can also be used, if Kant is right, to account for (some) knowledge of what is necessary—Kant’s own view being that all armchair knowledge, simply qua armchair knowledge, is knowledge of what is necessary.⁸ Indeed it can be used to account for (some) knowledge of what is necessary as necessary. Thus if some of the armchair knowledge in question is knowledge to the effect that whatever has the given form is of such and such a kind, then some of it is also, in Kant’s view, knowledge to the effect that whatever has the given form must be of such and such a kind. How to account for knowledge of what is necessary, as necessary, is another old philosophical puzzle. The puzzle this time arises from the sheer fact that we, finite contingent creatures that we are, can have epistemic access to all the ways things might have been. Many philosophers think that they can solve this puzzle, or at least that they can begin to solve it, by finding a grounding for necessity in contingency, where finding a grounding for necessity in contingency is more than simply discovering, with respect to some apparent necessity, that there is a contingency underpinning it—so that it no longer appears necessary. Doing that is not especially remarkable, nor does it have any great philosophical purchase. In fact it is an important part of growing up. ⁶ For the label ‘Platonist’ cf. Plato (1961a), 73–6. Whatever the exact nature of the relationship between this view and Plato, the fact that there is such a relationship helps to explain why W. D. Hart is emboldened to say, in Hart (1988), p. 158, that ‘we are all of us empiricists in our bones (even, or especially, Plato)’. Whether or not the view can be attributed to Plato, it can certainly be attributed to Russell: see Russell (1959), ch. 10. ⁷ Kant (1998), Bxvi. ⁸ See again the material in Kant (1998), B3–4, cited in n. 5; see also Axv and Bxii. I shall not, for the time being, query Kant’s view that all armchair knowledge is knowledge of what is necessary, although in section 7 I shall explore some of the implications of rejecting this view.

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Finding a grounding for necessity in contingency is doing something more delicate than that: it is discovering, with respect to some apparent necessity, that there is a contingency underpinning it without disrupting the appearance of necessity. This, if it can be done at all, does have philosophical purchase. And it is what Kant tries to do.⁹ Kant espouses a version of idealism whereby that part of the subject’s armchair knowledge which pertains to the given form is knowledge from a particular point of view.¹⁰ And a point of view, by its very nature, admits of alternatives. So the fact that there is knowledge from this point of view is the relevant contingency, the contingency in which the relevant necessity is grounded. But Kant does not think that the necessity is thereby compromised. For he does not think that there is anything in his idealism to preclude the subject’s continuing to have, and continuing to exercise, knowledge from the given point of view: it is just that such knowledge cannot itself include acknowledgement of the idealism. This explains why Kant’s idealism is, in his own terminology, ‘transcendental’ idealism rather than ‘empirical’ idealism. Kant uses these two terms to register a distinction between two views about the nature of space and time.¹¹ For our purposes, however, it will help to extend his usage and to work with a broader distinction. Let us refer to the dependence posited by the idealist—the dependence of the form of what the subject has knowledge of on the subject—as the i-dependence. Then we can construe transcendental idealism as idealism in which the i-dependence is not itself included in whatever has this form; and we can construe empirical idealism as idealism in which it is.¹² The reason why Kant’s idealism is transcendental is that it assigns contingency to the i-dependence (for it allows that there might not have been any such subject, nor therefore any such form depending on any such subject), and this contingency, simply qua contingency, must transcend the necessity attendant on whatever has the form in question.¹³ ⁹ On a popular reading of Descartes, it is what he tries to do too—by taking the necessity of any given necessary truth, say that twice four is eight, to lie in the contingent fact that human beings are incapable of grasping any other possibilities. If this is Descartes’s strategy, however, then all that he succeeds in doing is providing a very graphic illustration of why finding a grounding for necessity in contingency is such a delicate matter. For if it really is necessary that twice four is eight, then there are no other possibilities, hence no other possibilities for human beings to be incapable of grasping. I should add, however, that I am not persuaded that the popular reading of Descartes is correct, certainly not as a reading of his fully considered view: see Essay 3 in this volume, which is in turn indebted to Bennett (1994). ¹⁰ See Kant (2000), 5: 403. Cf. also Kant (1996a), 4: 452. ¹¹ See Kant (1998), A369. ¹² I have elsewhere construed the two doctrines slightly differently: see A. W. Moore (1997), p. 116, and (2012), ch. 5, appendix. But the differences, which are tailored to the demands of their specific contexts, are relatively insignificant. ¹³ We can extract, from these considerations, a general test for whether any given idealism is transcendental or empirical. Let TI be some version of transcendental idealism; let EI be some version of empirical idealism; let FTI be the form involved in the i-dependence that is posited in TI; and let FEI

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2. The Distinction between Analytic Armchair Knowledge and Synthetic Armchair Knowledge, and Two Associated Questions Now Kant famously distinguishes between two kinds of armchair knowledge: that which is analytic and that which is synthetic. This distinction is related to another that he draws, between intuitions and concepts. Intuitions are products of the subject’s receptivity: there is something passive about them. Concepts are products of the subject’s spontaneity: there is something active about them. It is by means of intuitions that the subject is given various objects of knowledge. It is by means of concepts that the subject thinks about these objects, as thus given.¹⁴ Any knowledge, at least if it has what Kant calls ‘content’,¹⁵ must make use of both. It must involve an exercise of concepts, whereby something is thought. But these concepts must in turn relate ultimately to intuitions, whereby what is thought has whatever content it has.¹⁶ What distinguishes analytic knowledge is that, in this case, the exercise of concepts does all the relevant work: the subject knows, simply by appeal to the concepts involved, and in particular by analysis of them, that what is being thought is true. By contrast, in the case of synthetic knowledge, the subject also appeals to the intuitions involved.¹⁷ Does it follow that no analytic armchair knowledge testifies to (ii), in other words that no armchair knowledge that is analytic concerns what is beyond the subject? It looks as though it does follow, because it looks as though what analytic armchair knowledge concerns, on this view—and here we are reminded of the empiricist view that I flagged in §1—is the subject’s conceptual repertoire. In fact, however, I do not think that we are forced to say this. There is a perfectly good be the form involved in the i-dependence that is posited in EI. Then, whereas an exposition and/or defence of TI is bound to reckon with the distinction between what transcends FTI and what has FTI, an exposition and/or defence of EI need not reckon with any such distinction concerning FEI. Moreover, there are family resemblances between claims about what transcends FTI and claims about what has FTI whereby it is natural to use the same language to express them—albeit not with exactly the same meaning—not least because we are liable to lack independent linguistic resources to talk about what transcends FTI. This means that, in practice, an exposition and/or defence of TI is liable, sooner or later, to include a claim that is to be deemed true when construed as a claim about what transcends FTI but as false when construed as a claim about what has FTI. (Paradigm cases include various claims that Kant considers whose truth is sensitive to the ambiguity in the expression ‘outside us’ that he notes in Kant (1998), A373.) The same is not true of an exposition and/or defence of EI. Here it might be objected that Berkeleian idealism, which is empirical if any idealism is, counts as transcendental by this (admittedly inconclusive) test because it does involve such equivocation, in particular where phrases such as ‘perceiver-independent’ are concerned: see e.g. Berkeley (1962), pp. 200–1. To pursue this matter is far beyond the scope of this essay, although it is worth noting that any problem about the application of this test to Berkeleian idealism may be a problem with Berkeleian idealism rather than a problem with the test. ¹⁴ Kant (1998), A19/B33. ¹⁵ Kant (1998), A51/B75. The significance of this qualification will be clear in due course. ¹⁶ Kant (1998), A50–1/B74–5. ¹⁷ Kant (1998), A47/B64–5, B73, and Kant (2002a), §2. (But see below, n. 49, for some complications in this connection.)

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sense of ‘concern’ in which the subject’s analytic armchair knowledge that all vixens are female, say, concerns vixens, not the subject’s concept of a vixen, nor any other part of the subject’s conceptual repertoire. There is a perfectly good sense of ‘concern’, in other words, in which it concerns what is beyond the subject. Indeed, my own view is that Kant allows for analytic armchair knowledge that lacks content, that is to say, analytic armchair knowledge in which the concepts involved do not relate to intuitions,¹⁸ and that even knowledge of this kind can, in the relevant sense of ‘concern’, concern what is beyond the subject. An example might be the subject’s knowledge that things in themselves are things irrespective of how they are given to us, knowledge which concerns things in themselves.¹⁹ But whether or not we adopt this attenuated sense of ‘concern’ and say that some analytic armchair knowledge concerns what is beyond the subject, two associated questions arise. We can begin to appreciate the force of these questions by noting that, to whatever extent it is appropriate to say that some analytic armchair knowledge concerns what is beyond the subject, to that extent it is likewise appropriate to regard such knowledge as part of the original puzzle— the puzzle that Kant tries to solve by invoking transcendental idealism, or rather, the puzzle part of which Kant tries to solve by invoking transcendental idealism. For, although Kant holds that transcendental idealism is needed to solve the puzzle with respect to synthetic armchair knowledge,²⁰ he also holds that it is needed to solve the puzzle only with respect to synthetic armchair knowledge.²¹ And, whatever his reasons for holding this, they were not apparent in anything I said in the previous section. The two questions are these. (1) Would Kant allow that transcendental idealism can be invoked to solve the puzzle with respect to analytic armchair knowledge too, even if it does not have to be? (2) What is it about synthetic armchair knowledge that makes Kant think that transcendental idealism must be invoked to solve the puzzle with respect to it?

3. Invoking Transcendental Idealism to Account for Analytic Armchair Knowledge Let us begin with question (1). Kant certainly thinks that the puzzle with respect to analytic armchair knowledge can be solved without recourse to transcendental ¹⁸ This is what I had in mind in n. 15. For arguments against the view that Kant would acknowledge any such knowledge see Kreis 2023, esp. §6. I remain unpersuaded. ¹⁹ See e.g. Kant (1998), A258–60/B314–15. Note that the distinction between knowledge and cognition that many Kantian exegetes draw is very pertinent to what I am suggesting here and may help to make what I am suggesting appear less exegetically contentious: see A. W. Moore (2012), ch. 5 n. 13. ²⁰ See Kant (1998), B41 and A92/B124–5. ²¹ See e.g. Kant (2002a), §5.

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idealism. It is enough, Kant thinks, to note that the subject can have such knowledge just by analysing the concepts involved.²² In effect, then, Kant is saying that, even if analytic armchair knowledge does concern what is beyond the subject, the sense in which it does so is sufficiently attenuated—it really just comes down to the fact that the concepts involved apply to what is beyond the subject²³—that there is no need, in accounting for such knowledge, to relate the form of what the subject has knowledge of to the subject in any way, still less to acknowledge the i-dependence, still less to acknowledge the i-dependence in such a way that some of the subject’s knowledge is to be seen as knowledge from a particular point of view that admits of alternatives. There is no need to do this. But the question is whether Kant would have any quarrel with a philosopher who, perhaps in an attempt to give a unified account of all armchair knowledge, does do this. Thus imagine a philosopher who urges that the form of what is beyond the subject, which depends on the subject and to which the subject’s armchair knowledge pertains, is not confined to those of its essential features that Kant famously fastens on—its spatio-temporality, its subjection to causal laws, et cetera—but extends to all those of its essential features that are in any way conceptual, such as the feature of being, if a vixen, female; and that the contingency of the i-dependence is no less a mark of the subject’s general conceptualization of things than it is of the subject’s spatio-temporal intuition of them. Such a philosopher does not have to disagree with Kant’s claim that, in order to have analytic armchair knowledge, the subject need only analyse the concepts involved: this extension of Kant’s transcendental idealism might be intended as an explication of that claim, not as a challenge to it. So to repeat: would Kant have any quarrel with such a philosopher? In fact he would. For Kant holds that the subject can have thoughts concerning things in themselves. I earlier suggested that some analytic armchair knowledge could serve as an example; but, even if that suggestion is open to dispute, there are uncontentious examples, such as the thought that we are free.²⁴ And a philosopher who holds that the subject’s conceptualization of things contributes as much to the contingency of the i-dependence as the subject’s spatio-temporal intuition of things must, in Kant’s view, hold that the subject’s thinking, no less than the subject’s intuiting, is always of appearances rather than of things in themselves. But to say that Kant would have a quarrel with such a philosopher is not to say that he would be justified in having it. The question remains what error, in Kant’s own terms, such a philosopher would be committing; and why Kant is not forced ²² Kant (1998), A7/B11. ²³ And it comes down to this, of course, only when the concepts involved do apply to what is beyond the subject. The subject’s knowledge that mermaids have fishes’ tails is arguably another example of analytic armchair knowledge and can arguably be said, in the same attenuated sense of ‘concern’, to concern mermaids. Cf. Kant (1998), A290–2/B346–9. ²⁴ Kant (1998), Bxxvi–xxx.

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to conclude, again in his own terms, albeit against his own conviction, that the subject’s thinking is always of appearances rather than of things in themselves. We can put it this way: when Kant argues from the existence of armchair knowledge concerning what is beyond the subject to the truth of transcendental idealism, at what point in his argument does he make crucial appeal to the fact that the armchair knowledge is synthetic and what, in his own terms, would preclude someone’s extending the argument to armchair knowledge that is analytic? This question is surprisingly unstraightforward. To be sure, there are, in Kant’s presentation of his argument, frequent appeals to the fact that the armchair knowledge is synthetic.²⁵ But if the argument were reworked to eliminate these appeals, it is not obvious what would prevent it from continuing to meet with success; or rather, it is not obvious what would prevent it from continuing to meet with whatever success it meets with in the first place.²⁶ Here are two responses that Kant might give at this point. First, he might say that his own argument for transcendental idealism is an inference to the best explanation (in fact, an inference to the only possible explanation) and that what would prevent it from meeting with the same success if extended to analytic armchair knowledge is the fact that, although it would still count as an inference to an explanation—of how we can have such knowledge—it would no longer count as an inference to the best explanation, because the simpler explanation involving nothing but the subject’s analysis of the concepts involved would still be available. Second, he might say that, not only does he want to allow for thoughts about things in themselves, which the extended version of the argument would rule out, but he is obliged to allow for thoughts about things in themselves, ‘otherwise there would follow the absurd proposition that there is an appearance without anything that appears’.²⁷ Neither of these responses is entirely satisfactory however. The first may beg crucial questions about the relative virtues of rival explanations. Why do the unity and the power of an explanation that applies to all armchair knowledge not count for more than the simplicity of an explanation that applies only to analytic armchair knowledge? To be sure, there would be an obvious answer to this question if the first response were buttressed by the second. But the second may beg crucial questions of its own about the coherence of Kant’s transcendental idealism: if the extended version of his argument leads to a contradiction, it may be a contradiction to which Kant is destined, eventually, to be led anyway. These are enormous issues. I shall say no more about them at this juncture (though I shall return to the issue of the coherence of transcendental idealism in section 5). Instead I want to take a slight detour that will bring us back to question (2). ²⁵ See e.g. Kant (1998), ‘Transcendental Aesthetic’, §8. ²⁶ See e.g. the summary of the argument in Kant (1998), Bxvi–xviii.

²⁷ Kant (1998), Bxxvi.

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4. Invoking Transcendental Idealism to Account for Synthetic Armchair Knowledge We have seen that one of the attractions of Kant’s transcendental idealism is that it accounts for (some) knowledge of what is necessary, as necessary, by locating a grounding for the necessity in contingency. But there are views other than transcendental idealism, indeed views that are not versions of idealism at all, that have title to the same claim. Consider, for instance, the view according to which the subject’s knowledge that vixens are female consists in command of a particular rule of representation, namely the rule that prohibits counting a creature as a vixen without also counting that creature as female. As before, there is an issue about whether such knowledge can be said to concern what is beyond the subject, in however attenuated a sense, or whether it is better described as concerning the subject’s conceptual repertoire or something of that sort. Be that as it may, there is certainly a sense in which it is knowledge of a contingency. For there might never have been any such rule (at least not if rules are conceived as social institutions, which is how I am conceiving them in this essay). The point, however, is that the necessity concerned is not thereby compromised. If there had never been any such rule, vixens would not have failed to be female. Rather, what sex vixens are would not have been an issue for anyone: no one would have thought in those terms. Vixens would not have failed to be female, because vixens must be female. And this ‘must’ is as hard as it either can or need be.²⁸ Now any view of this kind—any Wittgensteinian view, as I shall say²⁹—would be a variant of Kant’s view of analytic armchair knowledge. There would be differences, to be sure. Indeed there would be differences large enough for it to count as a rival view.³⁰ But there would also be a clear family resemblance which there assuredly would not be where Kant’s view of synthetic armchair knowledge is concerned. And, in exploring why not, we shall be helped on our way towards addressing question (2).³¹ On a Wittgensteinian view, there is a clear sense in which, given any item of knowledge to which the view applies, such as the knowledge that vixens are female, sheer familiarity with the concepts involved ensures that one can see the truth of what is known. (This is not because one can derive the truth of what is known from familiarity with the concepts involved. The order of derivation, in so far as there is one, is rather the reverse: one does not count as familiar with the ²⁸ Cf. Wittgenstein (1978), pt VI, §49, and McDowell (1993), pp. 282 ff. ²⁹ But I shall make no attempt to justify the exegesis here. For discussion see A. W. Moore (2019b), §1. ³⁰ For discussion of why it would count as a rival view, possibly even to the extent of having no truck with the notion of analyticity, and for some relevant references to Wittgenstein, see again A. W. Moore (2019b), §1. ³¹ Some of what follows, both in this section and the next, is based on A. W. Moore (2016), which is in turn a response to Baiasu (2016). I am grateful to Sorin Baiasu for the stimulus provided by his excellent essay.

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concepts involved unless one has command of the relevant rule.) But on Kant’s view of synthetic armchair knowledge, there is no clear sense in which, given any item of knowledge to which the view applies, sheer familiarity with the concepts involved ensures that one can see the truth of what is known; precisely not. Kant insists that one cannot see the truth of what is known, in such a case, without appeal to the intuitions involved. What Kant would accept, even in such a case, is that sheer familiarity with the concepts involved ensures that one can see how things must be for what is known to be true. In other words, it ensures that one can see, not the truth of what is known, but the truth conditions of what is known. This is in contrast to sheer familiarity with the logical form of what is known, which leaves the truth conditions of what is known undetermined.³² This in turn explains why, if one wanted to show that what is known is not an analytic truth, one could not avail oneself of any analogue of a procedure that would be available to show that what is known is not a logical truth. If one wanted to show that what is known is not a logical truth, one could specify a false proposition with the same logical form. If one wanted to show that what is known is not an analytic truth, by contrast, one would have to reckon with alternatives to that very truth. And this of course means that, as far as the concepts involved are concerned, there had better be alternatives to that very truth. Suppose, for instance, that the truth in question is that the sum of the angles in a triangle is equal to two right angles. Then there had better be alternatives in which the sum of the angles in a triangle is something other than two right angles. If no such alternative existed—if no such alternative existed, mind, not just if no such alternative were realized—then no such alternative would remain to be ruled out by anyone familiar with the concepts involved. And then there would be a sense, a clear sense, in which sheer familiarity with the concepts involved would ensure that one could see the truth of what is known. But now we are in sight of an answer to question (2), about why Kant thinks that transcendental idealism is needed to solve the original puzzle with respect to synthetic armchair knowledge. On Kant’s view, synthetic armchair knowledge, qua synthetic, is knowledge of a truth that admits of alternatives in the way just outlined; but, qua armchair, it is knowledge of a truth that in some sense admits of no alternatives. It is knowledge, somehow, both of a contingency and of a necessity. Now so too, as we have seen, is the subject’s knowledge that vixens are female, on the Wittgensteinian view. The difference is that, in the Wittgensteinian case, there does not even appear to be any conflict between the contingency and the necessity: the necessity attaches to the known truth itself, that vixens are ³² Here and subsequently in this paragraph, I am prescinding from the fact that, strictly speaking, there is no such thing as ‘the’ logical form of what is known: if what is known is a conjunction, for example, then, even so, it has as one of its logical forms simply ‘p’. Properly formulating the main point that I wish to make in this paragraph, so as to take this fact into account, would involve (tendentious) considerations about complete logical analysis that need not detain us now.

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female; the contingency attaches to the second-order truth that there is a rule in force whose statement consists in an enunciation of that first-order truth. The relevant alternatives are not alternatives to vixens’ being female; they are alternatives to that rule’s being in force, hence to anyone’s having thoughts about vixens. But on Kant’s view of synthetic armchair knowledge, the relevant alternatives are alternatives to the known truth itself: in the example considered above, they are alternatives to triangles’ having angles whose sum is equal to two right angles. So, in the Kantian case, there does appear to be a conflict between the contingency and the necessity. And the only way that Kant can see to resolve this apparent conflict is by introducing some appropriate relativization. He holds that the truth in question is necessary from a particular point of view, the very point of view that the subject’s knowledge is from, constituted, in part, by the intuitions to which the subject appeals in having the knowledge. But when the truth is not considered from that point of view—when a step back is taken to reflect on why appeal to these intuitions is necessary to have knowledge of the truth, which is done precisely by not appealing to them but rather by duly prescinding from them—then Kant thinks that the truth can be conceived as admitting of alternatives. And to make this work, in particular to explain how the subject can have armchair knowledge of what admits of such alternatives, Kant espouses an idealism whereby the necessity in question itself depends on the subject: this is the i-dependence. But the idealism has to be transcendental idealism. The i-dependence has to be conceived as transcending the necessity in question. For the i-dependence cannot so much as be entertained until that step back is taken from the original point of view. Both on Kant’s view and on the Wittgensteinian view, then, there is an attempt of sorts to ground necessity in contingency. But on Kant’s view, unlike on the Wittgensteinian view, the attempt assumes the form that it is peculiarly given by transcendental idealism, so as to allay what would otherwise be a simple conflict between a claim of necessity and a claim of contingency with respect to one and the same truth.

5. The Incoherence of Transcendental Idealism Kant is vindicated, then, as far as question (2) is concerned; or at least, he is vindicated up to a point. But that point comes quickly. In fact it is a point that I anticipated in the discussion of question (1) when I alluded to the possibility that Kant is destined, eventually, to lapse into contradiction, given the way in which he allows us thoughts about things in themselves. I think we are now witnessing such a lapse. For the step back that we have just been considering is not a shift from considering things from one point of view to considering them from another; it is a shift from considering things from one point of view to considering them from no point of view at all, from thinking about how things appear to thinking about

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how they are in themselves. (It involves thinking of how things appear as how they appear and, concomitantly, as different from, albeit dependent on, how they really are, something that we are required to do if we are to avoid ‘the absurd proposition that there is an appearance without anything that appears’.) This would be all very well if such thinking were only ever mere thinking. Kant is not involved in any internal inconsistency simply in allowing us thoughts about things in themselves. The problem is that, by Kant’s own lights, such thinking sometimes amounts to knowledge. Consider, for example, his own thought that there is synthetic armchair knowledge, which cannot be accounted for except by appeal to the distinctive i-dependence that is the hallmark of transcendental idealism. This, by his own lights, is a piece of knowledge—if only because he has arrived at it as a result of what he takes to be a decisive argument³³—and indeed not just a piece of knowledge, but a piece of synthetic armchair knowledge. For Kant would surely deny that it depends on sheer analysis of the concepts involved; but he would also surely deny that it depends on experience. The upshot is that Kant is, by his own lights, forced to accede to the very thing that it is his business to deny, synthetic armchair knowledge of how things are in themselves.³⁴ If this is right, Kant does eventually lapse into contradiction then. Not that this invalidates the argument that he advances from the existence of synthetic armchair knowledge to the truth of transcendental idealism. When I claimed earlier that, as far as question (2) is concerned, Kant is vindicated ‘up to a point’, precisely what I had in mind was the validity of this argument.³⁵ The crux, however, is that we can acknowledge the validity of this argument without lapsing into the same contradiction. For we do not have to conclude that transcendental idealism is true. Where Kant applies a modus ponens, we can apply a modus tollens instead and conclude that there is no such thing as synthetic armchair knowledge. This may be because there is no armchair knowledge at all, or because all armchair knowledge is analytic, or because there is something wrong with the very distinction between analytic armchair knowledge and synthetic armchair knowledge, or . . . But whatever the explanation, something, somewhere, must give.

6. A Priori Intuitions and Pure Concepts My own view is that what should give, first and foremost, is the Kantian thesis, to which I adverted in §4, that there are a priori intuitions, that is to say intuitions ³³ Kant himself uses the phrase ‘indubitably certain’ in this connection: see Kant (1998), A48/B66. Cf. Kant (1996b), 5:12. ³⁴ See A. W. Moore (2012), ch. 5, §§9–10, and A. W. Moore (2016), for further discussion and references. ³⁵ There is an important qualification to this claim that I shall pass over for now but to which I shall return in section 7.

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that the subject already possesses in the armchair, intuitions that are partly constitutive of the subject’s point of view. It is by appeal to such intuitions, Kant believes, and only by appeal to these, that the subject can have synthetic armchair knowledge.³⁶ He holds that space and time are two of them—and the form of all of them.³⁷ Suppose I am right to target this thesis—call it the Intuition Thesis. Suppose (in other words) that there are no a priori intuitions. There may yet be a priori concepts, that is to say concepts that the subject already possesses in the armchair. Of course, Kant himself believes that there are a priori concepts because he believes that there are concepts such as that of a triangle, or that of motion, which the subject possesses by virtue of possessing the two specified a priori intuitions. Importantly, however, Kant also believes that there are a priori concepts that are not thus dependent on those two intuitions. He labels these ‘pure’ concepts. The thesis that there are pure concepts—call it the Concept Thesis—could, as I have said, survive the Intuition Thesis. Kant has a long and elaborate story to tell about how pure concepts and a priori intuitions combine in structuring the subject’s point of view. This story is part of his transcendental idealism. But suppose we accept the Concept Thesis without the Intuition Thesis. Then we need have no truck either with transcendental idealism or with anything like it. Indeed this holds true even if we accept a variant of the Concept Thesis—call it the Relativized Concept Thesis—which allows for the possibility that different subjects, humans and extraterrestrials say, possess different pure concepts. I say ‘even if ’, because Kant expressly holds the corresponding variant of the Intuition Thesis—call it the Relativized Intuition Thesis— which allows for the possibility that different subjects possess different a priori intuitions,³⁸ and we might think that this is what makes his commitment to transcendental idealism mandatory. In fact, however, as I tried to indicate in §4, what makes his commitment to transcendental idealism mandatory is not any contingency that he endorses in the subject’s possessing such and such a priori intuitions; it is rather the contingency that he endorses in those intuitions’ being the way they are. As for any contingency in the subject’s possessing such and such pure concepts, this will be akin, in some respects, to the contingency in the subject’s abiding by such and such rules on the Wittgensteinian View. It will not force us to reckon with any relativization in any associated necessities that the subject acknowledges. Thus suppose that the subject’s possessing such and such pure concepts both ³⁶ For references—but also for discussion of some complications—see n. 49. ³⁷ See e.g. Kant (1998), A20–2/B34–6 and B73. ³⁸ See e.g. Kant (1998), B72. Note, however, that the only possibility to which Kant ever commits himself is an epistemic one. He is careful not to commit himself to what he elsewhere calls a ‘real’ possibility (e.g. Kant (1998), A244/B302). Quite what Kant means by a ‘real’ possibility is a large and difficult question, which does not matter for current purposes.

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involves and is involved in the subject’s acknowledging that things must be thus and so. It does not follow that things might not be thus and so for subjects not possessing those concepts. All that follows is that, for subjects not possessing those concepts, whether things are thus and so is not an issue: they do not think in those terms. Transcendental idealism is not beckoning. I have said that Kant expressly holds the Relativized Intuition Thesis. I have not committed myself one way or the other on whether he holds the Relativized Concept Thesis. I have described the latter as a variant of a thesis that he does hold, the Concept Thesis simpliciter, but I have left open the question whether he subscribes to the variant too. Does he? This is a fascinating exegetical issue in its own right, and I shall spend the rest of this section saying a little about how it might be addressed.³⁹ But I hope that this will also steer us back to some of the important questions about transcendental idealism and Kant’s argument for it that we have been considering. Concerning the exegetical issue, we can distinguish very roughly between textual evidence and philosophical evidence, that is between what Kant actually says and philosophical considerations that can be marshalled to make best sense of what he says.⁴⁰ The textual evidence is unexpectedly inconclusive. There are a few passages in which Kant suggests allegiance to the Relativized Concept Thesis. There are a few passages in which he suggests allegiance to its opposite—call it the Unrelativized Concept Thesis—that all concept-possessing subjects must possess the same pure concepts; or, more specifically, that all concept-possessing subjects must possess ‘our’ pure concepts, where by ‘our’ pure concepts Kant means the twelve fundamental pure concepts that he calls ‘categories’ together with all those that can be derived from them.⁴¹ For the most part, however, when he commits himself to the Concept Thesis simpliciter, he does so in such a way as to suggest nothing whatsoever about which of these variants he holds. Two of the passages in which he suggests allegiance to the Relativized Concept Thesis are Critique of Pure Reason, B145–6, and Prolegomena, 4: 350–1. In the former he refers to the ‘peculiarity’ of the subject’s conceptual faculty, that what it achieves by means of the twelve categories it achieves only by their means and only through such and so many of them; and he says that there is just as little prospect of explaining this peculiarity as there is of explaining the peculiarity of the subject’s intuitive faculty that its only two forms are space and time. In the ³⁹ I have been greatly helped in this part of my essay by discussion with Anil Gomes and Andrew Stephenson. See our joint essay, Essay 2 in this volume, for an extended discussion of the issue. Note that what follows in this section is not especially germane to the rest of the essay, whose main thread I pick up again in section 7. ⁴⁰ It is worth remembering, in connection with this distinction, Kant’s own claim that ‘it is not at all unusual to find that we understand [an author] even better than he understood himself, since he may not have determined his concept sufficiently and hence sometimes spoke, or even thought, contrary to his own intention’ (Kant (1998), A314/B370). ⁴¹ Kant (1998), A64–5/B89–90, A79–83/B104–9, and B306.

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latter he likens the ‘absurdity’ of thinking that there is only one possible way of understanding things, namely in accord with our pure concepts, to the absurdity of thinking that there is only one possible way of intuiting things, namely spatiotemporally. Neither passage settles anything, however. As far as the first is concerned, failure to admit of further explanation is not the same as contingency: some necessities are surd. And as far as the second is concerned, Kant seems to be adverting here, not to subjects whose understanding of things involves pure concepts other than ours, but to subjects such as God whose understanding of things does not involve concepts at all.⁴² The passage in which Kant most strongly suggests allegiance to the Unrelativized Concept Thesis is Critique of Pure Reason, B148. He there contrasts the limited range of space and time, which he says extend only to ‘objects of the senses’, hence only to objects capable of being given by means of spatio-temporal intuitions,⁴³ with the unlimited range of our pure concepts, which he says extend to any objects capable of being given by means of any intuitions whatsoever.⁴⁴ But this too fails to settle anything. True, Kant can be interpreted as saying that, for any subject S who is given objects by means of intuitions (whether these intuitions be spatio-temporal or not), S must, and may, use our pure concepts to think about those objects. But he can just as well be interpreted as saying that, for any subject S who is given objects by means of intuitions and who possesses our pure concepts, S must, and may, use those concepts to think about those objects. There is even an interpretation on which he is saying that, for any subject S who is given objects by means of intuitions, and for any subject S* who possesses our pure concepts, S* must, and may, use those concepts to think about those objects. Admittedly, if S’s intuitions are not spatio-temporal though S*’s are,⁴⁵ then there will not be much that S* is able to think about such objects; and there will be even less that S* is able to know about them. Even so, S* can use (for example) the category of causality-and-dependence to wonder whether every such object depends on some other such object as its cause; and the category ⁴² Cf. Kant (1998), B145. ⁴³ This is not quite the tautology that it appears to be, although it would not matter for current purposes even if it were. ⁴⁴ There is a phrase towards the end of this passage that is ambiguous in the original German and that translators render differently. The phrase is die jene allein enthalten. In Kant (1998) this is rendered as ‘which they alone contain’, while in Kant (1933) it is rendered as ‘which constitutes the whole [of their] content’—which I take to be equivalent to ‘which is all they contain’. The ‘which’ refers to ‘the synthetic unity of apperception’. The ‘they’ refers either to ‘the pure concepts of the understanding’, which Kant explicitly mentions in the previous sentence, or to the ‘forms of thought’ with which he subsequently identifies these. The first version (‘which they alone contain’) is the one that better supports the suggestion that he subscribes to the Unrelativized Concept Thesis. (For further discussion of this matter, see Essay 2 in this volume, §5.2.) ⁴⁵ That S’s intuitions are not spatio-temporal does not, I think, preclude S’s being spatio-temporal, or rather S’s appearing spatio-temporally, say as an inhabitant of some distant planet. Who is to say that S’s non-spatio-temporal intuitions may not facilitate the successful negotiation of spatio-temporal objects in S’s environment? For there may be some deep isomorphism between how the objects of those intuitions are and how those spatio-temporal objects are, perhaps based on some yet deeper isomorphism between how they both are and how things in themselves are.

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of negation to rule out any such object’s violating the principle of contradiction. And if S does not possess any of our pure concepts, then all that follows, as before, is that S does not think in these terms: S never confronts these issues.⁴⁶ So far, so inconclusive, then. What happens when we invoke philosophical evidence? One reason to think that we can make better sense of what Kant says by ascribing the Relativized Concept Thesis to him than by ascribing the Unrelativized Concept Thesis to him is this. If we ascribe the Unrelativized Concept Thesis to him, that is if we credit him with the belief that all conceptpossessing subjects must possess our pure concepts, then we confront awkward questions about the nature of this ‘must’. Given that it is (presumably) the kind of ‘must’ that is characteristic of armchair knowledge, what sort of armchair knowledge? Analytic? Or synthetic? Neither answer is entirely satisfactory. The ‘analytic’ alternative is unsatisfactory because it is in tension with Kant’s suggestion that the necessity in question is some kind of surd that fails to admit of further explanation. This alternative can also appear straightforwardly implausible. Could Kant really think that sheer analysis of the concepts involved would suffice to determine the truth of a thesis as recondite as this, whose very ascription to him is such a bone of exegetical contention?⁴⁷ If we opt for the ‘synthetic’ alternative, however, then we are in danger of opting for something too weak. There is supposed to be a contrast here between our pure concepts and our a priori intuitions. But when the ‘must’ is interpreted in the ‘synthetic’ way, it is no less true, for Kant, that all subjects that are given objects in intuition must be given them spatio-temporally:⁴⁸ the Unrelativized Concept Thesis no longer marks the requisite contrast.⁴⁹ ⁴⁶ I should however record an objection to this third interpretation (to which I paid insufficient attention when I endorsed the interpretation in A. W. Moore (2012), ch. 5 n. 40). The objection is this. There is a clause in the passage—‘as long as [the intuition] is sensible and not intellectual’—which, on this interpretation, amounts to an explicit refusal on Kant’s part to say the same thing about a subject such as God, whose understanding of things does not involve concepts, as he says about S. For by an ‘intellectual’ intuition Kant means an intuition of the kind that such a subject would possess, that is to say, an intuition that itself brings objects into existence rather than serving as the means by which objects are given (see Kant (1998), B71–2). But it is unclear why Kant would refuse to say the same thing about such a subject as he says about S; for would he not think that our pure concepts are as applicable to the objects of such a subject’s intuition as they are to the objects of S’s intuition? On each of the two other interpretations, by contrast, it is clear why such a subject has to be exempted from the claim that Kant is making: the claim that Kant is making is a claim about subjects that need to make use of concepts. All I can offer in response to this objection is the observation that, in a later work, namely Kant (2002b), Kant seems straightforwardly to contradict the offending clause and to say something much more conducive to the given interpretation: he writes, ‘[Our pure concepts] are merely thought-forms for the concept of an object of intuition as such, of whatever kind that may be, and even if it were a supersensible [i.e. intellectual] intuition’ (Kant (2002b), 20: 272, emphasis added; cf. Kant (1996b), 5: 54). ⁴⁷ This is a genuine question, not a rhetorical question. However implausible the view may appear, it is not outrageous. For further discussion see Essay 2 in this volume, §6.1. ⁴⁸ See e.g. Kant (1998), A24/B38–9 and A31/B46. ⁴⁹ Note: I do not deny that different kinds of necessity can answer to synthetic armchair knowledge. But unless and until we are given some reason to draw the relevant distinction between the two claims at stake here, that is the claim that all concept-possessing subjects must (in the ‘synthetic’ sense) possess

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None of this, however, is decisive against ascribing the Unrelativized Concept Thesis to Kant. And if we ascribe the Relativized Concept Thesis to him instead— that is, if we say that there is, for Kant, a contingency in the subject’s possessing ‘our’ pure concepts—then this in turn gives pause. For in so far as it is appropriate to talk about ‘the’ subject,⁵⁰ then surely the fact that the subject possesses these pure concepts counts as an item of armchair knowledge for Kant. And if it does, how can ascription of the Relativized Concept Thesis to him be reconciled with his belief, to which I referred in §1, that all armchair knowledge is knowledge of what is necessary?⁵¹ Note, however, that this may be a philosophical problem for Kant rather than an exegetical problem for anyone ascribing the Relativized Concept Thesis to him. True, the aim of the current exercise is to make best sense of what Kant says. But it would be setting the bar too high to insist that one cannot make best sense of what Kant says unless one absolves him of all internal inconsistency. And there are reasons to think that, even in his own terms, Kant ought not to believe that all armchair knowledge is knowledge of what is necessary. Indeed there are reasons to think that, inchoately and in spite of himself, he already recognizes counterexamples. Thus consider certain remarks that he makes concerning my consciousness

our pure concepts and the claim that all subjects that are given objects in intuition must (in the ‘synthetic’ sense) be given them spatio-temporally, then the former, in mimicry of the latter, cannot but allow for the possibility of concept-possessing subjects who do not possess our pure concepts. An advocate of the Unrelativized Concept Thesis needs to be able to rule out this possibility. That is to say, an advocate of the Unrelativized Concept Thesis needs to be able to rule out the possibility that what makes it true that all concept-possessing subjects must possess our pure concepts is some peculiarity of the understanding akin to the peculiarity of the sensibility that makes it true that all subjects that are given objects in intuition must be given them spatio-temporally. There is a further concern about the ‘synthetic’ alternative which relates to something that I said in the opening paragraph of this section, namely that it is only by appeal to a priori intuitions that Kant believes the subject can have synthetic armchair knowledge. For what appeal to a priori intuitions is being made here? However, I concede that an advocate of the ‘synthetic’ alternative has things to say in response. For one thing, there is room for doubt about whether it is Kant’s view that only by appeal to a priori intuitions can the subject have synthetic armchair knowledge. When I attributed this view to him, I had in mind such texts as Kant (1998), A47–8/B64–6, B73, A62/B87, A155–6/B194–5, and B289, and Kant (2002a), §2, final paragraph. (See also the second of the two notes added by Kant in his own copy of the first edition of Critique of Pure Reason at A158, mentioned by his translators in Kant (1998), p. 283 n. c.) But elsewhere Kant does in fact suggest the opposite, e.g. in Kant (2000), 5: 197 n. 1, and in connection with the fundamental law of morality in Kant (1996a), 4: 420, and Kant (1996b), 5: 31 and 42–3. This is all complicated by the fact that my very talk of synthetic armchair knowledge (as opposed to synthetic a priori cognition, or synthetic a priori propositions) already involves a departure from Kant’s own way of framing these ideas. These issues are far too large for me to address within these confines. I shall merely comment, specifically in connection with Kant (2000), 5: 197 n. 1, that Kant seems to be adverting to the distinction that he draws elsewhere between the way in which mathematicians achieve synthetic armchair knowledge, through the actual exhibition of relevant a priori intuitions, and the way in which philosophers do so, through appeal to the role that a priori intuitions play in the very possibility of experience: see Kant (1998), A713/B741 and A766/B794. ⁵⁰ See again n. 1 above. ⁵¹ This problem is somewhat mitigated by the fact that the contingency in question is epistemic—but only somewhat.

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that I exist,⁵² or concerning what he calls ‘the sole fact of pure reason’.⁵³ Consider, for that matter, his commitment to the Relativized Intuition Thesis. Not that any of these reasons is the primary reason for thinking that there may be a philosophical problem for Kant here. The primary reason for thinking that there may be a philosophical problem for him here is that the problem in question would be a version of the basic problem with transcendental idealism which I tried to identify in the previous section. The contingency in the subject’s possessing our pure concepts, like the contingency in the subject’s intuiting things spatiotemporally, would be one of the contingencies in which the transcendental idealist claims to find a grounding for the various necessities that stand in the relation of i-dependence to the subject. The transcendental idealist must acknowledge these contingencies—but cannot do so from the subject’s point of view. (From the subject’s point of view, there is, as I remarked above, no other way of intuiting things than spatio-temporally, nor any other way of understanding things than in accord with our pure concepts.) To acknowledge these contingencies, the transcendental idealist must therefore take a step back from that point of view. But to credit the transcendental idealist with the capacity to do this is to credit the transcendental idealist with the capacity to gain an insight into how things are in themselves of just the sort that Kant insists is impossible.⁵⁴

7. Accounting for Armchair Knowledge without Invoking Transcendental Idealism I said in §5 that I took Kant’s argument from the existence of synthetic armchair knowledge to the truth of transcendental idealism to be valid. But in the light of the discussion that we have just been having, it is worth emphasizing that I was presupposing a Kantian conception of armchair knowledge whereby such knowledge is always of what is necessary.⁵⁵ Kant’s argument crucially relies on this.⁵⁶ If we admit armchair knowledge of what is not necessary, that is if we admit armchair knowledge of what straightforwardly and unambiguously admits of alternatives, then the argument fails. This is clear as soon as we reflect on one of the contenders mentioned in the previous section. The sheer fact that I can be ⁵² E.g. Kant (1998), B157–9. Here (cf. n. 19) it is important that I am talking about knowledge, not about cognition. ⁵³ Kant (1996b), 5: 6, 31, 42, 43, 55, and 194, and Kant (2000), 5: 468. ⁵⁴ Question: Just as an advocate of the Unrelativized Concept Thesis can be pressed on the issue of what kind of necessity the thesis involves, cannot an advocate of the Relativized Concept Thesis be pressed on the issue of what kind of contingency the thesis involves? And might there not likewise be reasons for thinking that no answer—not even an ‘epistemic’ answer—is satisfactory? Answer: Maybe so, but then this too can be regarded as Kant’s problem. For he is already committed to just such a contingency in his endorsement of the Relativized Intuition Thesis. ⁵⁵ See nn. 8 and 35 above. ⁵⁶ See e.g. Kant (1998), ‘Transcendental Aesthetic’, §8, esp. A46–9/B64–6.

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credited with armchair knowledge of my own existence, if I can, clearly provides no support for transcendental idealism.⁵⁷ But my knowledge of my own existence does not concern what is beyond me. What happens if, at the same time as we admit armchair knowledge of what is not necessary, we prescind from all knowledge except that which concerns what is beyond the subject? Is there then a valid argument from the existence of synthetic armchair knowledge to the truth of transcendental idealism? Even then, I think not. I think it is possible to account for such knowledge by other means. By what other means? I have two suggestions. Each draws inspiration from the work of a great philosopher—albeit only in the sense of being a variation on one of the themes played out by that philosopher. In neither case is there any reason to think that the philosopher in question would be sympathetic to the suggestion. In neither case, come to that, is there any reason to think that sympathy for the suggestion is warranted. For my purposes, this does not matter. Neither suggestion need be plausible to ensure the invalidity of the argument in question, the argument from the existence of synthetic armchair knowledge (appropriately construed) to the truth of transcendental idealism. All that is required, in each case, is that the suggestion cannot be ruled out. My first suggestion, drawing inspiration from the work of Spinoza, is that there is some common feature of things, including the subject, which the subject can knowledgeably self-ascribe in the armchair and which, in Spinoza’s memorable phrase, ‘is equally in the part as in the whole’.⁵⁸ It is a difficult question what Spinoza himself means by this phrase. But what I mean by it is this: for a feature to be equally in the part as in the whole is for that feature to be like Euclidean spatiality in that its very possession by some given finite thing testifies to its possession by some larger thing of which that finite thing is a part. If there were such a feature, and if the subject could knowledgeably self-ascribe it, then the subject could, by extrapolation, have knowledge that some larger thing possessed the feature, and hence could have knowledge concerning what lies beyond the subject—all in the armchair.⁵⁹ This admittedly presupposes that the knowledge in question is not compromised by the existence of alternatives, that is by counterparts of non-Euclidean spatiality. For there must be such alternatives. If there were not—if the knowledge in question were knowledge of what is necessary, à la ⁵⁷ This is an apt point at which to signal another advantage that my use of the label ‘armchair knowledge’ has over the more familiar ‘a priori knowledge’. Among those who would be prepared to credit me with armchair knowledge of my own existence, there are many who would have qualms about classifying such knowledge as ‘a priori’. What rationale there would be for such qualms need not concern us here: I merely note that my use of the label ‘armchair knowledge’ ensures that I do not beg any questions in this regard. ⁵⁸ Spinoza (2002a), pt II, prop. 38. ⁵⁹ Cf. Spinoza’s argument in the proof of Spinoza (2002a), pt II, prop. 38. Not that I deny that there are important differences between Spinoza’s argument and mine.

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Kant—then there would once again be the problem of reconciling this lack of alternatives with the syntheticity of the knowledge. My second suggestion, drawing inspiration from the work of Aristotle, has three components: first, that the subject’s mind shares a form with certain things that are beyond the subject; second, that the subject thereby understands what it is for those other things to have this form and thereby has knowledge concerning those other things; and third, that none of this requires the subject to leave the armchair. Aristotle’s own view lacks the third component. He does not believe that the subject’s mind can share a form with other things in this way unless and until the subject has duly investigated those other things, and he believes that this does require the subject to leave the armchair.⁶⁰ But, even on Aristotle’s view, there is a sense in which the subject is eventually able to access knowledge concerning those other things through a kind of introspection. (Aristotle himself says that ‘the mind . . . is then able to think itself ’.⁶¹) All I am envisaging is that the subject’s mind has this form as a result of a pre-established harmony rather than as a result of investigation. As in the case of the first suggestion, this presupposes that the resultant knowledge is not compromised by the existence of alternatives in which some of the particularities of what the subject knows about the things that have the form do not hold. I referred just now to a pre-established harmony. This helps to highlight the most important difference between Kant’s own account of synthetic armchair knowledge and these two rivals. Kant, to repeat, thinks that such knowledge must be knowledge of what is necessary. Not only that; as I have more than once emphasized, he thinks that such knowledge must be knowledge of what is necessary as necessary. The necessity is revealed in the fact that the subject, at least while considering things from the point of view that the knowledge is from, cannot entertain alternatives. This means that the range of possibilities for how things are corresponds to the range of possibilities that the subject, considering things from that point of view, can entertain for how they are. And unless this is just some brute coincidence, there are only three possible explanations: that the former range is determined by the latter; that the latter range is determined by the former; or that the two ranges have some common determinant, in other words that there is a pre-established harmony. But only the first of these, that the former range is determined by the latter, serves to explain how the subject can have armchair knowledge of the coincidence; that is, armchair knowledge of the coincidence, of the fact that things not only are, but must be, thus and so.⁶² And to invoke the first of these, that is to accept that the range of possibilities for how things are is determined by the range of possibilities that the subject can entertain ⁶⁰ See Aristotle (1941d), bk3, chs 4–5. ⁶¹ Aristotle (1941d), bk 3, ch. 4, 429b8–9, emphasis in original. ⁶² Kant (1998), A92–3/B124–6 and B166–8.

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for how they are, is to appeal to the i-dependence that is characteristic of transcendental idealism. But this does not, I hardly need say by now, provide any ultimate vindication of Kant. Kant’s belief that the knowledge at stake is knowledge of what is necessary is precisely the thing which, as we have seen in the two previous sections, creates insuperable difficulties for him. My own view, as I intimated in §5, is that the primary casualty of these reflections is Kant’s very idea that there is synthetic armchair knowledge.⁶³

⁶³ I am very grateful to Anil Gomes, Andrew Stephenson, Robert Stern, and participants at a workshop on ‘Kant on A Priori Knowledge and the Necessity of the Categories’, at the University of Fribourg, for valuable discussion of the issues that I have pursued in this essay.

2 On the Necessity of the Categories written jointly with Anil Gomes and Andrew Stephenson

Abstract Kant holds that human cognition involves two faculties, sensibility and the understanding, and that each of these has pure forms: space and time in the case of sensibility, and the twelve categories in the case of the understanding. But Kant is also careful to leave open the possibility of there being creatures like us, with both sensibility and understanding, who nevertheless have different pure forms of sensibility. The question is raised whether he leaves open an analogous possibility in the case of the understanding, that is to say the possibility of creatures like us, with both sensibility and understanding, who nevertheless have different categories. It is argued that textual and systematic considerations do not determine an answer to this question. It is further argued that Kant might be neutral on the issue but that, if he is, the exact form his neutrality takes is subject to unexpected constraints.

Thus here is a case where the common saying holds, that no answer is an answer. —Immanuel Kant (A478/B506)¹

1. Introduction The Critique of Pure Reason aims to explain the possibility of synthetic a priori judgement. Kant’s explanation of this possibility involves certain claims about the structure of the mind. This much is straightforward. As too is the general shape of his explanation. It rests on the fact that there are two faculties to the cognitive

¹ References to Kant’s works are given by the Academy edition volume and page number along with abbreviations that are listed at the end, except for those to the Critique of Pure Reason, which take the standard A/B format. Translations are from the Cambridge edition listed at the end of the essay.

The Human A Priori: Essays on How We Make Sense in Philosophy, Ethics, and Mathematics. A. W. Moore, Oxford University Press. © A. W. Moore 2023. DOI: 10.1093/oso/9780192871411.003.0003

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mind, a passive faculty of sensibility and an active faculty of the understanding. Each has its own representations by means of which we relate to objects. Sensibility gives us objects by means of intuitions; we think of objects by means of concepts. Kant’s explanation of the possibility of synthetic a priori judgement turns on the claim that each of these faculties has its own a priori elements. Sensibility has pure intuitions, space and time; the understanding has pure concepts, the categories. It is these a priori elements to sensibility and the understanding—their pure forms—that explain that which Kant takes to require explanation. So far, so good. But Kant is careful to limit his claims about space and time to human sensibility. He is careful to leave open the possibility of creatures like us, with both sensibility and understanding, who nevertheless have different pure forms of sensibility. They would be finite rational beings and discursive cognizers. But they would not be human. And this raises a question about the a priori element to the understanding. Does Kant likewise limit his claims about the categories to human understanding? Specifically, does he leave open the possibility of discursive cognizers—cognizers with both sensibility and understanding, that is with sensibly conditioned intellects—who nevertheless have different intellectual forms from our own? These are the questions that we shall pursue in this essay. Their answers matter. Kant takes the world, in some sense, to depend on us, in some sense. But who is that ‘us’ on which the world depends? How much of our cognitive engagement with the world is shaped by our human nature, and how much by our discursive nature more generally? These questions get to the heart of the role humanity plays in Kant’s Critical philosophy. Many of Kant’s readers have taken him to be the great humanizing philosopher of the modern period. All of the questions of philosophy, he tells us, are contained within the question, ‘What is the human being?’,² and the Copernican turn looks to put human beings at the centre of knowledge and reality. Others have taken Kant’s commitment to a truly transcendental philosophy to require a withdrawal from the level of the human being to something more universal.³ The answers to our questions bear on these deeper issues. Our aim in this essay is to make our questions precise and to show the difficulty in answering them. We proceed as follows. In sections 2–4 we examine Kant’s claims about the possibility of discursive cognizers with other sensible forms and formulate structurally analogous and disanalogous claims about other intellectual forms. The result is a clear statement of two opposing positions that one might attribute to Kant. One holds that Kant leaves open the possibility of discursive cognizers with other intellectual forms; the other holds that Kant rules out such a possibility. In sections 5–6 we examine a number of textual and systematic

² JL 9: 25.

³ For discussion, see Tolley (forthcoming).

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considerations and argue that they do not settle the question of which view Kant endorses. The apparent inability of the textual and systematic considerations to settle our debate opens up an intriguing possibility: that Kant is neutral on the issue. We explore this option in sections 7–8. It indicates what would be an important asymmetry between Kant’s treatment of sensible and intellectual forms. For in the case of discursive cognizers with other sensible forms, Kant thinks we cannot know whether such beings are possible. But neutrality here would involve the second-order claim that we cannot know whether we can know whether discursive cognizers with other intellectual forms are possible. In section 7 we argue that such a view is compatible with Kant’s commitment to decidability in transcendental philosophy. In section 8 we argue that the exact form such neutrality can take is nevertheless constrained by Kant’s commitment to a kind of luminosity principle. To put it somewhat provocatively, neutrality demands silence.

2. Sensibility and Undecidability We start with sensibility. Sensibility is the capacity ‘to acquire representations through the way in which we are affected by objects’.⁴ This capacity is realized by means of intuitions, immediate and singular representations through which objects are given to us.⁵ Empirical intuitions relate us to objects through sensation. And they possess a form, a way of ordering the matter of intuition, which ‘must all lie ready for it in the mind a priori’.⁶ This is the pure form of sensibility. Kant makes a further claim: that the pure form of human sensibility subsumes two more specific forms, space and time. But this claim is explicitly limited to human sensibility. Consider the following passages:⁷ For we cannot judge at all whether the intuitions of other thinking beings are bound to the same conditions that limit our intuition, and that are universally valid for us.⁸ It is also not necessary for us to limit the kind of intuition in space and time to the sensibility of human beings; it may well be that all finite beings necessarily agree with human beings in this regard (though we cannot decide this).⁹

These passages restrict the claims about space and time to human sensibility. How should we understand Kant’s remarks? There is a sense in which he is acknowledging that forms of sensibility other than our own are possible: he is acknowledging that we cannot rule them out. This, however, is a purely epistemic matter. It would be a mistake to read Kant as saying that forms of sensibility other ⁴ A19/B33. ⁵ A19/B33 and A68/B93. ⁷ Cf. A26–7/B42–3, B71, B139, A230–1/B238.

⁶ A20/B34. ⁸ A27/B43.

⁹ B72.

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than our own are possible in any nonepistemic sense.¹⁰ Kant does not, here or anywhere else, assert any such thing. On the contrary, on the question whether forms of sensibility other than our own are possible in a nonepistemic sense, or more specifically whether they are possible in whatever nonepistemic sense is at stake in these passages, Kant remains explicitly and resolutely agnostic. He refuses to commit either way. ‘It may well be that all finite beings necessarily agree with human beings’—that is, nonspatiotemporal sensible forms may be impossible— ‘though we cannot decide this.’ And ‘we cannot judge at all whether the intuitions of other thinking beings are bound [presumably necessarily] to the same conditions that limit our intuition’. What Kant commits to in these passages is an undecidability thesis: we cannot know whether or not discursive cognizers with other forms of sensibility are, in whatever sense of possibility is at stake here, possible. Very well, what sense of possibility is at stake here? What kind of possibility does Kant want to leave open? Presumably not logical (i.e. conceptual) possibility. Kant says that we cannot judge or decide the matter. And presumably he thinks we can judge or decide whether other forms of sensibility are logically possible. For there seems to be no contradiction in the concept of a nonspatiotemporal sensibility. And there is no suggestion in Kant that the spatiotemporal form of our own sensibility is supposed to follow analytically from the defining features of sensibility as such—receptivity, passivity, and so forth. Kant would not think we need to remain agnostic about the logical possibility of a nonspatiotemporal sensibility. Perhaps, then, Kant means to assert that we cannot judge or decide whether other forms of sensibility are what we might call ‘formally possible’—that is, in agreement with our own sensible forms.¹¹ But this also cannot be right. For it is trivial, and so knowable, that other forms of sensibility are not in agreement with our own sensible forms. This reading likewise fails to explain why Kant thinks we are not in a position to settle the question. We take it, then, that Kant must rather be working with some notion of real possibility, a kind of possibility that approximates the contemporary notion of metaphysical possibility. There are questions, of course, about how exactly to characterize such a notion of possibility. We won’t pursue them here.¹² For our purposes we can leave our talk of possibility as a placeholder for whatever kind of real possibility is appropriate in the context. Kant’s claim is that we cannot know

¹⁰ For relevant interpretations, see e.g. Falkenstein (1995), p. 199; Carson (1997), p. 503; Van Cleve (1999), p. 40; Maddy (1999), p. 101; Maddy (2012), p. 488; and Marshall (2014), p. 550. ¹¹ A218/B265. ¹² For general discussion, see e.g. Chignell (2009); Chignell (2012); Stang (2016); and Leech (2017). For discussion of the issue in this context, see Abaci (2019), pp. 190–200; Gurofsky (2020); and Kohl (2021).

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whether or not discursive cognizers with other forms of sensibility are really possible. Unless relevant, we leave this qualification implicit in what follows. Kant holds that we cannot judge or decide whether discursive cognizers with other forms of sensibility are really possible. How should we understand these notions? We shall take them to be epistemic notions that entail our being unable to know whether discursive cognizers with other forms of sensibility are possible. But what kind of knowledge does Kant think we cannot have? There are a multitude of epistemic notions in Kant: analytic knowledge, synthetic knowledge, scientific knowledge (Wissen), cognition (Erkenntnis), perhaps others. Which of these is in play will depend in part on what kind of knowledge he thinks might otherwise be available for the possibility at issue. We cannot have analytic knowledge of real possibility.¹³ But if logical possibility is a condition on real possibility, then perhaps we can have analytic knowledge of real impossibility. The issue is delicate, and we return to it later. For present purposes we leave our talk of knowledge as a placeholder for whatever kind of epistemic relation (to whatever kind of real possibility) is appropriate in the context. We are now in a position to begin to articulate our target question. Modulo the preceding, Kant’s claim in the above passages is that we cannot know whether or not discursive cognizers with other sensible forms are possible. This undecidability thesis is not our concern in this essay.¹⁴ Instead our concern is whether Kant endorses a symmetrical undecidability thesis about the understanding: () We cannot know whether or not discursive cognizers with other intellectual forms are possible. This is one answer to our initial question. If Kant endorses Undecidability, then his view of discursive cognizers with other intellectual forms is perfectly on a par with his view of discursive cognizers with other sensible forms. Undecidability is equivalent to the conjunction of two claims: we cannot know that discursive cognizers with other intellectual forms are possible, and we cannot know that discursive cognizers with other intellectual forms are impossible. There will then be two ways of directly opposing the view, each corresponding to the negation of one of its conjuncts. First: () We can know that discursive cognizers with other intellectual forms are possible. While Undecidability leaves open the possibility of discursive cognizers with forms of understanding other than our own, Contingency positively affirms this ¹³ See e.g. Bxxvi and Prog. 20: 325–6. ¹⁴ It is the topic of Gomes and Stephenson, (forthcoming b).

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possibility; it thereby suggests that our own intellectual forms, although necessary for us, are in some deeper sense contingent for discursive cognizers as such. Second: () We can know that discursive cognizers with other intellectual forms are impossible. While Undecidability leaves open the possibility of discursive cognizers with forms of understanding other than our own, Necessity rules out this possibility; it suggests that our own intellectual forms are not only necessary for us but also in some deeper sense necessary for discursive cognizers as such. Contingency and Necessity are contrasting decidability theses. They are incompatible with Undecidability and with each other. There is an alternative way of carving up the terrain. Consider an epistemic conception of possibility according to which a claim is epistemically possible just in case it is compatible with what we can know. Both Undecidability and Contingency hold that it is epistemically possible that discursive cognizers with other forms of the understanding are really possible, while Necessity denies this. This alternative taxonomy offers a helpful framework for understanding an objection that goes back at least to Hegel: that Kant’s idealism threatens to collapse into a form of subjective idealism that precisely fails to rule out this epistemic possibility.¹⁵ We will not assess the merits of this objection here, and since for our purposes it will be important to distinguish Undecidability and Contingency (see section 4), we will continue with the current taxonomy. Our question is whether Kant endorses Undecidability, Contingency, or Necessity. Kant’s explicit undecidability thesis about the possibility of discursive cognizers with other sensible forms has provided us with the materials for distinguishing three views he might hold about discursive cognizers with other intellectual forms. But Kant’s complex conception of intellectual form means that each of these views can be further segmented. To complete the taxonomic space, we turn to the understanding.

3. The Understanding In his discussion of sensibility, Kant characterizes the understanding only negatively, as a nonsensible faculty of cognition. He later gives us a positive ¹⁵ See the Science of Logic, vol. 2, §§1312 and 1338; Encyclopedia, pt 1, §42; Lectures on the History of Philosophy, the section on Kant, in Hegel 1968–, vols 12, 13, and 30, respectively. For some especially relevant discussion, including of similar concerns in e.g. Reinhold and Fichte, see Horstmann (1995); Horstmann (2010); Ameriks (2000); Ameriks (2015); Förster (2002); Pippin (2005); McDowell (2009), ch. 4; and Houlgate (2015). See also Heidegger (1929), §§10–12.

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characterization of the understanding as ‘a faculty for judging’ or a ‘faculty for thinking’.¹⁶ Thinking is ‘cognition through concepts’.¹⁷ Concepts rest on functions, and a function is ‘the unity of the action of ordering different representations together under a common one’.¹⁸ Putting this together, we get the idea that the role of the understanding is to order different representations under common ones. The faculty’s ‘supreme principle’ is the ‘unity of apperception’.¹⁹ Kant’s identification of an a priori element to the understanding resides in the fact that he thinks there are only certain functions of unity by means of which we can order representations in judgement. These are the logical functions of the understanding in judgement, and they are listed in the Table of Judgement.²⁰ The table has four titles, each of which has three moments. Since ‘the same function that gives unity to the different representations in a judgment also gives unity to the mere synthesis of different representations in an intuition, which, expressed generally, is called the pure concept of understanding’,²¹ we are able to identify twelve pure concepts of the understanding, each corresponding to one of the logical functions of judgement and likewise organized into a table, the Table of Categories. These concepts comprise ‘all original pure concepts of synthesis that the understanding contains in itself a priori’.²² In section 2 we saw that Kant thinks we cannot know whether or not discursive cognizers with different pure forms of sensibility are possible. Kant’s distinction between the Table of Judgement and the Table of Categories gives us two levels at which we might formulate a symmetrical undecidability thesis about the intellectual forms. First, we might take it to be undecidable whether discursive cognizers with different functions of judgement are possible. Second, we might take it to be undecidable whether discursive cognizers with different pure concepts are possible. Intellectual form has two aspects and thus two ways in which it might vary among discursive cognizers. Intellectual form will be variable if either of its aspects is variable, or equivalently, it will be invariable only if both of its aspects are invariable. Thus those who think it is undecidable whether or not other discursive cognizers are possible with respect to both aspects of intellectual form are committed to Undecidability. Those who think that we can know both that discursive cognizers with other functions of judgment are impossible and that discursive cognizers with other pure concepts are impossible are committed to Necessity. Contingency says that we can know that intellectual form is variable, so it is equivalent to a disjunction: either we can know that discursive cognizers with other functions of judgment are possible or we can know that discursive cognizers with other pure concepts are possible.

¹⁶ A69/B94, emphasis in original. ¹⁷ A69/B94. ²⁰ A70/B95. ²¹ A79/B104–5, emphasis in original.

¹⁸ A68/B93. ²² A80/B106.

¹⁹ B136.

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Given the connection between Kant’s two tables, we will assume throughout that if the functions of judgment are variable, then so are the pure concepts. More fully, we will assume that, if it is possible for there to be discursive cognizers with different functions of judgement, then it is possible for there to be discursive cognizers with different categories. And for the majority of our discussion, we will also assume the converse: if it is possible for there to be discursive cognizers with different categories, then it is possible for there to be discursive cognizers with different functions of judgement. This assumption is warranted by Kant’s insistence on there being a kind of identity between these two aspects of intellectual form.²³ It ensures that they cannot vary independently of one another, leaving us with the same options as before. But someone might hold that the categories could vary while the functions of judgement remain fixed, and this could lead to mixed views. We consider that scenario explicitly in section 6. There is one further complication. In addition to distinguishing between the functions of judgement and the pure concepts, Kant also distinguishes between general logic and transcendental logic. General logic concerns ‘the form of thinking in general’, whereas transcendental logic ‘has to do merely with the laws of the understanding and reason, but solely insofar as they are related to objects a priori’.²⁴ The Table of Categories belongs to transcendental logic, as does the full Table of Judgement, which includes a moment of singular quantity distinct from the moment of universal quantity and a moment of infinite quality distinct from the moment of affirmative quality. But Kant sometimes talks as though a table of judgement without these third moments of quantity and quality would belong to general logic.²⁵ And while certain laws and inference rules, like noncontradiction, modus tollens, and excluded middle, clearly belong to general logic,²⁶ it is far less clear how exactly they relate to the Table of Judgement. These observations raise a set of interesting questions about the relation of general logic to the functions of judgement and the categories. For our purposes, and given that Kant thinks of general logic as an aspect of the pure form of the understanding,²⁷ it raises the question whether it provides a further level at which we can ask our question whether Kant endorses undecidability about other intellectual forms. Specifically: does Kant think that we can know whether or not discursive cognizers with other general logics are possible, or is he as resolutely agnostic about logical aliens as he is about sensible aliens?²⁸ We will not pursue this aspect of our topic here. It is relevant if one thinks that we can know that discursive cognizers with other general logics are possible, for then—given the connections between general and transcendental logic—we should also be in a position to know that discursive cognizers with other functions of judgement and other pure concepts are possible. Such a view on general logic ²³ A79/B104–5 and B143. ²⁴ A55–7/B79–82. ²⁵ A71–2/B96–7. ²⁶ JL 9: 51–3. ²⁷ A50–5/B74–9. ²⁸ For discussion, see Conant (1991a); Nunez (2018); and Miguens (2020).

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would all but entail Contingency. But the other stances that one might take on general logic leave open all the options on the other aspects of intellectual form. For instance, a necessity thesis about general logic does not obviously entail a necessity thesis about the categories—that would depend on the precise relation between general logic and the categories. Thus, although it is related, the question whether or not discursive cognizers with other general logics are possible is distinct from the question that is our concern here. We are now in a position to begin evaluation of the options. Our question concerns discursive cognizers with other intellectual—that is, judgemental and categorial— forms. We have introduced three views. Contingency holds that we can know that discursive cognizers with other intellectual forms are possible. Necessity holds that we can know that discursive cognizers with other intellectual forms are impossible. Undecidability holds that we cannot know whether or not discursive cognizers with other intellectual forms are possible. Which, if any, does Kant endorse?

4. Against Contingency We begin with a brief comment on Contingency. It holds that we can know that discursive cognizers with other intellectual forms are possible. Since Kant holds that we cannot know whether discursive cognizers with other sensible forms are possible, it is a view on which our knowledge is not symmetrically limited with regard to other intellectual forms. We do not think that Contingency is a plausible view. Our reasons for this will come out in the rest of the essay. In this section we just want to motivate our decision not to consider Contingency in detail alongside the other two views. Let us start by flagging a potential confusion. We have said that Kant commits to an undecidability thesis about our sensible forms: he holds that we cannot know whether discursive cognizers with other sensible forms are possible. However, it is not uncommon to ascribe to Kant the view that discursive cognizers with other sensible forms are possible. This would be an endorsement of contingency about the pure forms of sensibility. We think that the passages above do not admit of this reading. But if one did read Kant in this way, then (intellectual) Contingency would look to be the symmetrical view about the intellectual forms. And this is important, because some of those who have endorsed Contingency seem to be motivated by its supposed symmetry with Kant’s claims about sensibility. That is, they take Contingency to be the symmetrical claim to that which Kant endorses in the passages above about the sensible forms.²⁹ But this is incorrect. Considerations about symmetry do not support ascribing Contingency to Kant. ²⁹ For relevant interpretations, see Van Cleve (1999), p. 40; Buroker (2006), pp. 63 and 100; and Waxman (2014), pp. 285–6.

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Indeed, given Kant’s commitment to undecidability about the sensible forms, it seems to us that Contingency should look dubious from the outset. If we cannot know that discursive cognizers with other sensible forms are possible, how could we fare better when it comes to knowing that discursive cognizers with other intellectual forms are possible? There seem to us no good reasons to think that there is something that prevents us from knowing whether there could be discursive cognizers with other sensible forms but that nevertheless presents no obstacle to our knowing that discursive cognizers with other intellectual forms are possible. The converse does not hold: there is nothing on the face of it which is problematic about allowing that we can know more about the impossibility of discursive cognizers with other intellectual forms than we can about the impossibility of discursive cognizers with other sensible forms. It is only Contingency, not Necessity, that looks problematic when situated against Kant’s commitment to undecidability about other sensible forms. So if you think, contra Necessity, that Kant wants to leave epistemic room for the possibility of other intellectual forms, then Undecidability is the option for you. For these reasons, we will not consider Contingency any further.

5. Textual Considerations We are left with Undecidability, the view that we cannot know whether or not discursive cognizers with other intellectual forms are possible, and Necessity, the view that we can know that discursive cognizers with other intellectual forms are not possible. We begin in this section with textual considerations. Since Contingency is off the table, we assume in what follows that considerations that tell against Necessity tell in favour of Undecidability and vice versa.

5.1. Against Necessity We start with the following well-known passage, which has often been thought to tell against Necessity: But for the peculiarity of our understanding, that it is able to bring about the unity of apperception a priori only by means of the categories and only through precisely this kind and number of them, a further ground may be offered just as little as one can be offered for why we have precisely these and no other functions for judgment or for why space and time are the sole forms of our possible intuition.³⁰

³⁰ B145–6. Cf. Prol. 4: 318 and C 11: 51.

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The passage seems to say that, in the same way that we can give no explanation of why we have the forms of sensibility that we do, so too we can give no explanation of why we have the categories and functions of judgement that we do. If we can give no explanation of these facts, then one might think that we cannot rule out the possibility of discursive cognizers whose forms of sensibility and understanding differ. This looks to contravene Necessity, which holds that we can rule out the possibility of discursive cognizers with other intellectual forms.³¹ This is a challenging passage for the proponent of Necessity, but it is not decisive. The passage says that the sensible and intellectual forms are symmetric in that, in both cases, no further ground may be offered as to why we have them. But it is compatible with this symmetry that there is an asymmetry for which no further ground is available. In the case of the sensible forms, there is no further ground to be offered for why they are the forms of specifically human discursive cognition. In the case of the intellectual forms, it may be that there is no further ground to be offered for why they are the forms of discursive cognition in general. The passage allows such a reading. Moreover, since the claim is that no further ground can be offered, we ought to look to the broader context of the passage to see what grounds have been offered up to this point. It occurs in the Transcendental Deduction, §21, which is to say as a ‘Remark’ on the already completed first step of the Deduction. Not least, then, it occurs after the main discussion of the unity of apperception, and we may well expect that no further ground for the intellectual forms can be given at this stage in Kant’s argument. The Necessity theorist may hold that everything that has been said already about the nature of the understanding in general suffices to ground our possession of these forms and no others.³² We return to this passage in a moment. First, it will be instructive to consider another set of passages that might be thought to tell against Necessity. The following from the Prolegomena is an example: [I]t would be an even greater absurdity for us not to allow any things in themselves at all, or for us to want to pass off our experience for the only possible mode of cognition of things—hence our intuition in space and time for the only possible intuition and our discursive understanding for the archetype of every possible understanding—and so to want to take principles of the possibility of experience for universal conditions on things in themselves.³³

³¹ See Krüger (1968) for the classic statement of this reading. See also Buroker (2006), p. 100. ³² See Wolff (1995), pp. 177–81 for the classic statement of this response, where he points out that it also applies to two related passages (Prol. 4: 318; and C 11: 51). See also Allison (2004), pp. 135–6, and Allison (2015), p. 376. We return to the issue in §6. For further discussion of the Deduction in this context, see Gomes and Stephenson, (forthcoming a). ³³ 4: 350–1.

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This passage counsels against taking our forms of sensibility to be the only forms of sensibility and, correspondingly, against taking our form of understanding to be the only form of understanding. Again, the implication would seem to be that we should not rule out the possibility of cognizers with other forms of understanding. This would tell against Necessity. The Necessity theorist can provide an alternative reading of this passage. It certainly does counsel against our ruling out the possibility of beings with other forms of understanding. But it is a further step to say that it counsels against our ruling out the possibility of discursive beings with other forms of understanding. And it is this latter claim that is needed if it is to be used against Necessity. An alternative construal of the passage reads it in light of those texts in which Kant distinguishes discursive from nondiscursive cognition.³⁴ Kant is clear, across these texts, that we must allow for the possibility of nondiscursive cognizers, beings for whom cognition does not depend on the cooperation of two distinct faculties, sensibility and the understanding. If there could be any such beings, then our forms of experience will not be the only possible mode of cognition of things. This is what the passage from the Prolegomena tells us. But to allow that is not yet to allow that there could be discursive cognizers whose understanding has different forms. The passage from the Prolegomena is silent on this further question. This suggests a general strategy open to the Necessity theorist for reading any particular text in which it looks as though Kant is leaving room for the possibility of discursive cognizers with alternative forms of the understanding. The Necessity theorist can maintain that, properly contextualized, the passage in question only allows the possibility of nondiscursive cognizers. And the possibility of nondiscursive cognizers does not tell against the impossibility of discursive cognizers with alternative intellectual forms. With this strategy in hand, let us now return to the first passage we discussed, about the supposed ‘peculiarity of our understanding’ at B145–6. Here is the rest of the paragraph from which that passage is taken: In the above proof, however, I still could not abstract from one point, namely, from the fact that the manifold for intuition must already be given prior to the synthesis of understanding and independently from it; how, however, is here left undetermined. For if I wanted to think of an understanding that itself intuited (as, say, a divine understanding, which would not represent given objects, but through whose representation the objects would themselves at the same time be given, or produced), then the categories would have no significance at all with regard to such a cognition. They are only rules for an understanding whose entire capacity consists in thinking, i.e., in the action of bringing the synthesis of the ³⁴ E.g. B72, B138–9, B148–9, A252, A254–5/B309–10, and A286–8/B342–4; CPJ 5: 405–10; and C 10: 130–1.

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Far from telling against Necessity, this part of the passage can seem to tell in favour of the view. To be sure, Kant points out that the categories have ‘no significance’ for a nondiscursive understanding. But his contrast class—the class of understanding for which the categories do have significance—might be interpreted as the discursive understanding in general, not the human understanding specifically. And on this interpretation, the passage supports Necessity, the view that the categories have significance for the discursive understanding in general.

5.2. Against Undecidability We have considered passages that seem to tell against Necessity. Are there passages that seem to tell against Undecidability? Consider the following, from a little later in the Transcendental Deduction: The pure concepts of the understanding are free from this limitation [that is, the limitation of space and time, that they are conditions of the possibility of how objects are given to us and hence apply only to objects of experience] and extend to objects of intuition in general, whether the latter be similar to our own or not, as long as it is sensible and not intellectual.³⁶

Here Kant states that the pure concepts of the understanding extend to objects of sensible intuition in general, and one might read that as saying that any being that is given objects in sensible intuition must use the pure concepts of the understanding to think about those objects. This would seem to rule out the possibility of discursive cognizers with different pure concepts of the understanding, contrary to Undecidability. But this is an overreach. The passage says only that all objects of sensible intuition can be thought by means of the categories. It follows that there can be no sensible objects which could not be thought by creatures possessing the categories. It doesn’t follow that there could not be other creatures possessing different forms of the understanding who could similarly think all objects of sensible intuition, albeit by means of their own pure concepts.³⁷ Perhaps the same objects can be thought by means of different pure concepts. So the passage is compatible with Undecidability. ³⁵ B145, emphasis in original. ³⁶ B148. ³⁷ For a similar response, see A. W. Moore (2012), p. 122 n. 40, and Essay 1 in this volume, §6.

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A wider strategy open to Undecidability theorists can be brought out by considering a textual objection thrown up by the rest of the passage from which the above sentence is taken. Here is the whole passage: The pure concepts of the understanding are free from this limitation and extend to objects of intuition in general, whether the latter be similar to our own or not, as long as it is sensible and not intellectual. But this further extension of concepts beyond our sensible intuition does not get us anywhere. For they are then merely empty concepts of objects, through which we cannot even judge whether the latter are possible or not—mere forms of thought without objective reality—since we have available no intuition to which the synthetic unity of apperception, which they alone contain, could be applied, and that could thus determine an object. Our sensible and empirical intuition alone can provide them with sense and significance.³⁸

We have already considered the first sentence of the passage. But now consider what the passage says about the relation between the pure concepts of the understanding and the unity of apperception: that ‘they alone contain’ the synthetic unity of apperception. The German (‘jene allein enthalten’) is ambiguous between two readings. Either Kant means that only the pure concepts contain the unity of apperception, or he means that the pure concepts contain only the unity of apperception.³⁹ Both readings can seem to support Necessity and tell against Undecidability. Take the first: only the pure concepts contain the unity of apperception. This means there are no other concepts that contain the unity of apperception. So it looks as though there can be no discursive cognizers—cognizers with a discursive understanding, and thus with the unity of apperception—who have other concepts containing the unity of apperception. Since for something to be a pure concept of the understanding, it must contain the unity of apperception, these pure concepts must be the only pure concepts of the understanding. Take the second reading: the pure concepts contain nothing but the unity of apperception. Now if there are discursive cognizers with different pure concepts, their pure concepts too must contain nothing but the unity of apperception. But, for Kant, facts about containment relations set the individuation conditions for concepts.⁴⁰ In particular, if concept A contains only concepts C and D, and concept B contains only concepts C and D, then concept A just is concept B. It follows that the pure concepts of other discursive cognizers would be identical to our pure

³⁸ B148–9, emphasis in original. ³⁹ Paul Guyer and Allen W. Wood, in the translation from which we have been quoting, render it in the first way. Both Norman Kemp Smith and Werner S. Pluhar render it in the second. ⁴⁰ For discussion, see R.L. Anderson (2015), ch. 2.

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concepts, contrary to the hypothesis. On either reading, the passage seems to support Necessity. There might well be ways for the Undecidability theorist to reject one horn or the other of this dilemma. Perhaps, for instance, Kant doesn’t mean to invoke his notion of concept containment here, in which case the second reading might not support Necessity. But another plausible route would be to take the entire passage at B148–9 as involving a tacit relativization to our forms of the understanding. Note, for instance, the uses of the first-person plural throughout the passage, including in the final sentence. And this is important because it suggests a general strategy open to Undecidability theorists for reading any particular text in which it looks as though Kant is ruling out the possibility of discursive cognizers with alternative forms of the understanding and so endorsing Necessity. They can maintain that, properly contextualized, the passage in question only rules out the possibility of alternative forms of the understanding for us. And the impossibility of our possessing alternative forms of the understanding does not tell against the possibility of other discursive cognizers with alternative forms of the understanding. This leaves room for Undecidability, which entails that we cannot rule out such a possibility. An important application of this strategy would be to those crucial passages in which Kant suggests that the categories and the functions of judgement are complete and can be derived from a single principle or from the faculty of the understanding itself.⁴¹ These passages support Necessity if one reads them as proposing a complete derivation from the discursive understanding in general, or from a principle that applies generally to discursive cognizers as such. For if we can show that the categories and the functions of judgement are derived in this way, then we can know that discursive cognizers with other intellectual forms are impossible. If, however, the starting point for such a derivation is the human understanding specifically, then these passages are compatible with Undecidability, since they then indicate only the impossibility of human cognizers with other intellectual forms. We look at this issue in more detail in section 6. But, as a start, the Undecidability theorist can point to the many passages in the Transcendental Logic where Kant talks about the human understanding specifically, passages in which, by the Necessity theorist’s lights, we could reasonably expect him to be talking about the discursive understanding in general. Thus, right at the start of the Transcendental Analytic, Kant writes that he ‘will therefore pursue the pure concepts into their first seeds and predispositions in the human understanding’.⁴² As above, the thought is that those passages in which Kant makes his claims to derivation and completeness are more perspicuously read as operating under a tacit restriction to our own, human understanding. After all, the same restriction is ⁴¹ A68/B94 and A80–1/B106–7; Prol. 4: 322; MFNS 4: 476; and VM 29: 984. ⁴² A66/B91, emphasis added. Cf. B110, A85/B117, A119, A124, B135, B139, A297–8/B353–4, and A309/B366.

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often required but not explicit in Kant’s statements about sensibility, for example when he says, ‘In this investigation it will be found that there are two pure forms of sensible intuition as principles of a priori cognition, namely space and time’.⁴³ This bears on the relevance to our question of the various sophisticated attempts to account for the full details of Kant’s claims to derivation and completeness.⁴⁴ For it is not straightforward to determine whether these accounts should even be understood as aiming to establish Necessity, given that the distinctions that we have drawn in this essay have not been central to that debate. Michael Wolff, for example, claims that Kant’s derivation concerns a specifically human understanding.⁴⁵ On the face of it, this makes Wolff ’s account compatible with Undecidability. His reason for the restriction, however, is the recognition that Kant is clearly not concerned to provide an account that would also capture the forms of a nondiscursive intellect. That is surely correct, but our question in this essay is precisely whether Kant leaves room for something else here—namely, a discursive intellect that isn’t human. The import for our debate of Kant’s claims to derivation and completeness depend crucially on how we are to understand his starting point. The Undecidability theorist’s strategy here is the natural counterpoint to the one we saw above for the Necessity theorist. Where the Necessity theorist appeals to Kant’s frequent reference to the nondiscursive intellect, the Undecidability theorist appeals to Kant’s frequent reference to the human intellect. And while there will no doubt be other passages which are relevant to this debate, not least from the practical writings and the Critique of the Power of Judgement, we suspect that Undecidability and Necessity theorists will each be able to use versions of these general strategies for reconciling any problematic passage with their respective views. Textual considerations alone won’t settle the dispute, and we turn now to more systematic considerations. For reasons of space we restrict ourselves to the theoretical philosophy, though what we say in sections 7–8 will have broader implications.

6. Systematic Considerations 6.1. Against Necessity We begin with a challenge for Necessity. Necessity says that we can know that other intellectual forms are impossible. But what kind of knowledge is this? It is presumably a priori. But is it synthetic or analytic? The opponent of Necessity might claim that either option looks problematic. ⁴³ A22/B36. ⁴⁴ See esp. Reich (1992); Wolff (1995); and Schulting (2018). For critical discussion see e.g. Krüger (1968); Brandt (1991); Baumanns (1993); Longuenesse (1998); Longuenesse (2005); Allison (2004), pp. 136–56; and Lu-Adler (2018). ⁴⁵ Wolff (1995), pp. 20 and 58–60. See Reich (1992), p. 19, and Schulting (2018), p. 279, for similar moments in their accounts.

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Suppose first that the knowledge Necessity says we can have is synthetic a priori knowledge that other intellectual forms are impossible. One of the aims of the Critique of Pure Reason is to explain how it is that we can have synthetic a priori knowledge in mathematics, pure natural science, and metaphysics. But Kant’s explanation makes central appeal to the claim that we have two pure forms of sensibility, space and time, and he thinks that we cannot know whether these forms are common to discursive cognizers generally. This explanation, at least, could not account for our synthetic a priori knowledge that discursive cognizers with other intellectual forms are impossible. There therefore remains an issue about how we can have the knowledge that Necessity ascribes to us, if that knowledge is synthetic a priori. Suppose instead, then, that the knowledge Necessity says we can have is analytic knowledge that other intellectual forms are impossible. How could analysis demonstrate the truth of the Necessity thesis? Is it plausible, for instance, that Kant thought it part of the concept of discursive cognition as such that discursive cognizers have these and only these pure concepts? And are we sure that analysis can demonstrate an impossibility of the relevant kind—namely, a real impossibility? If the Necessity thesis can be neither synthetic nor analytic, then we have a line of reasoning in support of Undecidability. Can this dilemma be resisted? Start with the view that the knowledge involved in Necessity is synthetic a priori. Then we can have synthetic a priori knowledge that isn’t accounted for by Kant’s explanation, by appeal to our pure forms of sensibility, of how it is that we can have synthetic a priori knowledge. Is this a problem? There is plausibly a range of claims that Kant makes in the Critique that are synthetic a priori and yet that don’t seem to be accounted for by this explanation. Consider the claim that there is a distinction between intuitions and concepts. This doesn’t look to be analytic, nor is it known a posteriori. Yet it plays a central role in Kant’s theory. We need some account of how Kant thinks such claims are known. So if the knowledge involved in Necessity is synthetic a priori yet not subject to Kant’s explanation by appeal to our pure forms of sensibility, then it does not look isolated in this regard. This response is fine as far as it goes. But someone who takes this line either owes us an alternative explanation of the synthetic a priori knowledge that is not explained by Kant’s appeal to our pure forms of sensibility or owes us a criterion for distinguishing those synthetic a priori claims that require explanation from those that do not. Either would be a substantial commitment. But consider a particular and reasonably plausible candidate: that the synthetic a priori claims that either have an alternative explanation or do not require explanation are the ones that are exclusively about our mind’s representational structures.⁴⁶ This does

⁴⁶ For such an approach, see Marshall (2014), pp. 562–9.

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not obviously include the claim that discursive cognizers with other intellectual forms are really impossible. So the Necessity theorist either needs to explain why it is so included or needs to provide some other criterion for distinguishing the two classes of synthetic a priori claims. The synthetic horn of the dilemma for Necessity is not indefensible. But it requires further detail and explication of some of the central doctrines of the Critical system. Consider now the second horn, that the knowledge involved in Necessity is analytic. Would it be a problem to hold that it is analytic of discursive cognition as such that it involves just these and no other intellectual forms? On this view, the Necessity thesis might be thought comparable to the claim that discursive cognition involves sensibility and understanding or the claim that sensibility is passive and receptive while understanding is active and spontaneous—perhaps these claims simply articulate the concepts in question or in some other way proceed from sheer analysis of the faculties in question.⁴⁷ And perhaps it is similarly analytic of discursive cognition as such that it has these particular forms. Recall that Kant often suggests that the categories and the functions of judgement are complete and can be derived from a single principle or from the faculty of the understanding itself.⁴⁸ The thought would be that this derivation is supposed to be analytic and that it turns on features of the discursive understanding in general, such as the unity of apperception or the definition of discursive judgement. For instance, suppose that Kant thinks it analytic of discursive judgement as such that it be predicative and that its atomic form be categorical—that is, that it involve predicating one concept of another. If this were the case, it could then be analytic that the extension of ‘the subject is either wholly included in or excluded from the [extension] of the predicate or is only in part included in or excluded from it’.⁴⁹ Such an analysis would give us the first two moments of each of quantity and quality, alongside the first moment of relation, which together yield the Aristotelian square of opposition. Perhaps one can get from here to the full Table of Judgement and, from there, to the categories themselves—though it’s worth noting that what are most novel in Kant’s Table of Judgement are precisely those aspects that go beyond anything involved in the Aristotelian square of opposition: namely, his inclusion of the third moments of quantity and quality, his inclusion of a distinct variable for relation with just as many moments, and his treatment of modality.⁵⁰ ⁴⁷ Kant distinguishes concept analysis from faculty analysis at A65/B90. This may be important, for instance, because while Kant thinks we can never be apodictically certain that the analysis of a nonmathematical concept is complete (A728–30/B), matters might be different for the analysis of faculties. See Wolff (1995), pp. 6–7 and 177–8 for discussion. ⁴⁸ A64/B89, A69/B94, A80–1/B106–7; Prol. 4: 322; MFNS 4: 476; and VM 29: 984. ⁴⁹ JL 9: 102, original emphasis removed. ⁵⁰ See Tonelli (1966). The trichotomous aspect of Kant’s tables alone might suggest that they cannot be arrived at purely analytically, since analytic division proceeds according to the principle of

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Regardless, note that if this procedure is to support the claim that the knowledge involved in Necessity is analytic, it needs to be the case both that the starting point of the derivation be something about discursive cognizers as such and that the derivation proceed solely by analysis. We observed above that the Undecidability theorist might read the starting point for Kant’s derivation as restricted to the human understanding specifically. But they might also deny that the derivation proceeds solely by analysis. Kant, after all, allows for synthetic derivations, understood as ones that do not proceed solely in accordance with the principle of contradiction but ‘by some other principle’.⁵¹ Thus, even granting that the starting point for Kant’s derivation is the discursive understanding in general, if that derivation is synthetic and proceeds from a principle that, for all we can know, holds only for humans, the result would be compatible with Undecidability.⁵² To secure the analytic reading of the Necessity thesis, then, we would need to be sure both that Kant’s starting point is not restricted to the human understanding and that his derivation proceeds solely through analysis and not by appeal to our forms of sensibility or anything else that would render the result compatible with Undecidability. Again, the point is not that the analytic reading of the Necessity thesis is implausible, just that more has to be said about the nature of the analysis and the knowledge it provides. Would we also need some reassurance that, if such an analysis took the form of showing our inability to represent something, this inability had any bearing on real possibility? Isn’t Kant suspicious of any general entailments between the shape of our representational capacities and the real nature of things (save in the special case of appearances)? He is. But the only reassurance that we need in this context is that logical (or conceptual) impossibility entails real impossibility— an entailment that will strike many people as incontestable, both as a principle in its own right and as a commitment of Kant’s.⁵³ It is worth noting the bearing that this has on Stephen Engstrom’s appeal to Kant’s claim that we can’t represent any beings as thinkers except by ‘transference’ of our own consciousness to them.⁵⁴ Taking the categories to be conditions of such consciousness, Engstrom concludes that we can’t represent thinkers—nor, therefore, discursive cognizers—who possess categories other than our own. From this

contradiction alone and therefore only ever yields dichotomies, whereas trichotomies come from synthetic division by first dividing but then also uniting condition and conditioned (CPJ 5: 197). See Wolff (1995), pp. 163–70. for discussion. ⁵¹ Disc. 8: 229–30. ⁵² We discuss one such approach in §6.2, below. See Longuenesse (2005), ch. 1. ⁵³ This may not be as straightforward as it appears. At least one of us thinks there is room for doubt as to whether Kant endorses any such general entailment. For discussion which bears on the general issues here, see Bader (forthcoming), and for some relevant remarks, see e.g. A59/B84 and A291–2/ B348–9. ⁵⁴ Engstrom (1994), pp. 379–80, and A346–7/B404–5. Cf. A353–4.

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he infers Necessity.⁵⁵ This argument looks especially vulnerable to the concern that our inability to represent something is one matter, the real impossibility of that thing another matter entirely. For even if we admit that there are no categories other than our own such that we can represent discursive cognizers who possess them, mustn’t we keep an open mind on whether that’s because there is a real possibility here that we cannot (fully) represent? We must. But Engstrom’s argument, if successful, shows more than that. It shows that we can’t represent discursive cognizers who possess categories other than our own. If that’s right, and if it’s right because of what counts as an insight achieved through analysis—we are passing no judgement on whether Engstrom’s argument actually shows either of these things—then it’s logically impossible for there to be such cognizers and hence, given the entailment noted above, really impossible.⁵⁶ Return to the dilemma. The point we want to emphasize here is that both responses share a common form. The challenge to the Necessity theorist was that the Necessity thesis must be either analytic or synthetic and that neither option looks plausible. In each case, the response is to claim that what was posed as a dilemma for the Necessity theorist is really just the question of what status to give Kant’s fundamental claims about the mind. And this raises a final, more general issue about the knowledge involved in Necessity. Since it is a priori knowledge, we must be able to have it, in some sense, through reflection on the forms and activities of our own mind. But, the Undecidability theorist might press, how could reflection on the form and activities of our own mind put us in a position to know anything about other discursive cognizers? The Necessity theorist owes us an account of the source and nature of the knowledge they claim we can have.

6.2. Against Undecidability We turn now to a set of systematic considerations against Undecidability. Consider the unity of apperception. We assume that it is common ground in our dispute that this much is shared by all discursive cognizers, since it is constitutive of what it is to have a discursive understanding. Even Undecidability theorists must accept that there is a function of the understanding that is shared by all discursive beings—namely, that of ordering representations under common ones in accordance with the principle of the unity of apperception. For this is just

⁵⁵ See Kitcher (2017), pp. 169–70, and Nunez (2018), §6 for related discussion. ⁵⁶ The point can be made in terms of Kant’s distinction between positive and negative conceptions (e.g. at B307). Our merely being unable positively to conceive of something leaves us needing to keep an open mind as to whether there might nevertheless be a real possibility here that we cannot positively represent. Not so if we can’t even form a negative conception of something, for instance because ‘its concept cancels itself out’ (A292/B348), for then we simply have nothing in mind about which we should keep an open mind. See Nunez (2018), pp. 1170–2, for relevant discussion.

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an explication of what it is to possess a discursive understanding. And this is what we would expect a commitment to Undecidability to entail. For the same is true of sensibility: space and time might, for all we can know, be peculiar to us, but Kant thinks that there is a passive, receptive function to sensibility as such that is shared by all discursive beings, simply in virtue of possessing a faculty of sensibility. In taking there to be a function of the understanding that is shared by all discursive beings, the Undecidability theorist is simply treating the two faculties on a par. This already undermines some of the worries of the previous section. For if we can somehow know that any creature with a discursive understanding must enjoy the unity of apperception, then the same question will arise: is this knowledge analytic or synthetic? But put this to one side for a moment. We can instead ask: what is the relation between the discursive understanding in general and the forms of the understanding in particular? The Undecidability theorist accepts that there is a characterization of the understanding on which it is shared by all discursive cognizers: namely, the understanding as the unity of apperception. It is central to that characterization that it involves the function of ordering representations under common ones in accordance with the principle of the unity of apperception. And the Undecidability theorist also holds that there is a more specific characterization of human understanding that involves reference to our specific intellectual forms. The question that the Necessity theorist will push is: what explains our inability to know whether this more specific characterization applies to discursive cognizers more generally? The Necessity theorist holds that, just as we can know that the unity of apperception is invariant across discursive cognizers, so too we can know that our particular intellectual forms are invariant across discursive cognizers. For the analytic Necessity theorist, this is because our particular intellectual forms are analytically derived from the discursive understanding itself; for the synthetic Necessity theorist, it is because reflection on the mind’s structure somehow suffices to show the invariance. In contrast, the Undecidability theorist thinks that we can know that the unity of apperception is shared while not being in a position to decide whether our particular intellectual forms are invariable. But what could explain our being able to know the invariance of the former without being able to know the invariance of the latter? The challenge for the Undecidability theorist is to explain how our being able to know of invariance at the more general level is compatible with our being unable to know of invariance at the specific level. One way for the Undecidability theorist to respond to this worry is by noting that a symmetrical question can be posed about the pure forms of sensibility. There too we have a fundamental characterization that applies to any discursive cognizer: sensibility is a passive, receptive faculty. And we have too a characterization of our sensibility that, for all we can know, applies only to human beings:

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its pure forms are space and time. One might equally ask: what explains our inability to know that this more specific characterization applies to discursive cognizers more generally? The same issue arises for both sensibility and the understanding, so where’s the asymmetry? The Necessity theorist would be on stronger ground if there were a principled reason to think that there is an explanation for the undecidability of sensible variation that does not also apply to the understanding. Another, perhaps more satisfying way for the Undecidability theorist to respond would be to rise to the challenge. Can the Undecidability theorist explain how our being able to know that the unity of apperception is invariant among discursive cognizers is compatible with our inability to know whether the same is true of our particular intellectual forms? One natural way to provide such an explanation is to find a point at which our sensible nature enters into Kant’s account of those forms. This would provide an explanation of the discrepancy in the Undecidability theorist’s narrative. For if it is undecidable whether our sensible forms can vary, and if those forms are suitably implicated in an account of our intellectual forms, then it will likewise be undecidable whether these particular intellectual forms can vary even while it can be known that the unity of apperception is invariant. Is there any way to enact such an approach? Consider Béatrice Longuenesse’s claim that the categories ‘are a priori determined by “the subjective conditions of the spontaneity of thought” (the logical functions of judgment) together with the “first formal grounds of sensibility” (space and time)’.⁵⁷ The basic idea here seems to be that, since the categories are rules for the synthesis of the sensible manifold, just what these rules are will be determined, in part, by the kind of sensible manifold that is to be synthesized. This will be compatible with the a priori status of the categories, so long as the aspects of our sensible nature that enter in here are likewise a priori. But the salient point for our purposes is that, on the face of it, such a view promises to give us just the sort of explanation demanded of the Undecidability theorist. For if the pure forms of sensibility are involved in the a priori determination of the categories, and if it is undecidable whether the pure forms of sensibility can vary between discursive cognizers, then it will likewise be undecidable whether the categories can vary between discursive cognizers. Yet there are problems with appealing to Longuenesse’s view in defence of Undecidability in this way. First, note that such an approach would explain only why we are not in a position to know whether the categories can vary, not why we are not in a position to know whether the logical functions of judgement can vary. It would therefore not be a pure Undecidability view but rather a mixed view of ⁵⁷ Longuenesse (2005), p. 29, emphasis added. Cf. Longuenesse (2005), p. 20: ‘the cooperation of the understanding, as a capacity to judge, and sensibility, as a receptivity characterized by specific forms or modes of ordering, generates categories’.

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the kind mooted in section 3. Second, to the extent that the pure forms of sensibility are involved in the a priori determination of the categories but not the logical functions of judgement, the view faces questions about how to make sense of Kant’s insistence on there being a kind of identity between the two.⁵⁸ Finally, the view so employed cannot collapse into the claim that the pure forms of sensibility are involved only in the determination of the categories in so far as they are subject to a process of schematization. For the Necessity theorist will accept that our sensible nature enters into the determination of intellectual form at this point and can thus allow that we cannot know whether discursive cognizers with other schematized categories are really possible. In order for this approach to be of help to the Undecidability theorist, the claim must be that our sensible nature enters into an account of the categories even before they are subject to schematization.⁵⁹ This is not to say that Undecidability is indefensible. It may be that some other explanation can be given for our inability to know whether the particular intellectual forms can vary while being in a position to know that the unity of apperception cannot vary. But we can put these considerations together with those of the previous subsection to make a more general point about our debate. Both views hold that we can know that discursive cognizers without the unity of apperception are impossible. And both views hold that we cannot know whether discursive cognizers with other sensible forms are possible. They differ as to whether we can know whether discursive cognizers with other intellectual forms are possible. What the considerations adduced in the last two subsections draw out is that Kant is committed to there being some claim about the structure of mind that we are in a position to know is really impossible. And similarly, Kant is committed to there being some claim about the structure of mind that we are not in a position to know is really possible or really impossible. A satisfying defence of either Necessity or Undecidability must cohere with these other instances of and restrictions on knowledge. That is to say, any satisfying resolution to our debate will be but one part of a more general story about Kant’s account of the nature and limits of our knowledge of mind.

7. Second-Order Undecidability Our question is whether Kant thinks it is possible to know whether or not discursive cognizers with other intellectual forms are possible. Undecidability is the view that we cannot know whether discursive cognizers with other intellectual forms are possible. Necessity is the view that we can know that discursive ⁵⁸ See esp. A79/B104–5 and B143. ⁵⁹ For related discussion, see Sedgwick (2000); Allison (2012), ch. 2; and Schulting (2018), ch. 3.

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cognizers with other intellectual forms are impossible. We have argued thus far that textual and systematic considerations do not decide between these views. Each is either supported by or at least compatible with the relevant texts, each has systematic considerations that tell in its favour, and the systematic considerations that tell against each can be mitigated in plausible ways. The apparent inability of the textual and systematic considerations to settle our debate opens up an intriguing possibility. Perhaps Kant thinks we cannot know which of Undecidability or Necessity is true. That would explain the lack of decisive textual support for either reading. It would explain the contrast between Kant’s explicit statement of undecidability in the sensible case and his lack of explicit statement either way about the intellectual case. And it would explain why we have not identified a systematic consideration that decisively supports one view over the other. Such a view would amount to an undecidability thesis about Undecidability and Necessity themselves. It would be a second-order undecidability thesis. For Undecidability is an undecidability thesis and Necessity a decidability thesis. We suggested at the end of the last section that a satisfying resolution to our debate will be but one part of a more general story about Kant’s account of the nature and limits of our knowledge of mind. This holds true for the current proposal. If Kant thinks we cannot know which of Undecidability or Necessity is true, that too must fit within his more general account of our knowledge of mind. In this section and the next we draw on two such aspects of Kant’s account to explore the possibility that he was neutral on the issue. In the present section we argue that Kant’s commitment to decidability in transcendental philosophy does not rule out a second-order undecidability thesis concerning Undecidability and Necessity. In the next section we argue that Kant’s commitment to a kind of luminosity principle nevertheless precludes him from properly endorsing such a view. The result is not that Kant cannot remain neutral on the issue. It is that, if he does, then Kantian humility takes on a distinctive character. Before we proceed, it will be helpful to introduce some labels. Let Q-s be the question whether discursive cognizers with other sensible forms are possible. And let Q-i be the question whether discursive cognizers with other intellectual forms are possible. We saw in section 2 that Kant thinks we cannot know the answer to Q-s. Undecidability is the view that we cannot know the answer to Q-i. Let Second-Order Undecidability be the view that we cannot know whether Undecidability is true. And let Second-Order Decidability be the view that we can know this. Equivalently (given that Contingency is off the table), Second-Order Undecidability is the view that we cannot decide between Undecidability and Necessity, while Second-Order Decidability is the view that we can. Is there anything to be said in favour of Second-Order Decidability and thus against Second-Order Undecidability? There is an important section in the

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Critique that is directly relevant to this question—namely, the fourth section of the Antinomy of Pure Reason, titled ‘The transcendental problems of pure reason, insofar as they absolutely must be capable of a solution’.⁶⁰ Precisely what Kant provides in this section is a criterion for whether an issue is decidable, or, equivalently, for whether a question is answerable. Kant writes: [T]ranscendental philosophy has the special property that there is no question at all dealing with an object given by pure reason that is insoluble by this very same human reason; and that no plea of unavoidable ignorance and the unfathomable depth of the problem can release us from the obligation of answering it thoroughly and completely; for the very same concept that puts us in a position to ask the question must also make us competent to answer it, since the object is not encountered at all outside the concept.⁶¹

Roughly, a question is answerable if the sheer fact that we can ask it puts us in a position to answer it. Now, this criterion serves only as a sufficient condition. Failure to satisfy it doesn’t in general make a question unanswerable. However, there is reason to think that, in the specific case of Q-s and Q-i, Kant’s criterion can serve as a necessary condition too. For if either question fails to satisfy his criterion, that means that something more than whatever is accessible to us in the sheer posing of it would be required to answer it. But what could this be? To pose either question is already to exercise the understanding and invoke our concepts of cognitive forms. And it is clear that no exercise of sensibility, whether pure or empirical, could help us answer either question—in the one case because it is about the possibility of things’ being intuited in some quite different way, and in the other case because it is not about things’ being intuited at all. So if either Q-s or Q-i fails to satisfy Kant’s criterion, it is unanswerable. Given his commitment to undecidability about other sensible forms, Kant must hold that whatever is accessible to us in formulating Q-s does not put us in a position to determine whether other sensible forms are possible—he thinks that Q-s fails to satisfy his criterion and is thus unanswerable. If Q-i likewise fails to satisfy Kant’s criterion, then it is likewise unanswerable, which is to say that Undecidability holds. But if Q-i does satisfy Kant’s criterion, then it is answerable, which is to say that Necessity holds. This can seem to tell in favour of Second-Order Decidability and thus against Second-Order Undecidability. For surely we can know whether or not Q-i satisfies Kant’s criterion, in which case we can decide between Necessity and Undecidability. After all, Kant thinks this is true for Q-s—he thinks we can know that Q-s is a question about possibilities that exceed whatever is accessible

⁶⁰ A476–90/B504–18.

⁶¹ A477/B505, emphasis added. Cf. A763/B791.

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to us merely in asking it. But this would be too quick. In general, to have a criterion is one thing, to be able to know whether something satisfies it quite another.⁶² And we have nothing as yet to guarantee that we can know whether Q-i satisfies Kant’s criterion. So, as yet, there is nothing here to motivate Second-Order Decidability. And in fact nothing here can motivate Second-Order Decidability. For what is at issue is whether the answer to Q-i lies, in Kant’s words, ‘outside the concept’. If we cannot answer Q-i, the reason for this must be that, even though Q-i is a question about concepts, its answer does lie ‘outside the concept’—either because it concerns concepts other than those involved in posing it or because it concerns possibilities about the concepts involved in posing it that exceed what is thereby accessible to us. But to ask whether this is the case is simply to ask the original question whether Undecidability or Necessity holds—the second-order issue of whether Q-i satisfies Kant’s criterion collapses into the first-order issue about how to answer that question. For Necessity in effect just is the view that we can know that answering Q-i appeals to no more than is accessible to us in posing it— namely, our own intellectual forms. It just is the view that we can know that these forms are the forms of discursive understanding as such. And Undecidability just is the view that we cannot know these things. That is to say, precisely what is at issue between Undecidability and Necessity is whether or not Q-i is a question that poses one of those ‘transcendental problems of pure reason’ that, for the reason Kant gives, ‘absolutely must be capable of a solution’. There is no more guarantee here that we can tell whether or not Q-i satisfies Kant’s criterion than there is that we can bypass his criterion and tell straight off whether or not Q-i is answerable. What this shows is that Kant’s criterion for decidability does not rule out Second-Order Undecidability any more than it rules out (first-order) Undecidability or his commitment to undecidability regarding the possibility of other sensible forms. It thus remains open that Kant thought there was no way to settle which of Necessity or Undecidability is true.

8. Luminosity and Neutrality Second-Order Undecidability is the view that we cannot decide between Undecidability and Necessity. To endorse it would be to maintain a position of in-principle neutrality between the two views. So does Kant endorse it? In this section we will argue that Kant could not in fact do so. But this is not because ⁶² It is noteworthy that there is room for exegetical controversy concerning the application of this criterion to the very questions that Kant is concerned with in the broader context in which this section occurs—namely, the questions that generate the four antinomies. Graham Bird (2006, ch. 26) thinks that Kant thinks these questions are all unanswerable by his criterion; A. W. Moore (2019d) thinks that Kant thinks this is true only of the questions that generate the two dynamical antinomies.

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neutrality is not an option for him. It is because his views on philosophical knowledge constrain the form such neutrality can take. Our argument rests on the plausible claim that Kant would endorse a luminosity principle that states that, whenever we can have philosophical knowledge, we can know that we can. Call this principle Philosophical Luminosity. We give reasons for thinking that Kant endorses it below. First, we explain why endorsing the luminosity of any kind of knowledge precludes endorsing second-order undecidability concerning knowledge of that kind. Our talk of endorsement here is a placeholder for some generic relation to a claim that we cannot properly stand in without knowing the claim, for instance because knowing the claim is a constitutive norm of standing in the relation to the claim.⁶³ Thus the reason that endorsing the luminosity of any kind of knowledge precludes endorsing secondorder undecidability concerning knowledge of that kind is that luminosity and second-order undecidability are not jointly knowable. Together they form what Roy Sorensen calls a ‘knowledge blindspot’⁶⁴—the claims themselves are consistent, but knowing them is not. Here is why. Take knowledge whether p as a case in point. Luminosity entails that, if we can know whether p, then we can know that we can. But that is as much as to say that, given luminosity, the first-order decidability of an issue entails the corresponding second-order decidability, or equivalently, that the second-order undecidability of an issue entails the corresponding first-order undecidability. Now assume that someone knows both luminosity and the second-order undecidability of some issue. Then, by following the reasoning we just sketched, they can come to know the corresponding first-order undecidability. But no one can know both the first- and the second-order undecidability of an issue. For knowledge is factive, and the latter says that the former cannot be known. Thus our assumption was false and necessarily so. No one can know both luminosity and the secondorder undecidability of any issue. Since endorsement requires knowledge, endorsing the former precludes endorsing the latter. So far this is a purely formal result. It has application to our topic on the assumption that the knowledge Necessity says we can have—namely, knowledge that other intellectual forms are impossible—would be a case of philosophical knowledge. For this would mean that Philosophical Luminosity entails the following conditional: if Necessity is true, then we can know that it is true. Note that this holds regardless of whether or not Necessity is true and regardless of which view (if either) Kant endorses—all parties to our dispute can accept it. And it is enough to show that endorsement of Philosophical Luminosity precludes endorsement of Second-Order Undecidability. For Second-Order Undecidability entails that we cannot know that Necessity is true, which by the above conditional

⁶³ For discussion and defence, see Williamson (2000), ch. 11.

⁶⁴ Sorensen (1988), p. 52.

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would entail that it is not true. Thus no one can know both Philosophical Luminosity and Second-Order Undecidability, because that would allow them to know that Necessity is not true (via the above conditional), which Second-Order Undecidability itself says cannot be known. Kant’s endorsement of Philosophical Luminosity would preclude his endorsing Second-Order Undecidability. So does Kant endorse Philosophical Luminosity? Timothy Williamson’s celebrated anti-luminosity argument begins by connecting luminosity to the view that we have ‘a cognitive home’ in which ‘nothing is hidden from us’ and ‘everything lies open to view’.⁶⁵ He continues: To deny that something is hidden is not to assert that we are infallible about it. Mistakes are always possible . . . . The point is that, in our cognitive home, such mistakes are always rectifiable. Similarly, we are not omniscient about our cognitive home. We may not know the answer to a question simply because the question has never occurred to us. Even if something is open to view, we may not have glanced in that direction. Again, the point is that such ignorance is always removable.

Kant’s account of philosophical knowledge exemplifies, remarkably closely, the one sketched by Williamson. Kant does not think we are infallible in philosophy. Indeed, he thinks that mistakes in the form of transcendental illusions are not only possible but natural.⁶⁶ Nor does he think we are omniscient in philosophy. We may not know the answer to a question simply because it has never occurred to us, such as how synthetic a priori knowledge is possible.⁶⁷ But, as we saw in the previous section, he clearly thinks that philosophy is our cognitive home, where mistakes are always rectifiable and ignorance is always removable.⁶⁸ And Kant’s basic reason for this view is familiar: philosophy is our cognitive home because philosophical knowledge is a kind of self-knowledge.⁶⁹ The view that philosophy is our cognitive home—that nothing in philosophy must remain hidden from us so that we can always remove ignorance and error—is in effect just the view that philosophical issues are always decidable, that we can know the answer to any philosophical question. It is not yet a statement of Philosophical Luminosity, the view that, if we can have philosophical knowledge, then we can know that we can have it. But we can close the gap by attending to the fact that Kant thinks of philosophy as our cognitive home because he thinks of philosophical knowledge as a form of self-knowledge. He says, for instance:

⁶⁵ Williamson (2000), pp. 93–4. ⁶⁶ A293/B349 and Avii. ⁶⁷ B19, A764/B792, and A10; and Prol. 4: 270. ⁶⁸ A476–7/B504–5. ⁶⁹ Axi, Axiv, Axx, Bxviii, Bxxiii, and A12–13/B26; and MFNS 4: 472–3.

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    [T]hat such a system [the philosophy of pure reason] should not be too great in scope for us to hope to be able entirely to complete it, can be assessed in advance from the fact that our object is not the nature of things, which is inexhaustible, but the understanding, which judges about the nature of things, and this in turn only in regard to its a priori cognition, the supply of which, since we do not need to search for it externally, cannot remain hidden from us.⁷⁰

If the reason that we can always answer philosophical questions is that they have to do solely with the understanding and we do not have to search externally, the same will be true of questions concerning whether we can know the answers to philosophical questions. Otherwise put, if philosophical questions are answerable because they are in a certain way questions about ourselves, then questions about whether we can answer such questions will also be about ourselves in that very same way and so also be answerable. The same reasoning that motivates Kant’s view that philosophy is our cognitive home would also motivate him to endorse Philosophical Luminosity.⁷¹ It only remains to be shown that the knowledge Necessity says we can have— namely, knowledge that other intellectual forms are impossible—would be a case of philosophical knowledge. This assumption was required to show that no one can endorse both Philosophical Luminosity and Second-Order Undecidability. But the reasoning we have just outlined to support Kant’s endorsement of Philosophical Luminosity also makes clear why this should be true. For any such knowledge would be knowledge, of our own intellectual forms, that they are the forms of the discursive intellect as such, and hence would be knowledge of ourselves in just this crucial way. It would be a case of philosophical knowledge to which Philosophical Luminosity applied.⁷² If Kant endorses Philosophical Luminosity, then, he is precluded from endorsing Second-Order Undecidability—he cannot endorse the view that we cannot decide between Undecidability and Necessity. Moreover, since the same reasoning applies to higher orders, nor can Kant endorse any higher-order undecidability thesis. And this shows that Kant cannot remain neutral on the issue by endorsing Second-Order Undecidability or any higher-order undecidability thesis. ⁷⁰ A12–13/B26. ⁷¹ For further discussion and defence, see Stephenson (2021). ⁷² This can be helpfully related back to the discussion in the previous section. The reason why Undecidability and Necessity differ about whether or not our ignorance concerning the answer to Q-i is remediable is that they differ about whether we can know whether or not Q-i is a question about ourselves in the relevant way. If Necessity holds, our ignorance concerning the answer to Q-i is remediable because we can know that our knowledge of our own intellectual forms is ipso facto knowledge of discursive intellectual forms as such. If Undecidability holds, our ignorance concerning the answer to Q-i is irremediable because we cannot know whether or not our knowledge of our own intellectual forms is ipso facto knowledge of discursive intellectual forms as such. Note again that our argument is independent of which of these views holds or which (if either) Kant endorses—we require only the claim that, if Necessity holds, then Q-i is a question about ourselves in the relevant way, a claim that all parties can accept.

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What are the implications for the questions canvassed at the start of this essay? We have been motivated throughout by the recognition that Kant endorses an undecidability thesis about other sensible forms and the question whether he endorses a symmetrical undecidability thesis about other intellectual forms. We have defended four main claims: first, that textual and systematic considerations do not decisively settle this question; second, that addressing the question will require a more general story about the nature and limits of our knowledge of mind; third, that Kant’s criterion for decidability does not rule out neutrality on our debate; and fourth, that given his commitment to Philosophical Luminosity, Kant is not entitled to remain neutral by endorsing the claim that we cannot know which of Undecidability or Necessity is true. Where does this leave us? There is a notable contrast between Kant’s explicit statement of undecidability about discursive cognizers with other sensible forms and his lack of explicit statement about whether we can know whether or not there could be discursive cognizers with other intellectual forms. And this, combined with the inability of other textual and systematic considerations to settle the question, might motivate the thought that Kant simply did not know which of Necessity or Undecidability was true. For it is scarcely credible that he didn’t even consider the issue. And it is hardly more credible that he did consider the issue, satisfied himself that he knew which was true, but did not see fit to tell us. After all, he did both of these things for the sensible case. If Kant is ignorant about whether we can know whether or not there could be discursive cognizers with other intellectual forms, then we have an explanation for the inability of textual and systematic considerations to settle the issue. But any such ignorance is constrained by the formal results above. For Kant is not entitled to endorse the claim that we cannot know which of Necessity or Undecidability is true, given his commitment to Philosophical Luminosity. So if it is Kant’s ignorance that explains the inability of textual and systematic considerations to settle this issue, it is not the kind of ignorance that Kant can express by endorsing the claim that we cannot know which of Necessity or Undecidability is true. It can only be a commitment to the claim that we do not know which is true. Such contingent ignorance is not precluded by the above results, for we could recognize that we do not know which is true while remaining open to the possibility of a consideration arising that will settle the issue. For the time being we can only wait and see. Yet there’s something unsatisfactory about appealing to Kant’s contingent ignorance to explain the inability of textual and systematic considerations to decide the matter. After all, just as Kant does not explicitly commit to either Necessity or Undecidability, neither does he explicitly state that he doesn’t know which of them is true. So appealing to Kant’s contingent ignorance does little better at explaining his taciturnity on this point. And more generally, as much of the preceding discussion suggests, there looks to be something odd about

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contingent ignorance, or at least about self-conscious contingent ignorance, from the point of view of transcendental philosophy.⁷³ But there is another option. We have seen that Kant is not entitled to remain neutral on the issue by endorsing any higher-order undecidability thesis. In particular, he is not entitled to remain neutral by claiming to know that we cannot know which of Undecidability and Necessity is true. But this is because Philosophical Luminosity and Second-Order Undecidability cannot both be known. It is not because Philosophical Luminosity and Second-Order Undecidability cannot both be true. They can both be true. And that shows that someone could quite reasonably think, ‘Second-Order Undecidability probably holds. That is, we probably cannot in principle decide between Undecidability and Necessity. On pain of contradiction, I can never know that this is the case, given that I already know that Philosophical Luminosity holds. But still, I suspect that’s how it is!’ Such a person might even suspect that undecidability goes ‘all the way up’. This is important because it shows that there is a way in which Kant could maintain a position of in-principle neutrality on the question whether Undecidability or Necessity is true, not by endorsing the claim that it is undecidable which of them is true, but by adopting some attitude towards that claim that is not precluded by a failure to know it, or by simply saying nothing at all on the issue. To remain neutral in this way would still be to maintain an asymmetry between sensibility and the understanding, since Kant is explicit that we cannot know whether discursive cognizers with other sensible forms are possible. But this asymmetry would merely take the form of there being a limit to the claims that Kant could endorse with respect to the understanding that did not have any counterpart with respect to sensibility. Some will see such humility as deeply Kantian. Kant, after all, expressly sets out to draw a boundary to our cognition and knowledge, and part of that project involves drawing upon modes of assent that are not undermined by our being unable to know the claim in question.⁷⁴ Others will think that silence is the inevitable result of attempting to express the incoherence of discursive cognizers with other intellectual forms and that the difference between endorsable undecidability and unendorsable neutrality is simply the natural result of the difference between thinking about different forms of sensing and thinking about different forms of thought. This suggests another way to resolve our puzzle. Kant does not explicitly state whether we can know whether or not discursive cognizers with other intellectual forms are possible. Nor are there textual or systematic considerations that

⁷³ The contrast between contingent ignorance and necessary ignorance is one to which Kant is generally very sensitive, and of which he makes a great deal (e.g. A767–9/B795–7; and Prol. 4: 350–7). ⁷⁴ See e.g. B xxx, and Kant’s account of different modes of Fürwahrhalten at A820–31/B848–59; cf. JL 9: 66–75. For discussion, see e.g. Chignell (2007a), Chignell (2007b), and Buroker (2017).

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decisively settle the question. If these observations suggest that Kant was neutral on the question, such neutrality may simply register Kant’s ignorance on the topic, combined with an openness to some consideration coming along that decides the matter. But they may also register, either instead or as well, not the endorsement of the claim that we cannot know which of Undecidability or Necessity is true, but a recognition, however inchoate, of the fact that any statement of such in-principle neutrality can never be known, combined with a suspicion that it is nevertheless the truth of the matter. It may be that where endorsement gives out, we can only remain silent.⁷⁵

Kant Bibliography We have used the following abbreviations when referencing works in Kants Gesammelte Schriften (Berlin: De Gruyter and predecessors, 1900–). C = Correspondence CPJ = Critique of the Power of Judgment Disc. = ‘On a Discovery Whereby Any New Critique of Pure Reason is Made Superfluous by an Older One’ JL = Jäsche Logic MFNS = Metaphysical Foundations of Natural Science Prog. = ‘What Real Progress has Metaphysics Made in Germany since the Time of Leibniz and Wolff?’ Prol. = Prolegomena to Any Future Metaphysics that will be Able to Come Forward as a Science VM = Vigilantus Metaphysics Translations have been from the following works in the Cambridge Edition of the Works of Immanuel Kant (Cambridge: Cambridge University Press, 1992–): Lectures on Logic, ed. and tr. Michael J. Young (1992). Critique of Pure Reason, ed. and tr. Paul Guyer and Allen W. Wood (1998). Lectures on Metaphysics, ed. and tr. Karl Ameriks and Steve Naragon (1997). Correspondence, ed. and tr. Arnulf Zweig (1999).

⁷⁵ The authors’ names appear in alphabetical order; all authors contributed equally. The paper derives from one written by Anil Gomes and Andrew Stephenson which was presented in Cambridge and Berlin. The final version originates in discussion between all three authors at a seminar in Oxford and was presented in Potsdam and Fribourg. Our thanks to the audiences on those occasions. For further discussion and comments, our thanks to Ralf Bader, Max Edwards, Stefanie Grüne, Johannes Haag, Nora Kreft, Colin McLear, Paola Romero, Tobias Rosefeldt, Nick Stang, and Rob Watt. We are especially grateful to two anonymous referees for the Philosophical Review whose comments have greatly improved the paper. Anil Gomes acknowledges the support of a British Academy Mid-Career Fellowship. Andrew Stephenson acknowledges the support of a Research Fellowship from the Alexander von Humboldt Foundation.

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Critique of the Power of Judgment, ed. Paul Guyer, tr. Paul Guyer and Eric Matthews (2000). Theoretical Philosophy After 1781, ed. Henry Allison and Peter Heath, tr. Gary Hatfield, Michael Friedman, Henry Allison, and Peter Heath (2002). We have also referred to translations of the Critique of Pure Reason by Norman Kemp Smith (New York: Macmillan, 1993) and Werner S. Pluhar (London: Hackett, 1996).

3 What Descartes Ought to have Thought about Modality Abstract The starting point of this essay is the first section of James Conant’s essay ‘The Search for Logically Alien Thought’, in which Conant discusses Descartes’s views about necessity and possibility. Conant is especially interested in claims that Descartes makes, with respect to propositions that on Descartes’s own conception could not be true, that it would nevertheless be wrong to say that not even God could make them true. It is argued that Descartes ought not to have made these claims. The essay concludes with consideration of one particularly important case where Descartes not only says what (according to this argument) he ought to have said, in contradistinction to these claims, but makes crucial capital out of his entitlement to do so. In an appendix to the essay some of its central claims are clarified. And in a postscript written for the reprint of the essay there is a corrective to a basic misunderstanding of the essay on Conant’s part, exhibited in his published response to it.

Jonathan Bennett wrote two commentaries on Kant’s Critique of Pure Reason: Kant’s Analytic and Kant’s Dialectic. In the Preface to the latter he referred back to a review of the former: ‘I continue to be,’ he wrote, ‘in the words of an unhappy reviewer of my earlier work, “one of those commentators who are more interested in what Kant ought to have thought than in what he actually did think” ’.¹ Probably you, like me, find it hard not to sympathize both with Bennett and with his unhappy reviewer. What an author ought to have thought must surely be of interest to any historian of philosophy (especially if we accede to Bernard Williams’s distinction between a historian of philosophy and a historian of ideas²). In particular, what an author ought to have thought is of critical interest when the author was involved in some fundamental error that we now want to avoid. This is quite apart from the fact, famously noted by Kant himself, that ‘it is not at all unusual to find that we understand [an author] even better than he understood ¹ Bennett (1974), p. vii. ² See B. Williams (1978), p. 9, where he writes, ‘the history of ideas is history before it is philosophy, while with the history of philosophy it is the other way round’.

The Human A Priori: Essays on How We Make Sense in Philosophy, Ethics, and Mathematics. A. W. Moore, Oxford University Press. © A. W. Moore 2023. DOI: 10.1093/oso/9780192871411.003.0004

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himself, since he may not have determined his concept sufficiently and hence sometimes spoke, or even thought, contrary to his own intention’.³ To that extent our sympathy must lie with Bennett. But of course, determining what an author ought to have thought about some matter, in so far as it is a historical exercise, had better be constrained in some significant way by determining what the author actually did think about the matter: the former had better be consonant with the crux of the latter, or serviceable in making sense of the latter, or something along those lines. Otherwise determining what the author ought to have thought about this matter simply reduces to determining what it is right to think about it, and the author himself is liable to drop out as irrelevant. So the unhappy reviewer can at least be said to have signalled the correct starting point. And this is quite apart from the fact that subordinating what an author actually did think to what he ought to have thought runs the very real risk of producing history that is anachronistic and complacent, far less of a challenge to contemporary presuppositions than any worthwhile history of philosophy should be. I think these issues are brought into particularly sharp relief by the main focus of the first section of James Conant’s magnificent and endlessly thoughtprovoking essay.⁴ I have in mind Descartes’s treatment of necessity and possibility. It seems to me that, given various things that Descartes thinks about necessity and possibility, there are various other things that he ought to think about them but does not or ought not to think about them but does. My aim is to disentangle these. Descartes understands by the possible what, in his view, everyone commonly understands by the possible, ‘namely “whatever does not conflict with our human concepts” ’.⁵ There are various things that might be intended here. Even given some particular way of construing what sort of conflict is involved, and of identifying our human concepts, there is an issue about whether ‘is possible’ is to be understood as synonymous with ‘does not conflict with our human concepts’. An alternative would be a quasi-realist view, of the sort championed by Simon Blackburn, whereby a statement of possibility would express, rather than report, some lack of conflict between a given proposition and our human concepts.⁶ And even if some kind of synonymy is intended, there is an additional issue about whether the kind of synonymy in question is weak enough to allow the two expressions to be subject to different semantic behaviour in certain semantic contexts.⁷ On any ³ Kant (1998), A314/B370. ⁴ Conant (2020a)—of which this essay first appeared as a discussion. ⁵ ‘Second Set of Replies’ in Descartes (1984b), 107. Strictly speaking, Descartes commits himself only to a hypothetical: if this is how the possible is understood, then such and such consequences accrue. The context, however, makes clear that he has no stake in understanding the possible in any other way. ⁶ See Blackburn (1993). ⁷ Cf. the stipulation whereby the name ‘Julius’ refers to whoever invented the zip. This confers a kind of synonymy on the name ‘Julius’ and the description ‘the inventor of the zip’. Even so, these two

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interpretation, however, it seems that the impossibility of one plus two’s being anything other than three (say) ensures that not even God could make one plus two anything other than three. For if it conflicts with our human concepts that one plus two should be anything other than three, then surely it conflicts with our human concepts that God should make it anything other than three. And indeed we do find Descartes saying, in line with this, ‘I have never judged that something could not be made by [God] except on the grounds that there would be a contradiction in my perceiving it distinctly.’⁸ So far, you might think, so straightforward. Not so. We also find Descartes conveying the very opposite idea in some correspondence. Most notably we find him saying the following, in a letter to Arnauld: I do not think we should ever say of anything that it cannot be brought about by God. For since every basis of truth . . . depends on his omnipotence, I would not dare to say that God cannot make [it] . . . that one and two should not be three. I merely say that he has given me such a mind that I cannot conceive . . . an aggregate of one and two which is not three, and that such [a thing involves] a contradiction in my conception.⁹

My suggestion, bluntly, is that this is a lapse.¹⁰ It seems to me that Descartes has actually thought something here which, in his own terms, he ought not to think: in Kant’s words, he has thought ‘contrary to his own intention’. Given Descartes’s conception of necessity and possibility, I think that he should treat the claim that not even God can make one plus two anything other than three as being entirely of a piece with the claim that one plus two cannot be anything other than three.¹¹ In saying this, I am in a kind of exegetical disagreement with Conant. This is not, au fond, a disagreement about how to interpret anything that Descartes says. It is rather a disagreement about how to apportion significance to various things that he says. Conant sees the quotation from the letter to Arnauld not as an aberration but as Descartes’s fully considered view. He refers at one point to

expressions make a very different semantic contribution when inserted into the following context: ‘If Julius’s grandfather had anticipated his grandchild’s celebrated invention, then he would have been . . .’ (See further Evans (1982), pp. 50 ff.). ⁸ ‘Sixth Meditation’ in Descartes (1984a), p. 50, emphasis added. ⁹ Descartes (1991), p. 359. ¹⁰ Am I being too precipitate? Notice that Descartes is merely declining to say what I claim he should say; he is not denying it. Can this perhaps be attributed to a scholastic scruple of some kind? (I am indebted to Matthew Boyle for this suggestion.) I shall ignore this possibility: I shall presuppose the reading that is least conducive to my exegesis, the reading whereby Descartes’s reason for declining to say what I claim he should say is that he takes it to be false. If this reading is not correct, then so much the better for my exegesis. If it is correct, then I have no choice but to see Descartes as involved in a lapse. ¹¹ In arriving at this view, I have been influenced by Jonathan Bennett’s superb essay, Bennett (1994), to which I am much indebted. I am nevertheless less charitable to Descartes than Bennett is. In §VII of Bennett’s essay, he tries but fails, in my view, to justify the circumspection.

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‘Descartes’s unabashed willingness to indulge in such . . . assertions [as that God could have made contradictories true together]’.¹² And he suggests that Descartes’s claim about the grounds on which he has judged things to be uncreatable by God, noted above, is a claim merely about his own repugnance at what he finds impossible; not a claim about some conclusion that he takes himself to have reached on the strength of ‘a clear and distinct perception of the positive limits of God’s power’.¹³ I disagree.¹⁴ I see Descartes as acknowledging the existence of that which is absolutely impossible—‘even,’ as Conant nicely says we cannot help adding, ‘for [God]’.¹⁵ What Descartes says in the letter to Arnauld is in conflict with this, and he should not, in my view, have said it. There are two points that I need to make straight away in mitigation of this seemingly cavalier stance. First, the letter to Arnauld is a comparatively isolated case. It is not unique (see, for example, the passage from the letter to Mesland that Conant cites¹⁶). But the aberrations are few. They are also, to the best of my knowledge, confined to correspondence. Furthermore, there is one exceedingly important case, to which I shall return at the end of the essay, in which Descartes not only says precisely what I think he should say, in contradistinction to these aberrations, but makes crucial capital out of his entitlement to do so. The second point concerns a de re/de dicto contrast that is pivotal to this discussion. I have in mind the contrast between the two following claims. (1) Given any proposition that conflicts with our human concepts, Descartes should say that not even God could make it true. (2) Descartes should say that, given any proposition that conflicts with our human concepts, not even God could make it true. These are rough formulations of the two claims, designed precisely to highlight the de re/de dicto contrast between them. The formulations blur some important details which a full exploration of these matters would need to take into account, and I shall say something both about these details and about their importance in the appendix. For current purposes, however, these formulations of the two claims will suffice. It is (1) that I want to defend.¹⁷ I am not committed to (2). I see no harm in Descartes’s denying that the sheer fact that a proposition conflicts with our human ¹² Conant (2020a), p. 32 n. 8. ¹³ Conant (2020a), p. 50 n. 23. ¹⁴ Or at least I disagree provided that ‘limits’ is not understood in the sense of ‘limitations’: see further below. ¹⁵ Conant (2020a), p. 31. ¹⁶ Ibid. ¹⁷ Here already it is worth flagging a detail that has been blurred. I am construing the ‘should’ in such a way that it is impervious to what grasp, if any, Descartes has of the conflict in question. There are alternative natural interpretations of the ‘should’ whereby (1) is true only with respect to propositions whose conflict with our human concepts Descartes recognizes.

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concepts is a bar to God’s being able to make it true. For I see no harm in Descartes’s denying that it conflicts with our human concepts that God should be able to do such a thing. But it is a different matter where God’s making one plus two something other than three is concerned. Descartes cannot deny that it conflicts with our human concepts that God should do that. My claim, in line with (1), is that Descartes should be unabashed in saying that God could not do it.¹⁸ What then leads him astray? Well, no doubt one thing that leads him astray is this very contrast. The denial that a proposition’s conflicting with our human concepts is itself a bar to God’s being able to make the proposition true can readily issue in the denial, of a particular conflict between a particular proposition and our human concepts, that that conflict is a bar to God’s making that proposition true. It is hard to separate these. There is something structurally analogous, and intimately related, in some of Descartes’s thinking about indubitability. This too is something to which I shall return at the end of the essay, where I hope, at the same time, to cast light on how the separation is best effected. Another factor in Descartes’s being led astray is presumably a reluctance to declare anything to be beyond the power of God. But as far as this is concerned, he could and should have taken the same line as Aquinas in the lengthy passage from Summa Theologia that Conant quotes at the beginning of his essay.¹⁹ He could and should have said that, even though it is impossible for God to make one plus two anything other than three, there remains a clear, reasonable, and substantive sense in which nothing at all is beyond the power of God; for there is a clear, reasonable, and substantive sense in which there is no such thing as one plus two’s being anything other than three, the sense, namely, in which the domain of quantification consists of absolute possibilities. This is an appropriate domain of quantification in this context because the question of what is possible for a given being, granted its power, is appropriately restricted to what is possible in the first place. As Aquinas himself puts it, ‘power is said in reference to possible things’.²⁰ When we say that God could not make one plus two anything other than three, we do not describe any limitation on the part of God then.²¹ We make an anthropocentric claim. We advert to our own human concepts. We say that these concepts would be contradicted by God’s making one plus two anything other than three. Such, at any rate, seems to me the most robust and compelling account of these matters that is in accord with Descartes’s core conception of necessity and possibility. But there are some natural concerns that arise about this. These concerns are in part exegetical and in part philosophical. I shall proceed to explore them.

¹⁸ See ‘Third Meditation’ in Descartes (1984a), p. 25, for material that is highly pertinent to this de re/de dicto contrast. ¹⁹ Conant (2020a), pp. 28–9. ²⁰ Aquinas (1947), pt 1, question 25, art. 3. ²¹ This is what I had in mind in n. 14.

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The First Concern This is a concern that will occur to anyone with even a moderate knowledge of Descartes’s philosophy. In fact it is liable to occur to anyone who looks at the quotation from the letter to Arnauld. There we saw Descartes proclaim that ‘every basis of truth . . . depends on [God’s] omnipotence’. Does Descartes not famously insist that both the truth and the necessity of any necessary truth depend on God’s free choice? In fact, is this not the non-negotiable premise of everything else that he has to say about necessity and possibility? How then can it be right to suggest that, on the most natural and coherent development of his view, not even God could make one plus two unequal to three? Surely we relinquish any entitlement to the claim still to be talking about ‘his’ view if we say that.

Reply to the First Concern Descartes does indeed insist that both the truth and the necessity of any necessary truth depend on God’s free choice.²² But this can readily be reconciled with the claim that not even God could make one plus two unequal to three—the point being that dependence here need not be understood in terms of the exclusion of possibilities. That one plus two is three, and that it is necessary that one plus two is three—in other words, that our human concepts conflict with one plus two’s being anything other than three—can be regarded, for current purposes, as two data. Descartes’s view is that, like everything else, they depend on God’s free choice. The first holds because of how God has made the natural numbers; the second holds because of how God has made our human concepts. But we should not say that, in making the natural numbers thus, God has excluded other possibilities; nor that, in making our human concepts thus, God has prevented us from grasping other possibilities. For there are no other possibilities. To suggest that there are would simply be to violate the second datum: that it is necessary that one plus two is three. Admittedly, in saying that x depends on y, one is committed to saying that, had y been different in certain critical respects, x would have been different too. But the first of these is not a simple reformulation of the second. And the second is irrelevant if y could not in fact have been different in those critical respects (which would follow if x itself could not have been different). Thus the non-existence of a male barber who shaves all and only the men in his village who do not shave themselves depends on there being no man who both shaves himself and does not

²² ‘Sixth Set of Replies’ in Descartes (1984b), p. 291.

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shave himself; but this is not to say that, because there is no man who both shaves himself and does not shave himself, the possibility of a male barber who shaves all and only the non-self-shaving men in his village is unrealized. Or at least it is not to say this except in so far as the possibility in question is a harmlessly irrelevant epistemic possibility that has purchase only in so far as, and only for as long as, we have an imperfect grasp of the matter. (This is a possibility of a sort that has no less purchase, and no more purchase, where God’s decreeing the value of one plus two is concerned.²³) It is all a question of explanatory priority.²⁴ On Descartes’s view, one plus two’s being equal to three is explained by, but does not explain, God’s decreeing that one plus two is equal to three—which is in fact no different from God’s understanding that one plus two is equal to three.²⁵ Descartes takes the former to depend on the latter in just this sense.²⁶

The Second Concern In drawing the de re/de dicto contrast, I said that Descartes can allow for the possibility of God’s making a proposition true even though that proposition conflicts with our human concepts. Now this need not involve us in any error. God could change our human concepts so as to remove the conflict. But presumably He could also allow the conflict to remain. He would not do so, in ²³ Is the following an even better analogy: the reason why mermaids have fishes’ tails is that we have decreed that this is what mermaids are like; but there is, even so, a sense in which we could not have decreed otherwise, because our decree would not then have concerned mermaids? I think not. This is because I do not think that this is the right account of how we relate to mermaids. Granted that it is necessary that mermaids have fishes’ tails, then all that is explained by our decreeing that this is what mermaids are like is, not that this is what mermaids are like, but rather that we make use of the concept of a mermaid, or have introduced the concept of a mermaid, or some such. The difference between our decree, on my preferred conception of this matter, and God’s decree, on a Cartesian conception of this matter, is that, in the former case, the decree and its content are answerable to the necessity concerned, at least in so far as there is an answerability in either direction; whereas in the latter case, it is the other way around. ²⁴ Some people would put this in terms of grounding: see Fine (2001). ²⁵ Descartes (1985), pt 1, §23. ²⁶ Descartes’s view is in direct opposition to that of Suárez, which Conant discusses in nn. 6 and 7 of his essay. Suárez holds that God’s understanding that one plus two is equal to three is explained by one plus two’s being equal to three. A very interesting intermediate case, which Conant discusses in the third main section of Conant (2020a), is that of Leibniz. Conant represents Leibniz as a fierce opponent of Descartes; and so he is, as the passage from Leibniz’s Discourse on Metaphysics that Conant quotes on p. 50 makes clear. Leibniz denies, contra Descartes, that necessary truths depend on God’s will. Why then do I describe Leibniz’s case as an intermediate one? Well, as Conant notes (p. 51), although Leibniz denies that necessary truths depend on God’s will, he does not deny that they depend on God’s understanding. (Descartes, as I intimated in the main text, rejects the very distinction between God’s will and God’s understanding.) And because Leibniz holds that necessary truths depend on God’s understanding, it is Descartes rather than Suárez whom he follows on the issue of whether necessary truths would be true even if, per impossibile, God did not exist. He denies that they would: see e.g. Leibniz (1973), §43; cf. §§44–6. Part of what makes this especially interesting in the present context is that it furnishes another example of how x can be said to depend on y even though x could not have been different in relevant critical respects.

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Descartes’s view. To do so would be contrary to His benevolence.²⁷ But He could. And if He did, then there would be a proposition which, though impossible—or rather, though impossible for us—was nevertheless true. But how can Descartes accede to this possibility without compromising what I have said he should say de re? How, for example, can he deny that there is such a thing as one plus two’s being anything other three, if, as he now seems forced to concede, the impossibility of one plus two’s being anything other than three is at most an impossibility for us—an impossibility that does not even preclude its truth?

Reply to the Second Concern The first thing to note is that Descartes need not agree, and would not agree, that God could make a proposition true while allowing a conflict between that proposition and our human concepts to remain. For Descartes would deny that God could—not just that God would—do anything that was contrary to His benevolence.²⁸ But in any case, even if Descartes were to accede to such a possibility, this would not prevent his being resolute de re. The fact remains that it is not possible, in Cartesian terms, for one plus two to be anything other than three; and therefore it is not possible for God to make one plus two anything other than three. And ‘not possible’ here does not need to be relativized. Descartes can say, as I have suggested he should say, that there is no such thing as one plus two’s being anything other than three. This is nothing but a graphic way of adverting to the impossibilities in question, a way that removes whatever apparent threat they pose to God’s omnipotence.²⁹

The Third Concern This is a concern that arises in connection with a family of views that are structurally similar to that which I am now attributing to Descartes. I have in mind meta-ethical views whereby undesirability stands to our ‘pro’-attitudes in a relation of conflict akin to that in which, on the Cartesian view, impossibility stands to our human concepts. The concern is this: any view of this kind does seem to entail a sort of relativism. For instance, it seems to entail that, had we not set such store by truthfulness, say, then we would not have had to deplore lying in ²⁷ See ‘Fourth Meditation’ in Descartes (1984a). ²⁸ ‘Fourth Meditation’ in Descartes (1984a), p. 37. On Conant’s view, God’s (necessary) benevolence creates untold difficulties for Descartes; see n. 20 of Conant (2020a). I take it to be another advantage of my exegesis that I see Descartes as more or less immune to such difficulties. ²⁹ It follows from this reply to the second concern that Descartes is at liberty to acknowledge an absolute impossibility. It does not follow that he is not at liberty to acknowledge relative impossibilities as well. In resisting (2) he would be acceding to just such relativity (as the second concern itself makes clear). That is fine. The situation here is not unlike the situation in which someone acknowledges both unrestricted and restricted quantification.

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the way in which, in fact—and quite rightly—we do. If it does entail this, what makes the ethical case different from the modal case?

Reply to the Third Concern There are two contrasting points that might be made straight away in response to this third concern. First, it is not clear, on reflection, that such meta-ethical views do entail any such relativism. Secondly, and conversely, I do not in fact need to deny that Descartes’s view does. As far as the first of these points is concerned, we can turn to the work of Simon Blackburn, who defends a meta-ethical view of just the kind in question.³⁰ Blackburn strenuously denies that what he defends is in any relevant sense relativistic.³¹ The second point involves a distinction of levels. I have denied that Descartes’s view compromises the necessity of the proposition that one plus two is three. I have not denied that it compromises the necessity of the necessity of that proposition. For Descartes’s view to entail relativism of the relevant sort would be for it to do the latter. There seems to be no obstacle to my conceding that it does indeed do the latter. There seems to be no obstacle, in other words, to my conceding that, on Descartes’s view, had our human concepts been relevantly different, then we would have had to deny what we in fact rightly proclaim, the absolute impossibility of one plus two’s being anything other than three. I therefore appear to have an embarrassment of riches at this point: a choice between two quite distinct ways of sidestepping the third concern. Unfortunately, I am uncomfortable with both. Though Blackburn attempts to dissociate his own view from any relativism of the sort envisaged, I believe that he fails in his attempt: this is something that I have argued elsewhere.³² Conversely, though I have not yet denied that Descartes’s view entails relativism of the relevant sort, I do in fact want to deny it. For if a given proposition conflicts with our human concepts, then so does the proposition that this proposition should not conflict with our human concepts: our human concepts give us no more of a grip on our human concepts’ giving us a grip on one plus two’s being anything other than three than they do on one plus two’s being anything other than three. So I do, after all, need to confront the question of what pertinent difference there could be between the ethical case and the modal case. The third concern remains a live concern for me. I have a response which echoes my response to the first concern. Descartes can simply deny that our human concepts could have been relevantly different. Thus let us grant that it is not only necessary that one plus two is three but also necessarily necessary. Then, just as the natural numbers could not have been ³⁰ Blackburn (1998). ³¹ Blackburn (1998), ch. 9 passim and p. 314. ³² Essay 13 in this volume.

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fashioned in such a way that one plus two was anything other than three, so too our human concepts could not have been fashioned in such a way that we had to reckon with the possibility of one plus two’s being anything other than three. It seems hard to deny, on the other hand, that we could have had a different attitude towards truthfulness—or rather, more cautiously, it seems hard for a quasi-realist such as Blackburn to deny this. Anyone who sees the sort of connection between what is undesirable and our ‘pro’-attitudes that is envisaged here must, I would contend, acknowledge correspondingly different possibilities for what we should deplore.

The Fourth Concern This harks back to the two previous concerns and my replies to them. There seems to be a tension within those two replies. Consider first my reply to the second concern and, in particular, my stance on the de dicto issue. I said that Descartes can allow for the possibility of God’s making a proposition true even though that proposition conflicts with our human concepts. But I also said that Descartes cannot allow for the possibility of God’s doing such a thing without removing the conflict, since that would be contrary to God’s essential benevolence. How is this to be reconciled with my reply to the third concern? For in order to remove the conflict God would have to change our human concepts. But in my reply to the third concern, did I not have Descartes denying that our human concepts could be relevantly different?

Reply to the Fourth Concern There is a very crude point that deserves to be made straight away: it is simply not true that, in order to remove the conflict in question, God would have to change our human concepts; for God could remove the conflict in question by banishing us from the scene altogether. That said, I think there is an altogether more telling response that is available to the fourth concern, a response that would still be available even if our own presence on the scene were taken to be non-negotiable. This requires us once again to invoke a de re/de dicto contrast. The cardinal point in my reply to the third concern was essentially this: given any proposition that conflicts with our human concepts, Descartes should say that not even a change in those concepts could indicate how it might be true. The point that is relevant to my reply to the second concern is essentially this: Descartes should not say that, given any proposition that conflicts with our human concepts, not even a change in those concepts could indicate how it might be true. These two points are compatible.³³ ³³ There is a third point that is worth noting here. There is a sense that ought to be acknowledged by everyone, even by the most resolute anti-Cartesian, in which differences in our concepts could certainly have involved our acknowledging necessities and possibilities other than those which we actually acknowledge. They could have involved our recognizing as necessary or as possible, not propositions

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I retain my stance on (2) then. I agree with Conant when he insists that, for Descartes, God’s power had better not be limited to ‘that which is comprehensible to minds such as ours’³⁴—provided that this is understood in a suitably de dicto way.³⁵ Claim (1) remains another matter. Descartes can still say, and in my view should say, that, given the incomprehensibility to minds such as ours of God’s making one plus two anything other than three, it is impossible for Him to do such a thing. This is not to say that God’s power of creation is somehow reined in by our power of comprehension. The point is simply that our power of comprehension indicates what the possibilities are: the possibilities, that is, ‘even for God’. I made two promises earlier about issues to which I would return. I said that I would discuss a case in which Descartes not only says precisely what I think he should say but does so in a way that is crucial for his own purposes. I also said that I would discuss something in his thinking about indubitability that is structurally analogous, and intimately related, to what I claim to find in his thinking about necessity. I can fulfil these two promises in tandem. I begin with the analogy. There are many propositions that Descartes cannot doubt.³⁶ But this does not preclude his taking a critical step back and raising the following sceptical question: ‘Why should the sheer fact that I cannot doubt a proposition mean that it is true?’³⁷ So there is a contrast that is operative here that is not unlike the contrast between (1) and (2). Given any proposition that he cannot doubt, Descartes is impelled to say that it is true. But he is not impelled to say that, given any proposition that he cannot doubt, it is true—not yet, anyway. (Part of the project is to find something that will impel him to say this and thereby entitle him to say it.) As for the intimate relation to which I have referred, this resides in the fact that what makes certain propositions indubitable for Descartes is precisely that their falsity would conflict with our human concepts. Descartes famously addresses his own sceptical question by invoking God’s benevolence. But he cannot do this without availing himself of some of the indubitability that is at issue.³⁸ In particular, he needs to pass, in indubitable steps, from an assurance that he himself is thinking, to an assurance that he himself exists, to an assurance that God exists, to an assurance that nothing can occur except what God, in His benevolence, would allow to occur. It is when he takes the very first of these steps that he supplies the example to which I have been referring of his properly adhering to his own core conception of impossibility. which we actually recognize as respectively unnecessary or impossible, but propositions which, given our actual conceptual repertoire, we cannot so much as entertain. Cf. in this connection the material from Locke that Conant discusses in Conant (2020a), n. 25. ³⁴ Conant (2020a), p. 32. ³⁵ The material in n. 33 indicates another, quite innocuous sense in which Conant is right to insist on this. ³⁶ See ‘Second Set of Replies,’ in Descartes (1984b), pp. 103–4. ³⁷ See ‘Third Meditation,’ in Descartes (1984a), p. 25. ³⁸ This is the so-called Cartesian Circle. I shall not pause to consider the extent to which it is a problem for Descartes. For an excellent discussion, see B. Williams (1978), pp. 189–204.

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He does this near the beginning of the Second Meditation, when he is still toying with the sceptical possibility that there is, as he says, ‘a deceiver of supreme power’ who is constantly deceiving him. ‘Let [the deceiver] deceive me as much as he can,’ writes Descartes, ‘he will never bring it about that I am nothing so long as I think that I am something.’³⁹ That seems to me to be just what Descartes should say. And it is vital to his project that he should be able to say it. On Conant’s view, however, it appears to be an aberration. For if the deceiver really is ‘of supreme power’, then why, for Conant’s Descartes, should he not bring it about that Descartes is nothing even though he thinks that he is something? Surely, on Conant’s view, Descartes is not entitled to rule out the possibility of an omnipotent being’s bringing about an absurdity even as gross as this. Both Conant and I therefore seem forced to concede that Descartes sometimes speaks ‘contrary to his own intention’. The difference is that what Conant is forced to treat as aberrant is a cornerstone of Descartes’s entire edifice. If Descartes cannot even rule out the possibility of an omnipotent being’s making him think that he is something while in fact he is nothing, then his project is in ruins. By contrast, there is no comparable cost, if indeed there is any cost at all, in Descartes’s conceding, of any proposition that conflicts with our human concepts, that not even God could make it true.⁴⁰

Appendix I said that my formulations of (1) and (2) blur some important details. The purpose of this appendix is to provide (partial) rectification. The first thing to note is that the ‘could’ in (2) does a kind of double duty. There is really a nested modality here. Claim (2) is to be understood as follows: (2*) Descartes should say that, necessarily, for any proposition p, if p conflicts with our human concepts, then necessarily, God does not make p true. Claim (2) is a claim about how Descartes should say things must be. If it were not— if the first occurrence of ‘necessarily’ in (2*) were omitted—then (2) would be too close to (1) for my purposes. The distinction between them would remain, but there would be no good reason for accepting either that could not be converted into a good reason for accepting the other. But why does the first occurrence of ‘necessarily’ not suffice? This is because of complications concerning God’s benevolence. Descartes should say that, necessarily,

³⁹ ‘Second Meditation’, in Descartes (1984a), p. 17. ⁴⁰ I am extremely grateful to participants at a weekend retreat organized by Birkbeck College London, to participants at departmental seminars in the Universities of Warwick and Glasgow, and to Matthew Boyle, Penelope Mackie, Jean-Philippe Narboux, Simon Rippon, Charles Travis, Peter Sullivan, and especially James Conant, for comments that helped me to improve this essay.

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for any proposition p, if p conflicts with our human concepts, then God does not make p true: it would be contrary to God’s benevolence for Him to do otherwise. So if the second occurrence of ‘necessarily’ were omitted, (2) would not capture the falsehood—as I take it to be—that, for Descartes, had there been differences in what conflicts with our human concepts, then there would have been corresponding differences in God’s power. Neither can the need for the second occurrence of ‘necessarily’ be circumvented by inserting ‘actually’ before ‘conflicts with our human concepts’. That would once again leave (2) too close to (1) for my purposes. These remarks signal the second important detail that needs to be noted. Since both ‘conflicts with our human concepts’ and ‘necessarily’ occur within the scope of ‘necessarily’ in (2*), there is a question about the semantic behaviour of such expressions in such contexts. In particular, do they behave ‘flexibly’—that is, in such a way as to be sensitive to what would conflict with our human concepts if we and our human concepts were different in various ways? Or do they behave ‘rigidly’—that is, in such a way as to be sensitive only to what does conflict with our human concepts, given how we and our human concepts actually are? (This was the sort of issue that I had in mind towards the beginning of this essay when I suggested that, even if ‘does not conflict with our human concepts’ and ‘is possible’ enjoy a kind of synonymy, there may nevertheless be semantic contexts in which the semantic behaviour of one differs from the semantic behaviour of the other.) Throughout this essay I have been presupposing that ‘conflicts with our human concepts’ and its like behave flexibly, while ‘necessarily’, ‘possibly’, and their like behave rigidly. If there were no such difference in the semantic behaviour of these expressions, then (2) would once again fail to capture the falsehood—as I take it to be—that, for Descartes, had there been differences in what conflicts with our human concepts, there would have been differences in the propositions that God can actually make true; that is to say there would have been differences in the propositions whose being made true by God does not actually conflict with our human concepts. Note also that, given this difference in semantic behaviour between the expressions in question, there is scope both to deny that (1) is necessary, which is a prerequisite (or something close to a prerequisite) of denying that (2) is true, and to maintain that (1) is nevertheless a priori.⁴¹ This blocks another possible concern about my exegesis; namely, that I have no way of explaining how Descartes himself could join me in affirming (1), which my exegesis is presumably intended to allow, without thereby violating my denial of (2). I do have a way of explaining this. In as much as (1) is a priori, Descartes could have what in his own terms would count as a clear and distinct perception of its truth. This would not violate my denial of (2), because it would not require (1) to be necessary. But let us now return to the idea that there is a nested modality in (2). If this is so, then has it not been misleading for me to insist all along that the contrast ⁴¹ Cf. the claim that Julius invented the zip, where ‘Julius’ is understood in accord with the stipulation mentioned in n. 7. There is scope both to deny that this is necessary—Julius’s parents might never have met—and to maintain that it is nevertheless a priori—no empirical evidence is required to assent to it.

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between (1) and (2) is a de re/de dicto contrast? Is it not rather, despite my response to the third concern, something more like the contrast between a necessity and a necessary necessity? No. For there is a nested modality in (1) too. The full import of (1) and (2), and the contrast between them, are perhaps clearest when they are formulated as follows: (1**) For any proposition p, if p conflicts with our human concepts, then Descartes should regard the following as a necessary truth: necessarily, God does not make p true. (2**) Descartes should regard the following as a necessary truth: for any proposition p, if p conflicts with our human concepts, then necessarily, God does not make p true.

Postscript for the Reprint In the volume in which this essay originally appears, The Logical Alien: Conant and His Critics, there is a response by James Conant.⁴² The purpose of this postscript is to respond to Conant’s response. In particular, I wish to correct a basic misunderstanding of my position on his part. Conant himself distinguishes three positions: that which he adopts in his own original essay; that which he adopts in his response to me, which by his own account represents a significant change of mind; and mine. Moreover, he sees each of these as being fundamentally opposed to each of the other two. He uses the device of referring to himself as ‘Conant’ in the third person when referencing his own original position and using the first person when referencing the position that he adopts in his response to me. I shall distinguish between these by referring to ‘Conant₁’ and ‘Conant₂’ respectively. I shall not rehearse my disagreements with Conant₁ in this postscript, since it has been the burden of the current essay to do that. Rather, I wish to indicate my agreements with Conant₂. We differ, but we do not differ to anything like the extent that Conant₂ thinks we do. I should emphasize before I go any further that there is a vast amount, not only in Conant₂’s reply to me, but also in his replies to the other contributors to The Logical Alien, that I admire enormously, indeed that I take to be a paradigm of how the history of philosophy should be conducted, but about which I shall say nothing more in these remarks. My primary aim here is to correct the misunderstanding to which I have already referred. In his response Conant2 introduces two propositions which he labels (1) and (2). They are as follows:

⁴² Conant (2020b), §§I – VII, esp. §VII.

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(1) The eternal truths are freely created by God (2) The eternal truths are necessarily true.⁴³ By the eternal truths here are meant such necessities as the one that is the main focus of my essay, namely that one plus two is three. Conant₂ insists that Descartes affirms both (1) and (2) ‘with his eyes fully open to what he is thereby doing.’⁴⁴ I unreservedly agree. A little later Conant₂ says that the difference between Conant₁ and me is as follows: Conant₁ accepts Descartes’s commitment to (1) and thinks that Descartes therefore cannot really be endorsing (2), except in such a highly qualified way as to be effectively abandoning it, whereas I, by contrast, accept Descartes’s commitment to (2) and then water down what, if anything, is still to be meant by (1).⁴⁵ This is importantly wrong. If there were any justice to the complaint that the position that I attribute to Descartes involves any watering down of anything, then that complaint would have to be targeted at (2), not at (1). Recall that claims about necessity and possibility, on what I label Descartes’s ‘core conception’ of these notions, are anthropocentric claims. This might be thought to involve a watering down of (2). It cannot properly be thought to involve any watering down of (1). This error is related to Conant₂’s repeated appeal to the notion of priority in his characterizations of the view that I attribute to Descartes. Thus he has me attributing to Descartes the view that ‘these [sc. what we take to be all the possibilities there are] are all the possibilities there are; therefore they define the modal space of possibility, which limits even God’s free activity prior to His having created anything.’⁴⁶ Earlier he refers in a similar connection to ‘an explanation of what is necessary, which attempts to lodge it in a ground that is prior to God’s activity.’⁴⁷ There are countless further examples. I take it that Conant₂ intends the notion of priority non-temporally. It would be absurd to suppose that I attribute any such view to Descartes on a temporal understanding of the notion—even assuming that we can form any coherent idea of what such a view would be. But in fact it is only marginally less absurd to suppose that I attribute any such view to Descartes on a non-temporal understanding of the notion. I never use such language. And I would firmly resist doing so. I had hoped that this was clear in what I said in my ‘Reply to the First Concern’. True, I want to say that, for Descartes, what we take to be all the possibilities there are are all the possibilities there are. So does Conant₂.⁴⁸ I am also prepared to add—and here I differ from Conant₂, for reasons that I shall sketch shortly—‘even for God’. But this does not gainsay the fact that, for Descartes, the ultimate ground of these being all the possibilities there are is God Himself. And it certainly does not mean, on any reasonable interpretation of the notion of priority, that their being all the possibilities there are is prior to God’s having created anything. In particular, it does not mean that their being all the possibilities there are is, on any reasonable interpretation of that notion, prior to God’s having created us. Conant₂ says that ⁴³ Ibid., p. 471. ⁴⁴ Ibid., p. 471. ⁴⁵ Ibid., p. 478. ⁴⁶ Ibid., p. 567, emphasis added. ⁴⁷ Ibid., p. 475, emphasis added.

⁴⁸ Ibid., p. 567.

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‘Moore’s Descartes requires that our grasp of the truth of the proposition [that these are all the possibilities there are] does not rest on our grasp of any of the actualities there are.’⁴⁹ I say nothing that licenses that. On my view, Descartes requires that our grasp that these are all the possibilities there are involves reflection on our own concepts and what conflicts with them. Those are actualities. What Conant₂ wants to insist, in contrast to both Conant₁ and me, is that Descartes resists the very propriety of raising and answering questions about what God can or cannot do. Conant₂ takes this to explain why, in the letter to Arnauld, Descartes says, ‘I do not think we should ever say of anything that it cannot be brought about God,’ rather than the blunter, ‘There is nothing that cannot be brought about God.’⁵⁰ On Conant₂’s view, the impropriety of raising and answering such questions is of a piece with the dependence of everything on, and the non-priority of everything to, God.⁵¹ There is something very deep and very important going on here. I find myself in enormous sympathy with much of what Conant₂ says in the development and defence of his exegesis. I have learned a great deal from it too. Even so, I see nothing to dislodge me from my original position. I note first that there can be no denying that Descartes frequently and ineliminably makes claims that forestall any general prohibition against raising and answering questions about cognitive relations in which we human beings either do stand or can stand to propositions about God. Towards the end of the Fifth Meditation, for example, Descartes says that he himself ‘[has] perceived that God exists,’ that he ‘[has] understood that everything else depends on [God],’ and that ‘it is possible for [him] to achieve full and certain knowledge of . . . matters . . . concerning God.’⁵² On what I label Descartes’s ‘core conception’, claims about what God can or cannot do are nothing but a case in point. They are claims about how, by attention to our human concepts and what conflicts with them, we are able or are not able to determine the truth or falsity of certain propositions concerning God, for instance that God has not made it the case that one plus two is something other than three. And I still maintain both that Descartes makes such claims, indeed that he makes crucial capital out of making such claims, and that he should overcome his own reluctance, in other contexts, to do so. Such claims pose no more of a threat to the idea that everything depends on God than the claim that one of God’s creatures ‘has perceived that God exists’, or the claim that one of God’s creatures ‘has understood that everything else depends on God,’ or the claim that ‘it is possible for one of God’s creatures to achieve full and certain knowledge of matters concerning God,’ or—an additional example that I take to be especially instructive—the claim that one of God’s creatures can determine, by reflection on our human concepts, that God is not a deceiver. Why do I call this last example especially instructive? At one point Conant₂ states his position in the following terms: ‘Descartes insists that for any X we must not say ⁴⁹ Ibid., p. 567. ⁵⁰ Cf. n. 10. ⁵¹ See e.g. Conant (2020b), pp. 521–2. ⁵² ‘Fifth Meditation’, in Descartes (1984a), pp. 48–9.

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of God that He cannot do X.’⁵³ But in the Third Meditation Descartes does say that God ‘cannot be a deceiver’.⁵⁴ Something here is amiss. It is of course open to Conant₂ to insist that what is amiss is the face value of what Descartes says when we take it out of context. Thus Conant₂ has an instructive footnote in which he writes: [We] can speak of . . . what—in the light of what we know of the creation—God could and could not have done. This means that we must be careful . . . not to quote passages from Descartes out of context. If we are not careful about this, then it will not be a great challenge to find passages in a text such as Meditations . . . involving claims about what God could and could not have done.⁵⁵

Conant₂ may well say, indeed I presume would say, that my quoting Descartes’s claim that God cannot be a deceiver out of context is an instance of the danger against which he is warning us.⁵⁶ But the crucial point is that there is really no great difference, for current purposes, between what Conant₂ is committed to saying about the quotation from Descartes when it is seen in context—namely, that it adverts to what we can or cannot rule out by appeal to what we already know—and what I want to say about the quotation when it is taken, as I think it should be taken, at face value—namely, that it adverts to what we can or cannot rule out by appeal to our human concepts. On my view—to make this point one last time—claims about necessity and possibility are, on Descartes’s core conception, anthropocentric. It should be clear that this allows Descartes to affirm both that the eternal truths are freely created by God and that the eternal truths are necessarily true without any fear of conflict and ‘with his eyes fully open to what he is thereby doing.’⁵⁷ ⁵³ Conant (2020b), p. 484. ⁵⁴ ‘Third Meditation’, in Descartes (1984a), p. 35; cf. ‘Fourth Meditation’, in Descartes (1984a), p. 37. ⁵⁵ Conant (2020b), p. 540, n. 5. ⁵⁶ Something else that he might do is to invoke the contrast that he draws in ibid., p. 481, n. 23, between necessary truths whose necessity depends on God’s free activity and truths such as the truth that God is not a deceiver, that is truths directly concerning God Himself. This would have no purchase on the issue at hand, however, unless Conant₂ thought that, for Descartes, truths of the latter kind were also necessary—but necessary in a different way. And I take it that he does not think that (although it has to be said that the footnote is unclear in this regard and can be readily read as though he does). If Conant₂ did think that, then I would be out-Conant₂-ing him. For, on my view, there is no question but that, for Descartes, the necessity of every necessary truth depends on God’s free activity. ⁵⁷ I cannot resist a final incidental remark. In a revealing passage on pp. 544–5 of Conant (2020b), Conant₂ quotes from the part of my essay in which I claim that Descartes could and should have followed Aquinas in insisting that not being able to do the impossible is no limitation on the power of God. Conant₂ comments, ‘I find the casualness with which Moore is prepared to move Descartes over to that side of the theological debate fairly stunning,’ (p. 545). By ‘that side of the theological debate’ Conant₂ is referring to the view that what is necessary is prior to God’s activity. However, as I hope is clear by now, I have no interest in moving Descartes over to any such thing. The reason why I describe this passage from Conant (2020b) as revealing is that the extract from my essay that he quotes he quotes out of context. It is immediately followed by an italicized reference to the anthropocentrism that is so crucial to my view and that prevents my claim about what Descartes could and should have said from being the exegetical disaster that Conant₂ takes it to be. And indeed there is a passage in a letter to More, to which Sarah Patterson has subsequently drawn my attention, in which Descartes does say what I claim he could and should have said. Descartes writes, ‘[W]e do not take it as a mark of impotence when someone cannot do something which we do not understand to be possible . . . [W]e do not perceive it to be possible for what is done to be undone—on the contrary, we perceive it to be altogether impossible, and so it is no defect in power in God not to do it,’ (Descartes (1991b), p. 363).

4 Varieties of Sense-Making Abstract This essay was written for a special issue of Midwest Studies in Philosophy on the so-called ‘new atheism’. It is argued that part of what motivates the new atheism is an extreme naturalism, whereby the only way to make sense of things is the natural-scientific way. This naturalism is then discredited, partly on the grounds that the natural-scientific way to make sense of things is not the way to make sense of that very way of making sense of things. However, it is also argued that certain difficulties afflict any attempt, on the part of the theist, to resist the new atheism by invoking some quite different way of making sense of things: in particular, there are difficulties in ensuring that what is being invoked really is a way of making sense of things. The essay concludes with consideration of how due acknowledgement of the varieties of sense-making nevertheless provides a powerful response to the new atheism, and of how this bears specifically on one of the greatest challenges confronting the theist, namely the problem of suffering.

1. The New Atheism and the Naturalism that Underlies it Is there more than one way to make sense of things? This question is far too loose as it stands to have much philosophical purchase. But I take it that there are ways of tightening the question whereby there is room for genuine uncertainty concerning what to say, in as much as a philosophically substantive case can be mounted for each of the answers yes and no. And I further take it that at least one strand in the new atheism rests, in part, on a version of the view that the answer is no: the extreme naturalistic view that the only way to make sense of things is the way of natural science. For if we accept this view, if we accept that theism is an attempt to make sense of things, and if we accept that the methods and resources of the natural sciences cannot possibly vindicate theism, then we are bound to conclude that theism is, in its own terms, a failure, in which case we should either do as new atheists do, that is, reject it in favour of atheism, or dismiss the very choice between theism and atheism as one that lacks any genuine significance. Interestingly, those who are often invoked as being among the fiercest critics of theism, namely logical positivists, were not naturalists in this

The Human A Priori: Essays on How We Make Sense in Philosophy, Ethics, and Mathematics. A. W. Moore, Oxford University Press. © A. W. Moore 2023. DOI: 10.1093/oso/9780192871411.003.0005

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extreme sense. And although their own revolt against theism might be thought to be a paradigmatic example of this second approach—the approach of dismissing the very choice between theism and atheism as one that lacks any genuine significance—it was not straightforwardly an example of this approach at all. Logical positivists were always among the first to insist that there are varieties of sense-making. They were even prepared to acknowledge that the question that was of primary concern to them, namely what it is for an utterance of a declarative sentence to be meaningful (which itself hardly exhausts the question of what it is for an attempt to make sense of things to be successful), could be interpreted in more than one way. When they insisted that there was a whole class of utterances of declarative sentences that lacked meaning, what they meant was that these utterances lacked what they sometimes called ‘literal’ meaning: such utterances were not candidates for truth or falsity. But they could still have meaning of other kinds. They could still express feelings, prescriptions, or proscriptions, for example. This left open the possibility that utterances of declarative sentences involving religious vocabulary could express a distinctively religious way of making sense of things to which the methods and resources of the natural sciences were simply irrelevant. It is instructive in this connection to consider the following passage from A. J. Ayer’s Language, Truth and Logic: According to the account that we have given of religious assertions, there is no logical ground for antagonism between religion and natural science. As far as the question of truth or falsehood is concerned, there is no opposition between the natural scientist and the theist who believes in a transcendent god. For since the religious utterances of the theist are not genuine propositions at all [i.e. they lack literal meaning], they cannot stand in any logical relation to the propositions of science . . . An interesting feature of this conclusion is that it accords with what many theists are accustomed to say themselves.¹

Rudolf Carnap likewise recognized different ways of making sense of things. Even in his famous assault on the work of traditional metaphysicians, he acknowledged that they might defend themselves by denying that their work was an attempt to discover and state truths. They could, he admitted, insist that their work was an attempt to convey meaning of some other kind; and that it was less a scientific exercise than an artistic exercise. Carnap’s complaint about any such defence of traditional metaphysics was not that there was anything wrong with it per se. His

¹ Ayer (1971), pp. 154–6.

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complaint was rather that, in so far as traditional metaphysics was an artistic exercise, it was a third-rate artistic exercise. ‘Metaphysicians’, he wrote, ‘are musicians without musical ability’.² Be the artistic merits of traditional metaphysics as they may, it is unsurprising that many theists have reacted to the new atheism by protesting that, in effect, it inappropriately acknowledges only one way to make sense of things, the way of natural science. The protest has a familiar pedigree. Earlier versions of it include Wittgenstein’s remarks on Frazer’s Golden Bough,³ and indeed that old crude saw that science is concerned with ‘How?’ questions while religion is concerned with ‘Why?’ questions.⁴ Not that the protest can amount to much without a robust individuation of ways of making sense of things. It is no more than an opening move in the discussion. Even so, the idea that there are ways of making sense of things that are non-scientific without being unscientific does have clear potential to undermine at least some of the thinking behind at least some of the new atheism. An indication of what I have mind is provided by the following three quotations by Richard Dawkins. The first is taken from a 1992 debate with John Habgood at the Edinburgh Science Festival. The remaining two are taken from Dawkins’s book The God Delusion. You can’t escape the scientific implications of religion. A universe with a God would look quite different from a universe without one. A physics, a biology where there is a God is bound to look different. So the most basic claims of religion are scientific. Religion is a scientific theory.⁵ Either [God] exists or he doesn’t. It is a scientific question; one day we may know the answer, and meanwhile we can say something pretty strong about the probability.⁶ God’s existence or non-existence is a scientific fact about the universe, discoverable in principle if not in practice. If he existed and chose to reveal it, God himself could clinch the argument, noisily and unequivocally, in his favour.⁷

It is interesting to speculate in what way Dawkins thinks that God could ‘clinch’ the argument. What sequence of events does Dawkins think would leave us with no alternative but to acknowledge that God exists? It is not obvious to me what he has in mind. But that is largely because I do not believe that the question whether God exists is a scientific question and it is not obvious to me what other, scientific question Dawkins has confused it with. For now, this is less significant than his ² ⁴ ⁵ ⁷

Carnap (1959), p. 80. ³ Wittgenstein (1987). See also Wittgenstein (1980), pp. 30 ff. This is quoted in Johnston (2009), p. 46, emphasis in original. Dawkins (2007), p. 73.

⁶ Dawkins (2007), p. 70.

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sheer deafness, or what appears to be his sheer deafness, to the possibility that we ought not to think about an issue of this kind in scientific terms at all, perhaps not even in terms of truth or falsity. Dawkins does concede that there may be ‘some genuinely profound and meaningful questions that are forever beyond the reach of science’, adding that ‘maybe quantum theory is already knocking on the door of the unfathomable’.⁸ But he does not have in mind non-scientific questions. He has in mind scientific questions that are simply too hard for us to answer.

2. Against Such Naturalism Naturalism of the extreme kind that is of concern here seems to me to be subject to a damning criticism, quite apart from its relation to religion. It is not true that the only way to make sense of things is the way of natural science, because that is not the way, in particular, to make sense of that way of making sense of things: natural science is unsuited to providing a satisfactory account of how natural science itself is possible. W. V. Quine, who famously defends something like this extreme form of naturalism, equally famously attempts to forestall this very objection. He argues that we can provide a satisfactory account, in broadly natural-scientific terms, of how we make natural-scientific sense of things: such is the project of what he calls ‘naturalized epistemology’.⁹ The story that we are required to tell, on Quine’s view, is a story about ‘how we, physical denizens of the physical world, can have projected our scientific theory of that world from our meagre contacts with it: from the mere impacts of rays and particles on our surfaces and a few odds and ends such as the strain of walking uphill’.¹⁰ He believes that we can tell such a story by drawing on the relevant branches of natural science such as optics and neurophysiology. The problem with this, as many commentators have observed, is that the impacts of rays and particles on our surfaces, the strain of walking uphill, and suchlike, which are indeed well suited to appear in such a story, are unable, for that very reason, to stand in any but causal relations with what we subsequently do and say. They cannot stand in logical or rational relations with anything. They cannot act as evidence. So no story of this kind can do justice to the elementary way in which our natural-scientific sense-making is grounded in our evidence, and in particular in how things appear to us to be.¹¹ There is a further point. This grounding is not a deductive consequence relation. Theory is underdetermined by evidence. To arrive at a theory from evidence we need, among other things, to draw conclusions about the unobserved from

⁸ Dawkins (2007), p. 80. ⁹ Quine (1969). ¹⁰ Quine (1995), p. 16. ¹¹ See e.g. Stroud (1984), ch. 6., esp. pp. 250–4; and McDowell (1996), afterword, pt I, §3. One of the signal features of the phenomenological tradition is its recoil from such naturalism on such grounds: see e.g. Husserl (1981).

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premises about the observed. Making sense of how we make natural-scientific sense of things must therefore include some account of this process and of the normativity that attaches to it. This is another reason why it cannot itself be a simple exercise in making natural-scientific sense of things. It is not just that a simple exercise in making natural-scientific sense of things cannot adequately reckon with the normative, though that is certainly true. Neither can it adequately address whatever self-conscious doubts we have about its own justification, doubts exacerbated by the fact that some of the conclusions that it licenses concern that which cannot possibly have affected us, for instance because it lies in the future. The inferences involved cannot help looking like exercises of unjustified faith, as it may be faith in the uniformity of nature. In the present context this is doubly pertinent: it means that some of what is liable to appear to a naturalist as unjustified faith is a staple of the very form of sense-making that extreme naturalism proclaims is the only one there is. Ralph Walker has fastened on this fact in an attempt not only to remove some of the stigma from what appears to the naturalist as unjustified faith but also to urge a case for theism. ‘The world’, he writes, ‘keeps on meeting our expectations, from moment to moment and from year to year. Does this not need explanation? Does it not suggest that things have been arranged in our interest, with the independent world on the one hand, and our system of beliefs in [sic] the other, in a harmony continually and benevolently sustained?’¹² For Walker, it is not just that the naturalist needs to acknowledge the propriety of our drawing conclusions that strictly go beyond the evidence. He also takes the success that we enjoy in drawing these conclusions to be the basis of an argument for the existence of a benevolent designer. I make no comment here on the appeal of this argument. I simply note that to subscribe to it is to take an extra step. The first and less controversial step is just to usher the extreme naturalist out of his or her naturalism. Once this has been accomplished, the simple argument against a theistic attempt to make sense of things—that it lacks any natural-scientific vindication—lapses. But something more is clearly required (something of the sort that Walker himself provides) if theism is to be reinstated. The sheer fact that there are different ways to make sense of things, a fact that has in any case still not been properly explicated, does not sanction anyone’s making theistic sense of anything.

3. The Non-Hermetic Character of a Theistic Way of Making Sense of Things Indeed, in acknowledging that there are different ways to make sense of things, the theist incurs certain risks. One principal risk, which is incurred by non-theists too ¹² Walker (1989), p. 225, emphasis in original.

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if they do not subscribe to the extreme naturalism that we have been considering, is that of thinking—what is surely false—that a religious way of making sense of things can be somehow self-contained, impervious to any other way of making sense of things. Consider the following quotation from Stephen Jay Gould: The net, or magisterium, of science covers the empirical realm: what is the universe made of (fact) and why does it work this way (theory). The magisterium of religion extends over questions of ultimate meaning and moral value. These two magisteria do not overlap.¹³

Gould certainly seems to have fallen victim to the risk that I have in mind. Another casualty may be Peter Winch. He refers at one point to ‘those forms of life called ‘science’, ‘art’, etc.,’¹⁴—a reference that brings to mind Hilary Putnam’s famous complaint about the ‘fondness [of Wittgensteinians] for the expression “form of life”,’ namely that it ‘appears to be directly proportional to its degree of preposterousness in a given context’.¹⁵ At any rate, to distinguish two or more ways of making sense of things falls some way short of claiming that they are cleanly separable, a claim which, in the case of a religious way of making sense of things and a broadly scientific way of making sense of things, would surely be utterly implausible.¹⁶ Bernard Williams warns of the danger, to the (Christian) theist, of trying to effect a clean separation between ‘talk about God’ and other talk, in the following terms: If all talk about God were talk only about God, and all talk about the world talk only about the world, how could it be that God was the God of the Christian believer, who is a toiler in the world of men? Would not the views about the nature of God retire more and more away from the world of men[?] . . . And if that happened, it could not be of much concern whether he were there or not.¹⁷

Not that Williams is concerned to defend talk about God. On the contrary, having issued this warning, he straight away issues a second warning, this time of the opposite danger to the theist of allowing talk about God and talk about the world to mix. He writes: Although we must have some statement which says something about both God and the world, when we have it we find that we have something that we cannot properly say . . . [For] when we come to a statement that is about both God and

¹³ Gould (1999), p. 6. ¹⁴ Winch (1958), p. 41. ¹⁶ Cf. McGrath (2007), pp. 17–18. See also n. 27 below. ¹⁷ B. Williams (2006a), p. 14, emphasis removed.

¹⁵ Putnam (1970), p. 60.

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temporal events, it must be unsatisfactory; for if it were not, we should have adequately described the relation of the temporal events to God in terms appropriate only to the temporal events [which are the only terms we understand]: and this would mean either that we had described only the temporal events, and left God out, or had described God as a temporal being, which he is not.¹⁸

Williams sees a dilemma for the theist, then. This dilemma might be thought not to be a dilemma about mixing talk of God with other talk at all, but rather a dilemma simply about talk of God: the dilemma, namely, that we need to have some basic understanding of such talk for it to count as talk of God, whereas we need to lack any basic understanding of it for it to count as talk of God.¹⁹ In a way this is right. But the considerations about mixing such talk with other talk survive; for our having some basic understanding of such talk would consist partly in its being satisfactorily mixable with other talk that we understand, while our lacking any basic understanding if it would consist partly in its not being. Certainly a theist who insists that there is a distinctively theistic way of making sense of things needs to tread very carefully: it cannot be too distinctive or it will no longer count as a way of making sense of things; but it had better be distinctive enough for the point of insisting on its distinctiveness not to be compromised.

4. Theism and the Problem of Suffering As far as the point of insisting on its distinctiveness is concerned, I have so far focused only on the inadequacy of an atheism founded on the extreme naturalistic conviction that the only way to make sense of things is the way of natural science. But that is by no means the only reason we might have for insisting that there is a distinctively theistic way of making sense of things. Equally important is the bearing of this issue on what seems to me a much more reasonable atheism fuelled by the classic problem of suffering: atheism, in other words, founded on the conviction that the existence of a being that is both omnipotent and perfectly benevolent is incompatible with the existence of suffering. There is of course a huge amount to be said about this problem, indeed a huge amount that has been said about it, and the problem is not about to go away. Still, I think it is worthwhile, in this context, to make the following point: even if the existence of a being that is both omnipotent and perfectly benevolent is indeed incompatible ¹⁸ B. Williams (2006a), pp. 14–15. ¹⁹ A similar dilemma afflicted Descartes. He needed to have some basic grasp of his idea of God for it to count as a genuine idea; he needed to lack any basic grasp of it for it to be the kind of idea that he took it to be, an idea of something so great that only something that great could explain how he had come by it. See B. Williams (1978), pp. 143–5.

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with the existence of suffering, this only means that one cannot consistently accept that there is suffering and believe that there is such a being. It does not mean that one cannot consistently accept that there is suffering and be a theist. Nor do I have in mind here the simple point that our definition of God may not include omnipotence and perfect benevolence (though the simple point deserves to be made too). Rather, I have in mind the point that theism may be part of a distinctive way of making sense of things that does not consist in holding any particular truth-evaluable belief. This would mean that one could consistently be a theist no matter what one accepted. It is a further question, of course, whether one could reasonably be a theist no matter what one accepted, and to address that question we should need to provide some positive account of what exactly theism is. But there is at any rate not the direct route that there appeared to be from suffering to atheism. Very well; but what positive account of theism can we provide that distances it from the holding of any truth-evaluable belief? Well, people sometimes distinguish between belief in God and belief that God exists.²⁰ Perhaps we could equate theism with belief in God.—Yes, but what would be meant by belief in God?—One way of drawing the distinction between belief in God and belief that God exists would be on the model of the distinction between belief in justice and belief that justice exists. Someone can believe in justice in a given context, in the sense that he or she can believe that it is important for justice to prevail there, even while recognizing that it does not (yet). On this model belief in God would be, roughly, belief that it is important to promote and cherish whatever bespeaks God; belief that God exists would be belief that whatever bespeaks God speaks truly. One might even have the former belief alongside a compelling argument that whatever bespeaks God cannot be understood as speaking truly (or indeed, cannot be understood as making any sense at all). Here a quotation from Iris Murdoch is pertinent: No existing thing could be what we have meant by God. Any existing God would be less than God. An existent God would be an idol or demon . . . God does not and cannot exist. But what led us to conceive of him does exist and is constantly experienced and pictured . . . [It is] incarnate in knowledge and work and love . . . We experience both the reality of perfection and its distance away, and this leads us to place our idea of it outside the world of existent being as something of a different unique and special sort. Such experience of the reality of good . . . is a discovery of something independent of us . . . If we read these images aright they are not only enlightening and profound but amount to a statement of a belief that most people unreflectively hold.²¹

²⁰ E.g. Price (1966).

²¹ Murdoch (1993), p. 508, emphasis in original.

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But would this distance theism from the holding of any truth-evaluable beliefs? Not obviously. Belief that it is important to promote and cherish whatever bespeaks God certainly seems to be truth-evaluable. What it might do is to help with the problem of suffering, in as much as belief that it is important to promote and cherish whatever bespeaks God, even if truth-evaluable, is not obviously incompatible with belief that there is suffering. But for a way of extricating theism from the classic (intellectual) problem of suffering altogether we require something more radical. For this we can perhaps turn to Kant, and in particular to Kant’s notion of a regulative principle. By a regulative principle Kant means a rule directing us to act in accord with some given supposition.²² We can construe belief in God as the embracing of such a rule. To believe in God, on this construal, is to live one’s life as if God exists. But how can one live one’s life as if God exists if one has compelling reason to think that God does not exist? For that matter, what about the lesson, or the apparent lesson, of the quotations from both Williams and Murdoch, namely that we can have no understanding of what it would even be for God to exist? If we can have no understanding of what it would even be for God to exist—if the very idea that God exists is shot through with incoherence—then surely we have no prospect of living our lives as if God exists? This conclusion seems to me too precipitate. This in turn is for reasons that I have tried to advance elsewhere and that I shall take the liberty of sketching here. To live one’s life as if some given supposition holds is to exercise a kind of knowledge. But such knowledge is not knowledge that anything is the case. In fact it is not even expressible. It is knowledge of how to cope with situations in a certain way. It involves making a certain non-truth-evaluable sense of things. What then makes the exercise of it the living of one’s life as if that supposition holds? The fact that, if one were to attempt (unsuccessfully) to put the knowledge into words, then what one would do is to give voice to that supposition. In the current case, which I take to be a case in point, one would say that God exists. (It follows that the claim that God exists, without being an expression of the relevant knowledge, could, if made in the right way in the right context, help to celebrate, nurture, proclaim, or even impart that knowledge.) But this does not require that the idea that God exists should actually be credible. It does not even require that the idea that God exists should be intelligible. It requires only that the sentence ‘God exists’ should conjure up all sorts of relevant images and have all sorts of relevant associations and connotations.²³ These images, associations, and connotations may include notions of constancy, for example, where these in turn may

²² Kant (1998), A508–15/B536–43 and A669/B697 ff. ²³ For a full defence of these ideas see A. W. Moore (1997), esp. chs 7–9; ch. 10, §5; and pp. 277–8. See also A. W. Moore (2003a).

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correspond to certain hopes that can sustain us in our commitment to making this particular sense of things; hopes that can reinforce our confidence that the contingencies that enable us to make such sense are in some way necessary, just as the contingencies that enable the earth to carry on spinning on its axis are in some sense necessary: utterly steadfast, utterly to be relied upon. We saw earlier that Walker thought that he could argue from these contingencies to the existence of God. I do not myself believe that any such argument can be successful. But I do acknowledge the way in which these contingencies can put us in mind of God, and the way in which talk of God can put us in mind of them. In the present context, this is as much as is required of them. The relevance of belief in God to the problem of suffering may lie not in the capacity of the latter to refute the former, then, but in the capacity of the former to help cope with the latter. And indeed it is noteworthy how frequently extremes of suffering draw people into a belief in God rather than away from it. This is not just a matter of the consolations that such a belief affords. It is a matter of the conceptual shape that it helps people to impose on their suffering: the sense that it helps them to make of it. As Nietzsche insisted, it is not suffering that people find unbearable, but senseless suffering.²⁴ A quotation from Wittgenstein is also relevant: Life can educate one to a belief in God. And experiences too are what bring this about; but I don’t mean visions and other forms of sense experience which show us the ‘existence of this being’, but, e.g., sufferings of various sorts. These neither show us God in the way a sense impression shows us an object, nor do they give rise to conjectures about him. Experiences, thoughts,—life can force this concept on us.²⁵

It is a common complaint that the belief that God exists, construed as a truthevaluable belief, receives no vindication from considerations of this kind. Dawkins in particular has voiced this complaint. ‘Religion’s power to console’, he writes, ‘doesn’t make it true’.²⁶ Indeed it does not. But that is uncontentious common territory in this particular discussion. It merely reinforces what I hope this essay as a whole has gone some way towards showing: that there are grounds for a powerful response to the new atheism in a due acknowledgement of the varieties of sense-making.²⁷ ²⁴ Nietzsche (1967a), Second Essay, §7, and Third Essay, §28. ²⁵ Wittgenstein (1980), p. 86, first two emphases in original, third added. ²⁶ Dawkins (2007), p. 394. ²⁷ I should like to take this opportunity to commend Gabriel Citron’s outstanding doctoral thesis (Citron (2012)), in which he argues that not only are varieties of sense-making important to the philosophy of religion, but so too is what he calls their ‘messiness’. This allows for the various different kinds of sense-making exhibited in religious beliefs sometimes to mix with one another in indeterminate and fluid ways. It seems to me that due acknowledgement of this point makes the response to the new atheism towards which I have been gesturing all the more powerful.

PART II

HO W W E MA K E S EN S E I N PHILOSOPHY

5 Sense-Making from a Human Point of View Abstract This essay is an exploration of what it dubs the ‘artistic’ conception of philosophy. This conception has two components: first, the view of philosophy as a humanistic discipline that Bernard Williams advocates; and second, the view that the sense-making involved in philosophy is a creative exercise. The first of these components casts philosophy as anthropocentric, and stands opposed to scientism. The second casts philosophy as exploratory and open to radical innovation, and stands opposed to conservatism. In the course of the discussion, which proceeds via consideration of the distinction that is standardly drawn between ‘analytic’ philosophy and ‘continental’ philosophy, an internal tension in the artistic conception is identified. This is the tension between its anthropocentrism, which requires that philosophers make sense of things from a human point of view, and its anti-conservatism, which requires that they be prepared to make sense of things from beyond that point of view. An attempt is made to resolve the tension by arguing that the anthropocentrism in question need not be any more than provisional. In the final section of the essay some of Spinoza’s ideas are considered as a case study. The importance of proceeding with care in philosophy is emphasized, and it is urged that, even if the anthropocentrism in question need not be any more than provisional, it had better not be any less than that either: it had better not be abandoned altogether.

1. The Artistic Conception of Philosophy A view famously held by Bernard Williams is that philosophy is a humanistic discipline.¹ I entirely endorse this view—and the reasons he gives for it. I have tried to defend something similar elsewhere.² I shall not try to offer any further ¹ B. Williams (2006b). See also B. Williams (2014) for associated reflections on the nature of the humanities. ² A. W. Moore (2012), esp. pp. 602–4. Something similar but not the same, because my concern in that book is specifically with metaphysics and not with philosophy more generally. Still, my concern is with metaphysics on a somewhat idiosyncratic and very generous characterization of metaphysics, as the most general attempt to make sense of things (a characterization that many people would in fact

The Human A Priori: Essays on How We Make Sense in Philosophy, Ethics, and Mathematics. A. W. Moore, Oxford University Press. © A. W. Moore 2023. DOI: 10.1093/oso/9780192871411.003.0006

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defence here. For the purposes of this essay I shall take the view in question as a kind of datum. I am concerned with what follows from it, or rather with what follows from one particular embellishment of it, and with certain problems that this embellishment poses. But I must begin by saying something about what the view is; or rather, what it is not. It is not the view that philosophy is one of the human sciences. One might think that this barely needs saying. For one thing, is there not a familiar and well entrenched distinction, within academia, between the humanities and the human sciences? Maybe there is (although it is worth remembering that there are at least two disciplines, history and linguistics, which are standardly included in the humanities and which might also reasonably be classified as human sciences³). However, that is beside the point. For the point is not simply to classify philosophy as one of the humanities either.⁴ The point is rather, as Williams himself puts it, to signal ‘what models or ideals or analogies [we] should . . . look to in thinking about the ways in which philosophy should be done’.⁵ A slogan that helps to convey the point is this: philosophy, though it is not anthropological, is anthropocentric. That is to say, philosophy, though it is not the scientific study of human beings or of any of the peculiarities that mark their way of life, has a fundamental concern with human beings and with what it takes to be one and is properly pursued, at the deepest level, from a human point of view. Philosophy is an attempt, by humans, from their unique position in the world, to make sense both of themselves and of that position. But ‘make sense of ’ is a polymorphous term. One respect in which I would want to go beyond what Williams says is by urging that we take seriously the term’s overtones of invention rather than discovery in this context. I believe that the sense-making involved in philosophy, at least in philosophy of the best sort, is, quite literally, sense-making: not an exploration of something antecedently given, but a creation of something, most notably a creation of concepts by which to live (such as Kant’s concept of a kingdom of ends, or Nietzsche’s concept of eternal return, to pick two signal examples).⁶

take to be more appropriate for philosophy as a whole). And much of what this excludes within philosophy—such as aesthetics, ethics, political philosophy, and philosophy of religion—gives the discipline, if anything, an even greater claim to the title of being humanistic. ³ It is also worth remembering that Collingwood took metaphysics to be a branch of history: see Collingwood (1998). (That said, he did not take philosophy to be a branch of history: see Collingwood (2005), ch. 10, §3.) ⁴ Williams makes this clear right at the beginning of B. Williams (2006b). ⁵ B. Williams (2006b), p. 180. ⁶ This too is a view that I have tried to defend elsewhere, in relation specifically to metaphysics but with implications for philosophy more generally: see A. W. Moore (2012), esp. Intro., §7, and Concl., §4. I say that I am going beyond what Williams says. But am I in fact doing something more radical than that? Am I contradicting what he says? In particular, does the view that philosophy is creative in the sense indicated conflict with Williams’s insistence that ‘there has to be such a thing in philosophy as getting it right’ (B. Williams (2006d), p. 202, emphasis in original)? I do not think so. This is because

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Let us call the conception of philosophy on which it is both humanistic in Williams’s sense and creative in the sense just indicated the ‘artistic’ conception. (This is what I had in mind when I referred to an embellishment of Williams’s view.) Now if, as I hold, the artistic conception is correct, then we can straight away identify two things of which philosophers need to beware: one of these relates primarily to the element of humanism in the conception, the other to the element of creativity in it. The first thing of which they need to beware, the one that relates primarily to the element of humanism in the conception, is scientism. That is, they need to beware of the unwarranted appropriation of procedures that are suited to the natural sciences. Sometimes the appropriation of such procedures in the pursuit of philosophy is perfectly acceptable and not precluded by anything that I have said on behalf of the conception. For instance, among the many things in which philosophers can quite properly show an interest are the natural sciences themselves, these being (after all) a very significant part of human life; and such an interest may well include self-conscious engagement with them.⁷ But there can be no presumption that procedures suited to the natural sciences will in general serve philosophy well. The second thing of which philosophers need to beware, the one that relates primarily to the element of creativity in the artistic conception, is conservatism. If one of the purposes of philosophy is sense-making, understood quite literally as the production of something, then philosophers had better not be too beholden to extant forms of sense-making. They had better feel no compunction about modifying these, extending them in various ways, establishing new connections between them, supplementing them—or even challenging, disrupting, discarding, and replacing them.⁸ This I take to be an anti-Wittgensteinian idea. Wittgenstein is not in general a conservative with respect to sense-making.⁹ But he is a conservative with respect to sense-making in philosophy, which he famously says ‘leaves everything as it is’.¹⁰ For Wittgenstein, the purpose of philosophy is to cure us of the confusions that arise when we mishandle our own conceptual apparatus.¹¹ Innovation in our sense-making can only ever bring with it the risk of

I do not think that answering to something antecedently given is the only way of ‘getting it right’ (see my 2012, pp. 381 and 393–4). Certainly my view does not conflict with the idea that there is such a thing in philosophy as doing it well. ⁷ Cf. B. Williams (2006c), p. 182, and (2006d), p. 203. ⁸ Williams spends a great deal of time issuing warnings against scientism in philosophy (e.g. B. Williams (2006c) and (2006d), passim). Warnings against conservatism in philosophy, unsurprisingly, are less visible in his work; unsurprisingly, because the corresponding idea that philosophy is creative is not there. But such warnings are not absent from his work altogether. They hardly could be, given that there is a closely associated idea that is quite certainly in his work, namely that reflection must sometimes be allowed to disturb the concepts by which we live: see e.g. B. Williams (2006i), esp. chs 8 and 9. ⁹ See e.g. Wittgenstein (1967a), pt I, §132. ¹⁰ Wittgenstein (1967a), pt I, §124. ¹¹ Wittgenstein (1967a), pt I, §§89–133.

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new confusions whereas, on Wittgenstein’s view, philosophers should be looking to minimize that risk. That is contrary to the spirit of the artistic conception.

2. The Distinction between Analytic Philosophy and Continental Philosophy: A Problem for the Artistic Conception It is instructive, in the light of these twin dangers of scientism and conservatism, to consider how the artistic conception relates to the distinction that is standardly drawn between ‘analytic’ philosophy and ‘continental’ philosophy. Now I am, in common with many others, impatient both with the connotations that the drawing of this distinction typically has and with the absurd terminology that is used to draw it.¹² But I do not deny that such a distinction exists; nor do I see any great advantage in trying, at this stage, to promote new labels for it. More to the point, I think that the distinction has something to teach us about the artistic conception. It does this by creating a paradox vis-à-vis that conception. For, as far as actual practice is concerned, it is continental philosophers whom we might expect to be more sympathetic to the conception. They are the ones who seem more ready to engage with other humanistic disciplines, such as history and literary theory, and to do so, moreover, in such a way as to suggest some continuity with their own endeavours. They are the ones whose practice is on the whole more playful. Analytic philosophers are the ones who more often proceed as though they were mapping the features of something independent of the mapping, indeed independent of humanity altogether. They are the ones who are more likely to need reminding of the danger of scientism. On the other hand, as far as self-image is concerned, that is as far as the practitioners’ own conception of the scope and limits of philosophy is concerned, it is, if not exactly the other way round, then at least more nearly the other way round. Analytic philosophers are the ones who are liable to think that what they do is regulated by appeal to, or with reference to, some such fundamentally human phenomenon as language or discursive knowledge.¹³ It is among continental philosophers that we are more

¹² Cf. B. Williams (2006d), p. 201. ¹³ Cf. Dummett’s contention that ‘the philosophy of language is the foundation of all other philosophy’ (Dummett (1978f), p. 442), or Quine’s suggestion that ‘philosophy of science is philosophy enough’ (Quine (1966), p. 151), where by ‘science’ he means not much more than organized knowledge (see e.g. Quine and Ullian (1978), p. 3). Admittedly, these phenomena can be objects of (nonanthropocentric) scientific study. But their relevance to analytic philosophy is of a different ilk. If an analytic philosopher, reflecting on how (say) the word ‘causation’ is used, denies that there is any such thing as backward causation, then he or she is not announcing the result of an empirical investigation into the use of the word ‘causation’; he or she is enunciating a rule for its use. (Here I am betraying my Wittgensteinianism, my earlier complaint about Wittgenstein notwithstanding: cf. Wittgenstein (1967a), pt I, §383; cf. also Hacker (1996), esp. ch. 8, and Hacker (2007), pp. 7–11. For a sustained discussion of the relations between analytic philosophy and language from a deeply opposed

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likely to find the view that what philosophy is, first and foremost, is unmediated pursuit of the great questions of ontology.¹⁴ Part of my response to this paradox is simply to acknowledge a failing in the practice of some analytic philosophers, a failing which does indeed suggest that they have paid insufficient heed to the danger of scientism. It is not just that their practice is not true to philosophy as I conceive it, that is to the artistic conception. Their practice is not true to philosophy as they conceive it. Nor is it true to their own heritage. A significant part of that heritage is the aim, not only to make sense but to make clear sense, where clarity is a matter of presentation, and where presentation presupposes a potential audience. Analytic philosophers should be more self-conscious than they very often are, first about who their audience might be, and secondly about the need to make sense of things from some suitable point of view that they share with that audience. This shared point of view will typically be much more restricted than a human point of view. It will rarely, if ever, be less restricted. For one prominent example of the failing that I have in mind consider Derek Parfit’s book Reasons and Persons.¹⁵ In his conclusion to that book Parfit discusses the various kinds of argument that he has invoked. He says that these lie between two extremes (where ‘between’ is understood in such a way that this includes the two extremes themselves): what he calls ‘the Low Road’, which ‘merely appeals to our intuitions’, and what he calls ‘the High Road’, which ‘asks what is the meaning of moral language, or the nature of moral reasoning’.¹⁶ Both extremes, and the territory between them, involve a human element of the kind to which I have alluded. Yet Parfit’s conclusions in the book are notoriously detached from any relevant point of view that he might share with his audience. To be sure, his writing has the virtue of clarity prized by analytic philosophers, which shows that he shares with his potential audience whatever point of view is required to understand the philosophical issues themselves. But, given the message that he conveys, he incurs a further responsibility to that audience: to adopt whatever

perspective see Williamson (2007), passim.) What may be true is that the kind of attention that analytic philosophers pay language shows that they have not indulged in that suspension of our naturalscientific modes of thought which phenomenologists take as their starting point: what they call the epoché (cf. A. W. Moore (2012), p. 431; and for an explanation of the epoché see Husserl (1970), §35). But that is not, in itself, any offence to the artistic conception. There are all manner of ways in which philosophy might distance itself from the natural sciences without going as far as suspending their very modes of thought. (Some phenomenologists are perhaps insufficiently sensitive to this point: see e.g. Husserl (1962), §62.) Note that a yet different approach to philosophy has recently emerged, under the title ‘experimental philosophy’, which retains a broadly analytic interest in language but which also involves significant use of empirical investigation, notably the empirical investigation of people’s linguistic intuitions (see e.g. Knobe and Nichols (2008)). ¹⁴ Cf. Heidegger’s claim that ‘ontology and phenomenology are not two distinct philosophical disciplines among others’ but rather that they ‘characterize philosophy itself ’ and that ‘philosophy is universal phenomenological ontology’ (Heidegger (1962), p. 62). I shall have a little more to say about this shortly. ¹⁵ Parfit (1984). ¹⁶ Parfit (1984), p. 447.

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more restricted point of view is required, not only to grasp that message but to come to terms with it. For instance, he argues that there are good reasons to induce in ourselves dispositions that will subserve a certain ethical theory while keeping the content of that theory hidden;¹⁷ but he does little or nothing to say what this means from the point of view of those for whom, if he were right, the practicalities of inducing such dispositions (not least by suppressing reflective selfconsciousness of the very kind that his own argument has instilled) would constitute a real social and political problem.¹⁸ Is there a mirror-image failing in the practice of some continental philosophers? Is there a similar mismatch between their practice and their selfimage? That would afford an interesting symmetry, if it were so. In fact, however, there is no obvious reason to think that it is so—not if we focus on the anthropocentrism that is evidenced in their practice. The great questions of ontology can certainly be addressed in an anthropocentric way. Phenomenology provides the model. Heidegger at one point equates phenomenology with ontology (and each, in turn, with philosophy¹⁹); but he also insists that it should be executed with peculiar reference to the sort of being that each human being is, namely Dasein.²⁰ I said that we see no mirror-image failing in the practice of continental philosophers if we focus on the anthropocentrism that is evidenced in their practice. But what if we focus on the creativity that is evidenced in their practice—creativity being the other element in the artistic conception—and then reflect on the associated danger of conservatism? Is there perhaps, if not a tension between addressing the great questions of ontology and proceeding anthropocentrically, then a tension between doing both of those things and being radically innovative? What tension do I have in mind? Well, consider this. Why should the radical innovation that I have suggested is a feature of the best philosophy not be so radical that it brings us to a new conception of who ‘we’ are and of what it takes to be one of ‘us’; so radical, in other words, that it provides us with ways of making sense of things that leave our humanity behind? There are various things that might be at stake here. We might come to reassess the relations between human beings and other animals in such a way that the former no longer have the special significance for us that is required for there to be a distinctively human point of view. Or advances in technology might challenge the very application of the concept of a human being in such a way that we are eventually led to abandon ¹⁷ Parfit (1984), pt I, passim. ¹⁸ Parfit’s book may also contain another prominent example of what I have in mind. He argues that personal identity is nothing over and above certain impersonally understood facts of bodily and psychological continuity ((1984), pt III, passim). And he tries to draw conclusions about persons, in particular ethical conclusions, that can themselves be understood impersonally. But there is good reason to think that only from a certain point of view involving a set of values that run contrary to these conclusions can there be any meaningful talk of persons in the first place (see further A. W. Moore (1997), pp. 229–32). ¹⁹ See n. 14 of this essay. ²⁰ Heidegger (1962), pp. 61–2.

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the concept altogether. And there is indeed, in certain continental philosophers, a preparedness, if not an aspiration, to think beyond the human in this way. Foucault, Deleuze, and Guattari are among the clearest and most interesting cases in point.²¹ How can philosophy be pursued in a spirit such as this, while remaining resolutely anthropocentric? It is a good question. There is certainly a tension here. It is not, however, a tension that peculiarly afflicts any continental philosophers. I represented it above as a tension between three elements: the anthropocentrism evidenced in the practice of continental philosophers; the creativity evidenced there, where this is creativity of a kind that allows for radical innovation; and the pursuit of the great questions of ontology that many continental philosophers take to be their métier. But the third is not really relevant. The tension is already there between the first two. And this means that, if it afflicts anyone, it afflicts me. For those are precisely the two elements in the artistic conception. It is an urgent question for me, then, how this tension can be resolved.

3. Thinking beyond the Human in Philosophy Is the following a reasonable way of resolving the tension? We should indeed be open to the possibility of thinking beyond the human in philosophy. But what this means is that we humans should be open to the possibility of thinking beyond the human in philosophy. We should be open to the possibility that our philosophy will one day no longer need to be, or may even one day no longer have the proper resources to be, anthropocentric. Nevertheless, we cannot oversee its becoming non-anthropocentric except by overseeing its evolution from something anthropocentric. And ‘evolution’ is the right word here. Nothing can happen in a metamorphic flash. Quite apart from whatever gradual transformation may have to be involved in our coming to embrace non-human possibilities outside philosophy, there is a gradual transformation that will certainly have to be involved in our coming to embrace non-human possibilities within philosophy. We cannot come to make radically new philosophical sense of things save through a progressive piecemeal process. (This is a conceptual point, not an anthropological point. There is a limit to how drastic and how rapid an upheaval in our philosophical sense-making can be while still counting as an upheaval in our philosophical sense-making—as opposed to our being as it were magically transported to some new position on the philosophical landscape.) So for now our philosophy needs to be anthropocentric. That is the only way, for ²¹ See e.g. Foucault (2001), ch. 10, §6, and the discussion of ‘becoming-animal’ in Deleuze and Guattari (1987), ch. 10. The notion of ‘a body without organs’ which permeates the latter work is also relevant.

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now, in which we its practitioners can appropriate the sense that it helps us to make of things as distinctively ours, and the only way, therefore, in which we can recognize it as having the value and importance for us that it should. The tension between philosophy’s anthropocentrism and its creativity is resolved by our recognizing the former as provisional. I asked if this was a reasonable way of resolving the tension. And I think it is, at least in outline. Even so, I am uncomfortable with letting the matter rest there. For I also think we should be extremely wary of thinking beyond the human in this way. I have already remarked on Wittgenstein’s conservatism and Deleuze’s anticonservatism; and I have made it clear that my sympathies are with the latter. However, there is a further, related disparity between the two thinkers with respect to which my sympathies are more with the former. And this inclines me towards a conservatism of practice, if not of principle. The disparity that I have in mind turns on Wittgenstein’s and Deleuze’s different conceptions of the given. Wittgenstein says that what are given are forms of life.²² Deleuze says that what are given are differences.²³ It is not obvious that there is any conflict between these—not least because it is not obvious that they mean the same by ‘the given’. Even so, there is. For Wittgenstein, a form of life, which he relates very closely to a language,²⁴ provides a kind of framework within which sense is made of things. The limits of our form of life, we might say, are the limits of our world.²⁵ That would be an anathema for Deleuze, for whom all unity—including the unity of any framework of this kind—has to be constituted within multiplicity, that is to say within what he counts as the given,²⁶ and must itself, accordingly, be made sense of in the same way as everything else. I lean towards Wittgenstein in this conflict. There seems to me something fundamentally right in the idea that, for sense to be made of things, there must first of all be some such framework for it to be made within, a framework determining whose sense it is. And if there is something fundamentally right in this idea, then any disruption of the sort that would be required for us to think beyond the human in philosophy would have to be disruption to more than just our sense-making. There would have to be, beyond whatever new sense we made of things, a new ‘we’ making it. For Deleuze, too, there would have to be a new ²² Wittgenstein (1967a), p. 226. ²³ Deleuze (1994), p. 222. ²⁴ Wittgenstein (1967a), pt I, §§19, 23, and 241. ²⁵ This is of course an allusion to Wittgenstein’s early work, in particular to Wittgenstein (1961), 5.6, where he says, ‘The limits of my language mean the limits of my world’ (emphasis removed). The suggestion that some fundamental aspect of this idea survives into the later work is associated above all with B. Williams (2006h). B. Williams (2006b) is also relevant: it provides a compelling elucidation and defence of the idea that a human form of life is, for us humans, ‘given’ (though Williams does not himself put it in those terms). ²⁶ There is an echo of this in his attitude to empiricism. At one point he says that what marks a position out as empiricist is that it has an account of how the subject is constituted within the given (Deleuze (1991), p. 109); elsewhere he makes clear that he sees his own position as empiricist (e.g. Deleuze and Guattari (1994), pp. 47–8).

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‘we’. But, for Deleuze, this new ‘we’ would itself be a product of our sense-making, a sort of self-creation. The changes that would be involved in our thinking beyond the human, however extreme, would be of a piece with the changes that would be involved in our extending our sense-making in any other way. On a more Wittgensteinian conception, however, something more radical would be at stake. And the radicalness would be an ethical radicalness. For, in so far as the primary ethical question is the question of what it is for ‘us’ to live well, ethics itself would be called into question. None of this is a conclusive reason to eschew all such disruption. But it is a reason, an ethical reason, to tread extremely carefully.

4. Spinoza: A Case Study Let us take the case of Spinoza.²⁷ Spinoza might be reckoned an opponent of anthropocentrism in philosophy. One of his best known doctrines is that our supreme virtue involves our making sense of things sub specie æternitatis.²⁸ In fact, however—be the interpretation of that doctrine as it may—there is something profoundly anthropocentric in Spinoza’s philosophy. Nowhere is this more blatant than in his account of the difference between good and bad. Spinoza denies that these are anything ‘positive considered in themselves’.²⁹ Rather, they are ways we have of thinking of things, according to our desires. Thus, in Spinoza’s view, we judge a thing to be good because we desire it; we do not desire it because we judge it to be good.³⁰ This, of course, entails a kind of relativism, as Spinoza freely acknowledges.³¹ Nevertheless, because he believes that there is a ‘model of human nature that we all set before ourselves’, and because he is writing from a shared human point of view, Spinoza is able to bypass the relativism and define the good as ‘that which we certainly know to be the means for our approaching nearer to the model’ and the bad as ‘that which we certainly know prevents us from reproducing the said model’.³² Such anthropocentrism is striking in its own right. But it is striking also for a more indirect reason, highly pertinent to the caution that I am now urging. It counteracts what would otherwise be a disturbing and sinister aspect of Spinoza’s philosophy, itself a natural attendant of the relativism to which he is committed, namely the doctrine that the right of each thing extends as far as its power does.³³ Thus the right of a tiger on the loose extends as far as its power does; the right of a cancerous growth extends as far as ²⁷ One of the many reasons why Spinoza is worth considering in this connection is the high regard in which he is held by Deleuze, for whom he is ‘the ‘prince’ of philosophers’ (quoted in Joughin (1990), p. 11; cf. Deleuze and Guattari (1994), p. 48). ²⁸ Spinoza (2002a), pt V, props 29 ff. ²⁹ Spinoza (2002a), pt IV, pref. ³⁰ Spinoza (2002a), pt III, prop. 9, schol. ³¹ E.g. Spinoza (2002a), pt III, prop. 39, schol., and pt IV, pref. ³² Spinoza (2002a), pt IV, pref. ³³ Spinoza (2002b), ch. 16. §4. For discussion see Curley (1996).

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its power does; the right of a repressive regime extends as far as its power does. Spinoza’s anthropocentrism provides a bulwark here. As soon as we can see that an exercise of one of these powers is preventing us from reproducing the model of human nature in some way, we can see that it is bad; and we can accordingly resist it. To quote Max Stirner: ‘The tiger that assails me is in the right, and I who strike him down am also in the right. I defend against him not my right, but myself.’³⁴ By the same token, were we to start trying to extend Spinozism beyond the human, we would undermine the assurances that such anthropocentrism provides and exacerbate that which is troubling in Spinoza’s power-centred vision. We might, for instance, begin to take seriously a point of view from which the flourishing of individuals was subordinate to the flourishing of the state.³⁵ Admittedly, ‘assurances’ and ‘troubling’, like ‘good’ and ‘bad’, are to be understood from a human point of view. But that, in a way, is the point. The point is not that there is some neutral position from which to evaluate different forms of philosophizing; precisely not. The point is that our philosophizing and our evaluating are, at least for now, from a common point of view, a human point of view, and, as long as that is the case, we are bound to acknowledge the dangers, that is to say the human dangers, in its being otherwise. To be sure, the sheer fact that there is no Archimedean point means that it is equally important for us to acknowledge the dangers, perhaps the non-human dangers, in our remaining beholden to one particular philosophical paradigm.³⁶ But that does not gainsay the conclusion that philosophy is, at least for now, an exercise in making sense of things from a human point of view. It merely reinforces the conclusion that we must proceed with care when doing philosophy. Only, let us not underestimate the force of this conclusion. For if philosophy is an exercise in making sense of things from a human point of view—if, as I put it earlier, philosophy is an attempt, by humans, from their unique position in the world, to make sense both of themselves and of that position—then the care with which we must proceed when we are doing philosophy is the care with which we must proceed when our very humanity is in question.³⁷

³⁴ Stirner (1982), p. 128, emphasis in original. ³⁵ It is instructive at this point to recall Mao Zedong’s remarkable claim, quoted in Chang (1993), p. 293: ‘Even if the United States . . . blew [the earth] to pieces . . . [this] would still be an insignificant matter as far as the universe as a whole is concerned.’ There are times when trying to make more objective sense of things verges on the catastrophic. ³⁶ We might hear talk in this connection of ‘the human prejudice’, the phrase appropriated by Williams as the title for his essay on related themes (B. Williams (2006b)). ³⁷ I am very grateful to Yuuki Ohta for comments on an early draft of this essay.

6 Not to be Taken at Face Value Abstract This essay first appeared as a contribution to a symposium on Timothy Williamson’s book The Philosophy of Philosophy. Several concerns about the book are expressed, including a concern about what is seen as its scientism. It is argued that we can draw certain distinctions between scientific discourse on the one hand and mathematical or philosophical discourse on the other, despite Williamson’s arguments to the contrary. This in turn gives us licence not to take certain philosophical questions at face value. In particular, it gives us licence not to treat certain philosophical questions involving vague concepts as questions about what those concepts concern, but rather as questions about the concepts themselves or, equivalently, about the workings of associated words. Towards the end of the essay the following moral is drawn: that we sometimes cannot tell the import of a question, nor even whether it is a philosophical question, until after we have reflected on our answer to it.

[The] characteristic of a metaphysical question [is] that we express an unclarity about the grammar of words in the form of a scientific question. —Ludwig Wittgenstein It is a long time since I have admired a book as much as I admire this one.¹ It is a long time since I have disagreed with a book as profoundly as I disagree with this one. I hope this combination of reactions on my part has more than whatever limited biographical interest it has. I hope it helps to signal the combination of excellence and provocation that mark Timothy Williamson’s book, which is at once beautifully clear, forcefully argued, continually insightful, and, in my view, deeply wrong. One thing that I admire about the book is Williamson’s preparedness to reflect philosophically on the nature of philosophy, something that is surprisingly rare in a discipline that is marked by such a high degree of self-consciousness and in which

¹ Williamson (2007): the subject of the symposium in which this essay first appeared. All unaccompanied references will be to this book.

The Human A Priori: Essays on How We Make Sense in Philosophy, Ethics, and Mathematics. A. W. Moore, Oxford University Press. © A. W. Moore 2023. DOI: 10.1093/oso/9780192871411.003.0007

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there are so few scruples about reflecting on the nature of other disciplines. One thing that gives me pause is Williamson’s lack of serious engagement with the history of philosophy.² I am convinced that the philosophy of philosophy will always be handicapped if it is as dissociated from the history of philosophy as this. But I shall not dwell on that concern here, because Williamson is clear enough about where his interests lie and I would be in danger, if I did, of succumbing to that familiar absurdity of berating an author for not having written a numerically different book. What I will say is that Williamson’s failure to engage seriously with the history of philosophy seems to me indicative of a scientism about philosophy which, despite how much he does to motivate his conception of the discipline, is manifest in how much is left unargued in the background. In a telling paragraph very near the end of the book Williamson refers to ‘those who deny that philosophy is a theoretical discipline at all’, and he asks, rhetorically, ‘If they cannot argue convincingly that the long-run results [of disciplined theoretical work in philosophy] do not constitute progress, how is their opposition to philosophical theory any better than obscurantism?’.³ No doubt those who deny that philosophy is a theoretical discipline at all should think twice if they cannot do this. Even so, their opposition to philosophical theory, however erroneous it may be in itself, may still be better than obscurantism. It is not inconceivable that only a comparatively small and unimportant part of philosophy is theoretical in the relevant sense and that significant progress in various technical parts of philosophy is hampering progress of a quite different kind in other parts of the discipline that would contribute far more to its overall flourishing, rather as if an obsessive concern with etymology had come to dominate literary criticism. If so, then some erring in the other direction would be no bad thing. In alluding to this possibility, I am of course doing nothing to defend it. I claim only that there is something significant in Williamson’s having overlooked it. The phenomenon extends further than Williamson’s own work. Many people with views about the nature of philosophy, including many Wittgensteinians with views that are diametrically opposed to Williamson’s, have a tendency to treat philosophy in a monolithic way. I myself have no more sympathy for the idea that philosophy is nothing but conceptual therapy, say, than I have for what I find in Williamson. Certainly I would not want my opposition to what I see as Williamson’s scientism to suggest that I regard philosophy as a fundamentally practical exercise that does not in any important way involve the pursuit of truth. Much philosophical activity, if not most philosophical activity, consists in the production of declarative sentences, in application to which the question ‘Is this

² Footnote 22 on p. 69 is especially striking in this respect.

³ P. 291.

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true?’ is undoubtedly pertinent, registering one of the primary dimensions of assessment of the activity. But there are truths and there are truths. That is, there are fundamentally different ways of being true. I say this out of sympathy for certain ‘non-realist’ accounts of what it is for some utterances of some declarative sentences to be true,⁴ accounts for which Williamson not only fails to share my sympathy but has an antipathy that he spends much of his book motivating. I shall not now try to provide contrary motivation, though I shall in due course try to indicate some of the places where our opposition on this issue is reflected in our different conceptions of philosophy. But in connection with the semantic issue itself, I cannot resist a reference to something else that Williamson says near the end of the book, referring to his own assault on various intellectual vices which he believes can all too easily lead people to adopt the non-realist views that he abhors. ‘Unless names are invidiously named,’ he writes, ‘sermons like this one tend to cause less offence than they should, because everyone imagines that they are aimed at other people.’⁵ His sermon caused me no offence. I do indeed imagine that it was aimed at other people: it would be an arrogance, especially given how little I have written on these issues, to imagine otherwise. But his sermon did cause me some discomfort. For it made me acutely aware of how easily the vices in question can lead people to adopt the views in question. This is especially true of the vice of intellectual laziness. It is easy to think that gesturing in the direction in which a non-realist semantics would be located, if there were such a thing, is tantamount to going over there and searching hard for one. Still, intellectual laziness is one thing. Obdurate refusal to be moved by compelling arguments against one’s view is something far worse. Consider the following non-realist view of mathematical truth, which I shall call, without meaning to incur any exegetical debts, ‘the Wittgensteinian View’. The Wittgensteinian View There is a fundamental difference of kind between mathematical truths and empirical truths. In asserting a mathematical truth one is stating a rule, not saying how things are independently of any such assertion. And nothing but a mathematical proof can establish a mathematical truth. That is, nothing but a mathematical proof can determine the propriety of adopting the rule in question. Any proponent of the Wittgensteinian View should be prepared to think long and hard about how a mathematical truth earns its title of ‘truth’, about what constitutes having various propositional attitudes towards it, and about its ⁴ I intend the term ‘non-realist’ in a very broad sense, though I particularly have in mind what Simon Blackburn calls quasi-realist accounts. ⁵ P. 291.

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semantic relations with truths of other kinds. But more urgently, he or she should address apparently decisive objections to the view. Consider this variation on one of Williamson’s examples: One can specify in physical terms what it takes to be an inscription . . . in a given font of a proof of [Goldbach’s conjecture] in a given formal system of Peano Arithmetic; a physical theory could predict that an event of a specified physically possible type would cause there to be such an inscription.⁶ Would that not constitute a non-mathematical way of establishing a mathematical truth? In fact I think a proponent of the Wittgensteinian View can reasonably say no. This variation on Williamson’s example does indeed show what Williamson takes his own original to show, namely that scientific experiments can be relevant to mathematical questions—as can empirical considerations more generally. I might come to accept Goldbach’s conjecture because there is a physical theory of the kind described above. I might come to accept Goldbach’s conjecture because some internationally revered mathematician claims, with evident sincerity, to have proved it. At the limit I might come to accept Goldbach’s conjecture because there is, on these pages in front of me, an inscription of the following type: [and here follows an inscription of a proof of Goldbach’s conjecture].⁷ But still empirical evidence is only ever evidence for an empirical conclusion—albeit both the evidence and the conclusion may be described in a way that presupposes some mathematical truth and therefore in a way that needs to be ratified by a mathematical proof (‘He has proved Goldbach’s conjecture’, ‘If we attend his lecture and follow what he says, we shall see why Goldbach’s conjecture is true’, ‘There is, on these pages in front of me, an inscription of a proof of Goldbach’s conjecture’). In mathematical terms, the empirical evidence establishes nothing. Here it is instructive to turn to Williamson’s own example, which has ‘0 = 1’ in place of Goldbach’s conjecture. In this case there is, on the Wittgensteinian View, something more forthright to be said. If a physical theory had as a consequence that an event of some specified physically possible type would cause there to be an inscription of a proof in Peano Arithmetic of that, then the theory would stand refuted. There cannot be an inscription of a proof in Peano Arithmetic of ‘0 = 1,’ because there is no proof in Peano Arithmetic of ‘0 = 1.’ Why not? Because there is a proof in Peano Arithmetic of ‘0 ≠ 1’ and Peano Arithmetic is consistent. How, you might ask, do I know that Peano Arithmetic is consistent? It would be an answer—here I echo Wittgenstein’s response to a similar question⁸—to say, ⁶ Pp. 6–7. ⁷ For the sake of the example I am prescinding from whatever improbability attaches to my personally being able to follow such a proof. ⁸ Wittgenstein (1967a), pt I, §§5 and 6.

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‘I have learned Peano Arithmetic, and this puts me in a position to reflect on it and to see that it is consistent.’ ‘An’ answer; indeed an answer that a proponent of the Wittgensteinian View might give. But I am repeating Wittgenstein’s caginess not just because I want to remain non-committal regarding the Wittgensteinian View. I also intend to repeat the rhetorical effect of Wittgenstein’s own response: to indicate that the question itself can be taken in all sorts of ways and that it may need to be elaborated if the answer does not satisfy. Still, the answer might satisfy with respect to relevance though not with respect to truth. An opponent of the Wittgensteinian View might say, ‘You think you can “see” that Peano Arithmetic is consistent only because you take yourself thereby to be endorsing a rule that goes naturally with other rules that you accept. But consistency is not a matter of legislation. It is a matter of how things are in mathematical reality, independently of us.’ In that case there is a philosophical disagreement waiting to be resolved. More modestly, an opponent of the Wittgensteinian View might say, ‘Even if consistency is a matter of legislation, and even if we adopt a rule that guarantees the consistency of Peano Arithmetic, we must be sensitive to the possibility that we shall one day come under pressure, perhaps even pressure from some well attested physical theory, to rescind the rule.’ In that case the proponent of the Wittgensteinian View can just agree. I have dwelt on this issue, despite its merely tangential connection with Williamson’s concerns, because I think it illustrates well how those with a nonrealist view of a given region of discourse, even while remaining under an obligation to say what they envisage in place of a realist semantics for that region of discourse, could sidestep certainly apparently decisive objections to their view. I have not committed myself to the Wittgensteinian View because it is not to the main point in this context. But I shall hereby commit myself to a non-realist view of much of what would pass for philosophical discourse. In particular, I believe that many of the questions that it is appropriate to ask, and many of the assertions that it is appropriate to make, in connection with the problem of vagueness—a problem concerning which Williamson’s work is, of course, pre-eminent—are questions and assertions in the material mode about the workings of some of the vocabulary they contain. They are not, in other words, to be taken at face value. This sets me in direct opposition to Williamson, who entitles his second chapter ‘Taking Philosophical Questions at Face Value’, and who adopts as his study case a question ‘closely related to the problem of vagueness’ which he calls ‘the original question’:⁹ Was Mars always either dry or not dry? Williamson makes clear that by ‘the original question’ he means that interrogative sentence, as used in that context, a context that can presumably be construed ⁹ Pp. 23–4.

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widely enough to include his surrounding discussion. This spares us the trouble of considering ways in which the same sentence might be used in different contexts. It also means that the original question is Williamson’s own question. This gives him some authority on the issue of what is being asked. But not complete authority. As the discussion proceeds, we are well within our rights to say, ‘Unless you are asking this [which you deny you are], then it is not clear what you are asking.’ I myself cannot hear that interrogative sentence, as used in that context, except as a question about the workings of the word ‘dry’ or the concept of dryness. (For current purposes I do not distinguish these: I allow that the workings of the word ‘dry’ are equally the workings of the Serbian word ‘suv’.) When Williamson denies that he is asking any such thing, and insists that the original question is as much a question about Mars as the question whether Mars was always either uninhabited or not dry,¹⁰ I genuinely do not know what he thinks he is asking. That is not an objection. It is an autobiographical observation. As such, it enjoys a certain incorrigibility. But it scarcely advances the philosophical discussion. Nor can I hope to achieve much more in these confines. Certainly it would be silly for me to suppose that I can rebut the many arguments that Williamson advances, both here and elsewhere, for his own conception of these matters. But, as in the case of the Wittgensteinian View, I hope at least to show how to sidestep certain apparently decisive objections to my alternative conception.¹¹ For example, Williamson urges that the translation of the original question into Serbian would be:¹² Da li je Mars uvek bio suv ili nje bio suv? And this, he further urges, shows the original question not to be even implicitly about the English word ‘dry’. But in fact, to the extent that I do take the original question to be about the English word ‘dry’, I also take it to be about the Serbian word ‘suv’ (as indicated above). It is unmysterious that a translation into Serbian should make use of this second way of directing our attention to that to which the original question directs our attention. In any case, as Williamson is well aware, ‘translation’ covers a multitude of operations. Even when a sentence is explicitly about a linguistic item, it is not precluded that a good translation of that sentence into another language should render the linguistic item also. It would be a poor translation of Middlemarch into Serbian that left all the dialogue in the original English. For that matter, it would be a poor translation of, say, Ray Monk’s biography of Russell into Serbian that left

¹⁰ But see below for how the latter question may not be all it appears to be. ¹¹ I have tried to develop this conception elsewhere: see A. W. Moore (2019b), esp. §§5 and 6. ¹² P. 28.

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all the direct quotations from Russell’s letters in the original English. For that matter it would be a poor translation of Williamson’s book into Serbian that left the original question in the original English, notwithstanding Williamson’s clear indication that, in discussing that question, he means to be discussing an occurrence of that particular English sentence. The translation of philosophical works, like the translation of novels, needs to be sensitive to far more than what we might call ‘strict and literal truth conditions’.¹³ Again, Williamson considers simple arguments that the original question is implicitly about the word ‘dry’, or about the concept of dryness, which are based on the question’s equivalence, in some undemanding sense of equivalence, to questions that are explicitly about those things; and he insists that parallel reasoning would lead to the conclusion that the unphilosophical question ‘Was Mars always either uninhabited or not dry?’ is also implicitly about the word ‘dry’ or the concept of dryness.¹⁴ Quite. And this shows that the simple arguments had better not be all that an advocate of any alternative conception such as mine has to offer. What Williamson does here, very effectively, it has to be said, is to remind his opponents that there is an onus on them. His own view of the original question is certainly the default view. If there are reasons for departing from it, those reasons need to be specified clearly enough to show that they are not likewise reasons for departing from the default view in cases where the default view is clearly correct. Very well; why view the original question differently from its unphilosophical counterpart? Williamson argues convincingly that the sheer fact that we cannot adequately answer the original question without assessing rival theories about the nature of thought and language does not settle the matter. In his helpful analogy, determining the nature of a physical quantity may necessitate investigation of the experimental equipment being used. My own reason for viewing the original question in the way I do depends as much on what I take the answer to the question to be as it does on what I take the route to the answer to be. In essence: I take the answer to the original question to be, with various important qualifications which for current purposes we can ignore, no; but I do not take this answer to be a challenge to the law of the excluded middle; so I had better not think that the original question can be taken at face value. I think the answer to the original question is a suitably qualified no because there are times in the past with respect to which an utterance of the sentence ‘Mars was dry then’ would, given the workings of the word ‘dry’, be ¹³ At the risk of approaching this particular nut with a sledgehammer, I also invite you to reconsider the section from Wittgenstein (1967a) from which I quoted earlier—pt I, §381—where the original German has ‘Wie erkenne ich, daβ diese Farbe Rot ist?—Eine Antwort ware: “Ich habe Deutsch gelernt”,’ and where Elizabeth Anscombe’s translation has ‘How do I know that this colour is red?— It would be an answer to say: “I have learnt English”.’ ¹⁴ Pp. 26–30.

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neither true nor false: it would ‘say nothing’, to borrow Williamson’s own terminology from elsewhere.¹⁵ Williamson has several arguments against this view. One point that I am sure he would make straight away is that the view is inappropriately sensitive to the presupposed conditions on Mars: that ‘there was once plenty of water [there]’ and that ‘very gradually the water evaporated’.¹⁶ For let u be an utterance of ‘Mars was dry then’, with respect to a time at which Mars was neither clearly dry nor clearly not dry. Had there never been any water on Mars, u would certainly have been true. More significantly, it would have said something. More significantly still, on Williamson’s view, it would have said the very same thing as it actually says; for what an utterance of a sentence says cannot depend, at least not in that way, on external contingencies.¹⁷ But this last claim is precisely the claim that I would deny.¹⁸ I shall not try to argue the point now. More important, for current purposes, are the implications of my view at the next level up. I hold that the best way to view the original question is dictated, to a significant extent, by its answer. Similarly for any analogous question about the relation between Mars’s dryness and its inhabitation. Suppose there is someone who takes Mars to be both uninhabited and dry and who wonders whether there is a connection.¹⁹ And suppose this person asks, ‘Was Mars always either uninhabited or not dry?’ Suppose finally that research shows that Mars was never inhabited and clearly dry though it was at times inhabited and neither clearly dry nor clearly not dry. Then, on my view, this person has failed to pose the question they thought they were posing, since circumstances dictate that such a question does not, strictly speaking, arise. This person has, strictly speaking, and in the relevant sense, failed to pose any question at all (which is not to deny that the research might satisfy them as much as any answer to any question would have done). If they persist with that interrogative sentence, fully apprised of the research, they could be interpreted as posing a question, but now a philosophical question on a par with the original question and likewise, on my view, receiving the suitably qualified answer no—the point being that attention is now being directed to the workings of the word ‘dry’. All of this would obviously be anathema to Williamson. Whether he would count it, on its own, as a reductio ad absurdum of my view I am not sure. I suspect that he would be content to signal the lengths to which one must go in order to safeguard a view such as mine. These do not concern me. I think language is messy. I am quite happy to concede that there is no tidy, theoretically interesting way of specifying the meaning of the sentence ‘Was Mars always either

¹⁵ P. 187. ¹⁶ P. 23. ¹⁷ Cf. P. 196. ¹⁸ This crucial difference between us is also, arguably, a crucial difference between the author of the Tractatus and the author of the Philosophical Investigations. ¹⁹ Cf. p. 25.

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uninhabited or not dry?’ nor of saying what purposes the sentence can be used to serve in different circumstances. The significance of this in relation to the philosophy of philosophy is that I am obliged to say that whether someone is posing a philosophical question is, sometimes at least, drastically underdetermined by whatever interrogative sentence they are using; that it can also depend, in a non-systematic way, on context; and that we may not be able to tell the real import of the question, nor even that it is a philosophical question, until after we have reflected on our answer to it. These are all things that I am very happy to say anyway.²⁰ It seems to me that the art of philosophy has at least as much to do with the posing of good questions as it does with the supplying of good answers, and that this in turn can involve probing the situation that one is in by asking things of it, in it, and about it that can be clarified only in the very answering of them. Given how hostile Williamson is to the ideas that have led me to this point, he may deem it a seriously back-handed compliment if I say how well I think he practises this art. I certainly think he poses some excellent questions. Unfortunately, the question that seems to me in many respects the best of all is one that he does little to answer, at least in this book: he merely makes a few suggestive remarks in connection with it. I have in mind the question that structures Chapter 4, §7: ‘What binds together uses of a word by different agents or at different times into a common practice of using that word with a given meaning?’ It is because of his attitude to this question that Williamson allows for the possibility of someone’s using a word with a given meaning even if their uses of it would be regarded by most people who use the word with that meaning as violations of its meaning. It is also because of his attitude to this question that he thinks that correct uses of the English word ‘dry’ are able to fix a first moment at which it applied to Mars, even if nobody can know which moment that was (however much they know about the history of the distribution of water on Mars). It would be good to hear more from Williamson about his conception of these matters.²¹ My own view is that we cannot satisfactorily address his question without saying something about the shared grasp of rules, and therefore without saying something about rules. Some uses of a word may not, strictly speaking, be part of the relevant practice because they violate rules that govern the workings of the word; which is not to deny that they may be sufficiently closely related, in sufficiently relevant ways, to uses that are part of the relevant practice to gain their own admittance on a looser way of speaking. (An analogy: we may be quite happy to describe two people as playing chess even though they are oblivious to

²⁰ I do not, incidentally, mean to imply that Williamson would not be happy to say any of them. ²¹ He says a little more in Williamson (1994), §7.5, but what he says is still rather sketchy and it suggests that he is pessimistic about how much more can be said.

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the rule that precludes castling through check and even though they often violate that rule, nay even if they are aware of the rule and have agreed to ignore it; but strictly speaking, they are not playing chess.) Strictly speaking, when Graham Priest says, ‘There are true contradictions,’ he is violating rules that govern the workings of those words in the English language and is not using them with the meanings that these rules help to determine; which is not to deny that he may have discovered compelling reasons for changing the rules, nor to deny that, if he has, then asserting that sentence in the context of whatever else he says may be the best way of getting the rest of us to acknowledge those reasons, nor to deny that, if we change the rules as a result, then the words in question may retain their meanings on a looser way of speaking. (Another analogy: it was when the pawn was first allowed to move forward two squares, some time in the fifteenth century, that chess strictly speaking came into existence, though on a looser way of speaking chess had already existed for a long time and merely underwent a change then.) And note: I do not say that a strict way of speaking is the only correct way of speaking. What if Graham Priest vehemently denies, as indeed he would, that he is advocating any change of meaning, even ‘strictly speaking’?²² It is a fair question. I cannot claim any special authority on the rules that govern the workings of those words in the English language, nor do I have anything helpful to say, here at least, about how such rules are discerned.²³ Still, I think Priest would be wrong if he denied this. Here I agree with Williamson, that somebody’s philosophical views can lead him or her to deny the most remarkable things. As it happens, Williamson would not share my inclination to apply that thought to this case.²⁴ Nor, of course, would he share my inclination to apply that thought to some of the things that he himself denies, for instance that the original question is a question about the workings of the word ‘dry’. But that nicely allows me to finish on what I am sure is a note of consent between us: one of the joys of philosophy is to grapple with powerful arguments advanced by extraordinarily intelligent, serious individuals for conclusions that one takes to be preposterous.²⁵

²² ²³ ²⁴ ²⁵

Cf. pp. 89–90. I try to say something helpful in A.W. Moore (2017), §20 (‘Reply to Priest’). Cf. p. 126. My thanks to Corine Besson for her helpful comments on an earlier draft.

7 Carving at the Joints Abstract This essay first appeared as a critical notice of Ted Sider’s Writing the Book of the World. Sider argues that an optimal description of the world must capture its basic structure, and that both physicists and metaphysicians can contribute to this enterprise, although the latter have an additional concern with the nature of the enterprise itself. It is argued not only that this concern is more subjective and more parochial than Sider allows, but also, contrary to what he insists, that there is something subjective even about structure as he conceives it. It is further argued, in the same vein, that by allowing metaphysicians to have a concern with the nature of the enterprise Sider makes a significant concession to a conception of metaphysics to which he would be firmly opposed, namely the conception of metaphysics as a humanistic discipline. Sider’s own rival conception of metaphysics emerges as unduly scientistic, but also, revealingly, as containing elements that militate against just such scientism.

The world, according to Ted Sider in Writing the Book of the World,¹ has a basic structure. An optimal description of the world must capture this structure. It must also consist of truths. But these are two distinct requirements. We can produce more and more truths about the world and still not come any closer to capturing its structure. To do the latter we need to produce not just truths, but truths of the right sort. Now it has long been acknowledged that a mere assemblage of truths does not count for much. They might be uninteresting and insignificant truths. For that matter, they might be interesting and significant truths, but assembled in an uninteresting and insignificant way—without any attempt at systematization or explanation, without any attempt to establish connections. The idea of basic structure gives a further fillip to this familiar fact. It signals another respect in which, if we are to give the best possible account of the world, we need to do more than tell the truth. The concepts we use to couch the truth—the properties to which we advert, the sorts of thing we recognize as instantiating these properties, even the connectives we use to conjoin claims about such things—need to reflect

¹ Sider (2011)—of which this essay first appeared as a critical notice. All unaccompanied references will be to this book.

The Human A Priori: Essays on How We Make Sense in Philosophy, Ethics, and Mathematics. A. W. Moore, Oxford University Press. © A. W. Moore 2023. DOI: 10.1093/oso/9780192871411.003.0008

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the world’s basic structure. They need to ‘carve at the joints’. There is a privileged way to write the book of the world. In the course of defending this vision, Sider works through an impressive list of philosophical problems and shows in each case how the idea of basic structure can help us address them. I shall give just two examples. First, consider the fact that some questions seem intuitively insubstantial. For instance: is Snowdon high enough to be a mountain? Even if this question has a correct answer, it still seems insubstantial. We feel that nothing of importance separates us from a community whose understanding of what counts as a ‘mountain’ is sufficiently different from ours that they would answer their analogous question differently. By contrast, the question of whether Snowdon is high enough to prevent the planet Venus from being seen from a given location at a given time does seem substantial. It is surprisingly difficult, however, to articulate what the distinction consists in. Sider is able to give an account of it. On his view, a question is insubstantial if one of the expressions used to formulate the question (‘mountain’ in this case) has a range of equally good available meanings, on some of which the question is to be answered one way and on some of which the question is to be answered the opposite way, where what makes two or more meanings equally good is the fact that none of them does better than the others in carving at the joints. The difference between mountains and mere hills is not part of the basic structure of the world; it is just a matter of how we happen to carve things up. The second example concerns the philosophy of time. Participants in debates about the fundamental nature of time use various metaphors to illustrate their different conceptions. Thus we hear talk of a four-dimensional ‘block universe’. Sometimes this is supplemented with talk of a ‘moving spotlight’ that illuminates different parts of the block at different times. Sometimes the block is allowed to ‘grow’ over time. Then there is talk of the ‘flow’ of time. Other examples abound. But vivid though these metaphors are, it is irresistible, after a while, to wonder whether they represent genuine differences of view. There are two concerns. The less serious concern is that there is nothing genuinely separating the views to which these metaphors give expression. The more serious concern is that these metaphors do not give expression to genuine views in the first place. Sider is able to arrest such scepticism. He argues that there are indeed genuine views involved, and that there are genuine differences between them. Participants in these debates can be represented as differing about the basic structure of the world. Suppose that the expression ‘there are’ can be understood in such a way that it carves well at the joints. In these terms, some of the participants in the debates about time—those who are liable to express their view by saying that the past is as real as the present—can be represented as believing that there are, say, such things as dinosaurs. Others—those who are liable to express their view by saying that nothing is true unless it is true now—can be represented as believing that there are not.

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Now, I began by talking about what we need to do if we are to give an optimal description of the world. But who are ‘we’? This familiar philosophical question is always lurking in a discussion of this kind. In this case, it alerts us to some heady and important issues concerning whether various aspects of our attempt to write the book of the world—for instance, the criteria for the success of our attempt— may turn out to reflect certain peculiarities of ours. Is there something about the exercise that is fundamentally anthropocentric, for example? Or fundamentally ethnocentric? Or fundamentally androcentric? Perhaps what counts as an optimal description of the world for us rational primates is profoundly different from what counts for those rational beings, if such there be, with a quite alien biological constitution. Perhaps what counts for us modern Westerners is profoundly different from what counts for those in other times and places. Perhaps what counts for us Anglophones is profoundly different from what counts for those whose natural language is not English. These issues are of crucial significance to Sider. He is adamant that his enterprise exhibits no sensitivity to who ‘we’ are. Indeed, that is part of the point of his insisting that a description of the world is to be judged by how well it captures the world’s structure: the world’s structure, for Sider, is independent of us—whoever ‘we’ are. Before addressing these issues, however, I want to consider a much more modest interpretation of the question: ‘Who are “we”?’ That question need not be heard as an allusion to hidden biases or other elements of perspective in our attempt to write the book of the world. It can be heard as a question about disciplinary boundaries. Whose business is it to capture the world’s structure? Is it the business of physicists? Or is it the business of metaphysicians? Is it perhaps both? Or neither? When ‘we’ reflect on what ‘we’ need to do in order to give the best possible account of the world, are we simultaneously adopting a stance both in the arena of physics and in the arena of metaphysics? Or are we simultaneously adopting a stance both in the arena of metaphysics and in the arena of meta-metaphysics—where metametaphysics, on Sider’s helpful characterization, is ‘inquiry into the status of metaphysics’.² Or are we doing something quite different from either of these? Consider the three following claims: (1) e = mc² (2) Among the concepts that carve perfectly at the joints are those of physics (3) Metaphysics is concerned with the structure of reality On a broadly Siderian view, (1) is the sort of claim that physicists make; (2) is the sort of claim that metaphysicians make; and (3) is the sort of claim that metametaphysicians make. That (1) is the sort of claim physicists make is hardly open to dispute. That (2) is the sort of claim metaphysicians make is implied by Sider’s ² P. 6.

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actually making it in the course of sketching what he takes to be a plausible metaphysical worldview in the final chapter of the book. That (3) is the sort of claim meta-metaphysicians make is implied by (2)’s being the sort of claim that metaphysicians make. In these terms, when ‘we’ reflect on what ‘we’ need to do in order to give the best possible account of the world, what we are doing, at least in part, is reflecting qua metaphysicians on what we are up to qua physicists; and when ‘we’ register this fact, we are registering qua meta-metaphysicians something about the relationship between metaphysics and physics. To be sure, none of these boundaries is sharp. It would be out of the question for anyone seriously engaged in any of these endeavours not thereby to be engaged, to a significant extent, in one or more of the others. In particular, metaphysicians, on this conception, will not want to confine their efforts to explaining the special role that physicists play in writing the book of the world; they will want to muscle in and help to write it. (Sider is always clear that, whatever role physicists have to play in writing the book of the world, it is not their sole prerogative. There is also work to be done by metaphysicians—as indeed there is work to be done by logicians and pure mathematicians.) Moreover, these roles themselves are not sharply separated. There are, Sider writes, ‘laws of metaphysics concerning the behavior of the physical predicates. These are given to us by physics. (Perhaps these should not be called laws of metaphysics, since they are also laws of physics.)’.³ Be that as it may, we have here an indication of how and why, on Sider’s conception, metaphysics centrally involves deliberation on the nature of physics. This is significant. It already marks one way, indeed a central way, in which metaphysicians are not themselves engaged in writing the book of the world. For even if the concepts of physics carve perfectly at the joints, the concept of physics with which (on Sider’s conception) metaphysicians have to reckon assuredly does not. True, there is a section in Sider’s book in which he argues that the concept of basic structure itself carves perfectly at the joints, which, in somewhat looser terms, means that structure is itself structural. But even if that argument is successful, there is no analogous argument to show that physics is itself physical, in any relevant sense of ‘physical’. Physics is a science. It is the study of something, the pursuit of something. It has an essential connection with aims, interests, and values. These are not, on Sider’s view, part of the basic structure of the world. The idea that one of the central concerns of metaphysicians places them outside the project of writing the book of the world can be approached from a slightly different direction. Suppose we ask: ‘What is structure?’ On Sider’s conception, that is a metaphysical question. As a metaphysician, he tries to answer it. But he does not try to answer it by offering a definition. For he does not think that structure has a definition: he takes it to be a primitive notion. Instead, he tries to say enough about structure, for instance about its connection with other notions, ³ P. 293.

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about its workings, and about what would be a canonical way of representing it, to elucidate what he has in mind. Central to what he wants to say about it is something that he repeatedly does say about it and something that initiated my discussion here, namely that an optimal description of the world must capture the world’s structure. Here are some variations on the same theme: We think of scientific discovery as satisfying the aims of inquiry particularly well; why? Answer: it is because scientific discoveries are phrased in particularly jointcarving terms.⁴ It is better to think and speak in joint-carving terms.⁵ Regard as joint-carving the ideology that is indispensable in your best theory.⁶

These remarks further indicate how metaphysics, in Sider’s view, deals with aims, interests, and values, and thus with that which is not itself part of the basic structure of the world. In fact they indicate something more. They indicate how metaphysics, trafficking as it does in the notions of what is ‘particularly good’, ‘better’, or ‘best’, not only deals with aims, interests, and values, but betrays its own. Is this a problem? Not at all; but it does mean that, when we return to the original posing of the question, ‘Who are “we”?,’ on its headier and more ambitious interpretation, it seems to demand and deserve an answer. If these metaphysical deliberations of ours—about what is required of us if we are to give an optimal description of the world—are as steeped in evaluation as this (something that the very use of the word ‘optimal’ ought already to have suggested), then it seems to be legitimate and important to ask, whose evaluation is at stake and what difference it might have made if others had been doing the evaluating. As physicists we are interested in the behaviour of rocks and stars. As metaphysicians we are interested in the merits of being interested in the behaviour of rocks and stars. Is it not possible that the concerns and values that inform the second of these interests are every bit as subjective and parochial as the concerns and values that inform an interest in the behaviour of rock stars? For Sider, it would be an anathema to suggest that metaphysics exhibits any such subjectivity. The metaphysician’s concern with the structure of reality, in Sider’s view, has no more to do with the idiosyncrasies of one particular group than the physicist’s does. The difference between the physicist and the metaphysician is rather a difference in the levels at which they operate. And the superiority of thinking and speaking in joint-carving terms, which it is the metaphysician’s business to proclaim, is as objective, if not as metaphysically basic, as the joints themselves. It would be a yet greater anathema, for Sider, to combine the suggestion that metaphysics exhibits such subjectivity with the suggestion that the subjectivity it ⁴ P. 62.

⁵ P. 61.

⁶ P. 14.

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exhibits infects the very notion of structure. When he declines to define structure in favour of making various elucidatory claims about it, he does so precisely because he does not want the other notions to which he relates it to be understood as playing definiens to its definiendum. He does not even want them to be understood as enjoying some kind of explanatory priority over it. The reason why it is optimal to think in a way that reflects structure, in Sider’s view, is because that is what structure itself demands, not because structure is to be understood as that which is reflected by a way of thinking that can be independently recognized as optimal. To reverse the order of priority here, to think that the optimality concerned is subjective—on the grounds, say, that the very idea of objective optimality is incoherent—and to conclude that there is something correspondingly subjective about structure itself is to get things just about as badly wrong, in Sider’s book, as it is possible for any right-minded person to get them. There is a huge amount in Sider’s acute and brilliant discussion of these issues that I admire. There is much indeed with which I agree. Even so, I have a fundamentally different conception of metaphysics from his. And to the extent that I have any sympathy at all for his notion of structure, then I am in fact strongly inclined to get things—as Sider would see it—that badly wrong, and in just the way I have described. Admittedly, this is just autobiography. What is required at this point is argument. But the very fact that what is required at this point is argument is part of my reason for seeing things so differently from Sider. For it seems to me that what is required is not just argument, but metaphysical argument, indeed metaphysical argument from a point of evaluative engagement with the issues, including the issue of the status of just such evaluation. It is striking and significant that all Sider himself offers at this point is autobiography. Lots of it. Here is a sample: Speaking just for myself, [subjectivism about structure] is incredible.⁷ A certain ‘knee-jerk realism’ is an unargued presupposition of this book. Knee-jerk realism is a vague picture rather than a precise thesis. According to the picture . . . the world is ‘out there’, and our job is to wrap our minds around it. This picture is perhaps my deepest philosophical conviction. I’ve never questioned it.⁸ Knee-jerk realism requires recognising that there is something better about [thinking in physical terms than thinking in terms of some non-joint-carving rival to physics] . . . Knee-jerk realism further requires that the betterness be objective.⁹ If structure is subjective, so is this betterness. This would be a disaster . . . If there is no sense in which the physical truths are objectively better than the [non-jointcarving] truths . . . then the postmodern forces of darkness have won.¹⁰

⁷ P. 18.

⁸ P. 18.

⁹ P. 19, emphasis in original.

¹⁰ P. 65.

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I find some of this autobiography as incredible as he finds subjectivism about structure. Has he really never questioned his philosophical conviction? His very claim never to have done so suggests otherwise. (If I could be persuaded of the truth of that claim, then my incredulity would give way to despair for the current state of philosophy.) But that is not the point. The point is that Sider’s autobiographical observations divert our attention away from a more subject-centred conception of what metaphysics can and should do. Metaphysics can and should help us to make sense of ourselves. This can certainly include reflection on physics. Our capacity to engage in physics is one of our most distinctive features, serving as a kind of fulcrum between us and the rest of the world. But if the reflection exercised on physics in the course of our metaphysical deliberations is to be of value, then it needs to be of a very different sort from the reflection exercised in physics. This is not least because it needs to have a significant element of self-consciousness, making the identity of the ‘we’ who are engaged in reflecting an unavoidable focus of that reflection. We need to see metaphysics as an enterprise that is completely different from any of the natural sciences, an enterprise that, unlike them, not only betrays ‘our’ point of view but allows for exploration and refinement and creative development of that point of view. To be sure, there is fuel for Sider’s fire in these suggestions. For he could agree that the enterprise I am describing is a worthwhile one, which differs from what he himself is disposed to regard as metaphysics, but then dismiss the question of which of the two is really metaphysics as insubstantial, like the question of whether Snowdon is a mountain, whose answer is sensitive to how some key expression is to be interpreted, there being a range of available meanings for the expression that do equally well (or, more to the point, equally badly) in carving at the joints. But I am not especially concerned to deny that the question is insubstantial. For, as Sider observes, ‘many expressions that fail to carve at the joints are embedded in our conceptual lives in important ways’.¹¹ He goes on to cite an example about which he says that, in learning how best to answer the question involved in the example, ‘we are primarily learning something about ourselves’, then adds, ‘but we’re learning something important about ourselves’.¹² Quite. And this seems to me to be just the sort of thing that we should say in connection with what metaphysics is. Reflecting on how best to answer this question means reflecting on something important about ourselves. In particular, it means reflecting on something important about that extraordinary part of our heritage that includes the inventiveness, the visionariness, and the insight into what it is to be human of Spinoza, Kant, Nietzsche, and their like.

¹¹ P. 50.

¹² P. 50–1.

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Sider is guilty, as I see it, of a sort of scientism. He does not just want metaphysics to privilege physics. He wants it to ape it. He wants metaphysicians to play their own part in writing the book of the world. (This is one reason why it is important for him that structure should itself be structural; for the concept of structure is the one joint-carving concept whose exercise is a speciality of metaphysicians.) He knows full well that metaphysics cannot be entirely like physics. Its very privileging of physics, which is an arrogation that we do not find within physics itself, precludes assimilation of the two. But the more like physics metaphysics is, Sider thinks, the better. And this is because physics is supremely good at capturing the world’s structure. My own view is that metaphysics is not in the business of capturing the world’s structure. (This is not to deny that it has a concern with structure. Aesthetics has a concern with kitsch, but it is not in the business of capturing kitsch.) To do metaphysics justice, and to extend this part of our heritage in the most creative and the most effective way, we need to see metaphysics in the way in which Bernard Williams famously urged us to see philosophy as a whole: as a humanistic discipline. This is very different from the way in which Sider sees it. Nevertheless, there is much to applaud in this fascinating book. Even if it is not for metaphysicians to write the book of the world, there is no gainsaying their right to write books about writing the book of the world. Sider has done that very well indeed.

8 The Concern with Truth, Sense, et al.— Androcentric or Anthropocentric? Abstract In her book Re-visioning Gender in Philosophy of Religion, Pamela Sue Anderson discusses some of my ideas. In particular, she considers my views about a certain kind of philosophical nonsense. She argues that I am not interested in engaging seriously with such nonsense, and that my not being interested in engaging seriously with such nonsense betrays my gender. This essay is a response to Anderson’s discussion. It is argued that she is guilty of certain errors, both exegetical and philosophical. In the course of the argument some issues are raised about what we can aspire to as philosophers. These issues in turn bear on the relation between philosophy and the feminine, between philosophy and the masculine, and between philosophy and the human. Towards the end of the essay it is urged that the third of these relations—the relation between philosophy and the human—is of far greater significance than either of the other two.

1. Introduction It is a great honour to contribute to this collection of essays in memory of my friend Pamela Sue Anderson.¹ Of course the circumstances mean that the pleasure of doing so is mixed with sadness. Anderson and I spent many happy hours discussing philosophy together. It is impossible for me to convey how much I miss our discussions. In her written work Anderson generously paid a good deal of critical attention to my own. This is especially true of her last book Re-visioning Gender in Philosophy of Religion.² I hope it will not appear too self-indulgent if I use this occasion to respond to some of what she said. I believe that she was guilty of certain errors, both exegetical and philosophical. But only someone who was a

¹ This is a reference to Goulimari (2021), itself a republication of Angelaki, 25, a special memorial issue on the work of Pamela Sue Anderson in which this essay first appeared. ² P. S. Anderson (2012).

The Human A Priori: Essays on How We Make Sense in Philosophy, Ethics, and Mathematics. A. W. Moore, Oxford University Press. © A. W. Moore 2023. DOI: 10.1093/oso/9780192871411.003.0009

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stranger to philosophy, and indeed to Anderson herself, could think it the least paradoxical that I choose, as my tribute to her in this context, to dwell on what I take to be such errors. The discussion will eventually turn into a more general reflection on the nature of philosophy. What I have to say at that point will be more speculative. It will concern issues about which I am altogether less confident what the implications of Anderson’s views are and how, if at all, I would oppose them—though it speaks volumes about the richness and interest of Anderson’s work that it so much as leads us in that direction. Much of my own work has revolved around a number of dualities. In particular, much of it has revolved around the following five dualities: (1) (2) (3) (4) (5)

true/false absolute/perspectival effable/ineffable sense-possessing/sense-lacking finite/infinite.

Before I go any further, I want to make brief comments about the fourth and fifth of these. I begin with a comment about the fourth. This comment is largely terminological. I would like to have characterized this fourth duality by simply writing ‘sensical/nonsensical’. But standard English, notoriously and maddeningly, does not give us ‘sensical’. (Or at any rate, it does not yet give us ‘sensical’. I suppose it is just a question of time. People are increasingly availing themselves of it.) Would ‘meaningful/meaningless’ have suited my purposes here—or, for that matter, ‘meaningful/nonsensical’? No. This is because I like to distinguish between that which has sense and that which has meaning: the latter, for me, is a broader notion. For reasons that I am about to sketch, I believe that there are ways of putting language to use which, on the one hand, exploit the meanings of the words involved but which also, on the other hand, result in something that is strictly speaking nonsensical. And when language is put to such use, I see some rationale for classifying what results as having meaning but lacking sense. ‘Meaningful nonsense’ is therefore not, on my lips, oxymoronic. Concerning the fifth duality, I have a comment about the order of the two terms. Looking at how I have couched the other four dualities, you may suspect that I intend an alignment of sorts. In particular, you may suspect that I see some sort of priority of the first term over the second in each case, or some sort of superiority of the first term to the second. And you might then be surprised that I have not written ‘infinite/finite’ in the fifth case. This would be quite wrong. For one thing, I intend no such alignment in the first four dualities. Secondly, even if

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I did, it would be entirely unstraightforward how the finite and the infinite related to it. I shall come back to both of these points. Now Anderson discusses all five of these dualities, largely in the context of two others about which I say very little: (6)

male/female

and, perhaps differently, (7)

masculine/feminine.

Why perhaps differently? Well, here I must tread with caution. There are three reasons for this. First, and most basically, I have no expertise in this area. In particular, I have no expertise in feminism, where any distinction between the sixth duality and the seventh is liable to be especially significant. Secondly, and disconcertingly, I am nevertheless aware that practically nothing in this area is uncontroversial. Moreover, much of the controversy is not between feminists and those whose ideas they are critiquing, but among feminists. Thus it is controversial to what extent there really is any serviceable, robust, and objective distinction between the male and the female; to what extent there is any such distinction between the masculine and the feminine; and to what extent there is any such distinction between these two dualities themselves, or, more broadly, between sex and gender, of which this is supposed to be an instance.³ Thirdly, I find it hard to know where exactly Anderson situates herself with respect to these controversies or how this relates to her discussion of my own work. But I do know that, underlying Anderson’s discussion of my work, is a constant concern with how evaluation infects our philosophical discourse, and in particular with how it does so through the use of what Bernard Williams calls ‘thick’ concepts, that is to say concepts that have both an evaluative aspect and a descriptive aspect. (A standard example of a thick concept is the concept of infidelity. This has an evaluative aspect in as much as, in calling someone unfaithful, you censure that person. But it also has a descriptive aspect in as much as you are not entitled to call someone unfaithful unless that person has gone back on some relevant agreement.⁴) The question arises—for anyone, but particularly for anyone with Anderson’s interests—what sort of thickness, if any, there is in any of

³ See e.g. Fausto-Sterling (2000); Spelman (1988); Butler (1999); and Mikkola (2022). ⁴ See e.g. B. Williams (2006i), pp. 129–30 and 140–1. This idea of a thick concept structures much of ch. 6 of P. S. Anderson (2012): see e.g. p. 113.

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the concepts involved in these seven dualities. And this in turn obviously relates back to the issue of what sort of alignment there is among them. As I have already indicated, I assume no privileging of the first term in each duality over the second, not even when attention is confined to the first four dualities. In fact I assume no interesting alignment at all. This is not to deny that the concepts involved are thick. But, if they are, the thickness is in each case far from straightforward; and the relation between the thickness of each to the thickness of all the others is further still from straightforward.

2. How I View the Dualities I hope I will be forgiven if, as a prelude to amplifying on these remarks, I give a lightning sketch of some of my central views about the first five dualities.⁵ Directly relevant to all but the fifth is what I call a representation. By a representation I mean anything which has content and which, because of its content, is either true or false. (Representations thus include assertions, thoughts, judgements, theories, and the like.) Here already the true/false duality is in play. But only the most extreme of philosophical sceptics would deny that it has any right to be, or that it has any significant claim on our attention. Anderson would certainly not deny either of these things. Among representations—and this is where I invoke the second duality—I distinguish between those that are absolute and those that are perspectival. Absolute representations are not coloured by the feelings, concerns, or values of those who produce them. Nor does their content depend on their location, in any literal or metaphorical sense (in the way, for example, in which the content of a tensed representation depends on the time at which it is produced). In other words, absolute representations are not from any point of view. Perspectival representations are. Representations of both kinds, I argue, are possible. I also argue that it is representations of the former kind that scientists, and more specifically physicists, aspire to produce; and that, if ever anybody does produce such a thing, then it will have to be couched in scientific, and more specifically physical, terms. If ever anybody produces such a thing . . . I am not committed to the view that anybody ever actually has done so or will do so. The first two dualities cut right across each other. Absolute representations can be true or false. Perspectival representations can be true or false. Whatever privileging there may be of the true over the false, it does nothing to encourage a privileging of the absolute over the perspectival. Nor, in my view, does anything else. The aspiration to produce absolute representations has its rationale in a

⁵ What follows draws principally on A. W. Moore (1997).

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certain scientific context. Outside that context it has no rationale whatsoever. In particular—this is important for what will come later—it has no rationale in philosophy. Nor is the scientific context itself in any relevant sense privileged. The third duality is between the effable and the ineffable: this applies within states of knowledge. A state of knowledge is effable if and only if it is a representation, in other words if and only if it has content which makes it true or false—in fact true, given that it is a state of knowledge. But there are, I believe, states of knowledge, indeed important and familiar states of knowledge, that are not of this kind, states of knowledge that lack content and are therefore ineffable. Someone who has ineffable knowledge thereby knows how to do certain things, or knows what it is for certain things to be the way they are, or something of that sort; but he or she does not thereby know that anything is the case.⁶ If anyone were to attempt to put such ineffable knowledge into words, then the attempt would be a failure. But the result might be of interest for all that. And indeed I believe that, if the attempt were suitably executed—which admittedly raises some large questions about what would count as a suitable execution, though I shall not dwell on these questions now—then the result would serve to individuate the knowledge, and might even serve to convey it. Even so, the result would strictly speaking be a piece of nonsense: this is where the fourth duality is pertinent. (It is also what I had in mind when I spoke earlier of meaningful nonsense.) What would be an example? One example, I argue, would be the sentence: ‘Absolute representations are not possible.’ But why do I say that this would be a piece of nonsense, rather than simply false? Because, granted the soundness of my argument that absolute representations are possible, or in other words that it is false that they are not, this sentence could not serve its function without being hedged with qualifications that prevented it from being interpreted as that falsehood; and indeed, crucially, that prevented it from being interpreted at all. Another example would be the sentence: ‘The infinite exists.’⁷ Why would this be nonsensical? Because our very notion of the infinite precludes its existing. It is as if the infinite is too great for that.⁸ Or, in Wittgensteinian terms more conducive to the current point, it is as if the very ‘grammar’ of the word ‘infinite’ prevents it from directly characterizing anything in reality.⁹ Am I myself producing nonsense in peddling these ideas, and (in particular) in specifying the sort of nonsense that accrues from attempting to express the inexpressible? I do not think so. I think I can avail myself of the distinction ⁶ A possible exception is someone who has knowledge of a necessary truth: see my A. W. Moore (2019g), n. 16. I shall ignore that complication in what follows. ⁷ This is what Anderson is alluding to in P. S. Anderson (2012), p. 70, where she writes, ‘According to Moore, ineffability is “shown” in “images of infinitude” ’, though I would not express it like that. For clarification of my use of the terminology of ‘showing’, see A. W. Moore (1997), ch. 7, esp. §3. ⁸ Cf. Murdoch (1993), p. 508. ⁹ Cf. Wittgenstein (1975c), §XII.

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between discussing nonsensical uses of words and indulging in such uses. It is not necessary to talk nonsense in order to talk about nonsense: one can say, quite truly, ‘“Brillig” has two syllables.’ I aspire, in my discussion of these issues, to produce nothing but sense, and indeed to produce nothing but the truth. In a conference on the work of Derrida in which I participated and which is the focus of much of Anderson’s own discussion,¹⁰ I gave this some context by saying that the kind of philosophy that I practise, which I called ‘conceptual philosophy’, ‘has a commitment to the truth’.¹¹ But it is a commitment of a distinctive sort. Derrida, in his reply, pointed out that there are other, quite distinct sorts of commitment to the truth, including a sort that he saw as part of the kind of philosophy that he practised. One can be committed to the truth by questioning it, indeed by questioning its very possibility, and by taking nothing for granted about how it relates to language or to sense.¹² So be it. The key point, to repeat, is that I aspire, in my discussion of these issues, to produce nothing but the truth. Now I briefly relate all of this to the sixth and seventh dualities in my book Points of View.¹³ I admit there that current scientific theories may, in Sandra Harding’s phrase, ‘bear the mark of their collective and individual creators’, and that the creators in turn may ‘have been distinctively marked as to gender, class, race, and culture’.¹⁴ This does not trouble me. As I have already made clear, my argument is an argument for the possibility of absolute representations, not for their actuality. I also admit that some of the crucial concepts that I myself use to frame this discussion, including some of the concepts that appear in the first five dualities, contain an element of perspective.¹⁵ This is because they cannot be exercised except from some interpretative point of view. I would be happy to go further and admit that some of them contain an element of thickness, and that they cannot be exercised except from some evaluative point of view. But, if that is so, it has no implications for how I view either the sixth or the seventh duality. There is, as I have been at pains to insist, no presumed alignment in the first five dualities. A fortiori there is no presumed alignment that extends from them to the other two; or to the duality between what is to be valued and what is to be disvalued; or, heaven forefend, to all three. For, as I emphasize in my book, although there undeniably has been an ideological alignment of the true and the absolute and the scientific with what is to be valued and with what is either male or masculine (a kind of veneration of scientific practice as a detached, authoritative, rational domination of mother nature), I want no part of it. And I see no reason why the dualities themselves should not survive any such (false) ideology.

¹⁰ See P. S. Anderson (2012), ch. 4. ¹¹ A. W. Moore (2001), p. 59. ¹² Derrida (2001), p. 84. ¹³ A. W. Moore (1997), pp. 101–2 and 108–9. ¹⁴ Harding (1986), p. 15. ¹⁵ See A. W. Moore (1997), pp. 98–9.

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3. Anderson’s Critique Anderson, however, recoils. She attends to each of the first five dualities and suggests that there is, in my work, an implicit privileging in each of them of the first term over the second; and that this privileging, in fact my very concern with each of these dualities, betrays my gender, precisely what I am keen to deny. Before I go any further, I want to note straight away that Anderson’s discussion becomes somewhat muddied in the case of the fifth duality. This is not a criticism. It relates back to peculiarities of the infinite: the subject matter itself condemns whoever engages with it, at least in this sort of context, to a kind of unclarity. In fact I applaud what Anderson says in this connection. In a section of her book titled ‘The Infinite and Gender: An Ancient Question’,¹⁶ she discusses Grace Jantzen’s interesting classification of the urge for infinitude as ‘a masculine or male obsession’, and then notes that, at the beginning of Western philosophy, most notably among the Pythagoreans, it was the urge for finitude, or the privileging of finitude, that had the (far) better claim to that title. Indeed Anderson makes significant capital out of such ambivalence, capital that directly subserves her project. For, as she further says, ‘considering why the infinite is given this or that gender will give us new understandings for the re-visioning of gender’.¹⁷ This all strikes me as fundamentally correct—though I also think that the muddying of this part of Anderson’s discussion, despite how much of it depends on peculiarities of the infinite, should make us wary of expecting any engagement with the first four dualities to escape the same fate. Still, let us return to what Anderson says about the first four dualities and to my discomfort with it. Adverting to the conference on the work of Derrida that I have already mentioned, she comments on the all-male list of speakers and remarks that ‘the philosophical assumption seemed to be one of arguing from a genderneutral perspective, yet the maleness was conspicuous from another perspective’.¹⁸ Later there is a characteristically teasing passage where she suggests that, in my own talk of ineffability—of that which cannot be put into words—I may in fact mean no more than that which cannot ‘be put into the words of a particular (privileged) perspective within the dominant conceptual scheme’.¹⁹ And later still we find discussion of the fact that I, in contrast to Derrida, want to eschew nonsense in favour of what not only has a sense but is true: [Moore] . . . distinguishes himself from Derrida in insisting that the affirmation of truth is an ultimate concern . . . Moore is only willing to play with nonsense [i.e. Moore is only willing to engage in playful talk about nonsense—he is not willing

¹⁶ P. S. Anderson (2012), pp. 75–9. ¹⁸ P. S. Anderson (2012), p. 67.

¹⁷ P. S. Anderson (2012), p. 79. ¹⁹ P. S. Anderson (2012), p. 72.

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to produce it himself], while Derrida ironically seems far more serious about nonsense! . . . Moore does not aim to produce nonsense, or to engage seriously with it.²⁰

She expands on this in relation to the early Wittgenstein: Wittgenstein’s relation to . . . nonsense . . . as set out in his Tractatus begins to seem closer to Derrida’s relation to nonsense (since he takes it very seriously) than Moore’s detached play with nonsense.²¹

Much of chapter 6 of her book develops this idea, dealing as it does with the privileging of truth over falsehood and nonsense. But I demur. To anyone who knows how great my admiration for Wittgenstein is, this reference to his Tractatus²² should already give pause. As Anderson herself correctly insists later, ‘it could be that the negative connotations of “nonsense” should be ignored’.²³ Indeed they should. There is an irony here. The passage in which Anderson says this involves a rather spectacular failure of proofreading: it is presented as though it were a quotation from me, whereas it is Anderson’s own text, written in propria persona and supposedly contra me. The irony is that what she says here is something which in fact, as I have just indicated, I would be only too happy to endorse.²⁴ I shall come back to this. But there is more. It is not just that Anderson represents me as disparaging nonsense. She also thereby, however implicitly, represents me as disparaging the female and/or the feminine. For the whole thrust of the chapter from which I have been quoting is that the ineffable, and likewise the nonsensical that attends it, are to be thought of in female or feminine terms. Anderson talks about female mystics, and about the nonsense that they produce. She distances me from them. She writes: Moore does not intend to produce nonsense. Instead, he aspires to produce truth. Yet . . . doesn’t Moore belie his own gender bias, opting for the sense of effable knowledge, when the female mystic’s know-how and the wise woman’s tales fail to produce sense—and so truth[?]²⁵

²⁰ P. S. Anderson (2012), pp. 73–5, emphasis added. ²¹ P. S. Anderson (2012), p. 81. ²² Wittgenstein (1961). ²³ P. S. Anderson (2012), p. 84. ²⁴ I have another, entirely unrelated and frivolous reason for noting this error in Anderson’s text. It reminds me of a bizarre mistake of predictive texting in a message that Anderson once sent me, which resulted in much mutual hilarity. In response to my question whether she was able to accompany me to some event at short notice, instead of replying that she could not because she had a graduate student round helping her to proofread, she replied that she could not because she had a graduate student round helping her to procreate. ²⁵ P. S. Anderson (2012), pp. 84–5.

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She goes on to cite how Irigaray makes nonsensical play with images of infinitude; and how Jantzen recognizes the importance of stretching language to represent the ‘inexhaustible fecundity’ of the divine.²⁶ Finally, at the very end of the chapter, she says that we must move beyond ‘traditional answers to the philosophical question of ineffable knowledge’, which are ‘inadequate insofar as they have failed to acknowledge a necessary tension in our gendered relations to the finite and the infinite, as both corrupting and enabling’.²⁷ She continues: Philosophy of religion as practised by both those who aspire to produce truth and those who engage seriously with nonsense . . . can acknowledge this tension as the first step towards new ethical and epistemological relations between women themselves, men themselves, and women and men, of different material and social perspectives.²⁸

All of this, I confess, leaves me bemused. There is, in my discussion of these issues, no privileging of sense over nonsense. True, I aspire, in any such discussion, to produce only the former, as I have already said. In fact I aspire, whenever I write philosophy, to produce only the former. But this is not because I disparage the production of the latter, or at any rate not all of it. Some of it I disparage. Indeed most of it I disparage. So would Anderson. The production of nonsense that is effectively nothing but a botched attempt to produce sense is certainly to be disparaged. But that is not what Anderson is concerned with here. She is concerned with something of the sort that I described earlier: the creative production of nonsense in a suitably executed attempt to put into words knowledge that cannot be put into words, the sort of thing that may even serve to communicate the knowledge, the sort of thing that I have indicated I am prepared to count, despite its nonsensicality, as meaningful. I do not go in for that practice myself: I confine my engagement with such nonsense to quotations either of other people’s work or of certain very elementary examples. But this is not because I disparage the practice. It is because I take myself to be not very good at it. Other philosophers are very good indeed at it. Thus I believe that Wittgenstein’s Tractatus consists largely in the production of such nonsense; also that it is one of the great philosophical texts; and indeed, if you were to press me, that it is one of the great works of art. The reason why I aim in my own philosophical practice to produce nothing but sense, and more specifically nothing but the truth, is simply that such is the style of philosophy that I take to be my own métier. I do not deny that other styles are possible. And I do not say that other styles are inferior.

²⁶ P. S. Anderson (2012), p. 85. ²⁸ P. S. Anderson (2012), p. 87.

²⁷ P. S. Anderson (2012), p. 87.

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It seems to me, then, that Anderson is imposing an unhelpful grid here. In particular, I find her use of the categories of male and female unhelpful. In fact I think there is a real issue, ironically, about what, if anything, prevents us from classifying it as sexist. The thought that it is sexist connects, I believe, with the controversies to which I adverted earlier concerning the sixth and the seventh dualities. There is a threat of self-stultification hereabouts: the threat of meeting an unacceptable stereotyping in regard to such categories with more of the same. Evading this threat, for anyone who seeks to champion the cause of women, means striking a delicate balance. For one staple of championing the cause of women is, precisely, fighting an unacceptable stereotyping of women; but fighting it too hard, in particular fighting it hard enough to undermine any sort of essentialism concerning the category of women, may mean that there is nothing left to champion. To quote Mari Mikkola: ‘If feminist critiques of the category women are successful, then what (if anything) binds women together, what is it to be a woman, and what kinds of demands can feminists make on behalf of women?’²⁹ There are, to be sure, various tactics that suggest themselves for confronting this dilemma, including deliberately embracing a kind of sexism in a spirit of deconstruction. I certainly do not mean to suggest that I have found some insuperable objection to what Anderson is doing with the categories of the male and the female in this chapter of her book. I simply say that I do not find it helpful.

4. Philosophy as Anthropocentric (But Not as Androcentric) Now there is, of course—and this is something that I concede in Points of View³⁰— a very obvious, very basic, and potentially very damning objection to these attempts of mine to deflect Anderson’s critique, and it would be crass for me not to acknowledge it. I mean the objection that I am simply betraying my own male point of view once again. Anderson herself makes some telling points that pertain to this objection in a later chapter where she contrasts ways in which male thinkers manage to make themselves heard with ways in which female thinkers struggle to do so, implying that a certain lack of self-consciousness is integral to the former.³¹ Later she refers to ‘a keen sense of injustice which is often not noticed by a privileged thinker who seems to be reasoning about abstract matters’.³² Obviously it would be unacceptable for me simply to dismiss such thoughts. I would face my own threat of self-stultification if I did.

²⁹ Mikkola (2022). For discussion of some of the issues that arise here see: Stoljar (1995), pp. 261–93; and Alcoff (2006). ³⁰ See A. W. Moore (1997), p. 109. ³¹ See e.g. P. S. Anderson (2012), pp. 126 ff. ³² P. S. Anderson (2012), p. 137.

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Nor do I. Such thoughts constitute a crucial caveat to all that I have been saying. If I have been saying it from a male point of view, and if I have been doing so in such a way that I count as benighted, and if the situation is remediable, then I am, I hope, receptive to whatever remedy is available. If the situation is not remediable, and if this is not due simply to some limitation of mine, then that is itself of real philosophical significance. Either way, more needs to be said, even if it cannot be said by me. At any rate I do not think of myself as saying what I have been saying from a male point of view. Nor do I think that the kind of philosophy that I practise can be practised only from a male point of view. If I could be persuaded that this were the case, then it would force me to rethink my whole conception of the discipline. I want to close by saying a little more about that conception.³³ As I remarked much earlier, I deny that it is the business of philosophers to produce absolute representations. Philosophers have to pursue their discipline from some point of view. And I can see the appeal of the idea that they would do well to pursue it from a female point of view, at least to a greater extent than they currently do. Such an idea might even appear to sit well with my denial that the kind of philosophy that I myself practise can be practised only from a male of point. For this denial does not have to be understood as a claim to neutrality. It can be understood as the claim that the kind of philosophy that I practise can be practised just as effectively from a female point of view as it can from a male point of view, a claim that encourages practice of it from both. To repeat: I can see the appeal of such an idea. I think it contains an important insight: the insight, to put it in a way that I hope does not sound too flippant, that philosophy needs to be more in touch with its feminine side. Nevertheless, I do not believe that this insight is well expressed in terms of a female point of view. Talk of a female point of view, at least in this connection, still seems to me too close for comfort to what I find unhelpful in the grid that Anderson imposes. Admittedly, in trying to find a better way to express the insight, I am, in effect, confronting the very dilemma that I said Anderson confronts: that of finding a way to proclaim the feminine in philosophy without being sexist. I cannot, here and now, offer a satisfactory response to this dilemma. But let me sketch what I think such a response would look like. I think it would begin with the thought that what is really important here is not the relation between philosophy and the feminine, or between philosophy and the masculine, or between philosophy and either the male or the female, but rather the relation between philosophy and the human. I follow Bernard Williams in conceiving of philosophy as a humanistic

³³ One of Anderson’s own concerns is with the nature of philosophy. Her specific focus is the philosophy of religion; but it can scarcely be denied that this has, and is intended to have, repercussions for the discipline as a whole: cf. P. S. Anderson (2012), pp. 47–8.

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discipline.³⁴ What this means is that philosophy is in an important sense anthropocentric. It is an attempt, by human beings, from their unique position in the world, to make sense both of themselves and of that position. This is to be distinguished from the claim that philosophy is a branch of anthropology. Philosophy is not the scientific study of human beings, nor of any of the peculiarities that mark the way of life of any human beings. For it is not the scientific study of anything. It does nevertheless have a fundamental concern with human beings and with what it takes to be one. And it is properly pursued, at the most fundamental level, from a human point of view. In so far as this has purchase specifically on women, it is because the best philosophy reflects the varieties of human experience, both male and female. But the way in which it does this is by making truths about the varieties of human experience available to everyone. As far as the differences between the male and the female are concerned—or the differences between the masculine and the feminine, if these are different differences—that these are important, and in particular that they are important to philosophy, I do not dispute. Nor do I dispute that they are profound. But their importance and their profundity, within the ambit of the human, seem to me as nothing compared with the importance and the profundity of what unites the human, or with the importance and the profundity of the differences between the human and the non-human. In Spinozist terms, and hence in terms that I know would have been congenial to Anderson herself, the common notion ‘human’ seems to me incomparably more significant, to each and every one of us, than the common notion ‘male’ or the common notion ‘female’.³⁵ I take that to be a quite general truth. But I also take it to be a truth with a very particular and very significant application to philosophy, for the reasons sketched above. None of this, I barely need to say in conclusion, constitutes a decisive objection to anything that Anderson was doing in her work, still less to the general tenor of that work. I have rather expressed a certain dissatisfaction with what she was doing, and gestured at some of what I would do instead. Does this betoken a kind of vulnerability?³⁶ It leaves me feeling vulnerable, in particular by challenging my self-image as a philosopher. Anderson would certainly have been sensitive to this and would have wanted to accommodate it in our ongoing conversation about these issues. I only wish that she were here now to continue that conversation.

³⁴ See B. Williams (2006c). I explore an embellishment of Williams’s conception in Essay 5 of this volume. ³⁵ See Spinoza (2002a), pt II, prop. 37 and prop. 40, schol. 2. See also pt IV, props 35–7 (couched admittedly in what would now be classified as sexist terminology). ³⁶ Supplementary note: This reference to vulnerability is prompted by the context in which this essay was first presented, a conference on Anderson’s work titled ‘Love and Vulnerability: In Memory of Pamela Sue Anderson’. As the title of the conference indicates, vulnerability was one of its main themes.

PART III

HO W W E M A K E S EN S E I N ET HI C S

9 A Kantian View of Moral Luck Abstract Kant’s moral vision appears to be resolutely opposed to the very possibility of moral luck. But it is argued that, if we distinguish between good moral luck, whereby an agent deserves moral praise for something that is not in their control, and bad moral luck, whereby an agent deserves moral blame for something that is not in their control, then it would be at least Kantian, if not a commitment of Kant’s own, to accept the possibility of bad moral luck. Indeed, it is argued that, on a suitably Kantian view, bad moral luck is what evil consists in, an evil agent being an agent who culpably loses control. This signals an important asymmetry, since an agent who does not lose control, and who thereby exercises their freedom, is not a beneficiary of good moral luck: their virtue consists precisely in their being in control of what they are doing. These ideas are explored through a series of comparisons and contrasts, notably with the views of Plato and Aristotle and with the traditional Christian doctrine of grace whereby there is a kind of good moral luck in the form of justification through the death of Christ.

1. Kant, Aristotle, and their Differing Views Concerning Luck Some of the most interesting questions about Kant, and more particularly about his moral philosophy, arise when he is placed alongside the giants of antiquity. Where does he come together with Plato? Where with Aristotle? Where does he diverge from each? He comes together with Plato in a shared conception of Ideas. When he first outlines how he is using the term ‘Idea’ in his Critique of Pure Reason, he insists that he is using it in none other than its original Platonic sense; and he explains away certain discrepancies with the comment: ‘It is by no means unusual . . . to find that we understand [an author] better than he has understood himself. As he has not sufficiently determined his concept, he has sometimes spoken . . . in opposition to his own intention.’¹ For Kant Ideas are ‘concepts of reason’. They

¹ Kant (1933), A314/B370. The general discussion occupies bk 1, §1 of the ‘Transcendental Dialectic.’ Cf. Gadamer (1980), p. 38.

The Human A Priori: Essays on How We Make Sense in Philosophy, Ethics, and Mathematics. A. W. Moore, Oxford University Press. © A. W. Moore 2023. DOI: 10.1093/oso/9780192871411.003.0010

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have no direct application within the sphere of experience. But they do serve as paradigms against which we can measure, and thus in the light of which we can regulate, our conduct. As Kant puts it, they can be used to frame ‘regulative principles’. This means that they have a vital role to play when it comes to putting our reason to practical use—in other words, when it comes to recognizing what we ought to do and (thereby) doing it.² There is an otherworldliness about this, which both Kant and Plato recognize in the workings of reason—a kind of commerce with the transcendent. It connects with something else they share: the belief that our true worth, indeed our true being, is something isolable and pure which is not subject to the contingencies and vicissitudes of our empirical surrounds but is itself, to some degree, transcendent.³ This will be important later. My more immediate concern, however, is with the comparison between Kant and Aristotle. One common tendency has been to recognize them as the chief representatives of two great independent traditions in moral philosophy. And these traditions, along with some kind of utilitarianism, are frequently viewed as the apexes of a dialectical triangle.⁴ But there is at least as much to be said for viewing them as the two chief representatives of a single tradition. There are points of contact between them that cut every bit as deep as the differences—deeper, in my view.⁵ In fact it is not as easy as one might expect even to locate the differences, as I hope to indicate. Let us start by considering some basic points at which they come together. They both take themselves to be addressing an audience which already shares their moral or ethical convictions. For both of them the point is to provide a kind of self-understanding in terms of which those convictions make sense. And for both of them this involves a critique of practical reason: it is due exercise of practical reason which determines, and in fact ultimately is, our most fundamental moral objective. (It is perhaps the third of these, their shared belief in the

² E.g. Kant (1933), A316–17/B372–4. See further section 3 of this chapter. ³ See e.g. Plato (1961b), 523e–525a. For a fascinating suggestion as to how Kant and Plato might be even more closely related, see Walker (1989), p. 65. I think the suggestion is ultimately untenable, however. ⁴ This tendency is reflected to some extent in the structure of B. Williams (2006i): chs 3 and 4 are respectively concerned with an Aristotelian and a Kantian attempt to found ethics, while much of ch. 5 is concerned with utilitarianism. See also Lear (1988), pp. 152 ff. ⁵ Cf. Cooper (1975), pp. 87–8; and the work of John McDowell, e.g. McDowell (1978), in which Kantian and Aristotelian elements are brought together. I am very grateful to Philip Turetzky not only for first suggesting to me how close Kant and Aristotle are but for many valuable conversations on these issues.

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practicality of reason, which most clearly sets them apart from other traditions in moral philosophy.⁶) It will be said that Kant has a much more abstract vision of this than Aristotle. This is related to the earlier point about transcendence. It is also related, in a similar way, to another point often made in this context: that whereas Kant tries to attain moral self-understanding through an investigation of practical reason as such, Aristotle starts from a much deeper concern with our empirically determined human nature.⁷ Various charges of emptiness are often brought against Kant in this connection. They seem to me unfair. Or at least, they seem unfair if they are intended to highlight a gap between his project and the Aristotelian project of looking at what is distinctively human, and assessing human good accordingly. After all, what is distinctively human, for Aristotle, is a matter of our ergon. And by our ergon he means that essential part of our humanity which differentiates us from other animals, namely our ability to exercise practical reason, or, if you like, to act rationally.⁸ Furthermore, a good man, for Aristotle, is a man who does just that— acts rationally—to the highest degree (a claim that Kant would certainly want to endorse). And note that Kant too, just like Aristotle, takes basic justification for his project to lie in the fact that rationality is found in human beings but not in other animals in a way that prompts the question, ‘What is it for?’⁹ Conversely, Aristotle, just like Kant, shows that he can be guilty of leaving our humanity behind, when, having distinguished us from other animals, he veers to the other extreme of treating us as gods—beings that are purely rational—and argues that the most excellent human life would consist of isolated philosophical speculation.¹⁰ (In fact Kant is in much less danger than Aristotle of veering to such an extreme. He is always adamant to stress the primacy of practical reason over theoretical, or speculative, reason.¹¹) By the time we have taken stock of all of these points, the differences between Kant and Aristotle look, once again, much less striking than the similarities. A parenthetical point: even their different conceptions of ‘happiness’ and its role in ethics may not be very far apart. Jonathan Lear has argued that there is a crucial difference between them over the question of how far ‘human happiness’ is ‘a merely

⁶ On the first two points see e.g. Kant (1964a), pp. 60 and 79–80; and Aristotle (1941e), 1095a2–13, 1095a30–b13, and bk 1, ch. 7, passim. Cf. Lear (1988), p. 193; and Nussbaum (1986), ch. 8. On the third point cf. Korsgaard (1985–6). ⁷ See e.g. the Preface to Kant (1964a), esp. pp. 57–9 (though on p. 55 he writes, ‘[Moral philosophy] has to formulate its laws . . . for the will of man so far as affected by nature’). And see Aristotle (1941e), bk 1, ch. 7. ⁸ For a good discussion of how it is that our ergon might be shared by other, non-human beings (gods, say) see Nagel (1980), pp. 10 ff. ⁹ Kant (1964a), pp. 62–4. See also Kant (1960), bk 1, ch. 1, where he speaks of man’s Bestimmung. ¹⁰ Aristotle (1941e), bk 10, chs 7 and 8. ¹¹ See e.g. Kant (1956), pt I, bk 2, Ch. 2, §3. See also Kant (1978), beginning of §86. (Christine Korsgaard, incidentally, whose focus in Korsgaard (1985–6) is in fact this difference between them, thereby sees Kant as the one with the greater humanistic strain. She may well be right.)

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given end’. He writes, ‘Any attempt to achieve a merely given end—like human happiness—would, for Kant, count as heteronomy and thus be disqualified as morality.’¹² But Kant writes that, in a world of unadulterated morality, ‘freedom, partly inspired and partly restricted by moral laws, would itself be the cause of general happiness, since rational beings, under the guidance of such principles, would themselves be the authors both of their enduring well-being and of that of others’.¹³ Still, there are differences between Kant and Aristotle (though it is worth noting how far these are a matter of questions asked rather than answers given: that itself makes the convergence all the more significant). I want to focus in this essay on what I take to be one of the most vital—the difference over what we might call, simply, ‘good luck’.¹⁴ Here too the matter is not as straightforward as has often been supposed. But there is certainly a level, which I shall try to uncover, at which Aristotle believes in the possibility of good luck but Kant does not. (There is a more general difference at stake here, a difference between the ancient and the modern. But, as we shall see, it would be a mistake to attribute this—without qualification—to the intervention of Christianity.) In assessing this difference we again do well to start with some basic points of agreement. Kant and Aristotle both believe that there is a kind of good that can befall us as a matter of luck. (‘As a matter of luck’ here means: in a way that is beyond our control.) Indeed, for Kant it is an inescapable hope that we shall enjoy such ‘luck’—to the extent that we are virtuous. For what we hope is that our virtue will have its due reward, something other than its own intrinsic value.¹⁵ (As I noted above, however, it is Kant’s view that, if we were all completely virtuous, there would be no need to hope. I shall discuss the Kantian hope further below in section 3.) Conversely they both believe that there is a kind of good which cannot befall us as a matter of luck. Thus Aristotle takes it that ‘the greatest and finest thing of all’ should not be, as he says, ‘at the mercy of chance’.¹⁶ The difference is that for Kant the good which cannot befall us as a matter of luck is the only unconditioned good. It is the only thing that ultimately matters. It is also, crucially, the only thing for which we ourselves, or at any rate our lives, can be the objects of approval.¹⁷ For Aristotle, on the other hand,¹⁸ there are matters of

¹² Lear (1988), p. 155, emphasis in original. The discussion as a whole is on pp. 154 ff. ¹³ Kant (1933), A809/B837. ¹⁴ The whole question of the role of luck in morality has been the focus of much recent discussion. See esp. Nussbaum (1986) and the articles by Bernard Williams and Thomas Nagel, both titled ‘Moral Luck’, originally published together but now recast and appearing respectively as B. Williams (1981a) and Nagel (1979). See also Andre (1983); Lewis (1989a); and Richards (1986). I am grateful to Stephen Everson and Sabina Lovibond for helpful discussions on this, and to the former for directing me to a number of references. ¹⁵ Kant (1933), ‘Transcendental Doctrine of Method’, ch. 2, §2. (On the idea that virtue is its own reward see Wittgenstein (1961), 6.422.) ¹⁶ Aristotle (1941e), 1099b24. ¹⁷ Kant (1964a), pp. 61 ff. ¹⁸ At least in Aristotle (1941e). There may be important differences between Aristotle (1941e) and other ethical works of his: see Cooper (1985). The main Aristotelian references are in my n. 21.

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endowment, matters of education, and matters of personal relations which, though they are beyond our control, are all capable of determining, even at the most fundamental level, how far our lives are to be assessed as good. These things are to be distinguished, as Aristotle does distinguish them,¹⁹ from benefits that provide us with unsolicited opportunities to do good. Money is an example of the latter. So too, for that matter, is our existence. Even Kant can admit these (as, of course, he must). The point about benefits of this kind is that they determine the range of possibilities open to us. Whether we actually do good, or are virtuous, depends on which possibilities we realize (how we spend our money, say); and that is entirely within our control.²⁰ Where Aristotle and Kant part company, then, is over the question of whether things can count as going well with us at the most fundamental level (whether we can meet with approval) as a result of anything that is not within our control. And although the exegesis is extremely delicate, and the gap between them—even here—not blatant, I take it that Aristotle thinks they can.²¹ Kant thinks that the most fundamental level of approval is that of moral approval. So his position can be put succinctly as follows: there can be no such thing as moral luck. There is a problem, however, about using this particular formula to capture the contrast with Aristotle. Aristotle could well understand by the moral ‘the greatest and finest thing of all’ and then deny as vehemently as Kant that there could be any such thing as moral luck. That is, he could deny as vehemently as Kant that we could meet with approval of that—very distinctive— kind for anything that we had not voluntarily done. The contrast with Kant would then emerge on the question of the ubiquity of the moral—the question of whether moral approval is the only kind of approval, at the most fundamental level. In fact one could say that the link between the moral and the free, one aspect of which is tersely captured in the Kantian dictum that ‘ought implies can’, is, to the extent that anything is, analytic. That is, the expression ‘moral luck’ is, to the extent that anything is, a contradiction in terms. Even Bernard Williams, perhaps the best known exponent of the suggestion that morality is subject to luck, accepts that there is something deeply amiss in the notion; for in making the suggestion, he takes himself to be challenging the coherence of our moral conceptions.²² ¹⁹ Aristotle (1941e), 1099a31–1099b8. ²⁰ Aristotle (1941e), 1100b18–22. ²¹ See e.g. Aristotle (1941e) 1095b33–1096a2; bk 1, ch. 10; bk 2, chs 1, 2, and 4; and bk 10, chs 8 and 9. But John Cooper, in Cooper (1975), ch. 2, §2, argues that Aristotle can be interpreted in a much more Kantian way; cf. his Cooper (1985), p. 196. And contrast Martha Nussbaum’s exegesis in Nussbaum (1986), chs 11 and 12. See also on this issue Irwin (1985), and Kenny (1988). One of the bones of contention is the extent to which our own ‘most fundamental’ goodness is at the mercy of education (which is certainly beyond our control). For a somewhat Socratic view of education, as activating latent dispositions, see Kant (1964b), ‘The Ethical Doctrine of Method’, §2. However, in Kant (1963), pp. 46–7, we find something more Aristotelian. Finally, for an indication of where Plato stands on these issues—somewhat closer to Kant than to Aristotle—see Plato (1961e), 278e–282d and Plato (1961d), 87d–89a. ²² B. Williams (1981a), pp. 22, 39, and n. 11. (And see p. 38 for the point about ubiquity.)

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There is another problem with the formula, ‘There can be no such thing as moral luck.’ The expression ‘moral luck’ is ambiguous. The ambiguity turns on the fact that the word ‘luck’ sometimes, but not always, stands elliptically for good luck—as indeed the word ‘moral’ sometimes, but not always, stands elliptically for what is morally good. So far, the expression ‘moral luck’ has been used in this elliptical sense. But if we appropriate its other sense—as I propose from now on we do—then nothing that has been said up to now stops us from granting to Kant, or to any Kantian, a belief in moral luck. What we have been concerned with up to now has been the domain of moral approval. That may not be the same as the domain of moral assessment. Thus it may make sense, in moral terms, to disapprove of someone for what is beyond their control even if it would not make sense to approve of them for what is beyond their control. Anyone who believed that this was the case, and in particular any Kantian who believed that this was the case, would still be acknowledging the possibility of moral luck. The point is, roughly, it would be bad luck.²³ The notion of bad moral luck does strike a less jarring note with many people than the notion of good moral luck. To take a familiar example: an act of reckless driving which results in innocent deaths can seem, just because of its outcome, to have a kind of moral culpability not shared by an act of—equally—reckless driving which fortuitously results in no harm.²⁴ Many would regard this as an example of bad moral luck; and they would be hard pressed to say what a ‘good luck’ counterpart might look like. To be sure, in cases where it is a matter of how things turn out, as it is in this case, bad luck and good luck are to some degree subcontraries: if, while recklessly driving, you are not unlucky enough to kill anyone, then you are, ipso facto, lucky. But those who sense an asymmetry here are presumably recoiling from the idea that you can turn out to have been praiseworthy—rather than the idea that you can turn out not to have been blameworthy (in some specified respect). For a thoroughgoing Kantian, however, if there is such a thing as bad moral luck, then this is not the right kind of example. This is because the whole notion of good moral luck is such an anathema. And that includes not only the good moral luck of turning out to have been praiseworthy; and the more plausible constitutive good moral luck of simply being a certain kind of person; and the good moral luck of being in certain favourable circumstances;²⁵ but also, crucially, the negative complement of the reckless homicide’s supposed bad moral luck—which means that it cannot really be bad moral luck at all. Nothing about what is morally wrong with reckless driving can depend, for a Kantian, on its actual consequences.²⁶

²³ Cf. Adams (1985). ²⁴ See e.g. Nagel (1979), p. 29; and Andre (1983), p. 205. ²⁵ For these distinctions, and for discussion of them, see Nagel (1979) and B. Williams (1981a). ²⁶ Cf. Andre (1983) and Richards (1986).

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This radical stance, whatever its implications for the notion of bad moral luck, certainly threatens a lot of the moral praise we are inclined to mete out. One way to avert this threat, other than by severing the link between what we can be morally praised for and what is within our control, would be to develop a broader conception of what is within our control.²⁷ A more puritanical alternative would be to accept the threat and simply to urge scepticism about all the moral praise we are inclined to mete out. Another, related feature of Kant’s radical stance is the extent to which it undercuts a certain kind of utilitarian ‘moral scheming’. For example, it supports his well-known argument against the possibility of a morally justified lie, an argument that many people find to be both wanting and yet curiously appealing.²⁸ The argument takes as a (possibly question-begging) premise that if someone lies and terrible consequences accrue, then their action is ipso facto blameworthy. Given Kant’s stance, it follows that their action is blameworthy anyway: the moral justification for their lying cannot depend on their being lucky that terrible consequences do not accrue. However that may be, the notion of a certain kind of bad moral luck is inevitably as far beyond the Kantian pale as the notion of good moral luck; for the possibility of the former presupposes the possibility of the latter. But there may yet be bad moral luck of other kinds. Two important caveats are required before I proceed to defend this. First, I have been deliberately talking latterly about what is Kantian rather than about what is in Kant. It is unquestionable that some of the doctrines that I am about to portray as Kantian he would disavow.²⁹ They are Kantian in the sense that they accord with a deep and powerful strain in his thinking. Secondly, ‘moral luck’ has been serving, and will continue to serve, as something of a technical term, which means that its use in connection with these issues will sometimes sound strange—even allowing for the fact that it is not being used in its more familiar, elliptical sense. For someone to be subject to moral luck, in the sense that I intend, is for them to be subject to moral praise or blame for what is not in their control.

2. The Possibility of Bad Moral Luck on a Kantian View One route into these issues is via the notion of a moral conflict. Moral conflicts of a certain sort are impossible for Kant. Whenever there is a question about what someone ought morally to do, there is at most one answer. This connects with the

²⁷ Cf. in this connection Strawson (1974). Cf. also Kant (1933), A554–5/B582–3. Thomas Nagel, in Nagel (1979), p. 26, explicitly rejects this kind of expedient. ²⁸ Kant (1949). ²⁹ I can point already to Kant (1960), p. 35, and Kant (1933), A551/B579, footnote.

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idea that there is only one unconditioned value, and thus, for Kant, only one moral value, namely rational exercise of the will. In fact this cuts so deep that it effectively serves as a criterion of identity for acts, and makes moral conflicts (of the relevant sort) a kind of contradiction in terms: the unique act that is morally obligatory in any circumstances—there may be many that are morally forbidden—is the one that is rational. To recognize which act this is, and (thereby) to do it, is to exercise practical reason. It is impossible, in this scheme, for competing values to make each of two incompatible acts obligatory.³⁰ Another aspect of the Kantian view, related to this and very pertinent to our discussion, is that moral obligation is a matter of making oneself immune from luck, or at least from that kind of luck which is sometimes referred to as ‘incident’ luck.³¹ To exercise practical reason is to distance oneself from one’s circumstances and to do the one thing that can count as successful quite irrespective of how the circumstances pan out. For the act’s success is internal to it, and absolute. It does not have to be viewed from any special vantage-point. In particular it does not have to be viewed from a vantage-point of retrospection.³² (Nor does it have to be viewed from any individual’s vantage-point. This is why, for Kant, one is not acting rationally if one is motivated by a principle—any principle—that one could not accept as always apt to motivate others.³³ I say ‘any principle’ because several may be operative at once. All that is required for the act to be immoral is that one of them should be ‘non-universalizable.’) We see here the transcendent element in Kant’s moral philosophy, noted above in connection with Plato. Something of this kind has recurred throughout the history of the subject. (One of its most interesting recent manifestations is in the early work of Wittgenstein.³⁴) Always it has gone hand in hand with a view of ultimate value as unified, integrated, and hostile to the very possibility of internal conflict.³⁵ It is a different question, however, whether ultimate value is hostile to the possibility of external, or accidental, conflict. There is nothing in the view to say that it is. Nor, I shall argue, is a Kantian debarred from recognizing a corresponding kind of moral conflict. All that has been argued so far, on behalf of the Kantian view, is that not more than one act can be morally obligatory in any given circumstances. That is, not more than one act can count as an exercise of practical reason. But it may be that fewer than one can. There may be circumstances in which all the available acts are irrational because the circumstances have somehow interfered with the grounds of reason and so prevented its implementation: ³⁰ See the passage from Kant (1965), quoted at Nussbaum (1986), p. 31. Kant effectively equates the ‘ought’ of moral obligation with the ‘ought’ of practical deliberation. See B. Williams (2006i), pp. 174 ff. ³¹ See B. Williams (1981a), p. 20. ³² Cf. A. W. Moore (1988), p. 220. ³³ Thus the moral law is to ‘act only on that maxim through which you can at the same time will that it should become a universal law’ (Kant (1964a), p. 88). ³⁴ See e.g. Wittgenstein (1979), pp. 76–7 and 81. ³⁵ E.g. Plato (1961d), 330c ff.

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whatever the agent does will be morally forbidden. That would be a moral conflict of a (different) sort. (But note: if such a possibility is to be accommodated into the Kantian view, then the two sorts must not collapse into each other. So the twin notions of what is morally obligatory and what is morally forbidden must be allowed to act autonomously. In particular it must not follow from the fact that x and y are the only two possible acts in circumstances C, and that x is morally forbidden, that y is morally obligatory.³⁶) In terms of the main concern of this essay, the possibility of such circumstances— of a moral trap, as I shall say—may, indeed, look decisive. For if someone found themselves in a moral trap, through no fault of their own, and then acted in one of the forbidden ways, would they not be, paradigmatically, the victim of bad moral luck? This question is too hasty however. If there are to be any moral traps of this kind, then we must reject one of the following principles. (I have expressed them in a way that I hope is self-explanatory). (i) If it is morally forbidden for A to do x, then A ought not to do x. (ii) If A ought not to do x and A ought not to do y, then A ought not to do x-or-y. (iii) Ought not implies can help. (iii) is what the notion of bad moral luck challenges. But we could just as well retain that and give up (ii). (Rejecting any of them is something of an affront to intuition.) If we do give up (ii), and then go back to the case of someone in a moral trap of the kind we are envisaging, we do not have to think that there is anything that is both (a) their fault (in the sense that they are to blame for it) and (b) beyond their control. By hypothesis it is not their fault that they are in this situation. Nor, now that (ii) has been rejected, need we deem it their fault that they end up doing something—that is, something or other—forbidden. It may be their fault that they do the particular forbidden thing they do. On the other hand they seem to have control over that. Any link with bad moral luck must therefore still be teased out. Before we go any further, we must ask what an example of a moral trap would look like (that is, what it would look like for a Kantian). We are not interested in circumstances in which the question of acting rationally does not even arise. These include circumstances in which the agent is forced to make some quite arbitrary choice and can quite legitimately, indeed must, follow a whim. (Buridan’s ass provides an example of this.) In a moral trap, whatever the agent does must be irrational, not merely non-rational. In Kantian terms, this means that it must be a

³⁶ I am indebted here to B. Williams (2006i), pp. 174 ff. I hope that what I say in this essay does something towards answering the (semi-rhetorical) question that Williams poses in n. 2 of that discussion (p. 243).

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situation which, in the fullest sense, allows room for agency. (I said ‘follow a whim’, not ‘act on a whim’.) What the agent must do is to act according to some principle which he or she could not accept as a general principle for the motivation of others. He or she must succumb to some private inducement: the possibility of acting purely rationally has been ruled out. (Note that, on the rather stark view that is already beginning to develop here, every act is either morally obligatory or morally forbidden. There is no room for the merely morally permissible. Whatever is neither rational nor irrational is outside the jurisdiction of reason altogether. Regarding the caveat at the end of section 1, this certainly seems to be something that is Kantian rather than in Kant.³⁷ But the discrepancy may not be deep. It may turn on the semi-linguistic point of how finely we are prepared to carve acts: see the beginning of section 2.) What, then, would a moral trap look like? Take Herod. Having promised to give Salome, whose dancing has so delighted him, whatever she cares to ask for, he finds himself confronted by a request for the head of John the Baptist.³⁸ Nothing he can do in this situation is morally right. He can only follow whatever motive is strongest in the circumstances—fear, compassion, the desire to maintain his honour, the desire simply to keep his promise, whatever it may be. Many would insist that, on a Kantian view, he is morally obliged to keep his promise (always assuming that it is indeed a promise: there may be grounds for doubting the given description of the case³⁹). Kant himself might insist on that. But it is in the spirit of Kantianism, I think, to maintain only that he is morally forbidden to break it. The circumstances make his keeping the promise impossible as an act of duty because there can be no dutiful spilling of innocent blood. (Here I am presupposing a kind of extensionality which is itself, I think, in the spirit of Kantianism. An act is rational or not; it does not matter how it is described.) What this shows is that the promise was a rash one and should never have been made in the first place—which brings us to the main point of the example. It is not through no fault of his own that Herod is in this moral trap. He has got himself into it, doubtless because of his lust. And in so far as this is typical of such cases, the link between them and bad moral luck now looks even less straightforward. It is even less clear in Herod’s case what it is that is both (a) his fault, and (b) beyond his control. We seem to be able to scupper him as follows: Why Herod is Not the Victim of Bad Moral Luck Not only did Herod get himself into this situation as a result of his own free will, he is at perfect liberty to choose which of the available courses of action now to adopt—even if each of them is, as it happens, morally forbidden. For example, ³⁷ See Kant (1964a), p. 107, and Kant (1956), p. 69. ³⁸ Mark 6: 17–25. ³⁹ I am grateful to members of the Stapleton Society at Liverpool University for drawing my attention to this point.

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suppose he orders John’s beheading. That will not mean that he incurs moral disapprobation for anything beyond his control. He could have broken his promise. Nor is he morally unlucky that Salome has made the particular request she has, at least not in Kantian terms. For that would mean that he would have been morally lucky had she requested something innocent; and there is no such thing as good moral luck. So Herod’s case seems not to furnish an example of bad moral luck after all. Moreover, there really are reasons to think, on Kantian grounds, that it is typical of all moral traps.⁴⁰ The transcendent, world-distancing element in practical reason casts considerable doubt on the idea that anything in the world, that is anything outside the agent, should, by itself, be able to thwart the exercise of his or her reason. If it is going to be thwarted, this must be a result of some inner weakness. Thus it may be that, by an effort of reason, Herod can shrug off whatever the world throws at him, whereas he cannot cope with his own loss of resistance, his own irrational loss of control. Here he has got himself into a position of submission—irrational submission to whatever non-rational motive (impulse, desire) proves strongest. But consider: ‘submission’ is the operative word. Herod is subject to bad moral luck after all, and we can now see why. Because rational action has been precluded, he is bound to do something blameworthy; and the blameworthiness is going to be rooted in something beyond his control, contra what was said above. It is going to be rooted in whatever motive proves to be the strongest. He has no choice about that, now that he has lost rational control of the situation. This is why his moral trap serves as such a graphic example. (It is true that there are even more obvious ways of undermining one’s own opportunity to act rationally, for example getting very drunk. But although getting very drunk may itself be irrational, and blameworthy, it is not clear that, while one is very drunk, one acts irrationally—for it is not clear that, while one is very drunk, one acts. This presages various issues that will crop up later. As for Herod, what he in fact does, St Mark tells us,⁴¹ is to act ‘out of regard for his oath and for his guests’. Salome gets the Baptist’s head.) The point is this. On a Kantian conception nothing can be in Herod’s control— nothing can be subject to free exercise of his will—unless it is in his rational control. No act of his is truly free unless it is rational, that is unless it is what he is

⁴⁰ Bernard Williams urges (non-Kantian) scepticism about whether it is, in B. Williams (1973e), p. 179. At first blush it seems easy to construct examples to support Williams’s view; there is his own example on p. 180 of B. Williams (2006i). But in fact it is not at all clear what ice this example cuts. As Williams says, ‘to make the example realistic, one should put in more detail’. (Did you actually promise to visit your friend whatever might crop up? If so, why? Was that not rash?) ⁴¹ Mark 6: 26. I am using the New English Bible translation (Oxford University and Cambridge University Presses, 1970). I shall use this translation for all subsequent quotations from the Bible.

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morally obliged to do.⁴² Hence, not only is his doing what is morally forbidden compatible with his not being in complete control of what he does, it actually, in a deep sense, requires it. Herod is to blame for his loss of control. He is to blame for getting himself into a situation in which, and of which, he cannot be the master. And once he is in that situation, he is to blame for submitting to a non-rational impulse. To quote Kant: [A man] does not answer for . . . [incitements from desires and impulses] nor impute them to his proper self—that is, to his will; but he does impute to himself the indulgence which he would show them if he admitted their influence on his maxims to the detriment of the rational laws governing his will.⁴³

It is worth noting, in this connection, another point at which Kantians and Aristotelians (may) converge. Kantian morality allows scope for an important diachronic element (though this is obscured by Kant’s own emphasis on the rightness or wrongness of particular dated acts). An Aristotelian will certainly be interested in the overall shape of a person’s life.⁴⁴ But so too may a Kantian. Exercising practical reason can be viewed as giving one’s life a certain coherent shape, whose parts properly complement one another; or, more to the point, failing to exercise practical reason can be viewed as failing to do that. What happens in Herod’s case, for example, is that an initial act of irrationality traps him and bundles him along into further acts of irrationality. It is perhaps a more drastic and more radical version of this that underpins the Christian doctrine of original sin: we are so rooted in evil that we are inextricably caught in a grand, perpetual moral trap. (I say ‘inextricably’, but of course that is not really faithful to ⁴² Returning again to the caveat at the end of section 1, it is here that the question of a gap between what is Kantian and what is in Kant is most delicate. Such a radical conception of freedom is, so far as I know, nowhere explicitly embraced by Kant, though arguably it is implicit in Kant (1933), A538–41/ B566–9; in Kant (1933), ‘Transcendental Doctrine of Method’, ch. 2, §1; at the beginning and end of ch. 3 of Kant (1964a); at the beginning of the Introduction to Kant (1956); and in the discussion of ‘Problem II’ on pp. 28–30 of Kant (1956). Still, all that Kant strictly commits himself to is that a free will is a will subject to (its own) rational laws; it does not follow that for the will to be exercised freely is for it to be exercised in accordance with those laws. Indeed, elsewhere in Kant (1956), on p. 32, Kant clearly moves in the other direction and insists that an irrational act, though ‘pathologically affected’, is not ‘pathologically determined’ and is still free. (The Willkür/Wille distinction, which is often invoked in discussions of this, is especially prominent in Kant (1960). But there is much greater emphasis on it in the introductory essay, Silber (1960), by John Silber, than there is in Kant himself. Some of the issues that arise here—not all of them, by any means—are purely verbal.) Cf. in this connection Sidgwick (1907), bk 1, ch. 5, §1, and Allison (1986), esp. §6. In his discussion of ‘Theorem II’ on 20 ff. of Kant (1956), Kant suggests that an irrational act is one where self-love has got the better of the agent. Kant (1960), bk 1, §3, is very revealing in this respect. The suggestion that to succumb to temptation is to lose control of oneself, which is what we see here, is famously ridiculed by J. L. Austin, in ‘A Plea for Excuses’, in Austin (1970), p. 198, footnote. But Austin does not take due account of the fact that there are myriad ways of losing control of oneself. It is interesting to compare all of this with Aristotle’s discussion of akrasia, and related issues, in Aristotle (1941e), bk 3, chs 1 and 5, and bk 7, chs 3 and 4: Aristotle recoils from the idea that acting wrongly means acting involuntarily. ⁴³ Kant (1964a), pp. 125–6. ⁴⁴ See Aristotle (1941e), 1098a17–19.

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the Christian orthodoxy. I shall come back to this point is section 3.) On this view, increased awareness of how little is in our control need do nothing to challenge the moral blame we are inclined to apportion to ourselves. It may reinforce it.⁴⁵ Kant’s developed metaphysics in a way lends substance to this. For it famously incorporates the idea that exercise of our reason, or again, more to the point, the lack of it, is ultimately located in a transcendent realm outside space and time. This in turn allows room for the notion of some single primordial act of irrationality, blazoned across a life of unrelenting wickedness that is inescapable from any point within. (Again, the ‘inescapable’ needs qualification. Kant’s own view of these matters likewise demands that it be qualified. For he holds that we can, with a change of heart, initiate an escape from our wickedness. What he concedes is that the escape will never, at any point in time, be complete.⁴⁶ Of this too more anon.) There are other respects, however, in which Kant’s metaphysics merely serves to exacerbate some of the difficulties with this view. The idea that Herod is to blame for something beyond his control is all very well if he really is in a selfinflicted predicament. For the blame can be thought of as attaching ‘really’ to some ‘initial’ wrong-doing. (By ‘initial’ here I mean: not itself brought on by any moral trap.) But what are we to make of the blame incurred by that, if, as is required by this view, it too is for something beyond his control? The reason why this difficulty is exacerbated by Kant’s metaphysics is that there are now problems in understanding how there could even have been an ‘initial’ wrong-doing. It must have involved interference by the outside world with Herod’s use of reason. But Kantian doubts have already been cast on whether that sort of thing can happen. The doubts are now reinforced. For the interference now looks like interference by what is in space and time with some more ultimate, transcendent reality. These problems all come together in the question: How is primordial evil possible? (And note: although this question is of the form, ‘How is X possible?’, which is characteristically Kantian, it is not being asked in a characteristically Kantian way. It is being asked rather with what Jonathan Lear calls ‘a sceptical sneer’.⁴⁷ Note also: the question is separate from, though it is related to, the

⁴⁵ Kant’s own discussion of original sin occurs in Kant (1960), bk 1, §3. His views about overcoming it (extricating oneself from the trap) do, interestingly, reveal the importance to his thinking of the diachronic; see further below. Cf. Kant (1933), A316–17/B373–4. Cf. also in this connection the Aristotelian thought that one can become a morally bad person by repeatedly doing what is morally bad and getting into a habit; hence the importance of education (see Aristotle (1941e), bk 2, ch. 1). ⁴⁶ See Kant (1960), pp. 60–1 and 68. For discussion see A. W. Moore (1988). ⁴⁷ Lear (1984), p. 223.

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question, ‘How can one freely commit evil?’ On Kantian grounds, I have urged, one cannot—not truly.) Kant insists that primordial evil is possible, and he explicitly rebuts some of the challenges that have here implicitly been levelled against him.⁴⁸ At the same time he is as conscious as anyone of the need, within his system, to locate primordial evil; and of the impossibility of accounting for it.⁴⁹ Perhaps at least this much can be said on his behalf. There need not be any surrender of control on this view. There need only be something much less mysterious than that, loss of control. At some point, at some level, whoever acts irrationally has not stood firm enough. And as a result they have been deprived of their freedom, by something stronger. Can they help this? Can they help their weakness? No; but they are to blame for it. They are morally to blame for it.⁵⁰ That is their bad moral luck. Note the asymmetry though. Had they been stronger and stood firmer they would not have enjoyed good moral luck: they would have been exercising complete and perfect freedom. We must resist the idea that they could only have exercised freedom in making some anterior choice about whether to act rationally or not. Had they acted rationally, that in itself would have constituted their exercise of freedom. In not acting rationally they show themselves not to be free. This is a radical conception of freedom. But notice how well it dovetails with Kant’s utterly exigent, utterly puritanical morality. One incurs blame for not being free—that is, for not being steadfast, for losing self-control and becoming a slave to some passion or desire. A final point in connection with this: puritanical morality is liable to represent freedom itself as submission to the will of God. (Consider the familiar phrase ‘Whose service is perfect freedom’, which occurs in the second collect for Matins in the Book of Common Prayer.) Kant can accept this—but only to the extent that the will of God is viewed as the agent’s own rational will. The equivalence of freedom with autonomy is something which he takes to be absolutely paramount.

3. Consequences of the Kantian View There are various ways in which the Kantian view, as portrayed in the previous section, is counterintuitive—not least in the idea of bad moral luck itself. For example, the view allows us to say things of each of the following kinds.

⁴⁸ E.g. Kant (1978), p. 37, footnote. ⁴⁹ See Kant (1960), bk 1, §4. ⁵⁰ Cf. Romans 7: 14–25, to which we shall return. Cf. also Adams (1985), where, on p. 4, we find the examples of unjust anger, hatred, contempt for others, and lack of hearty concern for their welfare; Nagel (1979), pp. 32–3; and Schlossberger (1986). Aristotle’s contrasting view comes out in Aristotle (1941e), bk 2, ch. 5.

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(i) A is morally to blame for x, though x was not in A’s control. (ii) It is A’s fault that p, though nothing that A freely did is any part of the explanation for why p. (iii) A has acted in some way (perhaps in a way that makes it A’s fault that p), though what A did A did not do freely. Just how dissonant each of these schemata sounds to non-Kantian ears is something that need not detain us now.⁵¹ But the schemata do collectively highlight a special and urgent question that the Kantian must face. It manifests itself most graphically in (iii). If agency is not necessarily free agency, what then counts as an act? This question directly feeds into the problem of moral luck. For we can ask, analogously: if someone can be morally blamed for what is not in their control (for example, if they can be morally blamed for an act of theirs which is not a free act), what then is the domain of moral assessment? A non-Kantian need not be troubled by these questions. And normally we would not be. One of the things that normally inclines us to retract moral blame, as inappropriate, is the discovery that the person being blamed was not in control of whatever it is for which we are blaming them. The possibility of moral luck is not one that we ordinarily reckon with.⁵² But once that possibility has been acknowledged, we need some account of what it is that makes moral blame appropriate. What can a person be morally blamed for? Can a kleptomaniac be morally blamed for stealing? Can someone be blamed for what they do under duress? How much duress? Can they be morally blamed for being ugly? (Aristotle counts ugliness as something that detracts from a person’s eudaimonia.⁵³ But he also distinguishes between those who are naturally ugly and those who become ugly through lack of exercise and care, saying that it is only the latter that we blame.⁵⁴) The Kantian can say at least this. Someone can be morally blamed for something only if, at some point, at some level, and in some sense, they were free to do otherwise. There may seem to be some tension between that and what is being canvassed in this essay as the Kantian view. But there is not. In fact the one is entailed by the other. Someone can do something morally wrong only if their freely doing the morally right thing was (at some point, at some level, and in some sense) an alternative. Moral blame attaches to them for not being thus free. But that still leaves the question of when it is that they are not thus free; and of how we might tell when it is. This question is even more urgent for Kant himself than it is for most Kantians, because it points to a counterpart for moral praise. (Can someone be morally praised for being beautiful, say?) It is not that Kant

⁵¹ Cf. B. Williams (2006i), pp. 177–8 and 194. Much of this essay is meant as a response to Williams’s critique. ⁵² Cf. Nagel (1979), p. 25, and B. Williams (1981a), p. 21. ⁵³ Aristotle (1941e), 1099b3–4. ⁵⁴ Aristotle (1941e), 1114a23–5.

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allows moral praise to attach to what is not in a person’s control. Far from it. There has never been any question about that. But in Kant’s system, if we stick solely to the point of view from which we make moral assessments, we find ourselves having to admit that nothing is in anybody’s control: the concept of freedom has no application. This is why we are forced to think of freedom and its exercise as belonging to a transcendent realm outside space and time, a realm inaccessible (epistemologically) from any point of view that we can occupy. But since it is thus inaccessible, there has to be some prior, independent account of when moral praise is appropriate. And this is very much like the problem which I have just argued is faced by Kantians more generally with respect to moral blame. One possible response to this problem is the view that I shall call conservatism—the view, to put it in a very hazy way, that all the moral assessments that we ordinarily make should be allowed to stand (in the sense that they should be allowed to count as appropriate; they need not be allowed to count as correct). But does this help? What are ‘the’ moral assessments that we ordinarily make? Surely, if our moral assessments are ordinarily anything, they are ordinarily fluid and revisable. In particular, as I have already noted, they ordinarily get revised in the light of discoveries about what is in people’s control. If we adopt conservatism, then we are liable to find, paradoxically, that moral praise and blame are never appropriate. True, a particularly puritanical Kantian might be happy to accept this in the case of moral praise: not only does nobody ever deserve moral praise for anything, but nobody ever does anything that is even a candidate for moral praise. But the same conclusion cannot be extended to the case of praise and blame alike, without incurring what any Kantian must count as unacceptable scepticism. Somehow, then, both Kant and Kantians more generally need some way of registering when moral praise and blame are appropriate without the benefit of being able to say that they are appropriate in such and such cases because such and such cases are (antecedently recognizable as) cases of free agency. The way that Kant himself responds to this question is—in effect—by denying that our moral assessments are fluid and revisable. He then embraces conservatism. We must not forget that Kant, like Aristotle, takes himself to be addressing an audience which already shares his basic moral convictions. In Groundwork, he starts with those convictions—or, as he puts it, the ‘ordinary rational knowledge of morality’ (this phrase comes in the title of chapter 1)—and moves from there, via something more systematic, to a critique of practical reason, the aim being to provide a kind of self-understanding in terms of which the convictions make sense. The raw materials for this exercise, the convictions themselves, are not called into question. They provide some sort of datum on which everything else is allowed to hinge. Elsewhere Kant writes:

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I assume that there really are pure moral laws . . . I am justified in making this assumption, in that I can appeal not only to the proofs employed by the most enlightened moralists, but to the moral judgement of every man, in so far as he makes the effort to think such a law clearly.⁵⁵

And he does not think that the particular judgements we make are in fact sensitive, or need be sensitive, to subsequent discoveries about causes and determining conditions. The following quotation from Critique of Pure Reason is very pertinent: Let us take . . . a malicious lie . . . First of all, we endeavour to discover the motives to which it has been due, and then, secondly, in the light of these, we proceed to determine how far the action and its consequences can be imputed to the offender. As regards the first question, we trace the empirical character of the action to its sources, finding these in defective education, bad company, in part also in the viciousness of a natural disposition insensitive to shame, in levity and thoughtlessness, not neglecting to take into account also the occasional causes that may have intervened . . . But although we believe that the action is thus determined, we none the less blame the agent . . . Our blame is based on a law of reason whereby we regard reason as a cause that irrespective of all the abovementioned empirical conditions could have determined, and ought to have determined, the agent to act otherwise . . . In the moment when he utters the lie, the guilt is entirely his. Reason, irrespective of all empirical conditions of the act, is completely free, and the lie is entirely due to its default.⁵⁶

Our awareness of moral reality is fundamental then. It is through that awareness that we recognize ourselves to be free, freedom being, as we have seen, a sine qua non of morality.⁵⁷ Kant puts it like this: freedom is the ratio essendi of the moral law; the moral law is the ratio cognoscendi of freedom.⁵⁸ A revealing passage in Critique of Practical Reason highlights how Kant views this: Suppose that someone says his lust is irresistible when the desired object and opportunity are present. Ask him whether he would not control his passion if, in front of the house where he has this opportunity, a gallows were erected on which he would be hanged immediately after gratifying his lust. We do not have to ⁵⁵ Kant (1933), A807/B835. ⁵⁶ Kant (1933), A554–5/B582–3. (Cf. Kant (1956), p. 101.) It is only right to point out that some of the parts omitted from this quotation make Kant’s own position look somewhat further removed from the Kantian position being presented in this essay; see also, in this connection, the crucial disclaimer at A551/B579, footnote. The Kantian position is, however, adopted by Adams, in Adams (1985): see esp. p. 25. ⁵⁷ See e.g. Kant (1933), A468/B496. ⁵⁸ Kant (1956), p. 4, footnote. Cf. Kant (1978), §76, passim, and Kant (1960), p. 45, footnote.

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guess very long what his answer would be. But ask him whether he thinks it would be possible for him to overcome his love of life, however great it may be, if his sovereign threatened him with the same sudden death unless he made a false deposition against an honourable man whom the ruler wished to destroy under a plausible pretext . . . That it would be possible for him he would certainly admit without hesitation. He judges, therefore, that he can do something because he knows he ought, and he recognizes that he is free—a fact which, without the moral law, would have remained unknown to him.⁵⁹

Kant thinks that we can recognize two kinds of motivation: that such as lust, whose motivating force depends on there not being anything stronger to compete with it; and that such as the desire to act out of duty, which can annul anything else, and thereby show the agent to be genuinely free.⁶⁰ How do we know that we can be motivated in the second kind of way? By reflecting on our sense of duty, which leads us to recognize that we ought to be so motivated. Our sense of duty is at the same time our sense of freedom. Of course, if I do in fact act in accordance with duty, it remains an open question what is actually motivating me.⁶¹ In trying to determine this I could take up the test implicit in the passage just quoted and ask myself whether I would have done the same thing had there been various other motivations in play. But that question might be desperately difficult, perhaps impossible, to answer—even for me (or especially for me). As Kant says, the man in his example will unhesitatingly admit that he could overcome his love of life in order to do what is morally right, but ‘whether he would or not he perhaps will not venture to say’.⁶² Still, there is a moral insight that enables us to recognize at least how an ideal agent would be motivated. And it is this which leads us to see how it is possible for us to be motivated, though it cannot help us to determine how we are in fact motivated—how far we fall short of the ideal. Grasp of this ideal is, in effect, our awareness of moral reality. Our ideal is the concept of a perfect individual who always acts according to reason. It serves as a paradigm, or archetype, against which we can measure ourselves and in the light of which we can direct our lives. In so far as we see ourselves as falling short of the ideal we also recognize ourselves to be morally blameworthy. It is here that we see again the crucial point of contact between Kant and Plato. Our ideal is a Platonic Idea, personified.⁶³ In Kantian terms, it is a concept of reason. It can have no direct application in the realm of space and time. (Thus although we ought to conform to our ideal, and therefore can, no experience could ⁵⁹ Kant (1956), p. 30. ⁶⁰ Cf. McDowell (1978). ⁶¹ Cf. Kant (1964a), pp. 65 and 74–5, and Kant (1956), p. 74. ⁶² This is where the three dots of ellipsis come in the quotation above; the emphasis is added. Cf. again Kant (1933), A551/B579, footnote. ⁶³ Kant (1933), A567–8/B595–6. Cf. A314/B370 ff. and Kant (1960), pp. 54 ff.

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ever furnish decisive evidence that anyone had succeeded in doing so.⁶⁴) But we can pit our ideal, and related Ideas, against what we do experience in the world of space and time. And it is precisely when we do this that we activate our awareness of moral reality. It is then that we are able to recognize that things, including ourselves, are not as they ought to be; and that we are (therefore) to blame. Aristotle is famously sceptical about this idealistic strain in Platonic thought: it is too rarefied, too abstract.⁶⁵ It also carries a commitment to the transcendent, which, though it plays a vital and energizing role in both Plato and Kant, is absent, in that form, from Aristotle. Kant, for his part, must rebut this Aristotelian scepticism by arguing that Ideas have a real practical effect in the here and now; that they are not just the figments of some utopian fantasy, with which we make longing and ineffectual comparisons. Unless he can do this, the objection to his conservatism will be not that it is cavalier, or dogmatic, or question-begging— after all, he is not in the business of conversion—but that it is idle. Kant tries to rise to this challenge. He argues that proper exercise of our freedom involves implementation of our Ideas. Indeed, our concept of freedom is itself an Idea. And Kant argues that being truly free, in other words putting reason to practical use, is acting under the Idea of freedom.⁶⁶ But that is not the only respect in which Ideas have practical import. In cases where we cannot ourselves freely implement them, or do anything, by ourselves, to realize them, they can nevertheless give us direction and sustenance—through hope. Our hope is precisely that the Ideas will be realized.⁶⁷ We hope, for example, that virtue will be rewarded by happiness. (Here the Idea is of some transcendent connection between virtue and its reward.) We might also hope that, even if it is not a necessary truth that all moral traps are self-inflicted—that is, even if the Kantian reasons for thinking that they are are inconclusive—nevertheless it is a truth. (Here the Idea is of some transcendent safeguard against a certain kind of tragedy.⁶⁸) But is this not after all question-begging? Elsewhere in his philosophical system Kant is prepared to admit that, as soon as we bring Ideas to bear on our thinking about the here and now, we skirt illusion and incoherence. We have an Idea of the unconditioned whole, for example, which we mistakenly take to be directly applicable to what we are given in experience; and this gives rise to various conflicting beliefs about the infinitude of the physical world.⁶⁹ Should he not

⁶⁴ Cf. Kant (1933), A551/B579; cf. also Kant (1960), p. 56. ⁶⁵ See Aristotle (1941e), bk 1, ch. 6. ⁶⁶ See Kant (1964a), pp. 115–16. Such implementation of Ideas even involves them in a kind of causality: see e.g. Kant (1933), A550/B578. Cf., as part of his trying to rise to the challenge, Kant (1970). ⁶⁷ See Kant (1933), ‘Transcendental Doctrine of Method’, ch. 2, §2. ⁶⁸ I try to say a little more about how hope fits in A. W. Moore (2019a), pp. 231–2. ⁶⁹ See Kant (1933), ‘Transcendental Dialectic’, bk 2, ch. 2, §7; see also Kant (1933), A298/B354–5.

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have said the same sort of thing about our moral Ideas, and perhaps dismissed our pretensions to freedom as . . . well, as wishful thinking? Kant’s response to this charge is to distinguish between the theoretical and the practical. (This is related to the distinction between a constitutive and a regulative use of our Ideas.⁷⁰) No questions are being begged because no questions are, at the theoretical level, being addressed. In implementing our Ideas, or in grounding our hopes in them, we are not trying to apply them (as we unsuccessfully try to apply our Idea of the unconditioned whole, or, for that matter, as we successfully apply our concept of an apple when we recognize something as an apple): we are using them to guide and regulate our practice. And if it is said that we have no theoretical right to acknowledge any such thing as ‘our practice’, with all its connotations of freedom, then Kant will simply concur.⁷¹ There is still room for scepticism, however, about whether we really can use our Ideas in this way. The charge of question-begging again looks less urgent than the charge of futility. Consider the following argument: Why our Ideas Can No Longer Be of Use to Us Let us focus on the central Idea of rational agency. There is no harm, and much force, in the thought that we might have realized it. But the retrospection in ‘might have’ is crucial. For one thing, as was suggested earlier, it is possible that we are locked in a perpetual moral trap which precludes any further rational agency. But in any case, even if that is not so, the fact that, in Kant’s developed metaphysics, rational agency is ultimately atemporal means that any past wrong-doing on our part is already decisive. It already shows that we are fundamentally irrational (out of control). Even if we go on to act in accordance with duty, that cannot prevent the comparison of our lives with the Idea from being to the complete and lasting detriment of the former. We are beyond redemption. There can be no hope; there can only be contrition. I have tried to discuss this argument and Kant’s response to it elsewhere.⁷² Briefly, Kant thinks that our Ideas can continue to guide us (not just that they could have done) because he thinks that our past wrong-doing can be overcome. We can atone for our own past sins. But we can do so only if we have an infinite afterlife, because there is a kind of infinite gap to be made up. So if we are still to be guided by an ideal of rational agency, we must act as if we were immortal. Immortality is, in Kant’s phrase, a postulate of pure practical reason.⁷³ This is a striking view, not least because of what it reveals of Kant’s curious and ambivalent relationship to Christian orthodoxy. The grip that the orthodoxy has ⁷⁰ Cf. section 1 of this essay. Kant discusses the regulative use of our Ideas in e.g. Kant (1933), ‘Transcendental Dialectic’, bk 2, ch. 2, §8. ⁷¹ See Kant (1956), pt I, bk 2, ch. 2, §7. ⁷² A. W. Moore (1988). ⁷³ Kant (1956), p. 126.

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on him is never in question. It is certainly never questioned by Kant himself. Yet in his reluctance to admit that we are beyond self-redemption—redemption by our own efforts—he shows himself unable to come to terms with a cardinal (in its way, the cardinal) doctrine of Christianity.⁷⁴ It is notable how much of the position that I have been presenting in this essay as Kantian is also very Pauline. Consider the following familiar quotation from Paul’s letter to the Romans: I do not . . . acknowledge my own actions as mine, for what I do is not what I want to do, but what I detest. But if what I do is against my will, it means that I agree with the law and hold it to be admirable. But as things are, it is no longer I who perform the action, but sin that lodges in me . . . The good which I want to do, I fail to do; but what I do is the wrong which is against my will; and if what I do is against my will, clearly it is no longer I who am the agent, but sin that has its lodging in me. . . . In my inmost self I delight in the law of God, but I perceive that there is in my bodily members a different law, fighting against the law that my reason approves and making me a prisoner under the law . . . of sin . . . In a word then, I myself, subject to God’s law as a rational being, am yet, in my unspiritual nature, a slave to the law of sin.⁷⁵

For all that this is very Kantian, Paul’s response to his predicament is radically unKantian. In two verses omitted from the quotation above he writes: ‘Miserable creature that I am, who is there to rescue me out of this body doomed to death? God alone, through Jesus Christ our Lord!’⁷⁶ True, we can imagine Kant supplying an interpretation of this which would enable him to agree with it. But like so many of Kant’s interpretations, it would have to be pretty drastic and quite unlike what Paul intended. He would have to insist that Jesus Christ our Lord stood for the Idea of rational agency deep within us—so that, in the last analysis, the only person who can ‘rescue me out of this body doomed to death’ is I myself. That I should need help in this respect—that I should so much as be able to receive it— is precisely what Kant cannot accept. This is the stance from which much of this discussion originated: Kant refuses to acknowledge any such thing as good moral luck. We have come full circle.

⁷⁴ Cf. B. Williams (2006i), p. 195. ⁷⁵ Romans 7: 15–25. Kant comments on this passage in Kant (1960), pp. 24–5. Earlier, vv. 7–11 provide a fascinating critique of how the law of sin is able to get the upper hand in me; cf. Genesis 2: 15–17 and 3: 1–7. Later in Paul’s letter, 8: 18–25, we see how important hope is for him too. R. M. Hare makes much of the quoted passage in his discussion of ‘backsliding’ in Hare (1963), §5.1. Paul’s psychological model is famously rejected by Donald Davidson in part I of Davidson (1980b). ⁷⁶ Romans 7: 24–5.

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Christian orthodoxy commits one to a belief in good moral luck. Of course, that is a hopelessly misleading way of putting it. I would refer back to my earlier caveat about ‘moral luck’ serving as a technical term in this essay, except that I am now straying somewhat even from that usage. For it is not exactly that we are to be morally praised for anything beyond our control; in a way, Christian hostility to that kind of moral luck runs as deep as any other. But there is salvation, and indeed justification, through the death of Christ. Such is God’s grace. For our part, we must accept the grace through faith. Here is Paul again: ‘There is no condemnation for those who are united with Christ Jesus, because in Christ Jesus the lifegiving law of the Spirit has set you free from the law of sin and death.’⁷⁷ The Christian message is a remarkable one, for believer and unbeliever alike. It is that we can be cleansed of our sin and, through a divine gift, become as if blameless. The fundamental point of divergence between Kant and Aristotle, which was also the point of departure for this essay, thus turns out to have its counterpart (however assiduously Kant may try to hide it) in a fundamental point of divergence between Kant and the orthodox Christian. Kant cannot bring himself to accept the traditional doctrine of grace, because he cannot bring himself to accept that, at this most fundamental level, we could ever be the beneficiaries of good luck. He does concede, in some characteristic manoeuvring in the final section of Religion within the Limits of Reason Alone,⁷⁸ that there is ‘good we can do only with supernatural assistance’; and he is prepared to admit that God can, incomprehensibly, ‘perfect us’. But he will not yield his uncompromising insistence on our own virtuous efforts. In the final sentence of the book he writes, ‘The right course is not to go from grace to virtue but rather to progress from virtue to pardoning grace.’ In other words, for Kant—as indeed for anyone whose rationalism is as inveterate as his—the Christian message is, in the end, that bit too remarkable.⁷⁹

⁷⁷ Romans 8: 1–2, emphasis added. ⁷⁸ Kant (1960). ⁷⁹ For further discussion of the tension between Kant’s position and orthodox Christianity see Vossenkuhl (1987–8).

10 On There Being Nothing Else to Think, or Want, or Do Abstract The basis for this essay, which originally appeared in a volume of essays in honour of David Wiggins, is Wiggins’s idea of there being nothing else to think. It is argued that, if we construe Wiggins’s idea in such a way that there being nothing else to think but that p is equivalent to its being true that p, then it involves the following subsidiary idea: if it is true that p, then anyone who does not think that p pays a price. A proposal is made concerning what this price is, namely that of having at most a ‘conditioned’ thought about whether or not p, where a conditioned thought is defined in such a way that any such thought is always liable to be unsettled by critical self-conscious reflection. This proposal is then extended to each of volition and agency, thereby giving substance to the idea of there being nothing else to want but that p and the idea of there being nothing else to do but x. The role played by truth in the original proposal is (roughly) played by rightness in the proposal concerning volition and by categorical requirement in the proposal concerning agency. In the final section of the essay some sketchy remarks are made about how these proposals could combine with the idea that human beings are finite but have an aspiration to be infinite, to explain the value of the true, the right, and the categorically required.

1. The Idea in Wiggins of There Being Nothing Else to Think Four lines up on page 127 of ‘The Sense and Reference of Predicates: A Running Repair to Frege’s Doctrine and a Plea for the Copula’,¹ there is a sentence by David Wiggins which has the air of self-parody. It opens with a conjunction; it is annotated twice; it contains one pair of dashes and four pairs of parentheses (in one case nested); and it runs to 215 words. I mention it partly out of a sense of mischief but partly also because its characteristic excesses call to mind everything that makes Wiggins (in fact) such a pleasure to read, as well as a wonderfully

¹ Wiggins (1984).

The Human A Priori: Essays on How We Make Sense in Philosophy, Ethics, and Mathematics. A. W. Moore, Oxford University Press. © A. W. Moore 2023. DOI: 10.1093/oso/9780192871411.003.0011

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rewarding philosopher to study: his painstaking attention to detail; his delicacy of touch; his feel for instructive and salutary examples; the unhurried and unflustered way in which he describes familiar phenomena, injecting just enough theory here or there to cast philosophical light on them, never so much as to distort them; and the way in which he manages, through all the convolutions, to keep track of what is important. People say that he is hard to pin down. He is. But I cannot help thinking that he is hard to pin down not because he is so vague, but because he is so precise. I think his writing skills, and with them his unique brand of philosophical acumen, are especially well suited to his work in ethics. And it is an idea that has cropped up a few times in his work in ethics that I want to use as a basis for this essay. There will not be much exegesis. I want to remould the idea, then build on it. I hope that Wiggins, with his philosophically inquisitive instincts, will approve. If I end up with something quite different from what he originally had in mind, perhaps this will serve to illustrate the depth in his writing. If so, then I shall be satisfied that I have provided a fitting tribute. The idea (incongruous, some would say, in the context of such prodigious sentences, with their subtle qualifications and their labyrinthine structure) is that of there being nothing else to think. Here are two quotations: [Sometimes we can say that] there is nothing else to think . . . [And] sometimes, when we do, it will be possible for someone else to say that we think what we think (and that some of the others who think this may think this) not accidentally, but precisely because there is nothing else to think—with a ‘because’ that simultaneously vindicates and, by vindicating, explains.² [Suppose] one comes to believe that p precisely because p . . . [Then] the best full explanation of one’s coming to believe that p requires the giver of the explanation to adduce in his explanation the very fact that p. What follows from this is that the explanation will conform to the following schema: for this, that and the other reason (here the explainer specifies these), there is really nothing else to think but that p; so it is a fact that p; so, given the circumstances and given the subject’s cognitive capacities and opportunities and given his access to what leaves nothing else to think but that p, no wonder he believes that p.³

Now the example that Wiggins adduces most frequently is an arithmetical one: there is nothing else to think but that 7 + 5 = 12. This, combined with the fact that his ultimate concern is with ethics, suggests that he is trying to draw a boundary around the non-empirical (or the non-contingent). It suggests

² Wiggins (1987e), p. 348, emphasis removed. ³ Wiggins (1990–1), p. 66. Cf. Wiggins (1987c), §7; Wiggins (1987d), §10; and Wiggins (1991), §9.

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that he is trying to formulate something that sets arithmetical truths and ethical truths apart from those of geology, say, or a truth about where someone has left his spectacles. This in turn would chime with an interpretation of the sentence ‘There is nothing else to think but that 7 + 5 = 12’ whereby it meant that having any other view about whether or not 7 + 5 = 12 was literally impossible—and in the most stringent sense. For, arguably, though one can have all sorts of contingently mistaken thoughts, one cannot think something that is itself impossible. This idea has in fact had wide currency, in different forms. Wittgenstein, in the Tractatus, writes that ‘whatever is thinkable is possible too’.⁴ Quine writes that ‘when [a deviant logician] . . . tries to deny [a doctrine of logic] . . . he only changes the subject’.⁵ True, there is, in each case, a complicating subtext. Wittgenstein’s views make him equally hostile to the idea of thinking something necessary,⁶ and Quine is sceptical about the very idea of a necessary/contingent distinction.⁷ Still, it is not hard to envisage an argument to the effect that a necessity, unlike a contingency, is such that nothing would strictly count as believing its negation (or perhaps even as doubting its truth). And that would yield one particularly strong interpretation of the formula ‘There is nothing else to think but that p.’ It is clear, however, not least from the second of the two quotations above, that this is not what Wiggins intends. He wants the formula ‘There is nothing else to think but that p’ to be interpreted in a much weaker sense, and to have a much wider application.⁸ Furthermore, he intends it indexically.⁹ While there may be nothing else to think about a certain issue, here and now, there might once have been and there might still be in circumstances in which the issue presents itself in a murkier light and less is known about it. Perhaps, then, Wiggins intends the formula to apply, in any given context, not to simple necessities, but to the necessary consequences of whatever shared background knowledge can be presupposed in that context? That is nearer the mark. It would mean that Wiggins was essentially concerned with the drawing of conclusions, or with the ruling out of hypotheses. In fact, however, his idea is more subtle than that. The presupposed shared background is not just a range of propositions from which other propositions follow. It also includes commitments; sentiments; ways of understanding; canons of rationality determining what hangs together with what. In saying

⁴ Wittgenstein (1961), 3.02; cf. 5.4731. ⁵ Quine (1970), p. 81. ⁶ Wittgenstein (1961), 2.225–3.001. ⁷ Quine (1970), p. 9. ⁸ See e.g. Wiggins (1990–1), p. 68 n. 7. ⁹ Supplementary note: In the volume in which this essay originally appears, Essays for David Wiggins: Identity, Truth and Value (Lovibond and Williams (1996)), there is a response by David Wiggins in which he explains that this is a misinterpretation and that he does not intend the formula indexically: see p. 272. Fortunately, however, for reasons that Wiggins himself indicates, this does not have a significant bearing on anything I say in this essay.

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that there is nothing else to think about an issue but that such and such, one is presenting anyone who does not think that such and such with a challenge. The challenge is to come up with an alternative story, in full and coherent detail, without at any point straining intolerably against the bounds of credibility that must be acknowledged by anyone aspiring to keep a sane grip on the situation.¹⁰ There is much more to be said about this. But I want now to indulge in my first bit of remoulding. I want us to reconstrue Wiggins’s idea non-indexically. For whenever the formula correctly applies, it is true even outside the context that not having the thought in question means paying a price: in particular it means not assimilating, perhaps not being able to assimilate, whatever in the context makes having the thought mandatory. In so far as a single schematic account can be given of this,¹¹ there is a viable, elliptical, context-independent reading of ‘There is nothing else to think but that p’: there is nothing else to think that does not involve paying the price. The propositions to which the formula then applies are precisely those which, given a suitable presupposed shared background of the sort indicated above, come within the ambit of Wiggins’s conception. Viewing the matter in this way will, I think, direct us back to his own main interest in introducing the formula. Which, then, are these propositions? They are those whose acceptance can be given the kind of vindicatory explanation schematized in the second quotation above. (Both quotations, along with many other passages, reveal the direct connection that Wiggins recognizes between applying his formula and giving such an explanation.) Let us call such propositions ‘objectively true’. The question immediately arises whether objective truth, on this understanding, is (simply) truth. This is another of those issues on which Wiggins is maddeningly but justifiably hard to pin down. But we can be clear about two things. First, it is the ordinary, familiar, mundane concept of truth, in its full generality, that ultimately interests him. (One of his primary aims is to discover whether there is such truth anywhere in the realm of ethical thought.¹²) And secondly, he regards objective truth as at the very least the kernel of that concept.¹³ We shall not stray far from his concerns, then, and we shall in any case start down an interesting avenue of exploration, if we adopt the following policy: to treat the formula ‘There is nothing else to think but that p’ (on the context-independent reading of it¹⁴) as equivalent to ‘It is true that p.’ Hence any attempt to define it, or to explicate it, or to expand on it, must issue in a formula ‘T(p)’ such that every instance of the schema

¹⁰ ¹¹ ¹² ¹⁴

Cf. Wiggins (1990–1), §12. . . . which is not to say that ‘a price’ has wider scope than ‘whenever’, as I hope will become clear. See e.g. Wiggins (1990–1), pp. 64–5. ¹³ See e.g. Wiggins (1990–1), pp. 65 ff. Henceforth this qualification will be taken for granted.

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It is true that p $ T(p)¹⁵

is true.¹⁶ Here is another way of putting the same point. If we say that the formula ‘There is nothing else to think but that p’ denotes those propositions to which it applies, then our policy will be so to understand the formula that it denotes, quite simply, truths. Hence we must rule out a stringent interpretation whereby it denotes necessities. And, at a different extreme, we must rule out a psychological interpretation—‘It is psychologically impossible to think anything other than that p’— whereby the propositions denoted might include some falsehoods.¹⁷ Either of these, combined with an argument that the formula applied within the realm of ethics, would have opened up some interesting possibilities. The first would have opened up the possibility that (certain) apparent disagreements in ethics must really be linguistic disputes or disagreements about non-ethical facts. The second would have opened up the possibility that we are so psychologically constituted that we cannot face (certain) truths about how to live. But neither of these is an idea that Wiggins has bequeathed to us; and neither has anything especially to do with our understanding of the formula. Now in order to consolidate that understanding we must say some more about the ellipsis. What is the price that someone pays for not thinking the truth? This question can be turned round. For in saying that, when things are thus and so, anyone who does not think that things are thus and so pays a price, we are in effect saying that, when things are thus and so, anyone who satisfies a certain condition has no choice but to think that things are thus and so.¹⁸ The question can therefore be put in this form: what is this condition? Or rather: what is such a condition? (It is clear, or it should soon become clear, that more than one condition fills the bill.) Putting the question this way round can give shape to the project of specifying ‘T(p).’ We first assume that ‘T(p)’ takes the following form: (2)

Any thinker who satisfies condition C is bound to think that p.

¹⁵ Here and hereafter I use standard logical notation. ¹⁶ One might think that there were two ways in which the left-hand side of an instance of this schema could be false: as a result of ‘p’ being replaced by something false; and as a result of ‘p’ being replaced by something neither true nor false. The latter, at least if we prescind from issues of grammaticality, would raise some fascinating questions about how far (and how) there might be alternative things to think when truth was not involved, questions that are certainly pertinent to Wiggins’s concerns. However, I am simply going to bypass those questions by stipulating that ‘p’ must be replaced by a proposition, or by something that stands for a proposition; and I shall understand by a proposition something that is true or false. ¹⁷ Cf. where Wiggins distances himself from a certain kind of relativism in Wiggins (1987e), p. 348. ¹⁸ This is not, of course, to embrace the kind of relativism from which Wiggins wants to distance himself (see previous note), nor to disregard the difference upon which he thereby insists between saying that there is nothing else to think and saying that there is nothing else for us (or for me) to think.

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We then put our efforts into specifying the condition.¹⁹ I shall not myself defend any particular specification of it. I want to operate at a higher level of generality than that. But it is important for me to say something about the constraints that must be met if the project is to be carried out in a satisfactory and philosophically illuminating way. I list three such constraints. (i)

The guarantee that every instance of (1) is true must be non-trivial. We would be violating this constraint if we replaced ‘satisfies condition C’ by ‘knows whether or not p’.

Comment: ‘Non-trivial’ is a vague and slippery term, and (i) itself is correspondingly vague and slippery, as well as being context- and purpose-relative. But very roughly, how non-trivial the guarantee is will be inversely proportional to how quickly and with how few repercussions we are able to satisfy ourselves that we have it. I shall not try to say any more than that, beyond issuing the following warning. Nontriviality is quite separate from non-circularity. We would be involved in some sort of circle if, for example, we specified a condition that used the concept of truth. Whether we wanted to avoid that would depend on what our purposes were. If we were trying to analyse truth, we would want to avoid it. But, as Wiggins has continually reminded us, if we were trying to elucidate truth—as indeed he is—we would not.²⁰ (ii)

The guarantee that every instance of (1) is true must be (to the extent that we are prepared to think in these terms) a priori. We would be violating this constraint if, having discovered some oracle, we replaced ‘satisfies condition C’ by ‘arrives at a belief about whether or not p by consulting the oracle’.

¹⁹ But we should beware here of a hasty drive to formalization. It would be natural to cast ‘T(p)’ as follows: (2*) 8S 8t [X(S, t, p) ! S thinks at t that p], where ‘S’ ranges over thinkers, ‘t’ ranges over times, and ‘X(S, t, p)’ picks out the relevant condition. Cf. the formula in Wright (1988), p. 18. (Not that I intend any criticism of Wright in what I am about to say. He is certainly sensitive to the kind of problem that I raise in this footnote: see Wright (1988), p. 14, n. 26.) (2*), however, lacks the element of necessity in (2)’s ‘bound to’—in a way that matters. It leaves us at the mercy of false propositions with respect to which no thinker ever satisfies the condition, something which, pending an account of what the condition is, we must acknowledge as a possibility (unless, say, we believe in a supreme being who constantly satisfies the condition with respect to every proposition). In such a case, given that ‘!’ is purely truth-functional, the relevant instance of (2*) is true, and that of (1), therefore, false. On the other hand, we cannot reinstate the necessity by putting (2*) within the scope of a necessity operator. Given any reasonable elaboration of ‘X(S, t, p)’, that leaves us at the mercy of contingently true propositions, for which the relevant instance of (2*), thus supplemented, is false, and that of (1), therefore, again false. With sufficient ingenuity we could probably circumvent these difficulties. But we might as well stick with the informal version (2). What we must do is to take note from these efforts at formalization that (2) is not just an abbreviation either of (2*) or of its necessitation. ²⁰ See e.g. Wiggins (1987c), p. 142 n. 4, and Wiggins (1987d), p. 188 n. 4. (Cf. Johnston (1989), pp. 147–8, and also Wittgenstein (1961), 3.263.)

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Comment: The reason for (ii) is simply that the illumination we seek is philosophical illumination.²¹ (iii)

The specified condition must be such that everyone has a reason to satisfy it (or to satisfy any instance of it, subject to having a view about the matter in hand). We would be violating this constraint if we replaced ‘satisfies condition C’ by ‘has been sentenced by God to die as soon as she acquires a false belief, and has acquired a belief about whether or not p’.

Comment: The reason for (iii) is that we need to do justice to the idea that anyone who does not satisfy the condition, and who may thereby think something false, pays a price. This in turn reminds us of the alternative route into the main project, which I shall call the ‘contrapositive’ route: to focus on what someone must fail to be, or must fail to do, in order not to have the belief in question. Let me gesture towards four possible ways of carrying out the project, each of which seems to meet these constraints. (But note: although different, they are not incompatible.²² The constraints still do not force a unique answer.) (a) We could stay as close as possible to the genesis of our context-independent understanding of Wiggins’s formula. This means turning to the kind of thing whose assimilation, whenever it is available, makes having a true thought mandatory. And this in turn suggests replacing ‘satisfies condition C’ by some suitable embellishment of ‘can recognize the reasons for or against its being the case that p, attends to them, and forms a belief about whether or not p on that basis’. This would certainly fit with what Wiggins says in the second quotation above, where he talks about a subject’s ‘cognitive capacities’, ‘opportunities’, and ‘access to what leaves nothing else to think’.²³ (b) We could take a leaf out of Peirce’s book. We could replace ‘satisfies condition C’ by ‘has a belief about whether or not p that is “fated to be ultimately agreed to by all who investigate” ’.²⁴

²¹ But I concede that the matter is much more complex than this suggests. See Wiggins’s admonishment at the bottom of p. 79 of Wiggins (1990–1). ²² Nor are they by any means exhaustive. Any familiar theory of truth could be pressed into service here. We could replace ‘satisfies condition C’ by some embellishment of ‘has a belief about whether or not p that corresponds to reality’. Or we could replace it by some embellishment of ‘has a complete and coherent set of beliefs (about such and such a subject)’. ²³ Later (Wiggins (1990–1), p. 71) he makes comments that are more pertinent to the contrapositive route. He suggests that, sometimes anyway, thinking something false may mean ‘[opting] out altogether from the point of view that shall be common between one person and another’. This too might lend itself to suitable generalization and embellishment. ²⁴ Cf. Wiggins (1987b), p. 120. The phrase in quotation marks is extracted from the passage by Peirce that Wiggins himself quotes on the same page, n. 34.

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(c) We could take a leaf out of Ramsey’s book, as adapted by Hugh Mellor. We could replace ‘satisfies condition C’ by some suitable embellishment of ‘has a belief about whether or not p that, in conjunction with all his other beliefs, makes his desires cause actions that succeed in achieving what is desired’.²⁵ But more interesting than any of these, for my purposes, would be to explore some variation on the theme of the unconditioned. I deliberately use that word with its Kantian resonance. I have in mind a quasi-Kantian Idea,²⁶ a rational idealization of what we took from Wiggins in (a). An unconditioned thought would be a thought whose (objective) explanation and (subjective) vindication converged. It would be a thought whose best explanation included the fact that the subject’s own reason for having it, or continuing to have it, was a result of rational self-conscious reflection on its best explanation.²⁷ That this is an uncompromising ideal can be made clear by considering how a familiar kind of justified true thought will fail to exemplify it. Wiggins writes that all the children in his son’s class at school think that 7 + 5 = 12; and he says that the best explanation for their thinking this is that ‘(i) as can be shown by the use of calculating rules (and could in the end be rigorously demonstrated), it is a fact that 7 + 5 = 12 . . . [and (ii)] they are going by the calculating rules’.²⁸ I doubt this. At least I think I doubt it. A lot depends on how exactly Wiggins’s proposed explanation is to be expanded. But I suspect that these children first acquired the belief that 7 + 5 = 12, and have since retained it, less because of arithmetical reflection and more because of inculcation than a straightforward reading of Wiggins suggests. At any rate, there will be elements in the best explanation of their belief that play no part in informing any self-consciously formed reason they have for it. It would be absurd to call a typical 9 year old’s thought that 7 + 5 = 12 unconditioned, in the sense defined. (Wiggins could agree.) The same is true, I submit, of most of our thoughts. Unconditioned thought is a very demanding standard. Its complement, conditioned thought, is correspondingly commonplace. But one feature of conditioned thought is that critical self-conscious reflection is always liable ²⁵ Mellor (1988–9), p. 86. (This proposal satisfies the third constraint particularly clearly. To see why the others satisfy it one must first see why there is reason to believe what is true.) ²⁶ For the Kantian use of the term ‘Idea’, see Kant (1933), A312–20/B368–77 and A409/B435. ²⁷ The ‘loop’ in this definition is important. (It means that the definition could just as well have been turned inside out; an unconditioned thought would be a thought such that the subject’s own reason for having it, or continuing to have it, was a result of rational self-conscious reflection on its best explanation, where this explanation included the fact that that was what the subject’s own reason for having it, or continuing to have it, was a result of.) This loop is designed to ensure that the objective and the subjective are fully brought together. Cf. (but only cf.) certain currents of thought in Spinoza (1959), e.g. pt III, prop. 1, and pt V, prop. 6. Note: nothing in the definition guarantees that an unconditioned thought shall be responsive to ‘the facts’. It may be the result of a creative act of will on the part of what Kant would call an infinite being: see Kant (1933), B72 and B138–9. Indeed there are elements in Hegel that suggest that this is what it must be: see e.g. Hegel (1975), 4. 2, esp. §§44 ff. (Maybe there are such elements in Kant too: see again Kant (1933), B72 and the surrounding material.) ²⁸ Wiggins (1990–1), pp. 67–8.

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to unsettle it. For it is always liable to locate something in the explanation of the thought—some psychological, sociological, or cultural factor—whose discovery militates against the subject’s own original reason for having the thought (however mildly, however briefly, and however easily that reason can be reinstated).²⁹ Unconditioned thought, on the other hand, is already rationally reflective, which means that it is, by its very nature, impervious to critical self-conscious reflection. Now: (d) We could replace ‘satisfies condition C’ by ‘has an unconditioned belief about whether or not p’.³⁰ Here no doubt someone will protest that we would thereby be violating a fourth constraint that ought to have been mentioned earlier. The constraint is that the specified condition must be one that thinkers stand a good chance of satisfying, and typically do satisfy, otherwise we shall be departing from Wiggins’s original concern with actual explanation. But I do not share that concern, at least not in the same form. I have already indicated that I want to remould and build on Wiggins’s idea. I am interested in certain ideals of rationality which may thereby be illuminated. This interest in turn connects with an interest in parallel projects for volition and agency. It is to these that I now turn.

2. Counterparts of Wiggins’s Idea for Volition and Agency People certainly use formulae like ‘There is nothing else to hope for but that p’, ‘There is nothing for it but to do x’, ‘There is no other way to live but y’ (sometimes with ‘now’ or ‘any longer’ inserted at an appropriate point).³¹ There are numerous questions about how instances of these formulae would ordinarily be interpreted— what implicit relativization, for example, would naturally be assumed. But I want to ²⁹ Bernard Williams has argued (i) that reflection can locate something whose discovery actually prevents the thought from any longer being had, and (ii) that this can happen even when the thought in question constitutes knowledge: see B. Williams (2006i), pp. 148 and 167–9. I disagree with (i). (That is, I disagree if the thought in question is a genuine one, capable of being assessed as true or false. Of course reflection can expose incoherence in the case of a putative thought.) But I do not mind admitting a counterpart of (ii). That is, I do not mind admitting that reflection can unsettle knowledge in the way I am envisaging: see A. W. Moore (1991), appendix. [Supplementary parenthesis: For a subsequent more sympathetic discussion of Williams see A. W. Moore (2003b), in particular n. 20. See also Essay 12 in this volume. And for the significance of this change of view see the Introduction to this volume, the opening unnumbered section.] Cf. here the material from Peirce quoted by Wiggins in Wiggins (1987e), p. 342, and Wiggins’s own surrounding text. ³⁰ Is there a problem here if unconditioned thoughts include the creative acts of will of an infinite being, or indeed only include these (see n. 27 of this essay)? For such a being would have the power to decree that things stand differently from how they actually stand with respect to the proposition in question. In fact, this is not a problem. Cf. n. 19: the resultant schema is to be understood in such a way (and can be understood in such a way) that we may disregard these unrealized possibilities. ³¹ See e.g. Wiggins (1987a), end of §14.

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put these questions to one side. I shall simply appropriate the two formulae ‘There is nothing else to want but that p’ and ‘There is nothing else to do but x’ as (contextindependent) analogues of ‘There is nothing else to think but that p’, in a quest parallel to the one that we have just been inchoately conducting. My question is this. Is there any way of replacing ‘satisfies condition C’ in either (2w)

Any wanter who satisfies condition C is bound to want that p

(2a)

Any agent who satisfies condition C is bound to do x,

or

which, satisfying (counterparts of) the three constraints outlined above, renders every instance of a suitable analogue of (1) true; and, if so, what is the analogue (or more specifically, what is on its left-hand side)? I am particularly interested in what we might learn from exploring further variations on the theme of the unconditioned. (So my quest, though parallel to the earlier one, runs in the opposite direction.) What would it mean to say that a want, or an act, was unconditioned? Following the account given for thoughts, we can say that an unconditioned want is a want whose best explanation includes the fact that the subject’s own reason for having it, or continuing to have it, is a result of rational self-conscious reflection on its best explanation; while an unconditioned act is an act such that the best explanation for why the agent performs it, or continues to perform it, includes the fact that the subject’s own reason for performing it, or continuing to perform it, is a result of rational self-conscious reflection on that best explanation. And, as in the case of thoughts, we are led to the idea of wants, acts, habits, and suchlike that are by their very nature impervious to critical self-conscious reflection. All of this raises some very large questions. Where, for instance, are we allowed to pin the label ‘rational’? Can we talk about ‘rational wants’? How and why can recognizing certain constraints and influences on one’s habits mitigate them? How do wants that can withstand critical scrutiny interact with, or relate to, their cognitive counterparts? Does it make sense to talk of ‘false pleasures’?³² What does any of this have to do with Kant? Or with Aristotle? Some of these questions we shall come back to. But my immediate concern is this. With such an account in place, what does it mean to replace ‘satisfies condition C’ in (2w) with ‘has an unconditioned want as to whether or not p’, or in (2a) with ‘does x, or refrains from doing x, in an unconditioned way’? An extreme reaction would be to say that it means nothing; or rather, that in each case the result is a formula every one of whose instances is false. For a want,

³² See Lovibond (1989–90). Cf. Plato (1961f), 585d–587a.

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and mutatis mutandis an act, is not answerable to anything except the wanter; and no wanter is bound to want anything. But borrowing from Kant, and also no doubt from Aristotle, we have the wherewithal to construct a more interesting answer, as follows. (I shall put it in terms of wants. But it can readily be extended to acts.) There are certain norms of rationality to which any want is answerable.³³ But some wants, given their content, cannot answer to these norms. So some wants, given their content, cannot survive rational self-conscious reflection, let alone rational self-conscious reflection on what explains them. An unconditioned want as to how things should be in such a case would therefore be bound to have the opposite content. Some instances of the formula, then, are true. In our abbreviated form: there is nothing else, in such a case, to want. As for what this comes to, let us be bold and try out the equation of ‘There is nothing else to want but that p’ with ‘It is right that p’ or ‘It is wrong that ¬ p’ (as the case may be); and ‘There is nothing else to do but x’ with ‘There is a categorical imperative to do x’. These proposals, together with the original proposal concerning ‘T(p)’, I shall refer to as the Core Proposals. The Core Proposals ‘There is nothing else to think but that p’ is equivalent to ‘It is true that p.’ ‘There is nothing else to want but that p’ is equivalent to ‘It is right that p’ (if ‘p’ is true) or ‘It is wrong that ¬ p’ (if ‘p’ is false). ‘There is nothing else to do but x’ is equivalent to ‘There is a categorical imperative to do x.’³⁴ Before we assess the Core Proposals it will be worth ranging them against a strain in Kant’s thinking whereby unconditioned agency is the only true agency: anyone who does not obey a categorical imperative does not act rationally, and so does not act freely, and so does not, in the true sense of the word, act.³⁵ I do not ³³ If a wanter is essentially rational, this may not be incompatible with saying that the wanter’s wants are not answerable to anything except the wanter. There is in any case the question, to which we shall return, of what reason a wanter has to have wants that can answer to whatever they are answerable to. For the idea of wants that can answer to norms of rationality see further Spinoza (1959), pt IV, prop. 61. ³⁴ For the idea of there being nothing else to want (or to will) cf. Kant (1964a), pp. 91–2. Note that Kant says that there are two ways in which willing that things be thus and so can be literally, and nonelliptically, impossible: this is reminiscent of the extreme interpretation of Wiggins’s formula discussed near the beginning of section 1 and it connects with what I am about to say in the main text. As for my introducing the categorical imperative at this point, that connects with the idea that an unconditioned act would be an act whose vindication and explanation converged, which in turn seems to be what Kant has in mind when he talks about an act done from duty, as opposed to an act done merely in accordance with duty (the former would be bound to conform to any categorical imperative, the latter would only happen to); see Kant (1964a), p. 65. For a contemporary discussion of (if not the Core Proposals, then) proposals that have the same shape as the Core Proposals see Smith (1989), Lewis (1989b), and Johnston (1989). ³⁵ Cf. Essay 9 in this volume, section 2. Note: to say that someone, in acting, does not obey a categorical imperative may mean either that they disobey a categorical imperative or that they do something outside the jurisdiction of reason altogether.

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know of anywhere in Kant’s writing where he explicitly commits himself to this. Indeed there are places where he explicitly repudiates it.³⁶ But it may well be implicit in other things he says.³⁷ And it has the interesting consequence for the Core Proposals that, if there is a categorical imperative to do something, then there is literally, and non-elliptically, nothing else to do: any apparent instance of an agent doing something else shows the agent to be at the mercy of some passion (say) and out of rational control.³⁸ To put the same point somewhat differently, the proposed way of replacing ‘satisfies condition C’ in (2a)—‘does x, or refrains from doing x, in an unconditioned way’—contains a pleonasm.³⁹ There might even be a radical conception of thought along the same lines, whereby nothing less than an unconditioned thought counts as a thought; and similarly for wants. On that conception, we could revert to the extreme interpretation of ‘There is nothing else to think but that p’, discarded towards the beginning of section 1, and still see the formula as denoting truths, not just necessities. But to return to the main issue: how are we to assess the Core Proposals? Are there such links between the unconditioned (on the one hand) and the true, the right, and the categorically required (on the other)? Our philosophical heritage is replete with attempts to establish links of precisely this kind. But there is an obvious problem for such attempts. There is a circularity that threatens to make substantiation impossible. We cannot understand rationality, as it occurs in the definition of the unconditioned, except in terms of the true, the right, and the categorically required. (Likewise, conversely, we cannot understand what prevents certain conditions on a person’s thinking or wanting or acting—such as the weight of received opinion, the power of advertising, or the force of habit—from entering into a rational vindication, except in those same terms.⁴⁰) I commented in section 1 on the important difference between circularity and triviality. But I also commented that whether we want to avoid circularity depends on our purposes. If we are interested in putting these links to work in settling actual disputes, then we shall, it seems, want to avoid it. It is in that sense, in the sense that we seem not to

³⁶ E.g. Kant (1956), p. 32. ³⁷ See e.g. Kant (1933), A538–41/B566–9; Kant (1933), ‘Transcendental Doctrine of Method’, ch. 2, §1; Kant (1964a), the beginning and end of ch. 3; Kant (1956), the beginning and end of the Introduction, and Pt I, bk 1, ch. 1, §6. (All that Kant strictly commits himself to, however, is that a free will is a will subject to its own rational laws. It does not follow that for the will to be exercised freely is for it to be exercised so as to conform with those laws. Note: the Willkür/Wille distinction, which is often invoked in discussion of these issues, is especially prominent in Kant (1960), but there is a much greater emphasis on it in the introductory essay by John Silber (Silber (1960)) than there is in Kant himself.) ³⁸ Cf. Romans 7: 15–25. And cf. n. 34 above. ³⁹ A further interesting consequence is that deliberation about how to act can be understood on the model of (typical) deliberation about how to avoid a catastrophe—not as a matter of choosing between different ways of doing it, but as a matter of trying to find any way of doing it at all. ⁴⁰ Cannot? I think not. I think that if we tried to establish canons of rational vindication that did not beg any questions, e.g. suitable canons of induction, our guarantee that every instance of (1) is true would no longer be a priori—if indeed every instance of (1) were true. Cf. parts of Johnston (1989).

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be able to put the links to this kind of work, that we seem not to be able to give them any substance. Suppose, for example, that there is a dispute about whether it is right that something is going on. What progress can we make by considering what somebody with an unconditioned want would want? That will very soon become a matter of determining whether the thing is right or not. Nor is it likely to help to consider people’s actual wants about the issue. If we agree that these are conditioned in various ways, that will cut no ice: it is not precluded that somebody should have a conditioned want for what is right. If, on the other hand, someone claims to have an unconditioned want that this thing should be going on, then those who think this impossible are unlikely to get anywhere just by pointing out how and why they take the want to be conditioned: she may well refuse to accept their diagnosis, because of what her opponents will see as a similarly conditioned false belief. This is typical of how the two cases of belief and volition will interact, as indeed both will with that of agency. Such interaction may enlarge the circle, but it scarcely makes it any the less problematical.⁴¹ This example shows how the circle is liable to be traced when the truth of a particular thought is in dispute. Wiggins has shown how essentially the same circle is liable to be traced when the status of a whole family of thoughts as true or false is in dispute (in other words, when there is dispute about their very status as thoughts). His special concern is with evaluative thoughts. He has argued that we cannot set standards for having a true evaluative thought, and in particular, presumably, we cannot expound on what it would be to have an unconditioned evaluative thought, except, question-beggingly, in evaluative terms. For whether somebody is equipped to be a sound judge of evaluative matters is itself an evaluative matter. As Wiggins says, ‘the criterion for a good judge is that he is apt to get things right’.⁴² The circle will be a problem, then, if we want to use any of the Core Proposals to settle actual disputes. And if the disputes are deep and radical enough, as they very often are in ethics, it will reinforce that distinctive sense of vertigo which makes us wonder about the very possibility of grasping the true, the right, or the categorically required. But, again quoting Wiggins, ‘whatever difficulties there are in the possibility of irresoluble substantive disagreement, no position in moral philosophy can render itself simply immune from them’.⁴³ That distinctive sense of vertigo may be there to be reckoned with anyway.

⁴¹ Cf. B. Williams (2006i), pp. 40 ff. ⁴² Wiggins (1987d): the quotation occurs on p. 194. Note: one of Wiggins’s aims is to explore analogies between evaluative concepts and other concepts that involve human sensibilities, e.g. colour concepts; but as he and other writers have emphasized, the circularity here highlights an important disanalogy. See §6 of his essay. Cf. various currents in Blackburn (1985); McDowell (1985), p. 118; Wright (1988), esp. §5; and Johnston (1989), p. 144. ⁴³ Wiggins (1987d), p. 210.

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There are other purposes relative to which the circle will pose no threat. And one of Wiggins’s greatest achievements is the brilliant way in which he has developed some of these.⁴⁴ If we are less interested in conversion than in understanding (both of ourselves and of others), then there is enormous value precisely in tracing such circles, elucidating our concepts as Wiggins would say,⁴⁵ trying to cast philosophical light (where others, in different ways, have cast so much literary light) on some of the impediments to rational self-conscious reflection— impediments to the integration of the objective with the subjective—which, according to the Core Proposals, allow us to think what is false, to want what is wrong, and to do what we are categorically required not to do. And in the grip of actual disputes, we can continue to trace these circles, not, admittedly, as a way of settling the disputes, but as a prelude to seeing how a healthy combination of conviction and open-mindedness might bring us to such a settlement.⁴⁶ Where does all of this leave the Core Proposals then? Well, circularity at any rate is no objection to their correctness.⁴⁷ But I shall not now go much further than this in my ground-level assessment of them. Instead I want to step back up a level, and return to the question of how these proposals (or any others) might satisfy (iii), the third of the constraints that I outlined earlier. This was the constraint that the specified condition in (2)—likewise in (2w) and (2a)—must be one that everyone has a reason to satisfy. Investigating this question will lend further support to the Core Proposals. But it will also, more importantly, give us insight into the nature of the overall project and thereby, I hope, help us to say something in response to a very old philosophical puzzle. The puzzle is to explain the special force that attaches to the recognition that something is true, right, or categorically required. How can such a recognition impinge on us in the distinctive way in which it does? Wittgenstein alludes to this puzzle in the Tractatus when he says, ‘When an ethical law of the form, “Thou shalt . . . ”, is laid down, one’s first thought is, “And what if I do not do it?”’⁴⁸ What we need, he says, is some account of the ‘punishment’ that ‘resides in’ not doing as one ought—some account of the price one pays.⁴⁹ Providing this will be an integral part of showing that (iii) is satisfied.⁵⁰ ⁴⁴ See e.g. Wiggins (1987d), §4. ⁴⁵ See n. 20 of this essay. ⁴⁶ Cf. Wiggins (1990–1), §§7–8; and Wiggins (1991), §5. ⁴⁷ Again see n. 20 of this essay; and cf. Wiggins (1987c), pp. 188–9. ⁴⁸ Wittgenstein (1961), 6.422. Cf. Wiggins (1987c), pp. 198 ff. ⁴⁹ Wittgenstein (1961), 6.422. ⁵⁰ If, furthermore, we can show that it is satisfied in essentially the same way in each of the three cases, as of course we should have to if the Core Proposals are correct, then we may also be able to acknowledge an interesting degree of subsumption. Thus, at one extreme, there may be a categorical imperative to think the truth. That is, it may be that when there is nothing else to think, this is because there is nothing else to do but think it. (Do the Core Proposals already have this consequence? Not quite. They entail that there is a categorical imperative not to think what is false. Unconditioned agnosticism is not ruled out. This of course is more plausible—cf. the Cartesian view of error as involving a misuse of the will (Descartes (1970), ‘Fourth Meditation’, pp. 98–100)—though I am well aware that many will regard it as still not plausible enough to stave off the sense of a reductio ad absurdum. I cannot now enter into the many issues that would be involved in rebutting that sense. Suffice to ask: why should there not be a categorical imperative to attain an ideal?)

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3. Explaining the Value of the True, the Right, and the Categorically Required Specifying the condition by making some sort of reference to rationality, as we actually did and as others have effectively done, certainly helps in this connection. But there is still a question about what reason one has to be rational, or to avoid being irrational. (I am assuming that rationality is not defined in such a way that this question answers itself.) Again, suppose we were able to endorse the suggestion made earlier, that there is literally, and non-elliptically, nothing else to do but what there is a categorical imperative to do, so that agents who disobey a categorical imperative are forfeiting their own agency and so in some sense not being true to themselves. There is still a question about why they should mind.⁵¹ As soon as we start pressing questions of this kind, about people’s most basic reasons for doing things, we find ourselves in an area of fierce and familiar debate. The principal issue is this. Can there be a reason for anyone to do anything (whether adopted by them or not) which is not grounded in some element of their ‘subjective motivational set’, as Bernard Williams calls it, or in some conative state of theirs, for short?⁵² I seem to be prevaricating on this issue. For on the one hand, just by urging (iii), I am suggesting an affirmative answer. (I am suggesting that there is a reason that everyone has, or perhaps even must have, irrespective of what happens to motivate them.) On the other hand, by admitting that there is a question even about what reason one has to be rational, I am suggesting a negative answer. But I do not wish to prevaricate. In fact, I believe that the answer to the question posed is no.⁵³ (I shall say something below in response to the opposite intuition.) The point about (iii) is this. Everyone can share some reason if everyone shares some conative state. And that is what I think we should be looking for. This conative state may extend to all possible thinkers, wanters, and agents. Or it may be confined to those satisfying some condition that everyone (in fact) satisfies. Either way we can regard ourselves as back with our familiar pattern of enquiry, looking for some suitable way of replacing ‘satisfies condition C’ in (2s)

Any thinker, wanter, or agent who satisfies condition C is bound to be in conative state s.

But of course, this quest differs in two important ways from the earlier quests involving (2), (2w), and (2a). The first difference is the very fact that we now insist ⁵¹ Cf. B. Williams (1981a) and Smith (1987), pp. 42–3. ⁵² B. Williams (1981b), p. 102. Some contributions to the debate are: Hume (1888), pp. 457 and 462; Hume (1975), pp. 170–2 and 294; Foot (1972); McDowell (1978); Smith (1987) (pursued in Pettit (1987) and Smith (1988)); G. Thomson (1987), esp. pp. 50 ff.; Lewis (1989b); and Wiggins (1990–1), esp. §15. Note: many of the writers distinguish between ‘motivating’ reasons and ‘normative’ reasons (this is the terminology that Michael Smith adopts in Smith (1987), p. 38). The question I posed in the main text concerns normative reasons. ⁵³ Cf. B. Williams (1981b).

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that the condition be satisfied by everyone. (So we should immediately resist the temptation to specify the condition in the same way as we did before, with another reference to the unconditioned. Reference to the unconditioned should rather come when we specify the content of the conative state—if the Core Proposals are on the right lines.) We even allow for the limiting case, where ‘satisfies condition C’ is replaced by ‘is a thinker, wanter, or agent’ and the condition specified imposes no restriction at all. The second difference is that, unlike in the case of (2), (2w), and (2a), we are not interested in determining what the formula is equivalent to: we are interested in finding some suitably explanatory instance of it. So what is such an instance?⁵⁴ Our answer to this question will be more or less ambitious according to how much of a restriction we impose when we specify the condition. Most ambitious would be to argue for some instance of the limiting case, that is to replace ‘satisfies condition C’ by ‘is a thinker, wanter, or agent’. More interesting, perhaps, would be to argue for some instance of a case where it was a philosophically open question whether or not it was equivalent to the limiting case, for example by replacing ‘satisfies condition C’ by ‘is finite’. Into this category falls an answer that we can extract from Bernard Williams. He has argued that any rational agent must want to be free.⁵⁵ Here we replace ‘satisfies condition C’ by ‘is a rational agent’ and ‘be in conative state s’ by ‘want to be free’. Whether this case is equivalent to the limiting case depends on whether thinkers and wanters must also be agents, and on whether agents must be rational. However that may be, this answer fits the Core Proposals extremely well. Unconditionedness, as I understand it, just is a kind of freedom.⁵⁶ Another way of answering the question (perhaps equally ambitious) is by means of what we might call a ‘conative’ transcendental argument.⁵⁷ Its conclusion would be, say, that it is a necessary condition of being able to engage in the kind of reflective enquiry in which we are now engaged that one recognize the value of critical self-conscious reflection and that one be motivated, however minimally, to live in a way that can answer to it. This too would deliver a suitable instance of (2s) and would fit the Core Proposals extremely well. But I shall draw this essay to a close by gesturing—no more than that—at the answer, or the kind of answer, that attracts me most. Part of its appeal is its bearing on two important intuitions, one that I share and one that I do not. The first is the (very common) intuition that there is something about ultimate value,

⁵⁴ In Wiggins (1991), there is a beautifully worked out answer (of sorts) to this question. But it would connect with different outcomes to the earlier quests. I shall suggest some very different answers. ⁵⁵ B. Williams (2006i), pp. 55–8. Cf. Kant’s idea that a rational agent cannot act except under the Idea of freedom: see Kant (1964a), pp. 115–16. ⁵⁶ Cf. again the currents of thought in Spinoza: see n. 27. Cf. also Hookway (1990), ch. 8. And, of course, cf. Kant, e.g. Kant (1933), A419/B447. ⁵⁷ Cf. Essay 11 in this volume.

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the value of thinking what is true, wanting what is right, and doing what is categorically required—more specifically, there is something about its allure— that is beyond words.⁵⁸ This is the intuition that I share, and I shall make some very sketchy remarks about why. The second intuition (no less common) is the one that tells against what I said earlier. It is the intuition that ultimate value is a source of reasons that are not grounded in conative states, not even in conative states that are universally shared: ultimate value is ‘inherently motivating’, and therein lies the special force that I have been trying to explain. I disagree, and I shall try to combat the second intuition by saying something about why there is an impulse to think and talk in these terms. My basic idea, which I shall refer to as the Basic Idea, is as follows. The Basic Idea: Human beings are finite, but have an aspiration to be infinite.⁵⁹ Unconditionedness is a mark of infinitude.⁶⁰ So ‘satisfies condition C’ can be replaced by ‘is finite, but aspires to be infinite’ and ‘be in conative state s’ can be replaced by something like ‘want to exemplify unconditionedness’. This would fit the Core Proposals and it would be suitably explanatory. But how would it connect with the two intuitions? Via the Core Proposals, together with a development of the Basic Idea—again, I shall not now try to defend it—which runs as follows. Human finitude involves having insights that cannot be put into words. In fact such insights are acquired by rational selfconscious reflection.⁶¹ But rational self-conscious reflection being what it is, they include insights into its own nature (that is, into the nature of rational selfconscious reflection) and thus into the nature of the unconditioned. So there is something about ultimate value (the value of thinking what is true, wanting what is right, and doing what is categorically required) that cannot be put into words, namely whatever is revealed by these insights into the unconditioned. However, the aspiration to be infinite, and the attendant motivation to exemplify unconditionedness, involve an urge to express one’s inexpressible insights; for having insights that cannot be expressed is a mark of finitude.⁶² So if anyone did try to express their inexpressible insights, in particular their inexpressible insights into the essence of ultimate value—something that we all tempted to do—then ultimate value, or rather what gives ultimate value its allure, would itself be what

⁵⁸ See e.g. Wittgenstein (1961), 6.4 ff. Cf. also Wittgenstein (1965). ⁵⁹ I have tried to say more about this in A. W. Moore (2019a), ch. 15, and A. W. Moore (1992), esp. §3. Cf. Hegel (1977), pp. 104 ff. ⁶⁰ Cf. Kant (1933), A322/B379 and A411–20/B438–48. ⁶¹ This is something for which I try to argue in A. W. Moore (2019f), esp. §3, and A. W. Moore (1992), passim. ⁶² Again, I try to argue for this in A. W. Moore (1992), §3.

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was motivating them.⁶³ It would be motivating them to say something about it beyond the (mere, effable) fact that it was motivating them. They would be liable to cast it as inherently motivating.⁶⁴ The first intuition is vindicated, the second accounted for. These last few paragraphs have of course been highly schematic: I have left out all the arguments. Decency might dictate that I stop there. But I cannot resist giving these remarks a final twist. Trying to express the inexpressible means wanting unconditionedness, not only in the sense of being motivated to exemplify it but also in the sense of lacking it (at least in certain respects); for no one, in the full light of critical self-conscious reflection, could (continue to) attempt the impossible. So anyone trying to put into words their inexpressible insights into the essence of ultimate value would also to be liable to know, and to feel, that (at least in certain respects) they had not attained it. It would present itself to them as an unrealized ideal (very much as in Kant’s system, where the ‘I will’ of a purely rational being becomes the ‘I ought’ of a conditioned rational being⁶⁵). Our very reflection on how there can ever be nothing else to think, or want, or do is thus set to become an example of how guilelessly we can think, want, and do otherwise.⁶⁶

⁶³ And in so far as we can see ourselves as trying to express these insights, there is room here for another conative transcendental argument. ⁶⁴ But they might also see clearly enough to want, at the same time, to regard it as grounded in some conative state. Cf. Wittgenstein’s remarks on the ‘deeper conception of the essence of the Good’, in Wittgenstein (1965), p. 15, and Wittgenstein (1979), pp. 79–80. Such tensions are characteristic of attempts to express the inexpressible. ⁶⁵ Kant (1964a), pp. 122–3. ⁶⁶ Much of this I take to be a variation on a familiar biblical theme. See Genesis 3: 1–7; and cf. Paul’s letter to the Romans 7: 7–11.

11 Conative Transcendental Arguments and the Question Whether There Can Be External Reasons Abstract A characterization of transcendental arguments is proffered whereby they yield conclusions about how things are via intermediate conclusions about how we must think that they are. A variant kind of argument is then introduced. Arguments of this variant kind are dubbed ‘conative’ transcendental arguments: these yield conclusions about how it is desirable for things to be via intermediate conclusions about how we must desire that they are. The prospects for conative transcendental arguments are considered. It is argued that, although they can never be of practical use, they may nevertheless be of use in dissolving certain applications of the debate—initiated by Bernard Williams—about whether anyone can have an ‘external’ reason to do anything, that is to say a reason that is not grounded in some desire of the person’s, in a suitably broad sense of ‘desire’. The relevance of conative transcendental arguments to this debate is that they highlight desires that we cannot help having and with respect to which the debate lacks any suitable focus. In the final section of the essay five conative transcendental arguments deriving from the work of five moral philosophers are briefly considered.

1. Transcendental Arguments One fairly standard way of characterizing transcendental arguments would be this. Transcendental arguments are arguments of the form: (1)

p

(2)

It would not be possible that p if we did not think that q

∴(3) ∴(4)

We must think that q It is true that q,

The Human A Priori: Essays on How We Make Sense in Philosophy, Ethics, and Mathematics. A. W. Moore, Oxford University Press. © A. W. Moore 2023. DOI: 10.1093/oso/9780192871411.003.0012

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where what replaces ‘p’ is something about how we represent reality, either something minimal—such as that we have self-conscious experiences, or that we exercise concepts, or that we make assertions—or something more restrictive, such as that we do these things concerning some specified subject matter. By no means everything that has been given the label ‘transcendental argument’ fills this bill: the term has no single accepted usage. But the characterization is standard enough to capture a significant common core in the various uses of the term, and I shall appropriate it as a (stipulative) definition of my own use.¹ On this construal, there are certain objections to which any transcendental argument is vulnerable. There is the objection that unless (1) is interpreted in such a way that it carries question-begging presuppositions about what exactly we achieve in representing reality as we do, there is no reason to accept (2). There is the objection that (2) can in any case never be known, because there is never any ruling out our making (1) true in some hitherto unimagined way that does not involve our thinking, or even being able to think, that q. There is the objection that the modality involved in (2) and (3) is unclear, and that the argument as a whole cannot survive any clarification of it; more specifically, perhaps, that the modality involved in (2) and (3) is a conflation of conceptual and psychological modalities, and that there is no satisfactory way of disentangling these. There is the objection that the step from (2) to (3), or perhaps the double step from (2) to (4), confuses the necessity of a hypothetical with the necessity of its consequent. There is the objection that—Moore’s paradox notwithstanding²—the final step from (3) to (4) is invalid; or, a little less harshly, that the final step from (3) to (4) needs the support of some highly questionable metaphysical assumption, such as verificationism or idealism.³ Different transcendental arguments are obviously differently threatened by these objections, and generate different sorts of counter-objections. I shall not now try to answer the question whether any transcendental argument survives all

¹ Cf. Cassam (1987), p. 355; Harrison (1982), p. 211; Lear (1984), pp. 219 ff.; Walker (1978), p. 10; and esp. Stroud (1968), passim. Note: despite the obvious Kantian resonances of the term, and despite the affinities of so much of what has been written about transcendental arguments with Kant’s work, it is a term that he himself almost never uses. (The sole occurrence of it in his (1933) is at A627/B655— though see also A 589/B617. At A84–95/B116–29, he explains his use of the related term ‘transcendental deduction’: see esp. A85/B117.) ² Moore’s paradox has to do with puzzles such as this: although it is possible for me to think something false, there is an incoherence in my saying, of any given proposition, both that I think it and that it is false (see Sorensen (1988), ch. 1)). We would be involved in a similar incoherence if we claimed that (3) was true and that (4) was false. ³ For exploration of these and other objections, see Cassam (1987); Harrison (1982); and Stroud (1968).

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the objections. My purpose is rather to explore an unfamiliar variation on this familiar theme. I want to consider the prospects for what I shall call ‘conative’ transcendental arguments.

2. Conative Transcendental Arguments By ‘conative’ transcendental arguments I mean arguments whose form differs from that of (ordinary) transcendental arguments in three respects. The first is that ‘think’ in (2) and (3) is replaced by ‘desire’. (This results in a pair of schemata that have, for the most part, hopelessly unidiomatic instances. However, for an argument to be of this form it is not necessary that its actual wording conform to this pattern, only that the wording of some regimented paraphrase do so.) ‘Desire’ here is to be understood as a quasi-technical term denoting any conative state. Examples of desiring that q are hoping that q, wanting it to be the case that q, wishing it were the case that q, having a whimsical urge to make it the case that q, and thinking it only right and proper that q.⁴ The second respect in which the form of conative transcendental arguments differs from that of their non-conative counterparts is that ‘true’ in (4) is replaced by ‘desirable’. ‘Desirable’ here is to be interpreted in a similarly plastic way to ‘desire’, the idea being that the replacement for (4) should stand to the replacement for (3) in an analogous relation to that in which (4) stands to (3): a relation, very roughly, of vindication. Just as its being true that q means that we are right to think that q, so too its being desirable that q means that we are right to desire that q.⁵ The third and final respect in which the two forms of argument differ is that the original restriction on ‘p’, namely that what replaces it is something about how we ⁴ A phrase that could just as well have been used in place of ‘desire’ here, were it not so cumbersome, is ‘have a pro-attitude to its being the case’: cf. Davidson (1980a), pp. 3–4. Note: there are important questions, which I shall not address, about just what count as conative states. In particular, do needs? Here I echo Bernard Williams, who, having famously introduced his own broad notion of an agent’s ‘subjective motivational set’, raises the question of whether the agent’s needs belong to this set (B. Williams (1981b), pp. 102 and 105). He decides not. In similar vein I shall stipulate that conative states, on my understanding, do not include needs. For like Williams, I am interested in motivating states. And although I am more suspicious than he is about the possibility of non-motivating needs, I certainly do not claim to be able to rule the possibility out. ⁵ Strictly, I should say: ‘right in that respect’. (We could be right to think that q in respect of its being true that q, but wrong in some other respect, e.g. in respect of our limited evidence. Similarly, mutatis mutandis, in the case of its being desirable that q.) Note: to say that it is desirable that q, on this interpretation, is not to rule out the possibility that we have an overriding reason to ensure that it is not the case that q. For while we may be right. to desire that q, we may also be right to desire that r, where it is not possible that both q and r; and the latter desire may (rightly) prevail. For amplification of this point, and for further important qualifications concerning the interpretation of ‘desirable’, see section 4 of this essay.

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represent reality, is dropped. It would be natural to insist on a new restriction, namely that what replaces ‘p’ is something about how we desire reality to be. Conative transcendental arguments would then involve conation, so to speak, all the way down. I shall not insist on any such restriction however. My interest in conative transcendental arguments spans the different kinds of input they can have. I shall deliberately leave open what kind of claim the first premise is. In particular, I shall allow ‘p’ to be replaced by something that is neither (just) about how we represent reality nor (just) about how we desire reality to be but (however minimally and however indirectly) about both: for instance, ‘We are rational agents.’ It is worth noting, though, that given the various restrictions that might be imposed on ‘p’, there is scope for a finer grained classification of arguments than I am now effecting.⁶ These three differences issue in the following definition of a conative transcendental argument. A conative transcendental argument is an argument of the form: (1c)

p

(2c)

It would not be possible that p if we did not desire that q

∴(3c) ∴(4c)

We must desire that q It is desirable that q.

3. Good Conative Transcendental Arguments It is clear that, just as there are certain objections to which any transcendental argument is vulnerable, so too there are certain objections to which any conative transcendental argument is vulnerable. Indeed the objections listed above all have immediate analogues. Most obvious, and most significant, is the objection that the final step from (3c) to (4c) is invalid. This objection calls to mind Moore’s celebrated reproof of the step in Mill’s argument concerning which he said, ‘The fallacy in this step is so obvious, that it is quite wonderful how Mill failed to see it.’⁷ What Mill had said was:

⁶ Furthermore, given the distinction between arguments that proceed to the truth of something via our having to think it and arguments that proceed to the desirability of something via our having to desire it, we can extract a subsidiary fourfold classification that may be of some interest: into those arguments of this genre which, intuitively speaking, proceed from representation to representation; those which proceed from conation to conation; those which proceed from representation to conation; and those which proceed from conation to representation. The first of these are what I am calling transcendental arguments. The second and third are included in what I am calling conative transcendental arguments. The fourth I am not discussing at all, though they raise interesting questions in their own right. ⁷ G. E. Moore (1959a), p. 67. The argument in question, partially quoted below in the main text and quoted by Moore on his p. 66, is from Mill (1962), pp. 288–9.

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The only proof capable of being given that a thing is visible, is that people actually see it. The only proof that a sound is audible, is that people hear it: and so of the other sources of our experience. In like manner, I apprehend, the sole evidence it is possible to produce that anything is desirable, is that people do actually desire it.

Mill’s step is a fairly close cousin of the step from (3c) to (4c), And, prima facie at least, both steps do appear to involve the kind of mistake to which Moore was voicing his well-known opposition, the kind of mistake to which he famously gave the name ‘the naturalistic fallacy’.⁸ Much has been written about just what kind of mistake this is, and about whether Mill was really guilty of it. He had after all prefaced these comments with a reminder that ‘questions of ultimate ends do not admit of proof ’,⁹ and it is significant that in the comments themselves he had spoken not of the sole ‘proof ’ it is possible to produce that anything is desirable, but of the sole ‘evidence’.¹⁰ This relates to points that I shall be raising later, when trying to view conative transcendental arguments in a more favourable light. But whatever verdict we give on Mill’s step, the fact remains that the step from (3c) to (4c) does look likely to be the weak spot in any serious conative transcendental argument.¹¹ Before I proceed, and as a preliminary to trying to make this final step look more reasonable, I shall address another of the objections to which any conative transcendental argument is vulnerable (an objection that is likewise an analogue of one of those listed earlier, to non-conative transcendental arguments): the objection, namely, that the necessity of a hypothetical is being confused with the necessity of its consequent. The objection is that the ‘must’ in (3c) registers the conditional necessity of our desiring that q—conditional on its being the case that p—but is then treated as if it registered an unconditional necessity; or rather, that unless it is treated as if it registered an unconditional necessity, the final step from (3c) to (4c) is beyond redemption. For, the objection runs, however arguable the inference from our being unable to help desiring something to its being desirable, there is nothing to be said in support of the inference from our merely happening to desire something to its being desirable. How might this objection be met? Well, clearly, if it were necessary that p, the objection would lose its force: the necessity of the hypothetical would then yield the necessity of the consequent. The objection is likely to be met, in any given case, by arguing that the first premise is a necessary truth. However, the necessity here—and by ‘here’ I mean in the original objection as well as in this parry—must itself be conditional. There cannot be any absolute ⁸ G. E. Moore (1959a), p. 10. ⁹ Mill (1962), p. 288. ¹⁰ Cf. Hollis (1985), pp. 30–1, and Skorupski (1989), pp. 285–6. ¹¹ There does not even seem to be an analogous incoherence to that involved in our claiming both that we cannot help thinking something and that it is not true (see above, n. 2). There seems to be nothing wrong with saying both that we cannot help desiring something and that it is not desirable; nothing, that is, unless the step from (3c) to (4c) is valid after all. See further below.

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necessity about our desiring that q. There is no absolute necessity about our existing at all. And the objection was not that nothing less than this would prevent the final step from (3c) to (4c) from being beyond redemption.¹² What then will the necessity be conditional on? That will vary from argument to argument. Of greatest rhetorical interest will be cases in which the necessity is conditional on something that is in turn a necessary condition of our engaging with the argument in the first place, such as our being able to make judgements about what is desirable, our being rational, or the like. The most direct way for the first premise to instantiate this necessity is by being, quite simply, a claim to the effect that this condition holds: that we are able to make judgements about what is desirable, that we are rational, or whatever it may be. Suppose then that it can be shown that the first premise does indeed instantiate such necessity, and that the necessity is transmitted to our desiring that q. Then certainly the final step from (3c) to (4c) will look more reasonable. (This is indirectly related to the point that, although the desirability of a thing is not the same as the thing’s capacity to be desired, the former does require the latter.¹³) The final step will look more reasonable, I say; but still not indisputable. What further considerations can be brought to bear on it? At this point it helps to consider two comparisons between conative transcendental arguments and their non-conative counterparts.¹⁴ One of these comparisons is to the advantage of conative transcendental arguments. One is to their disadvantage. The comparison that is to their advantage concerns a basic instinctive realism that inclines us to doubt the step from (3) to (4) in non-conative transcendental arguments, a realism that inclines us to say: whatever the inescapability of our thinking that q, still what we think may be false. (Maybe we are constitutionally incapable of acknowledging the truth about this matter.) This worry can perhaps be met by a dose of verificationism or idealism.¹⁵ But then the verificationism or

¹² How can the necessity involved in the original objection be conditional, given that the objection itself made explicit reference to unconditional necessity? Well, here is an analogy. We can distinguish between the merely conditional necessity of Albert’s dying of cirrhosis—conditional on his not stopping drinking—and the unconditional necessity of his dying, period. But the necessity in both cases is conditional on the laws of nature. ¹³ Cf. G. Thomson (1987) passim. Note: if something’s desirability means that it ought to be desired, then the fact that the former requires the latter is a simple corollary of the principle ‘ought implies can’. ¹⁴ A good deal of what I have just said applies as much to non-conative transcendental arguments (whose analogous objection elicits an analogous sequence of parries and ripostes) as to conative ones. Thus it is true in their case too that the objection is likely to be met by showing that the first premise of the argument enjoys a certain necessity; that this necessity must be a conditional necessity; that, if the argument is to have a certain rhetorical force, then what the necessity is conditional on will be something that is in turn a necessary condition of our engaging with the argument in the first place; and that the first premise will often instantiate the necessity by being a claim to the effect that this condition holds. Cf. Harrison (1982), pp. 214–16. ¹⁵ There are two ways in which this might work. The first is by closing the gap between the best view that we can form about a matter and the truth of the matter: cf. Stroud (1968). The second is less direct.

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idealism is liable to give rise to a similar worry of its own. That is, the original impulse to realism is liable to remain unchecked. In the conative case, by contrast, there is no such impulse. There is not the same urge to say: whatever the inescapability of our desiring that q, still what we desire may be undesirable.¹⁶ Our thoughts, we are inclined to say, unlike our desires, answer to what is there anyway.¹⁷ The other comparison, the one that is to the disadvantage of conative transcendental arguments, is as follows. In their case, the conditional element in the necessity is still enough to be a real obstacle to the final inference. For even if what the necessity is conditional on is our very capacity to engage with such arguments, there is room for the question whether it is desirable that we have this capacity—in a way in which there is not room for the question whether it is true that we have it. To be sure, a very hardened sceptic may say, while engaging with one of these arguments, that he doubts his own capacity to do so. But it requires a much less hardened sceptic to say, while engaging with one of these arguments, that he doubts the value of his capacity to do so. He may suspect that we would all It proceeds from the assumption that the gap can be closed anyway, say by appeal to what it is for our view about a certain matter to be a view about that matter. Idealism is invoked to explain something which, on this assumption, would otherwise be a mystery, namely that, in some cases, there is only one view we can have. The thought is: it is only with respect to what has an ideal existence that there is only one view we can have (or in other words, that there is anything we must think). Cf. B. Williams (1973d), p. 128. ¹⁶ Cf. B. Williams (1972), p. 49. Here it is interesting to note that even Thomas Nagel, that most robust of realists, is not prepared to extend his realism to ethics: see Nagel (1986), p. 139. Part of what is going on here, of course, is the celebrated distinction between the two directions of fit of representation and conation: see Platts (1979), pp. 256–7; and cf. Anscombe (1957), §32. Note: the broad sense of ‘desire’ that I am using in this essay must not be allowed to slur over this distinction. One example I gave of desiring that q, namely thinking it only right and proper that q, is not wholly conative. It is also partly representative. But it is an example of desiring that q only in so far as it is conative. And only in so far as it is conative do the comments above apply to it. (In so far as it is representative it comes within the ambit of the impulse to realism.) One thing that the breadth of my notion of desire does do, however, is to accentuate the importance of the necessity in (3c). There may be narrower notions of desire whereby the inference from our contingently desiring something to its being desirable looks more reasonable. But these are not specially related to conative transcendental arguments, and they are not my concern. ¹⁷ Here again there is an allusion to Bernard Williams: see B. Williams (1978), p. 64. On the main point being made in this paragraph, cf. Harrison (1976), p. 42. On what is involved in our thoughts’ answering to what is there anyway, see my (1997), where I also try to say something about how we know that our thoughts answer to what is there anyway. A question that is related to these issues, and worth pausing to consider, is this. In desiring something, must one already regard it as desirable—omne appetitum appetitur sub specie boni—so that someone who professes both to desire something and not to regard it as desirable is involved in a kind of analogue of Moore’s paradox? Surely not (see my n. 11). In regarding something as desirable one is adverting to an additional objectivity, some kind of claim on the thing’s being desired from other than one’s own current point of view: cf. B. Williams (2006i), pp. 58–9. What is true is that one cannot desire something without feeling some impulse, however slight, to regard it as desirable. (This is related to, but different from, the point that G. E. M. Anscombe makes in her famous discussion of wants and desirability characterizations: see esp. her (1957), §37.) For Kant’s discussion of essentially the same issue, see his (1956), pp. 61 ff. One way to look at the matter would be this: in terms of any commitment beyond one’s current perspective, the relation between ‘It is desirable that q’ and ‘I desire that q’ does not correspond to the relation between ‘It is true that q’ and ‘I think that q’, but rather to the relation between ‘It is true that q’ and ‘It seems to me that q’.

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be better off if we could not so much as think about these things, in which case there would obviously be a question about the desirability of that which, given that we can think about them, we cannot help desiring. With these two comparisons in mind, I venture to draw the following conclusion from the discussion in this section: the first and most basic worry about any conative transcendental argument, the worry that the final step from (3c) to (4c) is invalid, can satisfactorily be met by showing that the first premise enjoys a conditional necessity, provided—and this is the lesson of the second comparison— that what the necessity is conditional on can itself be shown to be desirable. This may look like a convoluted way of saying that whatever is a necessary condition of something desirable is itself desirable. But it is not. And it is as well that it is not. That claim—the claim that whatever is a necessary condition of something desirable is itself desirable—is false, or at best dubious. Conative transcendental arguments take us beyond that claim because of their focus on what we cannot help desiring. Nevertheless, the force of the proviso is that no conative transcendental argument, as it stands, is sound. At best it is enthymematic, requiring the addition of a premise guaranteeing the desirability of its input. But if we say that a conative transcendental argument is ‘good’ when it can be converted into an argument that is sound by the addition of just such a premise, then at this stage there is no reason to think that there cannot be any good conative transcendental arguments.

4. Why Good Conative Transcendental Arguments Can Never Be of Practical Use Suppose there can. How might they be of use to us? Well, one note of caution needs to be sounded straight away. Suppose there were some impediment (of an empirical kind, perhaps) to its being the case both that q and that r. And suppose there were a good conative transcendental argument establishing¹⁸ that it was desirable that q. Even so, it might also be desirable that r. Indeed, there might be a good conative transcendental argument establishing that it was desirable that r. (Who knows but that we cannot help having conflicting desires?) In a way this note of caution has nothing specifically to do with conative transcendental arguments. Whenever we decide that something is desirable, we must reckon with the possibility that something else, incompatible with it, is also desirable, and perhaps more desirable. But in a way the note of caution is specially pertinent in the case of conative transcendental arguments. To see why, we need to ¹⁸ Relative to its being merely good rather than sound, that is. This qualification will be implicit whenever I talk in these terms in the remainder of the essay.

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address some important questions, hitherto shelved, about the interpretation of the schema ‘We desire that q.’ How exactly is this schema to be understood? Let us leave aside the issue of who ‘we’ are. (This is non-trivial; but it is not the main point of concern here.) The question we need to address first is whether the desire is a corporate desire, or a desire that each of us has severally. The latter is overwhelmingly the more natural interpretation. In fact it is not obvious what a corporate desire would be.¹⁹ True, there is a natural worry concerning the noncorporate interpretation: an awful lot seems to be required for each of us to desire that q. For instance, whatever attractions a thing may have, there is a certain basic level of rationality, attentiveness, knowledgeability, and so forth that any given individual must attain in order to be guaranteed an active interest in pursuing it. However, we should not forget the broad notion of ‘desire’ involved here—much broader than the notion of having an active interest in pursuing something—nor the narrow notion of who ‘we’ are that may also be involved. Once these two things have been taken into account, the claim that each of us desires that q will seem less demanding. Let us take for granted, then, that the schema does require each of us to desire that q. Now prima facie ‘q’ must be replaced by something that is true or false independently of this particular linguistic context. That is, it must not involve an anaphoric cross-reference. Thus consider the following instance of the schema: We desire that we are happy. This must, it seems, be understood in the sense that each of us desires that each of us is happy, not in the sense that each of us desires that he or she is happy. The reason for this is that ‘q’ also appears in the schema ‘It is desirable that q.’ A crossreference in the original case would lead, it seems, to incoherence in this. Thus, while it makes sense to say that each of us desires that he or she is happy, it makes no sense (or no self-standing sense) to say that it is desirable that ‘he or she’ is happy.²⁰ There is now a problem however. For if there are any desires that we cannot help having, then at least some of them, and arguably all of them, will be desires of the very kind just precluded, desires that each of us has concerning himself or herself (‘desires de se’, as we could call them). If good conative transcendental arguments are to exploit these desires, perhaps indeed if there are to be any good conative transcendental arguments, then we had better reassess the considerations that have brought us to this point.

¹⁹ This is not to say that there could not be such a thing. But it is not obvious what it would be. ²⁰ Cf. Thomas Nagel’s distinction between objective and subjective reasons in his (1970), pp. 90 ff. Cf. also B. Williams (1973f), pp. 260 ff.

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One possibility would be to revoke our assumption that the schema ‘We desire that q’ requires each of us to desire that q. We could specially construct a notion of corporate desire, whereby a group has the corporate desire that q if and only if at least one of its members desires that q; and we could understand the schema in accord with this. Thus, if each of us desires that he or she is happy, then what we desire, in this specially constructed corporate sense, is that I am happy, that you are happy, that he is happy, and so forth, one such proposition for each of us.²¹ The problem with this is that such gerrymandering would serve only to put further pressure on the final inference from (3c) to (4c): the mere fact that one of us cannot help desiring that q is a much weaker reason for concluding that it is desirable that q.²² This pressure would be relieved if the desirability in question could be construed as desirability relative to the individual concerned. But in that case there would be a far more satisfactory alternative. In fact there is a far more satisfactory alternative anyway. It is this: to construe (4c) as an abbreviation for: (4c*)

It is desirable for us that q,

allowing ‘q’ to contain an anaphoric cross-reference after all. ‘For us’ here acts as a universal quantifier whose scope is the entire schema; but it also picks up on any relativization to individuals that may be involved. Thus, suppose it makes sense to say that it is desirable for an individual that he or she is happy. Then one instance of the schema (4c*)—or of (4c), if understood as an abbreviation for (4c*)— would be: It is desirable for us that we are happy, meaning: for each of us, it is desirable for him or her that he or she is happy. But of course, this raises the spectre that what can be shown to be desirable in my case is in conflict with what can be shown to be desirable in your case, and that each of these is in conflict with what can be shown to be desirable in his case, et cetera: imagine, for instance, that what each of us cannot help desiring is that he or she is happier than everyone else. Here at last we come back to the point of concern from which this discussion took off. For while it would be coherent to claim that it is desirable for us that we are happier than everyone else (where the quantifier, remember, has widest scope—so this would mean that, for each of us, it is desirable for him or her that he or she is happier than everyone else, or perhaps even desirable tout court that he or she is ²¹ Even here there are familiar complications about de se desires that I am slurring over: John may desire that he is happy but not desire that John is happy, because he does not realize that he is John. Aficionados will notice further examples of my slurring over these complications at various points in what follows. ²² Cf. the discussion in n. 17 on the objectivity of desirability.

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happier than everyone else), this claim would nevertheless clearly involve the desirability of incompatibles, or the relative desirability. of incompatibles. Not that there is any reason to suppose that each of us does desire that he or she is happier than everyone else, still less that each of us cannot help desiring this. But once conative transcendental arguments are understood on this model, as it seems they must be, then it is a real possibility that the things which are shown to be desirable, or relatively desirable²³ are things which are (at least empirically) not compatible.²⁴ The seemingly bleak prospect to which the discussion so far points, then, is this. No good conative transcendental argument can ever be of practical use; or, a little more precisely, no good conative transcendental argument can ever constitute a piece of practical reasoning. Plainly no such argument can ever instil in us the desire for that which it shows to be desirable. This is for the reason (apart from any other) that the desire is one which, if the argument really is good, we must have anyway. But nor, we now see, can such an argument give us a decisive reason to implement our desire. Not only might the thing shown to be desirable conflict with other things that are desirable, it might conflict with other things that can be shown to be desirable by the very same argument. True, it would be a little premature to conclude at this stage that no good conative transcendental argument can ever be of practical use. There are some important intermediate possibilities, that is to say intermediate possibilities between our desiring something and our seeing that we have a conclusive reason to implement our desire. One of these is the possibility of our coming to recognize that we have the desire. Another is the possibility, precisely, of our concluding that the thing is desirable. There can obviously be practical significance in our doing either of those things. Even so, I suspect that no good conative transcendental argument can ever be of practical use. For instance, concerning the second of these possibilities, I suspect that, unless we already acknowledge the desirability of that which the argument shows to be desirable, then nothing in the argument itself has the jurisdiction to make us put more faith in the goodness of the argument than in the falsity of its conclusion. In particular, I suspect that there is never any foreclosing doubts about the suppressed premise, the premise guaranteeing the desirability of the argument’s input. However, I do not think that this need be viewed as a ‘bleak prospect’. Suppose we do already acknowledge the desirability of that which the argument shows to be desirable. There is still a significant role that the argument can play. It can provide a justification for our regarding the thing as desirable, a deduction to use

²³ Henceforth I shall tend to leave this rider about relative desirability implicit. ²⁴ It is an important question, which I shall not attempt to address here, how far this discussion applies, mutatis mutandis, to non-conative transcendental arguments; and, in so far as it does not or cannot, why not.

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Kant’s word.²⁵ And here at last we begin to glimpse the most exciting potential that conative transcendental arguments have. They have the potential to dissolve certain applications of the debate about whether there can be external reasons.

5. Why Good Conative Transcendental Arguments May Be of Use in Dissolving Certain Applications of the Debate about Whether There Can Be External Reasons This is a debate initiated by Bernard Williams. The question is whether there can be reasons for anyone to do anything which are not grounded in some element of the person’s ‘subjective motivational set’, that is in some desire of the person’s (in the broad sense of ‘desire’ that I have appropriated in this essay). External reasons are defined to be reasons of this kind. Williams’s own view is that there cannot be such reasons. Call this view internalism, and the opposite view externalism.²⁶ Now in order for somebody’s reason for doing something to be an external reason, it is not necessary that she actually lack any corresponding desire. (She may want to do what she has a reason to do anyway.) But it is clearly sufficient. And McDowell, in a recent defence of externalism, considers someone in just such a position, that is someone with a reason for doing something but without any corresponding desire. He asks how she could come to acquire such a desire. Not by correct reasoning, he admits. But he does not see why, as an externalist, he should be embarrassed by this. He is quite happy with the alternative, namely that she could only come to acquire the desire as a result of something that ‘would not count as [her] being swayed by reasons’; and as other possibilities, he cites, in partial echo of Williams, her ‘being persuaded by moving rhetoric’, her receiving ‘inspiration’, and her undergoing ‘conversion’.²⁷ Not that this is an exhaustive or even a fully representative list of all the possibilities there are: it is not intended to be. Presumably her receiving a blow to the head with a hammer, or her taking drugs, could give her a desire of the sort in question.²⁸ The possibilities cited are the ones that would merit a certain kind of approval (approval that would include, but not be exhausted by, approval of their outcome). Or, to put the point with an ²⁵ See n. 1. Kant’s own deduction of the categorical imperative can be found in his (1956), pt I, bk 1, ch. 1, §1. Here it is interesting to note that the categorical imperative was only ever intended to codify what each of us already knows anyway; see e.g. his (1933), A807/B835. (One way to view non-conative transcendental arguments, incidentally, would be as having a similar justificatory role.) ²⁶ See B. Williams (1981b). See also his (1995a) and (1995c), pp. 186–94, which is a response to McDowell (1995). For a further interesting contribution to the debate, see Hollis (1987), esp. ch. 6. Note: this use of the twin terms ‘internalism’ and ‘externalism’ is importantly different from, and indeed in some tension with, that introduced by Falk, in his (1947–8), and subsequently adopted by many other writers, e.g. Nagel in his (1970), ch. 2. See further Dancy (1993), app. I. ²⁷ McDowell (1995), p. 72; cf. p. 74. ²⁸ I say ‘presumably’. However, I concede that there is important room for (conceptual) doubt about this.

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element of caricature, the phrases ‘moving rhetoric’, ‘inspiration’, and ‘conversion’ are ‘hooray’-phrases used to register a certain kind of approval. Other people might register disapproval of the same processes by talking rather of corruption or manipulation or brainwashing. Not that this, in itself, is any threat to McDowell of course. After all, he could say that anyone who disapproves of someone’s coming to acquire a desire corresponding to an external reason, through a process that in fact merits approval, is simply in error. However—and this is something that McDowell would himself be the first to concede—he has no Archimedean point from which to justify saying this. He has no Archimedean point from which to arbitrate between those who have the ‘right’ desires and those who do not. And this immediately suggests the possibility that the same applies to his talk of external reasons; that the phrase ‘external reason’ is likewise a ‘hooray’-phrase, used by externalists to register a certain kind of approval of a certain kind of behaviour; and that there is no Archimedean point from which to arbitrate between those whose desires mean that they approve in this way of one kind of behaviour and those whose desires mean that they approve in this way of some quite different kind of behaviour. Very crudely, it looks as though externalists, when they say that someone has an external reason to do something, mean only that the person’s doing the thing would be desirable,²⁹ there being nothing to privilege their conception of what is desirable over its various rivals. But if this is so, then it surely constitutes a victory for internalism. For internalists can equally say that the person’s doing the thing would be desirable.³⁰ They may also point out that there is already a rich vocabulary of ‘hooray’-phrases that can be applied to her doing the thing, as indeed there is a rich vocabulary of ‘boo’-phrases that can be applied to her in so far as she lacks any desire to do it; and that, if talk of external reasons is somehow meant to enrich this vocabulary, then we are still owed an account of how. Thus Williams envisages a man of whom it may be said that he is ‘ungrateful, inconsiderate, hard, sexist, nasty, selfish, brutal, and many other disadvantageous things’. He then raises the question, ‘If [the form of words “He has a reason to be nicer”] is thought to be appropriate, what is supposed to make it appropriate, as opposed to (or in addition to) all those other things that may be said?’³¹ To be sure, we can imagine an externalist replying, ‘Nothing. That is, nothing is supposed to make this form of words appropriate as opposed to (or in addition to) ²⁹ The qualification ‘very crudely’ is important here. There is a clear respect in which externalists mean more than this. For instance, suppose that it is desirable that the person do the thing unintentionally. Then there is no question of this constituting her having any kind of reason to do it. Examples can perhaps also be contrived to indicate a respect in which externalists mean less than this. Thus externalists might say that Hitler’s mother had an external reason to feed him when he was a baby, although, given subsequent events, it was not desirable that she do so. The issues here are complex. For now, I am content just to advert to them. I do not think that they affect the substance of my argument. ³⁰ This too, of course, is subject to the qualifications signalled in the previous footnote. ³¹ B. Williams (1995a), p. 39.

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all those other things that may be said. For this man to be ungrateful, inconsiderate, and all the rest is for him to have a reason to be nicer.’³² Such an externalist may also protest against our calling this a victory for internalism. Surely it can just as well be viewed as a victory for externalism? Neutrals, meanwhile, may begin to suspect the debate of being a terminological one. There is, however, good reason to call this a victory for internalism if it really is the end of the debate. The point is this. According to what has been said so far, externalism grants no substance to the claim that somebody has an external reason to do something that is not grounded in somebody’s desires—‘ours’, if not hers. Thus whatever licence there may be to make this claim about her (that she has an external reason to do this thing), it has no purchase on her that cannot be offset by the purchase of an equal and opposite claim that she may make, with just as much right (if not more), by saying, ‘I have no reason to do this thing.’ If externalism is to secure a victory here, then it needs to do more than show how the actual words ‘She has an external reason to do this thing’ can have a legitimate use. It needs to establish an asymmetry between competing demands that can be placed on those words by that use. Thus the fact that McDowell does not have an Archimedean point from which to establish such an asymmetry is, it would now appear, a real problem for him. McDowell, of course, will deny that it is a problem. He will say that he does not need an Archimedean point; that all he needs is the point, or the ethical outlook as we might call it, from which things are seen ‘aright’.³³ From there, he will say, it is possible to establish the ethical primacy of certain desires over others, and, with it, the required asymmetry. If this is so, then the debate has a further twist and may yet culminate in a victory for externalism. What we have here is a familiar dialectical deadlock, where claims for the supremacy of a certain outlook, from within that very outlook, are pitted against claims for the parity of the outlook with its rivals, from within none of them—or supposedly from within none of them. I do not myself believe that there is a single correct way of resolving all such disputes. Different things need to be said in different debates about different kinds of outlook. In this particular case, it seems to me, the idea that there is a ‘right’ outlook is spurious, and I think that the ultimate victory lies where it originally seemed to lie, with internalism. But I shall not try to argue for that now. The point I want to make now is this. However the debate between internalism and externalism is resolved, there is one area in which it comes to nothing, namely, where there are no rival outlooks of the kind just described in the first place; or more to the point, where the existence of such rival outlooks is impossible because the existence of the relevant differences of desire is impossible. Thus, suppose there is something we cannot help desiring. And suppose we have ³² Cf. McDowell (1995), pp. 75–6. ³³ Cf. McDowell (1995), §4. Cf. also the comments about ‘the cosmic exile’ in his (1981), p. 248.

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corresponding reasons for doing things. Then with respect to those reasons the debate between internalism and externalism lacks any suitable focus. In particular, it is vacuous to ask whether we would have had those reasons even if we had not had the desire. Internalists can content themselves by saying that each of us has those reasons because he or she has the desire. Externalists can content themselves by saying that each of us would have had those reasons whatever desires he or she had had (whatever the make-up of his or her ‘subjective motivational set’ had been)—the point being that, whatever desires any of us had had, they would have included this one.³⁴ In fact there are many other characteristically externalist things that externalists can content themselves by saying, things which in other contexts would immediately register their externalism but with which, in this context, internalists need not disagree: for instance, that in so far as any of us refuses to acknowledge one of the reasons in question, this can only be because of a failure to see the matter aright. The reasons in question are reasons that we all have. They are reasons that we cannot help having.³⁵ It should now be clear why I said that conative transcendental arguments have the potential to dissolve certain applications of the debate between internalism and externalism. Precisely what a good conative transcendental argument would show is that there is a desire that we cannot help having. However, there are three important caveats that need to be entered straight away if this prospect is not to sound more exciting than it really is. First, ‘certain applications’ is the operative phrase. There is nothing here to suggest that the debate as a whole can be dissolved. If externalists want to insist that there are cases in which people do lack the desires corresponding to reasons they have, or even that there are cases in which people could lack the desires corresponding to reasons they have, then, at least with respect to those reasons, or putative reasons, the quarrel with internalists remains as real as ever. Quite how much impact conative transcendental arguments can have on this debate therefore depends on quite what externalists do want to insist. (As Williams complains, ‘one of the mysterious things about the denial of internalism lies . . . in the fact that it leaves it quite obscure when [the] form of words [“He has a reason to do this”] is thought to be appropriate’.³⁶) The second caveat concerns the modality involved in any good conative transcendental argument, the sense of ‘must’ in which the argument shows that we must have a certain desire. I have already talked at some length about this. In particular, I pointed out that the modality must be conditional—on our being rational agents, for instance. But even once it has been settled what the modality is conditional on, there remain questions, concerning which I have said virtually

³⁴ Cf. in this connection the sentence straddling pp. 186 and 187 in B. Williams (1995c). ³⁵ There are indirect connections here with what Korsgaard argues in her (1986), §6. ³⁶ B. Williams (1995a), p. 39.

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nothing, about what kind of modality it is, questions, in effect, about what kind of necessity attaches to (2c). Is it conceptual? Is it metaphysical? Might it even be biological? psychological? sociological? Now, as we shall see, the second caveat does not depend on any particular answer to these questions. Even so, it is worth digressing briefly to say something about how I would answer them. I have been assuming that the necessity is pretty ‘hard’. For example, when I talked about the necessity of (1c) being transmitted to the necessity of (3c), I took for granted that the necessity of (2c) was ‘hard’ enough to carry the transmission; that it did not, for instance, introduce some contingency about our animal nature that was lacking in the first premise. Again, when I said that we could satisfy ourselves about the validity of the step from (3c) to (4c) provided that we could satisfy ourselves about the truth of the suppressed premise guaranteeing the desirability of the argument’s input, I took for granted that the necessity of (2c) was ‘hard’ enough not to introduce some extra sully; that it did not, for instance, turn on some dark inability of ours to act except under conditions of self-deception. (Here one thinks of various familiar paradoxes of utilitarian thinking—paradoxes in which our maximizing happiness depends on our desiring something other than the maximization of happiness, something that may be positively undesirable on a utilitarian conception.) But taking these things for granted, which is something I shall continue to do, has only been a convenience. For clearly there is no effective difference between a ‘soft’ modality and a ‘hard’ modality that is conditional on whatever makes the ‘soft’ modality ‘soft’. There is no effective difference, for instance, between an argument in which the necessity of (2c) is ‘soft’ because it incorporates facts about our psychology and an argument in which the necessity of (2c) is ‘hard’ but the necessity of (1c) is conditional on those same facts.³⁷ These two features of a modality—how ‘hard’ it is and what it is conditional on—can be brought together under the single head of how fine its grain is, where the fineness of grain of a modality is a matter of how many possibilities it embraces. If one modality has a finer grain than another, then there are things that are possible in respect of the former but not in respect of the latter (for instance, there are things that are conceptually possible but not biologically possible) though not vice versa. Thus the ‘harder’ the modality, the finer its grain; and the more it is conditional on, the coarser.³⁸ ³⁷ But note: had I allowed the necessity of (2c) to be ‘soft’ in this way, then I would also have had to modify my definition of what it is for a conative transcendental argument to be ‘good’. I would have had to say that a conative transcendental argument is ‘good’ when it can be converted into an argument that is sound by the addition of a premise guaranteeing, not just the desirability of its input (which is all that was needed before), but the desirability of whatever makes the ‘soft’ modality ‘soft’. ³⁸ There is no obvious reason, incidentally, why there could not be good conative transcendental arguments in which the modality was fairly coarse-grained, conditional on facts about our biology say: cf. in this connection G. Thomson (1987), passim. I take it, however, that philosophical interest in conative transcendental arguments will tend to diminish as the grain gets coarser. (Certainly this is true in the case of (ordinary) transcendental arguments, although, as I observed at the outset, one of the objections to which any such argument is vulnerable is precisely that there is no satisfactory account of the modality, conceptual and psychological elements having got irreparably tangled up.)

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The second caveat is this. The capacity of any conative transcendental argument to dissolve some application of the debate between internalism and externalism depends entirely on whether the modality involved in the argument is as fine-grained as that which is supposed to give the application its focus. Thus even if there is a desire that we ‘must’ have, in some relatively coarse-grained sense, internalists and externalists can still disagree about whether we ‘must’, in some finer grained sense, have the corresponding reasons. There might, for instance, be a good conative transcendental argument showing that we cannot help having a certain desire in so far as we are rational (or granted that we are rational). But if so, this would settle nothing in a debate about whether the corresponding reasons are reasons even for those who are irrational.³⁹ At the limit there may even be externalists whose very point is that the necessity of our having certain reasons transcends the necessity of our having any desires. Conative transcendental arguments can do nothing to close the rift between them and internalists.⁴⁰ In sum then: conative transcendental arguments can dissolve applications of the debate between internalism and externalism only where the modalities involved are properly aligned.⁴¹ The third caveat is that, if we actually attempt to put a conative transcendental argument to this kind of use, then we are in danger of simply rehearsing parts of the original debate. Suppose, for instance, that the first premise of the conative transcendental argument is that we have reasons for our desires. Then the second premise will be that we could not have reasons for our desires unless, in particular, we desired that q. But whether that is true may just be a question of whether we have an external reason to desire that q. These three caveats notwithstanding, the prospects for conative transcendental arguments, in this particular application, look favourable. That is to say, if there are any good conative transcendental arguments, then their prospects of being put to effective use in this way look favourable. I am not talking about the prospects of there being good conative transcendental arguments in the first place, about which I have been non-committal. On this score, it is worth pausing to mention what seems to be an anomaly in my essay. As I pointed out earlier, the main obstacle to supposing that there can be any good conative transcendental arguments lies in concern about their final step. I tried to indicate ways in which this concern might be met. Yet that step appears

³⁹ Cf. B. Williams (1995c), n. 3. This connects with a point I made earlier, in parenthesis: it is nontrivial who ‘we’ are supposed to be in any given conative transcendental argument. ⁴⁰ What is required in their case, I believe, is diagnosis. See my (1997), ch. 11, §5. ⁴¹ This second caveat is also meant to cover the case where the debate between internalism and externalism is a debate about explanatory priority. Thus internalists and externalists may agree that we ‘must’ have certain desires and that we ‘must’ have certain corresponding reasons, in some more or less fine-grained sense, but disagree about which of these facts explains the other. This, I suggest, can be construed as a disagreement about whether we would still have the reasons even if—per impossibile, as we would naturally say—we did not have the desires. In other words, it can be construed as a disagreement about finer grained possibilities.

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irrelevant to what we have just been discussing. In other words, it appears irrelevant to what I am now suggesting is the most significant application that such arguments can have. Could we not have saved ourselves a lot of bother, therefore, by considering arguments that stopped, so to speak, at (3c)? No. The charge of irrelevance is ill-founded. Imagine a sound argument that did stop at (3c). Unless the extra step to (4c) were also valid—or rather, unless the step to (4c) would be valid given the addition of the relevant premise concerning the desirability of the argument’s input—then the necessity at work in the argument would not be of a kind fit for the argument to be applied in this way. It is our not being able to help desiring something in a way that makes it desirable which allows for the dissolution of the debate about whether our corresponding reasons are external. Remember: what a good conative transcendental argument supplies is a deduction, in Kant’s sense, of our having some desire. It shows that we are right to have the desire. An argument which did not show this—an argument which showed merely that we must have the desire (perhaps in a relatively coarsegrained sense of ‘must’ conditional on our having some feature that was itself undesirable)—would not be suitable for these purposes. Any attempt to apply it to the debate between internalism and externalism would straight away succumb to the worry expressed in the second caveat, the worry that the disregarded possibilities in which we lack the desire are precisely the possibilities that are pertinent to the debate.

6. Are There Any Good Conative Transcendental Arguments? That more or less concludes what I have to say in this essay about the prospects for good conative transcendental arguments. It would be perverse, however, to finish without doing something to indicate whether there are any. I shall cite five arguments which I think are interesting and serious candidates, each taken, with modifications, from the work of a notable moral philosopher. In each case I shall do little more than state the argument, which will in turn be a matter of stating what replaces ‘p’ and ‘q’ in the overall schema. I shall not attempt any defence. That would have to be a large and separate undertaking. In any case, I am not convinced that any of these arguments is good. (My non-commitment has been a result of genuine agnosticism.) The first argument, unsurprisingly, derives from Kant. Here ‘p’ and ‘q’ are replaced as follows: p:

We are finite rational agents.

q:

We are happy.

The ‘we’ in the second of these is intended to cross-refer (each of us desires that he or she is happy). The species of desire involved is hope. The idea is that it is

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impossible for finite rational beings to act except in the hope that they achieve happiness. In the specific case of those who act as they ought, and thereby make themselves worthy of happiness, the hope is that they achieve the happiness they deserve. Happiness is thus a good. It is not an unqualified good; indeed, in cases where those who enjoy it are not worthy of it, it is an apt object of disapprobation. But it is an indispensable part of the ‘highest good,’ an ideal in which virtue and happiness come together in joint reciprocal maximization.⁴² The second argument derives from Bentham. This time ‘p’ and ‘q’ are replaced as follows: p:

We make moral judgements.

q:

Happiness is maximized.

Bentham’s defence of this argument effectively rests on two claims: first, that in order to make moral judgements, we need to operate with a moral principle having objective and public criteria of application; and second, that the principle of utility, that is the principle that happiness is to be maximized, is the only principle that satisfies this condition. He does not use this argument as a proof of the principle (a proof that it is desirable that happiness is maximized). On the contrary, he claims that the principle is an ultimate principle that neither needs nor admits of proof. What he does do is, precisely, to use the argument as a ‘deduction’ of the principle, or rather of our use of the principle.⁴³ The third argument is adapted from an argument of Williams. Here ‘p’ and ‘q’ are replaced as follows: p:

We are rational agents.

q:

We are free.

As in the first argument, the second ‘we’ is intended to cross-refer (each of us desires that he or she is free). Williams’s own argument is not designed to show any more than what, as rational agents, we must want, namely to be free (and to have an adequate range of wants on the basis of which to exercise our freedom). He does not himself endorse the final step of the conative transcendental argument, the step to the conclusion that it is desirable (for us) that we are free— basically for reasons implicit in objections that I canvassed above.⁴⁴ The fourth argument is one that can be extracted from Wiggins, whose own purpose is to show that, with respect to certain fundamental questions concerning ⁴² See esp. Kant (1933), pt II, ch. 2, §2. See also Kant (1964a), pp. 61–2; Kant (1956), pt I, bk 2, ch. 2; and Kant (1978), §87. For an interesting if faint echo of these ideas in Wittgenstein, see his (1961), 6.422. ⁴³ Bentham (1962), ch. 1. And see again Harrison (1976), esp. pp. 25–8. Cf. the discussion earlier in this essay concerning Mill. ⁴⁴ B. Williams (2006i), ch. 4.

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the foundations of morality, ‘what makes all the difference between Kant and Hume . . . [has] the width of a knife-edge’. Here ‘p’ and ‘q’ are replaced as follows: p:

We are rational.

q:

We enjoy the solidarity of human beings qua human.

This time the second ‘we’ is not intended to cross-refer (each of us desires that we enjoy the solidarity of human beings qua human). Wiggins starts from the Kantian idea that, in so far as we are rational, we cannot help but aspire to belong to ‘the kingdom of ends, the systematic union of rational beings under common self-legislated rational laws’. This Wiggins describes as ‘the solidarity of rational beings qua rational’. He then asks rhetorically whether ‘one who rests morality on a solidarity of this kind is well placed to dismiss a theory that rests morality in another solidarity, . . . the solidarity . . . of human beings qua human’.⁴⁵ Fifthly, and finally, Korsgaard offers an argument which she herself calls a transcendental argument. In this case ‘p’ and ‘q’ are replaced as follows: p:

We act rationally.

q:

There is such a thing as our humanity.

It makes no effective difference in this case whether or not the ‘our’ cross-refers, since our humanity is shared. The species of desiring that is involved is valuing. ‘We desire that there is such a thing as our humanity’ is meant to stand proxy for ‘We value our humanity’, the idea being that our humanity is the source of any reasons we have for how we act and we cannot have those reasons without valuing that source. Now it is obviously a contrivance to represent our valuing our humanity as our having a propositional attitude in this way. Indeed some may think that it is a serious mistake to do so. But even if they are right, Kosgaard’s argument is at the very least closely related in form to a conative transcendental argument, and raises essentially the same issues. Most notably, as she expressly acknowledges, it raises the issue of whether, and how, our having to value something ensures that it is valuable.⁴⁶ I am sure that there are plenty of other equally compelling examples. Specially worthy of investigation, I think, are examples in which what replaces ‘p’ is something minimal about how we represent reality:⁴⁷ ‘We have beliefs’, say, with ‘q’ being replaced by ‘Our beliefs are true’. Also worthy of investigation are examples in which what replaces ‘p’ is the same as what replaces ‘q’, so that the ⁴⁵ Wiggins (1995), all quotations from pp. 326–8, emphasis in original. ⁴⁶ Korsgaard (1996), §§3.4.9–3.4.10. ⁴⁷ The first premise would then satisfy the condition that the first premise of a non-conative transcendental argument must satisfy.

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conclusion of the argument is at the same time (in effect) the suppressed premise whose truth is required for it to be good. (An argument of this kind would obviously have no suasive force. But suasion, as I have already urged, is not the point.) Finally, I myself have a particular interest in examples in which what replaces ‘p’ is something about our craving for infinitude.⁴⁸ But these are all matters to be addressed on another occasion.⁴⁹

⁴⁸ See Essay 10 in this volume, esp. section 3; and my (1997), ch. 11. ⁴⁹ For assistance of various kinds I am greatly indebted to participants at the conference in Sheffield at which I first presented this essay, esp. to John Skorupski. I should also like to thank Bill Brewer, Robert Frazier, Penelope Mackie, Derek Parfit, and Bernard Williams for their helpful comments on an earlier draft of the essay, and Robert Stern for inviting me to participate in the conference.

12 Maxims and Thick Ethical Concepts Abstract The starting point for this essay is provided by Kant’s notion of a maxim and the role that it plays in his formulations of the fundamental categorical imperative. This raises the question of what a maxim is, and why there is not the same requirement for resolutions of other kinds to play the same role. An answer to this question is proffered that draws on Bernard Williams’s notion of a thick ethical concept and that is intended neither in a spirit of simple exegesis nor as a straightforward exercise in moral philosophy but as something that is poised somewhere between the two. The aim is to provide a kind of rational reconstruction of Kant. In the final section of the essay it is argued that this reconstruction, while it manages to salvage something distinctively Kantian, also does justice to the relativism involved in what J. L. Mackie calls ‘people’s adherence to and participation in different ways of life’.

My starting point in this essay is Kant’s notion of a maxim, as it occurs in some of the cardinal doctrines of his moral philosophy.¹ But the essay is neither a straightforward exercise in Kantian exegesis nor a straightforward exercise in moral philosophy. It is poised somewhere between the two. My aim is to say something about maxims that is both sufficiently plausible to be at least serviceable in a rational reconstruction of Kant and sufficiently Kantian to be at least worth taking seriously in that role. But I shall certainly part company with Kant at various points. The notion of a maxim is one of two that are central to this essay. The other, which I shall introduce in section 3, is Bernard Williams’s notion of a thick ethical concept. But I shall part company with Williams too. I intend to put his notion to work in a way in which he himself never does.

1. Kant’s Notion of a Maxim To begin, then, with Kant’s notion of a maxim. A maxim, Kant tells us, is ‘the subjective principle of acting . . . in accordance with which the subject acts’, as ¹ I develop similar ideas in A. W. Moore (2003a). Some of the material in this essay is taken directly from that book, and I am very grateful to Routledge for permission to reuse this material.

The Human A Priori: Essays on How We Make Sense in Philosophy, Ethics, and Mathematics. A. W. Moore, Oxford University Press. © A. W. Moore 2023. DOI: 10.1093/oso/9780192871411.003.0013

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opposed to a (practical) law, which is ‘the objective principle valid for every rational being, in accordance with which he ought to act’.² The aim of this essay is to put some flesh on these bones. What, first of all, is a ‘principle of acting’, or a ‘principle’ for short?³ One way to broach this question is by considering the very idea of putting reason to practical use. For Kant, putting reason to practical use, if only in the formulation and implementation of hypothetical imperatives,⁴ involves actively determining what to do on the strength of one’s conative states. Actively determining what to do on the strength of one’s conative states contrasts with passively succumbing to their strength.⁵ It includes, as one vital component, adopting resolutions (however tacitly, however unselfconsciously, however retroactively, however extempore)⁶ and then acting on those resolutions. Principles, I suggest—and therefore maxims themselves—are resolutions of a certain kind. But of what kind? Given Kant’s distinction between maxims and laws, principles need to include not only resolutions which agents actually adopt, and which can therefore be invoked to explain (if not to justify) some of the things that agents actually do, but also resolutions which agents ought to adopt, and which can therefore be invoked to justify (if not to explain) some of the things that agents might do. One broadly Kantian proposal which meets this constraint is the following. A principle is a resolution which is not just designed to encapsulate and protect the (non-rational) conative states of whomever adopts it, or might adopt it, but which at least purports to have a claim on everyone. It is a resolution of such a kind that, simply by adopting it and allowing it to guide one’s behaviour, one is treating it as though it did have a claim on everyone—as though it were a resolution that everyone ought to abide by, irrespective of his or her (nonrational) conative states.⁷ This means that any principle that does not have a claim on everyone, and that cannot even be rationally treated as though it did, is ipso facto defective: it fails to satisfy one of the basic norms of being a principle. (It is in this same sense that a ruler purports to be straight: any ruler that is not

² Kant (1996a), 4: 421, emphasis in original. Cf. Kant (1996a), 4: 401, footnote; Kant (1996b), 5: 19; and Kant (1996c), 6: 225 and 389. ³ For guidance from Kant himself on this question see Kant (1996b), 5: 19–20. ⁴ For the distinction between a hypothetical imperative and a categorical imperative see Kant (1996a), 4: 414. ⁵ Of course, the suggestion that conative states are like alien forces by which we are beset is caricatural at best. It is well exposed in Blackburn (1998), ch. 8, esp. §3. The suggestion is there in what I have just said partly because I am providing a mere sketch of Kant at this point, though partly also because of Kant: see e.g. Kant (1996a), 4: 457–8. ⁶ For an insightful discussion of some of the complications covered by this parenthesis see Brewer (2002). ⁷ Cf. Kant (1996a), 4: 434 and 438.

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straight, and that is not even sufficiently close to being straight for us rationally to treat it as though it were, fails to satisfy a basic norm of being a ruler.) If we accept this proposal—if we accept that a principle, and therefore a maxim, purports to have a claim on everyone; and that abiding by a maxim means treating it as though it were a resolution that everyone ought to abide by—then ‘purports to’ and ‘as though it were’ are the operative phrases. For one thing, given the way in which the notion of a maxim is supposed to help elucidate and justify Kant’s idea of a fundamental categorical imperative,⁸ it would be question-begging to suppose that anything does have this kind of claim on everyone, irrespective of his or her (non-rational) conative states.⁹ But also, more significantly, even if we had an assurance that some principles do have this kind of claim on everyone, this would still leave room, and on Kant’s own conception would need to leave room, for the possibility of principles (specifically, maxims) that nevertheless do not have any such claim—principles indeed that could not have any such claim, either because it would be impossible for everyone to abide by them or, more modestly, because it would be impossible for anyone to will that everyone should abide by them.¹⁰ This seems to connect well with the contrast that Kant draws between maxims and laws. For we can say that, whereas a maxim purports to have a claim on everyone, a law does have: to abide by a maxim is to treat it as a law. It also seems to connect well with what is perhaps the most famous of Kant’s formulations of the fundamental categorical imperative. For if we accept that an agent who is purely rational must submit to the rule of law only where it makes sense to do so, then it seems to follow that such an agent must treat as laws only those maxims that really could be laws, compatibly with what agents in general are capable of willing. Such an agent must therefore not abide by any maxim which could not be a law, and indeed which he or she could not will to be a law. Hence the famous formulation in question: Act only in accordance with that maxim through which you can at the same time will that it become a universal law.¹¹

In summary, then, rational agents determine what to do partly by adopting resolutions (however tacitly, however unselfconsciously, however retroactively, however extempore) and acting on those resolutions. ‘Principles’ are resolutions that at least purport to have a claim on everyone. Of these, those that really do

⁸ E.g. Kant (1996a), 4: 440–1. ⁹ Cf. Kant’s claim that ‘if it is assumed that pure reason can contain within itself a practical ground, that is, one sufficient to determine the will, then there are practical laws; otherwise all . . . principles will be mere maxims’ (Kant (1996b), 5: 19, some emphasis removed). ¹⁰ Kant (1996a), 4: 421–3. ¹¹ Kant (1996a), 4: 421, emphasis in original.

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have a claim on everyone are laws: laws are resolutions that any rational agent ought to abide by. The principles that any such agent actually abides by are maxims. Some maxims may also be laws. But some are certainly not: they merely purport to be. This is reflected in the fact that the maxims in question could not be laws; or at least, if they could, the agents who adopt them could not will them to be laws. The fundamental categorical imperative is to abide only by maxims that one could also will to be laws.

2. How Are Principles to be Distinguished from Other Resolutions? Among the countless questions raised by this account, perhaps the most urgent is this. When does a resolution purport to have a claim on everyone? That is, how are we to distinguish principles from other resolutions? Since maxims are nothing but the principles by which agents actually abide, this is in turn a variation on that oft-posed question, ‘What is a maxim?’¹² Unless we can answer such questions, Kant’s moral vision will be severely compromised. For there need be nothing wrong with abiding by a resolution that one cannot will to be a law, if the resolution does not even purport to have a claim on others. People regulate their lives in all sorts of ways that are tailored to their own individual conative states—their own likes, dislikes, values, goals, ambitions, and the rest—as moulded by their own particular circumstances. And it is entirely reasonable for them to do so. Thus imagine someone who resolves to dine out each Friday evening at her favourite restaurant. She has adopted a resolution which patently not everyone ought to abide by, indeed which patently not everyone could abide by, and which is none the worse for that. Or again, imagine someone who resolves to pay off his credit card each month.¹³ He too has adopted a resolution which could not be a law: the institution of credit cards depends on there being people who do not do what he has resolved to do. Yet there is nothing wrong with his resolution, nor with his acting on it. If someone is criticized, on Kantian grounds, for abiding by a maxim that could not be a law, or that he could not will to be a law, what is to stop him from replying that what he is abiding by is not a maxim (not a principle) at all? Here is another way of putting the same concern. Kant’s moral vision invites us to ask the familiar question, ‘What if everyone did that?’ Sometimes it is perfectly ¹² This question has generated a large literature. One discussion of it that relates closely to what I shall be arguing in this essay is Herman (1993). Other related discussions include: Allison (1990), pp. 39–40 and 86 ff.; Beck (1960), pp. 70 ff., 80 ff., and 118 ff.; Brewer (2002); O’Neill (1989), esp. pp. 150 ff.; Walker (1998), pp. 33 ff.; T. C. Williams (1968), ch. 2; Wood (1970), pp. 44 ff.; and Wood (1999), pp. 51 ff. Cf. also Scanlon (1998), p. 53. ¹³ This example is taken from Blackburn (1998), p. 218.

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acceptable to deflect this question by responding, ‘Not everyone will do that.’ The two examples above illustrate this. But sometimes, for instance when someone has given false information on a tax return, the question can be asked in a way that makes this response inappropriate. The sheer fact that there would be something untoward about everyone’s doing the thing in question matters. Kant’s vision invites us to ask the question in this second way. The concern is that we do not yet have a good sense of when this second way of asking the question is available. The distinction between cases in which it is available and cases in which it is not—which is in effect the distinction between resolutions that are principles and resolutions that are not—cannot depend on how agents themselves view their resolutions. The tax fiddler who resolves never to mention any gratuities on his tax returns, and whose resolution the Kantian account should surely proscribe, precisely does not view his resolution as having a claim on others. He views it as a way of exploiting the greater honesty of others. Is it a question of generality, then? Does the resolution never to mention gratuities on one’s tax returns have a generality which means that, willy-nilly, it purports to have a claim on everyone, unlike the resolution to dine out each Friday evening at one’s favourite restaurant (whose underlying principle, if there is one, may be to allow room in one’s life for leisure activities)? No. The resolution to pay off one’s credit card each month is at least as general as the resolution never to mention gratuities on one’s tax returns. Lurking behind these concerns is another, which is absolutely basic for the whole Kantian enterprise. Not only must there be a way of distinguishing principles from other resolutions; there must be a way of doing so which shows why it is impossible for a rational agent to escape the force of the fundamental categorical imperative by not adopting any principles (any maxims) at all. What is to prevent a rational agent, when putting reason to practical use, from adopting only resolutions that are not principles, resolutions that serve merely as private recipes for organizing his own affairs? Kant says that ‘a rational being must always regard himself as lawgiving in a kingdom of ends’.¹⁴ But why?¹⁵ My aim in this essay is to answer such questions by providing an account of what a principle, and therefore a maxim, is.¹⁶ I hope that my account will constitute a partial defence of Kant. But the word ‘partial’ is crucial. This is for two reasons. First, as I have already indicated, I do not profess to be doing simple Kantian exegesis. Secondly, there are all sorts of objections to what Kant says about maxims, particularly to what he says about them in relation to the fundamental categorical imperative, on which my account has no bearing. Moreover, ¹⁴ Kant (1996a), 4: 434. ¹⁵ Cf. B. Williams (2006i), pp. 61–3. ¹⁶ I shall try to provide an account which is sensitive to what might be called the ‘sociology’ of maxims—an account, in other words, which is sensitive to where maxims come from, to how they are inculcated, and to why it is absurd to imagine any given individual either dispensing with them or, conversely, conjuring them up for himself or herself in complete independence of other people.

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among the objections on which it does have a bearing, there is one in particular on which the bearing it has is aggravating. I have in mind the following objection: that a person may sometimes quite properly abide by a maxim even though it could not be a law, in fact because it could not be a law, namely when she is concerned, not to satisfy her own conative states, but to subvert the institution or practice that (provisionally) makes the maxim possible. I shall return to this objection in section 5.

3. Williams’s Notion of a Thick Ethical Concept, and a Basic Proposition Concerning it I turn now to Williams’s notion of a thick ethical concept. By a thick ethical concept Williams means a concept such as infidelity or blasphemy whose applicability is both ‘action-guiding’ and ‘world-guided’. To apply a thick ethical concept in a given situation, for example to accuse someone of infidelity, is, in part, to evaluate the situation, which characteristically means either condemning or commending certain courses of action; but it is also to make a judgement which is subject to correction if the situation turns out not to be a certain way, for example if it turns out that the person who has been accused of infidelity did not in fact go back on any relevant agreement.¹⁷ I want to appropriate this notion of a thick ethical concept in answering the questions raised at the end of the previous section. To this end I shall take for granted the following proposition, which I shall call the Basic Proposition: The Basic Proposition: Anyone who embraces a thick ethical concept thereby has certain reasons for doing things. By way of illustration, anyone who embraces the concept of a promise thereby has a reason to keep any promise he or she has made; anyone who embraces the concept of privacy thereby has a reason to respect other people’s privacy; anyone who embraces the concept of blasphemy thereby has a reason not to blaspheme. These examples ought to give some indication of what the Basic Proposition means. But further elucidation is called for. In particular, I need to provide a gloss on ‘doing’ something, on ‘having’ a ‘reason’, and on ‘embracing’ a concept. To begin with the most straightforward of these: ‘doing’ something is to be understood very broadly. It is meant to include ‘omissions’ as well as ‘acts’.¹⁸ This

¹⁷ B. Williams (2006i), pp. 129–30 and 140–2. ¹⁸ For discussion of this distinction see Bennett (1995), passim; and B. Williams (1995b).

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is illustrated in the case of blasphemy: anyone who embraces the concept of blasphemy thereby has a reason not to blaspheme. Next, ‘having’ a reason is meant to fall short of acknowledging the reason. If someone has a reason, all sorts of factors, such as insensitivity, selfishness, and simple stupidity, may prevent him from acknowledging it.¹⁹ Concerning ‘reasons’, as they feature in the Basic Proposition, there are two points to be emphasized. First, they are meant to be normative. Second, they are not meant to be indefeasible. I shall expound each of these in turn. Normativity, first, contrasts with motivation. Reasons can be of either kind. A motivating reason is a matter of why someone actually does something: ‘Your only reason for listening to Beethoven is that you are afraid of appearing uncultured.’ A normative reason is a matter of why someone ought to do something: ‘The beauty of Beethoven’s music gives you a reason to listen to it.’ (This has clear resonances, of course, in Kant’s distinction between maxims and laws.) The Basic Proposition entails that anyone who embraces a thick ethical concept thereby ought to do certain things. Second, a (normative) reason for doing something is indefeasible when it can never be overridden by a (normative) reason for doing something else. The Basic Proposition does not require the reasons which someone has by virtue of embracing some thick ethical concept to be indefeasible. Thus even though anyone who embraces the concept of privacy thereby has a reason to respect other people’s privacy, there may, in certain circumstances, be some other overriding (normative) reason not to do so, say the need to gain information about someone that will save his or her life. Finally, I need to provide a gloss on ‘embracing’ a concept. This is something close to a term of art for me. To convey what I intend I need to draw a distinction. Thick ethical concepts can be grasped in two ways, an engaged way and a disengaged way.²⁰ To grasp a thick ethical concept in the disengaged way is to be able to recognize when the concept would (correctly) be applied, to be able to understand others when they apply it, and so forth. To grasp a thick ethical concept in the engaged way is not only to be able to do these things but also to feel sufficiently at home with the concept to be prepared to apply it oneself, where being prepared to apply it oneself means being prepared to apply it not just in overt acts of communication but also in how one thinks about the world and in how one conducts one’s affairs. What this requires, roughly, is sharing whatever beliefs, concerns, and values give application of the concept its point.

¹⁹ Note: this claim is neutral as regards the question whether all reasons are ‘internal’ in Williams’s sense: see B. Williams (1981b). ²⁰ This distinction is one that Williams frequently draws. See e.g. his (2006i), pp. 141–2, and his (1995c), p. 206.

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Take the concept of the Sabbath. Those who are not Jewish have no difficulty in grasping this concept in the disengaged way. A person who is not Jewish can understand perfectly well what someone means when she says that her birthday this year falls on the Sabbath. But only a Jewish person recognizing an obligation to keep the Sabbath can grasp the concept in the engaged way. We might say that such a person lives by the concept. To be sure, this distinction is one of degree, not of kind. Borderline cases can readily be constructed: think of the grasp that non-orthodox Jews have on the concept of the Sabbath. Furthermore, each of the two ways of grasping a thick ethical concept itself clearly admits of degrees. Thus a non-Jewish person may understand what somebody means when she says that her birthday this year falls on the Sabbath, but not quite what she means when she says that she always keeps the Sabbath: his grasp of the concept, even qua disengaged, is imperfect. And it is important to note that someone who grasps the concept in the disengaged way may yet apply the concept ironically, or as part of playing some kind of role, or as a pretence, or even in the process of attributing certain beliefs or values to someone else, who grasps the concept in the engaged way: we might call these vicarious applications of the concept. But none of these complications prevents the distinction from being a relatively robust one. To ‘embrace’ a concept is to grasp it in the engaged way. It is to enter into the spirit of the concept, to have whatever outlook gives the concept its point, to live by the concept as I put it above.

4. Using the Notion of a Thick Ethical Concept to Distinguish Principles from Other Resolutions With the Basic Proposition thus clarified, I can now proceed to my account of what a principle is. Given any thick ethical concept, let us say that the concept requires the practice of doing any of the things which anyone who embraces the concept thereby has a reason to do. Thus, for instance, the concept of a promise requires the practice of keeping any promise one has made. And given any resolution such that no one could adopt that resolution (however tacitly, however unselfconsciously, however retroactively, however extempore) without embracing a certain thick ethical concept, let us say that the resolution involves the concept. Thus, for instance, the resolution to keep any promise one has made involves the concept of a promise.²¹

²¹ Equivalently, a resolution r involves a thick ethical concept c when embracing c is a precondition of being in a position to adopt r—where being in a position to adopt r (which falls short of actually adopting r, as in the original version of the definition) is understood to mean being able to adopt r without changing one’s very outlook on the world. This version of the definition will be significant later.

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Finally, running these two ideas together, given any resolution which involves a certain thick ethical concept which in turn requires a certain practice, let us say that the resolution is answerable to the practice. Thus the resolution to keep any promise one has made is answerable to the practice of keeping any promise one has made. Then my proposal is this: a principle is a resolution to do something that either counts as observing some practice to which the resolution is answerable or, conversely, counts as violating some practice to which the resolution is answerable. The resolution to keep any promise one has made is clearly a case in point. That resolution is answerable to the practice of keeping any promise one has made. And it is a resolution to do precisely that. Hence it is a resolution to do something that counts as observing some practice to which the resolution is answerable. Hence, on my proposal, it is a principle. The resolution to exempt oneself from keeping any promise one has made when it is in one’s own interests to do so is a further case in point. That resolution likewise involves the concept of a promise.²² So it is likewise answerable to the practice of keeping any promise one has made. And it is a resolution to do something that counts as violating that practice. Hence, on my proposal, it too is a principle. On the other hand, the resolution never to make a promise to anyone whose own promises cannot be trusted is not a case in point. This resolution involves the concept of a promise,²³ as indeed it involves the concept of trust and the concept of a person. But it does not involve any concept that requires a practice that would be either observed or violated by acting on the resolution. Hence, on my proposal, it is not a principle. Nor, similarly, are the resolutions considered in section 2, whose seemingly unobjectionable non-universalizability prompted this discussion in the first place: the woman’s resolution to dine out at her favourite restaurant each Friday evening; and the man’s resolution to pay off his credit card each month. So far, so good. Now consider the resolution to exempt oneself from keeping any promise one has made if this will save someone’s life.²⁴ This is more interesting. Clearly this resolution is answerable to the practice of keeping any promise

²² It may differ in this respect from the resolution simply to break any promise one has made when it is in one’s own interests to do so (where this latter resolution makes no mention of exemption). Someone could arguably adopt this latter resolution even if his or her grasp of the concept of a promise was only disengaged. Such a person would still be able to apply the concept vicariously, and might be able to adopt such a resolution as a way of exploiting the engaged grasp that other people have of the concept. I am in fact deeply sceptical about how far this kind of pretence could go. But I mention this possibility here if only to explain why the resolution that I have specified in the main text takes the somewhat cumbersome form that it does. ²³ Unless, like the resolution considered at the beginning of the previous footnote (the resolution to break any promise one has made when it is in one’s own interests to do so), it is a resolution that someone could adopt even if his or her grasp of the concept of a promise was only disengaged, in which case there would be even less reason to regard it as a principle. ²⁴ Cf. the case of Herod and Salome in Mark 6: 17–25.

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one has made. And it is a resolution to do something that counts as violating that practice. On my proposal, then, it is a principle, and straightforwardly so. But this is only part of the story. To leave the matter there would be to suggest that there is something wrong with the resolution. It would indicate that anyone who adopts the resolution and acts on it thereby does something that he or she has a (normative) reason not to do. However, here we need to recall one of the points I emphasized in the previous section: that the reasons associated with any thick ethical concept, according to the Basic Proposition, may be defeasible. While it is true that anyone who embraces the concept of a promise ought to keep any promise he or she has made, the ‘ought’ here is a pro tanto ‘ought’. It may also be true that such a person has a (normative) reason, indeed an overriding (normative) reason, on occasion, to break some promise he or she has made. Not only that; a fuller exception-specifying reason may itself be one of the reasons associated with the concept of a promise. Thus we must allow for the possibility that the concept of a promise requires both the practice of keeping any promise one has made and the practice of exempting oneself from keeping any promise one has made if this will save someone’s life. Whether the concept does require the latter practice as well as the former is a matter of substantive debate. But suppose, for the sake of argument, that it does. Suppose that one is not being true to the concept of a promise if one accords a greater significance to promise keeping than one does to life itself. Then the resolution to exempt oneself from keeping any promise one has made if this will save someone’s life fits both ways of being a principle on my proposal. It is a resolution not only to do something that counts as violating a practice to which the resolution is answerable but also to do something that counts as observing a practice to which the resolution is answerable. Moreover, it is a resolution of both kinds with respect to one and the same concept. There is nothing awry in this. Nor does it indicate any incoherence in the concept. It simply registers the defeasibility of one of the relevant reasons. A concept may require two practices, then, one of which incorporates exceptions to the other. And a resolution may likewise be answerable to two practices, one of which incorporates exceptions to the other. But there is another possibility too. A resolution which is answerable to one practice may come to be answerable to a second practice, through suitable developments in the concepts it involves, where this second practice incorporates exceptions to the first. This raises the question of how, and how much, a concept could develop without loss of identity; and in what sense of ‘could’. For current purposes, we do not need to dwell on this question. It suffices to make two observations. First, a concept could certainly undergo some development without loss of identity. (Thus the concept of hearing has developed to apply to what we do to someone’s voice over the telephone: it would once have counted as a conceptual truth that someone’s voice cannot be heard unless he or she is within earshot. Again, the concept of democracy has

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developed to embrace suffrage for women: it would now count as a conceptual truth that denying women the vote is undemocratic.) Second, the relevant sense of ‘could’ is itself partly a conceptual matter, partly a matter of the ways of life that are open to those who embrace the concept, and partly a matter of the sociohistorical parameters within which the concept is situated. This possibility is relevant, as we shall see, to whether my proposal meets the following basic requirement: that it represent a principle as a resolution that at least purports to have a claim on everyone. Before I attempt to show that it does, I need one more definition. Given that a principle is a resolution to do something that either counts as observing some practice to which the resolution is answerable or counts as violating some practice to which the resolution is answerable, let us say that a law (echoing Kant) is a resolution that qualifies as a principle in the first of these ways. In other words, a law is a resolution to do something that counts as observing some practice to which the resolution is answerable. I shall now try to show that my proposal does indeed represent a principle as at least purporting to have a claim on everyone (where ‘everyone’ is to be understood as everyone in a position to adopt it²⁵). Let p be a principle. And let x be whatever p (which, qua principle, is also a resolution) is a resolution to do. Now either p is a law or it is not. Suppose it is. (An example is the resolution to keep any promise one has made.) Then anyone in a position to adopt p thereby has a (normative, defeasible) reason to do x. So if the question so much as arises for any given individual A whether she ought to abide by p or not, then the answer must be yes: she ought to. (I am taking for granted, incidentally, that resolutions, just like reasons, can be defeasible. Thus it is possible both for A to abide by p – which is to say, both for A to abide by the resolution to do x – and also for A to abide by a resolution to refrain, in certain circumstances, from doing x.) So p has a claim on everyone. Now suppose that p is not a law. Here again we must distinguish two possibilities. Either p is capable of becoming a law through suitable developments in the concepts it involves (the possibility to which I adverted above) or it is not. Let us suppose, first of all, that it is. (A putative example is the resolution to exempt oneself from keeping any promise one has made if this will avert a serious unforeseen risk to one’s own life.) If p is of this kind, then it is possible for a given individual A rationally to abide by p. If she does abide by p, she is acting in accord with her embracing of the concepts involved in p by treating them as if they had been refined in one of the many ways in which they could be refined. She is like someone who uses her discretion to apply some concept in a way that the concept neither demands nor precludes—as, for instance, when

²⁵ See n. 21. The importance of this qualification will surface in section 5.

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someone applies the concept of a child to a 14 year old. In treating p in that way, she is treating it as if it were a law, and hence as if it had a claim on everyone. And it is quite rational for her to do so. What, then, of the other possibility: that p is not capable of becoming a law through suitable developments in the concepts it involves? (An example is the resolution to exempt oneself from keeping any promise one has made when it is in one’s own interests to do so.) In that case, it is not possible for a given individual A rationally to abide by p. If she does abide by p, then any (normative) reason that she has to do x militates against some concept c that p involves in such a way that c could not so much as survive if everyone who embraced it came thereby to have such a reason. It must be a reason that is tailored to A’s own conative states, as moulded by her own particular circumstances. This means that A’s abiding by p involves her both embracing c and not being suitably beholden to her embracing of c in circumstances in which it suits her not to be. It is like A’s possessing the concept of a husband and thinking that the woman next-door is a husband because it fits some preconception that she has. It is irrational. But following resolutions is one hallmark of putting reason to practical use. A resolution that cannot be followed rationally fails to satisfy a basic norm of being a resolution. More particularly, a principle that cannot be followed rationally fails to satisfy a basic norm of being a principle. Hence p fails to satisfy a basic norm of being a principle if it is of this last kind. That is, p fails to satisfy a basic norm of being a principle if it not only does not have a claim on everyone but cannot even be rationally treated as though it did. But that is just to say that p at least purports to have a claim on everyone. That completes the argument. We can now graft this account back into Kant’s own conception. For suppose that some person A abides by some principle p—which is eo ipso one of A’s maxims. Then, given the account above, if A is being rational, p must either already be a law or be capable of becoming a law through suitable developments in the concepts it involves. But given the way in which the capacity to become a law is being understood here, as involving both conceptual and anthropological elements, this is at least akin to saying that, if A is being rational, then it must be possible for A (along with everyone else who embraces the relevant concepts) to will that p become a law. Here, then, is my reconstruction of Kant. Practical reasoning, on this reconstruction, includes a pure element: keeping faith with concepts. Theoretical reasoning also includes keeping faith with concepts. What makes it possible for keeping faith with concepts to have a practical dimension as well as its more familiar theoretical dimension is, ultimately, the fact that some concepts—thick ethical concepts—equip those who embrace them with certain reasons for doing things. And that just is the Basic Proposition, whose importance to this discussion ought now to be clear.

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5. Relating the Discussion Back to Kant How much of a vindication of Kant is this? Does it do anything to substantiate the idea of a fundamental categorical imperative? Does it, for that matter, do anything to substantiate the idea of a maxim whose adoption is a requirement of reason? It substantiates the idea of a maxim whose adoption is in accord with reason. But that is a weaker idea. Any maxim which is not itself a law but which is capable of becoming a law through suitable developments in the concepts it involves—the putative example given above was the maxim to exempt oneself from keeping any promise one has made if this will avert a serious unforeseen risk to one’s own life—is a case in point. The adoption of any such maxim is, on this account, in accord with reason. But its rejection would be in accord with reason too. Precisely what the account provides for in such a case is the possibility of rational discretion. Very well, then; what about a maxim which is already a law, say the maxim to keep any promise one has made? Is that an example of a maxim whose adoption is a requirement of reason on this account? Not if this means that any rational person is bound to adopt it. The most that can be said of the maxim, on this account, is that any rational person is bound to adopt it if he or she embraces the thick ethical concepts required to do so. (Recall that ‘everyone’, in the argument above, was to be understood as everyone in a position to adopt whatever principle was in question.²⁶) This leaves open the possibility of a rational person who rejects the maxim by rejecting the concepts themselves. I shall return to this possibility shortly. What the account does substantiate is the idea of a maxim whose adoption is contrary to reason, say the resolution to exempt oneself from keeping any promise one has made when it is in one’s own interests to do so. And that, arguably, is as much as Kant’s fundamental categorical imperative requires. (The famous formulation of this imperative considered in section 1 specifies a necessary condition for rationally abiding by a maxim, not a sufficient condition.) It is also, again arguably, as much as is available at the theoretical level. (It is not the case, for example, that anyone who is rational is bound to accept that all husbands are male, nor even that anyone who is rational and who has considered the matter is bound to accept this: it would be possible for someone to be rational yet not to accept this, because he or she abhorred the institution of marriage, say, and accordingly repudiated the very concept of a husband.) In light of this, we can see how my reconstruction of Kant might in fact be said to substantiate the idea of a fundamental categorical imperative. For, in as much as

²⁶ And see again n. 21.

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this reconstruction grounds the ethical ‘must’ in a requirement to keep faith with whatever concepts one embraces, it assimilates that ‘must’ to the logical ‘must’. The reason why I must not act on a maxim to exempt myself from keeping any promise I have made when it is in my own interests to do so is of a piece with the reason why I must not accept that the person next-door is both a woman and a husband. The alternative, in each case, would be to flout concepts to which I am committed. The fact remains that there is a significant element of hypotheticality in this categoricity. For there are all manner of questions, many of which are themselves ethical, about whether I do well to embrace the concepts I do. It would certainly be possible for someone who is rational, but who does not embrace the concept of a promise, not to acknowledge any value in promise keeping—and not just because he or she is incapable of even thinking in those terms; for it would be possible, more specifically, for someone who is rational, but who grasps the concept of a promise in the disengaged way, not to acknowledge any value in promise keeping. Such a person may think that the concept itself, and with it the whole institution of promising, is an anathema of some sort, a concept that we are better off not embracing. (Jesus thought this.²⁷) This echoes a very old concern about Kant’s own conception that goes back at least as far as Hegel.²⁸ The matter is made yet more complex by the fact that someone can both embrace a concept and, perfectly reasonably, want to be rid of it. She may have decided that there is something petty or degrading or pernicious about thinking in terms in which she herself still naturally thinks, and about conducting her affairs in ways in which she herself still naturally conducts them. She has not yet reached the detachment to which she aspires. And the best way to try to reach that detachment, and to try to get others to reach it too, may well be by adopting a tactic to which I adverted at the end of section 2, namely the tactic of being subversive—abiding by maxims which involve the concept but which, by design and without relevant extenuation, flout practices that the concept requires, maxims which therefore cannot be laws. A simple model of this would be someone’s resolving to shock people as often as possible by violating some taboo, with the aim of subverting the taboo and ultimately of depriving its violations of any capacity to shock. It is as if some local irrationality is being put to the service of some more global rationality, rather as an isolated dissonance in music can be put to the service of some more complete harmony, or again, rather as an inoculation in medicine can be put to the service of someone’s greater health.

²⁷ Matthew 5: 33–7. Jesus also had misgivings about traditional conceptions of the Sabbath, though these were not, I think, misgivings about the concept itself. It would not be too Procrustean, in my view, to say that what Jesus was urging was suitable development in the concept: see Luke 13: 10–17. ²⁸ See Hegel (1942), §135. And cf. Blackburn (1998), p. 222.

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That there is this element of hypotheticality in the categoricity afforded by my reconstruction of Kant does not in itself prevent that reconstruction from being reasonably faithful to Kant, for reasons that I have already indicated. (I am thinking once again about the fact that the famous formulation of Kant’s own fundamental categorical imperative specifies a necessary condition for rationally abiding by a maxim, not a sufficient condition.) But in any case, it wreaks less damage to the idea of a convergence between the ethical and the rational than it may appear to. I shall close this essay by giving a very brief indication of what I have in mind. What the element of hypotheticality signals is that there are issues about what concepts we are to embrace which are no less the concern of ethics than any issues about what maxims we are to adopt. But how are we to broach issues of the former kind? Clearly, what concepts we are to embrace depends on what concepts we are capable of embracing. And what concepts we are capable of embracing is partly a sociological matter, partly a political matter, partly a psychological matter, partly a biological matter, partly indeed a technological matter—and partly a matter of what concepts we already embrace (for we patently cannot come to have an outlook on the world that is not suitably accessible from the outlook that we already have on it). In order to broach such issues, we therefore need to exercise our imaginations within the various constraints set by each of these. We need to address questions of the form, ‘Would this work?’, ‘Could we live with that?’, ‘At what cost?’, ‘With what gain?’, ‘How can we get from here to there?’ We need to think about how the concepts that we currently embrace may yet develop, or evolve into others, or yield to others. That, however, is precisely the kind of thing which, on my reconstruction of Kant, we need to do when we broach issues about what maxims we are to adopt. For when we broach issues about what maxims we are to adopt, we need to think about whether various candidates are capable of becoming laws. And when we think about whether these candidates are capable of becoming laws, we need to think about whether the concepts involved in them could so develop that the maxims counted as observing practices to which they were answerable. Assessing our concepts is of a piece with assessing our maxims, then. They are both ways of trying to make sense. Or to put it another way, they are both ways of trying to be rational. It does not follow, of course, that Kant’s own original conception has been fully vindicated; nor even that it has been largely vindicated. But it does follow, I think, that Kant’s own original conception has been somewhat rehabilitated. In particular, we are now in a position to see how that conception can be stripped of some of its less appealing universalist garb without losing its fundamental message. A Kantian conception can acknowledge the diversity of thick ethical concepts that people embrace; it can understand such diversity as a reflection of what J. L. Mackie calls ‘people’s adherence to and

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participation in different ways of life’;²⁹ and it can understand this in a way that absolves anyone of any kind of error.³⁰ Where it will remain distinctively Kantian is in recognizing a non-relative requirement, within such diversity, for people’s adherence to and participation in different ways of life to make maximum possible sense.³¹

²⁹ Mackie (1977), p. 36. ³⁰ Cf. B. Williams (2006i), ch. 9. For an expression of a related relativism from a perhaps unlikely source see St Paul’s letter to the Romans 14. Cf. also David Hume, ‘A Dialogue’, in his (1975), esp. p. 333. ³¹ I should like to thank Myles Burnyeat for his comments on an earlier draft of this essay.

13 Quasi-Realism and Relativism Abstract This essay first appeared as a contribution to a symposium on Simon Blackburn’s book Ruling Passions. In this book Blackburn argues that the quasi-realism that he defends, whereby an ethic reflects the conative states of different people, does not entail that our own ethic is in any interesting sense ‘just’ ours, and so does not, in any interesting sense, involve any relativism. This essay is an attempt to rebut Blackburn’s arguments, while conceding their role in indicating both how carefully the relativism in question needs to be formulated and how important it is not to conflate metaphysical issues with ethical issues. Some comparisons and contrasts are drawn between the ethical quasi-realism under discussion and a modal quasi-realism concerning necessity and possibility. One lesson that emerges is that the relativism in question is partly a matter of the different ethical concepts by which people can live.

1. Introduction If it is true that ‘an ethic is the propositional reflection of the dispositions and attitudes, policies and stances, of people’, as Simon Blackburn says in summary of the quasi-realism that he champions in this excellent and wonderfully provocative book,¹ then it seems to follow that different dispositions, attitudes, policies, and stances—different conative states, for short—will issue in different ethics, each with an equal claim to truth; and this in turn seems to be one thing that could be reasonably meant by that slippery polyseme ‘relativism’. If such relativism does follow, a good deal remains to be said about how much force it has. At the limit it might do no more than signal the abstract possibility of an ethic rivalling that of humans. More potently, it might somehow legitimize the different ethics of different groups of humans in actual conflict with one another. But without the possibility of some such variability of ethic to match a possible variability of conative state, the quasi-realist’s claim that an ethic ‘reflects’ a particular combination of conative states appears hollow. ¹ Blackburn (1998)—the subject of the symposium in which this essay first appeared—p. 310. All unaccompanied references will be to this book.

The Human A Priori: Essays on How We Make Sense in Philosophy, Ethics, and Mathematics. A. W. Moore, Oxford University Press. © A. W. Moore 2023. DOI: 10.1093/oso/9780192871411.003.0014

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In the splendid final chapter of his book, and again in the appendix, Blackburn nevertheless tries to keep relativism at bay. Carefully distinguishing some of the many different things that ‘relativism’ can mean, he argues, with respect to each, either that he is not committed to it or that it does nothing to imply that our own ethic is in any interesting sense ‘just’ ours. But I want to suggest that Blackburn is committed to a form of relativism whereby our own ethic indeed is in some interesting sense ‘just’ ours (for some interesting value of ‘we’). The relativism in question is not the view that, had our conative states been different, different ethical standards might have applied; Blackburn has persistently and persuasively argued that he is not committed to anything like that. Nor is it the view that, had our conative states been different, we might have applied different ethical standards; that is a platitude (and scarcely merits the label ‘relativism’). The view is something lying subtly between these, namely that, had our conative states been different, we might have applied different ethical standards and it might have been right for us to do so; we might have had different ethical beliefs and those different ethical beliefs might have been true. But does this not collapse into the first view? I think not. I think it admits of an ‘opaque’ reading that keeps it separate from the first view. Here is an analogy. It is now 2 p.m. Had I caught that flight to Australia, it would still now have been 2 p.m. However, I would have thought that it was midnight and it would have been right for me to do so; my belief about what time it is would have been true. Its now being 2 p.m. is ‘just’ a feature of my current location.² (Later, I shall suggest a somewhat more refined model for the opacity. For now, I am content merely to identify the kind of relativism that I think quasirealism yields.)

2. Relativism as a Metaphysical View Blackburn is uneasy about the ‘just’ in the claim that our ethic is ‘just’ ours. He hears it as an invitation to accept the first of the views mentioned above, the view whereby different ethical standards might have applied (or worse still, that ‘over

² Purists may prefer to put it this way: had I caught that flight to Australia, I would have been prepared to assert the sentence, ‘It is midnight,’ and any such assertion would have been true. The worry then is that, while this makes the opacity plainer, it seems also to make the analogy more anodyne. Does it? Not really. It is not as if the meaning of the sentence is irrelevant. There is more transparency than that. The sentence I would have been prepared to assert in Australia is the very sentence I hereby deny in insisting that it is not midnight. (For an excellent exchange relating to some of the issues raised by this analogy, and by what I shall say later, see Perry (1986) and Blackburn (1986).)

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there’ different ethical standards do apply). This in turn is because he takes the relativist’s even-handedness in pitting our ethic against possible rivals as itself an ethical stance. He hears the relativist as saying that another ethic might have been (or worse still, is) as good ethically as ours; that it might have been ethically right for us to apply different ethical standards.³ But relativism of the sort I am envisaging no more involves adopting a particular ethical stance than relativism about what time of day it is involves adopting a particular location. It is a metaphysical view.

3. Ramsey’s Ladder Why does Blackburn not acknowledge the possibility of ‘metaphysical’ evenhandedness? Ironically, I think that he himself helps to give an effective diagnosis. Several times in the book he refers to what he calls ‘Ramsey’s ladder’. This is a series of propositions each of which, bar the first, looks as if it is on a higher level than its predecessor (in the sense of being substantially about its predecessor) though in fact they all have the same content; as Blackburn puts it, ‘Ramsey’s ladder is horizontal’.⁴ Thus if the first proposition in the series is that p, then further along are such propositions as that it is a fact that p, that it is true that it is a fact that p, and the like. Now Blackburn is quite right to say that Ramsey’s ladder is horizontal. However, he sometimes makes this sound like an exciting philosophical thesis,⁵ whereas it is really just a matter of definition. For obviously it is not impossible to produce a proposition of this general stripe that is on a higher level than the proposition that p; it is just that what is produced will not then count as being on Ramsey’s ladder. In particular, if the proposition that p is an ethical proposition, then it is not impossible to produce a meta-ethical proposition, off the ladder, about what makes it true that p. Blackburn himself insists on this, in opposition to Ronald Dworkin⁶—as of course he must, for quasi-realism itself involves producing just such propositions. (It would be bad news indeed for Blackburn if an opponent of quasi-realism, having first affirmed that p, went on to express his opposition to quasi-realism by insisting that it was ‘non-quasi-realistically’ true that p, and was able to justify this on the grounds that he was crawling along Ramsey’s ladder!) Blackburn’s chief concern about Ramsey’s ladder is the danger of our crawling along it while thinking that we are moving vertically. But there is also the danger of our moving vertically while thinking that we are crawling along

³ E.g. pp. 305 and 314.

⁴ Pp. 78–9 and 294–7.

⁵ E.g. p. 79.

⁶ P. 295.

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the ladder. It seems to me that Blackburn is so keen to avoid the first of these dangers that he succumbs to the second. And this is what prevents him from acknowledging the relativist consequences of his quasi-realism. He cannot hear the relevant debate about relativism as other than a ground-level debate; the relevant endorsements of relativism as other than attempts to deny propositions on Ramsey’s ladder.⁷

4. One Way for Blackburn Not to Try to Distance Himself from Relativism Very well, then; could Blackburn distance himself from the relativism that I am trying to pin on him by arguing that ultimately only one combination of conative states is genuinely possible? No. If such a thing could be argued at all, then either ‘genuinely possible’ would be doing ethical work, in which case this would be another ground-level failure to engage properly in the debate, or it would be doing meta-ethical work, in which case this would be an implicit rejection of quasirealism, in favour of some much more robust kind of realism. (Not that I am making any points here that Blackburn himself does not in effect make—and with effect too.⁸)

5. Ethical Quasi-Realism Compared with Modal Quasi-Realism But surely, someone might say, there is something untoward in my claim that Blackburn’s quasi-realism entails relativism, as we can see if we look at the analogue of Blackburn’s view for modality. (Let us call this analogue ‘modal quasi-realism’, as distinct from the ‘ethical quasi-realism’ that has been our concern up to now.) There are many forms that modal quasi-realism can take,

⁷ Pp. 295–6. There is a graphic illustration of this in Blackburn (1999), p. 217, where, having once again shown how quasi-realism enables us to crawl along Ramsey’s ladder, he continues, ‘And why does that not imply that divergent moral opinions are on all fours? Well, all I can hear that as meaning is that they are all equally good’ (emphasis in original). There is also an interesting echo of it elsewhere in the book, namely in his remarks on Bernard Williams’s well-known rejection of the possibility of external reasons (pp. 264–6). While substantially agreeing with Williams, Blackburn nevertheless urges that there is a ‘use of “external reasons” in which there are such things’ (p. 266), the use in question being a ground-level use, in contrast with the higher level use that is needed to state Williams’s view.—I am far from denying, incidentally, that there are some debates in philosophy which are spurious because they purport to involve a higher level use of certain expressions which is simply not available: the expressions are to be understood in ground-level terms if at all. (See further my (1997), esp. ch. 6, §1; and cf. Wittgenstein (1961), 5.64.) But as I have already intimated, Blackburn cannot dismiss the relevant debate about relativism as a case in point without endangering his own quasi-realism. ⁸ See e.g. pp. 302–3 and 308–10.

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but they all involve a core thesis to the effect that propositions about what is or is not possible reflect certain commitments that people have made—whether these be decisions to adopt certain linguistic conventions (a form I myself favour) or archived failures to make anything of certain ways of thinking (a form Blackburn favours) or something else besides.⁹ Now if my claim about ethical quasi-realism is correct, then a parallel claim about modal quasi-realism must be correct too; that is to say, modal quasi-realism must also entail relativism, in the form, roughly, that we might have acknowledged different possibilities and it might have been right for us to do so. However, while there may be room to deny that the relativism about ethics is itself an ethical view, there is no denying that this is a modal view, a view about what is possible. As such, it cannot help but compromise our modal commitments. Since modal quasi-realism is meant precisely to respect our modal commitments, the claim that it entails such relativism thus becomes the basis of a quick reductio ad absurdum of it. But this in turn casts doubt on the claim, because, whatever problems modal quasi-realism faces, it is too robust a doctrine to be dismissed as quickly as that.¹⁰ There is much that could be said in response to this worry. (Some would respond to it by saying that modal quasi-realism can indeed be dismissed as quickly as that; but not I, for, as I have already indicated, I favour a form of modal quasi-realism.) It is certainly not obvious that the relativism about modality compromises our modal commitments. At any rate we must beware of thinking that it does so for purely structural reasons. The mere fact that it is (in this context) a ground-level view does not mean that it automatically poses a threat to whatever other ground-level views we have. The corresponding ground-level ethical relativism would have compromised our conative states not just because it was (in that context) a ground-level view, but because it was the particular ground-level view it was: namely, that rival ethics are as worthy of our approval as our own, something we cannot concede without losing our grip on our own. But still, the worry persists. For just as we cannot concede the worthiness of rival ethics without losing our grip on our own, so too, surely, we cannot concede the possibility of rival arithmetics (say) without losing our grip on our own, or, more to the point, without losing our grip on its necessity. To be sure, we must not forget the opacity in the claim that rival arithmetics are possible. To say that they are—to say that we might have acknowledged different arithmetical possibilities and that it might have been right for us to do so—is not yet to say that any arithmetical falsehoods might have been true. Even so, it looks as if we cannot ⁹ For Blackburn’s views, see esp. his (1993); see also his (1984a), pp. 216–17. It is interesting to compare Blackburn’s views with those of Descartes, as superbly expounded by Jonathan Bennett in his (1994). [Supplementary note: See also Essay 3 in this volume.] ¹⁰ The worry expressed in this paragraph is related to the problems that Blackburn himself discusses concerning the enterprise of giving a naturalistic explanation of our modal commitments: see §6 of his (1993).

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dissociate saying the first of these from saying the second unless we adopt a very weak modal logic, denying, for instance, that what is possibly possible is eo ipso possible; and it is doubtful that modal quasi-realism on its own can force us to do that. (After all, could not our modal commitments include a commitment to S5?) In reply: this still betrays an insufficient grasp of the opacity. The opacity is not just a matter of iterated modalities resisting reduction. (If it were, then conceding the possibility of rival arithmetics would force us to adopt a modal logic that was even weaker than suggested; so weak, in fact, as to call into question our title to the claim to be talking about modality at all. For it would force us to deny that what is possibly necessary is eo ipso possible.) The point is this. Had we acknowledged different arithmetical possibilities, and had it been right for us to do so, this would have shown that we were using different concepts. For to say that we might have acknowledged different arithmetical possibilities is not, at least in this context, to say that we might have acknowledged the arithmetical possibility of propositions that we currently take to be arithmetically impossible. It is to say rather that we might have acknowledged the arithmetical possibility of propositions that we currently lack the concepts even to express.¹¹ We can concede that rival arithmetics are possible, then, without losing our grip on the necessity of our own, just as we can concede that non-Euclidean geometries are possible without losing our grip on the necessity of the proposition that between any two Euclidean points there is at most one Euclidean straight line. But in saying this, have I not deprived the relativism of any sting? Of course we might have had different concepts. If the ethical relativism that I am trying to pin on Blackburn amounts to no more than that—if it amounts to the claim that, had our conative states been different, we might have had different ethical concepts— then is it not after all a platitude that he can accept with equanimity? I am certainly happy to admit that this is, in part, what the ethical relativism I am trying to pin on Blackburn amounts to. (That was what I had in mind when I referred earlier to the ‘somewhat more refined model for the opacity’. We might have had different ethical beliefs, not just in the sense that we might have had

¹¹ Why in that case would our rival arithmetic have counted as an arithmetic? Because of a family resemblance. This shows up in the fact that if, starting from here, we were to come to adopt such an arithmetic, then we might well find it natural to use standard arithmetical vocabulary to couch it. For instance, we might find it natural to assert the sentence, ‘7 + 5 > 5 + 7’—although we would not then be using the terminology in what we currently recognize as a standard way. And how does this square with what I said above in note 2 about the relevance of the meaning of the sentence ‘It is midnight’ to what time I would have thought it was if I had been in Australia? Well, there certainly seems to be even greater opacity in this case. But the family resemblance ensures that there is still not total opacity. Suppose we did come to use ‘+’ to stand for a non-commutative function, in such a way that we were entitled to assert, ‘7 + 5 > 5 + 7.’ Our use of the symbol might nevertheless be sufficiently like our current standard use of it to warrant our saying that it still stood for an addition function. (We do after all talk naturally about the non-commutative ‘addition’ of transfinite ordinals—and we do find it natural to use the symbol ‘+’ to stand for this function. Think also of the way in which both Euclidean and non-Euclidean geometries are said to treat of points and lines.)

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ethical beliefs whose negations were of the same type as those we currently have, but also in the sense that we might have had ethical beliefs that we currently lack the concepts even to formulate.) But I deny that this is a platitude. The different ethical concepts that we might have had would have been the sort of thing that Bernard Williams has famously christened ‘thick’ ethical concepts. That is to say, they would have been concepts whose applicability was both ‘action-guiding’ and ‘world-guided’, of such a kind that even to think in those terms required having certain conative states.¹² (Blackburn himself more than once refers to concepts of this kind.¹³) It is no platitude to say that we might have had different concepts of such a kind, corresponding to our different conative states, and that it might have been right (in some non-ethical sense) for us to do so. On the contrary: it is a way of acknowledging that our own thick ethical concepts, and therewith the ethic that we use them to express, are in some interesting sense ‘just’ ours.

¹² See B. Williams (2006i), pp. 140–8. ¹³ See p. 303 and the last sentence of his answer to Q.9 on p. 314.

14 From a Point of View Abstract This essay first appeared as a critical notice of Derek Parfit’s On What Matters, Volume One and Volume Two. Its focus is Parfit’s repudiation in Volume Two of what he calls ‘naturalism’ about normativity. Parfit distinguishes between two versions of such naturalism: ‘analytic’ naturalism, whereby normative claims can be reduced to naturalistic claims; and ‘non-analytic’ naturalism, whereby any fact that can be stated by some normative claim can be stated by some naturalistic claim. Parfit rejects both. This essay mounts a (limited) defence of the latter. Towards the end of the essay it is argued ad hominem that this defence has a deep affinity with claims that Parfit himself makes in other contexts, and that it signals an irreducibly perspectival take that we have on reality, which in turn signals something fundamental about the human condition.

When a book ranges as widely as this one does—or rather, as widely as these two do, together adding up to not much less than 1500 pages—even a reviewer with a generous word allowance has to be savagely selective. Derek Parfit’s much-touted two-volume study of what matters¹ has already generated considerable discussion. A great deal of attention has understandably been directed towards Parfit’s defence of his core theory that ‘an act is wrong just when such acts are disallowed by some principle that is optimific, uniquely universally willable, and not reasonably rejectable’.² I shall not attempt to engage with this theory. Nor shall I attempt to address the many exegetical issues that arise in connection with Parfit’s contention that Kantians, contractualists, and consequentialists have all been playing variations on this single theme. I want to focus on some issues that arise in Volume Two about the very nature of normativity. The ignoring of everything else is serious, however unavoidable. Even so, I hope that my focus will signal something of critical importance to the whole project. Parfit considers various views about the nature of normativity. In particular, he considers a pair of views which he designates ‘analytical naturalism’ and ‘non-analytical ¹ Parfit (2011)—of which this essay first appeared as a critical notice. All unaccompanied references will be to this book. ² Vol. One, p. 413.

The Human A Priori: Essays on How We Make Sense in Philosophy, Ethics, and Mathematics. A. W. Moore, Oxford University Press. © A. W. Moore 2023. DOI: 10.1093/oso/9780192871411.003.0015

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naturalism’. Both these views accept that some normative claims are true. That is to say, both accept that some normative claims state facts, facts that can thereby be classified as normative facts. Analytical naturalism earns its title of ‘naturalism’ by insisting that these claims can nevertheless be reduced to naturalistic claims. Non-analytical naturalism earns its title of ‘naturalism’ in a quite different way. It denies that these claims can be reduced to naturalistic claims, but holds that, none the less, any fact that can be stated by some normative claim can also be stated by some non-normative naturalistic claim; or equivalently, that any normative fact is also, in that sense, a natural fact.³ My principal concern is with non-analytical naturalism. Patently, we cannot hope to assess non-analytical naturalism without, first, some account of what it is for a claim to be a normative or a naturalistic claim and, second, some account of what it is for two claims to state the same fact. Parfit has much to say about the first of these, rather less to say about the second. As far as the first is concerned, he gives us an initial steer by presenting a couple of lists of words. The first list contains (among others) ‘wrong’, ‘ought’, ‘good’, and ‘excellent’; the second list contains (among others) ‘kill’, ‘square’, ‘sister’, and ‘unexpected’.⁴ A normative claim, Parfit says, is one that can be expressed using words of the first sort, while a naturalistic claim is one that can be expressed using only words of the second sort. He makes the latter a little more precise when he goes on to cite, approvingly, a common definition of a natural fact (that is, a fact that can be stated by a naturalistic claim). According to this definition, a fact is natural ‘if facts of this kind are investigated or discussed by people who are working in any of the natural or social sciences’.⁵ Later he suggests how this can be made yet more precise. I shall not pursue these issues any further however. I am more interested in the second of the two accounts that is required: the account of what it is for two claims to state the same fact. At the beginning of §94 Parfit considers two senses in which two claims can be said to state the same fact. In one sense, which he calls the ‘referential’ sense, two claims can be said to state the same fact when they ‘refer to the same things and ascribe the same properties to these things’.⁶ In another sense, which he calls the ‘informational’ sense, two claims can be said to state the same fact when they ‘give us the same information’.⁷ There is a vast and familiar set of philosophical problems circulating here. What Parfit has provided, in effect, are two ways of individuating facts. But they clearly stand in need of individuation of their own. What counts as the same property? Or the same piece of information? Even a moderately full account of these matters would constitute a significant chapter in both the philosophy of mind and the philosophy of language, and no doubt in

³ This is encapsulated in the diagram in Vol. Two, p. 263. ⁴ Vol. Two, p. 265. ⁵ Vol. Two, p. 265. ⁶ Vol. Two, p. 336. ⁷ Vol. Two, p. 337.

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metaphysics too. Still, Parfit does not leave the matter there. He gives us some examples of what he has in mind. He juxtaposes the claim that Shakespeare was Shakespeare with the claim that Shakespeare was the writer of Hamlet. And he says that, in the referential sense, these two claims state the same fact, whereas, in the informational sense, they do not. Similarly in the case of the claim that water is water and the claim that water is H₂O. These examples are clearly meant to put us in mind of the Fregean distinction between Bedeutung and Sinn. And they do thereby give us a relatively clear idea of how we are meant to extrapolate. To be sure, they still leave countless questions unanswered. In fact I shall later raise doubts about whether one of these two ways of individuating facts is ultimately even coherent. Nevertheless, they give us enough to be going on with. Let us therefore bracket any reservations that we may have about what Parfit has provided and grant him, at least pro tempore, his two senses of what it is for two claims to state the same fact. The question now is: in which sense does non-analytical naturalism hold that any fact that can be stated by some normative claim can also be stated by some naturalistic claim? There seems to be a dilemma. The sense in question cannot be the referential sense, because that would not be enough to mark the view out as naturalistic: even the most resolute opponent of naturalism may acquiesce in the idea that, when facts are individuated as coarsely as this, any fact that can be stated by a normative claim can also be stated by a naturalistic claim. On the other hand, the relevant sense cannot be the informational sense either, because that would be in conflict with what is supposed to distinguish this kind of naturalism from analytical naturalism: to say that some normative claim ‘gives us the same information’ as some naturalistic claim is to say that the former can be reduced to the latter. In sum, the referential sense is too weak to do justice to non-analytical naturalism’s claim to be regarded as a species of naturalism, the informational sense too strong to respect its claim to be regarded as non-analytical. Parfit nowhere presents this dilemma in quite this form. But I hope that what I have just said is a fair representation of one strain in the complex set of arguments that we find at the beginning of Part Six of his book, where he mounts his opposition to various kinds of naturalism. I hope, in particular, that it captures his opposition to non-analytical naturalism. The issue that I wish to broach now is whether non-analytical naturalism can find some space between the horns of this dilemma. I believe that it can. Consider the following. (P) On any coherent way of individuating facts, any fact that can be stated by some normative claim can also be stated by some naturalistic claim. On the assumption that the informational way of individuating facts is coherent, this entails analytical naturalism. But it is open to the non-analytical naturalist to

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deny that assumption. (This is the doubt to which I referred earlier.) And, having denied it, the non-analytical naturalist can register his or her naturalism by endorsing (P). He or she can insist that, even if a particular normative claim and a particular naturalistic claim give us ‘different information’, there is no coherent sense of ‘different facts’ in which it follows that they state different facts. The point is not that facts cannot coherently be individuated any more finely than in the referential way. Clearly they can, as Parfit’s original examples help to show. The point is rather that they cannot coherently be individuated so finely that they are able to reflect all differences of information. In particular, they cannot coherently be individuated finely enough to reflect the differences of information involved in using normative and naturalistic vocabulary. Why might anyone mount this sort of challenge to the coherence of the informational way of individuating facts? To answer this question, let us return to the point at which we needed an account of what it is for a claim to be normative and of what it is for a claim to be naturalistic. Suppose that we were in a slightly different position. Suppose that some metaphysician had been talking, not about ‘normative claims’ and ‘naturalistic claims’, but about ‘A-claims’ and ‘B-claims’, these being two terms of art that he had introduced. Suppose further that this metaphysician gave us the same sort of initial steer as to what he intended by these two terms of art as Parfit gave us. That is to say, suppose he presented us with two lists of words and said that an A-claim was one that could be expressed using words of the first sort, while a B-claim was one that could be expressed using only words of the second sort. Suppose finally that the first list contained words like ‘past’, ‘future’, ‘now’, and ‘yesterday’, while the second list contained words like ‘earlier’, ‘later’, ‘simultaneous’, and ‘eve’. Assuming that this metaphysician was directing our attention towards the distinction between the tensed and the tenseless, an attractive view would be that: • some A-claims are true, that is to say some A-claims state facts; • no A-claim can be reduced to any B-claim; and: • on any coherent way of individuating facts, any fact that can be stated by some A-claim can also be stated by some B-claim. This view is of a kind that is familiar in the philosophy of time. It is a version of the so-called B-theory. The point of the third clause, which is obviously a direct equivalent of (P), is to repudiate the idea that there are ‘tensed’ facts, that is to say, facts individuated in the informational way and stated by tensed claims. The B-theorist believes that tense is a distinctive feature of certain claims, but not of

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any facts, and that nothing but confusion accrues from any attempt to cast it in the second of these roles. Before we return to the issue of normativity, there is an incidental ad hominem point that is worth making. Although Parfit does not himself engage with this debate in the philosophy of time, there is evidence that he would be sympathetic to this version of the B-theory. Consider the two following passages for example: What has happened is just as real as what will happen. Nor is the past less real than the present. Though people who are dead do not exist now, that is merely like the fact that people who live elsewhere do not exist here. And time’s passage is an illusion.⁸ It can seem meaningful to say that that we are moving through time into the future, or that nowness . . . is moving down the series of events from earlier to later, or that, every day, our death is getting closer. But such remarks, though they can seem deeply true, make no sense . . . Though our death is closer now than it was twenty years ago, that is merely like the fact that New York is closer to Washington than Boston is.⁹

Admittedly, these passages do not, strictly speaking, commit Parfit to the version of the B-theory in question. But little more is required to secure that commitment than the following enticing additional thought: that a tensed claim and a tenseless claim never give us the same information. For example, a claim to the effect that some football match will begin in a minute’s time gives us different information from a claim to the effect that that same football match begins (tenselessly) a minute later than 2.59 p.m. on 6 November 1971, even if the former claim is made at 2.59 p.m. on 6 November 1971. And what makes this additional thought enticing? Well, given a tensed claim and a tenseless claim, knowledge of the truth of one never suffices for knowledge of the truth of the other. I can know that the match will begin in a minute’s time without knowing what the date is. I can know that the match begins (tenselessly) at 3 p.m. on 6 November 1971 without knowing what the time is. And it seems appropriate to individuate pieces of information finely enough to ensure that, if one can know the truth of one claim but not another, then the two claims do not give us the same information. At any rate, there is at least as much reason to accept this additional thought as there is to accept that a normative claim and a naturalistic claim never give us the same information. But be Parfit’s own relation to the B-theory as it may, this version of the theory, if correct, signals how, in a world whose most finely individuated facts can always ⁸ Vol. Two, p. 427.

⁹ Vol. Two, p. 438.

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be stated by tenseless claims, occupancy of a temporal point of view allows for the possibility of using irreducibly tensed claims to state those same facts. Why then should we not accede to the idea of a world whose most finely individuated facts can always be stated by naturalistic claims but in which occupancy of a suitable point of view allows for the possibility of using irreducibly normative claims to state those same facts? In both cases there would be an irreducibly perspectival take on facts that were not themselves in any sense perspectival. And would not this latter idea deserve to be called ‘non-analytical naturalism’? An opponent of non-analytical naturalism might reply, ‘This is all very well. But all you have done is to indicate an abstract possibility, a position in logical space. You have come nowhere near a defence of non-analytical naturalism, and you will not have come anywhere near such a thing until you have further indicated what the relevant point of view could be.’ This is of course correct. Moreover, the best candidate for the relevant point of view would appear to be something involving conative states, which suggests that any non-analytical naturalist who takes this line will sooner or later be involved in defending some variation of the normative subjectivism that Parfit spends so much of his book attacking. It looks as if the discussion has not been advanced at all. It has not been advanced in the sense that it has not been brought any nearer to a satisfactory solution of any of the principal problems about normativity with which Parfit is concerned in his book. But it has been advanced in the sense that it has been brought back from a potentially unsatisfactory solution of some of those problems. The analogy with tense shows, I believe, that the material in Part Six of Parfit’s book on which we have been focusing has no independent force. Nonanalytical naturalists who hold that all normativity is grounded in conative states need have nothing to fear from the suggestion that, irrespective of what they take normativity to be grounded in, it is impossible to maintain both that all facts are in some substantive sense natural, which is what they need to maintain in order to count as fully paid-up naturalists, and that normativity is irreducible, which is what they need to maintain in order to count as a fully paid-up non-analytical naturalists. Furthermore, I believe that this kind of irreducibly perspectival take on facts that are not themselves in any sense perspectival signals something that is absolutely fundamental to the human condition. It sets the pattern for almost all of our engagement with reality, if not all of it. For Parfit to deny that the irreducibility of the normative is irreducibility of this sort, no matter what the relevant point of view may be, and in particular for him to deny that this is so no matter how deeply entrenched the relevant point of view may itself be in the human condition, is for him to make it increasingly difficult for his opponents to understand what he means by the normative. Here I want to revert once again to an ad hominem point. Parfit elsewhere famously defends a view about persons which he calls reductionism. This is the

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view, roughly, that a person’s existence and identity over time consist in the holding of facts that can be described ‘impersonally’, that is to say without explicit reference to the person and without presupposing his or her identity.¹⁰ His defence of this view is very powerful. But I believe that it can be supplemented in such a way as to make it a further instance of the pattern that I have been describing. What I have in mind is this. We can graft on to Parfit’s reductionism the idea that the concept of a person is an irreducibly perspectival concept, itself exercisable only from a point of view that is deeply entrenched in the human condition. This allows for the possibility of an alien being who does not share this point of view and who can thus never know how things are ‘personally’, however much it may know about how things are ‘impersonally’. Furthermore, the concept of a person has a crucial normative element to it. Part of what it is to recognize a person as such, or as the same as before, is to acknowledge him or her as making certain demands on one, and to acknowledge oneself as making certain demands on him or her. (Recall Locke’s famous observation that ‘person’ is a forensic term.) It matters very much whether something counts as a person, or as the same person—even if, as Parfit maintains, it matters less than we tend to think. This supplement to Parfit’s reductionism concerning persons therefore brings it very much into line with, and perhaps even turns it into an instance of, precisely the sort of non-analytical naturalism towards which I have been gesturing. I have described this as an ad hominem point, though there is clearly nothing here to embarrass Parfit. He need not have any truck with this proposed supplement to his reductionism. And there is nothing in the reductionism itself that stands in any obvious tension with his opposition to naturalism concerning the normative. Nevertheless, this supplemented reductionism does, I believe, give us a further sense of what is involved in this structure that I have been talking about— which I have already described as absolutely fundamental to the human condition and by whose means I have tried to indicate an important lacuna in some of the arguments that Parfit invokes to motivate his opposition to naturalism concerning the normative. Moreover, this supplemented reductionism—nay, the reductionism itself— reminds us of how easily the shoe could be on the other foot. Parfit admits that his reductionism concerning persons is ‘very hard to believe’.¹¹ He could easily find himself confronted with an opponent, as indeed I am sure he often has done, who simply finds the reductionism too hard to believe and insists, on intuitive grounds, that there are facts about a person’s existence and identity over time that are quite independent of any facts that can be described ‘impersonally’. I think Parfit would admit that he has nothing decisive with which to win over such an opponent. But when it comes to the issue whether there are normative facts that

¹⁰ Pp. 210 ff.

¹¹ P. 280.

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are quite independent of any facts that can be described naturalistically, Parfit is the one who adopts the non-reductive stance; it is he who criticizes his opponents’ views on the grounds that those views have consequences that are ‘hard to believe’.¹² And when one of his most notable opponents, Bernard Williams, finds mystery and obscurity in Parfit’s view, Parfit’s diagnosis is that Williams has simply failed to understand the view.¹³ The symmetry is striking. Not that there is anything here that need embarrass Parfit either. It is certainly not inconsistent for him to adopt a reductive stance in one of these debates and a non-reductive stance in the other. In any case the symmetry is not total. Consider the most extreme non-reductive view about persons, namely the Cartesian view. Parfit does not take even the Cartesian view to be mysterious or obscure. He describes possible circumstances in which there would have been evidence for its truth.¹⁴ He merely takes it to be false. He denies that any such circumstances obtain. I do not claim to have forced Parfit into some sort of corner then. I claim only that he would do well to look upon the debate about naturalism concerning normativity in the light of the debate about his own reductionism concerning persons and the debate about the B-theory concerning time. In all three cases, it seems to me, the more extravagant of the two principal available verdicts (that there are normative facts over and above any natural facts, that there ‘personal’ facts over and above any ‘impersonal’ facts, that there are tensed facts over and above any non-tensed facts) arises from a deeply engrained, perfectly natural, but ultimately doomed tendency to mistake a distinctive feature of some take that we have on the facts for a distinctive feature of the facts themselves. In the final paragraph of the Preface to the second edition of Kant’s Critique of Pure Reason Kant wrote that he was about to turn 63 and expressed the hope that others who were sympathetic to his project would help with its completion and dissemination. Kant’s book was one of the greatest philosophical works ever written, perhaps the greatest. Nevertheless, his hope was frustrated. The book’s impact, colossal as it was, was of a different kind. The book was more significant for the opposition that it provoked and the new lines of enquiry that it opened up than it was for any discipleship that it inspired. On p. 453 of Volume Two of On What Matters Parfit writes that he is 67 and expresses the hope that others who are sympathetic to his project will help with its completion and dissemination. I am certain that his hope will be similarly frustrated. But his book too is sure to provoke opposition and to open up lines of enquiry. I am glad to have been able to contribute to that process—if only by signalling what I take to be a deep error underlying the project.

¹² Vol. One, p. 81.

¹³ Vol. Two, p. 434–5.

¹⁴ Parfit (1984), §82.

15 Williams, Nietzsche, and the Meaninglessness of Immortality Abstract The first section of this essay considers the argument that Bernard Williams advances for the meaninglessness of immortality. It also considers various counter-arguments. An attempt is made to show that the more clearly these counter-arguments are targeted at the spirit of Williams’s argument, rather than at its letter, the less clearly they pose a threat to it. The focus of the second section of the essay is on Nietzsche, whose views about the eternal recurrence might appear to make him an opponent of Williams. It is argued that, properly interpreted, these views in fact make him an ally.

1. Williams 1.1. Williams’s Argument for the Meaninglessness of Immortality Bernard Williams’s untimely death, in 2003, added poignancy to one of his most forceful and most challenging essays, ‘The Makropolus Case’.¹ In this essay Williams had argued that a life without death would be meaningless; and, as his subtitle ‘Reflections on the Tedium of Immortality’ indicates, that it would be meaningless because it would eventually become tedious to the point of unendurability. Mortality, on Williams’s view, is something to be celebrated. His own death certainly posed a challenge to this view. Coming at a time when he had so much still to offer, it seemed to stand in defiant mockery of the idea that mortality is something to be celebrated and served rather as a stark reminder of what an evil death is. In fact, however, Williams had never denied that death is an evil. He makes clear from the very beginning of his essay that it is not his purpose to deny any

¹ B. Williams (1973c).

The Human A Priori: Essays on How We Make Sense in Philosophy, Ethics, and Mathematics. A. W. Moore, Oxford University Press. © A. W. Moore 2023. DOI: 10.1093/oso/9780192871411.003.0016

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such thing. Though he believes that all meaning and purpose must eventually drain away from life, and though he urges, in line with this, that death can come too late, he also concedes, what he says ‘many . . . need no reminding’,² that death can come too early. And, at least as things currently stand, the latter is the norm. Given that death’s coming too early is a matter of its depriving both the person who dies and others of goods, this furnishes a simple sense in which death is, at least normally, an evil. But there is a deeper structural point too. Williams distinguishes carefully (though he does not himself put it in these terms) between the idea that mortality is to be celebrated and the idea that death is to be celebrated. The idea that mortality is to be celebrated is the idea encapsulated in the very last sentence of his essay, that we are ‘lucky in having the chance to die’. Williams believes this because he believes that the alternative would be so much worse—indeed, to use an adverb whose use for once seems more than an organ of hyperbole, infinitely worse. If the alternative would furthermore be meaningless, as Williams insists it would be, and if life does in fact have some meaning, then we can even say that mortality is to be celebrated as a precondition of life’s having whatever meaning it has. But this is quite different from the idea that death is to be celebrated. Death, when it comes, can still deprive both the person who dies and others of basic opportunities to create and discover whatever meaning life might have. There is no logical conflict in death’s being a destroyer of meaning in life and mortality’s being a precondition of life’s having the very meaning that death destroys. There is no logical conflict, though there are undoubtedly conflicts of other kinds, and one of the many merits of Williams’s essay is how brilliantly he draws some of these out. Let us, however, stay with the logical point. It is a subtle point. Certainly there are many acute thinkers who have failed to grasp it, and in due course I shall cite Thomas Nagel as an example. The point emerges in Williams’s essay as follows. He explores the various ways in which death is an evil, or a ‘misfortune’ as he more frequently puts it, emphasizing that this is ‘other things being equal’.³ He then considers an apparent consequence, namely that ‘it would be not only always ² B. Williams (1973c), p. 100. ³ I am not sure that, strictly speaking, this is what he means. The use of the phrase ‘other things being equal’ in this context, if taken strictly, suggests that we can somehow treat death as a variable to be evaluated while other factors remain constant—as though there could be two situations whose only relevant difference was that one involved a death that the other did not—whereas Williams’s point is surely the point already adverted to, that death is an evil in circumstances which, at least as things currently stand, count as normal relative to its evaluation. This excludes, for instance, circumstances in which death brings longed-for relief from suffering that cannot be alleviated in any other way, circumstances which, as things currently stand, are comparatively rare. Still, granted that this is Williams’s point, then it would be cavilling to object to what he says: on a more colloquial use, the phrase ‘other things being equal’ is precisely what is required here, and I myself shall adopt that use in what follows.

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better to live, but better to live always, that is, never to die’.⁴ In as much as death is an evil, he says, ‘we seem committed to wanting to be immortal’.⁵ What is crucial here is the word ‘seem’: we seem committed to wanting to be immortal. The logical point to which I am referring emerges in Williams’s deliberate caution. This is echoed in the previous quotation by his use of the phrase ‘not only . . . but . . . ’: it would be not only always better to live but better to live always. The point is this: there is a logical gap between our always wanting something to be so, or its always being appropriate for us to want something to be so, and our wanting, or its being appropriate for us to want, this same thing always to be so. Put like that, the point sounds like a not very subtle point about a scope distinction. But the distinction is easy to miss, and Nagel for one misses it. In The View from Nowhere, published thirteen years after Williams’s essay, Nagel writes: ‘Given the simple choice between living for another week and dying in five minutes I would always choose to live for another week; and by a version of mathematical induction I conclude that I would be glad to live forever.’⁶ But that does not follow, either by a version of mathematical induction or by any other means. For one thing, the premise is concerned with choices I would make, whereas the conclusion is concerned with what I would be glad to do, which is a different matter. But also, more pertinently, the most that follows from the premise, as Nagel’s own reference to mathematical induction should have made clear, is that if, starting now, I were given a weekly choice between living for another week and dying in five minutes, then (since I would always choose to live for another week) my repeated choices would keep me alive for ever. This is not to say that I would ever actually choose to live for ever. I might have a clear preference not to live for ever, indeed I might be appalled at the prospect of living for ever, yet still never want these to be my last five minutes. I might never want to die, without wanting never to die.⁷ Now I have laboured this point partly just because of its intrinsic interest. It is not, however, Williams’s main point. Williams’s main point concerns the qualification that other things be equal, which is required even for Nagel’s premise to be true. Other things might of course not be equal. I might choose to die in five minutes because there was no other way of putting an end to some agony that I was suffering and that I would otherwise continue to suffer indefinitely.⁸ Williams’s point is that other things would eventually, and necessarily, not be

⁴ B. Williams (1973c), p. 89. ⁵ B. Williams (1973c), p. 89. ⁶ Nagel (1986), p. 224. ⁷ I have borrowed material in this paragraph from A. W. Moore (2019a), pp. xix and 226–7. Note: it might be said in defence of Nagel that my two criticisms cancel each other out; that the reason why his conclusion states what I would be glad to do, rather than what I would choose to do, is precisely that it adverts to an indefinite series of choices that I would make and not to a once-for-all choice. But the full context in which his argument occurs seems to me to belie any such interpretation. ⁸ See n. 3.

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equal. I would eventually, and necessarily, choose to die—never mind that I might in any case, independently, choose to be mortal rather than immortal. Williams’s reason for thinking this is that my life must eventually become, as I put it earlier, tedious to the point of unendurability. The kernel of his argument for this grim view is that the conditions that must be satisfied for my life to continue to count as mine militate against the conditions that must be satisfied for it to continue to be a life worth living. Conditions of the former kind demand a constancy, and conditions of the latter kind a variety, that cannot be reconciled. Towards the beginning of his essay Williams writes: ‘There is perhaps some profound temperamental difference between those who find consolation for the fact of death in the hope that it is only the start of another life, and those who equally find comfort in the conviction that it is the end of the only life there is.’⁹ Indeed. And it is surely just such a matter of temperament, as much as the forces of reason, that leads philosophers to disagree so trenchantly about the issues raised in Williams’s essay. Nagel, in the context from which I have already quoted, writes: ‘Perhaps I shall eventually tire of life, but at the moment I can’t imagine it, nor can I understand those many distinguished and otherwise reasonable persons who sincerely assert that they don’t regard their own mortality as a misfortune.’¹⁰ He then adds a footnote reference to Williams’s essay in which he asks, ‘Can it be that he is more easily bored than I?’ I dare say it can. Still, even those who find themselves on the Nagelian side of this temperamental divide can hardly fail to acknowledge the suasive and rhetorical power of Williams’s remarkable essay. Nagel says that he cannot imagine tiring of life. But Williams certainly shows that living indefinitely without tiring of life places heavy demands on the imagination too. (It would not be inconsistent, in fact, to claim that both things are unimaginable. For the precondition of both, namely living long enough for there to be an issue, might itself be unimaginable. It is worth noting in this connection that Williams’s conclusion is not in fact that immortality, conceived thus and so, would be meaningless, but rather that immortality, to the extent that it can be conceived, would be meaningless.) The question is, can imagination meet the heavy demands that Williams shows it to be under?

1.2. Counter-Arguments to Williams’s Argument Well, yes, in various ways it can—though not, as we shall see, in ways that address Williams’s real concerns. There are imaginable scenarios in which living for ever would be subjectively indistinguishable from living for just eighty years, say. I could for instance live for ever and, at the age of 80, suffer a permanent loss of

⁹ B. Williams (1973c), p. 83.

¹⁰ Nagel (1986), p. 224.

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consciousness. Now clearly that fails to address Williams’s concerns. Williams could quite reasonably say that such a loss of consciousness would be tantamount to death, and treat it as such. Or he could equally reasonably say that such a loss of consciousness would itself render my life thereafter meaningless. But there are subtler variations on the same theme, and, although these are likewise targeted at the letter of Williams’s argument rather than at its spirit, they cannot be dismissed in quite the same way. Consider, for example, the following case. The Decelerating Life Both I and everything in my local environment periodically start to function more slowly, and time accordingly seems to me to pass more quickly, in such a way that, whereas the first forty years of my life seem like forty years, the next forty seem like twenty, the next forty seem like ten, and so on ad infinitum. That is a scenario in which my endless life seems to me like a life of eighty years. Or consider this case. The Staccato Life I live normally for forty years, and these are followed by a trillion years of unconsciousness at the end of which everything reverts to the state that it was in at the beginning of that trillion-year period. I then live normally for twenty years, and these are followed a similarly undetectable trillion-year period of unconsciousness. I then live normally for ten years, and these in turn are followed by the same thing. And so on ad infinitum. Again, my endless life seems to me like a life of eighty years. In both these cases there are, at any point in my life, periods of subjectively normal life left for me.¹¹ That is why neither of them can be dismissed in quite the same way as the case in which I suffer a permanent loss of consciousness. The fact remains that, precisely because the endless lives in both these cases are indistinguishable to me from an eighty-year life, they are as far removed from Williams’s real concerns as the case in which I do permanently lose consciousness.¹² ¹¹ I am assuming that there is not a minimal period of consciousness. ¹² These two scenarios are adapted from two stories that I tell in A. W. Moore (2019a), pp. 227–8. For further discussion see Sorensen (2005). Concerning ‘The Staccato Life’, it is interesting to note that the crucial thing about the trillion-year bouts of unconsciousness is that they do not sum to a finite period. The story could just as well have been told, curiously enough, with intervals of a nanosecond; or, come to that, with a first interval of a nanosecond, a second and third interval of half a nanosecond each, a fourth, fifth, sixth, and seventh interval of a quarter of a nanosecond each, and so on; or indeed, for those who like the mathematical icing on their paradoxical cakes particularly thick, with a first interval of half a nanosecond, a second interval of a third of a nanosecond, a third interval of a fifth of a nanosecond, and so on through the reciprocals of the primes.

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Here is a case whose relevance to Williams’s concerns is less clear. The Repeating Life I live normally for eighty years, then lose consciousness. While I am unconscious I regress, both physically and psychologically, to the state that I was in when I was born and everything in my local environment reverts to the state that it was in at that time. I then regain consciousness and repeat the eighty years in exact detail, as does everything in my local environment, after which the same thing happens again. And so on ad infinitum. Here I live for ever, continually repeating my eighty-year life and never tiring of it (presuming, of course, that I do not tire of it the first time round). ‘The Repeating Life’ is somewhat reminiscent of ancient myths of an eternal recurrence in which there is a continually recurring cosmic cycle whereby I live out my life in exact detail again and again.¹³ There are, however, some potentially crucial differences. Two worries that arise about the very coherence of these ancient myths might be forestalled in the case of ‘The Repeating Life’. One of these is the worry about whether it makes sense to posit qualitatively indiscernible but numerically distinct cycles. It is clear why this worry arises in the case of the ancient myths, with their cosmic cycle. What might forestall the worry in the case of ‘The Repeating Life’, with its merely local cycle, is the possibility of non-recurrent developments elsewhere in the cosmos allowing numerically distinct cycles to be distinguished from one another in relational terms. The second worry is about whether it makes sense to identify the main actor in subsequent performances of this eighty-year drama with me. This worry, which actually tugs in the opposite direction to the first, in as much as it presupposes that there will be subsequent performances of the drama,¹⁴ arises in the case of the ancient myths for reasons that Williams himself has famously made graphic.¹⁵ How can anything make me identical to an atom-for-atom duplicate who is separated from me in time, granted that nothing can make me identical to an atom-for-atom duplicate who is separated from me merely in space? In the case of ‘The Repeating Life’, on the other hand, since there is bodily continuity between me and my doppelgänger, there might not be any such worry. There might not be. We would certainly need to hear more, and to think more, about the processes whereby I regress to babyhood before we could rest content on this point.¹⁶ If we wanted to vary the story in a way that made this worry look less severe, we could

¹³ See e.g. Simplicius’s ascription of this view to the Pythagoreans, quoted in Barnes (1987), p. 88. ¹⁴ The two worries accordingly reinforce each other: whatever appeases the one exacerbates the other. ¹⁵ See e.g. B. Williams (1973b). ¹⁶ Cf. another of Williams’s classic essays on these themes, B. Williams (1973a), p. 1.

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try letting the regress after my loss of consciousness take me back only as far as, say, my seventieth birthday, whereafter I keep repeating just the last decade of my life. However, if we did think that this made the worry look less severe, then it would be a real question, why. The idea, presumably, would be that the processes of rejuvenation required to take me from the state that I am in when I am 80 to the state that I was in when I was 70 are not radical enough to threaten the presumption that I retain my identity through them. But, as any Parfitian reductionist would be quick to point out, the relevant differences between these processes and the processes that take me all the way back to babyhood are differences of degree, not differences of kind: it is not at all clear why some quintessence of mine should be preserved through the former but not through the latter.¹⁷ Be any of that as it may, ‘The Repeating Life’ is not as immediately vulnerable to the charge of connecting only with the letter of Williams’s argument as either ‘The Decelerating Life’ or ‘The Staccato Life’. Does not a repeated eighty-year life afford more than whatever a one-off affords of whatever a one-off affords? This is a genuine question. There are those who would say that it obviously does. But there are others, in what may ultimately be another example of a deep temperamental divide, whose intuition is diametrically opposed and who would say that these recurring cycles are really only further testimony to the fact that living for ever can be subjectively indistinguishable from living for eighty years.¹⁸ A possible retort to those in the latter camp would be to posit cycles that are not exact repeats. Thus suppose that I keep playing out my eighty-year life except that I finish it differently each time: living that life would certainly not be subjectively indistinguishable from living the life that comprises just its first eighty years. The problem with this retort, at least in the present context, is akin to the problem with the variant on ‘The Repeating Life’ just considered. How much of my life is supposed to vary each time? Just the last decade? The point is this. However much is supposed to vary, there are surely nothing but differences of degree, where my identity is concerned, between this case and the case in which virtually the entire eighty-year period varies; and this in turn is pretty much the same as a case that Williams himself considers in his essay—a case in which there is a series of psychologically disjoint lives, and concerning which Williams, sceptical about whether personal identity is preserved through these lives, asks the following highly pertinent question: ‘[Could] this series of psychologically disjoint lives . . . be an object of hope to one who did not want to die[?]’¹⁹ The fact remains that the

¹⁷ See Parfit (1984), pt 3, passim. ¹⁸ Cf. Tanner (1994), p. 54. Roy Sorenson, in Sorensen (2005), argues that what matters to me in these various imagined scenarios is my ‘personal time’; but one of the problems, in the case of ‘The Repeating Life’, lies precisely in deciding whether or not my ‘personal time’ is the same as it would be if I lived for only eighty years. ¹⁹ B. Williams (1973c), p. 92; see further pp. 93–4.

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more clearly an imagined scenario connects with what is of real concern to Williams, the less clearly it poses a threat to his argument.²⁰

2. Nietzsche The cosmic cycle will put many in mind of Nietzsche. Nietzsche assigned very great significance to the idea of eternal recurrence: he described it as the ‘highest formula of affirmation that is at all attainable’.²¹ Now it seems clear from Nietzsche’s copious references to this idea that he saw in the eternal recurrence—the apparent countermeasure of all transience and irreversible loss—something to be greeted with joy. This makes him look like an enemy of Williams—at least if we bracket the various reservations that we have just been considering about the extent to which recurring cycles could really yield the kind of endless existence that Williams is concerned to deprecate. But we must proceed cautiously. How exactly Nietzsche understood this idea, and what use he made of it, are by no means obvious. I shall suggest that, properly understood, Nietzsche can in fact be seen as a significant ally of Williams, even without the bracketing. Let us first of all lay to rest any notion that Nietzsche wants to defend the idea of a recurring cosmic cycle as a theory about the actual nature of the universe. There are passages, to be sure, in which he toys with arguments concerning the play of finite resources in infinite time, arguments whose upshot seems to be that all of the finitely many states that the universe can be in it will be in, and will be in again, infinitely many times. They are unconvincing arguments, easily rebutted.²² But there are issues about what exactly Nietzsche is doing with these arguments. And in any case, the passages in question occur in The Will to Power,²³ about which Williams himself reminds us that it ‘is not a book by Nietzsche at all, but a selection from [his unpublished] notes tendentiously put together by his sister’.²⁴ If Nietzsche is concerned with the idea of a recurring cosmic cycle at all, then he is surely concerned with it as a thought-experiment, designed as a guide to living. Many people have interpreted Nietzsche in just this way.²⁵ So interpreted,

²⁰ That is in fact as far as I shall go towards defending Williams in this essay. I have not ruled out the possibility of an imagined scenario in which the balance is suitably struck. One type of case worth considering is that in which there is an upper limit to how far back my memory stretches at any given moment—rather as if I were a goldfish with a three-second memory span, perpetually circling a goldfish bowl and taking fresh delight in the castle each time it comes into view. ²¹ Nietzsche (1967b), ‘Why I Write Such Good Books’, and Nietzsche (1969), §1. Cf. Nietzsche (1990), ‘What I Owe to the Ancients’, §5. ²² Cf. Schacht (1983), pp. 263 ff. ²³ E.g. Nietszche (1967c), §§1062 and 1066. ²⁴ B. Williams (2006f), p. 319. Note: I myself shall frequently refer to Nietzsche (1967c) in subsequent footnotes; whenever I do, Williams’s caveat should obviously be borne in mind. ²⁵ I have: see A. W. Moore (2019a), ch. 7, §5. I try to explain below why I no longer accept such an interpretation. See also, for a brief corrective, A. W. Moore (2019a), ch. 16, §4, which comprises material added in the third edition of my book.

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Nietzsche is exhorting us to act out our lives as if there were such a cycle; to try to live in such a way that we could bear the infinite repetition. What might this involve? One thing that it might involve is striving to redeem our own pasts—not in a spirit of regret or remorse, but by living in such a way as to make sense of our pasts, as an integral part of our continuing biographies. Another thing that it might involve is striving to redeem the world’s past (to whatever extent it is the prerogative of any individual to do that). Williams, who himself interprets Nietzsche in this way, puts as follows what he takes Nietzsche’s exhortation to be: to confront, honestly and truthfully, all the horrors and all the suffering in the world; to acknowledge these as inextricably bound up with whatever we value (Williams makes memorable reference at one point to all the ‘dreadful [happenings] that [have] been necessary to create Venice’²⁶); to reject a Leibnizian cost– benefit analysis, or a Hegelian historical metaphysics, whereby it is all worth it; to reject, equally, a Schopenhauerian pessimism, whereby it is not all worth it; and, as the only way of sustaining both these rejections, without being crushed or choked by all the horrors and all the suffering, to refuse to assess the world at all, but rather to affirm it, and to mean the affirmation, that is to will the infinite repetition.²⁷ This interpretation finds support in Nietzsche’s claim that there cannot be any redemption of the past unless ‘It was’, which he describes as ‘the will’s teethgnashing and most lonely affliction’, is transformed into ‘Thus I willed it’.²⁸ It also finds support in Nietzsche’s frequent insistence that what is important about the eternal recurrence is, indeed, whether we can bear it.²⁹ But what is not clear, as Williams himself points out, is why, on this interpretation, the idea of the eternal recurrence should be the idea of an eternal recurrence. A thought-experiment involving just one repetition of the cycle would do the job as effectively as a thought-experiment involving infinitely many. As Williams puts it, ‘If you could overcome the “nausea” . . . of the prospect that [the past] . . . will come round again even once, and say “yes” to it, you would have taken the essential step: could willing all those further recurrences cost you very much more?’³⁰ ²⁶ B. Williams (2006f), p. 319. ²⁷ B. Williams (2006f), pp. 317–19. Cf. B. Williams (2006e), pp. 52–4. ²⁸ Nietzsche (1969), pt 2, ‘Of Redemption’. ²⁹ Nietzsche (2001), §§285 and 341; Nietzsche (1969), pt 3, ‘Of the Vision and the Riddle’, §2, and ‘The Seven Seals’; pt 4, ‘The Intoxicated Song’; Nietzsche (1967b), ‘Why I am So Clever’, final paragraph; and Nietzsche (1967c), §§1053–9. ³⁰ B. Williams (2006f), p. 319, emphasis in original. We might wonder about the force of the rhetorical question at the end of this quotation. Prima facie, if willing one recurrence could cost you anything, then willing all those further recurrences could cost you very much more. But we must not forget that what is at issue here is the cost of willing the recurrences, not the cost of enduring them. In order to will even one recurrence, you would already have to think in as much vivid detail as possible of all the horrors and all the suffering: it would already cost you as much as that. Could willing all those further recurrences cost you very much more? ‘Very well,’ an opponent might say, ‘but if enduring infinitely many recurrences could cost you very much more than enduring only one, then is that not itself a reason why, on this interpretation, the eternity of the eternal recurrence matters?’ No. This is

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For this reason, among others, I do not myself accept this interpretation— although I do think that Williams has captured very well many of Nietzsche’s true concerns. In particular he helps to highlight something that is unquestionably in Nietzsche: the idea that everything is knotted together in such a way that the recurrence of one thing is the recurrence of all things, the affirmation of one the affirmation of all.³¹ That this idea is unquestionably in Nietzsche is also, I think, a key to the correct interpretation of him. This is an interpretation whereby he wants neither to defend the idea of a cosmic cycle as a cosmological theory about how things actually are nor to use it as a heuristic thought-experiment: he is not concerned with it at all.³² Nietzsche says at one point in his notebooks, ‘It is simply a matter of experience that change never ceases’.³³ What the knotting together of things means is that the ceaseless change is a ceaseless change in everything, including everything that has been and everything that will be. The whole of the past and the whole of the future come together in each moment of change. This is the eternal recurrence: the eternal recurrence of all things, but ever different.³⁴ Here is Nietzsche again, in the words of Zarathustra: Behold this gateway . . . it has two aspects. Two paths come together here: no one has ever reached their end. This long lane behind us: it goes on for an eternity. And that long lane ahead of us—that is another eternity.

not, as we might think, because the costs in each case would have compensating benefits—indeed equivalently compensating benefits, so that determining whether the costs in one case would be outweighed by the benefits is tantamount to determining whether the costs in the other case would be. It is rather because, on this interpretation, affirming the world is not a matter of balancing costs against benefits at all. On the contrary, it is a matter of ‘[becoming] . . . well [enough] disposed . . . to life’ (Nietzsche (2001), §341) for its costs no longer even to constitute a weight. I am grateful to Robert Stern for a discussion that forced me to clarify my thinking about this matter. I am also grateful, incidentally, to Tom Stern for the following additional pair of suggestions about why, on this interpretation, the eternity of the eternal recurrence matters: first, it is an expression of the inextricability of all things; and second, it precludes our eventual escape from the cycle, whether into an existence of some other kind or into non-existence. Each of the observations is well taken in its own right, but neither seems to me to block Williams’s thought that, if you could will even one recurrence, you would have taken the essential step. The first is not really relevant to it; the second is still too closely tied to considerations of compensation. ³¹ Nietzsche (1996), bk 1, §208; Nietzsche (2001), §54; Nietzsche (1969), pt 3, ‘Of the Vision and the Riddle’, §2, and pt 4, ‘The Intoxicated Song’, §10; and Nietzsche (1967c), §§293, 331, and 1032. Cf. Nehemas (1985), pp. 6–7. (Note: Nehamas is someone else who accepts an interpretation of the kind I now reject. For a particularly powerful expression of his interpretation, see Nehemas (1985)i, pp. 167–8.) ³² I claim no originality for this interpretation. I am indebted to Deleuze (1983), esp. chs 2 and 5; and to Turetzky (1998), ch. 8, §§2–3. (I have also been greatly helped by discussions with Philip Turetzky, to whom I am extremely grateful.) That said, I am acutely aware of diverging in substantial ways from each of these texts. There is much in what follows that is quite out of accord with the interpretation that either Deleuze or Turetzky offers. ³³ Nietzsche (1967c), §688. ³⁴ In what ways the past and the future are ever different will, I hope, emerge in due course.

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They are in opposition to one another, these paths; they abut on one another: and it is here at this gateway that they come together. The name of the gateway is written above it: ‘Moment’. . . . From this gateway Moment a long, eternal lane runs back: an eternity lies behind us. Must not all things that can run have already run along this lane? Must not all things that can happen have already happened, been done, run past? And if all things have been here before: what do you think of this moment . . . ? Must not this gateway, too, have been here—before? And are not all things bound fast together in such a way that this moment draws after it all future things? Therefore—draws itself too? For all things that can run must also run once again forward along this long lane. . . . [M]ust we not return and run down that other lane out before us, down that long, terrible lane—must we not return eternally?³⁵

What I am suggesting is that for all things that can happen to have already happened, and to return again, is for everything to be, as Nietzsche elsewhere puts it, ‘chained and entwined together’.³⁶ What happens at any moment, on this interpretation, happens at every moment—albeit at some moments as future, at some moments as present, and at some moments as past. There are various more or less remote analogues of this in other thinkers. For example, in the Stoics there is the idea of Aion, or ‘incorporeal’ time, resulting from the continuous division of the ever-present Chronos, or ‘corporeal’ time, into past and future;³⁷ and in McTaggart there is the idea of an ever-changing Aseries, in which events are ordered as past, present, or future, in accord with the series’s continuous movement along a constant B-series, in which events are ordered as earlier or later than one another.³⁸ But one analogue that I find particularly interesting—in spite of some obvious limitations, not least of which is the fact that it is non-temporal—is that of Leibniz’s idea of a universe of monads, each affording its own different perspective on the whole, so that, in Leibniz’s own words, ‘it is as if there were as many different universes’.³⁹ Nietzsche likewise sees each moment as affording its own different perspective on the whole, its own different vantage-point from which to interpret the whole.⁴⁰ In his

³⁵ Nietzsche (1969), pt 3, ‘Of the Vision and the Riddle’, §2, all emphasis in original. See also Nietzsche (2001), §109; and Nietzsche (1967c), §617. ³⁶ Nietzsche (1969), pt 4, ‘The Intoxicated Song’, §10. ³⁷ For discussion and references see Turetzky (1998), ch. 3, §4. ³⁸ See McTaggart (1993). ³⁹ Leibniz (1973), §57; see also §§56 and 60. ⁴⁰ In Nietzsche there is the added twist that there is nothing to the whole beyond how it is interpreted: see e.g. Nietzsche (1967c), §§477, 481, and 567. Cf. also Nietzsche (2001), §374; Nietzsche (1973), §§14 and 16; and Nietzsche (1967a), essay 3, §12.

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notebooks, he writes that ‘the world . . . has a differing aspect from every point; its being is essentially different from every point’.⁴¹ This presents a vision of the eternal recurrence very different from that in each of the two interpretations that I have rejected. It has in common with one that it is a vision of a real feature of the universe. It has in common with the other—for reasons and in ways that I shall try to clarify—that it is a vision of something about which the most fundamental question, for us, is whether we can bear it. But it differs from each in excluding altogether the idea of a recurring cosmic cycle.⁴² The real importance of the eternal recurrence, for Nietzsche, lies in its relation to nihilism, which he defines as the conviction that there is no meaning in this ceaseless change: there is no transcendent atemporal structure by which it can be justified nor any telos towards which it is striving.⁴³ This, coupled with a refusal to forget all the suffering in the world, a refusal whose importance to Nietzsche Williams emphasizes, makes all the suffering seem unbearable. (Suffering with a purpose is one thing. Meaningless suffering is something else entirely.⁴⁴) And this in turn results in condemnation of the world, condemnation of this grievous senseless ceaseless change. Nietzsche’s question is how such nihilism can be overcome. How can we see all the suffering in the world as meaningless and not be broken by it? We might think that one way would be to accept everything, in a spirit of resignation, in other words to say ‘yes’ to everything. But Nietzsche is adamant that such passive and indiscriminate acquiescence would itself be meaningless and would leave nihilism entirely undefeated.⁴⁵ As Williams indicates, for nihilism to be overcome, the world must be affirmed. But affirming the world does not consist in saying ‘yes’ to everything. It consists in creatively and actively making sense of things—creatively and actively, because,

⁴¹ Nietzsche (1967c), §568. It is worth noting in this connection the use of the phrase ‘continual recurrence’ in §569.2. Another interesting comparison, incidentally—especially in light of how I shall develop this interpretation—is with the idea expressed by Wittgenstein in Wittgenstein (1961), 6.43: ‘If the good or bad exercise of the will does alter the world, it can alter only the limits of the world, not the facts . . . In short the effect must be that it becomes an altogether different world. It must, so to speak, wax or wane as a whole. The world of the happy man is a different one from that of the unhappy man.’ ⁴² Nietzsche rejects the idea of a recurring cosmic cycle in e.g. Nietzsche (1997), ‘Schopenhauer as Educator’, §1. Cf. also Zarathustra’s admonishment of the dwarf in Nietzsche (1969), pt 3, ‘The Convalescent’, §2. It is true that Zarathustra sees recurrence as a ring (Nietzsche (1969), pt 3, ‘The Seven Seals’; cf. his description of himself as the advocate of the circle in pt 3, ‘The Convalescent’, §1). But he sees it as the ring of what Nietzsche elsewhere calls ‘the eternal hourglass of existence [which] is turned over again and again’ (Nietzsche (2001), §341). ⁴³ See Nietzsche (1967c), bk 1, §1. That there is no such transcendent atemporal structure is due largely to the fact that we ourselves have destroyed it: we have killed God (Nietzsche (2001), §125). That there is no such telos is indicated by the very passage of time. For if there were such a telos, then the universe ought already to have reached it; there would be, as it were, no point to time (Nietzsche (1967c), §§55, 708, and 1067). ⁴⁴ See e.g. Nietzsche (1967a), essay 2, §7, and essay 3, §28. For a fascinating discussion see B. Williams (2006g). ⁴⁵ Nietzsche (1969), pt 4, ‘The Awakening’.

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granted nihilism, there is no sense or meaning already there to be discovered. (To overcome nihilism is not to refute it.) Making sense of things involves saying both ‘yes’ and ‘no’, ‘yes’ to some things and ‘no’ to others. This prevents the ‘yes’ of sense-making from being the meaningless ‘yes’ of acquiescence. The ‘yes’ of sensemaking is a ‘yes’ as opposed to . . . In particular, of course, it is a ‘yes’ as opposed to the ‘no’ that is directed at nihilism itself. Affirming the world does not mean saying ‘yes’ to everything, but it does mean saying ‘yes’ to the eternal recurrence of everything. There is a sense, as Nietzsche urges in his notebooks, in which the eternal recurrence presents the spectre of meaninglessness in its most extreme and most terrifying form, a form in which meaninglessness recurs and recurs and recurs, ad infinitum.⁴⁶ And yet this eternal recurrence is the very condition of sense-making. In its continual generation of new perspectives it allows for the continual generation of new interpretations and new evaluations. Through these the past can be continually transformed, so that, although it keeps returning, it keeps returning differently. The past can be continually lived, continually developed, continually cultivated. That is to say, sense can be continually made of it. This is how it is prevented from overcoming us. But the eternity of the eternal recurrence is vital. Nihilism can never be overcome once and for all. If ever the process were to cease, it would meet with an unanswerable ‘So what?’, and nihilism would stand undefeated.⁴⁷ At one level, sense-making is the business of individuals, acting out their own lives. But at the most fundamental level, sense-making is an activity of the will to power, a productive principle manifest in all change.⁴⁸ There is an issue, then, about what sense can be made of human life itself. More particularly, there is an issue about what sense can be made of the eventual termination of each individual human life in death. This is an issue which, in a Nietzschean context, is nonnegotiable. Death is simply there to be reckoned with, something that has to be affirmed if nihilism is to be overcome. Wishing it away is not only pointless; it is a major obstacle to the defeat of nihilism. It is part of that morbid inability to come to terms with who one is, and with how things are, and with the meaninglessness of how things are—that morbid inability to come to terms with nihilism—which Nietzsche calls ressentiment.⁴⁹ Little wonder, then, that Nietzsche’s texts abound with affirmations of death.⁵⁰ ⁴⁶ Nietzsche (1967c), §55. This is marvellously captured in Kundera (1984), pt I, §§1 and 2. ⁴⁷ For the material in the last two paragraphs, see: Nietzsche (2001), §373 and appendix, ‘Towards New Seas’; Nietzsche (1973), §56; Nietzsche (1967a), essay 3, §12; Nietzsche (1990), ‘What I Owe to the Ancients’, §5; Nietzsche (1967b), ‘Why I Write Such Good Books’, The Birth of Tragedy, §3; and Nietzsche (1967c), §§12–13, 567, 616, and 1067. Cf. Nehamas (1985), pp. 163–4. ⁴⁸ See e.g. Nietzsche (1967c), §§643, 676, and 1067. ⁴⁹ Nietzsche (1967a), essay 3, §14; and Nietzsche (1967b), ‘Why I am So Wise’, §6. ⁵⁰ Here is a sample: Nietzsche (2001), §365; Nietzsche (1969), pt 1, ‘Of Voluntary Death’ (whose opening sentence is, as we heard earlier, echoed in B. Williams (1973c): ‘Many die too late and some die too early,’); Nietzsche (1969), pt 4, ‘The Intoxicated Song’, §11; Nietzsche (1990), ‘Expeditions of an Untimely Man’, §36; and Nietzsche (1967a), appendix, ‘The Wanderer and his Shadow’, §322.

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Does this make him an ally of Williams? In a way, it seems to take him further than Williams; in a way, not so far. It seems to take him further than Williams in that it involves the affirmation of death, not just the affirmation of mortality. (This harks back to the distinction that I drew in §1.1 above.) In fact, however, it is not at all clear that Williams need balk at the affirmation of death. True, he does not celebrate death, in the way in which he celebrates mortality. Indeed he is prepared to regard death as, at least typically, a misfortune. But affirmation is not the same as celebration. One way to make sense of something is to make sense of it as, precisely, a misfortune. More interesting, for current purposes, is the way in which Nietzsche’s affirmation of death seems not to take him as far as Williams. For it seems to leave open the question—the profoundly un-Nietzschean question—‘Would immortality be preferable to mortality?’ The fact that this question is profoundly un-Nietzschean is of no small moment of course: it relates back to the point that death is simply there to be reckoned with. Even so, I want to close by suggesting that it has a Nietzschean answer of sorts, and that this answer does indeed make Nietzsche an ally of Williams.⁵¹ Central to this answer is the way in which that whose life is terminated by any given death, the ‘subject’ of the death, is itself (or himself or herself) a creature of interpretation, something begotten of the activity of sense-making. ‘The “subject” ’, writes Nietzsche, ‘is not something given, it is something added and invented and projected behind what there is’.⁵² This is an idea that he emphasizes time and again.⁵³ And it means that, even if there were natural processes that admitted of interpretation in terms of an immortal subject, ‘interpretation’ would be the operative word. There would still be an issue about what was to be said in favour of such an interpretation. In considering how this issue might be addressed, we should note first that there are two ways in which interpretation could beget an immortal subject, reminiscent of the scope distinction considered in §1.1. One way would be as follows: whenever the question arose as to whether the subject’s life had ended, it was answered negatively. Another way would be as follows: the question as to ⁵¹ There are, incidentally, various passages in Nietzsche that appear to indicate such an answer but do not really do so. Thus in Nietzsche (1967c), §765, he writes, ‘Christianity has accustomed us to the superstitious concept of the “soul,” the “immortal soul,” soul-monads that really are at home somewhere else and have only by chance fallen, as it were into this or that condition, into the “earthly” and become “flesh” . . . With this idea, the individual is made transcendent; as a result, he can attribute a senseless importance to himself.’ But this remark is exactly not targeted at the immortality of the ‘immanent’ individual, the immortality that concerns Williams (B. Williams (1973c), p. 96). Again, in Nietzsche (1967c), §676, Nietzsche writes, ‘In the long run, it is not a question of man at all: he is to be overcome’ (cf. Nietzsche (1969), prologue, §§3 and 4, and pt 4, ‘Of the Higher Man’, §3). But this too is another matter: the correct interpretation of this passage, and of related passages, lies far beyond the scope of the current essay. ⁵² Nietzsche (1967c), §481. ⁵³ The idea is especially prominent in his notebooks: see Nietzsche (1967c), §§370–1, 481–92, and 631; cf. also §§523–9, 546–50, and 715. See also Nietzsche (1973), §§17 and 54; and Nietzsche (1990), ‘ “Reason” in Philosophy’, §5.

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whether the subject’s life would ever end was answered negatively in advance, so that, thereafter, it never arose again. It is the second of these that would more directly correspond to a preference for immortality over mortality. And, in a Nietzschean context, it would be something close to a disaster. It would not only hinder but positively oppose the continual creation of new meanings and new values needed to overcome nihilism—not just because there was a question that it foreclosed, once and for all, but because the foreclosing of that question had such a stifling effect on the addressing of subsequent questions. It would both prohibit radical novelty of various kinds and remove even from what it did not prohibit that vital element of uncertainty and limitation needed to ensure that what was being invested, in the effort to make sense of things, was precious enough, and its investment risky enough, to yield real sense in return. This would be an interpretation to counter interpretation. For Nietzsche, rather as for Heidegger, a life in which life itself was not always at issue, that is to say a life in which death was not always a possibility, would be a standing invitation for meaninglessness to reassert itself.⁵⁴ Here, at least, there would be some sort of convergence between Nietzsche and Williams. But would it not be just as bad, in its own different way, to interpret the subject as having died, if such an interpretation were not mandatory? Would there not be a fundamental symmetry here, between foreclosing the question of the subject’s mortality with one answer (pronouncing the subject immortal) and foreclosing it with the opposite answer (pronouncing the subject dead)? And would that not provide a Nietzschean rationale for interpreting the subject as immortal in the first of the two ways indicated above, that is by never in fact pronouncing the subject dead—which would set Nietzsche apart from Williams after all? I do not think so. I see an asymmetry here. (Or at least, I see an asymmetry to the extent that I can prescind from the increasingly un-Nietzschean character of this discussion: we must not lose sight of the caveat entered above.) The asymmetry connects with the cardinal point of ‘The Makropolus Case’, which I earlier put as follows: the conditions of constancy that must be satisfied for a life to continue to count as mine militate against the conditions of variety that must be satisfied for it to continue to be worth living. There is, I suggest, a very similar tension here. It is due to what might be called the longeval law of diminishing returns. The preservation of the subject, beyond a certain point, and at the expense (let us not forget) of new subjects, would be counter-productive. Where allowing the subject to die, in favour of those other subjects, would open up new possibilities of narrative, new opportunities for sense-making, and new ways of defying nihilism, preserving the subject would impose restrictions and constraints on ⁵⁴ See Heidegger (1962), pt 1, div. 2, §1. Cf. Williams’s claim, in B. Williams (1973c), p. 82, that ‘some existentialists . . . seem to have said that death was what gave meaning to life, if anything did, just because it was the fear of death that gave meaning to life’.

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subsequent interpretation that would constitute an overall burden. To resist the destruction of the subject in spite of that—perhaps through nostalgia, perhaps through pity, perhaps through fear, perhaps through some craving for infinitude—would be of a piece with the nihilistic condemnation of change in favour of permanence, of becoming in favour of being.⁵⁵ It would represent a kind of petrification which, much like the cessation of all sense-making, would eventually meet with an unanswerable ‘So what?’ There is, I believe, an intimation of this line of thought in the following quotation from Nietzsche: Affirmation of life even in its strangest and sternest problems, the will to life rejoicing in its own inexhaustibility through the sacrifice of its highest types . . . , that is what I recognized as the bridge to the psychology of the tragic poet. Not so as to get rid of pity and terror, . . . but, beyond pity and terror, to realize in oneself the eternal joy of becoming—the joy which also encompasses joy in destruction . . . ⁵⁶

I conclude that, for Nietzsche, much as for Williams, immortality would nullify the very resources needed to overcome the sense of life’s meaninglessness.⁵⁷

⁵⁵ Nietzsche (1967c), §§12 and 617. ⁵⁶ Nietzsche (1990), ‘What I Owe to the Ancients’, §5, all emphasis in original. ⁵⁷ I presented an early draft of this essay as a lecture at the Central European University: I am very grateful to members of the audience for their comments. I am also very grateful to the editor of Mind for his comments on the same draft.

PART IV

HO W W E MA K E S EN S E I N MATHEMATICS

16 On the Right Track Abstract This essay first appeared as a critical notice of Crispin Wright’s Rails to Infinity. Its main focus is one of the main foci of Wright’s book: how self-consciousness about a ‘grammar’ of ours, to use Wittgenstein’s term, can lead to various sorts of scepticism about the acceptability of that grammar. This idea is considered principally in connection with mathematics. It is argued, in line with both Wright and Wittgenstein, that we can resist such scepticism provided that we also resist asking certain constitutive questions in connection with the grammar, which in turn involves acquiescing in the idea that we could not so much as operate with the grammar were it not for certain facts of (human) nature. The final section of the essay takes issue with Wright’s treatment of Wittgenstein’s argument against the possibility of a private language.

1. Introduction Suppose that everyone accepted the statement ‘Twice two is five’, simply because they used the numeral ‘five’ in the way that we currently use the numeral ‘four’. Would that make twice two five? It wants only a modicum of philosophical sophistication, combined with some awareness of the difference between mentioning an expression and using it, to answer no. Now suppose that everyone believed that twice two is five. Would that make twice two five? The second question seems much harder. This is not so much because it is unclear whether, if the supposition in question held, that would indeed make twice two five, but because it is unclear what the supposition in question is. What, as Wittgenstein once asked, would it be like for everyone to believe that?¹ How would it differ—how could it differ—from a mere notational discrepancy? We can imagine people who come to accept the statement ‘Twice two is five’ via an arithmetic just like ours, and who come to accept it, moreover, in the

¹ Wittgenstein (1967a), p. 226. It is something of a philosophers’ artefact, as Wittgenstein hints, to talk about everyone’s ‘believing’ a mathematical proposition in the first place.

The Human A Priori: Essays on How We Make Sense in Philosophy, Ethics, and Mathematics. A. W. Moore, Oxford University Press. © A. W. Moore 2023. DOI: 10.1093/oso/9780192871411.003.0017

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conviction that they have not merely changed their terminology but have, on the contrary, made some sort of cognitive advance, say because they are trying to accommodate some bizarre discovery in quantum mechanics, or because they are trying to accommodate some bizarre discovery in (for that matter) arithmetic, or because they have been brainwashed. Yet how clear is it, in any of these cases, what we are imagining, let alone that we are imagining something that merits the description ‘everyone’s believing that twice two is five’? After a while, it is hard to resist the conclusion that nothing would in fact count as everyone’s believing that twice two is five. And it is hard not to attribute this in turn to the (flagrant) impossibility of twice two’s being five. Indeed, whatever minimal commitment to realism we have already incurred—in answering no to the first question above—it is hard not to take a step beyond that to the following much more radical view. Our mathematical concepts are answerable to mathematical reality, not just in the sense that whether we count as exercising our mathematical concepts correctly depends on what mathematical reality is like, but in the sense that whether we count as exercising mathematical concepts at all depends on what mathematical reality is like. Unless our mathematical thinking were justified by mathematical reality, it would not be mathematical thinking. This is one of many views that might attract the label ‘Platonism’. What can be said about it? Well, one thing that can be said about it is that it is an anathema to Wittgenstein. At one point he asks whether our number system resides ‘in our nature or in the nature of things’, and he answers, ‘Not in the nature of numbers.’² Indeed many commentators would say that Wittgenstein accepts the converse answerability, the answerability of mathematical reality to our mathematical concepts. Some might even say, reverting to the second question above, that he thinks that everyone could believe that twice two is five, and that this would make twice two five. I disagree. There are times, certainly, when he totters on the brink of saying either that or something like that.³ But I think his considered view is rather the following. In accepting the statement ‘Twice two is four’, along with the other arithmetical apparatus needed to make us reject the alternative ‘Twice two is five’, we are endorsing certain rules of representation.⁴ If we accepted the statement ‘Twice two is five’, we would be endorsing different rules, using homonyms. This would have to be a notational discrepancy. No concept of ours would be the concept of five if we allowed it to be interchangeable with the concept of twice two. Twice two could not be five: such is our rule. And we could not ‘believe’ that twice two is five. ² Wittgenstein (1967b), §357, emphasis in original. ³ See again the passage referred to in n. 1. ⁴ Cf. his claim that ‘3 + 3 = 6’ is a rule as to the way in which we are going to talk, as quoted by G. E. Moore (1959b), p. 279.

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Very well; but if this is not expressive of the answerability of our mathematical concepts to mathematical reality, then must it not be expressive of the converse answerability after all? No doubt it is too crude to suggest that we could have made twice two five. But there does seem to be a constitutive link in this picture between the value of twice two and our rules. Does this not suggest that, unless it is because twice two is four that we have the rules we have, then it must be because we have the rules we have that twice two is four? I am sure that Wittgenstein would demur at saying either of these. He would urge scepticism about whatever metaphysical question appears to have these as its only two possible answers. Twice two is four. In saying this, we are expressing one of our rules. And it is contingent that we have the rules we have. We could quite easily and quite properly have had different rules. Had things been different in various specifiable ways, and in particular had we been different in various specifiable ways, we would have had different rules. But twice two would not then have been other than four. Rather, we would not have thought in those terms. Nor have we made twice two four. That twice two is four is a mathematical necessity, and if it has any explanation then it has a mathematical explanation.

2. Platonism and Cartesianism Wittgenstein’s rejection of the Platonism outlined in the previous section, and the question of what should replace it if he is right, are among the main concerns of Crispin Wright’s excellent new collection of essays Rails to Infinity.⁵ This collection includes previously published work by Wright in this area (though in some cases, it must be said, only on a rather liberal interpretation of ‘this area’) together with his previously unpublished Whitehead lectures on self-knowledge. There is also some new material in the four introductions written for the four sections into which the essays are organized, and in a pair of postscripts. The collection is a powerful reminder of how much there is to learn from Wright’s penetrating work on Wittgenstein or of broadly Wittgensteinian inspiration. There is far more in the collection than I can discuss here. I shall confine myself for the most part to what Wright says about mathematics, and indeed to some rather limited aspects of what he says about mathematics. But first I want to make some observations in connection with the other principal topic of the collection: the privacy of psychological phenomena. The privacy of psychological phenomena generates exactly the same dialectic as that which I described in the previous section in connection with mathematics. When we reflect on our psychological concepts, in which this privacy finds ⁵ Wright (2001), the subject of the symposium in which this essay first appeared. All unaccompanied references will be to this book.

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expression, we feel an urge to say that, unless it is because our psychological discourse is governed by the rules it is that psychological phenomena enjoy the privacy they do, which seems absurd, then it must be because psychological phenomena enjoy the privacy they do that our psychological discourse is governed by the rules it is: our psychological concepts must be answerable to psychological reality. And this, just like the corresponding view about mathematics, is an anathema to Wittgenstein. He believes it leads to all manner of confusion. Here is Wright: It is, for Wittgenstein, with the very craving for legitimising explanations of features of our talk about mind . . . or mathematics that we are led into hopeless puzzles about the status . . . of those discourses. Philosophical treatment is wanted, not to solve these puzzles but to undermine them—to assuage the original craving that leads to the construction of the bogus models and interpretations by which we attempt to make sense of what we do . . . The problem of self-knowledge is a signal example. It can have—I believe Wittgenstein’s [sic] holds—no solution of the kind we seek; for that very conception of a solution implicitly presupposes that there must be a something-by-virtue-of-which the distinctive marks of avowals are sustained. But those marks are part of ‘grammar’ and grammar is not sustained by anything. We should just say ‘this language game is played’.⁶

This parallel between the philosophy of mathematics and the philosophy of mind explains what would otherwise be a rather singular feature of Wright’s discussion: his use of the label ‘Platonism’ for what is standardly called ‘Cartesianism’ (e.g. p. 373). The exact historical suitability of either label for the views under discussion raises exegetical questions about Plato and Descartes that I cannot hope to address here. But I draw attention to this point because I think it is instructive to see how the converse appropriation, in other words the use of the label ‘Cartesianism’ for what I have been calling (in what I take to be a relatively orthodox way) ‘Platonism’, would also have some rationale. The view that I have been calling ‘Platonism’ allows us to derive conclusions concerning the form of mathematical reality from premises concerning the form of our thought about mathematical reality. A particularly significant case in point is the conclusion that mathematical reality is infinite, which we can derive from the premise that we exercise the concept of the infinite in the way we do. Whence comes the idea, we might ask (as of course Wittgenstein famously does⁷), that the beginning of a series is a visible section of rails invisibly laid to infinity? And the Platonist answers, ‘From the fact that that is precisely what it is. Nothing less could either ⁶ P. 372, emphasis in the original. ⁷ Wittgenstein (1967a), pt I, §218. This is the origin of the title of Wright’s book.

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justify or explain our conceiving the series in the way we do, as the product of the infinite applicability of a certain finite rule. We could not have our idea of the infinite unless that idea had its source in the infinite.’ It is clear why I say that the use of the label ‘Cartesianism’ for this way of thinking has some rationale. That we could not have our idea of the infinite unless that idea had its source in the infinite is, suitably construed, a familiar and vital precept of Descartes’s system.⁸ To see the Platonism here in this Cartesian guise is to see it in a guise that is frankly unattractive. Descartes’s principle that our idea of the infinite must be explained by something with at least as much reality as what the idea is an idea of, which fuels his argument for the existence of the infinite, has, for most people nowadays, very little appeal. And just as there are all sorts of alternatives to that principle, so too there are all sorts of ways of explaining how we have come to be able to participate in our finite mathematical practices (in terms of our natural capacities, our facility with manipulating symbols, our techniques of teaching and inculcation, the various applications that we make of our mathematics, and so forth) without alluding to anything infinite. But if we reject such Platonism, are we then forced into an equally unattractive scepticism? If we deny that there is any need to acknowledge the infinite to explain anything in our mathematical practices, must we doubt whether there is any need to acknowledge the infinite to explain anything at all? Must we doubt whether our concept of the infinite is so much as coherent? There are philosophers of mathematics who are prepared to take such scepticism extremely seriously.⁹ But most want to resist it. How can they?

3. Skolemite Scepticism One way to address this question is by focusing on a milder scepticism, familiar from discussions of the Löwenheim-Skolem theorem. I shall refer to this as Skolemite scepticism. Here the issue is not what explanatory project may or may not force us over the boundary between the finite and the infinite but what explanatory project may or may not force us over a boundary within the infinite, ⁸ See esp. Descartes (1984a), ‘Third Meditation’. Not that we have to go back as far as Descartes to find an example of a philosopher prepared to think in this way. Consider the following quotation: ‘To get [the idea of infinity], we need to be operating with the concept of numbers as the sizes of sets, which can have anything whatever as their elements. What we understand, then, is that the numbers we use to count things in everyday life are merely the first part of a series that never ends . . . Though our direct acquaintance with and designation of specific numbers is extremely limited, we cannot make sense of it except by putting them, and ourselves, in the context of something larger, something whose existence is independent of our fragmentary experience of it . . . When we think about the finite activity of counting, we come to realize that it can only be understood as part of something infinite.’ This is pure Platonism—pure Cartesianism. It is a quotation from Thomas Nagel’s most recent book, Nagel (1997), p. 71. (The whole of pp. 69–74 is worth reading in this connection.) ⁹ Wright himself is an example: see Wright (1982).

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between the countable and the uncountable. What the Löwenheim-Skolem theorem shows, according to Skolemite scepticism, is that, since there is no need to acknowledge the uncountable to explain anything in our mathematical practices—for everything we do and say can be quite satisfactorily interpreted in such a way that our quantifiers have only countably many things in their range—it follows that there is no need to acknowledge the uncountable to explain anything at all. (This is a kind of localized variant of the original scepticism—albeit localized to the transfinite—which is one reason why essay 12 of Wright’s collection, ‘Skolem and the Sceptic’, is not the incongruity which it may at first appear.) Neither the original scepticism nor this more modest, Skolemite version, both of which are targeted at our mathematical practices, can be allayed if we confine ourselves to an external view of those practices and then ask to what extent we need to indulge them in order to explain their success. Such, at any rate, is what the opponent of Platonism is bound to say. It is not clear, however, that this represents a victory for the sceptic. For where mathematical practices are concerned, in contrast, perhaps, to scientific practices, the very idea that they can be judged by how well they play this kind of explanatory role is already a concession to the sceptic (just as it is, of course, to the Platonist: indeed it is what makes Platonism and scepticism look like the only available options here). Furthermore, in the case of Skolemite scepticism, there is no motivating it if we confine ourselves to an external view of the relevant practices. For to acknowledge, say, that there is a model of Zermelo-Fraenkel set theory with a countable domain is already to wield some fairly heavy-duty set-theoretical machinery. Does this mean that Skolemite scepticism is self-stultifying? Does motivating Skolemite scepticism involve wielding enough set-theoretical machinery to drive Cantor’s familiar argument for the existence of the uncountable? Not necessarily. Suppose that the issue is whether there are any uncountable sets. At no point in the proof of the Löwenheim-Skolem theorem is the power set axiom used. A sceptic about whether there are any uncountable sets can withhold assent from this axiom. But if he does, then he is confronting the argument for the existence of uncountable sets directly, in its own terms. His scepticism is then simple mathematical scepticism, what we might call ‘ground-level’ scepticism as opposed to ‘meta-level’ scepticism. That is, it is scepticism that arises within our mathematical practices, scepticism about whether those practices are acceptable even in their own terms. It is not scepticism that arises at the prompting of philosophical reflections on our mathematical practices, scepticism about whether they are acceptable by some external standard. It is not Skolemite scepticism. What seems to follow is that Skolemite scepticism, even if it not self-stultifying, is unjustified. If anyone has doubts about the uncountable on mathematical grounds, preferring to work with a set theory that lacks the power set axiom, so be it. We must wait to hear what it is about the power set axiom that gives him

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pause. But at any rate his wariness had better be based on something other than the Löwenheim-Skolem theorem. In saying this, I take myself to be largely in agreement with Wright, whose helpful and incisive discussion of these matters, in essay 12, does much to mitigate the alarmism that so often accompanies the theorem. Similarly as far as the original scepticism is concerned. If anyone has doubts about the infinite on mathematical grounds, preferring to work with a set theory that lacks the standard axiom of infinity, so be it. We must wait to hear what it is about this axiom that gives him pause. But at any rate his wariness had better be based on something other than what we are impelled to say when we step back from our mathematical practices and can descry only their various finite features. There is nothing in that process to discourage us from reimmersing ourselves in the practices.

4. Ground-Level Mathematical Scepticism But still, we are surely not entitled to immerse ourselves in whatever practices we please. Surely not anything goes. For if it does, how are we to make sense of what Wright, in a related connection, calls ‘basic distinctions on which our ordinary ideas of objectivity [and] the growth of knowledge . . . would seem to depend’ (p. 5)? Well, our practices can be criticized for not meeting their own internal standards of acceptability. As I observed in the previous section, there is room for ordinary ground-level doubts about the uncountable. Wright himself voices one such doubt, towards the end of essay 12. He calls into question our grasp of the idea of an arbitrary subset of an infinite set, and more particularly our grasp of the idea of an arbitrary subset of the set of natural numbers. It is worth a brief digression to consider Wright’s strategy in doing this. He does not express his doubt in the most natural way. And he is wise not to. The most natural way to express the doubt would be to say that we have no guarantee, when we conceive of an arbitrary set of natural numbers, that our conception extends to all such sets. But here there is a threat of self-stultification. (No guarantee that our conception extends to all such sets?) What Wright does is to consider, in turn, various characterizations of the idea of an arbitrary set of natural numbers and to say in each case why he does not think it gives us a grasp of that idea. He concedes at the end of his essay that it is ‘to some extent a subjective business’ whether a given characterization does give us such a grasp.¹⁰ He is bound also to concede, as I am sure he would be happy to, that he may have overlooked a characterization which, even by his own subjective standards, succeeds where these fail.

¹⁰ P. 402.

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Nevertheless, in the last three or four pages of his essay, he presents an important challenge to those who uncritically talk about the power set of the set of natural numbers: the challenge, namely, to rebut the objection that they literally do not know what they are talking about. And this illustrates well how there can be ordinary ground-level scepticism about whether our mathematical practices meet their own internal standards. But if the Wittgensteinian response to the twin threat of Platonism and metalevel scepticism—the twin threat, in other words, of the view that we are entitled to our mathematical practices only in so far as they are justified by mathematical reality and the view that we are not entitled to them at all—is that we are entitled to them in so far as they do meet their own internal standards,¹¹ then is it not still too close for comfort to the principle ‘anything goes’?

5. Radical Scepticism and Grammar In order to see these issues from a slightly different angle, let us consider another doubt that we might have about the uncountable, more radical than Wright’s, namely the doubt about whether there is an inconsistency in the very idea; or, a little more precisely, about whether there is an inconsistency in the mathematics we use when reckoning with it. This is a doubt that we can do next to nothing to assuage. If someone prefers to work with a set theory that lacks the power set axiom simply because there is then one less potential source of inconsistency, all we can do is to note that he is more circumspect than we are. What is striking about this more radical scepticism is that, while it certainly counts as ground-level scepticism rather than meta-level scepticism, and while, relatedly, it receives no special impetus from the Löwenheim-Skolem theorem, it looks as if its fate is dependent on that of Platonism, or at least on that of Platonism in a specific respect. It looks as if nothing less than the actual consistency of our set theory can justify our accepting its consistency. If we explicitly affirm, for instance, that there is no way of deriving a contradiction from our settheoretical axioms, then surely there had better not be any way of deriving a contradiction from them. Surely this is not something about which we can simply legislate. This reinforces the thought that not anything goes. At most any consistent thing goes—consistency itself being an external feature of some practices and not of others, a feature determined independently of us by mathematical reality. Ground-level scepticism in general seems impervious to Platonism. But in the ¹¹ To repeat an earlier quotation from Wright, we should just say ‘this language game is played’ (cf. Wittgenstein (1967a), pt I, §654); or, to echo a familiar quotation from Wittgenstein himself, ‘without justification’ does not mean ‘without right’ (Wittgenstein (1967a), pt I, §289; cf. Wittgenstein (1978), pt VII, §40).

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specific case of ground-level scepticism about consistency (and about various related proof-theoretical features of our mathematical practices) it seems not to be. Is this the one respect, then, in which our mathematical practices cannot prop themselves up, and in which any Wittgensteinian philosophy of mathematics must be mitigated? My own view is that a thoroughgoing Wittgensteinian philosophy of mathematics both can and should resist this line of thought. In general, a Wittgensteinian philosophy of mathematics will emphasize the divide between the ground-level and the meta-level. Wittgenstein himself is adamant, for instance, that philosophers of mathematics have no business questioning or interfering with anything in mathematics.¹² Not even consistency, I think, should provide any kind of exception to this.¹³ But neither do I think that a Wittgensteinian philosophy of mathematics is hospitable to the principle ‘anything goes’. What it is hospitable to is something more like the principle ‘any “grammar” goes’.¹⁴ The problem, of course—a problem to which Wright adverts in the first of his postscripts, and which he discusses at the end of his Whitehead lectures¹⁵—is how to tell what counts as ‘grammar’.

6. Mathematical Practice and Grammar There is a graphic illustration of this problem in Wittgenstein’s own doubts about the uncountable. In Remarks on the Foundations of Mathematics he writes, ‘One pretends to compare the “set” of [sets of natural numbers] in magnitude with the set of [natural] numbers . . . I believe, and hope, that a future generation will laugh at this hocus pocus.’¹⁶ What is the reason for this scorn? Is Wittgenstein expressing simple ground-level scepticism of some sort? Earlier he voices his unease as follows: ‘The dangerous, deceptive thing about the idea . . . “The set [of sets of natural numbers] is not [countable]” is that it makes the determination of a concept—concept formation—look like a fact of nature.’¹⁷ Here there seems to be a confusion of levels of the very kind I have just said he abhors. What seems to be motivating him is his opposition to Platonism, his philosophy of mathematics.

¹² E.g. Wittgenstein (1967a), pt I, §124, and Wittgenstein (1978), pt V, §§52–3. ¹³ I cannot argue for this now. For the beginnings of an argument see Essay 20 in this volume, §§3 and 5; and for some relevant comments on consistency by Wittgenstein see e.g. Wittgenstein (1978), pt III, §§78 and 82 ff. See also Essay 18 in this volume, §3. ¹⁴ This may itself be part of the ‘grammar’ of ‘grammar’: see further Hacker (1986), ch. 7. ¹⁵ Essay 11, §7. ¹⁶ Wittgenstein (1978), pt II, §22. Wittgenstein’s own example concerns the set of real numbers rather than the set of sets of natural numbers, but it is plain that he would have said the same about both. ¹⁷ Wittgenstein (1978), pt II, §19.

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Yet somehow this has issued in doubts about a specific idea within mathematics. Surely, by Wittgenstein’s own lights, if there is any reason to criticize this idea, then it must be a mathematical reason. And surely, if there is no such mathematical reason, then we are entitled to give the same kind of impatient retort to the first of Wittgenstein’s remarks above (that one ‘pretends’ to compare one set in magnitude with another) as he himself might have given if the legitimacy of a more homespun measuring technique had been at issue: ‘One pretends no such thing. One does it.’ But matters are complicated by the fact that, even granted the divide between mathematics and the philosophy of mathematics, there remains an issue about where any given mathematical practice stands in relation to this divide. This is because, for all we know, some of our mathematical practices are themselves infected by certain philosophical pictures (cf. the first of Wright’s postscripts, §IV). Thus there are those, famously, who see our acceptance of the law of the excluded middle as symptomatic of a tacit Platonism.¹⁸ Who knows but that our acceptance of standard methods of comparing infinite sets in size is symptomatic of something similar? This is where the problem mentioned at the end of the previous section is manifest. Not everything in our mathematical practices reflects uncontaminated mathematical grammar. How are we to tell what does?

7. Limits of Explanation I have already suggested that, even in a mathematical context, ‘grammar’ is not simply a matter of internal consistency. Nor, once we reject Platonism, can we think of it as a matter of ‘external’ consistency (the consistency of our practices with something independent of them). The ‘great question’, as the blurb on the dust jacket of Wright’s book intimates, is what remains. But here we must prepare for disappointment. For if the general tenor of Wittgenstein’s later work is correct, then, whatever remains, and however we recognize it, we cannot hope to provide some general philosophical account of it. This is a clear lesson of Wright’s book. And, as Wright indicates, it is a lesson that he himself took a long time learning. He writes as follows about the closely related project of providing a general philosophical account of rule-following: Appreciating the problem [sc. the problem with providing this account] . . . does of course, depend upon a willingness to allow constitutive questions—What makes it the case that . . . ? What could constitute the fact that . . . ?—as legitimate ¹⁸ See e.g. Dummett (2000). And cf. Wittgenstein (1978), pt V, passim. Cf. also Wittgenstein (1967a), pt I, §254.

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philosophical currency, and hence implicitly credits philosophy with the power to provide satisfying, non-trivial answers to such questions . . . [It] only dawned on me much later that there is as much evidence in Wittgenstein’s text for impatience with this kind of question as argument . . . that the Platonist direction is a cul-de-sac.¹⁹

Philosophy, on a Wittgensteinian view, is not in the business of determining the constitution of ‘grammar’. Its business is rather to help us, in a piecemeal way, to keep as firm a grip as possible on specific grammars when reflection threatens to loosen that grip by tempting us into violations of them. Philosophy is not even in the business of determining the constitution of specific grammars. To ask, for example, what it is for twice two to be four, or what makes twice two four, if it is not to ask a mathematical question (for instance, about how that equation can be derived from the standard recursive definitions of addition and multiplication) nor a linguistic question (for instance, about our use of the statement ‘Twice two is four’) is to ask a mere pseudo-question. Similarly, to ask what makes it the case that a certain rule is infinitely applicable, if it is not to ask how we know an associated set to be infinite or something of that sort, is to ask nothing but a pseudo-question. And to think, not just that these are genuine questions, but that they are genuine questions whose answers have an explanatory role to play as far as our handling of each of the relevant grammars is concerned, is to aggravate the offence by lapsing back into Platonism. Nothing both makes twice two four and makes us think that twice two is four. Nothing both determines the steps that are to be taken in accord with a certain rule and determines us to take them. It would be something close to syllepsis to suggest otherwise. If there are rails to infinity, then they are part of mathematics. They cannot explain anything we do in the way in which physical rails can explain our movement through space.²⁰ None of this precludes our reflecting on the various contingencies that must obtain in order for us to operate with some specific grammar, the various facts of nature that make it possible for us to do so. Part of the reason why not anything goes for Wittgenstein is that not anything is of a type to be sustained by these contingencies, which include, most notably, the contingencies of our shared forms of life.²¹ Wright has much to contribute to the discussion of these matters. But I cannot resist taking issue, before I close, with the contribution that he makes in what I see as the least satisfactory essay in the collection.

¹⁹ P. 6, emphasis in original. ²⁰ Wright’s extremely fecund notion of ‘width of cosmological role’, to which he adverts in this book (p. 370) and which he develops further in Wright (1992), ch. 5, §5, can serve as a corrective against the temptation to expect explanatory work of ideas that are simply of the wrong sort to perform it. ²¹ See Wittgenstein (1967a), pt I, §§240–2, and p. 226.

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8. Private Language Famously, one of the ‘things’ that does not ‘go’ for Wittgenstein is a ‘private language’, a language enabling a person to describe his or her immediate private sensations in such a way that no one else can understand it.²² There can be no ‘grammar’ that is private in that sense. In essay 8 of his collection Wright considers whether, in sections 258–60 of Philosophical Investigations, Wittgenstein has a cogent argument for this view. He concludes, ‘Probably.’²³ But I do not think that he succeeds in substantiating this conclusion, either philosophically or exegetically. (One thing that should arouse our suspicion is the notably un-Wittgensteinian appendix to the essay, which seems completely out of place.²⁴ But that is a relatively minor consideration. For by his own admission, Wright pursues the technicalities in this appendix ‘somewhat as a jeu d’esprit’.²⁵) I shall not do anything here to register my exegetical qualms about whether Wright substantiates his conclusion. I shall confine my comments to my principal philosophical qualm, which is this: the argument for the impossibility of private language that Wright considers fails to satisfy the second of a number of constraints which he himself says at the outset any such argument must satisfy if it is to be ‘genuinely cogent’.²⁶ This is the constraint that the argument ‘must leave communal language alone’.²⁷ The argument that Wright considers is, in outline, both familiar and, it seems to me, unproblematically faithful to the text. (My exegetical quarrel is not with anything in the outline, but with the details.) It runs as follows. Suppose a would-be private linguist—let us call him A—resolves to make a daily record, in his private language, of whether a certain sensation has recurred. Then there can be no suitable gap between the case in which he thinks it is right to record a recurrence of the sensation, on any given day, and the case in which it is actually right to do so. But without such a gap, there is no ‘right’ or ‘wrong’. And this in turn means that there is nothing A is recording. His pretension to be using a private language is discredited. In section 7 of his essay Wright explicitly addresses the question whether this argument meets each of his constraints. He seems to have no difficulty in showing that it meets the second. For there does not seem to be any analogous problem for a public linguist, whose daily record of whether or not a certain type of event has occurred can in principle be confirmed or disconfirmed by other people. This

²² Wittgenstein (1967a), pt I, §243. ²³ P. 279. ²⁴ In this appendix Wright indulges in some ‘formal pyrotechnics’ (p. 218) to investigate the probability that a given number of sensation types will fall into a pattern which corroborates a theory of a certain specifiable kind. See later in the essay for the relevance of this to Wright’s argument. ²⁵ P. 218. ²⁶ P. 229. ²⁷ P. 229.

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creates a suitable gap between his own conviction that an event of that type has occurred and its actually having done so. But then Wright considers a counter-argument in favour of the possibility of private language. This counter-argument surfaces in the writings of Simon Blackburn and Ross Harrison, among others,²⁸ and Wright accordingly refers to the core idea of the counter-argument as ‘the Blackburn/Harrison proposal’.²⁹ The gist of the Blackburn/Harrison proposal is that there may be, for A—the would-be private linguist—a suitable analogue of the possibility of confirmation or disconfirmation by another person, namely the possibility of his own subsequent confirmation or disconfirmation, based on ‘well-established generalisations and theory’.³⁰ The bulk of Wright’s essay is concerned with establishing that this proposal is unsatisfactory. He argues that the ‘theory’ involved would have to satisfy certain criteria, and that these criteria would be exceedingly hard to satisfy. In particular, if A chose to record the occurrences of four kinds of sensations (not, by the way, five, as Wright says on p. 218 of the introduction to this section), then the probability of there being such a theory corroborated by these occurrences would, on Wright’s reckoning, be ‘a paltry 1 in 8,192’.³¹ It follows that whether or not a private language is possible is radically contingent in a way that Wright finds absurd. He writes: One who believes in the essential privacy of large parts of his mental life will surely want to suppose that his capacity to record its character in terms no one else can have reason to think he understands would be in no way contingent on the particular form of the patterns, if any, of concomitance which the various event types display.³²

Now I am no apologist for the possibility of private language; but it is not at all clear to me why someone who is will want to suppose any such thing. And here I think Wright is guilty of an error that would have been apparent to him had he thought more about whether the argument against the possibility of private language, as he is now construing it, still meets the second constraint. To be fair to Wright, he does consider this question. And he says that ‘it is wildly unlikely’ that we could have a suitable grasp of the notion of observational error ‘unless this grasp owed more to our membership in a language community in which we have faith in others’ judgements than to our engagement in theory-building’.³³ Well, perhaps; perhaps not. A good deal depends on how much is built into the notion of a ‘theory’. I doubt that the Blackburn/Harrison proposal needs to build

²⁸ See e.g. Blackburn (1984b), pp. 299–300; and Harrison (1974), p. 161. Cf. also Walker (1978), p. 115. ²⁹ P. 265. ³⁰ P. 217. ³¹ P. 218. ³² P. 279, emphasis in original. ³³ Pp. 270–1.

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anything like as much into that notion as the quotation from Wright suggests that he builds into it. (One thing that is extremely telling is what Wright says a little earlier about our usual criteria for observational error, which he cites as ‘discord with others’ reports, poor lighting, mislaid spectacles, and so on’: namely, that they ‘are, at least in part, of a largely non-theoretical sort’.³⁴) But that is not really the point. The point is how radical the contingencies are which underlie ‘our membership in a language community in which we have faith in others’ judgements’. A vast amount is required, not only by way of patterned occurrences in our social world but also by way of patterned occurrences in the natural world, for us to belong to any such community. That is itself, surely, a prime lesson of Wittgenstein’s later philosophy.³⁵ So when, at the very end of his essay, Wright summarizes his response to the Blackburn/Harrison proposal by complaining that it makes private language possible ‘only in very special, at best unlikely circumstances’,³⁶ surely this invites the following counter-response: ‘Yes; and it is only because our circumstances are very special, at best unlikely, that there is any communal language.’³⁷

9. Conclusion It would be misleading, however, to finish on such a critical note. Wright does much in this collection to guide us in a broadly Wittgensteinian direction away from Platonism. Through his stimulating combination of exegesis and philosophical exploration he helps us to a better understanding of the very idea of rails to infinity.³⁸ That there are rails to infinity is a natural picture. And it is a harmless one if we handle it properly. But we do not handle it properly if we think of these rails as constraining the human activities that make it possible for us even to think in such terms, the activities to which I have just alluded; nor if we think of them as fixing, independently of us, what rules we can have. They simply are rules we have, pictured in a certain way.

³⁴ P. 267. ³⁵ Cf. Wittgenstein (1967a), pt I, §§240–1. ³⁶ P. 279. ³⁷ In so far as it is a matter of conceptual necessity that there can be no language without the obtaining of certain natural conditions (see the end of section 7), this conceptual necessity is in no way compromised by the utter flesh-and-blood contingency of those conditions. ³⁸ Essay 6, §3 is particularly helpful.

17 Wittgenstein and Infinity Abstract The aim of this essay is to give an overview of Wittgenstein’s conception of the infinite. One focus of the essay is Wittgenstein’s rejection of what is dubbed a ‘realist’ model of our idea of the infinite. On this model our idea is the source of beliefs that we have about an independent reality. Another focus is the way in which Wittgenstein’s rejection of this model leads him to reject the idea of the infinite itself as it appears in certain mathematical contexts. It is argued that these two rejections can be uncoupled: abandonment of the realist model of our idea of the infinite is consonant with full endorsement of the use to which mathematicians put the idea. There remains scope for Wittgenstein to take issue, if not with the use to which mathematicians put the idea, then with their choice of language in doing so, something that he has reason to do precisely because this choice encourages adoption of the realist model.

1. Descartes’s and Nagel’s Realist Model Descartes, in his Third Meditation, famously argues that the only possible explanation for his having an idea of God, given his own finitude and given God’s infinitude, is that God actually exists; and that his idea of God is an innate idea placed in him by God, ‘as it were, the mark of the craftsman stamped on his work’.¹ The details of Descartes’s argument need not detain us now—except to comment that the scholastic elements in it put it more or less beyond the pale of contemporary analytic philosophy. Nevertheless, something strikingly similar, in broad outline, can be found in a book that is very much of our time and squarely within the analytic tradition, namely Thomas Nagel’s The Last Word.² Nagel reflects on our use of reason—’a local activity of finite creatures’³—to arrive at the idea of infinity. And, as against those who think that this both can and must be understood purely in terms of our finite resources, without appeal to infinity itself, he urges:

¹ Descartes (1984a), ‘Third Meditation’, p. 35.

² Nagel (1997).

³ Nagel (1997), p. 70.

The Human A Priori: Essays on How We Make Sense in Philosophy, Ethics, and Mathematics. A. W. Moore, Oxford University Press. © A. W. Moore 2023. DOI: 10.1093/oso/9780192871411.003.0018

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To get [the idea of infinity], we need to be operating with the concept of numbers as the sizes of sets, which can have anything whatever as their elements. What we understand, then, is that the numbers we use to count things . . are merely the first part of a series that never ends. . . . Though our direct acquaintance with and designation of specific numbers is extremely limited, we cannot make sense of it except by putting them, and ourselves, in the context of something larger, something whose existence is independent of our fragmentary experience of it . . When we think about the finite activity of counting, we come to realize that it can only be understood as part of something infinite.⁴

He goes on to say that ‘the description of what happens when we count must include the relation of that activity to the infinite series of natural numbers, since that is part of what our operation with the concept of number makes evident’,⁵ an interesting echo of Descartes’s claim that our idea of God is a ‘mark’ of God, providing us with an image of an infinite reality beyond us.⁶ To be sure, there are crucial differences between Descartes’s account and Nagel’s, even at this high level of generality. (Notice in particular that, where Descartes urges that we need to acknowledge the infinite if we are to explain how we have our idea of the infinite, Nagel urges that we need to acknowledge the infinite if we are so much as to characterize our idea.⁷ As regards explaining how we have our idea, Nagel takes seriously the possibility that this is something we cannot do.⁸) So it would be Procrustean at best to claim to find the same argument in both. But I am in any case less concerned with the argument that either of them gives than with a certain picture of their conclusion that they share. Having taken a critical step back from our idea of the infinite, and having concluded (rightly or wrongly) that a fully satisfactory account of that idea must involve further implementation of it, both Descartes and Nagel adopt a quasiperceptual model of the relation between our idea and that which it is an idea of. Both of them take our idea to be the source of beliefs that we have, which are answerable to how things are in an independent reality.⁹ The reason why I call this ⁴ Nagel (1997), p. 71. ⁵ Nagel (1997), p. 72. ⁶ There are other connections too. Nagel, in pondering the understanding of the infinite to which we finite creatures can attain, given that we can never fully grasp it, draws a distinction that Descartes draws: the distinction, in roughly Descartes’s words, between our understanding clearly that the infinite is infinite and our being able to grasp the infinite, qua infinite (Descartes (1984b), ‘Author’s Replies to the First Set of Objections’, AT VII, 112); or, in roughly Nagel’s words, between our seeing that there is something there that we cannot grasp and our being able to grasp it, in its entirety (Nagel (1997), p. 70). ⁷ There is an interesting comparison here with what Stroud argues, in Stroud (2000), with respect to our idea of colour. ⁸ Nagel (1997), p. 76. ⁹ Independent? Is it not a primary concern of both Descartes and Nagel to infer from the fact that we have certain beliefs about the infinite that those beliefs are true? Yes; but not because either of them fails

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model a quasi-perceptual model—the label ‘realist’ would be equally appropriate—is that, on one natural way of construing a perception, whenever I perceive an object, my perception is likewise a source of beliefs that I have (about the object) and these beliefs are likewise answerable to how things are in an independent reality (the reality of how the object is); and, we might add, my perception is a ‘mark’ of the object itself.

2. Wittgenstein’s Rejection of the Realist Model Very well; what are we to say about this quasi-perceptual, or realist, model? (There is an issue about how far any such model is adequate even in the case of perception. But let that pass.) One thing that we can say about the model is that it would be an utter anathema to Wittgenstein.¹⁰ What our ‘idea’ of the infinite comes to, on a Wittgensteinian conception, is our way of handling the concept of the infinite and related concepts; the set of rules that govern our use of the corresponding language. It is what Wittgenstein himself would call our ‘grammar’ of the infinite. What it leads us to ‘believe’— if ‘believe’ is the right word—are the various associated grammatical propositions, propositions to state which is to enunciate the rules of the grammar: for instance, that an infinitely long object is an object with no end;¹¹ or, to take an example close to Nagel’s concerns, that there are infinitely many natural numbers.¹² On the realist model, our acceptance of these rules is both explained and justified by the nature of reality. We preclude talk of a biggest natural number, for example, because we are sensitive to the fact that there is no biggest natural number. This fact is quite independent of us. It is something that we have discerned, like the fact that there is no planet between Mercury and the sun. It also of course means that we are right to preclude talk of a biggest natural number, and right, accordingly, to ‘believe’ that the series of natural numbers is infinite. For Wittgenstein, this model is utterly confused. Nothing justifies and explains our accepting whatever grammatical rules we accept, or at least not in the sense to see a logical gap here; rather because they both see a logical gap that they think they have the argumentative means to bridge. They both think that, having taken the critical step back, they can show that things are how, granted our idea, we believe them to be, that is to say there is actually a God (Descartes) or there is actually an infinite series of natural numbers (Nagel). But neither of them would have the least sympathy for the suggestion that, in having these beliefs, we make them true. (For a forthright expression of a similar realism see Aristotle (1941a), bk 3, ch. 8, 15–20.) ¹⁰ The early Wittgenstein or the late Wittgenstein? This essay is more concerned with the latter, though it treats of topics on which there is important convergence between the two. (It is worth noting in this connection that the majority of the material cited below is from transitional work.) Much of what follows, incidentally, derives from A. W. Moore (2019a), esp. ch. 9, §3. ¹¹ Wittgenstein (1974), p. 455. ¹² Wittgenstein (1974), p. 465.

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intended in the model. We might be justified in the sense that our rules fulfil some important function in our lives, and this too might explain why we have them. But that is not the sense intended in the model, for we have not thereby got anything ‘right’. Our rules do not answer correctly to some independent reality. As Wittgenstein memorably says in Zettel, after posing the question whether our number system resides in our nature or in the nature of things, ‘How are we to put it?—Not in the nature of numbers.’¹³ Does this mean that Wittgenstein rejects the realist model in favour of some sort of idealist alternative? For there does seem to be a constitutive link between the rules we accept and the essential features of reality that correspond to them. And if this is not due to the fact that the former answer to the latter, must it not be due to the fact that the latter answer to the former; so that, for example, the reason why there are infinitely many natural numbers is simply that we preclude talk of a biggest one? The relation between Wittgenstein and idealism is an exegetical quagmire in its own right and I am not going to try to wade through it now.¹⁴ Suffice to say that Wittgenstein is under no obvious pressure to acknowledge an answerability in either direction. True, in saying that a biggest natural number is impossible, we are adverting to the fact that talk of a biggest natural number is disallowed by one of our grammatical rules. And, as with any of our rules, this is a rule that we might not have had. But this is not to say that, but for us, there might have been a biggest natural number; or that, but for us, the series of natural numbers might have been only finite. It is to say rather that we might not have thought and spoken in those terms. We have not made the series of natural numbers infinite. That the series of natural numbers is infinite is a mathematical necessity. If it has any explanation, it has a mathematical explanation. The point is simply this. For it to be a mathematical necessity is for our stating it to be an enunciation of one of the rules of our mathematical grammar. ‘Essence’, as Wittgenstein puts it in Philosophical Investigations, ‘is expressed by grammar.’¹⁵

3. Wittgenstein’s Quasi-Aristotelianism Wittgenstein would recoil from a realist model of the relation between our grammar of the infinite and the infinite itself, then. But that is not because of

¹³ Wittgenstein (1967b), §357, emphasis in original. For helpful exegesis of Wittgenstein on the autonomy of grammar see Hacker (1986), ch. 7, §2. ¹⁴ I try to say something about it in A. W. Moore (2007). See also B. Williams (2006h). ¹⁵ Wittgenstein (1967a), pt I, §371, emphasis removed. Note that, just as Wittgenstein’s views prevent us from being said to legislate correctly, so too they prevent us from being said to legislate incorrectly. Thus suppose there were people whose grammatical rules demanded application of the epithet ‘biggest natural number’ to 999. This would not show that they took 999 to be the biggest natural number. It would show that they were legislating for a use of the expressions ‘biggest’ and ‘natural number’ that was different from their use in standard English. There are connections here with Wittgenstein’s early work: see e.g. Wittgenstein (1961), 3.02–3.03, 5.473–5.4732, and 5.5422.

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any peculiarity of the infinite. Wittgenstein would recoil from a realist model of the relation between any grammar and that which it is a grammar of. Even so, there are certain features of our grammar of the infinite that make the realist model especially compelling in this case. This is something that I hope to show in the next section. In this section, as a necessary preliminary, I shall focus on some of the details of our grammar, as Wittgenstein sees them. Note first that the word ‘infinity’ has, in Wittgenstein’s view, ‘many different meanings’.¹⁶ ‘Our grammar’ of ‘the infinite’, to the extent that there is such a thing, must therefore consist of—to use a metaphor that Wittgenstein himself uses elsewhere—many overlapping fibres.¹⁷ In what follows I shall only be able to unravel and examine what I take to be the most significant of these. Wittgenstein’s treatment of the infinite is part of a tradition whose roots lie in Aristotle. Aristotle insists that the infinite exists potentially, but not actually. What he means by this is something temporal: the infinite exists over time, but never in time. The endless ticking of a clock, for example, is potentially infinite, but never actually infinite.¹⁸ In a similar vein—I shall try to indicate shortly how similar—various medieval thinkers distinguish between a categorematic use of the expressions ‘infinite number’, ‘infinitely many’, ‘infinity’, and suchlike and a syncategorematic use of them. Roughly: to use one of these expressions categorematically is to convey the idea that there is something with a property that surpasses any finite measure; to use one of these expressions syncategorematically is to convey the idea that, given any finite measure, there is something with a property that surpasses it. With a little regimentation Aristotle’s distinction can be subsumed under this. For—and again this is rough—to use one of these expressions in order to refer to an actual infinite is to convey the idea that there is some time at which a given magnitude surpasses any finite measure, while to use it in order to refer to a potential infinite is to convey the idea that, given any finite measure, there is some time at which a given magnitude surpasses it. Suppose, for example, that I say, ‘An infinite number of people will be dead.’ If I am using ‘infinite number’ categorematically, I mean that there will come a time when the number of dead people exceeds any finite number: an actual infinity of people will then be dead. If I am using ‘infinite number’ syncategorematically, I mean that, given any finite number, there will come a time when the number of dead people exceeds it: a potential infinity of people will, each in his or her own time, be dead. And, just as Aristotle champions the potential infinite and repudiates the actual infinite, so too a number of these medieval thinkers hold that it is legitimate to use ‘infinite number’ and other such expressions syncategorematically but illegitimate to use them categorematically.¹⁹

¹⁶ Wittgenstein (1975c), p. 304. ¹⁷ Wittgenstein (1967a), pt I, §67. ¹⁸ For further discussion, and references, see A. W. Moore (2019a), ch. 2, esp. §3. ¹⁹ For further discussion, and references, see A. W. Moore (2019a), ch. 3, esp. §3.

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Wittgenstein insists on something very like this. In Philosophical Remarks he says: You could put it like this: it makes sense to say that there can be infinitely many objects in a direction, but no sense to say that there are infinitely many . . . The ‘infinitely many’ is so to speak used adverbially and is to be understood accordingly. That is to say, the proposition ‘Three things can lie in this direction’ and ‘Infinitely many things can lie in this direction’ are only apparently formed in the same way, but are in fact different in structure: the ‘infinitely many’ of the second proposition doesn’t play the same role as the ‘three’ of the first.²⁰

Note, incidentally, the three dots of ellipsis in this quotation. The omitted material deserves special mention. Having insisted that it makes sense to say that there can be infinitely many objects in a direction, but no sense to say that there are, Wittgenstein comments: ‘This conflicts with the way the word “can” is normally used. For, if it makes sense to say a book can lie on this table, it also makes sense to say it is lying there.’ This is a very clear echo of something we find in Aristotle. Aristotle, having insisted that the infinite has a potential existence, but not an actual existence, comments: ‘But the phrase “potential existence” is ambiguous. When we speak of the potential existence of a statue we mean that there will be an actual statue.²¹ It is not so with the infinite. There will not be an actual infinite.’²² The connections between Aristotle and Wittgenstein intimated here are in my view very deep. I said above that Aristotle’s distinction between the actual infinite and the potential infinite could be viewed as a result of one particular application of the more overtly grammatical distinction drawn by the medievals. In a way, Wittgenstein too is concerned with one particular application of that more general distinction. At one point he toys sympathetically with the suggestion that ‘infinity is an attribute of possibility, not of reality’; that ‘the word “infinite” always goes with the word “possible”, and the like’.²³ Linking the language of infinity with the language of possibility in this way, he in effect applies the categorematic/ syncategorematic distinction to possible situations—just as Aristotle, in effect, applies the distinction to times. Thus Wittgenstein sanctions uses of expressions such as ‘infinitely many’ to say that, given any finite measure, there is some possible situation in which a given magnitude surpasses it; but not to say that there is some possible situation in which a given magnitude surpasses any finite measure. This is illustrated in the quotation above. It is also illustrated by Wittgenstein’s view of a sentence such as ‘This stick is infinitely divisible.’ We ²⁰ Wittgenstein (1975c), §142. ²¹ Will be? Or only can be? For discussion of this point see below. ²² Aristotle (1941a), bk 3, ch. 6, 206a 18–21. ²³ Wittgenstein (1975c), p. 313.

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might have thought that this sentence was ambiguous, meaning either that, however many pieces this stick is divided into, it can be divided into more, or that this stick can be divided into infinitely many pieces. For Wittgenstein, as indeed for Aristotle, only the first of these is a legitimate interpretation.²⁴ Nor is this connection between Aristotle and Wittgenstein the purely structural affair that these remarks suggest. It is not just that, where Aristotle sanctions syncategorematic uses of the language of infinity (as opposed to categorematic uses of it) to characterize the variation of finite objects across time, Wittgenstein sanctions syncategorematic uses of the language of infinity (as opposed to categorematic uses of it) to characterize the variation of finite objects across the space of possibilities. Both Aristotle and Wittgenstein recognize a fundamental and intimate relation between time and possibility. For Aristotle, the question whether something is possible is, at least on one plausible reading, the question whether it has ever been so or will ever be so.²⁵ And for Wittgenstein, ‘infinity lies in the nature of time . . . [it] is an internal quality of the form of time’,²⁶ a remark that can be compared with his claim that ‘we all . . . know what it means to say that there is an infinite possibility and a finite reality, since we say . . . time [is] infinite but we can always only . . . live through finite bits of [it]’.²⁷ For Wittgenstein, then, the language of infinity has paradigmatic application in any situation that has written into it a never-ending series of nested possibilities, each involving an increase in some magnitude. (The case of an infinitely divisible stick is a clear case in point.) Somewhat more precisely, the language of infinity has paradigmatic application in any situation satisfying the two following conditions: first, something in this situation admits of a possibility, in which it admits of a second possibility, in which it admits in turn of a third possibility, and so on,²⁸ where each of these possibilities involves an increase in some magnitude over its predecessor; and secondly, something about the very form of the situation determines that these possibilities always arise. The reason for the second condition is that, unless the possibilities arise in some principled way, the language cannot get a grip. This relates to Wittgenstein’s frequent insistence that ‘the word “infinite” is always part of a rule’.²⁹ So does Wittgenstein reject entirely any use of the language of infinity to characterize the way things actually are, as opposed to the various interconnected ways they are capable of being? Not quite. Or at least, this is a misleading way to put it, as Wittgenstein himself warns.³⁰ After all, if a stick is infinitely divisible, that ²⁴ Wittgenstein (1975c), §139. ²⁵ Aristotle (1941b), bk 1, ch. 12. Cf. Aristotle (1941a), bk 3, ch. 6, 207b 10–14. (This addresses the concern raised in n. 21.) ²⁶ Wittgenstein (1975c), §143. ²⁷ Wittgenstein (1975c), §138. ²⁸ For interesting comments by Wittgenstein on the workings of the expression ‘and so on’, proffered in both his early work and his late work, see: Wittgenstein (1961), 5.25–5.2523; and Wittgenstein (1967a), pt I, §§208 and 229. ²⁹ Wittgenstein (1975c), p. 313, emphasis in original. ³⁰ Wittgenstein (1975c), p. 313.

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is a fact about how the stick actually is, just as, if a particular peg can fit into a particular hole, that is a fact about the respective sizes and shapes of the peg and the hole. Wittgenstein is not even unequivocally hostile to a use of the language of infinity to characterize a contingency about how an infinite region of space is filled. For instance, he would not dismiss out of hand, as entirely without sense, the claim that there are, as a matter of contingent fact, infinitely many stars to be observed in some particular direction—despite his apparent insistence to the contrary in the quotation above.³¹ What is true is that, even in this case, Wittgenstein would demand that the contingency be governed by some kind of rule. To say that there are infinitely many stars to be observed in some direction, where this is an inference from some well corroborated law of nature, would be one thing. To add that these stars differ from one another in some totally unprincipled way, for instance that they are all of different sizes and their sizes vary at random, would be something else entirely. The latter, on Wittgenstein’s view, would be without sense.³² The proposition that there are infinitely many stars to be observed in some direction is thus, on Wittgenstein’s view, very different in kind from the proposition that there are three.³³ The former proposition still, crucially, has something of the syncategorematic and the modal about it. (What it comes to, in fact, is this: however many stars have been observed in this direction, there is a nomically guaranteed possibility of observing more. The supplementary proposition about the stars varying randomly in size lacks any analogous paraphrase.) We should not think of infinity as like a natural number, only much bigger.

4. The Enticement of a Realist Model of the Grammar of the Infinite Very well; why might there be some special temptation to adopt the realist model in the case of the grammar of the infinite? One of the reasons why there is a temptation to adopt the realist model in the case of any grammar is the fact that when we take a critical step back from the grammar and raise questions about how we have arrived at it, why it takes the ³¹ Here perhaps it is salutary to remind ourselves of Wittgenstein’s caution that the word ‘infinity’ has many different meanings—although, as I hope to show, there is nothing yet that really forces us to revert to that. ³² See e.g. Wittgenstein (1978), pt V, §6; and Wittgenstein (1975c), §§145 and 147, and pp. 304–6. This compares with the following combination of doctrines in Kant: first, that, as a matter of necessity, every event has a prior cause (Kant (1998), A532/B560); and second, that there is no such thing as contingent infinite history (Kant (1998), A518–20/B546–8). ³³ Cf. Wittgenstein (1975c), pp. 306–7; and Wittgenstein (1974), p. 464. Cf. also Wittgenstein’s conviction that an infinite set is very different in kind from a set with three members; ‘set’, on Wittgenstein’s view, is hardly even univocal in the two cases. (This is discussed in Shanker (1987), ch. 5, §1, with references.)

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form it does, what it enables us to do, and suchlike, it is very natural to redeploy the grammar itself. For there is no more obvious toolkit for reflecting on the provenance, character, and power of some set of concepts that we use than that very set of concepts. Nor indeed is there any clear reason why we should not redeploy the grammar in this way. There need be no question of circularity (any more than there would be a question of circularity if we explained our facility with the concept of simplicity by appeal to the fact that it is a simple concept).³⁴ The point, however, is that when we do redeploy the grammar in this way, it is enormously enticing to take that crucial extra step: of using the grammar to explain our possession of it in accord with the realist model; of employing such explanatory schemata as the following: We have a rule that disallows talk of Xs because we have discerned that there are no Xs. This enticement exists in the case of any grammar. The peculiarity of the grammar of the infinite, which gives the enticement a special stimulus in that case, is this. Because the concept of the infinite applies whenever there is a process that admits of principled indefinite extension—because ‘the word “infinite” ’, to quote Wittgenstein again, ‘is always part of a rule’—that grammar, the grammar of the infinite, is one that we are liable to employ whatever critical step back we have taken, whatever other grammar we are investigating. As long as our own rules and their application to ever new possibilities constitute our subject matter, then we are liable to find a use for the language of infinity. (In particular, of course, we are liable to find a use for the expression ‘and so on’.) And as long as we are tempted to say that we have adopted those rules because they answer to essential features of reality that we have discerned, in other words as long as we are tempted to endorse the realist model, then one such use of the language of infinity will, apparently, be to provide us with the wherewithal to do so, that is to endorse the realist model. Thus if what is at issue is why we have adopted this or that particular rule, we might end up talking in terms that Wittgenstein himself familiarly canvasses and depict ourselves as having erected a sign post that points in the direction of rails that we have glimpsed, invisibly laid to infinity.³⁵ How much stronger, then, will the temptation to talk in such realist terms be when our very subject matter is the way in which our own sign posts give direction, that is to say when our very subject matter is, to that extent, the infinite? ³⁴ Contrast the circularity that Wittgenstein himself bemoans, specifically in connection with infinity, when he says, ‘It’s no wonder that time and again I can only explain infinity in terms of itself, i.e. cannot explain it’ (Wittgenstein (1975c), §138, emphasis in original). There is no critical step back in that case: the object of investigation there is infinity, not the concept of infinity. (Cf. Wittgenstein (1980), p. 10, second paragraph.) ³⁵ This is an allusion to Wittgenstein (1967a), pt I, §§85 and 218; cf. also §229.

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5. Wittgenstein’s Attempt to Give a Finite Account of the Grammar of the Infinite, and his Consequent Struggle to Maintain his Grip on the Grammar Now if there are no objections (in principle) to redeploying any given grammar in taking a critical step back from it, and if there are objections (in principle) to adopting a realist model of that grammar, then, whatever enticement doing the former provides for doing the latter, the enticement must be resistible—even in the case of the grammar of the infinite, where I have just been arguing it is peculiarly strong. Likewise, in order to rebut the realist account of our idea of the infinite which both Descartes and Nagel offer, it is not necessary to rebut their shared lemma: that a fully satisfactory account of our idea must involve further implementation of it. Nevertheless, for all we have seen so far, that shared lemma may be false.³⁶ Likewise, for all we have seen so far, it may be possible to take a critical step back from our grammar of the infinite and give a good account of it without redeploying it. It may be possible, in other words, to say how we have arrived at the grammar, why it takes the form it does, what it enables us to do, and suchlike, in thoroughly finite terms, that is in terms of our finite capacities, our facility with manipulating finite symbols, our skill in both giving and understanding finite instructions, the applications that we make of our grammar in characterizing finite objects, and suchlike. And in so far as it is possible to do all of this, then doing it will provide one particularly effective way of keeping the realist model at bay. It is scarcely surprising, then, that time and again we find Wittgenstein discussing our grammar of the infinite in just this way, with a selfconscious and pointed emphasis on its several finite features. Here is a representative sample of quotations. We learn an endless technique: that is to say, something is done for us first, and then we do it; we are told rules and we do exercises in following them; perhaps some expression like ‘and so on ad inf.’ is also used, but what is in question here is not some gigantic extension. These are the facts.³⁷ When one is a child, ‘infinite’ is explained as something huge . . . But if you say that a child has learned to multiply, so that there is an infinite number of

³⁶ It is worth noting in this connection that defending the lemma creates a distinctive quandary for both Descartes and Nagel. They need our idea to be clear enough to resist being dismissed as not a genuine idea at all, but not so clear as to be readily defined in simple finite terms, thereby leaving the lemma open to easy attack (cf. B. Williams (1978), pp. 144–5). This accounts for the distinction that each of them draws, to which I drew attention in n. 6. Cf. Nagel’s uneasy claim (Nagel (1997), p. 70) that ‘the infinity of the natural numbers is something we come to grasp through our recognition that in a sense we cannot grasp all of it’. ³⁷ Wittgenstein (1978), pt V, §19, emphasis in original.

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multiplications he can do—then you no longer have the image of something huge.³⁸ Suppose . . . I say ‘By “cardinal number” I mean whatever results from 1 by continued addition of 1’. The word ‘continued’ doesn’t represent a nebulous continuation of 1, 1 + 1, 1 + 1 + 1; on the contrary the sign ‘1, 1 + 1, 1 + 1 + 1 . . . ’ is to be taken as perfectly exact; governed by definite rules which are different from those for ‘1, 1 + 1, 1 + 1 + 1’, and not a substitute for a series ‘which cannot be written down’.³⁹ The expression ‘and so on’ is nothing but the expression ‘and so on’ (nothing, that is, but a sign in a calculus which can’t do more than have meaning via the rules that hold of it; which can’t say more than it shows). That is, the expression ‘and so on’ does not harbour a secret power by which the series is continued without being continued.⁴⁰ We reflect far too little on the fact that a sign really cannot mean more than it is.⁴¹ But how do we construct an infinite hypothesis, such as that there are infinitely many fixed stars (it’s clear that in the end only a finite reality corresponds to it)? . . . [It] can only be given through a law . . . It’s clear to us that no experience corresponds with [such a hypothesis]. It only exists . . in language . . . ⁴² ‘We only know the infinite by description.’ Well then, there’s just the description and nothing else.⁴³ Let us not forget: mathematicians’ discussions of the infinite are clearly finite discussions. By which I mean, they come to an end.⁴⁴

Not that the possibility of giving an account of our grammar of the infinite in these finite terms should occasion any kind of doubt about the grammar. We are at perfect liberty to reimmerse ourselves in it; and, when we do, we will of course talk once again in unashamedly infinitary terms. Wittgenstein himself, in one of

³⁸ Wittgenstein (1976), p. 255. ³⁹ Wittgenstein (1974), p. 284. Cf. Wittgenstein’s discussion of the notation for a recurring decimal in Wittgenstein (1974), p. 428. ⁴⁰ Wittgenstein (1974), p. 282, emphasis in original. ⁴¹ Wittgenstein (1975c), §144. The idea that a sign cannot mean more than it is may seem a curious idea: how, for instance, does it relate to the fact that the word ‘big’ is not big? But what Wittgenstein is expressing here is his view that ‘in mathematics, the signs themselves do mathematics, they don’t describe it’ (Wittgenstein (1975c), §157, emphasis in original) or again that ‘in mathematics everything is algorithm and nothing is meaning’ (Wittgenstein (1974), p. 468, emphasis in original). ⁴² Wittgenstein (1975c), §139, his emphasis. ⁴³ Wittgenstein (1975c), §135. ⁴⁴ Wittgenstein (1974), p. 483. Note: several of these quotations allude to a distinction that Wittgenstein frequently draws, between two kinds of dots of ellipsis: the ‘dots of laziness’, as he calls them, which occur in the expression ‘A, B, C, . . . .’, used to represent the alphabet; and the ‘dots of infinity’, as we could in turn call them, which occur in the expression ‘1, 2, 3, . . . ’, used to represent the series of natural numbers (see e.g. G. E. Moore (1959b), p. 298; Wittgenstein (1976), pp. 170–1; and Wittgenstein (1967a), pt I, §208). The former are an abbreviation. The latter are not: they are part of the mathematical symbolism with their own precise, specifiable use.

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his lectures, imagines an interlocutor whose reaction to the idea that we can give an account of our grammar without reference to any ‘gigantic extension’ is to say, ‘We aren’t talking of anything you would call big, and therefore not of anything infinite.’⁴⁵ Wittgenstein gives the following telling reply: ‘But as long as you try to point out that we are not treating of anything infinite, this means nothing, because why not say that this is infinite?’⁴⁶ This reply is just as we would expect. It is a familiar and cardinal precept of Wittgenstein’s philosophy that it is not our business, as philosophers, to challenge any given grammar. ‘Philosophy may in no way interfere with the actual use of language;’ he says, ‘it can in the end only describe it . . . It leaves everything as it is.—It also leaves mathematics as it is . . .’⁴⁷ The grammar of the infinite is simply there to be reckoned with, neither justified nor unjustified. All that we, as philosophers, need aspire to do is to gain a sufficiently clear view of it to be able to combat the confusion and perplexity that arise when it is mishandled.⁴⁸ And yet . . . So great is Wittgenstein’s concern to characterize the grammar of the infinite without redeploying it, in particular without mentioning any infinite subject matter, that he does at times visibly struggle to maintain his grip on the grammar. He comes close to saying the very thing that he criticizes his interlocutor for saying, namely that it lacks an infinite subject matter.⁴⁹ Thus at one point, in order to emphasize that there is nothing more to the impossibility of a biggest natural number than our precluding talk of such a number, he objects to our

⁴⁵ Wittgenstein (1976), p. 255, emphasis in original; and likewise for the next quotation in the main text. ⁴⁶ Cf. Wittgenstein (1978), pt II, §62. ⁴⁷ Wittgenstein (1967a), pt I, §124. ⁴⁸ This is perhaps an apt point at which to note that the question whether it is possible to give an account of our grammar of the infinite in finite terms, without redeploying it, is quite independent of the question whether Wittgenstein’s own quasi-Aristotelian account of that grammar is correct. There is room, at least in epistemic space, for all four combinations of answers to these two questions. In particular, there is room for each of the two following combinations of answers: (i) Wittgenstein’s account is correct but it cannot be given in finite terms because it cannot be given without adverting to the endless possibilities that the grammar of infinity affords; and (ii) Wittgenstein’s account is incorrect, because we can perfectly well talk about a non-rule-governed infinite contingency, but we can account for such talk in terms of the finite rules that govern the finite processes of extrapolation which make it possible. (This is highly schematic of course. I am not committing myself one way or the other on whether either of these stances would withstand much scrutiny.) ⁴⁹ The fifth and sixth quotations in the list above already testify to this. Of course, if the grammar of the infinite lacks an infinite subject matter, this means that we cannot talk about the infinite; for if we cannot use the grammar of the infinite to talk about the infinite, then what can we use to talk about it? And this in turn means that the grammar does after all need to be challenged; for if we cannot use the grammar of the infinite to talk about the infinite, then what can we use it for? To this last question, which is intended rhetorically, somebody might reply, ‘We can use the grammar of the infinite to generalize about the possibilities that finite things afford! Is that not the whole point of Wittgenstein’s critique?’ But ‘our talking about the infinite’ here is meant to include our generalizing about just such endless possibilities. And ‘the grammar’s having an infinite subject matter’ is meant to be simply its providing us with the wherewithal to do this. (A grammar with an infinite subject matter must therefore, among other things, enable us to invoke the infinite series of natural numbers—as in section 3 above, where I talked about an object’s admitting of a possibility in which it admits of a second possibility, in which it admits of a third possibility, and so on.)

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simply saying (what we do say when we are immersed in the grammar), ‘There is no biggest natural number.’ He writes: A proposition like ‘there is no [biggest natural] number’ is offensive to naive— and correct—common sense . . . The proposition ‘There isn’t a [biggest] one’ should rather be: it makes no sense to speak of a ‘[biggest natural] number’, that expression is ill-formed.⁵⁰

But there is no biggest natural number! We are not obliged to adopt the formal mode.⁵¹ It is revisionist to say that we are. Wittgenstein struggles in other ways too. He suggests that it is because there are not infinitely many ‘things’, by which he means ‘elements of knowledge’, that it makes no sense to talk as though there were. In the manuscript of Philosophical Remarks he writes: ‘The only reason why you can’t say there are infinitely many things is that there aren’t. If there were, you could also express the fact!’⁵² But that is precisely to adopt the realist model. Had Wittgenstein written, ‘The only reason why you can’t say there are infinitely many bananas is that there aren’t’, we could have understood him in a non-realist way: you ‘can’t say’ that because it is false. But the ‘can’t say’ in this context is the ‘can’t say’ of nonsense, not of falsity. (And this implies, incidentally, that there is an additional problem with the quoted remark, namely the self-stultification of saying that it makes no sense to say this or that.⁵³) Still, the remark is a pencilled revision to something that Wittgenstein never submitted for publication.⁵⁴ There may even be an issue, as there so often is in Wittgenstein, about how far the remark is intended in propria persona. At any rate it is unfair to dwell on it. Suffice to observe (for now) that Wittgenstein does not always find it easy to combine respect for his own philosophical principles, and in particular for his conservatism and his anti-realism, with maintaining a critical distance from our grammar. ⁵⁰ Wittgenstein (1974), p. 465. Wittgenstein himself uses the terminology ‘last cardinal number’, but this is an inessential difference. ⁵¹ This is an allusion to Carnap’s distinction between the material mode and the formal mode: see Carnap (1995), lecture II, §8. It is clear, of course, what Wittgenstein’s concern about the material mode is: to say, ‘There is no biggest natural number’ is to disguise the fact that one is stating a rule for how to speak, not a fact about how things are. But note that this is by no means a peculiarity of this example. On Wittgenstein’s view, to say ‘3 + 3 = 6’ is likewise to state a rule for how to speak: see G. E. Moore (1959b), p. 279. What is required is not a wholesale shift within mathematical discourse from the material mode to the formal mode—which would be utterly unworkable—but rather due circumspection vis-à-vis the ways in which the material mode can be philosophically misleading. ⁵² Wittgenstein (1975c), §147, footnote 1. ⁵³ Cf. A. W. Moore (2019g), §8. One way to avoid this problem, in terms once again of Carnap’s distinction (see n. 51), would have been to adopt the formal mode—to mention, not to use, the offending nonsense—which is what Wittgenstein does in Wittgenstein (1975c), §144. There he writes, ‘If I were to say, “If we were acquainted with an infinite extension, then it would be all right to talk of an actual infinite”, that would really be like saying, “If there were a sense of abracadabra then it would be all right to talk about abracadabraic sense-perception”.’ ⁵⁴ See Rhees (1975), p. 349.

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Elsewhere we find something more modest. We find Wittgenstein calling into question the ways in which we actually couch our grammar, and more specifically the vocabulary that we use to couch it, rather than the grammar itself.⁵⁵ (True, this is still revisionist, and we might think that it still sits ill with his claim that ‘philosophy may in no way interfere with the actual use of language’. But at least it respects the autonomy of the grammar.) The point is this. Wittgenstein thinks that there is, in the very language we use to express our grammar—in the metaphors we exploit, in the pictures we conjure up—something to encourage the realist model. This is especially true in mathematical contexts, where he thinks we talk as though we were engaged in some elevated species of geography. Thus, in the thrall of transfinite mathematics, we claim that the set of real numbers is ‘bigger’ than the set of natural numbers, just as we might claim that Mount Everest is bigger than Mount McKinley. Again, we claim that irrational points fill the ‘gaps’ on a line left by the rational points, just as we might claim that certain sediments fill the gaps in a rock left by the igneous material. Wittgenstein rails against this. ‘The dangerous, deceptive thing about the idea: [“the set of real numbers is bigger than the set of natural numbers”]’, he writes, ‘. . . is that it makes the determination of a concept—concept formation—look like a fact of nature.’⁵⁶ And again: ‘We are surprised to find that “between the everywhere dense rational points”, there is still room for the irrationals. (What balderdash!)’⁵⁷ Furthermore, it is Wittgenstein’s conviction that, if we only stopped talking in these terms, perhaps in favour of some purpose-specific mathematical jargon, then interest in transfinite mathematics would altogether wane. It would lose what Wittgenstein calls its ‘schoolboy charm’.⁵⁸ Hilbert famously said, referring to the work by Cantor in which transfinite mathematics was founded, ‘No one shall be able to drive us from the paradise that Cantor has created for us.’⁵⁹ Wittgenstein replies: ‘I wouldn’t dream of trying to drive anyone from this paradise . . . I would do something quite different: I would try to show you that it is not a paradise—so that you’ll leave of your own accord. I would say, “ ‘You’re welcome to this; just look about you.” ’⁶⁰ These, then, are some of the ways in which Wittgenstein tries to discredit the garb in which transfinite mathematics tends to be paraded. Yet ⁵⁵ Cf. the distinction that Wittgenstein draws between the ‘prose’ and the ‘calculus’ in mathematical discourse (Waismann (1979), p. 149). Note: calling into question the vocabulary that we use to couch our grammar is more modest even than calling into question the mode (material or formal) that we adopt to express it, which is what we saw Wittgenstein doing above. Even so, they are two variations on a theme, and the response to the latter given in n. 51—that what is required of us is not so much to find new ways of couching our grammar as to be on the lookout for the philosophical dangers inherent in the ways we already have—has some purchase here too. (The response given in the main text below is somewhat different. It fastens on the dangers that revision itself can incur.) [Supplementary note: For further discussion of the issues raised in this footnote see Essay 18 in this volume.] ⁵⁶ Wittgenstein (1978), pt II, §19. (I have modified Wittgenstein’s example, but inessentially: cf. Wittgenstein (1974), p. 287.) ⁵⁷ Wittgenstein (1974), p. 460; and see more generally §§40–1. ⁵⁸ Wittgenstein (1976), p. 16. ⁵⁹ Hilbert (1967), p. 376. ⁶⁰ Wittgenstein (1976), p. 103.

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here too he struggles. In challenging the terms in which the relevant mathematical results are couched, he is constantly on the brink of challenging the results themselves, in just the way that drives mathematicians to distraction—and in just the way that he himself claims to abjure. Thus he writes at one point, ‘One pretends to compare the “set” of real numbers in magnitude with that of [natural] numbers . . . I believe, and hope, that a future generation will laugh at this hocus pocus.’⁶¹ But this invites precisely the same impatient retort as he himself would give if the legitimacy of a more homespun measuring technique were at issue: ‘One pretends no such thing. One does it.’ ‘Comparing sets in magnitude’ may not be the most felicitous description of what mathematicians do, but that is what they do, and that is, for better or worse, its description.⁶² There is something almost paranoiac about his abhorrence of what he finds on the pens and in the mouths of mathematicians; and all too often, in spite of himself, he allows this to become an abhorrence of the mathematics.

6. Some General Issues about Grammar But is this perhaps unfair to Wittgenstein? Does it underestimate how revisionist his own principles allow him to be? I referred just now to ‘the mathematics’, as indeed I have been referring to ‘the grammar of the infinite’, as though these were entities that we had discovered and that now lay preserved in some philosophical analogue of a glass cabinet, rather than products of our own ongoing linguistic activity, which is of course what Wittgenstein takes them to be. But their being products of our ongoing linguistic activity allows for a crucial degree of slack. They cannot be straightforwardly read off from that activity. This is because, even on a Wittgensteinian conception, not all of our day-to-day linguistic activity is immune to philosophical criticism. Not everything that we say is expressive of, or in accord with, any genuine grammar; and not everything that mathematicians say is a contribution to proper unadulterated mathematics. ‘What a mathematician is inclined to say about the objectivity and reality of mathematical facts’, to quote Wittgenstein himself, ‘. . . [is] something for philosophical treatment.’⁶³ Moreover, it is not impossible for mathematicians systematically to mishandle their own grammars and to import conceptual confusion into their own discipline. Many people, Wittgenstein ⁶¹ Wittgenstein (1978), pt II, §22. ⁶² Likewise when Wittgenstein explicitly denies that the relation m = 2n can be used to correlate the set of natural numbers with one of its own proper subsets (Wittgenstein (1975c), §141). Surely it just can. Wittgenstein might reply that here too he is challenging, not the result, but how it is stated. (Later in the same section he talks about ‘ambiguous grammar’ and claims that ‘it all hangs on the syntax of reality and possibility’.) But if that is his reply, then he has not made himself sufficiently clear; and the fact that unclarity comes so easily in such matters is part of my very point. ⁶³ Wittgenstein (1967a), pt I, §254, emphasis in original.

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included, would cite the more or less unthinking commitment of mathematicians to the law of the excluded middle as a case in point, another symptom of an unwarranted realism.⁶⁴ So is Wittgenstein not at perfect liberty, even by his own lights, to remonstrate not just against the way in which certain mathematical results are presented but against the ‘results’ themselves? In the particular case of transfinite mathematics, whose subject matter is after all supposed to be the infinite, can he not insist that we have no business trying to use concepts whose proper function is to enable us to generalize about the possibilities that finite things afford to ascribe infinite magnitude to things such as the set of natural numbers (just as we would have no business trying to use expressions whose proper use is syncategorematic categorematically)? As he says in Philosophical Remarks, ‘when (as in set theory) [mathematics] tries to express [the possibility in its signs], i.e. when it confuses them with their reality, we ought to cut it down to size’.⁶⁵ That strikes me as an uneasy defence of Wittgenstein. With the possible exception of this last point about transfinite mathematics trying to express what it has no business trying to express, Wittgenstein’s complaint about such mathematics concerns confusion that it engenders, not confusion that it harbours. The complaint is first and foremost a philosophical complaint. It is not a mathematical complaint, a complaint internal to the theory, such as, for instance, the complaint of incoherence which was levelled in the seventeenth and eighteenth centuries at early theories of differentiation and integration. So it can only be, it seems to me, a complaint of presentation. (Admittedly, we should include as a feature of the presentation, and therefore as fair game, the use of the very language of infinity; the use, that is, of the same language as is used to characterize the infinitude of, say, time. Indeed, Wittgenstein does raise the question whether the word ‘infinite’ should be avoided in mathematics, and he says that it should ‘where it appears to confer a meaning upon the calculus; instead of getting one from it.’⁶⁶ Even so, this is still a complaint of presentation.) A different but related defence of Wittgenstein would be this. Not even when the deliverances of mathematicians are free of all confusion do they count as proper mathematics—in fact, do they count as proper meaningful activity at all— unless they have suitable application. Thus Wittgenstein writes:

⁶⁴ See Wittgenstein (1978), pt V, passim. Cf. Dummett (1978d); cf. also Wright (2001), First Postscript, §IV. ⁶⁵ Wittgenstein (1975c), §144, emphasis in original. (Why does Wittgenstein say, ‘when it confuses them with their reality’, rather than, ‘when it confuses their possibility with their reality’? This may of course be a slip. But I do not think it is. Earlier in the same section, and in the previous section, he contrasts the infinite, inexpressible possibilities contained in things, and in particular contained in signs, with the finite, expressible facts in which they participate—and which constitute their reality.) ⁶⁶ Wittgenstein (1978), pt II, §58.

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I want to say: it is essential to mathematics that its signs are also employed in mufti. It is the use outside mathematics, and so the meaning of the signs, that makes the sign-games into mathematics. Just as it is not logical inference either, for me to make a change from one formation to another (say from one arrangement of chairs to another) if these arrangements have not a linguistic function apart from this transformation.⁶⁷

As for transfinite mathematics, if indeed mathematics is what it is, there is a serious question about what use is to be made of it in non-mathematical contexts. We can of course say, unexcitingly enough, that a child who has learned to multiply can now do infinitely many multiplications. But at what point do we need to take the further step, which is the very hallmark of such (so-called) mathematics, of invoking all those distinctions of infinite size?⁶⁸ This too, however, strikes me as an uneasy defence of Wittgenstein. Unfortunately, it raises general issues in the philosophy of mathematics which go well beyond the scope of this essay, so all I can do in this context is to pose some questions (questions that may be targeted as much on Wittgenstein’s conception of mathematics as on its application to this case). First, can there not be parts of mathematics which do not have any application, but which nevertheless count as proper mathematics because of how they help to systematize other parts which do have application? Secondly, and relatedly, can there not be parts of mathematics which do not have any application outside mathematics, but which nevertheless count as proper mathematics because of how they apply elsewhere within it? And thirdly, what about a branch of mathematics that is not applied until well after its development—must we say that only then does it count as proper mathematics? All three of these questions, whatever general force they might have, have particular force in the case of transfinite mathematics, which is both vigorous in its relations to other branches of mathematics and relatively young. But whether or not I am right in my concerns about these two defences of Wittgenstein, which relate to specific issues about a specific grammar, it is worth reflecting in conclusion on a much more general problem to which such issues graphically draw our attention. How do we distinguish between those parts of our linguistic activity that are legitimate and those that are not; between those that implement some genuine grammar and those that do not? There is an apparent circularity: to be sensitive to any such distinction we must have a clear understanding of the grammars involved; to have a clear understanding of the grammars involved we must discern them in our linguistic activity; to discern them in our linguistic activity we must recognize which parts of our linguistic activity

⁶⁷ Wittgenstein (1978), pt V, §2, emphasis in original.

⁶⁸ Cf. Wittgenstein (1976), p. 253.

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implement them; and to recognize which parts of our linguistic activity implement them we must be sensitive to the original distinction.⁶⁹ I do not say that this apparent circularity is vicious. I am not even certain that it is real. (Each step in this sequence can be disputed.) Perhaps there is a distinctive discomfort and puzzlement occasioned by linguistic activity that is not properly in accord with any genuine grammar.⁷⁰ Perhaps there is—though even then, of course, ‘distinctive’ is the operative word, with its own threat of circularity (for mathematicians engaged in bona fide mathematics feel plenty of discomfort and puzzlement with respect to their mathematical problems). The point, however, is that, whatever the correct verdict on this apparent circularity is, it is a signal feature of the infinite and of Wittgenstein’s treatment of it that they put this problem, which I take to be one of the most fundamental problems in philosophy, into particularly sharp focus.⁷¹

⁶⁹ Cf. A. W. Moore (1997), p. 162: and Essay 16 in this volume, §7. ⁷⁰ Cf. Wittgenstein (1967a), pt I, §§54 and 123. ⁷¹ I am very grateful to Marie McGinn for some extremely helpful comments on an early draft of this essay.

18 Wittgenstein’s Later Philosophy of Mathematics Abstract The aim of this essay is to consider some aspects of Wittgenstein’s (later) philosophy of mathematics. The main focus of the essay is an apparent tension between his philosophy of philosophy, according to which it is not the business of philosophers to criticize actual mathematical practice, and his philosophy of mathematics, whereby he frequently seems to do precisely that. A defence of Wittgenstein is considered that exploits a distinction that he himself draws, within mathematical practice, between the ‘calculus’ and the surrounding ‘prose’: the defence is that the former is all that is strictly immune to criticism, while the latter, which harbours various philosophical confusions, is all that he ever criticizes. But concerns about this distinction are raised, partly in connection with how mathematics is applied. At the end of the essay it is suggested that, if there really is the tension between Wittgenstein’s philosophy of philosophy and his philosophy of mathematics that there appears to be, then the fault may lie with his philosophy of philosophy.

1. Introduction The philosophy of mathematics was of colossal importance to Wittgenstein. Its problems had a peculiarly strong hold on him; and he seems, at times, to have thought that it was in addressing these problems that he produced his greatest work. Thus Rush Rhees recounts that, in the mid-1940s, when John Wisdom had written a short paragraph on Wittgenstein for inclusion in a biographical dictionary, he (Wisdom) sent the paragraph to Wittgenstein for comments, whereupon Wittgenstein recommended just one change, namely to add at the end: ‘Wittgenstein’s chief contribution has been in the philosophy of mathematics’.¹ Yet Wittgenstein’s writings in the philosophy of mathematics stand in a curious relation to this self-assessment. By 1938 he had written an early version of

¹ See Monk (1991), p. 466; and Monk (2007), p. 273 and n. 2. The Human A Priori: Essays on How We Make Sense in Philosophy, Ethics, and Mathematics. A. W. Moore, Oxford University Press. © A. W. Moore 2023. DOI: 10.1093/oso/9780192871411.003.0019

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his masterwork Philosophical Investigations,² the second half of which was on the philosophy of mathematics. This material did not however survive into the version of Philosophical Investigations that was eventually published after his death. Instead it appeared, modified in various ways, along with notes that he wrote during World War II, as Remarks on the Foundations of Mathematics,³ another posthumous publication, assembled by his literary executors. Apart from this there are scattered remarks in other material that he had produced earlier, while his ideas were beginning to take shape, and there are notes taken by some of those who attended his lectures on the philosophy of mathematics.⁴ None of this was submitted for publication by Wittgenstein himself. And, be his own relation to this body of work as it may, its early reception, when it did appear (starting with Remarks on the Foundations of Mathematics in 1956), was largely dismissive, if not positively contemptuous. Michael Dummett, in a passage that was not at all unrepresentative, wrote: Many of the thoughts [expressed in Remarks on the Foundations of Mathematics] are expressed in a manner which the author recognized as inaccurate or obscure; some passages contradict others; some are quite inconclusive; . . . other passages again, particularly those on consistency and on Gödel’s theorem, are of poor quality or contain definite errors.⁵

My own view is that Wittgenstein’s reflections on the philosophy of mathematics, for all the disarray with which they have been passed on to us, can indeed be seen as incorporating some of his greatest insights; and that the opposition that they provoked when they first appeared, and that they have continued to provoke since, is due largely to the combined difficulty and radicalness of these insights. My chief concern in this essay is not however to substantiate that view. Instead I want to do something more oblique. I want to look at some questions of Wittgensteinian exegesis on which his philosophy of mathematics has a unique and critical bearing. These are questions in the first instance about his philosophy of philosophy.

2. Wittgenstein’s Precept that Philosophy Leaves Everything (Including Mathematics) as it is, and his Distinction between Calculus and Prose Wittgenstein famously says that ‘philosophy leaves everything as it is’.⁶ Immediately after saying this he makes the same point specifically in connection

² Wittgenstein (1967a). ³ Wittgenstein (1978). ⁴ See esp. Wittgenstein (1974), Wittgenstein (1975c), and Wittgenstein (1976). ⁵ Dummett (1978b), p. 166; see also Monk (2007), §4. ⁶ Wittgenstein (1967a), pt I, §124.

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with mathematics: ‘It also leaves mathematics as it is’.⁷ He is expressing his well-known conviction that the proper role of philosophy is to save us from the confusions into which we fall when we misconstrue the functioning of our own language. Philosophy should not try to modify our language, still less to take issue with anything that is said in the proper exercise of it. It should just guide us to a clear view of it. In particular, philosophy has no business challenging any of the developments that mathematics has undergone. Its business is to challenge the extra-mathematical deliverances of those who, when they reflect on the nature of these developments, or on the nature of any other part of mathematics, misperceive the workings of the concepts being exercised and then mangle them in their struggle to provide a coherent account of what is going on there.⁸ So far, so familiar. So far, one might think, so reasonable. But here is the rub. Wittgenstein himself, in his own reflections on the nature of mathematics, makes claim after claim to outrage the working mathematician. Sometimes the mathematician’s complaint would be that Wittgenstein misunderstands what it is to practise mathematics. Sometimes the complaint would be that he misunderstands the mathematics itself. Thus Gödel, in a letter to Abraham Robinson, dismisses Wittgenstein’s remarks on his (Gödel’s) famous incompleteness theorem on the grounds that they arise from ‘a completely trivial and uninteresting misinterpretation’.⁹ I do not myself believe that Wittgenstein misunderstood Gödel’s theorem.¹⁰ But even if I am right about that and Gödel is wrong, what about all the rest of what Wittgenstein says to give mathematicians umbrage? Time after time he seems, either through incompetence or by design, to violate his own philosophical precept that philosophy should leave mathematics as it is. But how credible is it that he should really have done so—so often, and so flagrantly? I hope I am not exhibiting undue deference to the master by registering my scepticism on this score. It is a question of how plausible it is that someone should be as steeped in such a distinctive conception of philosophy as this and then not be sensitive to ways in which his own philosophical work, including what may even be some of his greatest philosophical work, flies in the face of it. If Wittgenstein makes claim after claim to outrage the working mathematician, then the explanation had surely better not be either that he is simply oblivious to the fact or that, despite his own philosophical scruples, he is bent on reform of mathematical practice. In fact, of course, another explanation is available, and one that looks entirely consonant both with his conception of philosophy and with his practice of it. There is mathematics; and there is what people are inclined to say about mathematics. Wittgenstein’s target consists of confusions that beset the latter. And it ⁷ Wittgenstein (1967a), pt I, §124. ⁸ See Wittgenstein (1967a), pt I, §§89–133, and Wittgenstein (1974), p. 369. ⁹ See Dawson (1989), p. 89. ¹⁰ For a helpful corrective see Floyd (2001) and Kienzler and Grève (2016). See also Essay 20 in this volume, §5, for reflections of my own on the matter.

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would not be the least surprising if the people most prone to these confusions were mathematicians themselves. They are the people most likely to have opinions about the nature of mathematics, and there is no reason whatsoever why, in arriving at these opinions, they should be any less susceptible to the kinds of confusions that Wittgenstein is concerned to combat than the rest of us. Just the opposite in fact. ‘A mathematician is bound to be horrified by my mathematical comments’, Wittgenstein writes, ‘since he has always been trained to avoid indulging thoughts and doubts of the kind I develop.’¹¹ Again: ‘What a mathematician is inclined to say about the objectivity and reality of mathematical facts, is not a philosophy of mathematics, but something for philosophical treatment.’¹² That last comment has a quite specific target. In an article in Mind the celebrated mathematician G. H. Hardy wrote that ‘[the] truth or falsity [of mathematical theorems] is absolute and independent of our knowledge of them’, adding that ‘in some sense, mathematical truth is part of objective reality’.¹³ The same view has been stoutly defended more recently by another celebrated mathematician, Roger Penrose, who describes the way in which, in mathematics, ‘human thought [seems to be] . . . guided towards some eternal external truth—a truth which has a reality of its own, and which is revealed only partially to any one of us’, and who, in the old debate about whether mathematics is invention or discovery, accordingly places himself, with only minor qualifications, in the latter camp.¹⁴ This view is an anathema to Wittgenstein. ‘The mathematician is an inventor,’ Wittgenstein insists, ‘not a discoverer’.¹⁵ Not that Wittgenstein’s stance on this issue puts him at odds with all mathematicians. There are distinguished mathematicians who have been as keen as he is to reject the picture of mathematics as discovery.¹⁶ When mathematicians reflect on the nature of their own discipline, some of them incline one way in this debate, some of them the other. What can plausibly be said to put Wittgenstein at odds with all mathematicians, or at least with all but the most atypical of mathematicians, is not his stance on this issue, but the way in which he maintains it. Eschewing the picture of mathematics as discovery, he denies that the propositions of mathematics have a subject matter in anything like the way in which the propositions of physics or geography have a subject matter. Rather, in establishing the truth of a mathematical proposition, we are forming new concepts, establishing new ways of making sense of things, contributing to ‘a network of norms’.¹⁷ And, in asserting a mathematical proposition, we are not saying how things are, still less saying how things are independently of us; we are enunciating one of our rules of representation.¹⁸ It follows, for Wittgenstein, that we need to look to the proof of a mathematical proposition to find out what was being proved, and hence

¹¹ ¹³ ¹⁵ ¹⁷

Wittgenstein (1974), pp. 381–2. ¹² Wittgenstein (1967a), pt I, §254, emphasis in original. Hardy (1929), p. 4, emphasis in original. ¹⁴ Penrose (1989), pp. 95–6. Wittgenstein (1978), pt I, §168; cf. pt II, §38. ¹⁶ See e.g. Cohen (1967) and Davies (2003). Wittgenstein (1978), pt VII, §67. ¹⁸ See e.g. G. E. Moore (1959b), p. 72.

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that, in one sense of ‘understand’, we cannot really be said to understand a mathematical proposition unless we are in command of a proof of it.¹⁹ That is certainly at odds with what the typical mathematician thinks. Very few people are in command of the proof of Fermat’s last theorem, for instance, but most mathematicians would feel no compunction about crediting anyone who has mastered basic high-school mathematics with an understanding of what the theorem states, in any but an absurdly and unhelpfully demanding sense of ‘understanding’. Or consider Goldbach’s conjecture. At the time of my writing this, the conjecture has been neither proved nor disproved. Yet most mathematicians would not think twice about saying that the conjecture is already fully intelligible, again in any but an absurdly and unhelpfully demanding sense of ‘intelligible’. How else, the mathematician is liable to ask, can anyone be engaged in trying to settle it, something that plenty in the profession are, indeed, engaged in trying to do? (Not that Wittgenstein is unaware of this objection, incidentally.²⁰ He acknowledges the less demanding senses of ‘understanding’ and has characteristically helpful comments to make about these. Nevertheless, he continues to insist on the significance of the more demanding sense.) It is plain, then, that there is a crucial role to be played in Wittgenstein’s philosophy of mathematics by the distinction between what the mathematician says when strictly engaged in mathematical practice and what the mathematician says, however instinctively, and with however little sense of departure from such practice, when not so engaged. Wittgenstein notes that mathematicians themselves are alive to this sort of distinction. There is a reference in one of his lectures to the way in which mathematicians look upon ‘interpretations of mathematical symbols [as] . . , some kind of gas which surrounds the real process, the essential mathematical kernel’.²¹ Here he is once again echoing Hardy, who, in the article from which I have already quoted, defines ‘gas’ as ‘rhetorical flourishes designed to affect psychology, pictures on the board in the lecture, devices to stimulate the imagination of pupils’.²² Elsewhere, in a similar vein, Wittgenstein distinguishes between what he calls the ‘calculus’ and what he calls the surrounding ‘prose’.²³ ‘Prose’ is perhaps a more suitable term than ‘gas’, because it is not pejorative (or at any rate, not relevantly pejorative). Wittgenstein never suggests that there is anything wrong with such prose in itself. Nor should he. As the quotation from Hardy testifies, the prose that surrounds the calculus may play an indispensable heuristic role. The point, however, is that it is the prose that will harbour any confusions of the kind that Wittgenstein is concerned to combat. It is the calculus, and the calculus alone, that can be regarded as sacrosanct.

¹⁹ E.g. Wittgenstein (1978), pt V, §§42–6, and Wittgenstein (1974), pp. 369–76; cf. Wittgenstein (1967a), pt I, §578. ²⁰ See e.g. Wittgenstein (1978), pt VI, §13; and cf. Wittgenstein (1967a), pt I, §578. ²¹ Wittgenstein (1976), p. 1. ²² Hardy (1929), p. 18. ²³ Waismann (1979), p. 149.

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The question that will primarily concern me is how robust this distinction is. It is clear that Wittgenstein would count, say, a proof of the irrationality of √2 as part of the calculus and a claim to the effect that there is therefore at least one gap in ‘the everywhere dense rational points’ as part of the prose.²⁴ But why? By what criteria? We had better not say, what the word ‘calculus’ might encourage us to say, that authentic mathematics comprises all and only the formal proofs that belong to some formal system or other. That is both too broad and too narrow. It is too broad because some such proofs, indeed all but an infinitesimal minority of such proofs, while they may be of mathematical interest in their own right and thus apt objects of mathematical study, are too complex to have a place in real mathematical practice. In fact Wittgenstein even balks at dignifying them with the label ‘proofs’.²⁵ Here we see that Wittgenstein is not thinking of authentic mathematics as an idealization of whatever engages the working mathematician; it is itself what engages the working mathematician. It has no features that the working mathematician cannot in practice recognize it as having.²⁶ But this also helps to explain why authentic mathematics comprises not only less but more than is indicated in the characterization above. It is, as Wittgenstein puts it, ‘a  of techniques of proof ’.²⁷ It comprises the many varied procedures whereby mathematicians actually establish their results; and what survives in the formal proofs of any given formal system is liable to abstract from differences between these procedures. But this last point merely serves to reinforce the concerns we might have about how robust the distinction between the calculus and the surrounding prose is. For the differences in question, between these various proof procedures, might well be thought to include the flourishes, pictures, and other devices to which Hardy alludes. Or if not, why not?

3. Concerns about the Distinction between Calculus and Prose At one point Wittgenstein says something that may appear to settle the matter in a very simple and neat way: Mathematics consists entirely of calculations. In mathematics everything is algorithm and nothing is meaning; even when it doesn’t look like that because we seem to be using words to talk about mathematical things. Even these words are used to construct an algorithm.²⁸

²⁴ E.g. Wittgenstein (1974), p. 460 and, more generally, pp. 460–74. ²⁵ See Wittgenstein (1978), pt III, §§1–62, passim. ²⁶ Cf. Wittgenstein (1978), pt III, §1, and Wittgenstein (1967a), pt I, §126. ²⁷ Wittgenstein (1978), pt III, §46, capitalization in original. ²⁸ Wittgenstein (1974), p. 468, emphasis in original. Cf. Wittgenstein (1961), 6.2 ff., and Wittgenstein (1975c), §157.

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Here the suggestion is that anything that a mathematician says that is not a contribution to the establishment or implementation of an algorithmic procedure for manipulating symbols is a part of the accompanying prose. Moreover, anything that a mathematician says to intimate that there is more to authentic mathematics than that—in particular, that these symbols are related to an independent reality in the way in which the words and phrases in an empirical proposition such as ‘She walked to the station’ are related to an independent reality—is not just a part of the accompanying prose; it is a breeding-ground for all the confusions that beset the philosophy of mathematics and it can legitimately be challenged by the philosopher. This applies to the case considered earlier. Suppose a mathematician says, ‘We know exactly how things would have to be in mathematical reality for Goldbach’s conjecture to be true or false. What we do not know is which of the two it is.’ This is paradigmatic prose. Wittgenstein is well within his rights, on his own principles, to take issue with it. This all appears relatively straightforward. But now consider: must even talk of truth and falsity themselves count as part of the prose? Wittgenstein seems to think so. In his discussion of Gödel’s theorem, in appendix III to part I of Remarks on the Foundations of Mathematics, he urges that our ascription of truth or falsity to mathematical propositions rests on nothing more than their superficial grammatical similarity to other propositions, a similarity that we can readily imagine away: it is not an integral part of the mathematics itself.²⁹ And he further insists that, if we are going to ‘play the game of truth functions’ with mathematical propositions,³⁰ then we had better understand all ascriptions of truth or falsity to them as relative to some formal system. For a mathematical proposition can count as true or false only in so far as something can count as asserting it, and the only thing that can count as asserting a mathematical proposition is producing it as the result of a proof in such a system.³¹ Again the message seems clear: the concepts of truth and falsity have no purchase in mathematics beyond certain analogies that strike us when we compare mathematical practices with practices of other kinds; all that sustains application of the concepts within mathematical practice is the obtaining of certain proof-relations between mathematical propositions and formal systems. Yet what if we shift our attention from Gödel’s theorem to Tarski’s theorem— that arithmetical truth resists being defined in a certain way? The notion of truth involved here goes beyond provability in any given formal system. So is this not a case in which a fully fledged conception of truth has to be seen as part of the calculus itself, not just as part of the prose?³²

²⁹ ³⁰ ³¹ ³²

Cf. Wittgenstein (1978), pt IV, §§15–16, and pt V, §13. Cf. Wittgenstein (1978), pt I, app. III, §2, and Wittgenstein (1967a), pt I, §136. Wittgenstein (1978), pt I, app. III, §6. Cf. Wittgenstein (1974), pp. 366–8. For related discussion see Steiner (2001) and Floyd (2001), §3.

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Or consider the law of the excluded middle. We naturally acquiesce in this law when we consider mathematical propositions. To take the stock example: we naturally assume that any given sequence of digits either occurs somewhere in the decimal expansion of π or does not. Wittgenstein expresses reservations about this, which make clear that he sees this and other such assumptions as a contribution, not to any calculus, but to the accompanying prose.³³ He again reminds us that asserting a mathematical proposition is not a way of saying how things are; it is a way of stating a rule. This makes the claim that any given sequence either occurs somewhere in the decimal expansion of π or does not akin to the totally unwarranted claim that either ‘The opening move shall be a pawn move’ or ‘The opening move shall not be a pawn move’ is a rule of chess.³⁴ The problem for Wittgenstein is that, although some mathematicians have themselves had reservations about the law of the excluded middle, most notably Brouwer, it is hard to see why such reservations do not count as reservations about standard mathematical practice. Mathematicians standardly adopt classical logic, including the law of the excluded middle, when they are establishing and implementing their algorithmic procedures. How is this a fact about extra-mathematical prose and not a fact about—precisely—their establishment and implementation of algorithmic procedures? We can turn to the infinite for a third example. Wittgenstein is very uncomfortable with the way in which set theorists claim to have shown that some infinite sets are bigger than others, as though they were astrophysicists claiming to have shown that some distant galaxies are bigger than others. He writes: ‘The dangerous, deceptive thing about [such an idea] . . . is that it makes the determination of a concept—concept formation—look like a fact of nature.’³⁵ Again he would say that he is casting doubt on the prose surrounding the calculus. Again the concern is that he is casting doubt on the calculus itself. That some infinite sets are bigger than others would be accepted by any orthodox set theorist as an unassailable result of set theory. There is a fourth example, which might appear as compelling as any. In fact I think that Wittgenstein has ways of addressing it—at least in his own terms—that cannot be extended to the other three examples, though it is worth a digression to see why. The example concerns consistency. Like the concept of truth, the concept of consistency appears to have a substantive role to play in mathematics, a role ³³ E.g. Wittgenstein (1978), pt V, §§9–28. ³⁴ Of course, one could insist that the law of the excluded middle had application only to propositions whose assertions were a way of saying how things are. This would leave one free to say that Wittgenstein’s critique leaves the law completely unchallenged. There are times when Wittgenstein himself suggests that we should adopt this stance: see e.g. Wittgenstein (1978), pt V, §17; and cf. Wittgenstein (1975c), §173. But it would be little more than a terminological stance. It would not gainsay the fact that his critique does present a challenge to the law if the law is construed as having application wherever ‘the game of truth functions is played’—which is how I am construing it. ³⁵ Wittgenstein (1978), pt II, §19. Cf. Wittgenstein (1974), p. 287.

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whereby its ascriptions are answerable to the investigator-independent layout of mathematical reality. If a mathematician asserts that Zermelo–Frankel set theory is consistent, for instance, say as a prelude to proving that the continuum hypothesis is independent of it, then his or her assertion seems to be at the mercy of whether Zermelo–Frankel set theory is consistent; of whether there is in fact, quite independently of what he or she or any of the rest of the mathematical community might be disposed to say about the matter, a set-theoretical proposition that admits of both a proof and a refutation within the theory. For that matter, the very idea that mathematics consists of algorithmic procedures seems to entail that there is an issue for mathematicians, if not about the truth or falsity of their propositions, at least about the consistency or inconsistency of their procedures, where the consistency or inconsistency of a procedure is a mathematically investigable feature of it that is quite independent of mathematicians themselves. There seems, then, to be a notion at work within mathematics— within mathematics, not just within the surrounding prose—which embodies the very picture of mathematics, as answerable to an independent reality, that Wittgenstein is concerned to repudiate. As I said, I think Wittgenstein has ways of addressing this fourth example. What are they? They are largely a matter of his biting various bullets. Among these are bullets that he notoriously does bite and bullets that I think he would be happy to bite. To begin with the latter: I think he would simply accede to the idea that, when claims about consistency feature within mathematics, they are no more answerable to an independent mathematical reality than any other mathematical claims. Thus we are at just as much liberty to declare Zermelo–Frankel set theory to be consistent as we are to declare the successor function to be one:one. That declaration can serve as a piece of legislation, a contribution to an algorithmic procedure or to a family of algorithmic procedures. Of course it is natural to protest, ‘But what if Zermelo–Frankel set theory is not consistent?’ Here, however, Wittgenstein can precisely appeal to his distinction between calculus and prose. For there are two corresponding ways of taking this question. If it is taken as a question within mathematics, then there is plenty to be said in response to it, for instance that if Zermelo–Frankel set theory is not consistent, then it is finitely axiomatizable. This poses no threat whatsoever to Wittgenstein. Taken in this way, the question is just an invitation to do more mathematics, mathematics that can sit alongside whatever mathematics we might do on the strength of our declaration that Zermelo–Frankel set theory is consistent. If the question is taken as a contribution to the prose, on the other hand, then it adverts to the possibility that we shall one day acknowledge both a proof within Zermelo–Frankel set theory and a refutation within Zermelo–Frankel set theory of one and the same proposition. And this is where we find the bullets that Wittgenstein notoriously does bite. He is prepared to meet what he calls ‘the superstitious dread and veneration by mathematicians in the face of

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contradiction’³⁶ with a studied nonchalance. His stance, roughly, is that, as long as we do not find any such conflict in our procedures, we do not need to worry about the possibility, and, if ever we do find such a conflict in our procedures, then we can decide how to proceed.³⁷ He is even prepared to countenance our proceeding by simply circumventing the conflict. In one of his lectures he says: ‘If you can draw any conclusion you like from [a contradiction], . . . I would say, “Well then, just don’t draw any conclusions from a contradiction” ’.³⁸ This may seem literally laughable: it is reminiscent of the Tommy Cooper joke in which a patient tells his doctor that his arm hurts whenever he raises it and the doctor replies, ‘Well then, don’t raise it.’ But actually, Wittgenstein’s nonchalance does not seem untoward once we rid ourselves of the idea that mathematical propositions are related to an independent reality in the way in which empirical propositions are. As long as we think of mathematicians as establishing and implementing algorithmic procedures, then Wittgenstein’s nonchalance can simply be seen as his way of sanctioning mathematicians’ continued use and periodic revision of any given procedure until such time as it no longer serves their purposes. And lest it seem utterly fanciful to suppose that mathematicians should work with inconsistent procedures even while fully aware of the inconsistencies, worse still that they should do so by simply negotiating the inconsistencies as they see fit, let us not forget that this is precisely what they did in the seventeenth and eighteenth centuries when the notion of an infinitesimal difference, as both equal to zero and not equal to zero, still informed work on the differential calculus.³⁹ The fourth example seems to me not telling, then. But the other three remain— as no doubt do variants on them.

4. One Way to Meet These Concerns There is one obvious way for Wittgenstein to rise to this collective challenge. However robust the distinction between the calculus and the surrounding prose, the prose may infect the calculus; or, more strictly, the prose may infect how we couch the calculus. Thus in all three of the troublesome examples considered in the previous section Wittgenstein can say that the trouble lies, not in the calculus itself, but in our choice of certain vocabulary to express it: ‘true’, ‘false’, ‘either . . . or . . . ’, ‘bigger’. This vocabulary has a use in non-mathematical contexts that resonates loudly from there. And it harbours a certain view of the calculus that is strictly inessential to it. So the very fact that we say that some infinite sets are

³⁶ Wittgenstein (1978), pt I, app. III, §17. ³⁷ Wittgenstein (1978), pt VII, §§12 ff. Cf. Wittgenstein (1974), pp. 303–5. ³⁸ Wittgenstein (1976), p. 220. ³⁹ A.W. Moore (2019a), ch. 4, §§1–2.

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‘bigger’ than others, to take that example, is fair game for Wittgenstein’s animadversions on what goes beyond the calculus, however securely lodged within the calculus the result itself may be. There is even an issue about whether we should use the word ‘infinite’ in a strictly mathematical context. ‘Ought the word “infinite” to be avoided in mathematics?’ Wittgenstein’s interlocutor asks at one point. ‘Yes,’ Wittgenstein replies, ‘where it appears to confer a meaning upon the calculus; instead of getting one from it.’⁴⁰ Moreover—this is a separate point—it is Wittgenstein’s firm conviction that, if only we were to recast much of the mathematics that most captivates us, by removing the offending vocabulary in favour of some purpose-specific mathematical jargon,⁴¹ then interest in it would wane. It would lose what Wittgenstein calls its ‘schoolboy charm’.⁴² We feel a certain heady pleasure when we are told that some infinite sets are bigger than others. We feel considerably less pleasure when we are told that certain one:one correlations yield elements that are not in their ranges. Though Wittgenstein’s principal concern is to combat philosophical confusions attending mathematics, he does also see it as part of his mission as it were to cut mathematics down to size.⁴³ One might wonder how even this consists with Wittgenstein’s non-revisionary insistence that philosophy should leave mathematics as it is. But there is one marvellous remark in which he makes clear how they consist. The remark is proffered in response to Hilbert, who famously said in connection with the work by Cantor in which transfinite set theory was founded, ‘No one shall be able to drive us from the paradise that Cantor has created for us’.⁴⁴ Wittgenstein replies: ‘I wouldn’t dream of trying to drive anyone from this paradise . . . I would do something quite different: I would try to show you that it is not a paradise—so that you’ll leave of your own accord. I would say, “You’re welcome to this; just look about you.”’⁴⁵ Elsewhere he puts the point by saying: ‘What I am doing is, not to show that calculations are wrong, but to subject the interest of calculations to test’.⁴⁶

5. Renewed Concerns about the Distinction between Calculus and Prose Wittgenstein may appear vindicated then. Although he is keen to warn mathematicians about the dangers of transferring vocabulary from one context to ⁴⁰ Wittgenstein (1978), pt II, §58. ⁴¹ Cf. Wittgenstein (1974), pp. 468–9. ⁴² Wittgenstein (1976), p. 16. ⁴³ The two things are related. This is for reasons that we have just seen. As Wittgenstein nicely puts it at one point: ‘Philosophical clarity will have the same effect on the growth of mathematics as sunlight has on the growth of potato shoots. (In a dark cellar they grow yards long.)’ (Wittgenstein (1974), p. 381). ⁴⁴ Hilbert (1967), p. 376. ⁴⁵ Wittgenstein (1976), p. 103. ⁴⁶ Wittgenstein (1978), pt II, §62, emphasis in original. Cf. Wittgenstein (1976), p. 141.

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another, and although he knows, indeed intends, that heeding his warning will make them reconsider the value of some of their work, he does not himself want to issue a direct challenge to any of that work. There is still a problem for Wittgenstein, though. Mathematical use of the vernacular is never just a matter of transferring vocabulary from one context to another—or, if it is, that is not Wittgenstein’s concern. (No harm accrues from the fact that the word ‘exponent’ has a quite different use in mathematical contexts from the use it has in non-mathematical contexts.) When set theorists describe some infinite sets as ‘bigger’ than others, they are not just choosing an arbitrary label which happens to have a use elsewhere. They take themselves to be appropriating a concept with which we are already familiar and extending its application. In fact that is precisely what gives Wittgenstein pause.⁴⁷ But why does it give him pause? Wittgenstein himself urges that mathematics involves the formation of concepts.⁴⁸ Why should this formation of concepts not include the modification of concepts as well as their creation? And if it does, then the use of the relevant vocabulary will after all be essential to what the mathematicians are doing. To claim that that vocabulary can be peeled off from the underlying calculus is to issue a direct challenge to their work. For a clear example of what I have in mind, consider Wittgenstein’s reluctance, which we noted earlier, to dignify all the formal proofs of any given formal system with the label ‘proofs’. That cannot but be heard as a challenge, not only to each of the systems but also to proof theory, the branch of mathematics in which our informal notion of a proof is at once idealized and codified. Or consider this: Does the relation m = 2n correlate the set of all numbers with one of its subsets? No. It correlates any arbitrary number with another, and in that way we arrive at infinitely many pairs of sets, of which one is correlated with the other, but which are never related as set and subset.⁴⁹

Here Wittgenstein is balking at the standard way of couching the result that each natural number can be paired with its double. The standard way of couching this result makes reference to a one:one correlation between the complete set of natural numbers and one of its proper subsets, that which contains only the even natural numbers. But Wittgenstein refuses to sanction this use of the word ‘set’, if it is understood as involving application of a single concept to both the finite case and the infinite case: if we do talk of both ‘finite sets’ and ‘infinite sets’, then these two uses of ‘sets’ must be understood as having fundamentally different

⁴⁷ Wittgenstein (1974), p. 464. ⁴⁸ Wittgenstein (1978), pt VII, §67. ⁴⁹ Wittgenstein (1975c), §141, emphasis in original, ‘class’ replaced by ‘set’.

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grammars from each other.⁵⁰ This is certainly a bold stance. But the issue for us is not whether it is bold or not; nor whether it is justified or not; nor even whether it is a stance to which most set theorists would take exception. The issue, for us, is whether it is a direct assault on set theory. And surely it is. Wittgenstein could of course beg all the relevant questions and insist that the very challengeability of what he is challenging, in his capacity as a philosopher, ensures that it is not an essential part of any authentic mathematics. That would be uninteresting—save in so far as it highlights what may in any case be a circle that afflicts his philosophy. Wittgenstein believes that, qua philosopher, he is entitled to take issue with that which perverts or is in danger of perverting either mathematical thinking or thinking about mathematical thinking, but that he is not entitled to take issue with mathematical thinking itself. He can take issue with the prose, but not with the calculus. The apparent circularity is this: there is no way, in practice, of respecting this distinction without having a grasp of the calculus; and there is no way of acquiring a grasp of the calculus without being suitably sensitive to authentic mathematical practice; and there is no way of being suitably sensitive to authentic mathematical practice without knowing how to screen those parts of mathematical practice that do not constitute proper exercise of the calculus; and there is no way of knowing which parts to screen without already being able to respect the original distinction. I do not claim that this apparent circularity is vicious. I do not even claim that it is real. Each step in the sequence can be disputed. For example, Wittgenstein might say that we can tell which parts of mathematical practice to screen because there is a distinctive discomfort that eventually manifests itself when the prose gets out of control.⁵¹ Perhaps there is—though even then, of course, ‘distinctive’ is the operative word, with its own threat of circularity. (Mathematicians can display plenty of discomfort when they are wrestling with bona fide mathematical problems.) The point, however, is that whether the circularity is real or not, the distinction between calculus and prose is not just a piece of theory for Wittgenstein. It is a tool that he needs to be able to implement in practice, in his attempt to rid the philosophy of mathematics of the confusions that beset it. And the mere threat of such circularity is surely enough to disturb the confidence that proper handling of this tool requires. It is surely enough to call into question the very project of trying to approach the philosophy of mathematics with that selfconscious detachment which his philosophy of philosophy demands. In sum: there is a real practical issue for Wittgenstein about the effectiveness of this distinction that is so crucial to his philosophy of mathematics, the distinction between calculus and prose.

⁵⁰ Wittgenstein (1974), pp. 463–5.

⁵¹ Cf. Wittgenstein (1967a), pt I, §§54 and 123.

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6. An Issue about the Application of Mathematics Is part of the problem excessive censoriousness on Wittgenstein’s part? How would it be if his attitude were much more one of laissez-faire, so that, instead of regarding whatever could be seen as incidental to any given algorithmic procedure as part of the prose, he regarded whatever could be seen as a feature of some algorithmic procedure, however incidental, as part of the calculus? This would mean that set theorists could just be left to get on with their business, be the interest of the exercise as it may. The only point at which philosophers would need to get involved would be the point at which someone reflecting on the exercise began to mishandle the conceptual apparatus involved in it and got into a muddle as a result. The threat of circularity just considered would remain, but it would be mitigated by the fact that the distinction between calculus and prose would need to be drawn much less frequently: only when there was a troublesome uncertainty about how to proceed and the issue was whether it was an uncertainty calling for mathematical insight or an uncertainty calling for philosophical clarification. No doubt this would leave Wittgenstein himself feeling uneasy, but would it be contrary to the strict letter of anything in his philosophy of mathematics? Well, yes, it would. There is an issue about the application of mathematics that I have not mentioned at all so far. When Wittgenstein says that mathematics involves algorithms, he means to specify a necessary condition for something to count as part of mathematics, not a sufficient condition. He would not reckon the mere algorithmic manipulation of symbols a part of mathematics unless those symbols also had a use in non-mathematical contexts. Wittgenstein puts the point as follows: I want to say: it is essential to mathematics that its signs are also employed in mufti. It is the use outside mathematics, and so the meaning of the signs, that makes the sign-games into mathematics. Just as it is not logical inference either, for me to make a change from one formation to another (say from one arrangement of chairs to another) if these arrangements have not a linguistic function apart from this transformation.⁵²

Let us not pause to consider what tension there might or might not be between this demand that mathematical vocabulary have a use outside mathematics and the worries that we saw Wittgenstein express earlier about the mathematical use of any vocabulary that has a use outside mathematics. Of more immediate concern is the fact that what we have here is essentially an assault on the very idea of pure

⁵² Wittgenstein (1978), pt V, §2, emphasis in original; cf. pt V, §25.

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mathematics. And it immediately furnishes a new complaint for Wittgenstein to level against transfinite set theory. For, to date, there is no serious use of any of set theory’s heavy-duty measuring apparatus in non-mathematical contexts. This assault on the idea of pure mathematics, which is strictly independent of anything that we have considered hitherto, seems to me problematical for a number of reasons. Here are two. First, even if the algorithmic manipulation of symbols needs to have application to count as a proper part of mathematics and not just a game, it is not clear why it needs to have application outside mathematics as opposed to elsewhere within it. Second, a branch of mathematics often remains unapplied until well after its development, so that even those who think that the application of mathematics is what gives it its point should acknowledge the importance of allowing unapplied mathematics to have free rein. Both of these points, whatever general force they might have, have specific force in the case of transfinite set theory, which is both vigorous in its application to other branches of mathematics and relatively young. There is far more to be said about both points, obviously. But there is also a third point that needs to be made in this context, of even greater significance, namely that the assault on the idea of pure mathematics is an assault on mathematicians’ very self-image. Hardly any mathematician would agree with Wittgenstein that ‘it is essential to mathematics that its signs are also employed in mufti’. This of course brings us back to square one. For yet again Wittgenstein has made a claim in his philosophy of mathematics to which the typical working mathematician would take exception. This in itself need not worry him. What the typical working mathematician is prepared to count as mathematics was always going to be a clear candidate for classification as prose rather than calculus. But it is another stark reminder of how subversive Wittgenstein is prepared to be in his critique of what mathematicians themselves actually think and say; and of how hard he therefore makes it for himself to draw the distinctions that he needs to draw in order to maintain his precept that philosophy leaves mathematics as it is. There is much that lies deep in the territory within and around mathematics that Wittgenstein’s philosophy does not leave as it is. It has been one of the main burdens of this essay to show this. It remains for me to make one very brief but very significant final point. My aim has been to highlight a tension that I claim to have discerned between Wittgenstein’s philosophy of mathematics and his philosophy of philosophy. But even if I have succeeded in this aim, it is a further question where the fault lies. Despite the various reservations that I have voiced about Wittgenstein’s philosophy of mathematics, there is much in it that seems to me to embody insights of the most profound kind. I think that what we have been witnessing are, in large part, problems with his philosophy of philosophy. In particular, I think that we have been seeing manifestations of a continual struggle that his philosophy of philosophy has with its own highly distinctive brand of conservatism. But that, as they so often say, is a topic for another occasion.

19 A Problem for Intuitionism The Apparent Possibility of Performing Infinitely Many Tasks in a Finite Time

Abstract Intuitionism in the philosophy of mathematics, which is the position adopted by Brouwer, is founded on the conviction that the subject matter of mathematics is something that each of us can construct. One conclusion that intuitionists draw from this conviction is that, because construction takes time, there cannot be any mathematical structure whose construction would be infinite. Other positions in the philosophy of mathematics are considered whose advocates likewise repudiate any such structure, albeit for loosely related reasons rather than for exactly the same reasons. But there is a problem for intuitionists, which is that their reasons for repudiating any such structure seem to be undermined by the apparent possibility of completing an infinite construction in a finite time. This essay explores various ways of addressing this problem, drawing on the work of Aristotle and above all Wittgenstein. It is argued that a Wittgensteinian critique of the grammar of ‘infinity’ gives us a way of dismissing any story that appears to involve the possibility of completing an infinite construction in a finite time as strictly nonsensical.

1. A Route from Some Non-Intuitionistic Premises to Some Intuitionistic Conclusions One of the things said to have reawakened Wittgenstein’s interest in philosophy in 1928, after a long period of philosophical inactivity, was attending a lecture by Brouwer on the foundations of mathematics.¹ Whatever exactly the effects of attending this lecture were, it is well known that Wittgenstein came to embrace conclusions in the philosophy of mathematics that were very similar to Brouwer’s. In particular he came to suspect the idea of an infinite coincidence among the natural numbers, that is the idea that every natural number might happen to ¹ See Feigl (1980), p. 64.

The Human A Priori: Essays on How We Make Sense in Philosophy, Ethics, and Mathematics. A. W. Moore, Oxford University Press. © A. W. Moore 2023. DOI: 10.1093/oso/9780192871411.003.0020

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have some property even though no general reason could be given as to why. Correlatively with this he shared Brouwer’s mistrust of uncritical application of the law of the excluded middle.² The similarities between them ran deep. So too, no doubt, did Wittgenstein’s debts. Yet he was very far from accepting the whole of Brouwer’s intuitionism. His route to these intuitionistic conclusions was very different from Brouwer’s. One thing that helps to reinforce this point is the work of Michael Dummett. Dummett has done as much as anyone to trace out, if not exactly this route, then a route, from broadly Wittgensteinian considerations about language and meaning to the intuitionistic conclusions in question. What is remarkable is that it is a route that runs directly counter to much of what Brouwer himself believed. Brouwer’s own intuitionism is founded on the conviction that what mathematics answers to is the experience, enjoyed by each of us, of the pure structure of time. Starting with this, each of us is supposed to be able to ‘construct’ the subject matter of mathematics. For example, by separating time into parts (past and future, say) in our thought, and then seeing that this process can be indefinitely repeated over time, we are each supposed to be able to arrive at the fundamental idea of a progression—and thus of a natural number.³ But this is something that we have to do in isolation from one another. Our mathematical experience is essentially private and incommunicable. Dummett, by contrast, takes as his starting point the essential publicity and communicability of mathematical ideas. He connects this with the Wittgensteinian thought that the meaning of a mathematical expression must ultimately be a matter of how it is used in mathematics. Were it not something open to view in this way—were it the kind of thing that Brouwer takes it to be—it would be impossible for anyone ever to have learned it or, a fortiori, to demonstrate that they had done so, and this in turn would make communication impossible (as indeed Brouwer thinks it is—in any ideal form).⁴ So how is it that these different trains of thought converge? For Brouwer, the construction of any given number takes time. So one can never reach the point of having constructed them all, there being infinitely many of them. It follows that there is no way of determining, by brute inspection, that they are all alike in any given respect. But mathematical statements are supposed to derive their meaning from what we are capable of constructing. So the question arises: what is it for a generalization about the natural numbers to be true? It must be for there to be some way of seeing, in advance, why each of them has to have the property in question—some way of constructing a proof to that effect. This is why

² E.g. Wittgenstein (1974), pt II, §§35 and 39.

³ Brouwer (1983).

⁴ Dummett (1978d).

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Brouwer recoils from the idea of an infinite coincidence among the natural numbers. Dummett, for his part, has argued that, if meaning is open to public view in the way outlined, then no mathematical statement can be true in a way that essentially outstrips any capacity we have to recognize it or prove it to be such. If it were true in that way, its meaning would not be relevantly dependent on the kind of use to which we put it, or indeed could put it. Since a brute inspection of all the natural numbers is ruled out, as before, on the grounds that there are infinitely many of them and it takes time to inspect each one, the idea of an infinite coincidence among the natural numbers is again called into question. Dummett’s work thus shows how radically non-intuitionistic premises can sustain a radically intuitionistic conclusion. But the situation is not as bizarre as it may seem. We can in fact see a fundamental common feature in the two trains of thought. Each involves a special emphasis on the fact that we cannot survey (construct, inspect) the whole of an infinite totality in a finite time, combined with the idea, somewhat differently located in each of the two cases, that (mathematical) meaning depends on what we are capable of surveying (constructing, inspecting).

2. Strict Finitism But now consider: whenever there is a question of what we are capable of doing, there is a further question, ‘Capable in what sense, or capable in what respect?’⁵ Here the focus seems to be what we are capable of doing in respect of the fact that we are immersed in time. And this has prompted, from various quarters, the following question: if capability in this respect is to be taken so seriously in our philosophy of mathematics, then why not capability in more stringent respects? After all, there is a perfectly good sense of ‘can’ in which none of us can survey the whole of a finite totality that contains more members than the number of atoms in the known universe, or more members than the number of milliseconds that will have elapsed by the time the earth has been swallowed up by the sun. Why is this fact not to be taken every bit as seriously as the fact that we cannot survey the whole of an infinite totality? And if it is, cannot Brouwer’s and Dummett’s arguments, casting doubt on the idea of an infinite coincidence among the natural numbers, be extended to cast analogous doubt on the idea of a truth concerning some sufficiently large natural number? The position that would result from this is known as ‘strict finitism’.

⁵ Cf. Wittgenstein’s view that ‘can’ can always be rendered ‘can as far as . . . is concerned’. See Dummett (1978a), p. 151. (Dummett gives no citation, but cf. Wittgenstein, 1975b, pt I, §§44–9 and 59–66.)

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Wittgenstein is sometimes thought to have espoused something like this, though his exact relation to strict finitism poses problems of exegesis that lie far beyond the scope of this essay.⁶ Others have recently explored the position more or less sympathetically.⁷ But it finds no place in either Brouwer or Dummett. Indeed Dummett has argued that it is incoherent.⁸ The question of how to steer an intuitionistic middle course between strict finitism on the one hand and a classical view of mathematical truth on the other thereby assumes a special urgency.

3. The Problem for the Intuitionist An obvious way out of this quandary, for the intuitionist, would be to respond as follows. We are justified in prescinding from the kinds of physical limitations that would prevent us from working out whether some colossal number is prime, say, whereas we are not justified in prescinding from those of our capacities which relate to our immersion in time, because it is our experience of time that furnishes mathematics with its content: that, if you like, is the whole point. Such a response would not be available to Dummett. But since he has only ever wanted to maintain an open-minded scepticism about classical mathematics, he is not in quite the same predicament as the intuitionist anyway. Let us in fact put Dummett’s arguments to one side. It is on intuitionism that I want to focus. The response above falls prey to an intriguing objection, and I wish to devote the bulk of this essay to that objection. It runs as follows. A reason has been given, on behalf of the intuitionist, for why practical limitations (those that are not purely temporal) are not at issue. But once we register this, are we not forced to admit that we can, after all, construct all the natural numbers and inspect each one of them in a finite time? How? By starting with 0, then dealing with 1 twice as quickly, then dealing with 2 twice as quickly as that, and so on. Of course, this is not something that we can do in the more usual and more stringent senses of the term ‘can’. Sheer sluggishness will eventually thwart us—or, if not sluggishness, then cack-handedness, as we try to write smaller and smaller, or, if not these, then the minute constitution of the universe. But these are precisely the sorts of limitations that we are being allowed to ignore. One thing that will not thwart us, it seems, is our immersion in time. The obvious intuitionistic response to the suggestion that intuitionism collapses into strict finitism has, in effect, turned into a way of undermining intuitionism itself.⁹

⁶ For discussion see Dummett (1978b), Kielkopf (1970), and Wright (1979), esp. ch. 7. ⁷ E.g. van Dantzig (1955), Wright (1982), and George (1988). ⁸ Dummett (1978e). ⁹ I am grateful to Alexander George for drawing my attention to this issue. Bertrand Russell was aware of it. He famously declared, in Russell (1935–6), pp. 143–4, that the impossibility of performing infinitely many tasks in a finite time was merely ‘medical’. Cf. Weyl (1949), p. 42.

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4. One Way in which the Intuitionist Cannot Address the Problem I take it that there can be no solace for the intuitionist in the possibility, which I am ignoring, that time itself is quantized. That seems to be a matter for empirical investigation, and, until the relevant data are to hand, it is just as possible, and indeed compossible, that time is finite. But if time were both quantized and finite, and if what counted were modal notions that took features such as these into account, then there would certainly be a limit to how many natural numbers could be constructed—precisely what the intuitionist does not want. Presumably the intuitionist is interested only in what our immersion in time rules out on essentially a priori grounds. (It is, however, a nice question, which I shall simply record, what question-begging this might involve. Consider: is the intuitionist committed to believing that time is an a priori intuition in Kant’s sense? If so, then the sheer fact that its structure can be an object of empirical investigation already poses a problem that needs to be addressed. If not, then ignoring the possibilities in question makes it unclear what exactly intuitionism has to do with time.)

5. Others Who Face the Problem The problem here is not just a problem for intuitionists. It is a problem for anyone, from Aristotle to Wittgenstein, who believes that (i) there are infinitely many natural numbers, and (ii) there is therefore a sense in which they can never all be surveyed (constructed, inspected) which is fundamentally different from the sense in which all those with fewer than a million digits, say, can never be surveyed (constructed, inspected). I intend (i) in a fairly radical sense. Mayberry, in a recent review of a book on Cantor, writes: ‘Just as it is true to say, in mathematical analysis, that imaginary numbers are just as real as real ones, so, in Cantorian set theory, it is true to say that infinite sets are just as finite as finite ones.’¹⁰ The thought here is roughly as follows: precisely in being treated as determinate wholes and being subjected to mathematical scrutiny, sets that are infinite in the technical sense can be seen to enjoy a kind of finitude. The truly infinite, in Cantorian set theory, is that which is too numerous to be gathered together into a whole—that which is a many too big to allow itself to be thought of as a one. Cantor described totalities that are infinite in this more radical sense as inconsistent totalities. Nowadays set theorists talk of proper classes. I intend (i) in the sense, antithetical to Cantorian set theory, that

¹⁰ Mayberry (1986), p. 431.

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even the natural numbers constitute such a totality, in other words that their infinitude is a ‘true’ infinitude. This allows for various interesting ways of denying (i) and avoiding the problem. Particularly interesting would be to identify the truly infinite with the uncountable, or if you like the truly finite with the countable—perhaps for the very reason under consideration, namely that it is possible to perform countably many tasks in a finite time. (A totality is countable when it has no more members than the set of natural numbers.) This would be compatible with the view that there are infinitely many ordinals beyond the natural numbers. And it would be provably impossible to frame an analogous problem for that view: there is no way of performing uncountably many tasks in a finite time.¹¹ (Brouwer, incidentally, is an interesting example of someone who adopted this approach, or something like it. Unlike more extreme intuitionists, he acknowledged the existence of ordinals beyond the natural numbers—ω, ω + 1, and so on—and held that the truly infinite begins only with the uncountable, though he did not himself express it like that.¹² But he was still in trouble: rejecting (i) is not sufficient for avoiding the problem, it merely provides the wherewithal to avoid it in one particular way. The reason why he was still in trouble is that he still needed to say that it is more than just practically impossible to construct all of the natural numbers in a finite time.) Someone who denied (ii) could also avoid the problem. For them the difficulties concerning the infinite would be essentially no different from the difficulties concerning the very large. They might then go on to embrace strict finitism of course—though it is worth noting that strict finitism does not require the rejection of (ii). (Wittgenstein certainly held (ii), as should be clearer in a little while.) Anyone who believes both (i) and (ii), however, is, like the intuitionist, in trouble. What are the possible responses to this problem?¹³

6. One Way in which Some Others Can Address the Problem But the Intuitionist (Once Again) Cannot Consider first the following response: It is, admittedly, only a practical impossibility for us to survey all of the natural numbers in a finite time, just as it is a practical impossibility for us to survey all those with fewer than a million digits. But there is a crucial scope distinction between the two cases. In the former case we cannot perform the task specified ¹¹ See Clark and Read (1984). ¹² Brouwer (1983). ¹³ As we go on to address this question we shall find ourselves at the same time, indirectly, addressing some of the oldest paradoxes of the infinite, those due to Zeno: see Aristotle (1941a), bk 6, ch. 9. These paradoxes too turn on the possibility or impossibility of performing infinitely many tasks in a finite time.

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because there are limitations of a certain kind to which we are subject. In the latter case there are particular limitations of this kind such that we cannot perform the task specified because we are subject to them. (Schematically, it is the difference between a sentence of the form ‘P because ∃ xFx’ and a sentence of the form ‘∃ x½P because Fx’:) If we could work ten to the million times more quickly than we do, the latter task would no longer be beyond us. The former task, on the other hand, still would be. This is enough to justify regarding the difference between the two cases as a difference of kind, and thus to vindicate (ii)—and, for that matter, (i). This is a powerful response. We may wonder whether it provides rationale enough for refusing to treat the set of natural numbers as a determinate whole, subject to mathematical scrutiny, in other words whether it really does vindicate (i). But certainly it does justice to our sense that the extra difficulties involved in surveying all the natural numbers, as compared with surveying all those with fewer than a million digits, is not simply a matter of there being more of them. The problem, of course, is that although this will satisfy some of those originally threatened, it cannot satisfy the intuitionist. It involves a crucial concession: that our inability to survey all the natural numbers is not just a matter of our being immersed in time. (Aristotle, for the same reason, would not have been satisfied.¹⁴ Nor would Wittgenstein, as we shall see in due course.)

7. The Grammar of ‘Infinity’ There is, I think, a viable sense of possibility—perhaps the least stringent that is of any interest—in which it is possible to survey all the natural numbers in a finite time, in a minute say. What I have in mind is basically this: that it is at least logically consistent to tell a story in which this happens. Someone spends half a minute on 0, a quarter of a minute on 1, and so on. We can say how long it takes them to get to any given number (25, say); and we can say how long they spend on that number (in the case of 25, just under a microsecond). But between logical consistency and practicability there lies a multitude of different senses of possibility. It remains to be seen in what other ways this story may be incoherent.¹⁵ For it is hard to shrug off the feeling that stories in which infinitely many tasks are performed in a finite time are in fact incoherent. Too many of them raise too many embarrassing questions. Consider, for example, the story in which an infinitely divisible stick is cut in half at some point in time,

¹⁴ For Aristotle’s view on the infinite see esp. his (1941a), bk 3, chs 4–8, and bk 8, ch. 8. Further references are given below. ¹⁵ Cf. Wright (1982), p. 248.

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each half is cut in half half a minute later, each quarter is cut in half a quarter of a minute later than that, and so on. What will remain at the end of the minute? Infinitely many infinitesimally thin pieces? Can we really understand this? An infinitesimal slice of the stick would seem to be a surface that continues to exist even when whatever it is a surface of has been destroyed. It may be said that this is an unfair characterization, and that we can make sense of infinitesimal slices in terms of what happens in certain regions of space. But it is not obvious how. In any case we also need to make sense of the idea that infinitesimal slices can make up the stick. That remains deeply mysterious.¹⁶ But the fact that these stories may be incoherent does not—yet—help the intuitionist. For there seems to be no non-question-begging way of locating the incoherence if all that there is to appeal to is pure temporal structure, as it can be known a priori, together with the fact that we are immersed in time. Some independent leverage seems to be required. But what? It is not enough to throw out a challenge in the way that Wright does: how, if we were suddenly enabled to perform infinitely many tasks in a finite time, could we know that we were?¹⁷ For this invites the simple response, ‘By doing it’. Nor is it enough to point out that we cannot imagine doing it. This is all too easily explained. Our imagination is subject to the same kinds of limitations, and is sluggish in the same way, as our executive abilities. In fact fully to imagine constructing all the natural numbers would be tantamount to actually doing it, given Brouwer’s understanding of construction. Nor do I think there is any mileage to be got out of the fact that stories in which infinitely many tasks are performed in a finite time can give rise to a distinctive kind of indeterminacy. I said above that they can give rise to embarrassing questions. But I side with those who do not recognize any link between these. The kind of thing that I have in mind has been illustrated most graphically by Thomson.¹⁸ His well-known example runs as follows: There are certain reading-lamps that have a button in the base. If the lamp is off and you press the button the lamp goes on, and if the lamp is on and you press the button the lamp goes off. . . . Suppose now that the lamp is off, and I succeed in pressing the button an infinite number of times . . . making one jab in one minute, another jab in the next half-minute, and so on. . . . After I have completed the whole sequence of jabs, ie. at the end of the two minutes, is the lamp on or off? It seems impossible to answer the question.¹⁹

¹⁶ Cf. Aristotle (1941c), bk 1, ch. 2, 316a–317b. ¹⁷ Wright (1982), p. 248. ¹⁸ J. Thomson (1954). Although I take issue with Thomson in this essay, there is much in his article with which I agree, as should become clear. ¹⁹ J. Thomson (1954), p. 5.

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But there is no problem here, any more than there is a problem in the fact that we cannot say whether the lamp was on or off two minutes before the process began. The story simply leaves the question open. It is compatible with how things have been described—modulo the coherence of the description—that the lamp should be on at the end of the two minutes, equally compatible that it should be off, and equally compatible for that matter that it should have disappeared altogether. Exactly what state it is in is simply not written into the story.²⁰ We can reinforce this fact as follows. There is no question of the lamp’s being in any of these states (on, off, or not there at all) at any given instant, except derivatively, that is except by virtue of its being in that state throughout some period to which the instant belongs. But the instant that we are interested in—the instant at the end of the two minutes—neither occurs within, nor terminates, any period, however short, throughout which the lamp is in any of these states. The only way in which it can be on, say, at that instant, is if the instant initiates a period throughout which it is on. Similarly with the other states. And this does indeed serve to emphasize how the question posed goes beyond the scope of the original story: it concerns periods of time after the two-minute period with which the story is concerned.²¹ How else, then, are we to locate an incoherence in stories of this kind? One way in which a story can be logically consistent while remaining incoherent is by involving an abuse of ‘grammar’, to put it in Wittgensteinian terms. Such a story would not make proper sense. For example, it might be a story in which the average plumber moves in next door. Or it might be a story in which—to use an example of Wittgenstein’s—it is 5 o’clock on the sun.²² Or it might be a story in which—to use a more contentious example—two people swap bodies overnight. Wittgenstein’s own view is that stories in which infinitely many tasks are performed in a finite time are of precisely this kind. They involve a misappropriation of the language.²³ The correct use of terms such as ‘infinity’ is to characterize the form of finite things and, relatedly, to generalize about the possibilities that finite things afford. Such terms cannot be directly applied to anything we encounter in experience. Nor can they be used to describe anything as actually infinite. Thus, for example, it makes sense to say, ‘This stick is infinitely divisible’—this is a claim about the kind of physical object that the stick is, and we can say what would

²⁰ Geoffrey C. Beresford (1980) argues that it is physically necessary for the lamp to be on. But since we are already in the realms of the physically impossible, I think the argument loses much of its force. ²¹ There is a vast literature on this and related issues, much of it focused on Zeno’s paradoxes, and to much of which I am indebted. See e.g. in addition to work already mentioned, Benacerraf (1962); Benardete (1964), passim; Black (1950–1); Bostock (1972–3); Chihara (1965); Grünbaum (1968); Grünbaum (1973); Lear (1981); and Ryle (1954), ch. 3. Points of agreement and disagreement between each of these writers and me will emerge in the course of my essay. ²² Wittgenstein (1967a), pt I, §§350–1. See also Mackie (1981), which prompted a reply by Lyle E. Angene in Angene (1982). I side with Angene. ²³ E.g. Wittgenstein (1975c), pp. 307–8.

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count as evidence for or against it. But it does not make sense to say, ‘This stick is [has been] infinitely divided’.²⁴ A Wittgensteinian approach to our problem would certainly rescue the intuitionist. Think again about sentences such as, ‘I have just surveyed all the natural numbers’, or, ‘I spent half a minute constructing 0, and, each time I had finished constructing a number, I then spent half as long constructing its successor, until a minute had elapsed’. If these could be seen as (well-disguised) nonsense—despite the familiarity of the words and despite the correctness of the surface syntax— then the intuitionist could once again assert, with reasonable confidence, that there is nothing we can do, even in principle, to substantiate the idea of an infinite coincidence among the natural numbers. But is it not overly cavalier, or indeed straightforwardly question-begging, to say that these sentences are nonsense? Suppose someone were to insist that they make perfectly good sense, and that it is clear what sense they make. How to arbitrate? There is an important asymmetry here. If someone insists that a sentence makes sense, it is incumbent on them to describe the sentence’s use—to say, for example, in what circumstances the sentence could be recognized as true. To say that a sentence is nonsense, on the other hand, is to throw out a kind of challenge: a challenge to provide just such a description. There is justification in being cavalier so long as the challenge remains unmet. Moreover it is not enough, in trying to meet the challenge, to appeal to the agreed meanings of the sentence’s constituent expressions and simply to urge their extension to this context. That would be like saying: You surely know what ‘It is 5 o’clock here’ means; so you know what ‘It’s 5 o’clock on the sun’ means. It means simply that it is just the same time there as it is here when it is 5 o’clock.²⁵ In particular it is of no avail to say that ‘This stick has been divided into infinitely many parts’ means just the same as ‘This stick has been divided into three parts’ except that the three parts have been replaced by an infinitude. Still, it is clearly possible for me to spend half a minute constructing 0. It is also possible for me to spend three-quarters of a minute constructing first 0, then 1. Can we not envisage the continuation of this ad infinitum? We can, but only in the sense that we can envisage an endless series of possibilities. This is different from envisaging a possibility involving an endless series.²⁶

²⁴ Cf. Wittgenstein (1975c), p. 162. ²⁵ This is a quotation from Wittgenstein (1967a), pt I, §350. ²⁶ Cf. Wittgenstein (1975c), p. 159. Cf. also J. Thomson (1954), p. 4.

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But might there not just be, say, infinitely many stars—or some (natural) process whereby a particle, as a result of oscillating through successively shorter distances, managed to complete infinitely many oscillations in a finite time?²⁷ Perhaps. But again, this has to be understood in terms of the infinite possibilities that might be written into what we could encounter in experience. Let us suppose that space and time themselves provide endless possibilities of movement and reorientation. Then we can imagine empirical evidence to suggest that, if we were to travel further and further away from some point, or if we were to delve deeper and deeper into some small region of space, we should always be able to find certain specifiable phenomena (stars, the traces of a particle’s movement, or whatever). This is still not to envisage an ‘infinite reality’. It may seem obscure what more could possibly be required. But more is being presupposed when someone talks of performing infinitely many tasks, or, more specifically, surveying all the natural numbers, in a finite time; they are sanctioning the possibility of a kind of direct encounter with the infinite. We have yet to be presented with a situation, real or imaginary, where it would be appropriate to talk in these terms.²⁸ What it comes to is this. We have a reasonable grasp of how ‘infinity’ and its cognates are used in characterizing the endlessly nested possibilities that (finite) things afford. If something is completely homogeneous (that is, if all its parts can be treated in the same way as it can), then we say that it is infinitely divisible. Again, if it is always possible to count more natural numbers than have been counted at any given time, then we say that there are infinitely many natural numbers. And if, moreover, there are more natural numbers than can be gathered together into any mathematically investigable and determinate totality, then we say that there are infinitely many natural numbers in the ‘true’ sense. But these uses are all quite different from, and do not themselves confer sense on, the troublesome uses with which we are concerned. To be sure, sense can always be artificially conferred on these. And it is always possible that hitherto unimagined situations should present themselves in which, for one reason or another, these uses seem genuinely appropriate. But for the time being, there is point in rejecting them as nonsense. Nonsense, if not recognized as such, can issue in conundrums and puzzles. That there is a kind of nonsense here is borne out by the fact that there are indeed such puzzles. We have already looked at one, the puzzle about the stick, though we have also seen reason to be sceptical about what purports to be another, Thomson’s supposed puzzle about the lamp. I want now to present what I take to be another of these puzzles, which I think is as revealing as any.

²⁷ Cf. Bostock’s bouncing ball in Bostock (1972–3). Wittgenstein envisages such a proposal in Wittgenstein (1975c), pp. 166–9. ²⁸ Cf. Wittgenstein (1975c), p. 169. It is also instructive to compare Kant (1933), A503–5/B531–3.

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Suppose—what I am saying does not in fact make sense—that I have just surveyed all of the natural numbers in a minute, by spending half a minute on 0, a quarter of a minute on 1, and so on. Then it is surely possible that while I was doing this, my constantly increasing speed of performance meant that time seemed to be going more and more slowly to me; it seemed that I was dealing with the numbers at a steady rate. Yet nothing could now count for me as a retrospective grasp of that experience in its apparent endlessness. (I could not have an apparently endless experience apparently followed by further experience.) I must, now, have forgotten all but some part of an initial segment of it. How can this be? This puzzle, I think, simply highlights a problem with the original supposition. Nothing could now count for me as a grasp of this experience, because nothing could ever count for anyone as a grasp of an infinite reality. The grammar of ‘infinity’ is simply not geared to this. It did not make sense to suppose that I had just surveyed all of the natural numbers in the first place. Trying to ‘stretch out’ my experience of the survey in the way described merely made the incoherence more graphic. No genuine possibility was being described.²⁹

8. A Related Problem for the Intuitionist, and a Related Solution I am presenting these thoughts as Wittgensteinian. But they are also in large measure Aristotelian.³⁰ Aristotle came close to recognizing a completed performance of infinitely many tasks as a contradiction in terms—something like a traversal of the untraversable. This was part of what he meant by repudiating the actual infinite. It is instructive to invoke Aristotle at this point since his own views about the infinite presented one particularly acute problem which helps to cast further interesting light on our current discussion. Aristotle’s identification of the infinite with the untraversable³¹ seems all very well when our attention is focused on finite periods of time, or even on the future. But what about the past? Aristotle believed that the past was infinite, and indeed that there had always been motion (the revolution of the heavens).³² How could he reconcile this with the past’s now having run its course—with its, apparently, having been traversed?³³

²⁹ For an interesting echo of this puzzle see Geach (1977), appendix. (I am committed to regarding Geach’s suggestion as incoherent.) For a remark bearing on the strategy in the last two paragraphs of this section see Wittgenstein (1967a), pt I, §464. ³⁰ See the references in n. 14. ³¹ Aristotle (1941a), bk 3, ch. 4, 203b30–204a7. ³² Aristotle (1941a), bk 8, ch. 1, 251b10–28. ³³ For a presentation of this difficulty see Sorabji (1983), ch. 14.

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This problem suggests an analogous problem for the intuitionist, one which is interestingly related to the problem which has been the focus of this essay. The problem which has been the focus of this essay is the apparent possibility of surveying all the natural numbers in a finite time. The new problem is the apparent possibility of completing a survey of all the natural numbers in a quite different way—not in a finite time, but after having surveyed them at a steady rate, backwards, for the whole of past eternity. (This is reminiscent of something that Wittgenstein once said in a lecture. He invited his audience to imagine coming across a man who is saying, ‘. . . 5, 1, 4, 1, 3—finished!’ and, when asked what he has been doing, replies that he has just finished reciting the complete decimal expansion of π backwards.³⁴) The reason why this is a problem for the intuitionist is much as before: the impossibility of doing this can seem, prima facie, purely practical. That is, it can seem a purely ‘medical’ fact that no one has an infinite history.³⁵ (At any rate it does not look like a limitation of mere temporal immersion, unless past time is itself finite. But this, like the possibility that time is quantized, is something that, in this context, we can ignore.) We are liable to feel similarly about this problem as about the original problem: namely, that, although it may be logically consistent to talk about surveying all the natural numbers backwards, there is in fact something incoherent in the idea. But how do we locate the incoherence? We can once again adopt a Wittgensteinian approach.³⁶ It is an abuse of grammar to describe anything as being, essentially, the outcome of an infinite process, as it is to describe any process as being infinitely old. The most that we can say is that certain processes, by their very nature, could never have begun. We may also describe past time itself as infinite, but only in the following sense: however long ago any event occurred, other events might have occurred earlier. We must understand the infinitude of past time in the same way as we must understand the infinite divisibility of time, as a matter of possibilities. (Maybe this was how Aristotle understood it. If so, the problem described above need not have been serious for him.³⁷) Note: the past and the future are on a par here. We are more reluctant to think of a process as having no beginning than we are to think of it as having no end. But this is simply because the past leaves traces on the present in a way in which the future does not, for example in memory, so we are in greater danger, when we declare a process to have no beginning than we are when we declare it to have no end, of committing ourselves to some unacceptable nonsense about an infinite reality that we are now confronted with. (Thus it may well be that Wittgenstein’s ³⁴ See Bennett (1971), p. 135. Cf. Wittgenstein (1975c), p. 166. ³⁵ This is an allusion to Russell’s famous phrase. See above, n. 9. ³⁶ There are other things to be said too. The natural numbers being what they are, certain surveys of them, and indeed a construction of them, intuitionistically understood, are impossible backwards: each number can be dealt with only when all its predecessors have been dealt with. ³⁷ See Lear (1979–80) and Lear (1988), pp. 82–3.

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story is absurd, in a way in which the corresponding story about a man beginning to recite the decimal expansion of π and never stopping is not.³⁸) But in the case of both the past and the future we must avoid talking in terms of anything’s actually being infinite. Such is the grammar of our language.

9. Conclusion The upshot of the discussion is this. While it may be true that Wittgenstein’s philosophy of mathematics received a certain impetus from Brouwer, it is also true that Brouwer’s philosophy of mathematics is seen at its strongest when it receives a certain impetus from Wittgenstein. For a pivot of Brouwer’s philosophy of mathematics, and the crucial point of contact between his train of thought and Dummett’s, is that we cannot survey the whole of an infinite totality in a finite time. And this is best underpinned by a Wittgensteinian critique of the grammar of ‘infinity’. But is it more than an underpinning? Perhaps the Wittgensteinian critique is doing all the important work? After all, due attention to the grammar of ‘infinity’ can, and in Wittgenstein’s work does, lead directly to the intuitionistic conclusion that there cannot be an infinite coincidence among the natural numbers. To say that all the natural numbers share a certain property, on a Wittgensteinian conception, can only mean something like this: there is some principle whereby the process of generating successive natural numbers is constrained to carry on in a certain way. (It cannot just turn out that they are all alike in this or that respect. There is no ‘infinite reality’.³⁹) I do believe that, in some sense, a Wittgensteinian critique of the grammar of ‘infinity’ more directly achieves what Brouwer and Dummett are each trying to achieve. But it is not a question of bypassing, or rendering superfluous, their shared concern with temporal possibility—with what we can do in a finite time, given that we are ourselves inescapably immersed in time. That concern is itself concern with possibility of a deep kind. So it is already of a piece with a Wittgensteinian concern with language—our language. (‘Our language’ here does not mean, and has not meant in this essay, anything as straightforward as English. It is more like our way of understanding things. It is what determines the limits of comprehensibility—for us.⁴⁰) It is best, therefore, to think of it like this. The Wittgensteinian critique neither supplants, nor even supplements, what is common to Brouwer and Dummett. Rather it serves to draw it out.⁴¹ ³⁸ This is Jonathan Bennett’s point; see Bennett (1971) and Bennett (1974), ch. 7. Cf. Waismann (1982). Cf. also Kant (1933), A410/B437. ³⁹ See Wittgenstein (1974), pt II, §§35 and 39, and Wittgenstein (1978), pt V, passim. ⁴⁰ Cf. B. Williams (2006h), p. 369, and A. W. Moore (2019c). ⁴¹ I have benefited from discussions with many people on these topics. Especial thanks are due to Joseph Melia, Philip Percival, Andrew Rein, and Timothy Smiley.

20 More on ‘The Philosophical Significance of Gödel’s Theorem’ Abstract The starting point for this essay, which originally appeared in a volume of essays in honour of Michael Dummett, is provided by Dummett’s discussion of Gödel’s theorem. In his discussion Dummett considers the threat posed by Gödel’s theorem to the idea that meaning is use and argues that this threat can be annulled. In this essay an attempt is made to show that the threat is even less serious than Dummett makes it out to be. Where Dummett’s argument is in effect that Gödel’s theorem does not prevent us from capturing the truths of arithmetic, the argument of this essay is that the idea that meaning is use does not require that we be able to capture these truths anyway. Towards the end of the essay this argument is related first to Dummett’s concept of indefinite extensibility and then to some of Wittgenstein’s (notorious) remarks on Gödel’s theorem.

1. The Idea that Meaning is Use and the Threat Posed to it by Gödel’s Theorem Among Michael Dummett’s many signal contributions to philosophy has been a sustained critique of the idea that meaning is use. This idea is, of course, both rough and programmatic: Dummett is as clear about this as anyone.¹ Even so, there is enough in it to impose a variety of non-trivial constraints on the theory of meaning, and Dummett has done a colossal amount, through his critique, to articulate, elaborate, explore, and apply these constraints. At the same time, as someone who is fundamentally sympathetic to the idea, he has been concerned to defend it. This is what he sets out to do in his celebrated essay on Gödel’s theorem.² ¹ See e.g. Dummett (1978c), pp. 188–9. ² Dummett (1978c), the article from which my title derives. See also his (1994). Note: the idea that meaning is use is standardly attributed to Wittgenstein. Dummett follows this practice, e.g. in his (1993), p. 139. In fact, Wittgenstein’s endorsement of the idea is altogether more circumspect than the orthodoxy would suggest. See e.g. his (1967a), pt I, §§43 and 139; cf. Hacker (1996), pp. 244–9, and Rundle (1990), ch. 1. Nevertheless, I take it that the idea does embody, in however rudimentary a way, an important precept in Wittgenstein’s thinking.

The Human A Priori: Essays on How We Make Sense in Philosophy, Ethics, and Mathematics. A. W. Moore, Oxford University Press. © A. W. Moore 2023. DOI: 10.1093/oso/9780192871411.003.0021

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Gödel’s theorem entails that no formal system can contain all and only the truths of arithmetic. At least, it seems to entail that. There are philosophical issues, going beyond Gödel’s purely formal result, about what counts as a truth of arithmetic. I shall return to these issues shortly. But meanwhile, let us take for granted this standard conception of the purport of the theorem. Now if we also take for granted (a) some constitutive connection between the meaning of arithmetical vocabulary and the truths of arithmetic that is strong enough for each to determine the other, and (b) some constitutive connection between our use of arithmetical vocabulary and our acceptance of some formal system that is likewise strong enough for each to determine the other, then Gödel’s theorem poses a clear threat to the idea that meaning is use. Dummett’s aim is to show that this threat can be annulled. Given the little that I have said so far, this may not appear to be a very ambitious aim. For while there is no denying the threat, there do seem to be, at this superficial level, numerous potential ways of averting it. For a start, neither (a) nor (b) is entirely uncontroversial. Neither, come to that, is entirely clear. But Dummett takes the discussion down to a much deeper level. And in the course of doing so, he makes the threat look much more serious.³ My sympathies are very much in line with Dummett’s. I too see something worth defending in the idea that meaning is use and nothing in Gödel’s theorem that is incompatible with it. But I disagree about how far down the defence of the idea needs to go. Very roughly, I shall try to argue that the threat posed by Gödel’s theorem is less serious than Dummett makes it out to be. I shall suggest that he exaggerates the extent to which the threat can be sustained before it dissolves. There is an important caveat that I must enter before I proceed. When I talk about the threat posed by Gödel’s theorem, I mean the threat posed peculiarly by Gödel’s theorem. I dare say that there would be all sorts of difficulties in working out how exactly to marry Gödel’s theorem with the idea that meaning is use; and I dare say that these would enforce all manner of clarification, modification, and qualification of the idea. But if so, this would be because of general features of language that the theorem served to illustrate, features that could just as well be illustrated by non-technical facts about items of vocabulary from some quite different area of discourse. My watchword in this essay will be: ‘What is it that we can do with non-arithmetical expressions such as “thin”, “yesterday”, and “freely” that we cannot do with arithmetical expressions such as “0”, “+”,

³ As a result, when he eventually argues that the threat can be annulled, he seems to reap a double benefit: he seems not only to safeguard the idea that meaning is use, but also to attain a correspondingly deeper understanding of what the idea comes to.

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and “prime” that makes special trouble, where the latter are concerned, for the idea that meaning is use?’⁴

2. Annulling the Threat Let us return to the prima-facie tension between Gödel’s theorem and the idea that meaning is use. As I said, this comes about granted two connections: (a) the connection between the meaning of arithmetical vocabulary and the truths of arithmetic; and (b) the connection between our use of arithmetical vocabulary and our acceptance of a formal system. There is a great deal to be said about (a). Much of this relates back to the issues I mentioned earlier, concerning what counts as a truth of arithmetic. There are some radical views whereby nothing counts, because no arithmetical statement is a proper candidate for truth or falsity.⁵ On other, less radical views, nothing of the kind that Gödel’s theorem shows to lie outside any standard formal system of arithmetic counts, either because nothing of that kind counts as a truth⁶ or, more specifically, because nothing of that kind counts as a truth of arithmetic.⁷ Again, there are views according to which there is no notion of arithmetical truth except relative to a formal system.⁸ Myself, I am inclined to demur from all such views and acquiesce in the intuitive notion that any statement which is couched entirely in arithmetical vocabulary, and which we can prove, is a truth of arithmetic. But of course, this raises large questions about each of ‘arithmetical vocabulary’, ‘can’, and ‘prove’ (and perhaps also about ‘we’), questions that are entirely germane to the current discussion. All of this bears on (a), and puts pressure on it. Much of it, however, also puts pressure on my original informal gloss on Gödel’s theorem, and does little to relieve the threat that the theorem poses to the idea that meaning is use. If anything, it exacerbates the threat. For that reason, among others, it is more appropriate in this context to focus on (b). What motivates (b)? Presumably something like the following train of thought. (1) If our use of arithmetical vocabulary is to determine the meaning of that vocabulary (which is a simple corollary of the idea that meaning is use), then it must be possible to gain access to the feature of it in virtue of which it does so without (yet) understanding the vocabulary, which means, among other things, that this feature must be describable in a finite and non-question-begging way;

⁴ ⁵ ⁶ ⁷ ⁸

Cf. A. W. Moore (2019a), ch. 12, §6. This essay is an elaboration of the argument sketched there. This is the position that Wittgenstein adopts in his (1961): see 6.2 ff. This is the position that Hilbert adopts in his (1967). This is the position that Isaacson adopts in his (1996). This is the position that Wittgenstein adopts in his (1978): see further section 5 below.

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otherwise the meaning of the vocabulary could never be grasped, which would be intolerable. But (2) the only feature of our use of arithmetical vocabulary that would be accessible in this way, and that would be strong enough to determine its meaning, would be our acceptance of a formal system.⁹ This train of thought points to a contrast between arithmetic and chess, say. In the case of chess there is a finite set of clearly statable rules that determine what counts as winning. We are relatively comfortable with the idea that someone who has been exposed to these rules and consciously assimilated them should know how to play the game. In the case of arithmetic there is nothing analogous that anyone can assimilate that will similarly serve to explain their ‘knowing how to play the game’. Or at least, such seems to be the lesson implicit in Gödel’s theorem. And we find this puzzling. The train of thought above helps to locate our puzzlement. Ultimately, we are attracted to the idea that meaning is use. And we are inclined to construe this as a demand, in the case of arithmetic, that there should indeed be some analogue of the rules of chess. This is essentially what step (1) in the train of thought comes to. But the only thing that seems capable of acting as such an analogue is a formal system. So we think that ‘knowing how to play the game’ must derive from accepting such a system. This is what step (2) in the train of thought comes to. Both steps can be challenged. The heart of Dummett’s essay, as I understand him, is his challenge to step (2). And hence my quarrel with him, such as it is. For although I too have reservations about step (2), I do not think that we need to delve as deep as that. Our puzzlement is better checked, it seems to me, higher up, by challenging step (1). This is not to suggest that Dummett accepts step (1). My point is rather that, unless he does, he could have made Gödel’s theorem look less threatening by saying why he does not. I shall begin, however, by saying something about step (2). Step (2): There are questions about what it is for us to accept a formal system. These are targeted as much at the idea of a formal system as at the idea of our acceptance of one. In order for step (2) to be relevant to Gödel’s theorem, a formal system needs to be construed as a recursively axiomatizable set of sentences. But it is simply not clear that only a recursively axiomatizable set of sentences is capable of acting as a suitable analogue of the rules of chess. (In general, we must beware of undue reverence towards the axiomatic method in mathematics. After all, people were engaged in arithmetic for thousands of years before any attempt was made to supply it with an axiomatization.) It is instructive to compare step (2) with another step which takes us further in the same direction and which has a certain superficial appeal, but which we surely want to reject: namely, the step whereby the only thing capable of acting as a ⁹ Cf. Dummett (1978c), pp. 187–90. Cf. also Wright (1994), p. 200; in particular, compare Wright’s (A) with step (1) and his (B) with step (2).

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suitable analogue of the rules of chess is, not a recursively axiomatizable set of sentences, but a recursive set of sentences. Call this step (2*). (Step (2*) leads to worries, not in the first instance about Gödel’s theorem, but about Church’s theorem.¹⁰) We are happy not to take step (2*). Why should we not be similarly happy not to take step (2)? What is it that we think only a formal system can do for us? ‘Capture’ the truths of arithmetic? But why think that only a formal system can do that? Perhaps a formal system, supplemented by some general precept that allows us to infer, from any given set of premises, various ‘metaconclusions’ about the truth of those premises, can do it. This seems to be the kind of thing that Dummett has in mind when he refers to ‘the general principle according to which any precise formal characterization [of the use of the expression “natural number”] can always be extended’.¹¹ By a ‘precise formal’ characterization Dummett means the kind of characterization that a formal system provides. What he is envisaging here is a characterization that is not a ‘precise formal’ one, a characterization in which essential use is made of a formulation of the general principle to which he refers. Such a formulation, if it could be given, would be just the sort of supplement that I have in mind.¹² To be sure, Dummett himself does not give such a formulation. And we are no more entitled to assume that it can be given (in a way that is finite and non-question-begging) than we are to assume that it cannot. But as long as this matter remains unresolved, then step (2) is in doubt. But what if the doubt could be removed? What if there were some way of showing conclusively that nothing less than a formal system could capture the truths of arithmetic? Then Dummett’s argument would presumably be impugned. But his main contention—that, as far as the idea that meaning is use is concerned, Gödel’s theorem can be defused—I think would not be. This is because I think that step (1) can be rejected, as I shall now try to argue. Step (1): Suppose it turned out that we could not capture the truths of arithmetic, and thus that our use of arithmetical vocabulary could not be described in a finite and non-question-begging way. Why think this would matter? Here it is important to remember my watchword in this essay. I am not interested in an answer to this question that presents a problem for arithmetical vocabulary if it presents exactly the same problem for any other vocabulary and perhaps even forces us to reassess the idea that meaning is use. Thus consider a simple word like ‘thin’. This word has an infinite range of possibilities woven into its meaning. It can be applied in indefinitely many ways, for indefinitely many purposes, and to indefinitely many effects, whether literally or metaphorically, ¹⁰ Church’s theorem entails that the set of logical truths, in any suitably rich first-order language, is non-recursive. ¹¹ Dummett (1978c), p. 198. ¹² Cf. Wright (1994), §9, where, on p. 201, he makes a similar though more specific proposal. But see also Dummett’s reply, in Dummett (1994), pp. 335–8.

‘     ¨  ’   ’

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precisely or loosely, prosaically or poetically, straightforwardly or ironically, strictly or metonymically. There is no legislating in advance for all of these possibilities. For instance, there is no legislating in advance for the success of metaphorical applications, some of which may be contrived to describe situations completely unlike anything that anybody has ever encountered before. So even if it is our use of the word ‘thin’ that determines all these possibilities, there may well be no way of describing that use which is not either, per impossibile, an infinite list of all these possibilities or something effectively question-begging such as: ‘We use “thin” to denote thin things.’ Why then would it matter if it turned out that the same was true of our use of arithmetical vocabulary; that there was no way of describing our use of arithmetical vocabulary that did not either consist of some infinite list, perhaps of arithmetical truths or perhaps (less crudely) of ways of supplementing any given incomplete formal system of arithmetic, or take the form: ‘We use “0” to refer to 0, “+” to represent addition, and so on’? Of course, in spite of my watchword, it would be unsatisfactory to let the matter rest there and not to say something in response to the intuitions that motivate step (1). If in fact Gödel’s theorem makes graphic a problem for the idea that meaning is use which would otherwise have been much harder to see, then there is a sense in which it poses a special threat to the idea. And that special threat remains to be addressed. But I think it can be. I think the intuitions that motivate step (1) can be exposed as untenable reductive intuitions. Those intuitions, to paraphrase what I said earlier, come together in the following thought: if some feature of our use of a given set of expressions determines the meaning of those expressions, then that meaning cannot be grasped unless the feature is in some sense independently accessible. But this thought is simply unwarranted. In order for some feature of our use of a given set of expressions to determine their meaning, what is required is that we manifest our understanding of the expressions through our use of them; that we communicate with one another through our shared access to such use. To come to recognize the feature is to come to understand the expressions. This we manage through suitable training. And no doubt part of what makes us responsive to the training is the happy circumstance that we have some minimal range of shared interests and shared sensibilities (partly innate and partly inculcated). In the case of arithmetical expressions, this happy circumstance might include a shared sense of how self-conscious reflection on basic principles that we already accept can inform the exercise, at a meta-level, of a concept of truth, thereby leading us to accept new principles. Thus it might be a fact, more basic than any fact about what arithmetical expressions mean, that there is a certain level of consensus among us about what counts as a correct way to supplement any given incomplete formal system of arithmetic. But if so, there is nothing mysterious in that. After all, even in the case of chess, with which arithmetic seemed earlier to brook an unfavourable comparison, the rules would come to nothing without a certain level of consensus among us about what counted as implementing them.

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(But note: in neither the case of arithmetic nor the case of chess do we have any guarantee that we shall not find ourselves in circumstances that lead to irresoluble disagreement, or at the very least to unclarity, about how to carry on. I have seen the rule for castling formulated in such a way as to leave it unclear whether White is allowed to castle with a rook on e8, provided the rook is there as the result of a pawn promotion. It is easy to imagine the rule only ever having been formulated in this way.¹³)

3. An Additional Complication Concerning Consistency I deny, then, that in order for the idea that meaning is use to remain intact, arithmetical truth needs to be capturable. Our propensity to react to new mathematical problematics in the same way as one another can do the work that any capture of the truths of arithmetic was supposed to do. It can sustain the meaning of the arithmetical vocabulary we use. And it can explain how our exposure to usage, though this falls short of use, can give us as much as we need to grasp that meaning.¹⁴ So I claim. But there is an obvious and important objection that I need to address. This objection turns on the crucial idea that we have a propensity to agree about what counts as a proper extension of our mathematical procedures. Even someone thoroughly sympathetic to the familiar line of thought that I have been advocating will want to resist any crude reduction of the normative to the nonnormative. Indeed the line of thought is supposed to be, precisely, a non-reductive one. Thus our propensity to reach such agreement sustains the meaning of the arithmetical vocabulary we use only given various normative presuppositions about the circumstances of the agreement. (It would be immaterial what agreement we reached in circumstances in which we were not suitably knowledgeable, competent, clear-headed, and the rest.) We have to be entitled to whatever we arrive at through agreement. But, the objection runs, in the case at hand, where the agreement involves our affirming the consistency of one of our formal systems of arithmetic, these normative presuppositions must include the presupposition both that the system is consistent and that we can know it to be. If the system is not consistent, then any propensity on our part to say that it is is by the by (although ironically, of course, there may then no longer be any threat to the idea that meaning is use, either because the arithmetical vocabulary we use is meaningless, or because it is not such as to support any notion of arithmetical truth, or because the notion of arithmetical truth that it supports can, trivially, be captured). If, on

¹³ For the ideas in these two paragraphs I am, of course, leaning heavily on Wittgenstein’s later work on meaning and rule-following, esp. in his (1967a), passim. ¹⁴ For the distinction between usage and use, see Hacker (1996), pp. 248–9.

‘     ¨  ’   ’

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the other hand, the system is consistent, then any propensity on our part to say that it is is still by the by unless reinforced by the entitlement that comes of our knowing it to be so. And if we do know it to be so, this is a substantive cognitive achievement. So there are real questions that remain to be addressed, first about whether we have the knowledge, and secondly about whether, even if we do, there will not after all be some threat to the idea that meaning is use in any satisfactory account of how we come by it. That is the objection. Before I address it, I should mention an apparent anomaly. Crispin Wright, in his own discussion of Dummett’s essay, explores these questions as thoroughly and as carefully as anyone.¹⁵ One of his aims is to show that Gödel’s theorem poses a threat to the idea that meaning is use only in so far as we can know that the formal systems of arithmetic we accept are consistent. The dialectic here suggests rather the opposite; that only in so far as we cannot know that the formal systems of arithmetic we accept are consistent does Gödel’s theorem pose a threat to the idea that meaning is use. Thus let S be such a system. And suppose that, even though it is consistent, we cannot know it to be. Then Gödel’s theorem seems to entail that the meaning of the vocabulary in which S is couched reaches beyond anything we do with it, or indeed anything we can do with it, to ensure that certain statements are, unknowably to us, true. That is where the threat to the idea that meaning is use seems to lie. On closer inspection, however, the apparent anomaly disappears. The supposition that, even though S is consistent, we cannot know it to be already poses a threat to the idea that meaning is use. We do not need Gödel’s theorem to see that this supposition involves the existence of statements that are, unknowably to us, true. (It is important to note that Wright adds as an express qualification to his discussion: ‘unless we accept Bivalence’.¹⁶) Furthermore, as the objection above indicates, if we can know S to be consistent, then something still needs to be said about how we can know this, and it remains to be shown that this can be done in a way that harmonizes with the idea that meaning is use. Where I do part company with Wright is that I have no qualms about saying that we do indeed know various standard formal systems of arithmetic to be consistent, and that such knowledge is in fact part of the solution to any problem here, not part of the problem. This may suggest that I am closer to Dummett than to Wright. But I suspect that what I say in response to the objection above will not be very appealing to either. First, here is a quick response that I am tempted to give. As before, let S be some formal system of arithmetic that we accept. And suppose that, for one reason or another, we are not entitled to affirm its consistency. Then, given the relationship between consistency and truth, neither are we entitled to affirm the truth of its

¹⁵ Wright (1994).

¹⁶ Wright (1994), p. 174.

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axioms and the truth-preservingness of its rules. But that is tantamount to saying that we are not entitled to affirm the axioms and the rules themselves: we are not entitled to accept S in the first place. However, this is a possibility that we cannot take seriously unless we are prepared to entertain some rampant scepticism about meaning, which violates my watchword in this essay. This response is not, I think, completely without merit. But it is too quick. Even given the relationship between consistency and truth, the fact is that the transition from working with S to working with all the relevant meta-arithmetical apparatus involved in affirming its consistency introduces a particular reflective conception of S, as faithful to some unified subject matter, that does create an additional burden. It is not obvious that our entitlement to accept S extends over all this apparatus, in the way indicated, to an entitlement to affirm S’s consistency. (Indeed it is worth remembering that even our entitlement to accept S, as a whole, is more than the sum of our entitlements to accept its several components.¹⁷) My more considered response to the objection is to insist that we can know formal systems such as S to be consistent but to deny that this is a ‘substantive cognitive achievement’ in any sense that is the least bit problematical for the idea that meaning is use. In particular I deny that our taking S to be consistent requires any sanction that is external to our use of S and its vocabulary. For there is no gap between our taking S to be consistent and its being consistent, or at least not the same gap as there is between, say, our taking Jupiter to have moons and its having moons. Our taking S to be consistent is more like our taking the successor function to be one:one. It is a piece of legislation. To be sure, there is an important difference between our taking S to be consistent and our taking the successor function to be one:one. In the former case we get there via reflection on the nature of truth, derivability, and the like. In the latter case it scarcely makes sense to say that we ‘get there’ at all: it is axiomatic that the successor function is one:one, at least in any typical system of arithmetic. But in both cases we are dealing with something whose truth is fixed by our practices. In saying this, I am, of course, presupposing a highly controversial view of mathematics, one whose defence would be well beyond the scope of this essay. But I do not mean to be presupposing anything about how our taking S to be consistent keeps faith with any earlier mathematical practices on our part. Of course we think it does. And of course this needs to be accounted for. But this is just an instance of the fundamental philosophical problem about norm-governed practices, a problem that has no special bearing on mathematics, still less on Gödel’s theorem. Just the same problem arises when we look at something in perfect viewing conditions and classify it as green. In that case too there are questions about how our shared reactions keep faith with earlier classifications of ours. Hence, given my ¹⁷ There are various familiar paradoxes that illustrate the non-additive character of entitlement: see Sorensen (1988), §1.II.C.

‘     ¨  ’   ’

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watchword in this essay, we ought to use the advance from accepting S to affirming its consistency, not to cast doubt on our conception of what counts as keeping faith with earlier practices (and therewith on the idea that meaning is use), but to enrich it.¹⁸ But what if S is inconsistent? We surely cannot legislate that possibility away. And if S is inconsistent, then we surely cannot know it to be consistent. Am I therefore simply taking for granted that S is not inconsistent, and grounding my claims that we know it to be consistent in a kind of externalism about knowledge? Not at all. Externalism about knowledge is precisely what I am resisting here. (This is not to deny that externalism about knowledge may be perfectly acceptable in other contexts, for instance where certain kinds of empirical knowledge are concerned.) The question ‘What if S is inconsistent?’ needs to be handled very carefully. In particular we need to separate the mathematical from the empirical. Certainly, at a purely mathematical level, saying that we know S to be consistent, or—more directly—affirming S’s consistency, commits us to denying its inconsistency. That is just a matter of elementary propositional logic. Again, at a purely mathematical level, there is much we can say in response to the question ‘What if S is inconsistent?’ For instance, if S is inconsistent, then it is finitely axiomatizable. But to say such things is to make moves within mathematics. It does not address the original worry. So the question ‘What if S is inconsistent?’ should probably have been understood as adverting to an empirical possibility: what if we later come to acknowledge an inconsistency in S? But to this the proper response can only be, ‘We must cross that bridge when we come to it.’ In general, there is no gainsaying the fact that our laying down certain rules here and now is compatible with our coming to view them differently later, perhaps even rejecting them later. Of course it is. But in laying them down here and now, we can be certain that, for now, they are indeed our rules. (So there is a sense in which we can know S to be consistent even if it is inconsistent: we can have a rule that S is consistent even if ¹⁸ It is worth comparing what I say in this paragraph with what Wright says in the extremely interesting n. 16 to his (1994). There he discusses the possibility that ‘the soundness of our arithmetical thought is something which we are entitled to assume on no other basis than its intuitive cogency’, and later he adverts to the strategy of ‘trying to make out that the belief in the consistency of intuitively sound systems of arithmetic is one in which we are groundlessly justified’ (emphasis in original). Dummett, addressing these issues on p. 333 of his (1994), asks, rhetorically, whether our understanding of basic arithmetical concepts only has the power to induce us to commit ourselves to certain axioms, or whether it supplies us with reasons for accepting those axioms. I am happy to deny the latter. To echo Wittgenstein, ‘without reasons’ does not mean ‘without right’ (Wittgenstein (1978), pt 7, §40). Peacocke, in his (1993), §4, also addresses these issues. Later in the same essay he claims, in effect, that if our affirmation of S’s consistency helps to determine the meaning of our arithmetical vocabulary, rather than being determined by it, then ‘it is a mistake to look for something present in our previous uses of . . . [this] vocabulary from which we can soundly argue that [S is consistent]’ (p. 184). I disagree. I think we can see our affirmation of S’s consistency both as helping to determine the meaning of our arithmetical vocabulary and as supported by our previous uses of that vocabulary. (Peacocke’s claim is on a par with the claim that, if our reaction to an arrangement of lines on a piece of paper helps to determine that there is a picture of a face there, rather than being determined by it, then it is a mistake to look at the piece of paper itself to see whether there is a picture of a face there.) See Diamond (1991).

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our rules also lead us, further down the line, to acknowledge that S is inconsistent. Obviously, were this to happen, the inconsistency in S would be matched by an inconsistency in its meta-system, namely the inconsistency that S both is and is not consistent.) This is an apt point at which to mention an empirical argument that someone might adduce against the inconsistency of S: namely, that if S were inconsistent, then we would probably have found the inconsistency by now. (Arguments of this kind have been mooted by Hartrey Field.¹⁹) Both Dummett and Wright have criticized this kind of argument, for independent but similar reasons. Both, indirectly, have suggested that no argument of this kind can work without the sort of non-empirical input that, if it were available, could be applied directly in showing that S is not inconsistent.²⁰ I am entirely in agreement with them in this. But I think that the real anomaly of the argument, which Dummett’s and Wright’s criticisms serve to highlight, is that, as it stands, it is an empirical argument for a mathematical conclusion. On the conception that I have been advocating, there cannot be such a thing. And when one tries to think of an empirical argument for an empirical conclusion in this territory, it is hard to conceive of anything that is not either clearly question-begging or clearly ill-grounded. There is no good reason, for instance, to think that if we were going to repudiate the concepts that we exercise in accepting S, then we would probably have done so by now. Concepts get repudiated for all sorts of reasons, sometimes after very long and distinguished histories. This connects with what I said above. We have no guarantee that we shall not some day want to turn our back on S for one reason or another. But in the meantime we accept S; and we have a concept of truth which, when we apply it in the light of our acceptance of S, leads us to acknowledge that S is consistent; and this in turn leads us, through application of techniques used in Gödel’s theorem, to accept an extension of S. Although there are various interesting and distinctive things going on here—although, as we might say, our arithmetical concepts have their own particular physiognomy²¹— there is nothing, so far as I can see, that poses a peculiar threat to the idea that meaning is use.

4. Dummett’s Concept of Indefinite Extensibility and its Relevance to Gödel’s Theorem So far I have not referred explicitly to Dummett’s concept of indefinite extensibility, which plays a crucial role in his discussion of Gödel’s theorem and which

¹⁹ See e.g. Field (1989), pp. 88 and 232. ²⁰ Dummett (1991), p. 311; and Wright (1994), pp. 179–80. ²¹ The allusion to Wittgenstein here is deliberate: see his (1967a), pt I, §568, and p. 218.

‘     ¨  ’   ’

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has appeared in several other contexts too.²² I shall use this section to explore the concept and to try to relate it to some of the things that I have been arguing. Although I have not referred to it explicitly, I did refer to it implicitly when I talked about Dummett’s idea that there is a principle according to which, given any precise formal characterization of our use of the expression ‘natural number’, the characterization can be extended to another. Precisely what this is is an instance of indefinite extensibility. In general, a concept is indefinitely extensible, according to Dummett, if there is a principle such that, for any definite totality of things falling under the concept, this principle can be applied to yield another definite totality of things falling under the concept more inclusive than the first. To see how this relates to the Gödel case we need only equate giving a precise formal characterization of our use of the expression ‘natural number’ with specifying a definite totality of things falling under the concept well-defined property of all the natural numbers. Dummett’s point, as I tried to indicate, is that if, in the Gödel case, we formulate the relevant principle of extension, then we can arrive at a characterization of our use of the expression ‘natural number’ which is not indeed a ‘precise formal’ one, but which is enough to safeguard the idea that meaning is use.²³ Now the general account of indefinite extensibility given above cries out for clarification of the notion of a ‘definite totality’. There is a clue as to what this notion comes to in the very fact that a ‘precise formal’ characterization is a characterization of the kind that a formal system provides; but it is not much of a clue. It suggests that a totality is definite when we can use precise mathematical tools to define it. But that cannot be right. That includes the totality of ordinal numbers, as against Dummett’s insistence that the concept ordinal number is a prototypical example of an indefinitely extensible concept. For we can define the totality of ordinal numbers as the totality of transitive sets well-ordered by membership. Come to that, we can exploit the fact that the principle of extension for the totality of ordinal numbers can itself be given a precise mathematical characterization and define the totality of ordinal numbers as the smallest totality that contains any transitive set of its own members.²⁴ Unless we acknowledge that ²² See pp. 195 ff. of Dummett (1978c). One of the other contexts in which it has appeared is Dummett (1991), pp. 316 ff. ²³ Note: there is no implication here that the principle allows the least latitude in what counts as a proper extension of any precise formal characterization. I find it puzzling, therefore, that Wright, expounding these same ideas, says, ‘Accepting [the Gödel sentence for some formal system of arithmetic that we accept] as a further arithmetical truth need not be a fully determinate obligation of [our] arithmetical understanding’ (Wright (1994), p. 170); and that Dummett, in his reply (Dummett (1994)), does not demur. There may be a misapprehension on Wright’s part: see further below, n. 25. ²⁴ Of course, these definitions, like the discussion in which they are embedded, presuppose that there is a totality of ordinal numbers in the first place—whereas, on at least one natural way of extending Dummett’s conception, the only (genuine) totalities are definite totalities. But even then, the concept ordinal number shows that there is more to the notion of a definite totality than definability in precise mathematical terms. (My own view, incidentally, is that in so far as there is in some sense a totality of ordinal numbers, this is what might be called a regulative sense: cf. A. W. Moore (2019a), pp. 149–50.)

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there is more to Dummett’s notion of a definite totality than definability in precise mathematical terms, we shall see any attempt to provide a precise mathematical characterization of the principle of extension for an indefinitely extensible concept as inherently self-stultifying—which it need not be.²⁵ It is in his book Frege: Philosophy of Mathematics²⁶ that Dummett provides his clearest account of what it is for a totality to be definite. He writes that ‘a definite totality is one quantification over which always yields a statement determinately true or false’; he also says that ‘for a totality to be definite in this sense, we must have a clear grasp of what it comprises’, and later suggests that this is something that admits of degrees, so that different concepts will count as indefinitely extensible according to how rigorously the standards are set.²⁷ Thus a certain kind of finitist will say that the concept of a natural number is indefinitely extensible, while others, exercising less rigorous standards, will deny this. But no one, Dummett thinks, can deny that the concept of an ordinal number is indefinitely extensible, on pain of contradiction.²⁸ What interests me—and this is what I was trying to indicate in various things I said in section 2—is the question whether, on a very natural extension of these ideas (itself, perhaps, illustrative of a kind of indefinite extensibility), there is any concept that is not indefinitely extensible. Thus suppose we think in terms of intensions rather than extensions. Then is there not a principle, indeed a whole family of principles, such that, for any concept, as soon as we have a definite conception of what falls under the concept, we can exercise this principle, or any one of this family of principles, in order to attain a definite and more inclusive conception of what falls under the concept? What I have in mind are principles of metaphorical extension, analogical extension, ironical extension, hyperbolical extension, metonymical extension, jocular extension, loose extension, and so forth. Any concept, it seems to me, enjoys a kind of infinitude. It carries infinite possibilities of application, tailored for an infinite variety of imaginable cases, in such a way that we can form no definite conception of what falls under it that does not immediately yield, with the help of a little imagination, to another that is more inclusive. There is, of course, much more to be said about this. In particular there are questions that need to be addressed about whether non-strict, nonliteral application is really a kind of application; whether, for instance, being metaphorically thin is a way of being thin.²⁹ But for my current purposes it is

²⁵ It is possible that Wright fails to appreciate this point, and that this goes some way towards explaining the sentence I quoted in n. 23. See the concluding section of Wright’s essay; and see Dummett’s reply, in Dummett (1994), pp. 336–8. (This is the exchange I referred to in n. 12.) ²⁶ Dummett (1991). ²⁷ Dummett (1991), pp. 316–17. ²⁸ Cf. Lear (1977). There are also important connections with the Reflection Principle in set theory: see Rucker (1982), pp. 50, 203, and 255 ff. ²⁹ See e.g. Goodman (1981), pp. 68–71.

‘     ¨  ’   ’

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enough to highlight similarities between this general semantic phenomenon and the phenomenon that Dummett has drawn to our attention. I am not claiming that they are the same phenomenon, though I do think that the similarities between them would look more and more striking the more one considered them. (For example, can we not see it as an analogical extension of the concept of addition to acknowledge the addition of infinite ordinals? Is proof literally a relation between natural numbers?) The point is simply that, with concepts in general, as with what Dummett calls indefinitely extensible concepts, there is no determinately circumscribing all that their application can involve. Where this more general kind of indefinite extensibility is concerned, however, there is surely no prospect of our formulating the relevant principles of extension. To see what is involved in a previously excluded non-strict, non-literal application of a concept, attention to the particular case, whether real or imaginary, is surely always required. This is why I claimed that, even if it is our use of language that determines how a concept can be applied, there may be no finite, non-questionbegging description of that use. And this in turn is why I claimed, in view of my watchword in this essay, that, in order to protect the idea that meaning is use from the threat posed by Gödel’s theorem, it is more appropriate to challenge step (1) in the §2 train of thought than step (2). The upshot of all of this is that we can see Gödel’s theorem as providing a graphic illustration of features of language that are pervasive—and fundamental. We should not so much be concerned about the threat that the theorem poses to the idea that meaning is use as welcome the theorem for what it can teach us about the idea.

5. The Bearing of Wittgenstein’s Ideas on Gödel’s Theorem, and its Bearing on them I want to turn finally, and briefly, to Wittgenstein. Clearly, much of what I have been urging in this essay is Wittgensteinian. I think it would be useful to make some explicit connections. In particular I want to say something about the one place where Wittgenstein talks at any length about Gödel’s theorem, the notorious appendix III to part I of his Remarks on the Foundations of Mathematics.³⁰ Here, ironically, he appears to adopt views that are not in line with what I have been urging. I use the adjective ‘notorious’ in acknowledgement of the fact that many people, including many people who are otherwise sympathetic to Wittgenstein’s work in the philosophy of mathematics, see this appendix as particularly weak, perhaps

³⁰ Wittgenstein (1978): see also pt VII, §§21–2.

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even as containing straightforward errors.³¹ I do not myself share this view. Certainly I do not think that Wittgenstein is guilty of any technical errors in the appendix. What he is doing, it seems to me, is lending support to ideas that are familiar from elsewhere in the Remarks: for instance, that, superficial grammatical similarities notwithstanding, mathematical statements have a very different function from empirical statements; that they do not represent how things are in the same way; that they have no meaning in abstraction from how they are proved, nor, therefore, from all the relevant canons of proof. But I shall not try to argue for any of this now.³² Rather I want to challenge what I see as an unfortunate error of emphasis in the appendix, which does indeed lead Wittgenstein to adopt views that are somewhat out of line with what I have been urging. But I shall suggest that they are also somewhat out of line with what he himself urges elsewhere. In §8 of the appendix Wittgenstein considers the Gödel sentence for the system of Principia Mathematica, ‘Russell’s system’ as he calls it.³³ He designates this sentence ‘P’ He considers the standard informal argument that P, though not provable in Russell’s system, is true. He then asks, ‘ “True” in what system?’ For on his view, truth is as system-relative as provability. It looks as if the answer cannot be ‘True in Russell’s system.’ But, we may ask, does this matter? Is it not perfectly acceptable to answer, ‘True in an extension of Russell’s system obtained by introducing a suitable truth-predicate and adding the axioms necessary to prove that Russell’s system is consistent’? Well yes, it is perfectly acceptable to answer in this way, even on Wittgenstein’s view. Nor does he suggest otherwise. But what he does suggest is that Russell’s system and the extended system are essentially unrelated; that they are like two separate games that happen to share some terminology, rather as there can be reference to kings and queens in both chess and bridge.³⁴ The proof of P in the extended system means nothing, on this conception, as far as the status of P in Russell’s system is concerned—except, of course, in so far as it can be seen as a proof that P is unprovable in Russell’s system. On this conception, ‘P’ is not univocal in the two systems, any more than the word ‘king’ is univocal in chess and bridge. And that seems to me needlessly counterintuitive. The problem, as I see it, is that Wittgenstein is placing undue emphasis on the axiomatic method. He seems to be guilty of just the same kind of mistake as those who accept step (2) in the section 2 train of thought. For Wittgenstein’s conception makes sense only in so far as ‘system’ is understood as denoting something like a recursively axiomatizable set of sentences, that is to say, only in so far as

³¹ Cf. Dummett, in his (1978b), p. 166. ³² For a full discussion of Wittgenstein’s remarks on Gödel’s theorem, see Shanker (1988). ³³ By the Gödel sentence for a given system I mean the sentence which, if we directly apply Gödel’s techniques together with some specific Gödel numbering, we can show to be unprovable in that system. By the system of Principia Mathematica I mean, of course, the system developed in Whitehead and Russell (1927). ³⁴ Cf. the comment in parentheses at the end of §8 of his appendix.

‘     ¨  ’   ’

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‘system’ is understood in the same kind of way as ‘formal system’ has been understood throughout the rest of this essay. If ‘system’ is understood in a looser way, as picking out something like a family of techniques and proof-procedures bound together by what we could question-beggingly call ‘subject matter’—or less question-beggingly, ‘principles of mutual relevance’—then there is no reason why the transition from working with the axioms of Principia Mathematica to working with the axioms necessary to prove P should not be seen as a transition within a system. There is no reason, that is to say, why ‘Russell’s system’ should not be seen as already embracing what I referred to above as ‘the extended system’, making it more than whatever is caught in the axiomatic net of Principia Mathematica. We could then accept both that mathematical truth was relative to a system and that P was true in Russell’s system.³⁵ Moreover, such an approach ought surely to be very appealing to Wittgenstein. His regard for ‘the motley of mathematics’³⁶ ought surely to make him amenable to the idea that mathematical practice cannot always be represented as the derivation of theorems from recursive sets of axioms. Should he not welcome, as an example of a distinctive mode of mathematical reasoning, all that is involved in establishing P on the strength of accepting the axioms of Principia Mathematica? After all, that is what mathematicians themselves are prone to do, at least when they are not being self-consciously philosophical. And Wittgenstein is adamant that philosophers have no business interfering with mathematical practice; that it is the business of philosophers only to observe mathematical practice and carefully to describe it.³⁷ To be sure, there is a familiar problem here, as my qualification ‘when they are not being self-consciously philosophical’ indicates. Mathematical practice in this context has to be construed normatively. Some of the things that real working mathematicians actually do, even in their capacity as real working mathematicians, may be open to legitimate censure, even on Wittgenstein’s view.³⁸ This is related to the point I made towards the beginning of section 3 when I said that our propensity to agree about certain things cannot sustain the meaning of the arithmetical vocabulary we use unless the circumstances of the agreement satisfy various normative conditions. When I say that there is a familiar problem here, I mean the problem of how to identify authentic mathematical practice; of how to distinguish between what mathematicians ought to do and what they do do, there being no way of attaining such knowledge except by exposure to, precisely, authentic mathematical practice. But whatever may be

³⁵ There is a somewhat different approach to these issues in Azzouni (1994): see esp. pp. 134–5. But I think that many of the differences, if not perhaps terminological, are differences of emphasis. Certainly I find Azzouni’s approach very Wittgensteinian: see A. W. Moore (1996). ³⁶ Wittgenstein (1978), pt III, §48. ³⁷ See e.g. Wittgenstein (1967a), pt I, §§124–5. ³⁸ Cf. Wittgenstein (1974), §254. And think about Wittgenstein’s own reservations about classical logic, say, or about the development of transfinite mathematics: see Wittgenstein (1978), pts V and II, respectively.

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the right thing to say about this problem, I see no reason why the techniques and procedures involved in exploiting a formal system of arithmetic to establish the truth of its Gödel sentence should not count as part of authentic mathematical practice. At any rate, the onus is on whoever thinks otherwise to say why, and they had better come up with something more than blank deference to the axiomatic method. Gödel’s theorem, on the simple understanding whereby it entails that no formal system can contain all and only the truths of arithmetic, so far from being a threat to anything in Wittgenstein’s philosophy of mathematics, can actually be used to cast light on some of the darker elements in it. It can be used to cast light on Wittgenstein’s idea that advances in mathematics are a matter of making decisions, these being in a way spontaneous and in a way constrained.³⁹ For there is a sense—indeed a precise technical sense—in which our acceptance of a formal system of arithmetic leaves us free either to acknowledge its consistency or to deny its consistency. Yet only one of these makes proper sense to us. We feel compelled to go in one direction rather than the other, at least in so far as we are trying to keep faith with our earlier understanding of things.⁴⁰ Again, Gödel’s theorem can be used to cast light on Wittgenstein’s idea that in mathematics we ‘win through’ to our decisions.⁴¹ For it takes a distinctive kind of reflection to acknowledge the system’s consistency. We first accede to the truth of each of its axioms and the truth-preservingness of each of its rules: this is a way of self-consciously endorsing the system. We then proceed from there to a proof of consistency. Furthermore, because of the arithmetization of syntax which is integral to Gödel’s theorem, we are able to see the proof of consistency as a proof that every natural number has a certain complex arithmetical property. And this can be used to cast light on Wittgenstein’s idea that often, in mathematics, one can say, ‘Let the proof teach you what is being proved.’⁴² The statement that every natural number has the given arithmetical property has a very different sense for us once we have seen it proved, once we have seen it as a statement that the formal system is consistent.⁴³ Finally, and most notably—as I have tried to argue in this essay, in homage to Dummett, though by taking some of Dummett’s lines of thought further than he himself would want to take them—Gödel’s theorem can be used to cast light on the idea, itself broadly Wittgensteinian,⁴⁴ that meaning is use.

³⁹ Wittgenstein (1978), pt VI, §24. ⁴⁰ Cf. Wittgenstein’s remark, ‘Here it is difficult to see that what is at issue is the fixing of a concept. A concept forces itself on one,’ in his (1974), p. 204 (emphasis in original). And cf. Wittgenstein (1978), pt I, §§116–17. ⁴¹ Wittgenstein (1978), pt III, §27. ⁴² Wittgenstein (1974), p. 220. ⁴³ Cf. Wittgenstein (1978), pt I, app. III, §§16–17. ⁴⁴ See n. 2.

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Williamson, Timothy 1994. Vagueness. London: Routledge. Williamson, Timothy 2000. Knowledge and its Limits. Oxford: Oxford University Press. Williamson, Timothy 2007. The Philosophy of Philosophy. Oxford: Blackwell. Winch, Peter 1958. The Idea of a Social Science and its Relation to Philosophy. London: Routledge & Kegan Paul. Wittgenstein, Ludwig 1961. Tractatus Logico-Philosophicus, tr. D. F. Pears and B. F. McGuiness. London: Routledge & Kegan Paul. Wittgenstein, Ludwig 1965. ‘A Lecture on Ethics.’ Philosophical Review, 74. Wittgenstein, Ludwig 1967a. Philosophical Investigations, tr. G. E. M. Anscombe, 3rd edn. Oxford: Blackwell. Wittgenstein, Ludwig 1967b. Zettel, ed. G. E. M. Anscombe and G. H. von Wright and trans. G. E. M. Anscombe. Oxford: Blackwell. Wittgenstein, Ludwig 1974. Philosophical Grammar, ed. Rush Rhees and tr. Anthony Kenny. Oxford: Blackwell. Wittgenstein, Ludwig 1975a. The Blue Book. In The Blue and Brown Books: Preliminary Studies for the ‘Philosophical Investigations’. Oxford: Blackwell. Wittgenstein, Ludwig 1975b. The Brown Book. In The Blue and Brown Books: Preliminary Studies for the ‘Philosophical Investigations’. Oxford: Blackwell. Wittgenstein, Ludwig 1975c. Philosophical Remarks, ed. Rush Rhees and tr. Raymond Hargreaves and Roger White. Oxford: Blackwell. Wittgenstein, Ludwig 1976. Lectures on the Foundations of Mathematics, Cambridge, 1939. from the notes of R. G. Bosanquet, Norman Malcolm, Rush Rhees, and Yorick Smythies, ed. Cora Diamond. Brighton: Harvester Press. Wittgenstein, Ludwig 1978. Remarks on the Foundations of Mathematics, ed. G. H. von Wright, R. Rhees, and G. E. M. Anscombe and tr. G. E. M. Anscombe, 3rd edn. Oxford: Blackwell. Wittgenstein, Ludwig 1979. Notebooks: 1914–1916, ed. G. H. von Wright and G. E. M. Anscombe and tr. G. E. M. Anscombe. Oxford: Blackwell. Wittgenstein, Ludwig 1980. Culture and Value, ed. G. H. von Wright and Heikki Nyman and tr. Peter Winch. Oxford: Blackwell. Wittgenstein, Ludwig 1987. Remarks on Frazer’s Golden Bough, ed. Rush Rhees and tr. A. C. Miles and Rush Rhees. Doncaster: Brynmill. Wolff, Michael 1995. Die Vollständigkeit der Kantischen Urteilstafel. Frankfurt am Main: Vittorio Klostermann Verlag. Wood, Allen W. 1970. Kant’s Moral Religion. Ithaca, NY: Cornell University Press Wood, Allen W. 1999. Kant’s Ethical Thought. Cambridge: Cambridge University Press. Wright, Crispin 1979. Wittgenstein on the Foundations of Mathematics. London: Duckworth. Wright, Crispin 1982. ‘Strict Finitism.’ Synthese, 51. Wright, Crispin 1988. ‘Moral Values, Projection and Secondary Qualities.’ Proceedings of the Aristotelian Society, supp. vol. 62. Wright, Crispin 1992. Truth and Objectivity. Cambridge, Mass.: Harvard University Press. Wright, Crispin 1994. ‘About “The Philosophical Significance of Gödel’s Theorem”: Some Issues.’ In The Philosophy of Michael Dummett, ed. Brian McGuiness and Gianluigi Oliveri. Dordrecht: Kluwer Academic Publishers. Wright, Crispin 2001. Rails to Infinity: Essays on Themes from Wittgenstein’s Philosophical Investigations. Cambridge, Mass.: Harvard University Press.

Index For the benefit of digital users, indexed terms that span two pages (e.g., 52–53) may, on occasion, appear on only one of those pages. analytic knowledge 27–32, 34, 38, 48, 59–64, 153 analytic philosophy 7–9, 110–12, 140, 143, 145 Anderson, Pamela Sue 9, 135–46 Angene, Lyle E. 314n.22 Anscombe, Elizabeth 123n.13, 195n.16 anthropocentrism: and the human point of view 1–2, 4–9, 11–12, 15–16, 45–6, 58–9, 62, 65–6, 81, 91, 93, 108, 111–13, 115–16, 145–6, 177n.23, 238–9 in ethics 9–15, 151, 183n.42, 207–8, 226 in mathematics 9–10, 15, 269, 325–30, 15 in philosophy 7, 9–10, 71–2, 110n.13, 111–16, 129, 133, 144–6 see also philosophy as a humanistic discipline a priori: concepts 1–2, 5–6, 16, 24n.3, 34–40, 50–1 see also Kant’s notion of a category intuitions, Kant’s belief in see Kant’s belief in a priori intuitions knowledge 1, 6–7, 23–4, 41n.57, 59–63, 72, 89, 176–7, 182n.40, 310, 313 see also armchair knowledge sense making, its systematic pursuit see systematic pursuit of a priori sense-making Aquinas, St Thomas 81, 90n.57 Archimedean point 116, 200–2 Archimedes see Archimedean point Aristotle 42, 61, 149–53, 160–1, 160n.42, 162n.50, 163–4, 167, 170, 180–1, 274n.9, 277–9, 284n.48, 310, 312, 317–18 Arnauld, Antoine 79–80, 82, 92 armchair knowledge 23–43 see also a priori knowledge art see philosophy as an artistic exercise atheism: new see new atheism see also theism Austin, J.L. 160n.42 axiomatizations and formal systems 19, 120–1, 296–7, 299–300, 302, 320–32, 331n.23, 334–6 see also mathematical proof Ayer, A.J. 95 Azzouni, Jody 335n.35

Baiasu, Sorin 31n.31 basic concerns of ethics 13, 114–15, 152–5, 193, 207–8, 224–5 Basic Model 2–5, 12–13, 15–16, 222 Beethoven, Ludwig 216 benevolence, God’s see God’s benevolence Bennett, Jonathan 26n.9, 77–8, 79n.11, 230n.10 Bentham, Jeremy 207 Beresford, Geoffrey C. 314n.20 Berkeley, George 26n.13 Bird, Graham 69n.62 Blackburn, Simon 14–15, 78–9, 85, 119n.4, 211n.5, 226–32, 271–2 blame and blameworthiness 10–11, 154–5, 157, 159–67, 170, 233 see also evil blameworthiness see blame and blameworthiness Bostock, David 316n.27 Boyle, Matthew 79n.10 Brewer, Talbot 211n.6 Brouwer, L.E.J. 298, 306–9, 311, 313, 319 Buridan, Jean 157–8 Cantor, Georg 264, 286–7, 301, 310–11 Carnap, Rudolf 95–6, 285nn.51,53 Cartesianism 240, 262–3 see also Descartes categorical imperative see Kant’s notion of a categorical imperative categories see Kant’s notion of a category Christianity 11, 99, 151–2, 155, 160–2, 164, 168–70, 254n.51 see also Jesus Christ, St Paul, religion, theism, theodicy Church, Alonzo: his theorem see Church’s theorem Church’s theorem 323–4 Citron, Gabriel 103n.27 cognition: Kant’s notion of see Kant’s notion of cognition Collingwood, R.G. 108n.3 Conant, James 6–7, 77–93

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conative states: ethics as grounded in see ethics as grounded in conative states shared by everyone 185–7, 196–200, 202–3 conative transcendental arguments 12, 186, 188n.63, 189–209 concepts: a priori see a priori concepts mathematics as a formation of see mathematics as a formation of concepts the distinction between an engaged grasp of them and a disengaged grasp of them 3–4, 13, 215–25 see also thick concepts thick see thick concepts see also Kant’s distinction between sensibility and understanding, Kant’s notion of a category conditionedness and unconditionedness 152–3, 167–8, 178–88 conservatism 7–8, 17–18, 109–10, 112, 114, 150–1, 164, 167, 267, 284–7, 292–3, 301–5, 335–6 consistency and inconsistency 120–1, 125–6, 266–8, 298–300, 310–12, 314–15, 317–18, 326–30, 332, 334, 336 continental philosophy 110–13 see also phenomenology contingency: necessity and possibility grounded in see necessity and possibility grounded in contingency Cooper, Tommy 300 countability and uncountability 263–8, 286, 289, 298, 300–3, 311 see also Skolemite scepticism, transfinite mathematics Davidson, Donald 169n.75 Dawkins, Richard 96–7, 103 death 154, 165–6, 169–70, 237, 241–5, 247–8, 253–6, 277 see also human finitude, immortality Deleuze, Gilles 112–15, 115n.27, 250n.32 Derrida, Jacques 140–2 Descartes, René 6–7, 11, 14, 16n.21, 26n.9, 77, 100n.19, 184n.50, 230n.10, 262–3, 273–5, 282 see also Cartesianism disengaged grasp of concepts see concepts—the distinction between an engaged grasp of them and a disengaged grasp of them Dummett, Michael 19, 110n.13, 291–2, 307–9, 319–24, 327, 329n.18, 330, 336 his notion of indefinite extensibility 323–4, 330–3 Dworkin, Ronald 228

education 152–3, 153n.21, 161n.45, 165, 178, 263, 282–3, 289, 325–6 empiricism 24–5, 27–8, 114n.26 engaged grasp of concepts see concepts—the distinction between an engaged grasp of them and a disengaged grasp of them Engstrom, Stephen 62–3 eternal recurrence 246, 248–50, 252 Nietzsche’s idea of see Nietzsche’s idea of eternal recurrence ethics: anthropocentrism in see anthropocentrism in ethics as grounded in conative states 14, 84–5, 115–16, 226–32, 238 see also externalism versus internalism basic concerns of see basic concerns of ethics Kant’s see Kant’s ethics truth in see truth in ethics Euclid see Euclidean geometry Euclidean geometry 32–3, 41–2, 231 evil 10–11, 160–2, 168–70 see also blame and blameworthiness experimental philosophy 110n.13 externalism versus internalism 185–8, 199–200, 206–9, 216n.19 Bernard Williams on see Bernard Williams on externalism versus internalism see also ethics as grounded in conative states Falk, W.D. 200n.26 Fermat, Pierre de: his last theorem see Fermat’s last theorem Fermat’s last theorem 294–5 Field, Hartrey 330 finitism: strict see strict finitism finitude: human see human finitude see also infinity, strict finitism formal systems see axiomatizations and formal systems Foucault, Michel 112–13 Fraenkel, Abraham: his set theory see Zermelo-Fraenkel set theory Frazer, James 96 freedom 82, 91, 93, 162, 170, 186, 207 Kant’s notion of see Kant’s notion of freedom Frege, Gottlob 171–2, 234–5 Geach, Peter 317n.29 gender 9, 129, 137, 140–6 George, Alexander 309n.9

 God 6–7, 11, 36–7, 38n.46, 55–6, 77–93, 96–7, 99–103, 143, 162, 169–70, 176n.19, 177, 252n.43 benevolence of 83–4, 86–9, 98, 100–1 infinitude of 178n.27, 179n.30, 273–5 see also God’s omnipotence omnipotence of 79–82, 80n.14, 84, 87–8, 90n.57, 100–1 see also God’s infinitude see also Christianity, Judaism, new atheism, religion, theism, theodicy Gödel, Kurt 293 his theorem see Gödel’s theorem Gödel’s theorem 17, 19, 292–3, 297, 320–36 Goldbach, Christian: his conjecture see Goldbach’s conjecture Goldbach’s conjecture 120, 294–5, 297 Gomes, Anil 5–6, 36n.39, 75n.75 Gould, Stephen Jay 98–9 grammar: Wittgenstein on see Wittgenstein on grammar Guattari, Félix 112–13 Guyer, Paul 57n.39 Habgood, John 96 Harding, Sandra 140 Hardy, G.H. 294–6 Hare, R.M. 169n.75 Harrison, Ross 271–2 Hart, W.D. 25n.6 Hegel, G.W.F. 49, 178n.27, 223, 248–9 Heidegger, Martin 111n.14, 112, 255 Herman, Barbara 213n.12 Herod 158–61, 218n.24 Hilbert, David 286–7, 301, 322n.6 history of philosophy 77–8, 90, 117–18 hope 152–3, 167–8, 169n.75, 179–80, 191, 206–7, 244, 247–8 human: finitude 1–2, 5–6, 12, 15, 25–6, 45, 186–8, 206–7, 273–5, 282, 308–9, 311–13, 318–19 see also death point of view see anthropocentrism and the human point of view, see also anthropocentrism in ethics, anthropocentrism in mathematics, anthropocentrism in philosophy, see also post-human Hume, David 207–8 Husserl, Edmund 97n.11, 110n.13 idealism, 25–6, 31–3, 45, 49, 190, 194–5 transcendental see transcendental idealism

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immortality 168, 254–6 Bernard Williams on see Bernard Williams on immortality see also death inconsistency see consistency and inconsistency indefinite extensibility see Dummett’s notion of indefinite extensibility ineffability 6n.9, 102–3, 136, 139–43, 186–8, 288n.65 inexpressibility see ineffability infinity 12, 15, 17–19, 136–7, 139, 141, 143, 167–8, 187–8, 208–9, 242, 248–9, 253, 255–6, 262–5, 272–5, 277, 306–19, 324–5, 332–3 of God see God’s infinitude Wittgenstein on see Wittgenstein on infinity see also countability and uncountability, eternal recurrence, human finitude, immortality, transfinite mathematics internalism see externalism versus internalism see also ethics as grounded in ethical states intuitionism 18–19, 287–8, 298, 306–19 intuitions: Kant’s belief in a priori see Kant’s belief in a priori intuitions see also Kant’s distinction between sensibility and understanding Irigaray, Luce 143 Isaacson, Dan 322n.7 Jantzen, Grace 141, 143 Jesus Christ 169–70, 223 see also Christianity John the Baptist 158–9 Judaism 217, 223n.27 Kant, Immanuel 1–2, 77–9, 102, 133, 178, 180–1, 190n.1, 195n.17, 199–200, 206–7, 240 his belief in a priori intuitions 5–6, 15–16, 18–19, 29, 34–40, 44–9, 53, 58–61, 64–6, 310 see also Kant’s conception of time his belief that pure reason can be practical 10, 13, 39–40, 149–51, 155–62, 164, 166–9, 181–2, 186–8, 212, 212n.9, 214, 221–5 see also Kant’s notion of a categorical imperative, Kant’s notion of freedom his conception of philosophy 38n.49, 45–6, 59–69 his conception of time 5–6, 15–16, 18–19, 26, 29, 34–8, 40, 44–7, 53–4, 56, 58–60, 63–5, 161, 163–4, 166–8, 310 see also Kant’s belief in a priori intuitions his distinction between sensibility (the faculty of intuitions) and understanding (the faculty of concepts) 27–8, 31–2, 34–40, 44–6, 48–50, 55–7, 60–1, 74

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Kant, Immanuel (cont.) his ethics 10–13, 108, 149–70, 188, 207–8, 210–15, 220–5, 233 see also Kant’s notion of a categorical imperative, Kant’s notion of a maxim his metaphysics and epistemology 4–6, 10, 12, 15–16, 18–19, 23–76, 149–50, 161, 168, 280n.32 his notion of a categorical imperative 38n.49, 156, 165–6, 181–8, 200n.25, 211n.4, 212–15, 222–5 see also Kant’s belief that pure reason can be practical, Kant’s ethics, Kant’s notion of a maxim his notion of a category 36–8, 44–76 his notion of a maxim 156n.33, 160, 210–16, 221–5 his notion of cognition 28n.19, 38n.49, 40n.52, 48–50, 55–6, 60–1 his notion of freedom 11, 29, 153, 159–70, 181–2, 186nn.55,56 see also Kant’s belief that pure reason can be practical his notion of things in themselves 27–30, 33–4, 37n.45, 40, 54, 161, 163–4, 168 his philosophy of mathematics 10, 15–16, 18–19, 32–3, 35, 38n.49, 60, 61n.47 see also transcendental idealism Kemp Smith, Norman 57n.39 knowledge: analytic see analytic knowledge a priori see a priori knowledge armchair see armchair knowledge ineffable see ineffability synthetic see synthetic knowledge see also Kant’s notion of cognition Korsgaard, Christine 151n.11, 208 Kundera, Milan 253n.46 Lear, Jonathan 151–2, 161–2 Leibniz, G.W. 83n.26, 248–9, 251–2 Locke, John 86n.33, 238–9 logic 32, 47–8, 50–2, 62–3, 97, 130, 172–3, 222–3, 230–1, 242–3, 268, 287–9, 298, 300–1, 304, 306–7, 312, 314–15, 318, 324n.10, 327, 329–30, 335n.38 logical positivism 94–5 Longuenesse, Béatrice 65–6 Löwenheim, Leopold see Löwenheim-Skolem theorem Löwenheim-Skolem theorem 263–6 see also Skolemite scepticism luck 149–70, 242 luminosity 45–6, 59–67 Mackie, J.L. 224–5, 314n.22 Mao Zedong 116n.35

Mark, St 159 mathematical proof 19, 119–20, 264, 266–7, 294–300, 302, 307–8, 322, 328–9, 333–6 see also axiomatizations and formal systems, the necessity of mathematics mathematical truth and its characteristics 172–3, 178, 259–61, 289, 294–5, 297–301, 307–9, 321–8, 330, 331n.23, 334–6 see also mathematical proof, mathematics, the necessity of mathematics mathematics 130, 243, 245n.12 anthropocentrism in see anthropocentrism in mathematics as a formation of concepts 16, 19, 267–8, 286, 294–5, 298, 302, 307–8, 336 its applications 263, 289, 304–5 its necessity 77–93, 230–1, 260–1, 276, 307–8 see also mathematical proof, mathematical truth and its characteristics Kant’s philosophy of see Kant’s philosophy of mathematics transfinite see transfinite mathematics Wittgenstein’s philosophy of see Wittgenstein’s philosophy of mathematics maxim, Kant’s notion of see Kant’s notion of a maxim Mayberry, John 310–11 McDowell, John 200–2 McTaggart, J.M.E. 251–2 meaning 95, 100–3, 111–12, 119, 121–6, 128, 133, 136, 139, 143, 283, 283n.41, 288–9, 296, 304, 307–8, 315, 320–36 as value 15, 99, 241–2, 244–5, 252–3, 255–6 see also nihilism, value how truth relates to it see how truth relates to meaning see also nonsense Mellor, D.H. 178 Mesland, Denis 80 metaphysics 8–9, 95–6, 107n.2, 108nn.3,6, 117, 127–34, 227–8, 234–8, 248–9, 261 see also Kant’s metaphysics and epistemology Mikkola, Mari 144 Mill, J.S. 192–3 Monk, Ray 122–3 Moore, G.E. 192–3 his paradox 190, 193n.11, 195n.17 More, Henry 90n.57 motivating reasons see normative reasons versus motivating reasons Murdoch, Iris 101–2

 Nagel, Thomas 195n.16, 197n.20, 200n.26, 242–4, 263n.8, 273–5, 282 natural science 7–8, 60, 94–8, 100–1, 109, 110n.13, 133, 234 see also naturalism as a view about normativity naturalism: as a view about normativity 226–9, 233–40, 326–7 see also naturalism as equivalent to scientism as equivalent to scientism 7–9, 94–5, 97–101, 109–11, 118–19, 134 see also naturalism as a view about normativity necessity and possibility 5–6, 11–12, 24, 31–3, 36–7, 40–3, 49, 54, 57, 60–3, 66, 74n.73, 102–3, 139n.6, 153, 172–3, 175, 190, 193–6, 278–80, 287–8, 287n.62, 314–16, 318, 324–5, 332–3 grounded in contingency 15–16, 25–6, 31–3, 35–6, 78–9, 81–3, 86, 229–31, 269, 272, 272n.37, 276, 284nn.48,49 how they are construed 6–7, 14, 46–8, 77–93, 203–5, 229–32, 278–80, 308–13, 314n.20, 318–19 see also the necessity of mathematics Nehemas, Alexander 250n.31 new atheism 7, 94–104 Nietzsche, Friedrich 15, 103, 133, 248–56 his idea of eternal recurrence 108, 248–56 nihilism 252–3, 255–6 nonsense 95, 136, 139–43, 237, 278–80, 285, 312–19, 326–7 see also meaning normative reasons versus motivating reasons 185n.52, 216, 329n.18 normativity see normative reasons versus motivating reasons, value naturalism as a view about see naturalism as a view about normativity objectivity and subjectivity 45, 57, 65, 116n.35, 131–3, 137–8, 174, 178, 184, 195n.17, 197n.20, 198n.22, 207, 210–11, 238, 244–5, 247–8, 265–6, 293–4 see also points of view omnipotence, God’s see God’s omnipotence Parfit, Derek 14–15, 111–12, 233–40, 246–7 Patterson, Sarah 90n.57 Paul, St 10–11, 169–70, 188n.66, 225n.30 Peacocke, Christopher 329n.18 Peirce, C.S. 177, 179n.29 Penrose, Roger 294 Perry, John 227n.2 personal identity 112n.18, 238–40, 244, 246–8, 254–6, 314–15

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phenomenology 97n.11, 110nn.13,14, 112 see also continental philosophy philosophy: analytic see analytic philosophy anthropocentrism in see anthropocentrism in philosophy as a humanistic discipline 7–9, 107–10, 134, 145–6, see also anthropocentrism in philosophy as an artistic exercise 95–6, 107–13, 125, 143 continental see continental philosophy experimental see experimental philosophy its history see history of philosophy Kant’s conception of see Kant’s conception of philosophy Wittgenstein’s philosophy of see Wittgenstein’s philosophy of philosophy see also metaphysics, phenomenology physics 9, 96, 129–34, 138, 294–5 see also natural science Plato, 23n.2, 25n.6, 149–50, 153n.21, 156, 166–7, 262–3 see also Platonism Platonism 24–5, 260–4, 266–9, 272 see also Plato Pluhar, Werner S. 57n.39 points of view 14–15, 26, 28–9, 32–5, 40, 42–3, 129, 132, 136, 138–41, 145, 195n.17, 237–8, 251–3 see also anthropocentrism and the human point of view, objectivity and subjectivity possibility see necessity and possibility post-human 7–8, 112–16, 254n.51 Priest, Graham 125–6 private language 270–2, 307 Wittgenstein on see Wittgenstein on private language promising 158–9, 215, 217–23 proof: mathematical see mathematical proof pure reason 11, 67–9, 72, 273 Kant’s belief that it can be practical See Kant’s belief that pure reason can be practical see also rationality Putnam, Hilary 99 Pythagoreans 141, 246n.13 quasi-realism 14, 78–9, 84–6, 119n.4, 226–32 Quine, W.V. 97, 110n.13, 172–3 Ramsey, F.P. 178 his ladder 228–9 rationality 1–2, 10–11, 45, 97, 129, 140, 151, 169, 173–4, 178–88, 191–2, 194, 197, 200–1, 203–8, 210–12, 220–5 see also Kant’s belief that pure reason can be practical, pure reason

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reason: pure see pure reason see also Kant’s belief that pure reason can be practical, rationality reasons: normative versus motivating see normative reasons versus motivating reasons relativism 14–16, 19, 32–40, 83–6, 115–16, 175nn.17,18, 179–80, 198–9, 224–33 religion 94–9, 103, 107n.2, 143, 145n.33 see also Christianity, Judaism, theism, theodicy Rhees, Rush 291 Robinson, Abraham 293 rules of representation 31–3, 35–6, 110n.13, 119, 121, 125–6, 229–30, 260, 262, 275–6, 279, 281, 283, 285n.51, 294–5 Russell, Bertrand 25n.6, 122–3, 309n.9, 334–5 Salome 158–9, 218n.24 Schopenhauer, Arthur 248–9 science 48, 95–7, 99, 108, 130–1, 138–40, 145–6, 234, 264 natural see natural science see also naturalism as a view about normativity, naturalism as equivalent to scientism, physics scientism see naturalism as equivalent to scientism sense-making, varieties of see varieties of sense-making sensibility: Kant’s distinction between it and understanding see Kant’s distinction between sensibility and understanding Shakespeare, William 234–5 Sider, Theodore 8–9, 12–34 Silber, John 160n.42, 182n.37 sin see blame and blameworthiness, evil Skolem, Thoralf see Löwenheim-Skolem theorem, Skolemite scepticism Skolemite scepticism 263–5 Socrates 153n.21 Sorensen, Roy 70, 247n.18 Spinoza, Benedictus de 41–2, 115–16, 133, 146, 178n.27, 181n.33, 186n.56 Stephenson, Andrew 5–6, 36n.39, 75n.75 Stern, Robert 249n.30 Stern, Tom 249n.30 Stirner, Max 115–16 Stoics 251–2 strict finitism 263, 308–9, 311 Stroud, Barry 274n.7 Suárez, Fransisco 83n.26 subjectivity see objectivity and subjectivity suffering 100–3, 242n.3, 243–4, 248–9, 252

synthetic knowledge 27–35, 38, 40–3, 48, 59–61, 63–4, 71 systematic pursuit of a priori sense-making 1–4 Tarski, Alfred: his theorem see Tarski’s theorem Tarski’s theorem 297 tense see time and tense theism 7, 94 see also Christianity, new atheism theodicy 100–3 thick concepts 2–5, 13, 108, 109n.8, 137–8, 140, 215–25, 231–2 Bernard Williams on see Bernard Williams on thick concepts see also concepts—the distinction between an engaged grasp of them and a disengaged grasp of them things in themselves: Kant’s notion of see Kant’s notion of things in themselves Thompson, Michael 1–2 Thomson, James 313, 316 time 18–19, 91, 97–100, 128, 245–52, 252n.43, 277–9, 280n.32, 288, 307–13, 311n.13, 318–19 and tense 128, 138, 236–8, 240, 250–2, 317–19 Kant’s conception of see Kant’s conception of time see also immortality, Nietzsche’s idea of eternal recurrence transcendence 26, 32–3, 95, 150–1, 156, 159, 161, 163–4, 167, 205, 252 transcendental arguments 189–91, 192n.6, 193–6, 199n.24, 200n.25, 204n.38 conative see conative transcendental arguments transcendental idealism 25–6, 28–36, 40–3, 45, 49 transfinite mathematics 231n.12, 264–5, 286–9, 298, 300–5, 310–11, 333, 335n.38 see also countability and uncountability, Skolemite scepticism truth 1, 24n.3, 125–6, 132, 136, 138–9, 195n.17, 237, 325–9 how it is construed 2–3, 19, 118–20, 171–9, 181–3, 297–9, 334–5 how it is valued 95–6, 127–8, 139–43, 174–5, 177, 184–8, 208–9 how it relates to meaning 32, 95, 227n.2, 231n.12, 321–2, 326–7 in ethics 174, 176, 226–9, 233–4 mathematical see mathematical truth and its characteristics

 where it applies 14, 96–7, 100–3, 118–20, 123–4, 128, 175n.16, 179n.29, 183, 322, 326–7, 333–4 Turetzky, Philip 250n.32 unconditionedness see conditionedness and unconditionedness uncountability see countability and uncountability understanding: Kant’s distinction between sensibility and it see Kant’s distinction between sensibility and understanding utilitarianism 150, 155, 192–3, 204, 207, 233 vagueness 121–4, 128, 133, 176, 220–1 value 9, 130–3, 137–8, 140, 155–7, 183, 186–8, 191, 195–6, 208, 213, 216–17, 223, 233–40, 248–9, 253, 255 meaning as see meaning as value see also naturalism as a view about normativity varieties of sense-making 1, 3–4, 7, 95–9, 108, 111–14, 119, 173–4, 224–5, 248–9, 252–6, 294–5, 319, 336 Walker, Ralph 98, 102–3, 150n.3 Whitehead, A.N. 261, 267, 334n.33 Wiggins, David 2–3, 13, 171–9, 183–4, 186n.54, 207–8 Williams, Bernard 7–8, 77–8, 99–100, 102, 107–10, 114n.25, 116n.36, 134, 145–6, 150n.4, 153, 159n.40, 186, 207, 248–50, 252–3 on externalism versus internalism 185, 191n.4, 200–3, 229n.8, 239–40 see also externalism versus internalism on immortality 15, 241–4, 254–6

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on thick concepts 2–3, 137–8, 179n.29, 210, 215, 216n.20, 231–2 Williamson, Timothy 8, 71, 110n.13, 117–26 Winch, Peter 99 Wisdom, John 291 Wittgenstein, Ludwig 31–3, 35–6, 96, 99, 103, 109–10, 110n.13, 114–15, 120–1, 123n.14, 124n.18, 142–3, 156, 172–3, 184, 188n.64, 207n.42, 252n.41, 271–2, 306–7, 308n.5, 318–19, 320n.2, 326n.13, 329n.18, 330n.21, 333, 336 his philosophy of mathematics 16–19, 119–22, 259–61, 266–8, 283nn.41,44, 284, 285n.51, 286–305, 307, 309, 319, 322nn.5,8, 333–6 see also mathematics as a formation of concepts, Wittgenstein on infinity his philosophy of philosophy 17–18, 109–10, 114, 117–19, 262, 267–9, 284–5, 287–8, 292–7, 303–5, 335–6 on grammar 117, 139, 262, 267–70, 275–7, 280–2, 284–90, 302–3, 314–15, 317–19 see also Wittgenstein on infinity on infinity 139, 262–3, 267–9, 275–90, 298, 300–3, 306–7, 310–12, 314–15, 317–19 see also Wittgenstein’s philosophy of mathematics on private language 270 Wolff, Michael 59 Wood, Allen W. 57n.39 Wright, Crispin 16–17, 176n.19, 261–72, 313, 327, 329n.18, 330, 331n.23, 332n.25 wrongdoing see blame and blameworthiness, evil Zeno of Elea 311n.13 Zermelo, Ernst: his set theory see Zermelo-Fraenkel set theory Zermelo-Fraenkel set theory 264, 298–300