Aspect Perception After Wittgenstein: Seeing-As and Novelty 9781138840393, 9781315732855

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Aspect Perception After Wittgenstein

This volume brings together new essays that consider Wittgenstein’s treatment of aspect perception in relation to the idea of conceptual novelty broadly construed; that is, the acquisition or creation of new concepts and the application of an acquired understanding in unfamiliar or novel situations. Over the last 20 years, aspect perception has received increasing philosophical attention, largely related to applying Wittgenstein’s remarks on the phenomena of seeing-as, found in Part II of Philosophical Investigations (1953), to issues within philosophical aesthetics. Seeing-as, however, has come to occupy a broader conceptual category, particularly in philosophy of mind and philosophical psychology. The essays in this volume examine the exegetical issues arising within Wittgenstein studies, while also considering the broader utility and implications of the phenomenon of seeing-as in the fields of aesthetics, philosophical psychology, and philosophy of mathematics, with a thematic focus on questions of novelty and creativity. The collection constitutes a fruitful interpretative engagement with the later Wittgenstein, as well as a unique contribution to considerations of philosophical methodology. Michael Beaney is Professor für Geschichte der analytischen Philosophie, Institut für Philosophie, Humboldt-Universität zu Berlin, Germany, and Professor of Philosophy, Department of Philosophy, King’s College London, UK. He is editor of the British Journal for the History of Philosophy and The Oxford Handbook of the History of Analytic Philosophy (Oxford University Press, 2013). Brendan Harrington holds a doctorate in Philosophy from the University of York (UK), and currently manages and facilitates group work within various mental health units of the UK prison system. Dominic Shaw holds a doctorate in Philosophy from the University of York (UK).

Routledge Studies in Contemporary Philosophy For a full list of titles in this series, please visit www.routledge.com

91 Philosophical and Scientific Perspectives on Downward Causation Edited by Michele Paolini Paoletti and Francesco Orilia 92 Using Words and Things Language and Philosophy of Technology Mark Coeckelbergh 93 Rethinking Punishment in the Era of Mass Incarceration Edited by Chris W. Surprenant 94 Isn’t That Clever A Philosophical Account of Humor and Comedy Steven Gimbel 95 Trust in the World A Philosophy of Film Josef Früchtl 96 Taking the Measure of Autonomy A Four-Dimensional Theory of Self-Governance Suzy Killmister 97 The Legacy of Kant in Sellars and Meillassoux Analytic and Continental Kantianism Edited by Fabio Giron 98 Subjectivity and the Political Contemporary Perspectives Edited by Gavin Rae and Emma Ingala 99 Aspect Perception After Wittgenstein Seeing-As and Novelty Edited by Brendan Harrington, Dominic Shaw, and Michael Beaney

Aspect Perception After Wittgenstein Seeing-As and Novelty

Edited by Brendan Harrington, Dominic Shaw, and Michael Beaney

First published 2018 by Routledge 711 Third Avenue, New York, NY 10017 and by Routledge 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN Routledge is an imprint of the Taylor & Francis Group, an informa business © 2018 Taylor & Francis The right of the editors to be identified as the author of the editorial material, and of the authors for their individual chapters, has been asserted in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data A catalog record for this title has been requested ISBN: 978-1-138-84039-3 (hbk) ISBN: 978-1-315-73285-5 (ebk) Typeset in Sabon by Apex CoVantage, LLC

Contents

1

Introduction

1

BRENDAN HARRINGTON

2

Wittgenstein, Seeing-As, and Novelty

29

WILLIAM CHILD

3

Gombrich and the Duck–Rabbit

49

ROBERT BRISCOE

4

Gestalt Perception and Seeing-As

89

KOMARINE ROMDENH-ROMLUC

5

Aspect-Perception and the History of Mathematics

109

AKIHIRO KANAMORI

6

Seeing-As and Mathematical Creativity

131

MICHAEL BEANEY AND BOB CLARK

7

Prospective versus Retrospective Points of View in Theory of Inquiry: Toward a Quasi-Kuhnian History of the Future

153

THOMAS NICKLES

8

Vision, Norm, and Openness: Some Themes in Heidegger, Murdoch, and Aristotle

173

DENIS McMANUS

List of Contributors Index

199 201

1

Introduction Brendan Harrington

1. The contributions to this collection explore the philosophical terrain surrounding Wittgenstein’s treatments of seeing-as, and the connection of this terrain to novelty in experience and concept use. This is a wide field to cover, and intentionally so; the aim is to give an overview of how a family of connections play out in different ways, without being limited by exegetical concerns, or by the bounds of any one particular domain of philosophy. As we will see in the course of this introduction, what ought to count as appropriate uses of ‘seeing-as’, and what, if anything, gives appropriate uses their unity, are difficult questions in their own right. As a way into these questions, consider the sections of Philosophy of Psychology—a Fragment (hereafter PPF1) where Wittgenstein contrasts two ways in which we talk of seeing:2 111. Two uses of the word “see”. The one: “What do you see there?”—“I see this” (and then a description, a drawing, a copy). The other: “I see a likeness in these two faces”—let the man to whom I tell this be seeing the faces as clearly as I do myself. What is important is the categorial difference between the two ‘objects’ of sight. 112. The one man might make an accurate drawing of the two faces, and the other notice in the drawing the likeness which the former did not see. 113. I observe a face, and then suddenly notice its likeness to another. I see that it has not changed; and yet I see it differently. I call this experience “noticing an aspect”. Roughly, the “catergorial difference” (ibid.) here is between: (a) talk of seeing features which one could point to, or present to another through a description or depiction which represents the spatial relations and/or colours one is experiencing; (b) talk of seeing relations between things, or seeing aspects of things, which do not fall under (a). For this reason, when an aspect dawns upon one, one cannot point to a change within the visual scene experienced,

2 Brendan Harrington but one is seeing it differently; and whilst one might reasonably expect that any other seer can see whatever one sees in sense (a), one could not reasonably expect any other seer to see whatever one sees in sense (b). Aside from the two senses of ‘see’ just mentioned, we use ‘see’ in a wide variety of ways. For example, I might: see a red door; see the door to my home; see that I am home; see how my partner is feeling about my arriving home; see that my partner’s feelings were caused by how I acted earlier on; see myself in a particular light; then, see a solution to this problem. We also use ‘seeing-as’ in a variety of ways: we might see someone’s behaviour as virtuous; see a pattern as evidence for a theory; see a poem as belying a certain attitude in its author. This collection takes a liberal view regarding uses of ‘seeing-as’ which are not directly about visual experience. Contributors apply and explore the application of ‘seeing-as’ within the fields of mathematics, ethics, and scientific practice, as well as the more usual fields of perception and aesthetics. One reason for taking this liberal view is that some of these applications constitute novel uses of ‘seeing-as’; uses in which what is familiar in one domain of language comes to find application in another. In being so, these uses of ‘seeing-as’ feed back into this collection’s underlying themes: how we think and talk about conceptual innovation, and the relation between canonical and novel language use. Another reason, which I hope to make clear in this introduction, is that presuming that there is a single set of phenomena, or domain, to which it is appropriate to apply the term ‘seeing-as’—to, as it were, police the bounds of sense3—is to miss a central point of Wittgenstein’s treatments of seeing-as. Before starting to fill out the reasons for the broad scope of this collection, though, we ought to address the following: why should we think there is any interesting connection between seeing-as and novelty? Or, if there is not an interesting connection, what, if anything, is illuminating about considering why we might think that there is? Then, only if the relations (or lack thereof) are worth examining, should we here concern ourselves with the reasons for doing so in a way which is broad, and with what unites the uses of ‘seeing-as’ which are in question (if they are indeed united).4 In sections 2 through 10, I aim to justify interest in, and establish a coherent background for, this collection by offering some brief explorations of the above questions and concerns. After this immediate need has been met, I address the particular contributions of each chapter to the core themes of this collection. This is not to say that the contributions to this collection stand in agreement or harmony with each other. Rather, this introduction presents an outline of the relations between seeing-as and novelty, which can be elaborated, or opposed, in a wide variety of ways, and inasmuch as this is the case, the following sections are only intended to serve as useful points of comparison when exploring this terrain. The core motivation of this collection, then, is the intuition that whilst seeing-as and novelty may not be necessarily related, nor straightforwardly

Introduction 3 explanatory of one another, considering cases of seeing-as is important to our understanding of novelty. It is important with regard to both the descriptive project of elucidating novel language use and more general language use, on the one hand, and the project of illustrating how concepts, such as those of experience and thought, are not as homogeneous and distinct from one another as they might appear to be when we consider paradigm cases, on the other hand.5

2. One strong source of the intuition that there is an interesting connection between seeing-as and novelty is the following. When we acquire a new concept, we acquire a way of picking out features of the world hitherto unavailable to us. We thus also acquire the capacity to see—in some sense of the word ‘see’—sections of the world previously otherwise available to us6 as instances of the novel concept. Some cases of novel concept acquisition— most obviously those involving features it is possible to identify visually— may thus seem to induce seeing-as as a consequence. It seems trivial, however, that we cannot experience something as a kind of thing until we can somehow identify instances of that kind. It also seems trivial that whilst some things are identifiable visually, some are not. Whilst in this vicinity there are interesting questions about the limits of human visual capacities, and how we become visually sensitive to features which we have learnt of in non-visual ways, it is not obvious that these questions ought to compel us to investigate the connection between seeing-as and novelty.7 If we consider the possibility of the aspect-blind (or even the aspectpartially-sighted) then the intuition, as it stands, seems even less compelling.8 The aspect-blind are those who cannot experience a change in aspect, or multiple aspects, and yet can still function in much the same way as an aspect-seer.9 The problem is that if it is possible for the aspect-blind to do almost everything an aspect-seer can do, then even if a change in experience (or experience of a change in experience) is a consequence of concept acquisition, it seems like a change with little consequence. Thus, unless cases of experiencing seeing-as are interesting for other additional reasons, the intuition that seeing parts of the world as instances of newly acquired concepts is a result of novel concept acquisition, is not of much interest for our purposes. So we ought to focus upon whether aspect-seeing affords those who can have such experiences further possibilities for thought and action. Let us now turn to another intuition which is about seeing-as affording just such further possibilities.

3. A more interesting connection between seeing-as and novel concepts might be thought to run in the opposite direction to the intuition expressed in 2; that is, a connection from an ability to see aspects to the acquisition of

4 Brendan Harrington concepts. For seeing-as and novelty to be connected in this way it could be the case that seeing-as is a necessary condition for our acquisition of novel concepts—that all concepts we acquire originate in instances of seeing-as; that seeing-as is a necessary condition for some concept acquisition; or, that seeing-as is merely contingently, but significantly, involved in the acquisition of some concepts. So how might an instance of seeing-as allow us to either gain some way of thinking of the world which we did not have before, or to apply some concept to the world in a way in which we hitherto had not? Let us approach these questions by first thinking of a very basic case of the application of a concept. Consider, for instance, how we make sense of the underlying network of resemblances which underwrite our individuation of things like footballs. To individuate an instance of a football which has not yet been recognized as a football we must be able to recognize that the part of the world we are considering is similar to the parts of the world we already recognize as footballs; to be identified as a football that ball must resemble footballs more than any other things. Yet, whilst it may seem in some sense natural to say that footballs resemble each other more than they do, say, bricks, it is far from clear that there is any absolute sense in which this is so; for resemblances might be drawn in respect of many different features and combinations of features of whichever respective parts of the world we are comparing. In this sense resemblances come cheaply (and it is important not to forget that unless the aforementioned features of parts of the world are in some sense given to us, their individuation is problematic in the same way as the football’s individuation is). Narrowing down the options for what gets associated with what, then, is a matter of first having some criterion of comparison—something which sets the standard for resemblance. There are, however, an indefinite number of potential criteria with which to relate most parts of the world that we would seek to compare. To make sense of resemblance talk, then, we already need some framework of salient relations of resemblance, set by the aforementioned criteria, which fix the ways in which to compare the relata in question. Such a framework, however, seems to be articulable only by recourse to just such relations between relata as we are here hoping to identify the grounding for by finding a criterion for being a football. Thus, to individuate footballs, it seems that, in the above sense, we need to already know what a football is—but how might we gain this knowledge? One might think that the very experience of seeing-as could provide the requisite information for the acquisition of a new concept, or for the novel application of an established concept. For instance, in the case of the football, we might think it is in an experience of seeing-as that our novel application of ‘football’ is determined. The very idea is, however, potentially deeply problematic, for it would be easy to take from this idea the impression that an experience can provide us with the rules for its own correct interpretation. Thinking as much would be to ignore the deep problems

Introduction 5 brought out by Wittgenstein’s considerations of rule-following, representation, and interpretation.10 It would be tantamount to accepting the possibility of self-intimating experience: an experience which not only presents us with something but also tells us what it is presenting us with. If this section’s intuition is to hold, then, it seems the role of seeing-as in bringing about novel concept acquisition must, rather, be part of the following sort of account. Experiences of seeing-as may make tentative connections and associations available. Finding uses for, and then choosing to exploit these connections and associations in particular ways, however, is down to us (that is, what we do with the experience is not determined by the seeing-as). For this reason, talk of ‘concept acquisition’ may be misleading. For if the intuition is to hold in this way, it must pertain to seeing-as providing us with the materials with which to go to work in any number of possible ways, and not to seeing-as providing us with a set of instructions. We might, then, better think of coming to have new concepts by way of seeing-as as more akin to invention, and less akin to discovery, though not clearly as a case of either.

4. Notwithstanding the possibility of the intuitions of sections 2 and 3 being red herrings, it is still possible that considering seeing-as can help to elucidate something about conceptual novelty. If considering seeing-as is an essentially useful tool for this purpose, or even a very useful tool as a matter of contingency, we can satisfyingly answer the questions and concerns with which this introduction started. So how might we think of seeing-as playing an elucidatory role in our understanding of novel concept use? We will now explore this idea by connecting it to Wittgenstein’s considerations concerning transitive and intransitive language use; that is, language use of which the sense depends upon the possibility of comparison to some kind of standard and language use which does not depend upon comparison in this way. The thought we will explore by appealing to this distinction is that both novel language use and descriptions of seeing-as are instances of intransitive language use. This is so because neither the sense of descriptions of experiences of seeing-as, nor the sense of novel uses of language, depends upon accord with criteria. Seeing-as thus serves as a fruitful object of comparison when thinking about how we bring existing concepts to bear upon novel terrain. We will also see that considering seeing-as alongside novel language use may to serve to elucidate confusions in how we think of visual perception. As this line of thought is being unpacked in sections 5 through 10, we will find an acceptable form of the intuition expressed in 3 developing; namely, that instances of seeing-as may form a precondition (necessarily or contingently) of a given novel concept’s possibility.

6 Brendan Harrington

5. Let us now look to how seeing-as, novel language use, and the idea of the transitive and the intransitive fit together in Wittgenstein’s work. In the Brown Book the discussion of seeing-as follows a discussion of transitive and intransitive language use; respectively, language use which gets its sense from comparison with some criteria, and language use which does not (for instance, expressing, ordering, stipulating rules). Drawing this distinction allows articulation of a “characteristic situation” which we “find ourselves in when thinking about philosophical problems” (BrB 160). The characteristic situation is that a word has a transitive and an intransitive use, and that we regard the latter as a particular case of the former, explaining the word when it is used intransitively as a reflexive construction. (ibid.) Take the word ‘meter’ for instance, within the sentence ‘That is a meter long’. ‘That is a meter long’ has a very common transitive use in which we are reporting a measurement. The criterion for correctness for the measurement is whether, upon comparison with a standard meter, the two are the same length. The truth or falsity of ‘That is a meter long’ in this context, then, depends upon the possibility of this comparison; in this way, there being a standard meter is a prerequisite of the sense of such statements. This brings us to an intransitive use of ‘meter’. When creating the unit of measurement, a meter, we might point to any given length of material and stipulate ‘that is a meter long’. In doing so we are not appealing to any point of comparison, we are rather laying the way for future comparisons of this kind through an act of creation. What, then, would it be to regard the stipulation of being a meter long “as a particular case of” transitive use, by somehow construing the stipulation as a “reflexive construction” (ibid.)? It would be to assume that when we say of the standard meter that it is a meter long, this is itself a statement which is made true or false by comparison. As this meter is the standard, the comparison would need to be to itself; it is in this sense that the comparison is reflexive. However, the “reflexive construction” (ibid.) is then, at best, contentless in a manner akin to the proclamation of the man in the following imagining:11 Imagine someone saying: “But I know how tall I am!” and laying his hand on top of his head to indicate it. (PI 279) At worst, believing that criteria for correct use are available in such situations: (a) buys us into the idea that there can be criteria for correct application independently of a system of use—a standard for correctness that not

Introduction 7 only serves as a point of comparison, but somehow does so self-evidently; (b) encourages us to go on a metaphysical quest to look for these mythical self-evident criteria. Wittgenstein’s articulation of this situation, then, is aimed at showing a confusion with the following structure: we have a strong expectation for the availability of criteria for correctness for all language use, but no such criteria could be available in all cases of language use, so we end up thinking of criteria in a way which disconnects them from the actual use of language. The further thought is that we can see that no such criteria need be available by attending to our ordinary language use, and observing that the grammar surrounding novel word use—like the stipulation of a new measurement—is just not aimed at truth and falsity in the same way as transitive language use is. The considerations surrounding this confusion have a wide application to general language use: they discourage an idea of language use as homogeneous, with all of its uses fitting a model on which meaning obtains in virtue of accord with correctness conditions. If we do succumb to thinking of language as homogeneous in this way, actual language use appears to present a problem. Looking to novel language use makes the foregoing point particularly clear. Take, for instance, ‘Barry is the pilot of the plane’. Consider the use of this sentence which pioneered the transition from the prior use of ‘pilot’ (the person who steers a boat) to a more current use of ‘pilot’ (the person who flies a plane). The user would be maintaining that Barry is related to the plane as, hitherto, pilots had been related to ships. To understand this would require ignoring the relations previous uses of ‘pilot’ would have borne to their contexts which, if taken to be essential to the concept pilot, would be in conflict with the possibility of attributing the concept pilot to the flyer of an airplane. If the two uses of ‘pilot’ are taken to be uses of the same concept, then, being in command of a vessel must be seen to be internally related to, or essential to, the concept, whilst the hitherto exclusive application of the term to water vessels would have to be taken to be accidental (along with many other relations that hitherto would have always coincided with the use of ‘pilot’).12 So, the problem novel language use throws into a particularly clear relief is this: what facts about past use could possibly provide a criterion for the novel use? Nothing in the use of the word so far can do so with any determinacy, and so, if we are compelled to look, we end up in something like the confusing situation Wittgenstein characterized. One way in which we might re-orientate ourselves amidst this confusion is to look around for other uses of language which it does not typically make sense to think of as transitive, and chart their relations to transitive uses. The language surrounding seeing-as fits this bill well, and we will see examples of this in the next two sections. However, to chart the relations between transitive and intransitive language use, whilst rejecting the homogeneity of language use, it is important to keep the following in mind: even though the criteria to which our

8 Brendan Harrington transitive language use are accountable cannot provide the criteria for the application of a novel use of a concept, we do not want to say that the nexus of correctness conditions surrounding the canonical use of a concept plays no role in making novel uses of that concept available. Rather, we want to say that the new use depends upon the architecture surrounding the old use, but that nothing about that architecture determines how we then go on in using the familiar phrase. The point of the discussion so far, then, is threefold: (A) novel extensions could not be made if the normative context—that being whatever is determining what we ought and ought not to do with a word in a certain circumstance—surrounding the canonical usage had not been so; (B) the use of a concept in a novel situation requires counting certain relations which the concept had previously borne to the world and to other concepts as accidental or external, and some as essential or internal; and (C) the normative context surrounding the canonical usage cannot justify a particular novel deployment of a concept. A asserts an internal relation between the canonical and novel concept use; B asserts that there is normativity at play (for the concept to be understood in the appropriate way, certain relations ought to be taken to be essential and certain relations ought to be taken to be accidental); C asserts that the connection is arbitrary (nothing about the normative context, or the empirical world, could necessitate which relations must be taken to be essential, and which relations must be taken to be accidental). For our present purposes it is worth noting that if we lose sight of C, the situation is ripe for confusion: the internal relation of A may then be taken to be an indication of a justificatory link rather than a mere precondition—for example, we might think the prior use of the word indicates to us how to extend it; and the source of the normativity in B may be taken to be determined by any number of candidates for external criteria. Novel rule application, then, epitomizes a problem for any theory of the relation between concepts and their criteria for application in the following way: even if we had a contingently correct principle relating all of the hitherto canonical cases of rule application, this would not, in principle, apply to novel cases. For such a principle to work, it would have to be not only contingently applicable to all of the previous cases of rule use but also applicable to any possible future case.13 This difficulty is only intensified when we note that there is no obviously clear way to distinguish between the novel and the canonical: because every instance of concept use is made in a different context to any prior use, we must perpetually pick out which relations are salient and which are not. So there is no hard-and-fast distinction to make between the canonical and the novel. Rather, all we have are varying degrees of novelty at each instance of a diachronic relation between uses of the same word. Thus, in novel language use, we have a clear example of a more general feature of language: that because language does not work according to a

Introduction 9 uniform set of rules, language is not homogeneous. Rather, we utilize such rules when modelling certain aspects of language; we might, for instance, imagine a language in which every language act is an act of comparison in order to highlight similarities and differences to what we actually do when using language transitively. We ought to keep this order of priority in mind: to presume such rules always apply is to risk confusing the object of investigation (actual language use) with the method of investigation (making models of language use). Considering this way of becoming confused also serves to help us avoid a similar danger when thinking of seeing. It helps us to remind ourselves that seeing is not characterizable as a particular kind of representation; rather, we appeal to representations like pictures when modelling particular aspects of seeing.14

6. Similarly to the Brown Book, in PPF seeing-as appears as part of the exploration of the lack of semantic homogeneity in words so central to our conceptual schemes that it seems to us that they must have a single particular essence. In PPF, as well as demonstrating a lack of homogeneity, Wittgenstein illustrates the way in which changes in how we allow sense to uses of such words can come about. We can see this in 87 through 110 of PPF, when just before the treatment of seeing-as in section xi, Wittgenstein considers Moore’s paradox.15 The paradox is that sentences of the form ‘I believe it is raining, but it is not’ seem like nonsense, though those of the form ‘I believed it was raining, but it was not’ make perfect sense.16 Recognizing this difference requires seeing that the same word—‘believe’—appears to have radically different correctness conditions when used first-personally in the present and past tenses, respectively. Wittgenstein’s immediate diagnosis of Moore’s paradox shares a structure with the diagnosis of the confusion between transitive and intransitive language, which he made explicit in the Brown Book (discussed above). Namely, that the paradox is only a paradox if we expect the same word to function in the same way in every context; that is, it is only a paradox if we presume semantic homogeneity. If we let go of this presumption and examine the use, we can see that whereas the first-person present-tense indicative use of ‘belief ’ counts as an avowal, the corresponding past-tense expression reports a belief in the manner one would if it came from someone else’s mouth. Here, then, we have two expressions about one’s own beliefs which have dramatically different criteria for correct use. For our purposes, it is crucial to note where the line of thought goes after the diagnosis. Wittgenstein expresses the thought that talking about one’s belief as if it were another’s (that is, applying how we typically speak of others to ourselves) opens up a new possibility. One can now say sentences such as ‘Judging from what I say, this is what I believe’. Realizing this possibility,

10 Brendan Harrington in turn, allows sense to be given to a whole new domain of language; a domain where one can think of oneself as an empirical subject which may be, in some ways, opaque to the first-person point of view. Once we have made all of these moves, we are in a position to endow Moore’s paradoxical statement with sense: ‘It is raining, but I don’t believe it is’ can be heard in the same way as ‘My ego believes this, but it isn’t true’. Here the point is twofold: (1) again, we see that at each stage the use of ‘believe’ is dependent on the prior architecture of the word’s use, but at no point is a new use determined by this architecture, and, in fact, the way ‘believe’ is used in the end could not share all criteria with the original use; (2) at each stage in the development of the use something like seeing-as is implicated—we must see ourselves as we see others to make sense of a basic difference in grammar for one of our most fundamental words (the past- and present-tense uses of ‘believe’), and then see ourselves as we see others in a more radical sense in order to extend the use further. In this example, then, we have a potential example in support of the intuition of 3: an instance of seeing-as opening a hitherto unavailable realm of conceptual possibility but not determining any particular way of proceeding. A further point of contact between seeing-as and novel language use arises if we compare the grammar surrounding the use of seeing-as to that surrounding novel word use. Whilst it would usually seem odd to say of oneself that one is seeing a familiar object as the object it is,17 it would usually not seem odd to say it of someone else, or as part of an explanation of one’s own past behaviour. Similarly, too, once we come to extend ‘seeing’ in such a way as to include the perception of aspects, we open up a way of talking about experience which allows us to express how our experience might otherwise be (for example, one might say of a Necker cube, ‘I just can’t see it as inverted again’). This possibility depends upon already having a grasp of a primary sense of ‘see’, but being willing to use it without heeding the usual criteria. Comparing the two grammars also highlights that, in a similar way to which it would be odd to talk about seeing a familiar object as the object it is—for example, ‘I am seeing the table as a table’—it would be odd to talk about perception in terms of continuously seeing-as; likewise, to talk about continuously believing something after coming to believe it, whilst not obviously incorrect, is odd for the same kinds of reason.

7. In summary, in both the Brown Book and PPF much attention is paid to the difference between novel cases and canonical cases, and to the lack of homogeneity in experience and in the use of even our most fundamental concepts. Furthermore, the ability to see-as looks to play a role in our ability to make novel use of concepts. In both texts the discussion of seeing-as appears in this wider context. For this reason alone, there are rich exegetical questions about quite how

Introduction 11 Wittgenstein saw the relation between seeing-as and novelty, which are worth addressing both on their own terms and in relation to the efforts of other philosophers engaged with similar questions. Of course, much more of relevance goes on in each text than I have given credence to here; for instance, the descriptions of many different ways of using ‘seeing-as’ itself, and illustrations of how we might push particular ways of using ‘seeing-as’ to the limits of their sense. My intention has, however, only been to establish these tentative links between concept extension and treatments of seeing-as. As we saw when we just touched upon the connection to transitive and intransitive language use, these aspects of seeing-as also connect with the deeper diagnoses Wittgenstein made: they pertain to how we ought to think of rule-following—and, more particularly, cases in which the idea of accord with criteria applies, and cases in which it does not. Shortly, in considering these connections further in 8 and 9, the elucidatory use of considering seeing-as will become clearer. A preliminary summary of where we are with the aim set in 1, then, is this: we are tempted to think of novel concept use as being possible because of the application of some pre-existent criteria; but, in such cases—as in the extension of ‘pilot’ and ‘belief ’—the obvious candidates for criteria cannot fit the bill (where the obvious candidates are whichever rules we have found to fit canonical applications). So in novel cases we may feel the utmost temptation to think meta-logical theories to be necessary (‘meta-logic’ is used here in Wittgenstein’s idiosyncratic sense: the level of the prerequisites of the possibility of a logical order, which, of necessity, sits in harmony with the order of the world). We feel this temptation because, at least in part, if we think correctness conditions are necessary for sense, and criteria for judgement are a prerequisite for correctness conditions, then we must believe the criteria are out there somewhere, and available in advance of our novel uses. We have seen that a remedy to this sort of thinking may be to look and see how word use changes and that seeing-as is arguably part of this process: as we found when considering the changes in use of ‘believe’ and ‘pilot’, an ability to see-as can be implicated in the movement from canonical to novel usage. Each change in use required treating as essential some particular aspect of how the concept had hitherto been used whilst also embracing new possibilities.

8. If the foregoing is correct, appeal to cases of seeing-as in which novelty is implicated is useful in the task of elucidating how, and to some extent why, in other areas of our conceptual lives we might be looking for criteria where we ought not—that is, considering novelty is useful to the wider project of grappling with our “tendency to sublime the logic of our language” (PI 38). In the next section, before turning to the contributions to this volume, I would like to bring out the links between seeing-as and intransitive language

12 Brendan Harrington use by way of an example which also brings out some other key features of our talk of seeing-as. Before we do so, though, I want to look at how both seeing-as and novel language use tempt us to “sublime the logic of our language” (ibid.) by considering the following points: •

•

•

•

For transitive talk to make sense, there must be an object of comparison providing a standard for a correct or incorrect comparison; making comparison according to such a criterion is, in such cases, a prerequisite of talk of truth and falsity with regard to the comparison. Just as we must already have some criterion in order to make a comparison at all, for such a criterion to have application we must be able to separate out the respect in which we are comparing our object of investigation to our criterion. When talking intransitively, it is a confusion to apply the concepts of truth and falsity in the sense in which we might discover truths and falsities by acts of comparison of objects of investigation to separate criteria. Talk of seeing-as may seem to be transitive: we seem to compare what it is we are experiencing with something else. However, when we try to articulate the comparison, we cannot. Changes in aspect are, for instance, not reflected in any objective description of the object’s features, or by any copy we might make of it (cf. PPF 111, quoted in 1).

This last point is made a number of different times, and in a number of different ways by Wittgenstein, and very clearly at PPF 129–131: 129. The change of aspect. “But surely you’d say that the picture has changed altogether now!” But what is different: my impression? my attitude?—Can I say? I describe the change like a perception; just as if the object had changed before my eyes. 130. “Ah, now I see this”, I might say (pointing to another picture, for example). This has the form of a report of a new perception. The expression of a change of aspect is an expression of a new perception and, at the same time, an expression of an unchanged perception. 131. I suddenly see the solution of a puzzle-picture. Where there were previously branches, now there is a human figure. My visual impression has changed, and now I recognize that it has not only shape and colour, but also a quite particular ‘organization’.—My visual impression has changed—what was it like before; what is it like now?—If I represent it by means of an exact copy—and isn’t that a good representation of it?—no change shows up. To connect this idea—that changes in aspect are not reflected in any changes in a description of the object itself—with the idea of the intransitive, and

Introduction 13 the idea of criteria for correct usage not applying, consider Wittgenstein’s treatment of a schematic face in the Brown Book: One feels that what one calls the expression of the face is something that could be detached from the drawing of the face. It is though we could say: ‘This face has a particular expression: namely this’ (pointing to something). But if I had to point to anything in this place it would have to be the drawing I am looking at. (BB II 16 p. 161) Later on, Wittgenstein makes this summary of the confusion, alluding to the form of the confusion between the transitive and the intransitive: And in this way ‘seeing dashes as a face’ does not involve a comparison between a group of dashes and a real human face; and, on the other hand, this form of expression most strongly suggests that we are alluding to a comparison. (BrB II 16 p. 164) And of the feeling that we are making a genuine comparison when seeing-as, or rather the feeling that compels us to look for a comparison, Wittgenstein says further, But no such mold or comparison enters into our experience, there is only this shape, not any other to compare it with, and as it were, say ‘Of course’ to. As when in putting together a jig-saw puzzle, somewhere a small space is left unfilled and I see a piece obviously fitting it and put it in the place saying to myself ‘Of course’. But here we say, ‘Of course’ because the piece fits the mold, whereas in our case of seeing the drawing as a face, we have the same attitude for no reason. (BrB II 16 p. 166) So whilst it is understandable that descriptions of seeing-as are taken to express empirical content (for the surface grammar is the same), on looking for a criterion for correctness for the content, we see that they do not. This is the sort of confusion that Wittgenstein points to again and again to bring out the sort of intransitivity we find in hidden in apparently transitive descriptions; for instance, of what it is to have a particular thought, belief, experience, expectation, particular impression, and so on.

9. To make the foregoing discussion more concrete, let us now apply these comparisons to an example of seeing-as. I am driving through a mountain range in Ireland and the scene passing me by, although striking, is composed of things, and combinations of things,

14 Brendan Harrington appearing in a way I am used to. To my left, rock rises in jagged outcrops piled upon one another, and where the piles meet the grass of the lower land, it is mottled with the greens and greys of moss and lichen. To my right bobbles of tree tops blend into shimmering blue water by ebbing out at the end of long elegant fingers which seem interlocked with like fingers of the body of water they meet. As I reach the brow of the hill, however, and the vista ahead is revealed, I see a small range of hills illuminated by the late afternoon sun as a section of the body of a muscular animal—short-haired and lying down. I see the compact valleys and gullies as the creases and folds in the skin in the places in between those where it is stretched and defined by the firm bulges of its articulated mass. This experience is new to me—it is novel—and somewhat shocking. ‘It’s a dog!’, I say. I find it hard to describe the familiar part of my experience without appeal to metaphor and simile—‘elegant fingers’, ‘tree tops blending into water’— and to some extent, I could not give as good a description without doing so. These devices put emphasis on the qualities of the scene which I pick up upon in a way that it seems less florid language could not. In the case of the seeing-as, however, the description is in a way constitutive of my experience: I could not describe this experience without expressing it like that. Wittgenstein makes something like this point in the following way in PPF: 138. I look at an animal; someone asks me: “What do you see?” I answer: “A rabbit.”—I see a landscape; suddenly a rabbit runs past. I exclaim: “A rabbit!” Both things, both the report and the exclamation, are expressions of perception and of visual experience. But the exclamation is so in a different sense from the report: it is forced from us.—It stands to the experience somewhat as a cry to pain. 139. But since the exclamation is the description of a perception, one can also call it the expression of thought.—Someone who looks at an object need not think of it; but whoever has the visual experience expressed by the exclamation is also thinking of what he sees. 140. And that’s why the lighting up of an aspect seems half visual experience, half thought. Similarly, in being both the ‘description of a perception’ and the ‘expression of a thought’ my use of ‘it’s a dog!’ is intransitive, but appears to be transitive. It appears so, in part, because of the use of the empirical concept of a dog. Here is our first point of connection to intransitive novel language use: although an expression of seeing-as seems like an act of comparison, it is not. Of course, I do not take it that the massive mass ahead of me actually is a section of a body of an animal. To do so would surely be an error of judgement. Considering this instance of seeing-as, however, may serve as a useful starting point for further thoughts and associations; and these thoughts and associations may occur in virtue of some feature of the landscape sharing

Introduction 15 more than a mere resemblance with the body of an animal. Perhaps the grasses, in order to expel heat and moisture, arrange themselves into a less compact configuration, as hair does (and, of course, perhaps not); perhaps the areas where creases form, being less exposed to the regular changes in the elements which the rest of the surface is well adapted to, are richer in natural variations and more prone to some sorts of fissure, as skin is (and of course, perhaps not). Importantly, then, although an instance of seeing-as may provide the basis for my thinking novel and empirically correct thoughts about the world in the sense that it may enable me to think them, it does not ground these thoughts in the justificatory sense (cf. p. 5, 8). If seeing-as has a place in the contingent etiology of my particular line of thought (and perhaps sometimes, even a place which could not be otherwise filled), it does not, in virtue of it forming part of my experience, necessarily give me reason to think any further empirical thought is true. Here is our second point of connection to intransitive language use: just as the modal and normative relations surrounding canonical language use enable but do not in any way force or entail novel usage, neither do the features of an experience of seeing-as force any judgements about the empirical world upon us. Similarly, focusing now on how we are using ‘see’, although the experience I have is not something I could articulate independently of articulating my visual experience, it is not obvious that when I describe my ‘seeing-as’ using ‘see’, I use ‘see’ in the common-or-garden sense. When describing the experience of my surroundings using the common-or-garden sense of ‘see’, the object of my descriptions are the distal objects in the environment I share with others. When I describe my seeing-as, however, it cannot be the common-or-garden seeing of these objects that I am describing, for any description or depiction of the visible portions of the distal objects seen would be identical to that of a description given of the same objects when not seen-as anything at all. We can say, then, that descriptions of seeing in the first sense are objective in just the following sense: if something is seen, then an observer with all of their faculties, at the same point of view, could see the same thing. Descriptions of seeing-as are not objective in this way. Despite the lack of objectivity, however, I mention my seeing-as to my passengers, and they to come to see-as in a like way; furthermore, they are able to share and elaborate upon the thoughts and associations I express. We thus do feel it appropriate to call on some intersubjective capacity when we do see-as, but do not have anything to appeal to if no one will share our experience. So whilst we may be incredulous if someone, say, fails to see a cloud as a bus in quite the way we do, and we may consider them in some respect impoverished for this lack, we could not rightly infer that there was something wrong with their eyesight, or in their command of the concepts cloud and bus.18 What would I do if my passengers could not appreciate what I was seeing the mountains as? As we have noted, if I drew an accurate picture of what I could see, or made a model, I would just be presenting my passengers with

16 Brendan Harrington what we already had a common experience of. My best bet would be to perhaps draw something symbolic (say, by including a depiction of an animal’s head in an appropriate place and proportion within my drawing of the landscape), or better still, finding a picture of a muscular short-haired animal lying down, or a bull lying down in a nearby field, pointing to it, and saying ‘like this’. (This again emphasizes a point of contact between seeing-as and novel concept use: nothing about the particular similarities and differences necessitates sharing in that taking or movement.) Here we can also start to appreciate the difference between the two senses of ‘see’ which are at play, and why the distinction turns on the particular sense of objectivity briefly outlined a few paragraphs ago. Although I might give voice to my experience by using what appear to be sentences with empirical content, they do not in fact have such content. We can see they do not if we ask ourselves what the criteria would be for the correct use of these empirical concepts, or ask ourselves how we could show someone our experience. In asking these questions we realize that there is nothing to appeal to because there is no separation between the object of the experience and the experience. If we now accept that descriptions of seeing-as do not necessarily make empirical claims—and accept the associated thought: that they are not necessarily instances of transitive language use—then we may have a neat account of the uncanniness of our ability to share in experiences of seeing-as. Taking such descriptions to be empirical claims makes it seem as if we are discovering something that is already there; the fact that others can immediately share in this experience compounds the idea that we are triangulating upon something common; and yet, when pushed to look for the criteria for our sharing of the same experience, we ought not expect that there is something we could point to.19 Here we can see how considering seeing-as exposes a confusion (of a like form to the confusion we saw Wittgenstein expose in 5) in thinking about two fundamental forms of contact with the world: language and perceptual experience.

10. In comparing seeing-as and novel language use, then, we can see, in cases of seeing-as, similar tendencies to confusion as those which are present in treatments of intransitive language use as special cases of transitive language use. In cases of seeing-as, we see this confusion afflicting, at a very basic level, our understanding of experience of the world. As with the confusion in language, we might trace this confusion in understanding to a want for our various modes of experience to be somehow homogeneous. As with other fundamental concepts (like thought and belief ) the remedy to our confusion might be to look and see how our concept of perceptual experience is more plastic than examining paradigm cases of perceptual experience would suggest.

Introduction 17 We have also seen how seeing-as may be a prerequisite of making certain movements from canonical to novel usages, such as in the cases of ‘believe’ and ‘pilot’; and extending this thought further, we can talk of seeing-as in situations in which experiencing different aspects serves as a precursor to exploring hitherto unavailable possibilities through creative acts. With these ideas as a backdrop many avenues of exploration become available; we will see as much when we survey the contributions to this collection in the next section. It is not obvious, however, and perhaps not even true, that the way of thinking of seeing-as developed so far (or some similar way) is a good one. If it is not, then seeing-as and novelty presents us with (or it at least throws into sharp relief) a genuine puzzle about the possibility of novel content in both experience and in our conceptual lives. That is, if the puzzles discussed so far are not the results of confusions, then the search for the criteria for correctness for novel language use and descriptions of seeing-as is a real search. In this case how to give a meta-logical explanation for the sense of novel language use is a real problem; for it would seem that in using novel language correctly we are somehow discovering hitherto hidden facets of the world, but it is not clear how the harmony between logic and the world which would make such discovery possible could be coherently realized. It is in this problematic at least that the interest in the connection between seeing-as and novelty would reside if the intuitions explored so far do not hold. With this cursory survey of the purpose of this collection in mind, let us now look to the contributions which make it up.

Contributions William Child asks if there is ‘any special connection between seeing-as and novelty’ and finds this connection in cases of novel concept acquisition in which ‘there is not merely a change in the way I experience things. There is also an experience of change: an awareness of a change in the way I am understanding or conceptualizing things’ (p. 36). Once identified, the experience of undergoing a change in outlook upon the world comes under further scrutiny: ‘Is the special connection between seeing-as and conceptual or theoretical novelty simply a matter of doing descriptive justice to the facts about our experience? Or does seeing-as in some way help to explain conceptual innovation?’ (p. 36). That is, even if special, is the connection in any way important? In asking and answering these questions William Child gives a thorough and precise overview of some of the key relations seeing-as and novelty bear to one another and situates them amidst Wittgenstein’s wider thought. In doing so, the kinds of intuitions sketched in 2 and 3 and, to some extent, 4 are filled out and explored in their differing aspects. Through this process the reader is guided away from some tempting confusions and towards a clearer view of some of this collection’s key questions:

18 Brendan Harrington how ought we think of the experiential nature of seeing-as, and what bearing has this upon the wider use of ‘seeing-as’? What difference would it make to come to a new concept by way of an experience of seeing-as, rather than by other potential routes? Does aspect-perception play any essential role in our coming to grasp new concepts? More generally, emphasis is placed on the relation between different senses in which we use and connect ‘experience’ and ‘understanding’ in relation to the various ways in which we might grasp novel concepts. This emphasis serves to locate the plurality of ways in which we may use ‘seeing-as’ amidst pluralities in the use of other concepts which are fundamental to the descriptions we give of our mental lives. Robert Briscoe takes up the question of what a particular kind of experience is; namely, that of seeing a two-dimensional (2D) surface and experiencing a three-dimensional (3D) pattern or scene. Brisoce’s view is that pictorial experience, is not a kind of seeing-as: we do not see the 2D pictorial surface as a 3D scene. He seeks to elaborate and defend this view by modifying, and then defending, what he terms Gombrich’s Continuity Hypothesis. In Gombrich’s formulation, the continuity hypothesis was of roughly this form: when we see a 2D surface as a 3D scene, we are having the same kind of experience as we would if really seeing a 3D scene. Further to this, Gombrich took it that, as what we are experiencing in such circumstances is not a real 3D scene, such experiences must be non-veridical. Briscoe’s modification is to make the ‘same kind’ claim present in Gombrich’s formulation more specific. He claims that pictorial experience is of the same psychological kind as normal, 3D perception in the following sense: in each case the same mechanisms are brought to bear when engaging with the interpretation of visual information. However, Briscoe also emphasizes ways in which pictorial seeing is nonetheless distinguishable from veridical visual experience of 3D scenes: although in each case common mechanisms are brought to bear upon the available information, ‘pictorial experience is non-committal about the reality of its objects because, unlike ordinary, non-pictorial visual experience, it fails to specify their locations at certain absolutely scaled distances in depth’ (p. 56). In Briscoe’s contribution, then, we have a clear example of just the sort of work alluded to at the end of 10. Once one subscribes to this kind of project—that of saying what constitutes a particular kind of experience— one ought to engage thoroughly in the project of considering different ways in which that experience might be constituted. Briscoe engages in this task by aiming for a hypothesis consistent with both current empirical research in vision science and the phenomenology of looking at pictures (specifically, that we experience pictorial scenes as having properties like depth whilst, at the same time, not experiencing those properties as veridical). Komarine Romdenh-Romluc also examines the relations between different modes of experience—veridical and non-veridical, illusory and non-illusory— and the body. She seeks to show that, whilst one might suppose that what is

Introduction 19 commonly called ‘seeing-as’ and what Merleau-Ponty calls ‘Gestalt perception’ are the same, they, in fact, are not. It is argued that whilst Merleau-Ponty uses the idea of Gestalt perception to provide a solution to the problem of illusion, the idea of seeing-as cannot be employed for this purpose.20 Key to this difference, Romdenh-Romluc argues, is that whilst Gestalt perception aims to provide the perceiver with veridical information about the world, seeing-as experiences are imaginative seeings and thus make no direct claim about how the world is. This assessment speaks to the features of seeing-as discussed earlier in the introduction: seeings-as are cast as novel ways of experiencing the world which, in being so, make various forms of human creative practice possible. In picking out this feature of seeing-as, this contribution speaks to the concerns of other contributors: the sense in which William Child finds it acceptable to talk of seeing-as explaining novel concept use (p. 40); and the more general sense in which, although what we experience when seeing-as may enable us to use concepts fruitfully in a novel way, it does not determine that we do. This more general thought features in every subsequent chapter in this collection in one form or another. An interesting correlate to the claim about the differing statuses of seeing-as and gestalt perception is that whilst it is claimed that the former involves no bodily commitment, it is claimed the latter does. Here we have an interesting link to, and contrast with, Briscoe’s concluding remarks concerning the continuity of physiological and psychological mechanisms, and their underpinning of both seeing a 3D scene and seeing a 2D scene which represents a 3D scene. Akihiro Kanamori seeks to show that aspect-seeing is ‘inherent in mathematical activity’ (p. 112), with emphasis placed on the idea of mathematical proofs. He considers this to be the case in at least two respects: first, in our gaining understanding of the logic surrounding a proof’s operation ‘aspect-perception provides language, and so a way of thinking, for discussing and analyzing concepts, proofs, and procedures—how they are different or the same, how they can be compared or correlated’ (p. 112). Secondly, aspect-perception plays a role in widening the context in which we take particular mathematical proofs to function; both their immediate logical context and their context within the history of mathematics. These two ways of seeing-as being inherent within mathematical thinking are, for Kanamori, complementary: new aspects are objective inasmuch as they are ‘there to be seen’ (p. 112) by both the first person to come to see some particular proof and by subsequent learners of the logical moves which are inherent in coming to appreciate the proof’s significance. Once established, these aspects present views on ‘the logical space of possibilities’ (p. 113), and the appreciation of this space serves as a prerequisite for future explorations. This inherency of seeing-as in mathematical activity is brought out through the examination of two examples: aspects of the irrationality of square roots

20 Brendan Harrington (variously, ‘geometric and algebraic, ancient and modern’; p. 113); and, the use of aspect-seeing in moving beyond an apparent puzzle found in the mathematics surrounding the derivative of the sine function. In each case Kanamori examines the role of aspect-perception in making the passage from one understanding of relevant proofs to another, and in the movement from problems within the system of current proofs to new proofs. In each example we follow the widening of the logical sphere of the subject of the proof, and of the proof itself, by coming to terms with a ‘“new” result “found” by the author’ (p. 114). Further to this—as is implied by the scare quotes in the previous quotation—the author intends the reader to see in the new result ‘that creativity is belied to a substantial extent by context’ (p. 114). This intention expresses the tension between creation and discovery we first touched upon in 4, and explored in relation to Wittgenstein’s transitive/intransitive distinction. It also leads us to revisit the idea that instances of seeing-as can provide a prerequisite for conceptual novelty; and, to ask again quite how this could be possible. The emphasis on objectivity, and on the appreciation of the context given by the logical space of possibilities belying the application ‘creativity’ to descriptions of progression in mathematics, would seem to place Kanamori’s views as pulling towards the idea of aspect-perception being a vital tool in the conceptual discovery of what is already out there; whilst the idea of mathematics as a conceptual construction, might seem to indicate a pull towards creation. Whilst there may be a tension here between creation and discovery (and, of course, given an appropriate understanding of terms like ‘objectivity’ and ‘logical space’, there may not), the thought that the logical context surrounding a proof, when seen under a particular aspect, can open up the possibility of conceptual creation, is examined in the full detail of its mathematical complexity. The progressions in logical space charted in this contribution thus serve as a point of comparison to Wittgenstein’s treatment of ‘belief ’, which we briefly surveyed in 6. Making this comparison both serve to demonstrate the multifarious use we can make of ‘seeing-as’ and to enrich our understanding of the common aspects of these uses. The picture of logical space existing, and our understanding expanding to encompass more of it through our engagement with different aspects, is one way in which one might be tempted to unpack what is happening in cases of the novel application of existing concepts. This picture, and the implication that communities of mathematicians are using aspect perception to navigate logical space (and then leading others to do the same), offers a rich point of comparison, and counterpoint, to the next two contributions. Michael Beaney and Bob Clark offer an account of the fundamental importance of seeing-as to Wittgenstein’s engagements with mathematics, and this as an instance of his more general methodology. This role of seeing-as is demonstrated by surveying circumstances in the history of mathematics—from ancient Greece to the near present—‘where we see the way that criteria for the relevant mathematical concepts come apart [. . .],

Introduction 21 enabling different aspects of those concepts to be seen, which then allows something like a spontaneous choice to be made as to which aspect to take as primary’ (p. 133). We thus see the application of ‘seeing-as’ being stretched, not only to a domain of application other than the visual but also from the singular realization of a subject who sees-as to the plural realization of a community choosing to go on in one way rather than another. In doing so the authors aim to establish that what is going on in novel mathematics is best described neither as ‘discovery’ nor as ‘invention’ of something entirely new. There are facts to be revealed, and creativity to be exhibited, but what is crucial is the opening up of different aspects of something, the perception of which prompts a choice that sooner or later ‘catches on’ in a mathematical community and proves fruitful. (p. 133) Here we see a key connection both with the intuition featured at the end of 3 (and thereafter), and more particularly with Kanamori’s ideas concerning aspect perception and the logical space of possibility. In contrast to Kanamori, however, emphasis is laid more heavily on the manner in which aspect perception can make options for going on in a different way available to us, and, inasmuch, present us with a choice. This emphasis makes more explicit, and opens up for exploration, the manner in which the choice between these differing aspects may be weighted by factors outside of the immediate logical sphere of a particular problem. In this way, what may seem in retrospect to be a necessary movement in logic is also seen to be under the influence of historical accident, social influence, and pragmatic considerations (a point taken up in a different way in Thomas Nickles’s contribution). Treating movements in mathematics in this way allows us to see movements in criteria within mathematical practice as analogous to those Wittgenstein made with regard to movements in the sense of ‘belief’ when exploring Moore’s paradox (p. 9). Similarly, by following the genealogies of mathematical problems laid out by Beaney and Clark, we can see seeing-as playing a role in the progression of mathematics in a manner akin to that described at the end of section 6—that of opening one up to a variety of conceptual options. Thomas Nickles examines something like the communal sense of ‘seeing-as’ developed in the previous contribution by describing in broad outline Kuhn’s views upon progress within science. Particular emphasis is placed on the relation of Kuhn’s mature views on shifts in scientific understanding and practice to the following elements of his thought (which stand in close relation to Wittgenstein’s thought—or, at least, to interpretations of it): •

Kuhn’s early views upon visual experience cast it as (a) theory laden and (b) not explicable in terms of a level of raw visual information and

22 Brendan Harrington

•

interpretation of that information in terms of some theory. (Kuhn’s early claims that scientists with differing beliefs occupied radically different experiential worlds arose from this belief). Although Kuhn moved away from the thought that scientific observation was a special species of sensory observation, he retained the idea that observation was, as it were, theoretical all the way down. Kuhn’s view of the activity of representing the world as one way or another centre around the use of exemplars as points of comparison, where ‘exemplars are models, paradigm cases. Students and junior investigators deeply internalize the exemplars of their field and eventual specialty field in such a way that they come to see the world through the filter of this set of exemplars’ (p. 156).21

Kuhn drew from these thoughts implications which maintain the close relation to Wittgenstein’s thought. The two strands in Kuhn’s thought, above, interrelate in roughly this way: observation is always theoretical because when we observe we must be doing so by way of comparison with some exemplar or other, and exemplars just are what constitute theories, or models of how things work. Through exemplars, then, insofar as we can make good sense of it, we see the world as always being already ordered in a particular way. Kuhn thus maintains the idea that scientists operating using different exemplars occupy different realms, and that evolutions in scientific practice are constituted by changes in exemplars used. As exemplars are prerequisites of description of the world, they cannot be justified prior to use, and changes in exemplar can only be argued for by appeal to other exemplars and practical usefulness. In this sense, ‘Kuhn demoted logic and promoted its classical opponent, rhetoric, by emphasizing analogy, metaphor, and simile in problem-solving work’ (p. 157). Once established, Nickles uses Kuhn’s views to make a point about epistemic humility: there is an unfounded whiggishness implicit in the assumption ‘that scientific research should be regarded as getting us ever closer to the metaphysical truth’ (p. 156). To show this, whereas Kuhn highlighted large scale shifts (revolutions) in what counts as an exemplar within a particular scientific field, which allegedly happened in the past (an approach which may be accused of relying on over-dramatization), Nickles imagines how we could justify thinking of the deep future of science as related to the present state of science in a linear manner. He claims that, even if there are no large-scale revolutions in the future, to suppose that many small iterative changes in exemplars over time would not have an effect sufficient enough to give us grounds to reject a linear view of science (that of its progressing further and further towards accurate, fully worked out, representation), would be folly. In this chapter, then, we find clear parallels to Wittgenstein’s thought about objects of comparison, world views, picturing, and the role of persuasion as the main agent of change in these bedrock features of our lives.22 Furthermore,

Introduction 23 there is a link being drawn between these ideas and our propensity to, as with exemplars, see parts of the world in terms of other parts of the world. We thus are led to consider deep ways in which we might connect seeing-as with the process of coming to see the world in radically novel ways. Denis McManus extends the use of ‘seeing-as’ further into the moral sphere by examining ‘what it is for us to be properly responsive to, and properly responsible in the face of, the world in its concrete heterogeneity and novelty’ (p. 174). McManus does so by finding common ground between Heidegger’s thought about authenticity and Murdoch’s criticisms of this thought; this common ground centres around Aristotle’s notion of phronēsis—being able to see the right thing to do in a particular situation—and it is argued that authenticity of a kind common to Heidegger and Murdoch is implicated in this ability. The examination of what it takes for a moral agent to respond to particular situations authentically thus yields an account of the relationship between being an authentic agent and aspect-perception. McManus argues that an ability to see-as is a prerequisite of being an authentic agent in roughly the following way: authenticity requires genuine expression; genuine expression requires that one sees oneself as expressing norms inherent in a particular style of conducting oneself (as a teacher, mother, doctor, etc.); seeing oneself as expressing requires more than mere accord with norms, it requires awareness of such accord; awareness of accord requires that that one particular way of going on can be distinguished from other possible ways; and, making such distinctions requires access to a host of aspects in order that one might distinguish between them. Thus, for McManus, aspect perception is a prerequisite of our appreciation of the world in all its concretia, and so a prerequisite of our owning the particular role we play within it (rather than merely blindly playing it). McManus claims seeing something as the correct way of going on for a mother, doctor, villain, etc., but also (through seeing-as) having a capacity to appreciate that to be bound by norms for such roles is to be bound by a particular set of norms which could be otherwise, allows one to accommodate Heidegger’s insights without conceding to Murdoch’s criticisms of existentialism. For, on this reading we need suppose neither that to be authentic we must be constantly re-inventing and questioning our norms, nor that each thing we do must be a radical choice made in a void of norms. Rather, we must hold ourselves responsible for, or own, the overarching standpoint we own by attending to how—in its light and in each new situation we find ourselves—we are called to act.23 Underlying this discussion we have a contrast between making decisions which are: (a) based on ‘an openness to the concrete nature of a situation in all of its presently available aspects’; and (b) based upon some criterion of correctness, set by some external, or prior, circumstances or rules, which ought to hold for any given situation. The link between seeing-as and novelty thus plays out a nuanced version of the intuition featured at the end of 3, and connects with the themes which arose in our discussion of transitivity,

24 Brendan Harrington intransitivity, and novel concept use. What also comes out clearly is that—as in the case of novel word use—an ability to see-as makes available for us a set of possibilities which allow that we might go on in a different way to before by making use of what has gone before.

Notes 1. In translations previous to Hacker and Schulte’s 2009 edition, PPF was referred to as Philosophical Investigations Part Two. 2. Seeing-as features most prominently in the Brown Book and in Philosophy of Psychology—A Fragment. Wittgenstein neither prepared nor intended these texts to be published as they now are (if indeed at all). We thus ought not to take either text to be in any sense definitive or authoritative. These facts, however, do nothing to detract from the central role which we can see examples of seeing-as playing within Wittgenstein’s thought and method; furthermore, exegesis aside, even if the relations between instances of seeing-as and other concerns which appear in these texts were not intended to be expressed—or even thought of— this has no bearing on whether such relations obtain or not. 3. I borrow this phrase from Katherine Morris’s introduction to Wittgenstein’s Method Neglected Aspects: Essays on Wittgenstein by Gordon Baker (2004). Morris uses this phrase to describe an attitude towards the purpose of the practice of philosophy which Baker held and would later reject: “to replace theorizing with platitudinous descriptions of ‘grammar’ and to police the borders between sense and nonsense, issuing tickets to those philosophers, psychologists and linguists who transgressed the bounds of sense” (ibid.: 1). 4. These two concerns, however, will only stand if reasons do not arise whilst considering the relation between seeing-as and novelty which address, or obviate, them. I hope the reader will agree that such reasons do arise. Namely there is no strict homogeneity to how we use the phrase ‘seeing-as’ and that examining particular cases of seeing-as is especially useful in elucidating that this is the case, not only for uses of ‘seeing-as’ but also for general language use. 5. Consider the following sections of PPF, for instance: 245. Is being struck looking + thinking? No. Many of our concepts cross here. 246. (‘Thinking’ and ‘talking in the imagination’—I do not say ‘talking to oneself’— are different concepts.) 247. The colour in the visual impression corresponds to the colour of the object (this blotting paper looks pink to me, and is pink)—the shape in the visual impression to the shape of the object (it looks rectangular to me, and is rectangular)—but what I perceive in the lighting up of an aspect is not a property of the object, but an internal relation between it and other objects. 248. Do I really see something different each time, or do I only interpret what I see in a different way? I’m inclined to say the former. But why?—To interpret is to think, to do something; seeing is a state. 6. Perhaps available to us under some other conceptual guise, or perhaps otherwise available; I leave these possibilities open. 7. For example, whilst I may be able to see a shape as a parallelogram after acquiring the concept, it is unlikely that I could recognize a chiliagon (a 1,000-sided shape) as such after acquiring that concept. Although of philosophical interest, questions such as this fall more obviously under the purview of psychology. 8. This idea arises in PPF 257–260 and is explored in this collection in a variety of ways (in particular, cf. pp. 34–46).

Introduction 25 9. We might want to be wary of quite how we think of aspect-blindness; and especially of presupposing that when thinking of the ‘aspect-blind’ we are thinking of a clear and coherent possibility. For instance, once we start comparing the experiences of the aspect-seers and the aspect-blind, we are on decidedly shaky ground—the distinction may lead one to presuppose that there is a thing respectively present and absent to experiencing subjects who see and do not see aspects. Taking on this presupposition both puts us in danger of not heeding, or at least demands engagement with, Wittgenstein’s warnings against the reification of mental states, and, by looking to ground seeing-as talk in such things, lends a misleading sense of conceptual-unity to talk of seeing-as And above all do not say “Surely my visual impression isn’t the drawing; it is this—which I can’t show to anyone.” Of course it is not the drawing, but neither is it something of the same category, which I carry within myself.” (PPF 132)

10. 11. 12.

13.

14.

It may well be that there are other more useful ways of thinking of aspectblindness. We might, for instance take talk of aspect-blindness to be only a rhetorical device; or, as a way of talking to an interlocutor who suffers from some confusion or other on their terms; or, in a way free of any associations with talk of the content of experience. In particular, consider PI 138–242. Diamond (2001) makes this link to PI 279, and draws it and its implications out in clear and full detail. Many examples of this kind can be found in Wittgenstein’s work. Take, for instance: variations in and extensions of the use of ‘in’ highlighted by comparing having a gold tooth in one’s mouth and a pain in one’s mouth (Blue Book pp 49–52); the discussion of the extension of the practice of naming from people to rivers given at Philosophical Grammar 202–207. Such a problematic is familiar from Wittgenstein’s consideration of rule following (PI 138–242). In this way, Wittgenstein’s rule following considerations, in part, serve as a reductio ad absurdum, warning against confusion of transitive and the intransitive language use. Cf. PPF 133–134: 133. The concept of an ‘inner picture’ is misleading, since the model for this concept is the ‘outer picture’, and yet the uses of these concept words are no more like one another than the uses of “numeral” and “number”. (Indeed, someone who was inclined to call numbers ‘ideal numerals’ could generate a similar confusion by doing so.) 134. Someone who puts the ‘organization’ of a visual impression on a level with colours and shapes would be taking it for granted that the visual impression is an inner object. Of course, this makes this object chimerical, a strangely vacillating entity. For the similarity to a picture is now impaired.

Also, cf. Wittgenstein’s use of ‘Object of Comparison’ PI 130–131 (quoted in part in note 20 to this chapter). 15. Wittgenstein first heard this paradox in Certainty a lecture given by Moore to the Cambridge Moral Sciences Club in 1944. Written problems of this form first feature in Moore’s Reply to My Critics (1942). 16. Here I am ignoring differences in Moore’s formulation and Wittgenstein’s (cf. PPF 87), and the wide debate around the paradox, possible resolution of the paradox, and where Wittgenstein fits into all of this. For more detailed engagements, cf. Baldwin (1983), Clark (2017), Green and Williams (eds.) (2007), Heal (1994), Sorenson (1988), and Williamson (1996).

26 Brendan Harrington 17. Consider PPF 122–123: 122. It would have made as little sense for me to say, “Now I see it as . . .” as to say at the sight of a knife and fork “Now I see this as a knife and fork”. This utterance would not be understood. Any more than “Now it is a fork for me” or “It can be a fork too”. 123. One doesn’t ‘take’ what one knows to be the cutlery at a meal for cutlery any more than one ordinarily tries to move one’s mouth as one eats, or strives to move it. 18. Here we have a milder worry in the same vein as the worry about aspect-blindness raised in 2. 19. Though, in very simple cases, one might think that an aspect could be so basic as to be shared by any mature observer who allowed themselves the time to appreciate it, or had it pointed out to them (my thanks to Michael Beaney for this point). 20. The claim of this contribution is actually more specific. It is that the idea of one of the many different phenomena that could be considered instances of seeing-as— namely the experience of a picture like Jastrow’s duck–rabbit under one of its possible aspects—cannot be employed to deal with the problem of illusion. 21. Here there is a clear link to Wittgenstein’s use of ‘object of comparison’: we can avoid unfairness or vacuity in our assertions only by presenting the model as what it is, as an object of comparison—as a sort of yardstick; not as a preconception to which reality must correspond. (PI 131) 22. Cf. PI 217: “How am I able to follow a rule?”—If this is not a question about causes, then it is about the justification for my acting in this way in complying with the rule. Once I have exhausted the justifications, I have reached bedrock, and my spade is turned. Then I am inclined to say: “This is simply what I do.” (Remember that we sometimes demand explanations for the sake not of their content but of their form. Our requirement is an architectural one; the explanation a kind of sham corbel that supports nothing.) 23. Whether, though, we can own one particular role without needing to do further work in considering why we do, or if we ought to, is, however, a question which threatens to bring back all of the same worries about the coherence of the idea of authenticity at this (as it were) meta-level. Whilst this is not an issue which is addressed here, cf. McManus (2015).

Bibliography Baker, G. P. (2004). Wittgenstein’s Method: Neglected Aspects: Essays on Wittgenstein. Oxford: Blackwell. Baldwin, T. (1983). G. E. Moore: Selected Writings. London: Routledge. Clark, B. (2017). Wittgenstein, Mathematics and World. London: Palgrave-Macmillan. Diamond, C. (2001). How Long is the Standard Meter in Paris? In T. McCarthy and S. C. Stidd, (eds.), Wittgenstein in America. Oxford: Clarendon Press. Green, M. S. and Williams, J. N. (eds.). (2007). Moore’s Paradox: New Essays on Belief, Rationality and the First Person. New York: Oxford University Press. Heal, J. (1994). Moore’s paradox: A Wittgensteinian approach. Mind 103(409): 5–24.

Introduction 27 McManus, D. (2015). Anxiety, choice and responsibility in Heidegger’s account of authenticity. In D. McManus (ed.), Heidegger, Authenticity and the Self. London: Routledge. Moore, G. E. (1942). A reply to my critics. In P. A. Schilpp (ed.), The Philosophy of G. E. Moore. Chicago: Open Court. Sorenson, R. (1988). Blindspots. Oxford: Clarendon Press. Williamson, T. (1996). Knowing and asserting. Philosophical Review 105: 4 489–521. Wittgenstein, L. (1972). The Blue and Brown Books: Preliminary Studies for the ‘Philosophical Investigation’. Oxford: Blackwell. ——— (2009). Philosophical Investigations, 4th edition (trans. Hacker and Schulte). Oxford: Wiley-Blackwell.

2

Wittgenstein, Seeing-As, and Novelty William Child

1. Introduction What is the relation between novelty, creativity, and innovation, on the one hand, and the phenomenon of seeing-as, on the other? And what if anything does Wittgenstein’s work on seeing an aspect teach us about that relation? It is common to say that someone who acquires a new set of concepts or a new system of belief comes to see things in a new way. In that vein, Thomas Kuhn famously likens paradigm shifts in science to Gestalt shifts in perception.1 The shift from a geocentric to a heliocentric cosmology, he thinks, or from Newtonian physics to Einsteinian physics, is like the shift from seeing the duck aspect of the duck–rabbit figure to seeing its rabbit aspect; when we shift from one paradigm to another, we come to see the world in a new way—as containing things and properties that we did not see before. Similarly, religious conversion is often described as involving a transformation in the convert’s way of seeing the world. In one sense, everything remains as it was before; but in another sense, everything is seen differently. Or again, the development of the concept of sexual harassment is said to have changed people’s way of seeing things; a remark that would formerly have been seen as a joke, for instance, came to be seen as an instance of harassment. In these and other cases it is extremely natural to describe the acquisition of new concepts, or a new theory, or new knowledge, as involving coming to experience the world in a new way. But how seriously should we take that idea and what should we make of it? I shall consider three issues. First, is there any special connection between seeing-as and novelty: a connection that is absent from the case of grasping familiar concepts, theories or knowledge? Second, is there an explanatory relation between seeing-as and novelty? Is conceptual or theoretical innovation explained by the fact that the innovator sees things in a new way? Third, whatever association there is between seeing-as and novelty, is it important? Why, if at all, does it matter that someone who grasps new concepts experiences things in a new way? Would she be any worse off if she simply operated with the new concepts in a way that left her experience of the world unaffected?

30 William Child Before proceeding, I note a preliminary point about the notion of novelty. We can distinguish between two kinds of case in which someone acquires a new theory or set of concepts. On the one hand, there is the case of genuine theoretical or conceptual innovation: the case where a person devises a theory or set of concepts that no one has grasped before. On the other hand, there is the case where she acquires an existing theory or set of concepts: the theory she acquires is new to her; but it is already understood by others. That is an important distinction; it is more difficult, and requires more creative effort, to formulate an original theory than to learn an existing theory from other people. But with respect to the links between seeing-as and novelty, the distinction seems less important. In both cases, I move from not understanding to understanding; in both cases, the theory that I come to grasp is new for me; in both cases, I come to see phenomena, patterns, or connections that I did not see before. In exploring the connections between seeing-as and novelty, therefore, I shall generally use the notion of novelty in an inclusive sense: for the purposes of this discussion, theories or concepts are novel if they are new for the person grasping them, even if they are not novel from the perspective of humanity as a whole.2

2. Is There a Special Connection Between Seeing-As and Novelty? Is there a special connection between seeing-as and novelty: a connection that is absent from situations where someone applies familiar concepts in straightforward and familiar cases? The following argument might suggest that there is no special connection. Suppose we accept that coming to grasp a new theory or set of concepts involves coming to see things in a new way. If conceptual innovation involves seeing things in a new way, there must have been some way in which the innovator saw things before: an old way of seeing, associated with the old system of concepts, just as the new way of seeing things is associated with the new system of concepts. So, if there is a connection between seeing-as and novelty, it is hard to avoid the conclusion that grasping any set of concepts involves seeing-as: seeing things in some particular way. According to this argument, there is a quite general connection between the concepts one grasps and the way one sees the world; the connection between grasping new concepts and seeing things in a new way is simply a product of that more general connection.3 There is some plausibility in that argument. But Wittgenstein’s discussion of seeing an aspect suggests a more nuanced position. On this more nuanced position, there is indeed a general connection between seeing-as and grasping concepts. But there are also links between seeing-as and novelty that go beyond that general connection. We can work up to that position by considering some details of Wittgenstein’s discussion. Does Wittgenstein think there is a special connection between seeing aspects and novelty? Or does he think that seeing-as is a perfectly general

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phenomenon? On the one hand, he draws particular attention to a range of phenomena that specifically involve novelty or change: noticing an aspect, experiencing a change of aspects, an aspect’s dawning or lighting up.4 On the other hand, he talks about the ‘continuous seeing’ of an aspect, which seems independent of any change or novelty (see PPF §118). What, in his view, is the connection between these two kinds of phenomena? On one interpretation, Wittgenstein takes continuous aspect-perception as the basic case; the cases involving novelty and change (noticing an aspect, experiencing a change of aspects, etc.) are merely upshots or manifestations of that basic phenomenon.5 On a different interpretation, change and novelty are basic to Wittgenstein’s conception of seeing-as: those are the phenomena he is really interested in, and in his view, the phenomenon of continuous aspect perception occurs only in the context of some possible change.6 Which interpretation best captures Wittgenstein’s discussion? Before addressing that question, we should register a note of caution. It would be a mistake to try to organize the whole range of phenomena that Wittgenstein discusses as instances of seeing an aspect in terms of a single distinction between the continuous seeing of an aspect, on one hand, and noticing an aspect or experiencing a change of aspects, or an aspect’s lighting up, on the other hand. It would be similarly mistaken to press the question, ‘Exactly what does Wittgenstein take seeing-as to involve?’ In his view, seeing-as is not a single, homogeneous phenomenon. Part of his point is that ‘there is an enormous number of interrelated phenomena and possible concepts’ in this area (PPF §155). He notes that the concept of what is seen is ‘very elastic’ (PPF §147). And he insists that ‘there is not one genuine, proper case’ of describing what is seen (PPF §160). Thus, if we apply what Wittgenstein says to the case where someone sees, say, a portrait of Elizabeth I, we can distinguish at least four senses or levels of seeing that may be involved. First, the person may see the picture without being conscious of its qualities at all.7 Second, he may see it and be aware of the colours and shapes it contains, without seeing it as a picture of Elizabeth I. And, as Wittgenstein says, that is consistent with his knowing that it is a picture of Elizabeth I and being able to work out various things about her from the picture. Third, he may see it as a picture of Elizabeth I: the person sees her in the picture; and in Wittgenstein’s phrase, he views the picture as the person it represents (PPF §197). Fourth, he suggests, there is a further concept of seeing-as: [the concept] of a seeing-as which occurs only while I am actually concerning myself with the picture as the object represented. I could say: a picture is not always alive for me while I am seeing it. ‘Her picture smiles down on me from the wall.’ It need not always do so, whenever my glance lights on it. (PPF §§199–200)

32 William Child What all that suggests is something that we would anyway expect on general grounds. Wittgenstein’s discussion explores a large number of related but different perceptual phenomena. He does not aim to explain or analyse the phenomena in terms of anything more basic. (So, for instance, he rejects the assumption that the various phenomena of seeing-as can be explained in terms of a bipartite distinction between experience on one hand and thought on the other.8) Nor does he hope to formulate a general theory about seeing-as. On the contrary, he explicitly eschews such ambitions: (In giving all these examples, I am not aiming at some kind of completeness. Not a classification of psychological concepts. They are only meant to enable the reader to cope with conceptual unclarities.) (PPF §202) In the light of that, we should be cautious about the idea that Wittgenstein thinks of one kind of phenomenon (continuous seeing of an aspect, say, or noticing an aspect) as the basic case or that his main interest is focused on one class of cases rather than another. That point is well taken. But, having acknowledged it, we can still consider our question about Wittgenstein’s attitude to the relation between seeing-as and novelty. As we have noted, he is happy to talk about the continuous seeing of an aspect. And he is happy to say that someone who only ever sees the duck–rabbit picture as a picture-rabbit is a case of seeing-as: he sees the figure as a picture-rabbit (PPF §121). In such a case, seeing-as seems to have no essential connection with novelty: with noticing an aspect, experiencing a change of aspect, or an aspect’s lighting up. And visual experience seems quite generally to involve seeing-as in this sense: I see the object on my desk as a mug (rather than merely knowing that it is a mug), I see the mouthpiece of my French horn as a mouthpiece (and do not have to work out that it is a mouthpiece), I see the thing in the shop window as a bicycle, and so on. However, someone who thinks that the real focus of Wittgenstein’s interest is this inclusive notion of seeing-as must explain why he gives so much attention to the more limited and specific phenomena of novelty and change of aspect. At least two explanations might be suggested. One suggestion is that the reason why Wittgenstein emphasizes examples of aspect-change and so forth is simply that they are particularly good cases for establishing a point he wants to make about visual experience in general. The general point, on this view, is that perception is theory-laden: our awareness of the world does not present us with a mere mosaic of shapes and colours, which we have to interpret as experience of objects and properties around us; on the contrary, the objects and properties are built into the intrinsic nature of the experience. The phenomenon of aspect-change is then said to provide compelling support for this general point. For, it is said, it is simply obvious that aspectchange is an experiential phenomenon and not a merely cognitive one; the idea that our experience remains the same throughout, and that what changes

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is just the interpretation we put on it, obviously falsifies the phenomenology. But once that point is appreciated for the special case of aspect-change, the more general point can be seen to apply to all visual experience. A second putative explanation for Wittgenstein’s focus on cases where aspects change or light up starts from Wittgenstein’s insistence that it is only appropriate to say, ‘S sees x as an F’ (or even, that the utterance only makes sense) if there is some contrast or ambiguity in question. In a straightforward case where I recognize a familiar object—a cup, say—for what it is, and have no thought of its being anything else, it would, he thinks, be inappropriate (or even senseless) for me to say, ‘I see the cup as a cup’; that utterance ‘would not be understood’ (PPF §122). Similarly, if I see the duck–rabbit figure and have no idea that it is an ambiguous figure, it would be inappropriate for me to say, ‘I see it as a picture-rabbit’; I should simply say, ‘I see a picture-rabbit’. That makes it natural for Wittgenstein’s discussion of seeing-as to focus on instances of aspect shift or aspect dawning, where—in the nature of the case—there is always a relevant contrast, which guarantees that talk of seeing-as will always be appropriate. But, on the current suggestion, it is compatible with that to hold that Wittgenstein takes the phenomenon of seeing-as to be a ubiquitous one. For even if I am not aware of the ambiguity, so cannot appropriately say, ‘I’m seeing the figure as a picture-rabbit’, it is still true that I do see it as a picture-rabbit; and others who are aware of the ambiguity can appropriately say of me, ‘He is seeing it as a picture-rabbit’ (PPF §121). So, on this view, Wittgenstein means to be discussing a quite general phenomenon; but there is a good reason why his discussion of seeing-as tends nonetheless to focus on the narrower category of cases where aspects change or light up. What should we make of these arguments? Wittgenstein certainly does employ a notion of seeing-as that is intended to apply very generally— including in cases where there is no kind of novelty in play. But it would be wrong to conclude that the reasons for his particular focus on cases of novelty are exhausted by the considerations we have just sketched. For, as he presents things, the character of the experiences involved in cases where aspects change or light up goes beyond anything that is intelligible in terms of the more general phenomenon. In a shift of aspects (say, the shift from seeing the rabbit-aspect of the duck–rabbit picture to seeing the duckaspect), there is not just a change in our experience: a change from seeing things one way to seeing them another way. There is also an experience of change: an experience of something changing (or seeming to change). And that experience of change itself has a distinctive character: the person who experiences a change of aspects is aware of things changing in one respect but being unchanged in another. As Wittgenstein puts it: I observe a face, and then suddenly notice its likeness to another. I see that it has not changed; and yet I see it differently. (PPF §113)

34 William Child Or again: The expression of a change of aspect is an expression of a new perception and, at the same time, an expression of an unchanged perception. (PPF §130) That feature of a change in aspects is plainly not present in the general case of seeing-as. Nor is it explicable in terms of anything that is. Parallel remarks apply in the case of the dawning of an aspect. Consider the case where I am suddenly struck by the likeness between someone and his father (PPF §111–13, §§239ff.). When I am struck by the resemblance, there is a change in my experience. But the change is not just a shift from one way of seeing things to another. Being struck by a likeness—like being aware of a change—is itself an experience. And it is an essentially transitory experience; it is not something that could be a permanent feature of my awareness of anything. As Wittgenstein puts it: ‘The likeness strikes me, and its striking me fades’ (PPF §244); ‘There is a physiognomy in the aspect, which then fades away’ (PPF §238). So in this case, as with the case of aspects changing, there is something present in the novelty-involving case that is not a feature of the general case of seeing an aspect and cannot be accounted for in terms of that general case. Wittgenstein is certainly interested in a general phenomenon of seeing-as. But his discussion of the novelty-involving cases highlights features that go beyond the more general phenomenon. A further issue is suggested by Wittgenstein’s discussion of aspect-blindness. As we have just seen, the general phenomenon of continuous aspect-perception is distinct from the particular phenomena of aspect-change and aspectdawning in this sense: the particular phenomena cannot be accounted for in terms of the more general phenomenon. But is the general phenomenon distinct in a further sense: can the general ability to see something as something exist without the ability to have experiences of aspect-shift, aspect-dawning, and so forth? Empirical evidence suggests that it can: that there is a stage in human development at which children do have continuous seeing-as without the ability to see changes in aspect.9 For instance, they react to pictures as we do; they see the rabbit aspect of a picture-rabbit and do not merely interpret the picture as a representation of a rabbit. And, with an ambiguous picture, they can see it in one way in one context and in a different way in a different context; they see the duck–rabbit picture as a picture-rabbit when it is surrounded by other pictures of rabbits and as a picture-duck when surrounded by other pictures of ducks. But they cannot see the ambiguous picture shift from one aspect to the other; nor, while seeing something as a picture-rabbit, can they try to see it as something else, or acknowledge that it could be something else. If that is the right way of understanding the data, then the ability to see a picture shift from one aspect to another is indeed a separate

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ability in the further sense defined earlier; someone can have the ability to see aspects without being able to experience shifts of aspect.10 Where does Wittgenstein stand on this question? The evidence is equivocal. In a remark originating in notebooks from February 1948, he writes: ‘Of course it is imaginable that someone might never see a change of aspect, the three-dimensional aspect of every picture always remaining constant for him, for example. But this assumption doesn’t interest us’ (RPP II §480). The first sentence of that remark might suggest that Wittgenstein agrees that someone can see a schematic cube as a three-dimensional cube, say, without being able to experience a shift from one aspect to the other. But the second sentence cancels that suggestion. Someone may, as a matter of empirical fact, never see a change of aspect. Nonetheless, Wittgenstein seems to suggest, that empirical possibility is consistent with there being a necessary connection between the ability to see the schematic cube as a cube and the ability to see aspects change, even if that latter ability is never in fact triggered. A remark from the slightly later discussion in Philosophy of Psychology: A Fragment, however, might suggest a different view. Here Wittgenstein is discussing the double cross figure, which can be seen as a white cross on a black ground and as a black cross on a white ground; he calls these ‘the A aspects’ (see PPF §§212, 215). He writes: The aspect-blind man is supposed not to see the A aspects change. But is he also supposed not to recognize that the double cross contains both a black and a white cross? So if told ‘Show me figures containing a black cross among these examples’, will he be unable to manage it? No. He is supposed to be able to do that, but not to say: ‘Now it’s a black cross on a white ground!’ (PPF §257) It is explicit here that the aspect-blind person cannot see the change of aspect. But can he see the double-cross figure as a black cross on a white ground and, separately, as a white cross on a black ground? That depends on how we understand Wittgenstein’s use of ‘recognize’.11 If ‘recognizing’ that the figure contains a black cross involves seeing it that way (rather than merely knowing, or working out, that it contains a black cross), then this will be a case where Wittgenstein allows that a person can see something as something without being able to see the relevant aspects changing. But if ‘recognizing’ that the figure contains a black cross is consistent with not seeing it that way, then Wittgenstein’s example provides no evidence for his acceptance of that possibility. I shall not try to resolve here the question of exactly what patterns of deficits Wittgenstein takes to be intelligible in this area. As I have said, the evidence about his view seems equivocal. That is perhaps unsurprising, since Wittgenstein’s discussion of aspect-blindness is deliberately open-ended and exploratory. He is not setting out to describe an actual phenomenon with which we are all familiar. Rather, he uses the

36 William Child imagined phenomenon of aspect-blindness as a way of probing the familiar phenomenon of seeing aspects. As he puts it, ‘I form a concept and ask myself how one might follow through with it consistently. What we feel would deserve to be called that’ (RPP II 491). It is time to take stock. We started this section with the question, whether there is a special connection between seeing-as and novelty, or whether the relation between seeing things in a new way and grasping new concepts or theories is simply a manifestation of a more general phenomenon of seeingas. The lessons we have drawn from Wittgenstein’s discussion of seeing aspects cast light on that original question. Suppose we accept that grasping new concepts or theories involves coming to see things in a new way. That is, in part, a manifestation of a more general phenomenon: grasping any system of concepts, new or old, involves seeing things in a certain way. But there is something special about cases of novelty that goes beyond that general phenomenon. When I come to accept a new theory I shift from an old way of seeing things to a new way. But in some cases, at least, there is not merely a change in the way I experience things. There is also an experience of change: an awareness of a change in the way I am understanding or conceptualizing things. There may also be a sense of things that had previously been puzzling suddenly making sense. And so on. Such experiences are part of the phenomenon of conceptual or theoretical novelty. They go beyond a mere shift from an old way of seeing things to a new way. And they have no analogue in the general case of applying familiar concepts. To that extent, then, there really is something special about the specific connection between seeing-as and coming to grasp new concepts or theories. Even if we accept that point, however, we may still wonder about the significance of the connection. Is the special connection between seeing-as and conceptual or theoretical novelty simply a matter of doing descriptive justice to the facts about our experience? Or does seeing-as in some way help to explain conceptual innovation?

3. Does Seeing-As Explain Innovation? It may be tempting to think that seeing-as can help to explain creativity or innovation. It is her ability to see things in new or unorthodox ways, we may think, that explains the innovator’s success in discovering or devising new theories. Relatedly, it can be tempting to think that seeing-as helps to explain our grasp of new concepts. Indeed, it might be argued that the only way of explaining how someone can come to grasp a completely new system of concepts is in terms of the idea that she comes to see things in a new way. For, the thought would go, if the concepts she acquires really are new ones, then she cannot come to understand them by grasping a definition or explanation formulated in terms of concepts she already understands; what makes the new concepts new ones is precisely that they cannot be captured in that way. But there must be some way in which she comes to grasp these

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new concepts and categories. And the idea that the learner comes to see things in a new way seems to fit the bill: it recognizes that the new concepts cannot be captured in terms of the old ones; but, on the current view, it avoids leaving our ability to acquire new concepts as a complete mystery. A version of the same idea is expressed in the appealing thought that a person’s grasp of a rule ultimately depends simply on her seeing things in the right way. As Wittgenstein says, there must be ‘a way of grasping a rule which is not an interpretation’ (PI §201). And, on the current proposal, what someone has when she grasps a rule without an interpretation is a kind of perceptual capacity; the person sees this, this, and this as correct applications of the rule at the relevant points. I said that it may be tempting to think that we can appeal to seeing-as to explain theoretical innovation or the acquisition of new concepts. If we follow Wittgenstein, however, it is a temptation we should resist. He himself shows no inclination at all to appeal to seeing-as to give such explanations. And, as we shall see, though his discussion leaves room for seeing-as to figure in a certain kind of explanation, of limited scope, it does not leave room for the kind of explanation that it might be tempting to envisage. Wittgenstein asks, ‘How can a language-game suddenly become clear to a child?’ (LW I 873). The particular case he is considering is the ‘languagegame’ of ascribing pain to people: a language-game that acknowledges the possibility of pretence, and includes expressions of both belief and doubt. In learning this practice, Wittgenstein says, one ‘learns the use of the expression “to be in pain” in all its persons, tenses, and numbers, but also in connection with negation and the verbs of opinion’ (LW I 874). How does a child come to grasp that practice? Today for the first time he said ‘I believe she’s in pain’. But that’s not enough. So I must assume that in what followed he showed that he hadn’t simply repeated somebody’s words. In short, that his utterance was the beginning of a game, and that he was able to continue with it. Today, so it seemed, the game had become clear to him. (LW I 873) Then comes the question quoted earlier: how can the language game suddenly become clear to the child? Wittgenstein responds: God only knows—One day it starts doing something. An analogue might be the child learning a board game which he sees played daily. (LW I 873) Wittgenstein does not say that the reason why the practice suddenly becomes clear to the child is that he suddenly sees things in the right way. On the contrary, there is in his view no informative answer to the question, how the practice suddenly becomes clear. At least, there is no informative answer

38 William Child at the level of personal-level, common-sense psychology, which is, for him, the level at which philosophy operates. The fact is that the practice does suddenly become clear to the child: at one point, he has not grasped it; at another point, he has. That he has grasped it consists in his ability to do the right thing: to use ‘She’s in pain’, ‘I believe she’s in pain’, ‘I doubt whether she’s in pain’, and so on appropriately. And that is all we can say. We cannot explain how the practice becomes clear to the child: how he suddenly comes to master the language-game; how he moves from not understanding to understanding. As Wittgenstein puts it in another context, ‘there is no how’ (see RPP I 428). Of course, there is in principle an account to be given of what went on in the child’s brain and nervous system when he acquired the new ability. But, in Wittgenstein’s view, such an account cannot give us the sort of explanation we wanted when we asked how a practice can suddenly become clear. The remark from LW I 873 concerns a particular case. But the view Wittgenstein expresses—that there is no informative answer to the question, how a practice suddenly becomes clear to someone—is quite general. In particular, Wittgenstein does not suggest that we can explain someone’s suddenly grasping a practice by appealing to the fact that she suddenly saw things in a new way. That might seem puzzling. For doesn’t Wittgenstein himself use the vocabulary of seeing in connection with concept-acquisition: specifically, in connection with grasping a rule? In teaching someone a rule, he says, we start by explaining how the rule is to be applied. If that explanation is not understood, we can switch to another, or supplement the first explanation with a second. But at some point, we can give no further explanation. And at that point, all we can say is ‘Don’t you see. . . ?’ (see e.g. PI §185, Z 302). The person who has grasped the rule sees how to go on; she sees what the rule requires at each successive step. The person who does not grasp the rule, by contrast, does not see how to go on; she does not see the regularity or pattern involved in the correct applications of the rule. But in that case, we might think, we surely can appeal to facts about how someone sees things in order to explain her grasp of a rule. Why did Anne grasp the rule for developing the series 2, 4, 6, 8 . . . while Bob did not? Because she saw the pattern in the series while he did not; she saw 1000, 1002, 1004 . . . as the correct continuation of the series, while he did not. So, at least, one might be tempted to think. In Wittgenstein’s view, however, the putative explanations are illusory. The fact that Anne sees 1000, 1002, 1004 .  .  . as the correct continuation of the series is not something that explains her grasp of the rule for developing the series; it is part of what her grasp of the rule consists in. Similarly, Bob’s failure to see the pattern in the series is not something else about him, distinct from the fact that he does not grasp the rule for developing the series. On the contrary, his failure to see the successive steps of the series as the correct continuation of the series just is his failure to grasp

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the rule for developing the series; so we cannot use the former to explain the latter. What about the broader suggestion that seeing-as helps to explain creativity or innovation: that the conceptual or theoretical innovator’s achievement in devising her new concepts or theory can be explained in part by reference to the fact that she sees things in a new way? The assessment of that suggestion depends on what the new way is in which the innovator sees things. We can distinguish two kinds of case. In cases of the first kind, seeing things in the new way is simply a matter of seeing them in terms of the new concepts or theory. In cases of the second kind, the new way of seeing is independent of grasp of the new concepts or theory. Seeing-as can help to explain innovation in the second kind of case, but not in the first. Suppose we accept that, in coming to grasp a new system of concepts or a new theory, one thereby comes to see things in a new way. On this view, seeing things in the new way is seeing them in terms of the new concepts or theory. But in that case, the fact that someone sees things in this new way cannot explain how she formed, or was able to form, her new concepts or theory; on the contrary, her seeing things in the new way presupposes her grasp of the new concepts or theory. Conceptual or theoretical innovation may involve seeing things in a new way; but the innovator’s achievement is not explained by her seeing things that way. In the second kind of case, by contrast, one can see things in the new way without already grasping the new concepts or theory. Suppose someone sees a new pattern in the phenomena; or she sees similarities between different cases that she had not previously noticed. That prompts her to look for something that explains the pattern. The person devises her new theory to provide such an explanation. In a case of that sort, the innovator’s new way of seeing things really can help to explain the formation of the new theory. What makes room for explanation is the fact that seeing the pattern was distinct from grasping the theory; after all, she saw the pattern before formulating the theory. For example, suppose an experimenter suddenly sees the pulses of electromagnetic radiation emanating from a particular point in the sky in a new way; she sees them as resembling the flashes of a lighthouse. Seeing the pulses in that way suggests a particular theory about their source; the pulses, she hypothesizes, are produced by the rotation of a star that emits a constant beam of radiation, just as the flashes of a lighthouse are produced by the rotation of a lamp that emits a constant beam of light. But the new way in which she saw the pulses—as resembling the flashes of a lighthouse—was distinct from the theory she subsequently devised to explain the phenomenon; she saw the pulses in that way before she devised the theory. In that circumstance, the person’s seeing the pulses as she did really does helps to explain her formulation of the new theory. We asked whether seeing-as can explain conceptual or theoretical innovation. The discussion of this section has yielded the following points. First,

40 William Child Wittgenstein himself does not appeal to seeing-as to explain the acquisition of new concepts or the mastery of new practices. In his view, philosophy cannot explain how we make the transition involved in coming to master new concepts or practices. Second, the idea that the innovator’s success in discovering new theories is explained by the fact that she sees things in new or unorthodox ways can be developed in two different ways. In one version, the idea should be rejected; if the new way of seeing itself involves the new theory, it cannot explain the development of the theory. In a different version, the idea contains an important truth: in cases where the new way of seeing is independent of the new theory, it may help to explain the development of the theory; but even in such cases—if Wittgenstein is right—there is bound to remain something basic or inexplicable in the innovator’s act of creation.

4. Is Seeing-As Essential to Understanding? In the previous sections we have been supposing, for the sake of argument, that when someone comes to grasp new concepts or a new theory she comes to see things in a new way. But is that supposition true? Couldn’t someone grasp the new concepts or theory without any difference at all in the way she sees things? After all, you grasp a word or a concept if you use it in the right way: or, at least, if you are a sufficiently fluent participant in the practice of using it.12 That is a matter of employing it appropriately, responding appropriately to its use by others, and so on. If you do that, you grasp the word or concept. The way you experience things seems irrelevant. Saying that someone who grasps a new theory sees things in a new way may be a good metaphor. But is there any reason to think that it is literally true? Wittgenstein raises a closely related point. He writes: The importance of [the concept of aspect-blindness] lies in the connection between the concepts of seeing an aspect and of experiencing the meaning of a word. For we want to ask, ‘What would someone be missing if he did not experience the meaning of a word?’ What would someone be missing, who, for example, did not understand the request to pronounce the word ‘till’ and to mean it as a verb— or someone who did not feel that a word lost its meaning for him and became a mere sound if it was repeated ten times over? (PPF §261) And similarly: When I supposed the case of a ‘meaning-blind’ man, this was because the experience of meaning seems to have no importance in the use of language. And so because it looks as if the meaning-blind could not lose much. (RPP I 202)13

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Wittgenstein’s comments focus on experiences of a particular kind, which he calls ‘experiences of meaning’. But the question he is asking can be generalized: if understanding a word is a matter of grasping its use, how can the way we experience things have any essential connection with whether or not we understand the word? And why should grasping a new word or concept have any important connection with coming to experience things in a new way? One response to those questions is to accept that the talk of seeing in this context is simply a façon de parler; acquiring new concepts or theories does not literally involve a change in the way we see things. We find it natural to say that the person who grasps a new theory or system of concepts sees things in a new way. But we could equally well, and more accurately, say that the person with a new theory or system of concepts conceives or understands or describes things in a new way. The natural association between grasping new concepts and seeing things in a new way is, on this view, an example of the general way in which our talk of thought, belief, and understanding is suffused with visual metaphors: ‘I don’t see what you mean’, ‘She saw the solution’, ‘He was blind to the implications’, and so on. In each case, it is claimed, the phenomena in question are essentially cognitive, not experiential. How should we react to that suggestion? One possibility is to concede the basic point—that there is no general connection between grasping new concepts and experiencing things in a new way—but to insist that in some cases the person who grasps new concepts really does thereby come to see things in a new way. There is, as Wittgenstein points out, a difference between, on the one hand, merely knowing that something is an F, or interpreting it as an F, and, on the other hand, seeing it as an F. And though in some cases the person who acquires new concepts merely comes to conceive or describe things in terms of the new concepts, in other cases she does more than that; in a perfectly literal, non-metaphorical sense, she comes to see things as falling under those concepts. In short, if it is a mistake to say that acquiring new concepts always involves seeing things in a new way, it is equally mistaken to say that it never involves a new way of seeing. That is an important corrective to the suggestion of the previous paragraph. But does Wittgenstein’s treatment of seeing-as suggest any stronger response? I explore two lines of thought that might be developed from Wittgenstein’s discussion. The first line of thought adds modal force to the point just made: it is not just that a person who grasps a system of concepts may experience things in terms of those concepts; for every person, there must be some concepts for which that is true. The second line of thought focuses on what it takes for someone to understand a concept and argues that we can distinguish different standards, or notions, of understanding. By one standard, we can agree, someone who applies a set of concepts appropriately without seeing things in terms of those concepts counts as understanding the concepts in question; by another standard, however, she does not. We can consider these two lines of thought in turn.

42 William Child Consider, first, the case of linguistic understanding. It is a fundamental insight of Wittgenstein’s that there must be some cases in which one understands words or utterances immediately, without interpretation (see PI §201). Understanding a word or an utterance can sometimes require an act of interpretation: as when I have to consult a dictionary in order to understand some word or phrase in Chaucer’s Canterbury Tales. But it could not be the case that all understanding involved interpretation in that sense. If I could not understand the dictionary’s definitions of Chaucer’s words without reference to a second dictionary, and I couldn’t understand the second dictionary’s definitions without reference to a third dictionary, and so on, then I could never understand Chaucer at all. According to the first line of thought, a parallel argument shows that there must be some cases of seeing-as: cases in which one sees an object as falling under a given concept. Identifying something as a such-and-such sometimes requires an act of interpretation: as when I work out that an unfamiliar object is a bottle-opener without seeing it that way. I detect various properties of the object; I interpret the object as a bottle-opener on the basis of its possession of those other properties. But it could not be the case that all concept application worked like that. Suppose that the only way to identify an object as an F was to interpret it as an F on the basis of its being G, and the only way to identify the object as being G was to interpret it as being G on the basis of its being H, and so on. Then we could never identify objects as falling under concepts at all. So there must be some cases in which one simply sees an object as falling under a concept rather than merely interpreting the object that way.14 To the best of my knowledge, Wittgenstein does not present that argument explicitly in his discussion of seeing and seeing-as. But he does explicitly argue that seeing-as cannot be understood in terms of interpretation (see PPF §§248–9). And the argument just sketched is certainly suggested by Wittgenstein’s argument about linguistic understanding and interpretation. Wittgenstein argues that there is a way of grasping a rule that is not an interpretation (PI §201). Similarly, on the present argument, there is a way of identifying an object as falling under a concept that is not an interpretation: namely, seeing it as falling under the concept. The second line of thought focuses on the idea of understanding. We began this section with a question: If understanding a concept is a matter of using it appropriately, how can the way someone experiences things be relevant to whether or not they understand? One response to that question is to ask whether it is true that understanding a concept requires no more than using it appropriately. At least, even if there is a legitimate sense in which one understands a concept provided only that one uses it appropriately, isn’t there a different sense in which understanding a concept requires, in addition, that one should experience things in a particular way?15 We can start with a different case. In Remarks on the Philosophy of Psychology volume II, Wittgenstein raises the question, ‘Can [a human being] understand what “fearing” is without knowing fear?’ (RPP II 26). ‘Knowing’

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here is a translation of the verb kennen. So what Wittgenstein is asking is whether someone can understand what ‘fearing’ is without being acquainted with fear: without having felt fear. On one view, there is no reason at all why someone should have to have felt fear in order to understand the word ‘fear’. All that is needed is the ability to use the word ‘fear’ appropriately: to apply it to cases in appropriate ways; to make appropriate connections with other words; and so on. And one could perfectly well do that without ever having felt fear. After all, doctors and psychiatrists can successfully identify sensations and emotions that they have never experienced themselves. They plainly have concepts of those sensations and emotions. The same, it may be said, is true for the simple concept of fear. But it is possible to take a different view. If someone has never felt fear, we could say, she does not really know what fear is; she does not fully understand people’s talk of fear. But why should we say that? Wittgenstein approaches that question by considering what the person who has never felt fear is thereby unable to do: The question is: What kind of language-games can someone who is unacquainted with fear eo ipso not play? (RPP II 27; see also Z 267) He highlights a deficiency in the person’s mastery of the ‘language-game’ of ascribing fear to others: One could say, for example, that he would watch a tragedy without understanding it. And that could be explained in this way: When I see someone else in a terrible situation, even when I myself have nothing to fear, I can shudder, shudder out of sympathy. But someone who is unacquainted with fear wouldn’t do that. We are afraid along with the other person, even when we have nothing to fear; and it is this which the former cannot do. Just as I grimace when someone else is being hurt. (RPP II 27) That suggests that the ability to relate someone else’s fear to feelings with which one is oneself familiar is an essential part of understanding what she is feeling as fear. Someone who has never felt fear might learn to apply the word ‘afraid’ to others in circumstances in which they are indeed feeling fear. But she could not mean by ‘fear’ and ‘afraid’ what we mean by those words. For the meaning of those words on our lips, the suggestion goes, depends in part on our sympathetic response to others’ fear: ‘we are afraid’ along with the other person. And the person who is unacquainted with fear can have no such sympathetic response to others’ fear. One response to this suggestion is that it fails to distinguish between understanding the word ‘fear’, on the one hand, and understanding the emotion of fear, or people who feel fear, on the other. Having felt fear, on this

44 William Child view, is not a necessary condition for understanding the word ‘fear’. It may be a necessary condition for understanding the emotion or the people who experience it. But that has nothing to do with understanding the word. Wittgenstein himself sometimes puts things in just that way. Consider this case, for instance: It is . . . important as regards our considerations that one human being can be a complete enigma to another. One learns this when one comes into a strange country with entirely strange traditions; and, what is more, even though one has mastered the country’s language. One does not understand the people. (PPF §325, first emphasis added) In Wittgenstein’s example, I understand the language of these people whose lives and traditions I do not share; I know what their words mean. What I don’t understand are the people themselves: I cannot see the point or attraction of thinking and behaving as they do; I cannot make sense of them in the way that I can make sense of people whose values and traditions are closer to my own.16 Wittgenstein seems, then, to offer us two different ways of thinking about the person who has never felt fear. It is common ground that such a person lacks the kind of sympathetic response to others’ fear that we have: she does not understand what fear is like for the person who feels it. But there are different ways of describing this person’s situation. On one view, she understands the word ‘fear’ but doesn’t understand people’s experience of fear. On a different view, she does not understand the word ‘fear’ precisely because she has no sympathetic understanding of those who are afraid. It is natural to ask which of those views is correct. Wittgenstein’s position, however, is that there is no need to choose between these two positions. There is, in his view, no hard-and-fast distinction between understanding words, on the one hand, and understanding people, on the other. There are numerous different aspects, or strands, of understanding, and correspondingly many different standards of what it takes to understand a use of language. Those different standards are equally legitimate. No one standard is more basic or theoretically significant than the others. It is just part of the concept of understanding that it has these different aspects, or strands. He writes: We speak of understanding a sentence in the sense in which it can be replaced by another which says the same; but also in the sense in which it cannot be replaced by any other. (Any more than one musical theme can be replaced by another.) In the one case, the thought in the sentence is what is common to different sentences; in the other, something that is expressed only by these words in these positions. (Understanding a poem.)

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Then has ‘understanding’ two different meanings here?—I would rather say that these kinds of use of ‘understanding’ make up its meaning, make up my concept of understanding. For I want to apply the word ‘understanding’ to all this. (PI §§531–2) Wittgenstein would say the same thing about the case we have been considering. There is a use of ‘understand’ on which one can understand the word ‘fear’ without knowing what fear feels like. There is another use of ‘understand’ on which one cannot. But that does not mean that the word ‘understand’ is ambiguous. Rather, these different kinds of use of ‘understand’ make up its meaning. What does this discussion of the connection between understanding the word ‘fear’ and experiencing fear suggest about the connection between grasping concepts and seeing-as? What we have seen in the case of ‘fear’ is that there is a sense in which one only understands the word ‘fear’ if one has felt fear oneself. Is there, similarly, a sense in which one only understands the concept F if one experiences things as Fs? There is a range of concepts, at least, for which it is very plausible that there is indeed such a sense of ‘understand’. That certainly seems true for a range of simple perceptible properties. Someone who is colour-blind might learn to distinguish red things on the basis of their physical properties, without ever seeing them as red. In one sense, she grasps the concept red. In another, she does not. (Alternatively: she grasps one concept of red; but she fails to grasp a different concept of red.) The same is true for many aesthetic properties. Someone might learn to distinguish minor chords from major ones on the basis of their structural characteristics, without ever hearing a chord as minor or being able to pick it out as minor on the basis of its sound. She might learn to distinguish pairs of words that rhyme from pairs that do not without ever hearing two words as rhyming. She might even learn to distinguish beautiful landscapes from dreary or ugly ones (even if only in a way that was parasitic on other people’s classifications), without a landscape ever looking beautiful to her. There is a sense in which such a person grasps the concepts minor, or rhyming, or beautiful. But there is also a sense in which she does not, or does not fully, grasp those concepts. The same is true for a range of concepts that apply to social interactions. Someone may have a theoretical understanding of what it is for a remark to be insulting or patronizing, and he may be able to classify a wide range of remarks as insulting or patronizing (or not), without ever experiencing a remark as insulting or patronizing. There is a sense in which he grasps the concepts ‘insulting’ or ‘patronizing’. But there is a sense in which he does not; just as the person who is not acquainted with fear does not ‘shudder out of sympathy’ when she sees someone else in a terrible situation, so the person who has never experienced a remark as insulting does not feel a sense of sympathetic indignation or resentment when he sees someone else being insulted.

46 William Child However, this kind of link between grasping a concept and experiencing things as falling under it does not extend to every case. Consider the concept cancerous. Oncologists and pathologists can see a cell as cancerous; most laypeople cannot. But should we say that there is a sense of ‘understand’ in which understanding the concept cancerous requires the ability to see things as cancerous? It is not clear that we should, because it is not clear why it matters that someone cannot see cells as cancerous but must rather identify cancerous cells in some less immediate way; it is not clear what that person is ‘eo ipso unable to do’. But, for present purposes, we need not determine the limits of the link between grasping a concept and experiencing things as falling under it. It is sufficient to have shown that, for some concepts at least, there is a sense of ‘understanding’ in which understanding the concept requires the ability to experience things in terms of that concept. That provides one answer to Wittgenstein’s question, why the way someone experiences things should matter for her understanding of words and concepts; in some cases, at least, seeing-as really is essential to understanding. We began the current section with a question: is it true that someone who comes to grasp a new theory or system of concepts thereby comes to see things in a new way? If understanding a concept is a matter of using it appropriately, why should there be any essential connection between one’s grasp of concepts and the way one experiences the world? We have explored two responses to that question, developed from strands in Wittgenstein. The first response was that, for every thinker, there must be some concepts that figure directly in the content of her experience; it could not be true that applying concepts always required interpretation. The second response distinguished different notions of understanding and argued that there is a sense of ‘understand’ in which more is required for understanding a concept than simply applying it appropriately; one must also enjoy appropriate experiences. Earlier in the chapter, I quoted Wittgenstein saying that, when he gives his examples of seeing-as and aspect-perception, he is ‘not aiming at some kind of completeness’; nor is he aiming at ‘a classification of psychological concepts’; his examples ‘are only meant to enable the reader to cope with conceptual unclarities’ (PPF §202). The point of his discussion is not to advance a comprehensive or explanatory theory of seeing-as: it is to explore the phenomena in a way that will show by example how to achieve the kind of ‘overview’ or ‘surveyable representation’ (PPF §122) that brings philosophical understanding. The current chapter is a contribution, in that spirit, to understanding the connections between seeing-as and novelty.17

Notes 1. See Kuhn (1962: ch. 10). 2. Margaret Boden makes similar comments (though without the specific focus on seeing-as), distinguishing between historical creativity and psychological

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3.

4. 5.

6. 7. 8. 9.

10.

11. 12. 13. 14. 15.

16.

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creativity and observing that psychological creativity is the philosophically more interesting phenomenon (see Boden 2004: 2). There are important views of perception on which the character of a subject’s perceptual experience is completely independent of the concepts, if any, that he or she possesses. On some views, the representational content of experience is entirely non-conceptual. In other views, experience itself has no representational content at all (see e.g. Campbell 2002: ch. 6; Travis 2004; Brewer 2011: ch. 5). These views certainly require discussion. For present purposes, however, I simply assume that seeing-as involves experiences with conceptual content—in order to focus on the special issues concerning novelty. For noticing an aspect, see PPF §113; for experiencing a change of aspects, see, for example, PPF §§129, 130, 135, 152, 257; for the dawning or lighting up of an aspect, see, for example, PPF §§118, 140, 207, 237, 247. See, for instance, Mulhall (2001: 255): ‘On my account, [experiences of aspect dawning] constitute only one striking manifestation of continuous aspect perception; and it is this concept—and the general attitude it characterizes—which is Wittgenstein’s real concern.’ See, for example, Glock (1996: 36ff.) and Johnston (1993: 243–244). For this case, see PPF §242. See, for example, PPF §§143–44 and 245. For more on this point, see Child (2011: 185–187). I owe this point to Naomi Eilan, who discussed it in her presentation ‘AspectSeeing and Thinking’, at the ‘Seeing-As and Novelty’ conference at York in 2012. Eilan drew attention in particular to the developmental findings reported in Rock et al. (1994), Doherty and Wimmer (2005), and Wimmer and Doherty (2011). Stephen Mulhall takes the contrary view: ‘Someone who cannot, for example, see a schematic drawing of a cube first one way then another is someone who cannot continuously see the schematic drawing as a cube either way—she could not stand to such a picture of a three-dimensional object as she does to that object itself’ (Mulhall 2001: 254). The German has ‘erkennen’ which, like the English ‘recognize’, can be used in both a perceptual and a non-perceptual, merely cognitive way. For the ‘sufficiently fluent participant’ formulation, see Williamson (2007: ch. 4). See also Wittgenstein’s remark about a person who says that to him the words if and but feel the same: ‘If he used the words “if” and “but” as we do, wouldn’t we think he understood them as we do?’ (PPF §40). The same caveat that was mentioned in note 3 above is needed here; a full defence of this argument would need to consider views of perception on which perceptual experience does not have conceptual content at all. An alternative way of developing essentially the same idea would set things up differently. On this alternative view, understanding a concept never requires more than using it appropriately; but using the concept appropriately may require that one experiences things in a particular way. Compare LW I 198: ‘He is incomprehensible to me means that I cannot relate to him as to others.’ Wittgenstein gives an example: ‘he gets angry, when we see no reason for it; what excites us leaves him unmoved.—Is the essential difference that we can’t foresee his reactions?—Couldn’t it be that after some experience we might know them, but still not be able to follow him?’ (LW I 192) For related comments, see RPP II 568. An earlier version of this chapter was presented at the conference on ‘Seeing-as and Novelty’ at York in 2012. I am grateful to the participants in that conference, and especially to Michael Beaney, for their very helpful comments.

48 William Child

Bibliography Works by Wittgenstein LW I. (1982). Last Writings on the Philosophy of Psychology, Vol. I (eds. G. H. von Wright and H. Nyman, trans. C. Luckhardt and M. Aue). Oxford: Blackwell. PI. (2009). Philosophical Investigations, 4th edition (eds. P. M. S. Hacker and J. Schulte, trans. G. E. M. Anscombe, P. M. S. Hacker, and J. Schulte). Oxford: Blackwell. PPF. Philosophy of Psychology: A Fragment. In Philosophical Investigations, 4th edition. (In previous editions, this material appears as Philosophical Investigations, Part II. References in the text are given by the § numbers used in the 4th edition. For a table of correspondences between these § numbers and the page numbers of earlier editions, see Day and Krebs 2010). RPP I. (1980). Remarks on the Philosophy of Psychology, Vol. I (eds. G. E. M. Anscombe and G. H. von Wright, trans. G. E. M. Anscombe). Oxford: Blackwell. RPP II. (1980). Remarks on the Philosophy of Psychology, Vol. II (eds. G. H. von Wright and H. Nyman, trans. C. Luckhardt and M. Aue). Oxford: Blackwell. Z. (1981). Zettel, 2nd edition (eds. G. E. M. Anscombe and G. H. von Wright, trans. G. E. M. Anscombe). Oxford: Blackwell.

Other Works Boden, M. (2004). The Creative Mind: Myths and Mechanisms, 2nd edition. Abingdon: Routledge. Brewer, B. (2011). Perception and Its Objects. Oxford: Oxford University Press. Campbell, J. (2002). Reference and Consciousness. Oxford: Oxford University Press. Child, W. (2011). Wittgenstein. London: Routledge. Day, W. and Krebs, V. (2010). Seeing Wittgenstein Anew. Cambridge: Cambridge University Press. Doherty, M. J. and Wimmer, M. C. (2005). Children’s understanding of ambiguous figures: Which cognitive developments are necessary to experience reversal? Cognitive Development 20: 407–421. Glock, H.-J. (1996). A Wittgenstein Dictionary. Oxford: Blackwell. Johnston, P. (1993). Wittgenstein: Rethinking the Inner. London: Routledge. Kuhn, T. (1962). The Structure of Scientific Revolutions. Chicago: University of Chicago Press. Mulhall, S. (2001). Seeing Aspects. In H.-J. Glock (ed.), Wittgenstein: A Critical Reader. Oxford: Blackwell. Rock, I., Gopnik, A., and Hall, S. (1994). Do young children reverse ambiguous figures? Perception 23: 635–644. Travis, C. (2004). The Silence of the Senses. Mind 113: 57–94. Williamson, T. (2007). The Philosophy of Philosophy. Oxford: Oxford University Press. Wimmer, M. C. and Doherty, M. J. (2011). The development of ambiguous figure perception. Monographs of the Society for Research in Child Development 76(1): 1–130.

3

Gombrich and the Duck–Rabbit Robert Briscoe

An oil painting caught and held him. . . . There was beauty, and it drew him irresistibly. He forgot his awkward walk and came closer to the painting, very close. The beauty faded out of the canvas. His face expressed his bepuzzlement. He stared at what seemed a careless daub of paint, then stepped away. Immediately all the beauty flashed back into the canvas. ‘A trick picture’, was his thought . . . —Jack London, Martin Eden (1909/1982)

1. Introduction A picture is a marked or otherwise patterned, two-dimensional (2D) surface that, when present to sight, elicits the experience as of an absent, threedimensionally (3D) organized scene.1 How ought we to understand the nature of this experience? There are a number of distinct suggestions in the philosophical literature: A. Pictures elicit a 3D-scene-representing experience of the same psychological kind as the experience of seeing face-to-face (Gombrich 1961/2000, 1972, 1982; Briscoe 2016). Since the represented, 3D scene is absent from sight, however, this experience is non-veridical. B. When we look at a picture, we enjoy an experience as of the depicted, 3D scene. This experience, however, is always fused with awareness of the superficial pattern on the pictorial surface (Wollheim 1987, 1998, 2003). In this respect, pictorial experience has two different dimensions or ‘folds’ of representational content. Accounts (A) and (B) maintain that the appearance of depth and 3D structure ‘beyond’ the 2D pictorial surface is essential to pictorial experience. When we look at a suitably patterned surface, Richard Wollheim writes, we are typically aware of ‘something in front of or behind something else’ (1998: 221). As John Kulvicki puts it, ‘there is a strong sense in which depicted scenes seem to recede from the canvas’ (2009: 391). Other prominent accounts, by contrast, attach theoretical priority to seeing properties

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of the pattern or design visible on the pictorial surface. They agree with Malcolm Budd that the way to capture the experience of seeing a picture as a depiction of its subject is not by adding any other visual experience to the visual awareness of the picture-surface—either as a separate experience or by fusing it with the visual awareness of the picture-surface—but by specifying the nature of the visual awareness of the picture-surface when you see what the picture depicts. (1992/2008: 203–204) There are two familiar approaches to characterizing the nature of this surface-awareness: C. To experience a 3D scene in a picture’s surface is to see the latter as resembling the former in certain respects (Budd 1992/2008, 1993/2008; Hopkins 1998, 2006). For example, to experience a cubical object oriented in depth when looking at a drawing of the Necker cube is to see the pattern of lines that make up the drawing as resembling such an object. D. Pictures function as props in ‘visual games of make-believe’ (Walton 1990, 2008). In looking at a picture, the viewer imagines of her experience that it is a seeing of whatever the picture portrays. ‘In the case of picture perception, not only does looking at the picture induce us to imagine seeing an ox, we also imagine our actual visual experience, our perceiving the relevant part of the canvas, to be an experience of seeing an ox’ (Walton 2008: 118).2 In the philosophy of art, E. H. Gombrich is by far the best-known proponent of option (A), according to which pictorial experience is psychologically continuous with the experience of seeing face-to-face. Pictures, Gombrich says, have the power to arouse in us a ‘visual experience of a kind that we know from our encounters with reality’ (1982: 181). In what follows, I will refer to this view as the Continuity Hypothesis. The Continuity Hypothesis comprises three main claims. First, a picture is a patterned, 2D surface designed to elicit a non-veridical experience as of depth and 3D structure—‘something akin’, as Gombrich puts it, ‘to a visual hallucination’ (1972: 208). Crucially, this hallucination-like experience represents its intentional objects as located on the far side, as it were, of the patterned, pictorial surface. Gombrich, as we have already seen, is not alone in characterizing the spatial phenomenal character of pictorial experience in this way. ‘The first intention of the painter’, Leonardo da Vinci tells us in his Treatise on Painting, ‘is to make a flat surface display a body as if modeled and separate from this plane’ (quoted by Kemp 1989: 15). A picture, as J. J. Gibson puts it, ‘is both a scene and a surface, and the scene is paradoxically behind the surface’ (1979: 281). When we look at The Peasant Wedding (Figure 3.1), for example, we do not merely perceive an array of colours

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Figure 3.1 Pieter Bruegel the Elder, The Peasant Wedding (1567) Oil on panel, 124 cm × 164 cm, Vienna, Kunsthistorisches Museum.

located on a single plane of depth. We also experience a complexly organized, 3D scene in which voluminous objects participate in a ‘recessional movement’, to borrow Wölfflin’s (1929) phrase, from the pictorial point of view. Following standard usage in art history, aesthetics, and perceptual psychology, I will refer to the virtual, 3D space in which we visually experience shapes, sizes, colours, textures, orientations, and other features when we look at a picture as pictorial space (Wölfflin 1929; White 1967; Pirenne 1970; Kubovy 1986; Rogers 1995, 2003; Koenderink 1998, 2012; Hecht et al. 2003; Thompson et al. 2011: ch. 12). The second claim is that phenomenological and representational continuities between pictorial experience and seeing face-to-face reflect their underlying, psychological continuity. Both experiences result, Gombrich says, from an unconscious, inference-like process of ‘guided projection’. On the one hand, the process is said to be guided because it relies on certain nonconscious expectations and assumptions about the structure of the visible environment in order to interpret the message conveyed by the retinal image. These assumptions perform two closely related functions: they are used by the perceptual system to generate a ‘hypothesis’ about the most probable cause of the image in the distal environment and, in addition, to test that hypothesis by generating predictions concerning, among other things, the sensory effects of bodily movement: ‘every message sets up a set

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of expectations with which the incoming flow can be matched to confirm correct assessments or to modify and knock out false guesses’ (Gombrich 1978: 158; see also 1961/2000: 274–275). Gombrich refers to the contribution of these background assumptions to perception as the ‘beholder’s share’. On the other hand, the process is said to involve projection because its conscious end product is underdetermined by the structure of the light sampled by the eye. A given retinal image could be caused by the light reflected from (or emitted by) many different 3D scenes. (This is the so-called inverse optics problem in vision science.) Contemporary Bayesian ‘predictive coding’ models of visual processing replace talk of implicit assumptions with talk of prior probabilities and likelihoods (for discussion, see Hohwy 2013; Clark 2015; Rescorla 2015), but in other key respects, they bear strong affinities to the Gombrichian account. On both approaches, perception is a process . . . in which we (or rather, various parts of our brains), try to guess what is out there, using the incoming sensory signal more as a means of tuning and nuancing the guessing rather than as a rich . . . encoding of the state of the world. (Clark 2015: 27) The following passage from Art and Illusion so perfectly captures the kinship between the Gombrichian account of perception and the more recent predictive coding framework that it is worth quoting at length: The experience of the radio ‘monitor’ confronted with indistinct speech and that of the sailor confronted with indistinct shapes on the horizon are not incommensurate. We must always rely on guesses, on the assessment of probabilities, and on subsequent tests, and in this there is an even transition from the reading of the symbolic material to our reaction in real life. When we wait at the bus stop and hope the Number Two is coming into sight, we probe the indistinct blot that appears in the distance for the possibility of projecting the number ‘two’ into it. When we are successful in this projection, we say we now see the number. This is a case of symbol reading. But is it different with the bus itself? Certainly not on a foggy night. Nor even in full daylight, if the distance is sufficiently great. Every time we scan the distance we somehow compare our expectation, our projection, with the incoming message. If we are too keyed up, as is well known, the slightest stimulus will produce an illusion. Here as always it remains our task to keep our guesses flexible, to revise them if reality appears to contradict, and to try again for a hypothesis that might fit the data. But it is always we who send out these tentacles into the world around us, who grope and probe, ready to withdraw our feelers for a new test. (Gombrich 1961/2000: 178–179)

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Like the predictive coding approach, the Gombrichian account of perception turns a traditional, input-dominated view of how vision works on its head. Visual experience is not a passive imprinting of the world on the mind but, rather, an active, hypothesis-generating, and hypothesis-testing perceptual process: ‘The world never presents a neutral picture to us; to become aware of it means to become aware of possible situations that we can try out to test for their validity’ (1961/2000: 275). The central role of prediction and projection in perception is masked, Gombrich suggests, by the rapid, automatic, and seemingly effortless way in which the 3D structure of a visible scene is usually revealed to us. This point is illustrated in a discussion of a natural history engraving of some plants, insects, and animals displayed on an unstructured, white background: Looking at Jacob Hoefnagel’s plate . . . , we always supply the appropriate ground to the figure: the lizard sits on a slope, while some insects, throwing shadows, are imagined against a flat ground and others are seen as flying. Without knowing it, we have carried out a rapid succession of tests for consistency and settled on those readings which make sense. (1961/2000: 231) The pictorial arts, as this passage makes clear, provide an especially useful arena for probing the role of the beholder’s share in perception. The information in the light available to the eye, as Gombrich emphasizes, is ‘immeasurably richer .  .  . when we move around in the real world’ (1961/2000: 274) than when viewing an etching, drawing, or painting of a 3D scene. In consequence, pictures afford special opportunities for teasing out the relative contributions of ‘top-down’ projection and the ‘bottom-up’ sensory signal to perceptual processing. The more ambiguous or incomplete the signal from the environment, the more prominent the role played by the beholder’s share in the process of visual hypothesis formation. ‘The deliberately blurred image, the sfumato, or veiled form . . .’, for example, ‘cuts down the information on a canvas and thereby stimulates the mechanism of projection’ (1961/2000: 175–176). Finally, it is central to Gombrich’s project in Art and Illusion that the ambiguity of the environment’s image on the retina ‘can never be seen as such’ (1961/2000: 249). The visual system selects only one consistent, 3D-scene interpretation of the image at a time, even when more than one such interpretation can be made to ‘fit’.3 In support of this claim, Gombrich appeals to ambiguous or ‘multistable’ figures in which figure-ground assignments, shapes, orientations, groupings, or other organizational properties appear to alternate with prolonged inspection. Whether side abcd or side efgh appears closer in depth varies from one moment to the next, when we look at a drawing of the Necker cube (Figure 3.2), but we never experience both organizations at once. The neuroscientists David Leopold and

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Figure 3.2 Ambiguous figures: the Necker cube and the duck–rabbit

Nikos Logothetis refer to this familiar property of multistable perception as ‘exclusivity’: Exclusivity, or uniqueness, ensures that conflicting visual representations are never simultaneously present [in conscious awareness]. That only a single perceptual solution can exist at once is likely to have its origins in the structure of the sensory machinery itself; that is, uniqueness is a fundamental encoding principle among neurons in the visual cortex. (Leopold and Logothetis 1999: 260) Gombrich argues that the exclusivity of visual processing—the requirement of a single, consistent, 3D scene interpretation—has important implications for understanding the nature of pictorial experience. When an observer looks at The Peasant Wedding (Figure 3.1), for example, two conflicting sets of depth cues are typically available to her visual system. At close range, binocular disparity, convergence, and accommodation specify the orientation of the canvas and its distance in depth. The structured array of light reflected from the painting’s surface, however, is also a vehicle for sources of spatial information that jointly specify the layout and properties of objects in a (virtual) three-dimensionally organized scene. These monocular or ‘pictorial’ depth cues, as they are sometimes called, include, but are not limited to, occlusion, texture gradients, shadows, reflections, relative size, linear perspective, atmospheric haze, height in the visual field, and the horizon ratio (for useful reviews, see Ames 1925; Cutting and Vishton 1995; Palmer 1999; and Thompson et al. 2011). In consequence, a picture may elicit either

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of two very different experiences: an experience that attributes properties to the patterned, 2D pictorial surface or an experience that attributes properties to objects in phenomenally 3D pictorial space. We can look at The Peasant Wedding, or we can look into it. Gombrich argues that while it is possible to alternate or ‘switch’ between these different experiences, it isn’t possible to enjoy both of them at once: But is it possible to ‘see’ both the plane surface and the battle horse at the same time? If we have been right so far, the demand is for the impossible. To understand the battle horse is for a moment to disregard the plane surface. We cannot have it both ways. (1961/2000: 279) In this respect, Gombrich suggests, there is an analogy with the experience of looking at an ambiguous figure. Just as noticing the duck aspect in a drawing of the duck–rabbit (Figure 3.2) excludes noticing the rabbit aspect at the same time, visually experiencing the pattern on an opaque, pictorial surface excludes visually experiencing the way objects are arranged ‘beyond’ that surface in pictorial space. In both cases, Gombrich suggests, ‘We are not aware of the ambiguity as such, but only the various interpretations. . . . We can train ourselves to switch more rapidly, indeed to oscillate between readings, but we cannot hold conflicting interpretations’ (1961/2000: 236). The now standard objection to this analogy is that while the two interpretations in the first case (duck vs rabbit) do indeed genuinely conflict, the two interpretations in the second case (2D surface vs 3D scene) are merely different. Richard Wollheim complains: But by what right does Gombrich assume that we can no more see a picture as canvas and as nature, than we can see the duck-rabbit figure as a duck and as a rabbit? Because—it might be said—canvas and nature are different interpretations. But if this is Gombrich’s argument, it is clearly invalid. For we cannot see the duck-rabbit figure as duck and as rabbit, not because these are two different interpretations, but because they are two incompatible interpretations. (Wollheim 1963: 29) A similar objection has been voiced by Dominic Lopes: the duck–rabbit figure undermines Gombrich’s intended use of it. . . . Switches between the two contents (as of duck and rabbit respectively) are not analogous to switches between the figure’s design, on the one hand, and either of its contents, on the other. That duck cannot be seen simultaneously with rabbit fails to show that duck cannot be seen simultaneously with design or design simultaneously with rabbit. (Lopes 2005: 31)

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My primary purpose in the remainder of this chapter is to clarify, develop, and defend the Gombrichian account of pictorial experience, drawing on resources from contemporary vision science. Section 2 provides motivation for Gombrich’s use of the duck–rabbit analogy by situating it in the context of his attack on a historically influential conception of the appearance–reality distinction in visual perception. According to the conception in question, everyday visual experience comprises two distinct ‘layers’ or ‘folds’ of representational content. The content of the first layer represents an object’s viewpoint-dependent appearance or ‘look’, where this is supposed to be phenomenally flat or 2D in character. The content of the second layer, by contrast, is supposed to represent the object’s intrinsic, viewpointindependent properties as well as its distance in depth. Pictorial perception is arguably the best case for this ‘dual content’ theory of visual experience, since a picture really is a flat patchwork of colours that, when present to sight, elicits the impression of depth and 3D structure. One of Gombrich’s main aims in Art and Illusion is to demonstrate that the theory fails even as an account of pictorial experience. The dual content theory of pictorial experience is seemingly paradoxical. It maintains that when we look at a picture, the same solid angle in the visual field is represented twice over—once as filled by an opaque, twodimensionally organized surface and once as containing a three-dimensionally organized scene. Section 3 argues that two of the better-known attempts to eliminate the appearance of paradox at the heart of the dual content theory of pictorial experience are unsuccessful. The two putative layers of representational content in pictorial experience, it concludes, contrary to Wollheim, aren’t merely different. As Gombrich insists, they are also incompatible. Section 4 defends and constructively elaborates on the Continuity Hypothesis. In particular, it argues for an account of the structure of pictorial experience that I refer to as weak onefoldness. Pictorial experience is onefold in the sense that its content reflects a single, consistent, 3D-scene interpretation of the retinal image. Pictorial experience is only weakly onefold, however, in that it typically attributes certain combinations of properties to the 2D pictorial surface and to objects in phenomenally 3D pictorial space at the same time. Having the experience of virtual depth and 3D structure, when looking at a picture, I argue, excludes representing some, but not all, of a picture’s surface properties. A second aim of section 4 is to reconcile the claim that pictorial experience and seeing face-to-face are psychologically continuous with the observation that the former experience does not typically dispose the perceiver to believe that its objects are real. A recent account of stereopsis from Dhanraj Vishwanath (2014), I propose, makes such reconciliation possible. According to Vishwanath, pictorial experience does not dispose the perceiver to believe that its objects are present to sight because, in contrast with ordinary, non-pictorial visual experience, it fails to specify their locations at certain absolutely scaled distances in depth. This contrast, I argue, however,

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is best understood as a difference at the level of representational content rather than a difference at the level of psychological kind.

2. Pictorial Experience and the Denial of Dual Awareness Gombrich, as we have seen, denies that it is psychologically possible to experience the properties of the pattern visible on a 2D pictorial surface, while simultaneously experiencing depth and 3D structure in that surface. These two interpretations or ‘readings’ both fit the configuration of light intensities on the retina but are mutually exclusive. This view is central to argument of Art and Illusion. Gombrich goes so far as to identify our inability to experience the ambiguity of the incoming retinal signal as the ‘theme song’ of the book (1961/2000: 313). Gombrich, it is important to emphasize, doesn’t just reject a dual content theory of pictorial experience. He also rejects of a dual content theory of ordinary, non-pictorial seeing. The primary target of Gombrich’s criticism in Art and Illusion is, in fact, a historically influential conception of the appearance-reality distinction in everyday visual perception and its application to the special case of perceiving pictures. According to the conception in question, the representational content of visual experience divides into two ‘layers’. The first, putative layer of visual content represents the viewpoint-dependent appearance or ‘look’ of a perceived object, where this is supposed to be phenomenally 2D in character, much like a flat projection of the object’s shape on the frontal plane. In addition, the first layer of content is frequently alleged to be phenomenologically, epistemically, and/or developmentally more basic than the second layer.4 ‘What we really see’, according to one prominent version of the dual content theory, ‘is a medley of colored patches such as Turner paints’ (Gombrich 1961/2000: 296). The 19th-century art critic John Ruskin provides an especially clear expression of this outlook in The Elements of Drawing: The perception of solid Form is entirely a matter of experience. We see nothing but flat colours.  .  .  . The whole technical power of painting depends on our recovery of what might be called the innocence of the eye; that is to say, of a sort of childish perception of these flat stains of colour, merely as such, without consciousness of what they signify—as a blind man would see them if suddenly gifted with sight. (Ruskin 1856/1971: 27)5 The second, putative layer of visual content, by contrast, is supposed to represent an object’s intrinsic, viewpoint-independent properties as well as its distance and orientation in depth. It results from the application of perceptual constancy mechanisms, spatial organizational principles, and learned associations to the content of the first layer. Gombrich variously associates versions of this dual content theory of visual experience with the British

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Empiricist tradition in philosophy, the introspectionist movement in psychology, and Impressionism’s ‘discovery of appearances’ (Fry 1934). The dual content of theory of visual experience is not just a historical curiosity. Contemporary advocates in philosophy include William Lycan (1996, 2008), Jonathan Cohen (2010), and Berit Brogaard (2012). Perhaps the most influential recent proponent of the theory, however, is Alva Noë (2004, 2005): Perceptual constancy—size and shape constancy—coexists with perspectival nonconstancy. Two tomatoes, at different distances from us, may visibly differ in their apparent size even as we plainly see their sameness of size; a silver dollar may look elliptical—when we view it from an angle, or when it is tilted in respect of us—even though it also looks, plainly, circular. Perceptual experience presents us with the world (the constancies) and it presents us with how the world perceptually seems to be (the nonconstancies). A satisfying account of perception must explain how the silver dollar can look both circular and elliptical, how the tomatoes can look to be the same in size and yet different in size. Perceptual experience is two-dimensional, and this needs explaining. (Noë 2005: 235) Noë identifies an object’s non-constant, visually apparent shape with the shape of the patch that would perfectly occlude the object on a plane perpendicular to the line of sight. He refers to this as the object’s ‘perspectival shape’ (P-shape). An object’s non-constant, ‘perspectival size’ (P-size), in turn, corresponds to the size of the patch that would occlude the object on the same plane. Non-constant P-properties are ‘perceptually basic’ (2004: 81), according to Noë, because in order to see an object’s constant spatial properties it is necessary both to experience its P-properties and to understand how these would undergo transformation across changes in one’s point of view. Seeing is thus a ‘two-step’ process: ‘How they (merely) appear to be plus sensorimotor knowledge gives you things as they are’ (Noë 2004: 164). Although we typically attend to the content of the second, post-constancy layer in our everyday interactions, the content of the first layer, according to dual content theorists, is supposed to remain introspectively accessible, if only we know how to look for it. ‘There is a sense . . .’, Noë writes, ‘in which we move about in a sea of perspectival properties and we are aware of them (usually without thought or notice) whenever we are perceptually conscious. Indeed, to be perceptually conscious is to be aware of them’ (Noë 2004: 167). We normally attend to the circular shape of an obliquely viewed coin, but this doesn’t mean that it ceases to look elliptical when we do. As paradoxical as it may sound, a silver dollar can look circular and elliptical at the same time.

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The Gestalt psychologists referred to the assumption that pre-constancy visual appearances co-exist alongside our post-constancy representations of the world as the doctrine of ‘unnoticed sensations’ (Köhler 1913/1971). In The Ecological Approach to Visual Perception, J. J. Gibson characterizes this doctrine in following way: It has been generally believed that even adults can become conscious of their visual sensations if they try. You have to take an introspective attitude, or analyze your experience into its elements, or pay attention to the data of your perception, or stare at something persistently until the meaning fades away. I once believed it myself. I suggested that the ‘visual field’ could be attended to, as distinguished from the ‘visual world’, and that it was almost a flat patchwork of colors, like a painting on a plane surface facing the eye . . . (Gibson 1979: 286) Using Gibson’s terminology, the first layer of visual representational content posited by the dual content theory represents the structure of the twodimensionally organized visual field, while the content of the second layer represents the layout of the three-dimensionally organized visual world. Gombrich, like Gibson, is deeply sceptical of the doctrine of unnoticed sensations. Perceptual constancy mechanisms, he argues, do not operate on a platform of conscious, but normally inconspicuous, 2D appearances or P-properties. The first, introspectively accessible product of sub-personal, perceptual information processing is an experience that represents the disposition and properties of objects in the 3D visual world. Gombrich writes: Presented with a circular disk, for instance, we are well aware of the fact that it might be fairly large and far away, or small and close by. We also may remember intellectually that it might be a tilted ellipse, or a number of other shapes, but we cannot possibly see these infinite possibilities; the disk will appear to us as an object out there, even though we may realize, as students of perception, that another person may guess differently. One must have experienced these effects to realize how elusive they make the idea of ‘appearance’ as distinct from the object itself. The stimulus school of psychology and the phenomenalists talked as if the ‘appearance’ of the disk, the stimulus pattern, were the only thing really ‘experienced’ while all the rest was inference, interpretation. It sounds like a plausible description of vision, but it is untrue to our actual experience. We do not observe the appearance of color patches and then proceed to interpret their meaning. . . . To see is to see ‘something out there’. (Gombrich 1961/2000: 260) It is even harder to see the world as a two-dimensional field than it is to see one’s own image on the mirror’s surface. Our belief that we can

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Ruskin’s ‘flat stains of color’ and Noë’s P-properties, on this view, are not a foundational, pre-constancy layer of conscious experience on which a further layer of post-constancy, visual representational content is constructed. Instead, perceptual processing begins with texture gradients, edge junctions, and other properties of the retinal image that are predictive of properties in the environment and, on this nonconscious evidential basis, constructs a representation of the most probable, three-dimensionally organized scene— a ‘possible configuration in space and light’ (Gombrich 1961/2000: 327). Conscious visual experience, in other words, begins only after perceptual constancy mechanisms and Gestalt organizational principles have done their work, that is, with the Gibsonian visual world as opposed to the Gibsonian visual field (Briscoe 2008). The eye is never innocent in Ruskin’s sense. The Gombrichian account, as already observed, has a close counterpart in contemporary vision science. The central challenge faced by the visual system, according to recent Bayesian models of perception (for overviews, see Knill and Richards 1996; Mamassian et al. 2002; Clark 2013, 2015; Hohwy 2013; Rescorla 2015), is to infer the most probable cause of the retinal image on the basis of two sources of ‘evidence’: (1) the various depth cues present in the image itself, as well as (2) learned or innate assumptions about the statistical properties of the natural environment and the image formation process. The content of the perceptual state formed in response to a particular pattern of retinal stimulation—the brain’s operative ‘hypothesis’ about the structure of the impinging environment—is the cause to which the highest probability is assigned given all the available endogenous and exogenous evidence. In general, this will be one of many different possible three-dimensionally organized scenes: the hypothesis space for causal inference in vision is a 3D-scene space, in which different hypotheses correspond to different possible arrays of objects at a distance from the perceiver’s eyes.6 (One such hypothesis picks out the very scene in front of the reader now.) Crucially, each interpretation of the retinal image contained in the hypothesis space for vision is a post-constancy interpretation. Interpretations in the hypothesis space for vision, this is to say, range over possible states of the Gibsonian visual world rather than Gibsonian visual field.7 To see is always to see, as Gombrich says, ‘something out there’. Neither Gombrich nor Gibson would deny, of course, that there are contexts in which an observer will experience an obliquely viewed disk as an ellipse in the frontal plane. When the disk is poorly illuminated, or far away, or seen under experimentally contrived ‘reduced cue’ conditions, such

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non-veridical perception is entirely possible. It seems clear, however, that the possibility of visual illusion under informationally impoverished viewing conditions by itself provides no support for the dual content theorist’s claims about the way the world appears to us under normal, informationally rich viewing conditions (Briscoe 2008; Hopp 2013). No more, say, than the possibility of mistaking Sarah Palin for Hillary Clinton on a dark night provides support for the claim that Palin looks like Clinton in broad daylight. For present purposes, there are three important points. First, that a disk may look elliptical in the contexts mentioned in the last paragraph does no work when it comes to motivating the claim that there is a conscious, preconstancy layer of perceptual representational content. On the contrary, the non-veridical, ellipse-in-the-frontal-plane interpretation of the retinal image is a post-constancy interpretation, in particular, the post-constancy interpretation that is most probable in light of the (meagre) information available to the visual system. The disk appears to be intrinsically elliptical in shape— full stop. There is just one layer of post-constancy, visual representational content. Second, the two interpretations, viz., disk-slanted-in-depth and ellipse-inthe-frontal-plane, are clearly incompatible. A single, opaque surface cannot look to be intrinsically circular and slanted in depth and intrinsically elliptical and at a right angle to the line of sight at the same time (or at different times, holding sources of optical information and viewing conditions fixed). Finally, far from having a developmentally, phenomenologically, or epistemically privileged status, the elliptical appearance, in this context, is simply evidence of a breakdown in perceptual constancy. Veridical shape perception is not guaranteed when the visual system operates outside of informationrich, ecologically normal viewing conditions. The real contrast between the two interpretations of the retinal image, in short, is not that only one of them is ‘innocent’ but, rather, that only one of them also is accurate. Recent empirical work on the role of depth in perceptual organization is helpful for purposes of further elucidating Gombrich’s view. Roland Fleming and Barton Anderson (2004) divide ‘legal interpretations’ of a luminance edge in the retinal image into two main classes. Interpretations in the first class represent a surface event in which both sides x and y of the edge are located at the same distance in depth from the perceiver. Examples include reflectance edges, cast shadows, and creases on a single, opaque surface. Interpretations in the second class represent the contour of a backgroundoccluding object and, hence, a difference in depth at the edge: one side of the edge, either x or y, is located at the depth of the object while the other side of the edge lies at the more distant depth of the partially occluded background (Fleming and Anderson 2004: 1287). In this case, the edge is said to be ‘owned’ by the object (Nakayama et al. 1995). In terms of this framework, Gombrich’s view is that, when looking at a picture, the visual system cannot legally interpret a given edge or set of edges in the retinal image in both ways at the same time. What the visual system

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can do, he suggests, is alternate or switch between a ‘surface event’ interpretation of an edge and an ‘object contour’ interpretation of the edge. These different interpretations do not correspond to different, simultaneously accessible layers of content in one’s experience of the picture but, rather, to two, temporally distinct experiences. An analogous Gombrichian claim holds for how we experience non-spatial properties, such as lightness (albedo) and colour, when looking at a picture. Consider the lightness illusion by Barton Anderson and Jon Winawer (2005) reproduced in Figure 3.3. In the figure, the disks on the light and dark surrounds are photometrically identical, but the disks on the light surround appear as uniformly black objects visible behind a semi-transparent, light haze, whereas the disks on the dark surround appear as uniformly white objects visible behind a semi-transparent, dark haze. This illusion, Anderson and Winawer propose, is caused by photometric and geometric relationships in the figure that modulate the (non-veridical) perception of transparency inside the disk regions: whether a given disk in the demonstration looks white or black depends on the way the visual system uses these relationships for purposes of decomposing the contrasting luminances that define the texture inside the disk into surfaces or layers at different distances in (virtual) depth. In consequence, it is not possible to experience the lightness illusion, while simultaneously experiencing the disks as textured patches on a 2D surface, that is, as co-planar with their surround. Similar remarks can be made with respect to the apparent colours of objects in pictorial space across different virtual illumination conditions. Consider a variant of Dales Purves and Beau Lotto’s Rubik’s cube colour illusion, in which one cube appears to be lit by a yellow light source and the

Figure 3.3 Lightness illusion Reproduced with permission from Anderson and Winawer (2005).

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other by a blue light source (Purves et al. 2002: 241).8 What is surprising, when we look at the image, is that although some tiles on the top side of the first cube look blue and some tiles on the top side of the second cube look yellow, the corresponding regions on the pictorial surface are physically identical. Indeed, they are precisely the same shade of grey. This is perceptually evident when a mask is superimposed upon the demonstration, covering the other tiles. The key observation, in the present context, is not just that context affects colour constancy in both physical and pictorial space (Azzouni 2013: 91), but that experiencing the colours of the tiles in phenomenally 3D pictorial space as chromatically different (yellow vs blue) excludes seeing the regions on the 2D pictorial surface in which the tiles are displayed as chromatically identical (grey). It is not possible to enjoy both experiences at the same time. The reader is invited to give it a try!9 Critics of the dual content theory have sometimes charged its adherents with over-analogizing everyday visual experience to looking at paintings, photographs, and other flat media (Gibson 1979; Smith 2000; Schwitzgebel 2006; Briscoe 2008; Snowdon 2015). Nevertheless, the best case for the dual content theory, it might be thought, is the experience of looking at a picture. For one thing, a picture really is a flat patchwork of colours that, when present to sight, elicits the experience as of depth and 3D structure. In addition, it is not clear that criticisms of the dual content theory of everyday visual experience canvassed above straightforwardly apply to a dual content theory of pictorial experience. Indeed, there are two disanalogies between the experience of looking at a voluminous object ‘in the flesh’ and the experience of looking at an image of such an object that arguably make a dual content theory of pictorial experience more promising. The first disanalogy concerns the different kinds of optical information available to the visual system when viewing pictures and real-world scenes, respectively. In most cases, sources of information concerning the layout of a real-world scene are rich and consistent enough to support the selection of a single interpretation from the hypothesis space for vision. This explains why it is not possible, holding sources of optical information fixed, to switch from a disk-slanted-in-depth interpretation of the retinal image to an ellipsein-the-frontal-plane interpretation in the above example. (In contemporary Bayesian parlance, although both interpretations or hypotheses may have roughly the same prior probability, binocular disparity, texture gradients, vergence cues, and other sources of sensory evidence make one interpretation much more likely than the other, and this means that it will be assigned a proportionately higher posterior probability.) When you look at a picture, by contrast, two conflicting sets of depth cues are typically present in the light reflected (or emitted) to your eye. The first set of cues, as already mentioned, includes sources of optical information that enable your visual system accurately to recover the properties of the opaque, 2D pictorial surface. When you look at The Peasant Wedding (Figure 3.1), for example, there is ‘surface event’ information that specifies

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the colours and elliptical shapes of the patches of paint that Bruegel used to depict some dishes of food in the bottom-right quadrant of the canvas. The second set of cues, by contrast, includes sources of monocular spatial information that support the experience as of depth and 3D structure in pictorial space. For example, when viewing Bruegel’s painting, there is information that specifies the voluminous shapes, relative sizes, and orientations of the virtual objects that are intended to resemble dishes. Since there is ample information for both of these 3D-scene-interpretations when you look at the painting, it could be argued that both interpretations may be simultaneously reflected in the content of your experience. There is a second point of disanalogy. One objection to the dual content theory, as we saw, had to do with the fact that it allows for simultaneous attribution of incompatible determinates of the same determinable to the same object. A disk viewed at an oblique angle is supposed to look both circular and elliptical as well as both tilted and upright at the same time. This is what Charles Siewert calls the ‘Problem of Contradictory Visual Appearances’ (Siewert 2006: 5). Matters are rather different, it could be argued, when looking at a picture. Here, one interpretation of the retinal image (I1) attributes certain properties to the 2D pictorial surface, while the other interpretation (I2) attributes certain properties to a virtual object in phenomenally 3D pictorial space. In terms of feature-binding theory (Treisman 1996), different features are bound to the surface in I1 than are bound to the object of pictorial experience in I2. So there is no contradiction internal to the content of an experience that simultaneously reflects both I1 and I2. How might a Gombrichian respond? With respect to the first putative disanalogy, it is important to observe that there are contexts in which the sources of optical information in the light received from a real-world scene are highly ambiguous, even when the constraining assumptions of the ‘beholder’s share’ are brought to bear on its uptake. In Bayes-speak, there are real-world contexts in which different visual hypotheses have about the same posterior probability. For example, just as it is possible to experience a reversal in orientation when looking at a drawing of the Necker cube (Figure 3.2), it is also possible to experience a reversal in 3D orientation when looking at a cubical wireframe face-to-face (especially when one eye is shut). Another example is the phenomenon of binocular rivalry. In binocular rivalry, the image presented to the left eye is different than the image presented to the right eye, for example, an image of a face and an image of a house (Blake and Logothetis 2002; Alais and Blake 2015). In most cases, however, only one of these objects is visible at a time: the other is suppressed from visual awareness. The important point is that when sources of optical information in real-world contexts are highly ambiguous, we do not experience different 3D-scene interpretations at the same time. Instead, we alternate between competing interpretations. In relevant cases, the visual system does not select a conjoint or ‘blended’ hypothesis concerning the distal causes of proximal sensory stimulation. We do not experience the cubical wireframe as having

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two different orientations at the same time, nor do we experience a blending of the face and house. This makes good sense, from a Bayesian perspective, since conjoint hypotheses such as these have extremely low prior probabilities (Hohwy et al. 2008). The Gombrichian, in short, can maintain that what holds when the information in the light received from a real-world scene is equivocal—that is, when it does not support the selection of a single 3D-scene interpretation— also holds when the information in the light from a patterned, pictorial surface is equivocal. The visual system, in both cases, switches between competing interpretations that ‘fit’ the available sensory evidence.10 It does not generate an experience that reflects both interpretations at the same time. With respect to the second putative disanalogy, the Gombrichian can concede that the two interpretations attribute properties to different objects, viz., the opaque, 2D pictorial surface (I1), on the one hand, and an object in phenomenally 3D pictorial space (I2), on the other. This doesn’t mean, however, that the two interpretations are consistent. On the contrary, there is a very straightforward reason to think that they are mutually exclusive: I1 represents the presence an opaque, 2D surface S in front of the perceiver, while I2 represents (non-veridically) an array of objects receding in depth behind S. Human perceivers, however, do not have the capacity see through an opaque surface, for example, a sheet of canvas covered with paint, to the world on the other side. If a given solid angle in the visual field appears to contain a non-transparent, 2D surface located at some distance in depth D, then it cannot, at the same time, also appear to contain an array of 3D objects at locations more distant in depth than D.

3. The Seeing-In Theory Richard Wollheim is a prominent critic of the Gombrichian account of pictorial experience (1963, 1987). According to Wollheim, when we look at a picture, we not only experience the organization of a virtual 3D scene; we also experience the design present on the 2D pictorial surface. In this sense, pictorial experience—or ‘seeing-in’, as Wollheim calls it—has two ‘aspects’ or two ‘folds’ of representational content: Seeing-in is a natural capacity we have—it precedes pictures, though pictures foster it—which allows us, when confronted by certain differentiated surfaces, to have experiences that possess a dual aspect, or ‘twofoldness’, so that, on the one hand, we are aware of the differentiation of the surface, and, on the other hand, we observe something in front of, or behind, something else. (1993: 188) Wollheim argues that while the two ‘folds’ of content in pictorial experience are clearly different, they are not, contrary to Gombrich, incompatible

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(Wollheim 1963: 29). As paradoxical as it may sound, the same solid angle in the visual field, when we look at a picture, is represented twice over: once as encompassing an opaque, two-dimensionally organized surface and once as encompassing a (virtual) three-dimensionally organized scene. Two different two ways of dispelling the appearance of paradox at the heart of the seeing-in theory have been proposed. The first appeals to the idea that pictorial space and physical space are experienced as entirely unrelated. Wollheim writes that ‘there are two distinct dimensions here along which “on,” “level with,” and “behind” are values: a physical dimension and what we might call a pictorial dimension’ (1974: 27). Hence, it would be a phenomenological mistake to describe pictorial experience as representing an opaque, 2D surface in front of a 3D scene. This description conflates apparent physical and pictorial depth relations. The problem with this suggestion is that pictorial space and physical space, while experienced as distinct, are not experienced as entirely unrelated (Hopkins 1998: 195–196; Kulvicki 2009; Lopes 2010). For one thing, objects in pictorial space not only appear to be positioned in certain directions relative to the pictorial point of view, they also appear to be positioned in certain directions relative to the viewing subject’s location in physical space. When you watch a film, Lopes notes, ‘there is a rich and systematic overlap in the two viewpoint-determining contents—that is, between the represented directions from the camera to depicted objects and directions the objects appear to lie in from the picture viewer in normal viewing conditions’ (Lopes 2010: 76). In other words, we can be aware of the direction of an object in pictorial space in both picture- and observer-based frames of reference. A second, related consideration has to do with the observation that objects in pictorial space sometimes seem to ‘follow’ the viewer (Gombrich 1972; Goldstein 1979; Koenderink et al. 2004; Newall 2015). When the viewer moves in relation to the famous British Army recruiting poster depicting Lord Kitchener, for example, the virtual object she experiences in pictorial space curiously appears to rotate toward her (Gombrich 1961/2000: 113). Koenderink et al. write: Does this mean that the pictorial object ‘rotates along with the observer’ as the observer assumes a series of oblique viewing positions by walking along the painting on the wall? . . . Yes in terms of the physical space containing the scene, picture, and observer: the pictorial object always squarely faces the observer, it thus looks or points into the observer’s visual direction. As the observer changes the visual direction with respect to the picture plane, the looking or pointing direction of the pictorial object in physical and visual space has to rotate with it. (Koenderink et al. 2004: 526) The main point is that we not only experience the pictorial object’s constant orientation in pictorial space vis-à-vis the picture plane, we also experience

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its orientation as changing vis-à-vis our own location in physical space. Together, these considerations (but see Kulvicki [2009] for others) suggest that the ‘disjoint space’ view fails to dispel the appearance of paradox at the heart of Wollheim’s account of pictorial experience. John Kulvicki (2009) enlists our capacity to see through transparent media in the service of an alternative account of how twofold pictorial experience is possible. In familiar cases of transparency perception, the visual system distributes light-altering properties, such as colour, transmittance, and glossiness, to surfaces at different distances in depth along the same line of sight. This process, which results in a ‘layered’ representation of the distal scene, is known as scission (Kanizsa 1979; Metelli 1970, 1974; Fleming and Anderson 2004; Anderson and Winawer 2008). When such distribution is unnecessary either because the overlying layer is completely opaque or completely transparent, scission does not occur. As the vision scientist Fabio Metelli writes, [i]f all the color goes to the transparent layer, it becomes opaque. If all the color goes to the underlying surface, then the transparent layer becomes invisible. Transparency is perceived only when there is a distribution of the stimulus color to both the [overlying] layer and the [underlying] layer. (1974: 94) Kulvicki argues that human capacities for transparency perception allow for the possibility of twofold pictorial experience: the luminance intensities in the retinal image of a picture are used by the visual system to assign properties to the opaque, 2D pictorial surface and, at the same time, to 3D objects that appear to recede in depth behind the surface. It does not matter to this proposal, he says, that this kind of layering typically happens when one is confronted with semitransparent objects. . . . What matters is that the visual system has the resources for representing objects and qualities at different distances in one and the same direction, as the literature on transparency perception strongly suggests. (2009: 393) The basic problem with this proposal is that the process of scission involves the distribution of light-altering properties to surfaces at different distances in depth along the same line of sight. The fundamental question for the visual system, as Fleming and Anderson put it, is, ‘How much of the light is due to reflectance of underlying surface, and how much is due to the properties of the overlying layer?’ (2004: 1294). However, if, by hypothesis, the pictorial surface is experienced as opaque in twofold pictorial experience (in ‘one and the same direction, at a given time, one sees two rather different opaque surfaces, one behind the other: the picture plane and

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the receding content’ [Kulvicki 2009: 394]), then this just means that no such distribution of light-altering properties can have taken place. Instead of attenuating the paradox at the heart of Wollheim’s theory of pictorial experience, the proposal that twofoldness is made possible by our capacity to perceive through transparent media only aggravates it. Proponents of the seeing-in theory have put forward several reasons to think that twofoldness, however paradoxical it may sound, is a necessary feature of pictorial experience. One reason, Wollheimians maintain, is that twofoldness is required for the aesthetic appreciation of pictures (Wollheim 1980; Nanay 2010). If true, however, this would only entail that we experience certain art pictures in a twofold way, not that twofoldness is necessary for pictorial experience in general (Lopes 1996: 48). More generally, it could also be argued that twofoldness isn’t even necessary for purposes of aesthetic evaluation. Why isn’t rapidly shifting attention between properties of the pattern on the pictorial surface, for example, the texture of the brushstrokes on a painting, and properties of the virtual object in pictorial space sufficient? A second reason has to do with evidence that when a picture is viewed from an oblique angle, we do not experience objects in pictorial space as significantly distorted in shape. In other words, shape constancy in pictorial space obtains across changes in perspective. Wollheim argues the best explanation of such constancy is that the viewer not only is aware of the object in pictorial space but also compensates for the orientation of the pictorial surface. Such compensation, he suggests, involves awareness of the ‘surface qualities of the representation’ (1980: 215–216). There are three lines of response. First, as Lopes points out, even were it true that surface awareness is necessary for pictorial shape constancy, this wouldn’t entail that it is necessary for pictorial experience in general (1996: 49). Second, the compensation process could take place an entirely subpersonal level, using nonconscious information about the orientation of the pictorial surface: Wollheim provides no reason to suppose that compensation depends on conscious perception of surface orientation. Third, there are alternative explanations of pictorial shape constancy in vision science. ‘A quite different (and also quite common) view,’ as the perceptual psychologist Jan Koenderink and co-authors write, is that observers simply don’t care, that is to say, disregard (but not in any active sense) the distortions of the retinal image, since the monocular cues as to the structure of the pictorial space are rich enough anyway. In that case, a subsidiary awareness of the picture surface is irrelevant. (Koenderink et al. 2004: 515) Summarizing the evidence, they conclude that ‘there appears to be some (weak) consensus that no “correction” is applied to pictorial space due to obliquely viewed pictures’ (Koenderink et al. 2004: 526).

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According to the seeing-in theory, it is possible to experience the same solid angle in the visual field both as completely filled by an opaque, 2D surface at a single distance in depth and as containing an array of 3D objects at different distances in depth. I have shown that two of the main attempts to render this view non-paradoxical confront serious objections, and this speaks in favour of the analogy Gombrich draws between pictorial experience and the experience of looking at an ambiguous figure like the duck– rabbit. In both cases, we ‘can train ourselves to switch more rapidly, indeed to oscillate between readings, but we cannot hold conflicting interpretations’ (1961/2000: 236). Before proceeding, however, I would like to make two additional points. First, Gombrich suggests that it is typically possible to alternate at will between awareness of the configuration of colours, textures, and marks on a 2D pictorial surface and awareness of the properties and layout of objects in phenomenally 3D pictorial space, much as it is possible to alternate at will between the different ‘aspects’ of the Necker cube (Gombrich 1961/2000: 236, 280). There are good reasons, however, to question the generality of this assumption. Whether such voluntary switching is possible when looking at a picture depends, among other things, on the range of depth cues in the light reflected from its surface to the eye. It is quite hard, in fact, to ‘unsee’ 3D structure in line-drawings that contain appropriate T-junctions, linear perspective, and other depth cues under ordinary, binocular viewing condition (Kennedy 1974; Zeimbekis 2015).11 It is even harder when standard, surface-specifying cues are eliminated. Looking at a painting through a narrow, monocular aperture often results in a robust impression of depth and ‘stereopsis’, that is, solid form and immersive space (Kubovy 1986: ch. 12; see the next section). A similar effect is obtained when looking at a single picture through a zograscope (Koenderink et al. 2013).12 These devices eliminate cues that specify the presence and properties of the 2D pictorial surface and, so, render a Gombrichian switch effectively impossible. Viewing distance also matters. As suggested by the epigraph from Martin Eden, it may not be possible to enjoy an experience as of virtual depth and 3D structure when viewing certain paintings at close range.13 By contrast, it is exceedingly difficult to elude that experience when watching a film from the back row of a movie theater. These considerations suggest that whether or not it is possible to perform a Gombrichian switch depends on both the sources of information in the light reflected from a pictorial surface as well as the conditions under which the surface is viewed. If this is right, however, then the real problem with Gombrich’s deployment of the duck-rabbit analogy, contrary to Wollheim, isn’t that it underestimates, but rather that it overestimates our ability to see the superficial pattern on a picture’s surface as such. The second point is that when a Gombrichian switch between competing interpretations of the retinal image of a picture is possible, iconic and/or visual short-term memory of ‘pre-switch’ interpretations may support the

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illusion of simultaneity. That is, instead of simply oscillating between an experience as of a pattern of brushstrokes on a 2D canvas (I1) and an experience as of an object or scene in phenomenally 3D pictorial space (I2), as Gombrich suggests, the viewer may oscillate between I1 + a sensory, visual memory of I2 and I2 + a sensory, visual memory of I1. In consequence, she may deceptively seem to enjoy both visual experiences (I1 and I2) at once. Also at work may be a kind of “immanence” (Minsky 1986) or “refrigerator light” illusion (Block 2001), in which the viewer mistakes the potential presence in visual consciousness of a certain set of properties for their actual presence. As an example of the refrigerator light illusion, here is Eric Schwitzgebel’s (2008) explanation of why naïve introspectors tend to overestimate their visual acuity outside the central, foveal region of their visual field: Here’s the root of the mistake, I suspect: When the thought occurs to you to reflect on some part of your visual phenomenology, you normally move your eyes (or “foveate”) in that direction. Consequently, wherever you think to attend, within a certain range of natural foveal movement, you find the clarity and precision of foveal vision. It’s as though you look at your desk and ask yourself: Is the stapler clear? Yes. The pen? Yes. The artificial wood grain between them and the mouse pad? Yes—each time looking directly at the object in question—and then you conclude that they’re all clear simultaneously. (Schwitzgebel 2008: 255; also see Dennett 1969: 139–140) Similarly, I would suggest, our ability to switch attention without significant delay between the colours, textures, and other superficial properties visible within some pictorial surface region R, on the one hand, and the properties of the virtual 3D object or scene displayed by R, on the other, may contribute to the illusion that both sets of properties are present in visual consciousness at the same time. Consider in this connection the view that certain pictures elicit a twofold experience that Dominic Lopes refers to as ‘design seeing’ (2005: 28). When we engage in design seeing, according to Lopes, we see the configuration of design features visible on a picture’s 2D surface as ‘undergirding’ or as ‘responsible for’ the very experience of depth and 3D structure that those features elicit in us. Pictures that elicit design seeing ‘wear the process of depiction on their sleeves’ (Lopes 2005: 52). This view is controversial, however, not only because it takes the possibility of twofoldness for granted but also because it presupposes that the contents of visual experience are rich enough to represent pictorial design features as such. That is, it presupposes that visual experiences (and not only visually based beliefs) have capacity to represent the high-level property pictorial design feature. I do not want to take a stand here on the dispute between ‘rich’ and ‘thin’ theories of visual representational content (but see Prinz 2006; Siegel

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2010, 2016; Briscoe 2015; and Byrne 2016). For present purposes, the point is only that the visual-memory/attentional-switching proposal provides resources to explain why looking at a picture may sometimes deceptively appear to involve design seeing. Because, in relevant cases, we have the capacity to switch attention between the set of design features visible within some pictorial surface region R and properties of the virtual, 3D object or scene displayed by R and, in addition, because our experience of the design features visible within R may be, so to speak, coloured by a concurrent, visual memory of that virtual object or scene, it may seem to us as though our experience of those design features represented them as such—and not merely as a superficial configuration of marks, colours, textures, and other low-level properties.

4. Toward a Weakly Onefold Theory of Pictorial Experience Dominic Lopes (1996) refers to the view that twofoldness is essential to pictorial experience as ‘strong twofoldness’ and, correspondingly, to the view that twofoldness is merely consistent with pictorial experience as ‘weak twofoldness’. Gombrich, by contrast, is sometimes interpreted as defending what might be called a strongly onefold conception of pictorial experience. According to strong onefoldness, it is not psychologically possible to experience any properties of the 2D pictorial surface while simultaneously experiencing properties of a virtual scene in phenomenally 3D pictorial space. To the extent that we are aware of virtual depth and 3D structure when viewing The Peasant Wedding (Figure 3.1), for example, we are effectively blind to the superficial pattern on the painting’s canvas. Strong onefoldness is a core component of the so-called illusion theory of pictorial experience. The illusion theory goes far beyond the claim that pictorial experience and seeing face-to-face are experiences of the same psychological kind (the Continuity Hypothesis). It further maintains that pictures elicit experiences that cannot be introspectively distinguished from experiences of seeing the objects that they depict face-to-face. According to the illusion theory, Lopes writes, ‘one sees O in a picture when and only when one’s experience as of O when looking at the picture is phenomenally indistinguishable from a face-to-face experience of O’ (2005: 30). But, if the experience of looking at a picture of an object is supposed to be phenomenologically on all fours with the experience of seeing an object with the same properties face-to-face, then the former experience evidently cannot involve seeing the properties of the picture’s surface. Not surprisingly, the paradigm case of pictorial experience for the illusion theory is the experience of seeing a trompe l’oeil painting from the appropriate station point. The possibility of pictures that ‘fool the eye’ indicates that pictorial experience is, at least sometimes, strongly onefold. Putting trompe l’oeil paintings to the side, however, most pictures do not elicit experiences that are introspectively difficult to distinguish from experiences of actually seeing

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their depicta. As Gombrich points out, ‘we rarely get into situations in which the eye is actually deceived’ (1961/2000: 246). One reason is straightforward. When we look at a picture of some high-level kind of F, for example, a woman, or a tree, or a clock, the properties attributed by our visual system to the object we experience are very often different from those that would be attributed to an actual F, when seen from the relevant point of view. Indeed, the intentional object of our experience may appear to have properties that no actual F could have (think of Magritte’s surrealist paintings) and/or to lack properties that no actual F could lack (think of Picasso’s highly abstract line drawings). These considerations present a serious challenge to the illusion theory, but they do not threaten the Continuity Hypothesis. That the content of the experience caused by a picture of a horse typically does not match the content of an experience that might have been caused by an actual horse does not conflict with the claim that they are experiences of the same psychological kind. In this respect, after all, the experience elicited by a picture of a horse is completely on par with the experience of seeing a 3D sculpture or model of a horse in the flesh. I have indicated one reason why pictorial experience is typically nondeceptive: the properties attributed by pictorial experience to its intentional object are in many cases different from those that the depicted object would be seen to have when confronted face-to-face. There is a more profound respect, however, in which the experience elicited by a picture is normally distinguishable from the experience of actually seeing the object that it depicts. Robert Hopkins writes, In some way, when I see a woman in a painting, I am visually aware of a woman. . . . I am presented with a woman, but not so as to suggest that that is what is really there. Unlike perceptual consciousness, this awareness is non-committal about the reality of its objects. In this respect, if no other, pictorial consciousness is like visualizing. (Hopkins 2012b: 434) Similarly, Michael Martin observes that when we look at pictures, we experience objects that manifestly lack ‘solidity’ and a ‘self-standing appearance’. The objects of pictorial experience, in contrast with objects seen face-toface, are experienced as ‘mere visibilia’ (Martin 2012: 342). Other authors in this connection refer to a feeling of ‘presence’ or ‘reality’ that accompanies seeing face-to-face but is conspicuously absent when looking at a picture (Sartre 1940/2004; Michotte 1960; Matthen 2005; Wiesing 2010). That the experience elicited by a picture of an object and the experience of seeing the depicted object face-to-face do not typically match in content, as we saw, does not threaten the Continuity Hypothesis. That pictorial experience typically does not dispose us to believe that its intentional object is real—‘something out there’—by contrast, presents a serious challenge. As

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Hopkins points out, if the claim that pictorial experience is an experience of the same psychological kind as the experience of seeing face-to-face is to be informative, then the respects in which former is phenomenally distinguishable from the latter must be limited to differences in the properties—the shapes, sizes, colours, and so forth—that they respectively attribute to their intentional objects: The move from presenting [the intentional object] O as real to no longer doing so hardly fits that bill. One central difference between seeming to see something and visualizing it is that the former necessarily presents its object as real. Failure to do that secures that [pictorial experience] cannot be [a] visual experience as of O, and opens up the possibility that it is visualizing. (Hopkins 2012a: 654–655) These considerations not only put pressure on the illusion theory, they also put pressure on the Continuity Hypothesis. If everyday visual experience typically disposes its subject to believe that its intentional objects are real, but pictorial experience does not, then this is seemingly a good reason to suppose that we are dealing with experiences that not only contrast in representational content but also in psychological kind. The remainder of this section has two aims. The first is to motivate an account of the structure of everyday, non-deceptive pictorial experience that I refer to as weak onefoldness. Everyday pictorial experience, according to the account, is onefold in the sense that its content reflects a single, consistent 3D-scene interpretation of the retinal image. It does not represent the same solid angle in the visual field twice over—once as filled by an opaque, twodimensionally organized surface and once as filled by a three-dimensionally organized scene. Everyday pictorial experience, however, typically is only weakly onefold in the sense that it attributes properties to the pictorial surface and to objects in pictorial space at the same time. It represents a single scene with both real and virtual constituents—a scene that straddles the boundary between physical and pictorial space. On this approach, surface representation in pictorial experience is not an all or nothing affair. Having the experience as of virtual depth and 3D structure, when looking at a picture, I argue, excludes representing some, but not all of its superficial properties. The important consequence, in the present context, is that is possible to embrace Gombrich’s requirement of a single, consistent, 3D-scene interpretation while rejecting strong onefoldness and the illusion theory. The second aim is to reconcile the claim that pictorial experience and seeing face-to-face are psychologically continuous with the observation that the (virtual) objects of pictorial experience normally do not appear to be present in the physical environment. An empirically motivated account of stereopsis developed by Dhanraj Vishwanath (2014), I argue, makes such reconciliation possible.

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In order to motivate weak onefoldness, it is necessary to demonstrate that although pictorial experience is structured by just one layer or ‘fold’ of representational content, that content can consistently attribute certain combinations of properties to the 2D pictorial surface and to objects in phenomenally 3D pictorial space at the same time. There are at least two different ways, I propose, in which this is possible. First, while it may not be possible simultaneously to experience the same solid angle in the visual field as filled by an array of voluminous 3D objects that recede in depth and by a single, non-transparent 2D surface, it does seem possible, when looking at a picture, to divide attention between the region of pictorial space contained within some solid visual angle θ and the distribution of pictorial surface properties contained within some different solid visual angle φ. For example, we can divide our attention between the 3D shape of an object displayed in one region of a canvas and the facture or colour of the paint laid down in some other. If so, then we may be visually aware of certain pictorial-space properties and certain pictorial-surface properties at the same time. None of the considerations adduced earlier in support of Gombrich’s account of pictorial experience rule out distributing our attention and the content of our experience across the pictorial space/ physical space frontier in this manner. Second, it is possible to experience regions on a pictorial surface as opaque and as partially occluding objects in pictorial space. Words printed on a pictorial surface, for example, may be visually experienced as occluding figures on a more distant, three-dimensionally organized background, and when this is the case, we experience properties of the real-world surface and virtual scene at the same time. Posters, movies with subtitles, advertisements, and other pictures that incorporate textual elements elicit experiences that plausibly fit this description. They provide an existence proof of the possibility of weakly onefold pictorial experience. As an example, consider the poster for the Buster Keaton film Sherlock, Jr. reproduced in Figure 3.4. Some edges, including those belonging to the letters in the title of the film are represented by the visual system as surface events and, so, are assigned to locations on the pictorial surface. Other edges, in contrast, are assigned to locations beyond the surface in pictorial space. The contour of the virtual object resembling Keaton’s head, for example, appears to occlude the lowermost edges of some of the letters in the name ‘Keaton’, with the result that the entire name is dragged back in depth. It appears to float inside the virtual, 3D scene displayed by the poster, while the title Sherlock, Jr. rests on the plane of the poster’s surface. Contradiction is avoided because no edge is interpreted in both ways at once. The overall experience elicited by the poster, I would suggest, is one in which certain textual elements appear partially to occlude objects that recede in depth from the plane of the pictorial surface. There are constraints, of course, on which experienced combinations of 2D surface properties and virtual, 3D scene properties are psychologically

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Figure 3.4 Poster for Sherlock, Jr. (1924)

possible. Consider once again a drawing of the Necker cube (Figure 3.2). There are eight vertices or ‘y-junctions’ present in the retinal image of the drawing, each of which can be interpreted in at least three different ways: as convex, that is, pointing toward the viewer; as concave, that is, pointing away from the viewer; or as co-planar, that is, as the intersection of three line segments on the same plane of depth as the pictorial surface. Each local interpretation constrains how the visual system interprets the remaining seven y-junctions in the image. Interpreting y-junction a as convex, for instance, constrains the visual system to interpret b, c, d as pointing in the same direction with the result that face abcd is experienced as in front of face efgh. Interpreting a as concave results in the reverse depth ordering. Interpreting a as co-planar with the pictorial surface results in a globally ‘flat’ interpretation, in which neither abcd nor efgh is experienced as in front

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of the other. Each local interpretation, in short, constrains how the visual system interprets the other edges present in the retinal image (Albert and Hoffman 2000; Cooper 2008). For present purposes, the main point is that the visual system processes 3D structure in a holistic way. In the pictorial case, this means that whether a given edge in the retinal image is interpreted as an event on the 2D pictorial surface or as the contour of a (virtual) 3D object typically has consequences for how other nearby edges in the retinal image are interpreted. I have argued the visual system can consistently attribute certain properties to the pictorial surface and to objects that appear beyond the surface in pictorial space at the same time. When it does, the experience elicited by a picture is weakly onefold. It represents a single, integrated scene containing both real and virtual elements. If this is right, however, then it possible to embrace the Continuity Hypothesis and Gombrich’s requirement of a single, consistent, 3D-scene-interpretation while rejecting strong onefoldness. I turn, now, to the observation that pictorial experience does not typically dispose the viewer to believe that its intentional object is real, that is, present to sight in physical space. Does this observation, as Hopkins suggests, speak against the Continuity Hypothesis? It is helpful to begin by reflecting on some of the very different reasons why visualizing and pictorial experience do not tempt the subject to believe that she is perceptually related to an object in the physical environment. The primary reason that visualizing is ‘non-committal’ about the real presence of its objects is that its content is subject to the will (Sartre 1940/2004; Wittgenstein 1953; McGinn 2004).14 Visualizing, like thinking, is something we do, not something that happens to us. Accordingly, we do not adopt an attitude of observation toward its intentional objects. This is not to deny that we can sometimes learn about the world by engaging in visualization: a subject, for example, could imagine what her flat would look like if all the walls in it were transparent and thereby come to realize for the first time that certain windows were opposite each other.15 The right point is that the contents of visualization are either imported from the subject’s memory and belief systems (as when she imagines the layout of the flat) or derived from certain manipulations of that information (as when she counterfactually imagines that the walls in the flat are transparent). It is in this sense that the objects of visualization are ‘not in a position to feed new information into our cognitive system’ (McGinn 2004: 20). The contents of pictorial experience in contrast with the contents of visualization are not in general subject to the will. And, like the experience of seeing face-to-face, the content of pictorial experience is systematically guided by the flow of information from the external environment. A pictorial surface will elicit an experience as of an object with a specific 3D shape Σ, for instance, only if it reflects light to the eye of the same type as would elicit experience as of a Σ-shaped object when reflected from a real-world scene.16 ‘[P]erceiving depth in pictures and perceiving depth in

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the real world’, as the psychologist James Cutting puts it, ‘are cut from the same informational cloth’ (2003: 236). Pictures, in short, can elicit experiences containing genuinely novel, visual content. They can surprise us and invite what McGinn refers to as an attitude of cognitive openness. In this respect, experiencing an object with certain properties in pictorial space is fundamentally unlike visualizing an object with the same properties. Nonetheless, pictorial experience doesn’t so much as tempt the perceiver to judge that its object is really there in front of her. Why is this the case? In order to answer this question, it is necessary to begin by introducing two distinctions. First, following Vishwanath (2014), we need to distinguish between having the experience as of depth and 3D structure, on the one hand, and having the experience of stereopsis (from the Greek for solid and appearance), on the other. Both pictures and real-world scenes elicit the former experience. Vishwanath characterizes the experience of stereopsis, in contrast, as the ‘vivid impression of tangible solid form, immersive negative space and realness that obtains under certain viewing and stimulus conditions’ (Vishwanath 2014: 153). Pictures, as already observed, typically elicit the former experience without eliciting the latter. When we look at The Peasant Wedding, for example, we experience the layout of a complexly organized, 3D scene, but we are hardly disposed to judge that the objects we experience are really there in front us.17 Second, it is also necessary to distinguish between the sources of relative depth information and sources of absolute distance information available to the visual system (Landy et al. 1995; Banks et al. 2011). Independently variable sources of relative depth information include, but are not limited to, occlusion, texture gradients, shading, the kinetic depth effect, height in the visual field, and binocular disparity. These relative cues specify, among other things, depth ratios between objects in a visible scene, for example, that object a is twice as far away as object b, and simple ordinal relations, for example, that a is behind b but in front of c. They do not, however, specify the absolute distance of an object from the eye in either conventional units such as meters or non-conventional units such as eye heights (Sedgwick 1986; Bennett 2011). Sources of absolute or ‘metric’ distance information comprise motion parallax, defocus blur, and the oculomotor cues of convergence and accommodation. In order to integrate these two different types of spatial information for purposes of representing the 3D layout of a visible scene, relative cues must be scaled or ‘promoted’ by absolute cues so that the depth values they provide are commensurate (Landy et al. 1995: 392). For example, the convergence angle of the eyes is a source of absolute information about fixation distance, which, when combined with an estimate of the intraocular separation, can be used to promote binocular disparity to an absolute distance cue.18 According to what Vishwanath calls the absolute depth scaling hypothesis (ADSH), the experience of stereopsis obtains under real-world, binocular viewing conditions because both sources of relative depth and absolute distance

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information typically are available to the visual system. Only relative depth information, however, is typically available for objects in pictorial space, and this explains why pictorial experience is not accompanied by the impression of stereopsis: When a picture is viewed binocularly, distance cues such as binocular convergence, vertical disparity and the accommodative state of the lens specify the distance of the visible picture surface, so there are no optical distance cues that specify the distance of pictorial objects. Pictorial depth cues, such as shading, perspective, and interposition, can specify the 3-D shape and relative layout of objects in the depicted scene, but without distance information, these cues cannot be scaled to derive absolute depth or size. (Vishwanath 2014: 158–159) A number of philosophers have observed that pictorial experience does not represent its objects as standing in fully determinate, egocentric spatial relations to the observer herself (Carroll 1996; Cohen and Meskin 2004; Matthen 2005; Nanay 2014). In consequence, although the observer may be aware of depth and detailed 3D structure when looking at a picture, the virtual scene it displays does not ordinarily appear to be a potential arena for performing visuomotor actions. The ADSH provides a psychophysical explanation of why pictures are, as Cohen and Meskin (2004) put it, ‘spatially agnostic informants’: sources of absolute distance information in the light received from a pictorial surface are assigned to the 2D pictorial surface, rather than to objects located in phenomenally 3D pictorial space.19 One source of evidence for the ADSH is its ability to explain why the experience of stereopsis is successfully induced when paintings, photographs, and other types of pictures are viewed with one eye through a peephole (for others, see Vishwanath 2014). Under these conditions, neither vergence nor disparity is available as a source of information for the absolute distance of the pictorial surface. Accommodation-based distance information, however, remains, and, in the absence of a visible pictorial surface, Vishwanath proposes, it is reassigned to objects in pictorial space, enabling the visual system to compute estimates of their absolute distance in depth and intrinsic size. The ADSH predicts that the reassignment of accommodation-based distance information to pictorial objects should make them appear to be relatively close to the observer—that is, as located at roughly the actual distance of the pictorial surface from the eye—and, hence, as smaller than the real-world objects they are intended to resemble. (Indeed, accommodation is an effective range-finder only for objects in nearby space [Cutting and Vishton 1995].) Subjective reports concerning the effects of monocular aperture viewing bear this prediction out (Vishwanath and Hibbard 2013). The cars and buildings in a photograph of a cityscape, when viewed through

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a narrow peephole, often have the appearance of being miniaturized and toy-like. This experience is sometimes referred to as the ‘diorama effect’.20 As Hopkins argues, the claim that pictorial experience and seeing face-toface are experiences of the same psychological kind is informative only if the respects in which the two experiences are phenomenally distinguishable are at the level of representational content: they must be limited to differences in attributed visual appearance properties. On Vishwanath’s account, however, the reason that pictorial experience does not dispose the observer to believe she is seeing a real object is that it fails to specify the object’s location at some absolutely scaled distance in depth. But, if this right, then the relevant contrast with seeing face-to-face is best understood as a difference at the level of representational content rather than a difference at the level of psychological kind. Just as visual experiences can be more or less determinate with respect to properties such as shape, size, and colour (Tye 2003)—an object present in peripheral vision, for example, may be experienced as dark in colour or as elongated in shape but not as instantiating a specific dark colour or a specific elongated shape—they can also be more or less determinate with respect to the distance of an object vis-à-vis the observer herself. In the limit, conscious vision simply may not offer comment on an object’s absolute distance in depth. In short, if the ADSH is correct, then the fact that the objects of pictorial experience do not appear to be present in the same space as the surface in which they are displayed does not speak against the psychological continuity of pictorial experience with seeing face-to-face.

5. Conclusion I will conclude this chapter with a speculative proposal. According to Gombrich’s exclusivity requirement, the content of visual experience always reflects a single, consistent, 3D-scene interpretation of the retinal image. In the last section, I argued that weakly onefold pictorial experiences satisfy the exclusivity requirement, and I developed this claim with two examples. Here is a third possible way in which pictorial experience could be weakly onefold. The ADSH maintains that, when we look at a picture, absolute distance in depth estimates are only computed for the picture’s surface. Conscious vision is silent when it comes to the absolute distance of an object positioned ‘beyond’ the surface in pictorial space. In this respect, the content of our experience when we look at a picture is significantly less determinate than the content of our experience when we engage in seeing face-to-face. Now, one way in which the experience elicited by a picture could represent the presence of the picture’s surface and the configuration of objects in pictorial space within the same solid angle in the visual field at the same time would be if it also represented the surface in a highly indeterminate way, in particular, if it represented the viewer-relative absolute distance and orientation of the surface, as well as some of its intrinsic geometric properties (curvature, shape, size), but did not attribute any colour properties to

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it. Just as the visual system does not offer comment on the absolute distance of an object embedded in pictorial space—this explains why the object never appears closer to us when we move in its direction (Matthen 2005: 316–317)—it may not offer comment on the colour(s) of the surface in which the object is displayed. This description, I would suggest, phenomenologically fits the experience of looking at certain photographs and naturalistic paintings, especially at a distance, as well as the experience of watching movies. In ‘the normal case of watching a movie’, Colin McGinn writes, ‘we don’t focus our attention on [the] fleeting patterns of light—we, as it were, look right through them’ (McGinn 2005: 17). But this does not mean that visual awareness of the surface drops out entirely. The colour properties we experience when we watch a movie are assigned by our visual system to virtual objects on the ‘other side’ of the screen, but we are still aware of the screen’s presence and typically have no difficulty estimating its distance from us or its orientation. This proposal is not as counterintuitive as it may sound. In a recent treatment, Fiona Macpherson (2015) critically evaluates the traditional, Aristotelian view that visual experience as of an object necessarily represents the object’s colour(s). Drawing on evidence concerning tactile-visual sensory substitution, amodal completion, and type 2 blindsight, she argues that there is no good reason to deny that visual awareness of ‘pure distal form without colour’ is possible. Contrary to philosophical tradition, colour may not be a structural feature of object-representing visual experience. Certain cases of picture perception, if I am right, put another arrow in Macpherson’s quiver. To be clear, I am not proposing that in relevant cases we experience the pictorial surface as transparent to the virtual 3D scene. Recall Metelli’s constraints on transparency perception: If all the color goes to the transparent layer, it becomes opaque. If all the color goes to the underlying surface, then the transparent layer becomes invisible. Transparency is perceived only when there is a distribution of the stimulus color to both the [overlying] layer and the [underlying] layer. (1974: 94) In the experience that I am characterizing, however, there is no distribution of colour across layers. Instead, all the colour goes to the underlying layer, that is, to the object (or array of objects) in pictorial space. But this does not entail that the pictorial surface is invisible, like a perfectly clear pane of glass. Instead, we may visually experience the surface in a highly indeterminate way: as present at some absolute distance in depth and as having such-andsuch orientation relative to our line of sight but not as having any colour properties of its own. If so, then there would be no inconsistency internal to an experience that represented both the surface and (non-veridically) a 3D scene receding in depth behind it. Michael Newall develops a similar view in a recent treatment. According to Newall, when we look at certain photographs, ‘all the picture surface’s

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visible properties are attributed to the subject matter and the viewer loses all visual awareness of the picture surface’ (2015: 144). His example involves the photographic image of a glass of milk that appears on the cover of some editions of the novel A Clockwork Orange: The milk is depicted by a white, or slightly grey, colour. Here my claim is that this colour is wholly attributed to the subject matter—the milk— which appears as if a little behind the picture surface. While we have this experience, the picture surface appears as if it lacks all its colour properties. It appears, one might say, transparent to the point of invisibility. (Newall 2015: 145) We can, to be sure, also experience the white colour as belonging to the pictorial surface, Newall allows, but this requires a Gombrichian switch. My proposal in this concluding section owes a lot to Newall’s discussion of this example. It differs from his own proposal, however, in that it denies that in relevant cases the pictorial surface is completely invisible. Instead, I claim that we may be aware of the surface but in an uncharacteristically indeterminate way: the surface may be represented as having a certain absolute distance in depth and orientation, as well as certain intrinsic, geometric properties, but not as having any colour properties. This seems phenomenologically more plausible than the claim that the surface is altogether absent from visual awareness since we can attend to the changing distance or orientation of a photograph like the one discussed above, while simultaneously paying attention to the virtual 3D object or scene that it displays.21

Notes 1. Two clarificatory remarks. First, there are many different types of pictures and any given picture can elicit many different types of mental states in its viewers. Philosophers of perception and vision scientists, however, are specifically interested in those pictures that elicit an experience as of depth and 3D structure (for a useful, interdisciplinary overview, see the essays collected in Hecht et al. 2003). Accordingly, my focus is on pictures that do elicit this distinctive experience. Although I do not argue for the claim here, I think that they constitute a fairly unified artefactual kind. Second, the point of using the ‘experience as of’ locution is to bracket the question concerning whether the experience in question is veridical. A visual experience as of a Σ-shaped object, for example, may be an experience in which the perceiver actually sees a Σ-shaped object or an experience in which she only seems to see a Σ-shaped object. 2. There are ways of developing Walton’s imagining seeing account on which it is an elaboration of (A). Looking at Kasimir Malevich’s Suprematist Painting, for example, may elicit a non-veridical visual experience as of a yellow rectangle in front of a green rectangle in front of a black trapezoid, which, in turn, may prompt the viewer to imagine that she is actually seeing a three-dimensionally organized scene with the properties represented in the former non-imaginative experience (Walton 1990: 56). In this sort of case, the content of the experience of imagining seeing is asymmetrically dependent on the content of an experience of the type described in (A).

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3. What about Escher lithographs such as Belvedere, the Devil’s Tuning Fork, and other ‘impossible figures’? These images display 3D structures with locally consistent parts that are assembled in a globally inconsistent way. It is far from clear, however, that we visually experience the global inconsistency of an impossible figure as such. Instead, we may see that the building depicted by Escher’s Belvedere is impossible—in Dretske’s (1969) epistemic sense of ‘seeing that’—by serially inspecting its locally consistent parts and drawing the conclusion that they cannot be coherently assembled (Gombrich 1978; Bayne 2010: 54–55). Julian Hochberg’s work is especially helpful here (see the essays and commentaries collected in Peterson et al. [2007]). According to Hochberg, locally consistent regions of an impossible figure registered across multiple fixations are not normally compared to each other directly. Instead, we notice global inconsistencies when the internal model or ‘schema’ of the virtual object’s 3D structure set up by the depth cues available in one region of the picture is found to conflict with information about the object’s structure available in some other subsequently fixated region. In keeping with this account, perceivers are much more likely to experience an impossible figure as a flat 2D form when globally inconsistent regions are close enough on the pictorial surface to be compared at a single glance. 4. Hatfield (1990) and Smith (2000) provide helpful historical overviews. 5. Compare Hume’s claim that ‘all bodies, which discover themselves to the eye appear as if painted on a plain surface’ (1740/1978: 56). 6. Although multistable perceptual experiences, as we have seen, can occur in which the selected hypothesis alternates from one moment to the next, depending on the allocation of attention and other factors. A clear example here is the flip in depth assignments when viewing a drawing of the Necker cube (Figure 3.2). 7. The affinities described in this paragraph are no accident: Gombrich cites as major influences on his account of pictorial experience Tolman and Brunswik’s ‘The Organism and the Causal Texture of the Environment’ (1935) and Hayek’s The Sensory Order (1952), both of which anticipate ideas central to the Bayesian approach. 8. An online demonstration is available here: www.lottolab.org. 9. Might consciously experiencing the chromatic properties instantiated by a picture’s surface nonetheless have epistemic or computational priority relative to perceiving the way chromatic properties appear to be distributed in pictorial space? Recent metacontrast-masked priming experiments conducted by Liam Norman and colleagues (2014) provide empirical evidence that speaks against this assessment. 10. This is not to deny, of course, that it is usually possible to falsify one (or more) of these interpretations through movement and active exploration in real-world cases (Gombrich 1961/2000: 274–275), that is, by ‘sculpting the flow’ of incoming information from the environment (Clark 2015: ch. 4.) 11. ‘Everyone knows’, Wittgenstein remarks, ‘that a cube which is clearly drawn will be seen three-dimensionally. One might not even be able to describe what one sees in anything other than three-dimensional terms’ (1980: 85). 12. This is sometimes referred to as paradoxical monocular stereoscopy (Schlosberg 1941; Koenderink et al. 1994; Koenderink et al. 2013). 13. Compare Kenneth Clark’s story about ‘stalking’ Velázquez’s painting Las Meninas, discussed in the introduction to Art and Illusion (Gombrich 1961/2000: 6). 14. This is not to claim that visual imagining in general is subject to the will. See Briscoe (Forthcoming) for discussion. 15. I am grateful to Fiona Macpherson for this example. 16. Many different token light arrays, it should be emphasized, can convey optical information about the same spatial attribute. The arrays of light respectively reflected from a cubical wire framework seen face-to-face, a photograph of the framework, and a line drawing of the framework, for example, can convey

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substantially the same information about 3D shape and orientation despite the numerous photometric differences between them (Gibson 1979; Kennedy 1974). Stereopsis is often taken to require (or even to be identical with) the computation of depth from binocular disparities. Vishwanath points out that this is an error since the experience of stereopsis can be readily induced under monocular viewing conditions: when you close one eye, objects in front of you do not seem to be less real or ‘out there’. There is significant debate when it comes to specifying the units in terms of which visual experience ‘measures’ absolute distance and size. Robert Schwartz writes: ‘It has long been recognized that the pictorial cues cannot indicate absolute spatial measures. . . . Of the traditional cues only the nonvisual motor cues of convergence and accommodation might seem to vary directly and unambiguously with distance.  .  .  . Still, in order to evaluate absolute distance it is not enough to have a cue K that varies directly and unambiguously with distance. In addition, we need a scheme for assigning absolute-distance meaning to the values of K. We must know how much distance goes with so much K’ (Schwartz 2006: 23–24). One empirically well-motivated view is that visual experience represents absolute distance in terms of the height of the perceiver’s point of view from the ground plane, that is, in eye-height units. For discussion, see Bennett (2011), Firestone (2013), and Briscoe (2015). Several philosophers including Mohan Matthen (2005: ch. 13) and Bence Nanay (2010, 2014) have proposed that objects in pictorial space are represented by the ventral visual information processing stream in the brain, but are not normally represented by the action-guiding, dorsal processing stream. This, they argue, explains why the ‘feeling of presence’ does not typically attend pictorial experience. The ADSH should be distinguished from the Matthen-Nanay proposal. That absolute distance information required for programming motor actions is not available for the objects of ordinary, non-deceptive pictorial experience, does not entail that the dorsal stream is completely blind to depth and 3D structure in pictorial space (see Briscoe 2016). As Kubovy (1986) points out, however, depth of field is inversely proportional to peephole size. In consequence, a picture seen through a very narrow aperture ‘would be nicely in focus even if the eye accommodated so that its focus plane would be at the distance one might expect the walls of a real room to be’ (1986: 36). If this is right, then the reassignment of accommodation-based distance information to objects in pictorial space, under peephole viewing conditions, may place them at locations significantly further away from the perceiver than the actual location of the 2D pictorial surface. The location of a peephole relative to the pictorial surface and to the physical ground plane also matters. In her study of 17th-century Dutch perspective boxes, Susan Koslow writes, ‘One of the most surprising effects of a perspective box according to Samuel van Hoogstraten was that a figure only a finger’s length appeared as large as life. This illusion was achieved because the figure was viewed from its own eye-level. As a consequence, the spectator felt himself included within the projected space of the perspective box’ (Koslow 1967: 38). I am grateful to David Bennett, Derek Brown, Clotilde Calabi, E.J. Green, Fiona Macpherson, Lisa Mosier, Paul Noordhof, Paolo Spinicci, Tom Stoneham, and Alberto Voltolini for helpful discussions of the ideas presented in this chapter.

References Alais, D. and Blake, R. (2015). Binocular rivalry and perceptual ambiguity. In J. Wagemans (ed.), The Oxford Handbook of Perceptual Organization. New York: Oxford University Press.

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4

Gestalt Perception and Seeing-As Komarine Romdenh-Romluc

‘Normal’ visual experience (i.e. that of an adult human perceiver with no injury or damage to the visual system) immerses the perceiver in a rich, meaningful world. The perceiver sees what appear to be items and features of particular sorts that occupy a surrounding spatial environment. This all-pervasive feature of our experience plays a central role in Merleau-Ponty’s phenomenology. He uses the notion of the Gestalt to capture this aspect of human perception, where this is the idea of a meaningful whole that is composed of its parts in such a way that it cannot be reduced to them—‘it is impossible, as has often been said, to decompose a perception, to make it into a collection of sensations, because in it the whole is prior to the parts’ (1964a: 15). Normal human perception also encompasses a second, very similar phenomenon, illustrated by Jastrow’s (1899: 312) famous duck–rabbit. When a normal human perceiver sees this figure, she does not experience it as a mere collection of colours and shapes; she sees it as having a meaningful form. Indeed, she sees it as either a duck or a rabbit. Typically, the figure will ‘flip’ between the two meanings so that the perceiver sees it first as a duck and then a rabbit (or vice versa). Let us follow tradition and call this ‘seeing-as’. Note that many discussions of seeing-as use the label to refer to more than just this phenomenon.1 However, I use it here to refer to the way in which a figure like the duck–rabbit is typically seen. One might, on the face of it, think that Gestalt perception (as understood by Merleau-Ponty), and seeing-as are one and the same. This impression is seemingly confirmed by the fact that the ‘flip’ in perceptual meaning from duck to rabbit (and vice versa) is known as a Gestalt switch, suggesting that the flip simply involves a change from one Gestalt perception to another. In this chapter, I investigate these two phenomena from the perspective of Merleau-Ponty’s philosophy. I argue that, contrary to first appearances, they are in fact distinct phenomena, and whilst Merleau-Ponty does not provide a thematic discussion of seeing-as experiences, they are important to his understanding of artwork and images, and so I show that a rich account of them can be drawn from his work. My investigation reveals that crucially, Gestalt perceptions are capable of being true or false whilst seeing-as experiences are not truth-apt. This difference is reflected in the way that

90 Komarine Romdenh-Romluc Merleau-Ponty takes Gestalt perception to characterize normal human perception in general, whilst he understands seeing-as to be a distinctive sort of imaginative seeing, which underlies the human capacity for certain kinds of creativity and artistic expression.

Merleau-Ponty and Gestalt Perception: The Problem of Illusion I want to begin by considering Merleau-Ponty’s solution to the problem of illusion. My reason for focusing on this is because—as will become apparent—it serves to highlight an important difference between Gestalt perception and seeing-as. Illusions are misperceptions. A distinction is often drawn between illusions and hallucinations. Broadly speaking, illusions are defined as misperceptions of existing things, whereas hallucinations are perception-like experiences of non-existent entities. They are sometimes treated as being essentially the same sort of state, differing in the degree to which they fail to correctly present the world. However, it is clear from Merleau-Ponty’s discussion in Phenomenology of Perception (Merleau-Ponty 2013) that he takes them to be different in kind. Hallucinations are extraordinary states, often pathological, associated with conditions such as schizophrenia and the ingestion of drugs such as mescaline. Illusions, in contrast, are mistakes made in the ordinary course of perception.2 My discussion here focuses on his treatment of illusion. There is more than one way that illusions raise difficulties for accounts of perception. But the problem that concerns Merleau-Ponty can be understood as follows. Like many other theorists, he holds that perception puts the perceiver in contact with the world, and a satisfactory account of perception should capture this fact. One way to do so is to hold that the content of perception just is the world (or, to be more precise, that part of the world currently in the perceiver’s visual reach). This claim seems to go hand in hand with a second thesis: the content of perception does not represent or depict the world; it merely presents it. An example may clarify what is being suggested here. Imagine I have an empty picture frame that I hold up in front of me. The frame encloses a ‘picture’. However, unlike a painted portrait, the ‘picture’ is not a depiction or representation of the world; it is not distinct from the chunk of the world it ‘shows’. It just is the bit of the world enclosed by the frame. The claim that the world itself is the content of perception is the claim that visual experiences simply present a chunk of the world in a roughly analogous way. The second claim (that visual experiences have no representational content) seems to follow from the first (that the content of perception is the world itself) because it seems that representations must be distinct from the items they represent. For example, the sentence ‘Billy is bored’ represents my dog Billy being in a state of boredom. But this sentence and my dog are

Gestalt Perception and Seeing-As 91 clearly distinct entities—one is a linguistic item whilst the other is a canine being. Moreover, the bored canine just is; he does not represent or depict his being bored. Thus, if perceptions represent the world, it seems that they must be distinct from it. If so, then it is natural to conclude that perceptions are ‘inner’ items, contained within the perceiver’s mind, and if they are within the mind, they could exist there irrespective of how things are with the world. In this way, the contact with the world is lost. Thus, it seems that to capture the claim that perception puts the subject in contact with the world, we must take the world itself to be the content of experiences of it, which subsequently must be thought of as presenting the world, rather than representing or depicting it. Illusions pose a significant problem for this view. As stated above, they are misperceptions: perceptual experiences that show the world incorrectly. If they show the world incorrectly, then it seems they are false—indeed, the latter claim just looks like an alternative way of saying that they are presentations of the world that are incorrect. But to say that something is false necessarily presupposes that it has truth-conditions—it is false by virtue of the fact that these conditions have not been met. To be capable of being false (or true, for that matter) just is to have truth-conditions. By definition, something that has truth-conditions is a representation. It follows that illusions must have representational content and cannot simply present the world to the perceiver. It is then very natural to suppose that perceptions are veridical experiences of the world, that is, experiences whose truth-conditions have been met, which means that they too must be representations. The existence of illusions thus appears to threaten the claim that in perception, the subject is in direct contact with the world. There are three central strategies that have been employed in response to this problem. The first is to claim that whilst illusions are subjectively indistinguishable from perceptions, they are, in fact, different sorts of experiential state. A second strategy is to argue that illusions are not experiential states at all but false judgements or beliefs about what the world is like. Both of these options seemingly allow one to maintain that whilst illusions represent the world as being a certain way, perceptions simply present the perceiver with a chunk of the world and have no representational content. A third strategy is to try to somehow reconceive the content of perception to both retain the contact between perceiver and world and to account for illusions as being the same sorts of states as veridical perceptions. Merleau-Ponty employs the third strategy. He does not consider the first option, but there are prima facie reasons for thinking that it is unsatisfactory. When one undergoes an illusion, it is fairly common for only part of one’s perceptual experience to be illusory (insofar as illusions are defined as partial misperceptions, this is so as a matter of definition). For example, I may correctly perceive my dog sleeping in his bed next to the radiator whilst misperceiving the knot of blanket underneath his head as a ball. It seems very odd to claim that for the bulk of this experience, its content is

92 Komarine Romdenh-Romluc simply a chunk of the world, but the small part that looks to me like a ball is a (false) representation. It is not obvious to me how to make sense of the idea that the content of a single experience can be constituted by both the world itself and representations of it. One could perhaps contend that my perception of my dog sleeping in his bed is not a single experience after all, despite showing a scene that instantaneously occupies my visual field. It is instead a patchwork of smaller experiences, which includes both perceptions and illusions. However, if one takes this line, then it becomes unclear how perceptual experiences should be individuated. Is my experience of my dog’s tail a single experience? Or does it consist of two numerically distinct experiences of his white tail tip and the black part of his tail? Some principled means of deciding is required, but it is unclear what that might be. Moreover, it is often the case that illusions ‘make sense’ to the perceiver, in that once the illusion has been revealed as an illusion (and Merleau-Ponty contends that this is always possible, rejecting as oxymoronic the idea of an illusion that cannot be revealed as such), the perceiver will often recognize why she made the mistake she did. For example, I can see—once my blanket-ball illusion has been corrected—that the knot of blanket looks like a ball, and so it makes sense to me that I previously saw it as a ball. The illusion thus involves some sort of contact with the world. But the motivation for claiming that illusions and veridical perceptions are different sorts of state is that we want to preserve the idea that there is contact between the perceiver and the world in veridical perception, which seems to be incompatible with the idea that such experiences have representational content. If illusions both represent things being thus-and-so, and put the subject in contact of some sort with the world, then the motivation for claiming they are different sorts of state from veridical perceptions is lost. We are instead left with the problem of trying to reconceive the content of perception (and illusion) to both retain the contact between perceiver and world whilst accommodating the fact that these states have representational content, as per the third strategy. Whilst these difficulties may not be insurmountable, they provide motivation for rejecting option one if a better solution can be found. The second option claims that illusions are not experiences at all but judgements. One might initially suppose that this position fares better than the previous one. It can, for example, make sense of the idea that illusions involve some sort of contact with the world. When I ‘misperceive’ my dog’s blanket as a ball, what actually happens is that I perceive his blanket—and so am in contact with the world—but mistakenly judge that it is a ball. Moreover, this judgement makes sense, since his blanket bears some resemblance to a ball. However, Merleau-Ponty offers a significant objection to the claim that illusions are really judgements. He argues that a proponent of this view—which he categorizes as ‘intellectualist’—must say whether or not the judgement is based on perception, but in answering they face a dilemma. If it is based on perception, then the experience must itself be illusory, in

Gestalt Perception and Seeing-As 93 which case the judgement plays no explanatory role, and an account of illusion is still required. If it is not, then it is unclear why the perceiver makes the judgement, and—perhaps more importantly—why it seems to her to be connected with her perceptual experience (indeed, it seems to the perceiver that it is a perceptual experience). Merleau-Ponty’s objection applies to even fairly sophisticated contemporary accounts that attempt to retain the claim that the content of perception is the world itself, whilst taking illusions to be judgements. One example is the view offered by Travis (2004). His account is interesting in the present context as he can be read as sharing some of Merleau-Ponty’s commitments. Moreover, his explanation of illusions as judgements seems at first to escape Merleau-Ponty’s dilemma. However, on closer inspection, Travis’ account does, in the end, fall foul of this objection. A grasp of where Travis goes wrong will aid us in understanding Merleau-Ponty’s alternative. It is thus to Travis’s account that I now turn. Travis holds that perception is directly of the world and so cannot have any representational content. He takes illusions to be false judgements about the world, which are based on the subject’s perceptual experience but in a way that does not require that experience to be illusory or represent anything. He develops his account by appealing to a distinction between two different senses of the word means. In one sense, talk of what something means is talk about what it conventionally signifies, that is, its representational content. For example, if I say the French sentence ‘il pluit’ means it is raining, I am saying that ‘il pluit’ represents this state of affairs. But there is another sense of means that is not about representation. This is what Grice (1989) calls ‘natural’ meaning.3 One item or event means another in this sense if they are connected such that they generally occur together. For example, the sentence ‘smoke means fire’ should not be read as claiming that smoke has the representational content ‘fire’. Instead, this sentence says that smoke and fire usually occur together. Similarly, ‘the presence of this virus in someone’s bloodstream means the person has measles’ says that a particular virus generally co-occurs with measles. Again, it does not imply that the virus has any representational content. (Indeed, it’s not clear that it makes sense to talk of a virus representing anything.) I use ‘means’ to refer to the former sort of meaning, and ‘means-n’ to refer to natural meaning. Travis then claims that an illusion is an erroneous judgement about what some perceived item or event means-n. For example, a perceiver sees smoke. Knowing that smoke is generally accompanied by fire, she judges, ‘this smoke means-n there is a fire’. However, smoke does not always co-occur with fire. Instead, smoke can be produced by a smoke machine or by a group of people enjoying cigars. If the smoke the perceiver sees is accompanied by either of the latter, rather than by a fire, her judgement will be false. Travis claims that in this case, the perceiver undergoes an illusion. His account seems to sidestep Merleau-Ponty’s dilemma in the following way: the first horn of Merleau-Ponty’s dilemma holds that if the judgement is based on the subject’s

94 Komarine Romdenh-Romluc perceptual experience, then the experience must itself be illusory. The second horn states that if the experience is not illusory, then the judgement has no basis in experience. But on Travis’s account, as the smoke example above makes clear, there is nothing illusory about the subject’s experience. It simply presents her with a chunk of the world. Yet the erroneous judgement that constitutes the illusion is based on the subject’s perceptual experience—it is a judgement about what usually co-occurs with the perceived item or event. Thus Travis’ account seemingly allows him to hold on to the idea that the content of perception is the world itself and does not represent anything, whilst providing a connection between the perception and the erroneous judgement that constitutes the illusion. However, as it stands, this is inadequate as an account of perceptual illusion. Perceptual illusions surely involve error about something that is currently seen. But at least some judgements about what things mean-n concern things that are not currently in view. Suppose, for example, that I see Fred eating a lot of mangos and judge it means-n that he will feel ill later tonight. On Travis’s account as I have expounded it so far, I will have undergone a perceptual illusion when I saw Fred eating a lot of mangos if Fred does not feel ill later. But it is deeply counter-intuitive to suppose that what does or does not happen to Fred later determines whether or not my earlier experience was illusory. In fact, Travis does not claim that all false judgements about what something means-n count as illusions. It is instead only a subset of such judgements—those that are motivated by the ‘look’ of what one sees, that is, the way that it appears. Thus, it is judgements such as ‘this “look” means-n p’ that are candidates for perceptual illusions. But this restriction is still not sufficient; some judgements about what a ‘look’ means-n will still concern things that are not perceived. Thus, some false judgements about what a particular ‘look’ means-n will not count as perceptual illusions. I might, for example, judge ‘this “Fred-eating-a-lot-ofmangos look” means-n that Fred will feel ill later tonight’. Intuitively, the failure of Fred to feel ill later does not render my current experience illusory. To deal with this problem, the subset of candidate judgements needs to be further restricted. They must be judgements about what a ‘look’ is of. An example is ‘this “look” means-n that there is a cow in front of me’, where this can be paraphrased as ‘this “look” is that of a cow’. On Travis’s analysis, this judgement states that a particular cow-ish ‘look’ is accompanied by the presence of a cow in the same way that ‘this smoke means-n fire’ states that a particular instance of smoke is accompanied by the presence of fire. The claim is that given this analysis of judgements about what a ‘look’ means or is of, allows us to hold that ‘looks’ have no representational content. Just as smoke does not represent anything, neither do ‘looks’. However, now Travis’s account faces a problem. It’s not at all obvious that ‘looks’ do not represent things as being thus-and-so. On the contrary, to identify an appropriate subset of judgements as candidates for perceptual illusions, Travis has to appeal to what a ‘look’ is of, and to say that a ‘look’

Gestalt Perception and Seeing-As 95 is of something is surely to say that it depicts things as being a certain way. Moreover, Travis’s analysis of judgements that are motivated by ‘looks’ as judgements about what a ‘look’ means-n does not rule out this possibility, because we can make such judgements about things that have representational content. Suppose, for example, that I watch a documentary about the Vietnam War and judge that its content means-n that Kissinger bribed the producer. My judgement is based on my belief that documentaries with such content tend to co-occur with instances of producer-bribing by Kissinger. It is thus analogous to my judgement that the smoke I see means-n there is a fire in the vicinity. But unlike the smoke, the documentary has representational content—it depicts events relating to the Vietnam War—and it may be true or false. Thus, even if Travis is correct in claiming that the judgement ‘this “look” means-n there is a cow in front of me’ is analogous to ‘this smoke means-n fire’, there is still scope for arguing that ‘looks’ have representational content. Travis must rule out this possibility to defend his thesis that perception does not represent the world as being thus-and-so. Travis offers the following argument to establish that ‘looks’ do not represent. People can and do have perceptual illusions about all sorts of things. These illusions consist in false judgements about what a certain ‘look’ is of. One judges that a particular ‘look’ is accompanied by the presence of object a, when in fact—on this occasion—it is accompanied by the presence of object b. Why might someone make a false judgement of this sort? It must be, according to Travis, because the particular ‘look’ in question is sometimes accompanied by the presence of b. Thus, what this mistake reveals is that just as smoke can mean-n either the presence of fire or the presence of a dry-ice machine, so, too, can the same ‘look’ be of/mean-n either a or b. Since a great variety of perceptual illusions are possible, it seems that the same ‘look’ can be of/mean-n an indefinite number of things. But it is essential to a representation that it shows one particular way the world might be. It follows that a ‘look’, which can be of an indefinite number of things, cannot represent things as being thus-and-so. Suppose, for example, that I misperceive a cement mixer as the back of a horse. On Travis’s account, my illusion consists in the following false judgement ‘this “look” is of (means-n the presence of) a horse’. I make this false judgement because the same ‘look’ can be accompanied by either cement mixers or horses. To put the matter more simply: a cement mixer—in a certain light, from a certain angle, and so on—looks indistinguishable from a horse. Moreover, since one may misperceive the cement mixer as a good many other things—a cow, the back of a sheep, the front of a tractor, and so on—the same ‘look’ can be of/means-n a cement mixer, a horse, a cow, the back of a sheep, the front of a tractor, and so on indefinitely. The ‘look’ thus has no determinate content. It does not represent anything. It should now be clear that Travis has not escaped Merleau-Ponty’s dilemma. The claim that ‘looks’ have no determinate content means that the way things look in no way determines for the perceiver what it is she

96 Komarine Romdenh-Romluc perceives. In fact, Travis’s account of ‘looks’ as indeterminate entails that things do not look a certain way to the perceiver—they look no way at all. It follows that the perceiver’s judgement that a certain ‘look’ is of object a rather than object b receives no support from the way that things look. (Indeed, it is not clear that she could even pick out a certain ‘look’ in order to make this judgement in the first place.) Rather than being based on perception, judgements of this sort are wild stabs in the dark. It is thus unclear why the perceiver makes the judgement and why it seems to her to have any connection with her perceptual experience. This is not a satisfactory account of illusion. More generally, Travis’s theory of perception, which claims that how things look in no way determines for the perceiver what it is she perceives, is inadequate as an account of perception. Merleau-Ponty’s solution to the problem of illusion can be usefully set against the failures of Travis’ account. Our discussion of Travis helps bring home the point that illusions cannot be explained away as false judgements about perception. They are false (or non-veridical) perceptual experiences, which means that they must have representational content. Moreover, since, according to Merleau-Ponty, perceptions and illusions are not different types of experience, it follows that perceptions, in general, must have representational content. As we saw above, it is tempting to think that once perception is claimed to have representational content, a link with the world is lost. The challenge Merleau-Ponty thus faces is to conceive of perception’s representational content in such a way as to maintain the contact between the perceiver and the world. Merleau-Ponty does this by using the notion of the Gestalt. Typically, human visual experience is not a disorganized mass, but is arranged into meaningful wholes: Gestalten, or what Merleau-Ponty sometimes calls ‘physiognomies’. Merleau-Ponty holds that experience has a Gestalt structure ‘all the way down’. Even the simplest visual experience has a physiognomy. Consider, for example, my current visual perception. The scene as a whole has a certain form. I am presented with a room. But if I focus on just one part of the room, such as the stool in front of me, that, too, has a form. Similarly, if I focus on just one part of the stool, such as its leg, that also has a form. If I focus on just one part of the stool leg, my visual experience is again meaningfully structured so that I am presented with a textured surface, and so on. At no stage will I reach an element of experience that is unstructured. Gestalten have, for Merleau-Ponty, the following important feature: they are composed of their parts in such a way that they are nothing over and above the sum of their parts, yet they are not reducible to them. We can gain a better grasp of this claim by considering the experience that Merleau-Ponty describes in the following passage: If I am walking on a beach towards a boat that has run aground, and if the funnel or the mast merges with the forest that borders the dune, then there will be a moment in which these details suddenly reunite with the boat and become welded to it . . . I merely felt that the appearance of the

Gestalt Perception and Seeing-As 97 object was about to change. . . . The spectacle was suddenly reorganized, satisfying my vague expectation. (Merleau-Ponty 2013: 17–18) Here, Merleau-Ponty describes what is known as a Gestalt switch. He first has a perception characterized by a trees-Gestalt; that is, he is presented with what appear to be trees. His perception then abruptly changes so that he is presented with what appear to be a ship’s masts; that is, his perception takes on a ship’s-masts-Gestalt. Notice that the components of what is seen do not alter with the change in perceptual meaning—he continues to see the same arrangement of vertical elements. Since the same elements can be unified as different Gestalten, the scene’s physiognomy as either trees or ship’s masts cannot be simply reduced to those elements. Nevertheless, the scene’s physiognomy as trees or ship’s masts is not something extra to the elements of the scene. No trees or masts remain if the vertical structures are taken away. Thus, the Gestalt is nothing over and above its parts, but it is composed of them in such a way that it cannot be reduced to them. Merleau-Ponty accounts for these two features of Gestalten—that they are nothing over and above their parts; but that they cannot be reduced to them—in the following way. He argues that they are uniquely perceptual phenomena, brought about when consciousness makes perceptual contact with the world. They are the forms the world takes on in perception. They do not exist in the world-in-itself, as he makes clear in remarks such as this—‘form is not a physical reality, but an object of perception . . . form cannot be defined in terms of reality’ (Merleau-Ponty 1963: 143). Gestalten are nothing over and above their parts because they must be realized in some sensuous matter. The perceived form does not exist without this matter because it just is the form taken on by the matter in perception: It is the notion of an order of meaning which does not result from the application of spiritual activity to an external matter. It is, rather . . . an earthy and aboriginal sense, which constitutes itself by an organization of the so-called elements. (1964b: 77) Merleau-Ponty holds that Gestalten are realized in the matter of the world itself. They are not, however, reducible to this matter because Gestalten are not something the world possesses independently of consciousness. The vertical structures in Merleau-Ponty’s example remain constant throughout the switch in perceptual form from trees to ship’s masts. What changes is the way the perceiver sees the structures. Yet the perceiver does not ‘add’ Gestalten to worldly matter from nothing. Instead, she picks out or discerns a pattern or form in it. Think about, by way of analogy, seeing pictures in the clouds. One may look up at the sky and see rabbits in the clouds. The clouds in themselves are not arranged as a rabbit. They simply occupy a certain spatial location

98 Komarine Romdenh-Romluc in the atmosphere above the earth. They only have this physiognomy when someone looks at them. But the person looking does not create the rabbit ex nihilo; she finds it in the configuration of the clouds. It is usual to think that a visual perception is correct insofar as its content faithfully copies or reproduces the part of the world that is perceived, and incorrect insofar as it does not. Since the Gestalten that characterize visual experiences are not mind-independent features of the world, Merleau-Ponty cannot account for the correctness or incorrectness of visual experience in this way. He offers the following alternative. The Gestalten that characterize perceptual experience refer to the perceiver’s abilities to act. The entities I come across in the world are each correlated with a certain set of possibilities for action that serve to define them as the sorts of entities they are. A football, for example, allows for a particular range of kicking, holding, throwing, slipping out of hands when wet, deflating, inflating, and bursting actions (there are also many other things one might do with a football). The actions I can perform when I encounter some entity on a particular occasion are further circumscribed by the location of that entity within the environment and its relations to other objects in the surrounding world. A football coming towards me from a certain direction offers me opportunities to perform a range of kicking actions, a diving-on-the-ground-to-stop-it action, and so on. Thus, when someone perceives the world, she experiences it as inviting her to interact with it in various ways. Merleau-Ponty then claims that a perception is correct if the opportunities for action it offers to the subject are ones that she can really, in principle, take up. The perceiver’s experience of the football is correct if it really could be kicked. A perception is incorrect if the opportunities for action it offers the perceiver could not be taken up. Merleau-Ponty describes the experience of seeing what is in fact a patch of sunlight as a stone, lying in one’s path. One’s perception presents the stone as something that one could walk on, pick up and skim across the water, kick against a nearby tree trunk, and so on: I see the illusory stone in the sense that my entire perceptual and motor field gives to the light patch the sense of a ‘stone on the lane’. And I already prepare to sense this smooth and solid surface beneath my foot. (Merleau-Ponty 2013: 310) However, as one gets closer, one realizes that the ‘stone’ turns out to be a patch of sunlight. The possibilities for action are not ones the perceiver can really take up, and so the experience of the stone is an illusion. We are now in a position to see how Merleau-Ponty solves the problem of illusion. Illusions are misperceptions; that is, they are incorrect experiences. For them to be incorrect, they must have truth conditions, and if they have truth conditions, then they are representations. Conversely, veridical experiences are correct experiences. For them to be correct, they must have truth conditions, and so they must also be representations. The problem is that if

Gestalt Perception and Seeing-As 99 visual perceptions are representations, then it seems we can no longer hold on to the idea that perception puts the perceiver in direct contact with the world. This is because it seems that representations must be distinct from what they represent. Just as a sentence that represents my dog being bored is a separate linguistic entity, which is distinct from my dog and his boredom, so, too, it seems that a visual representation of some scene is distinct from that scene. It is then natural to suppose that visual perceptions are representations that are contained within the mind. Merleau-Ponty aims to solve this problem by conceptualizing the content of perception in such a way that it can be right or wrong (and so have representational content) whilst maintaining the direct connection with the world. He does this by employing the notion of the Gestalt. As we have seen, he holds that visual experiences have a Gestalt form. The form is one that offers the perceiver opportunities to act. An experience is veridical if she can really take up these opportunities and illusory if she cannot. The forms taken on by the world in perception are the result of the perceiver discerning patterns or Gestalten in worldly matter. The Gestalten are realized in the matter of the world itself and cannot exist without being so realized. In this way, visual experiences—whether they are veridical or illusory—are essentially world-involving. Their content is the sensuous matter of the world, characterized as having a certain form. The perceiver is thus still in direct contact with the world when she undergoes an illusion. Merleau-Ponty writes, ‘[A] mescalin intoxication can give animal appearances to objects and make an owl out of a clock without any hallucinatory image whatsoever’ (Merleau-Ponty 1963: 168).

Seeing-As Experiences For Merleau-Ponty, normal human perception always has a Gestalt form. The perceiver is presented with a rich world of meaningful things and places, rather than a disorganized mass of sense-data. Normal human perception also encompasses another phenomenon that I refer to here as seeing-as. These are experiences typified by a normal perception of Jastrow’s duck–rabbit, which—as the name suggests—appears to the perceiver as either a duck or a rabbit, and tends to ‘flip’ between the two meanings or Gestalt forms. One might initially suppose that seeing-as just is Gestalt perception. Both Gestalt perceptions and seeing-as experiences have a physiognomy or meaningful form—in neither case is the subject presented with a disorganized collection of impressions. Moreover, the ‘flip’ in meaning that one typically experiences when looking at something like the duck–rabbit is known as a Gestalt switch, which suggests that the flip simply involves a change from one Gestalt perception to another. Finally, Merleau-Ponty appeals to Gestalt switches when discussing how visual illusions are corrected in experience. In the example given above, Merleau-Ponty experiences a Gestalt switch from the illusion of the ship’s masts as trees to the veridical perception of them as masts. The ‘flip’ in perceptual meaning seems to be exactly the same sort of

100 Komarine Romdenh-Romluc abrupt change one experiences when the look of Jastrow’s figure alters from a duck to a rabbit. However, whilst Gestalt perception and seeing-as clearly share some features, there also seems to be an important respect in which they differ. Gestalt perceptions are capable of being true or false. As we saw above, this plays a pivotal role in Merleau-Ponty’s solution to the problem of illusion. The peculiar nature of Gestalten allows him to account for perceptual experiences as having representational content whilst maintaining the idea that they put the subject in direct contact with the world. In contrast, it does not seem that seeing-as experiences can be true or false. There is something odd about claiming that someone’s experience of the duck–rabbit as a duck (or rabbit) is veridical, or that her perception of it as a rabbit (or duck) is illusory. One might respond by claiming that this is because the duck–rabbit is precisely an ambiguous figure that admits of two perceptual meanings. To see it under just one meaning is thus not entirely correct, but neither is it exactly an illusion. Yet this does not quite get at the oddness of claiming that a perception of the duck–rabbit as either a duck or a rabbit is veridical or illusory. The strangeness of this claim can be further illustrated by noting that whilst most people can see the duck–rabbit as either a duck or a rabbit, and they can see the meaning of the figure as switching between these alternatives, we would not say of someone who failed to see the figure in this way that his or her experience was incorrect or illusory. Suppose we accept this intuition and take Gestalt perception and seeing-as to share some features but differ in this one important respect. How should we explain their similarities whilst accommodating the difference between them? More specifically, how might we do this from within Merleau-Ponty’s framework? To see this, we need to know more about his account of perception. We saw above that he takes perceptual experience to be characterized by Gestalt forms that refer to the perceiver’s capacities for action. In other words, the perceiver is presented with a world that offers her opportunities to engage with it in various ways. A tree may be perceived as for-climbing, a bicycle as for-riding, a cliff edge as for-avoiding, and so on. Each object is correlated with a particular set of possibilities for action that it offers to human perceivers. These serve to define it as being an entity of a particular type. Merleau-Ponty—as is well known—argues that experience has this character due to the connection between perception and the perceiver’s motor skills. He tends to call the latter ‘habits’ (Merleau-Ponty 2013). Motor skills are developed by practising the activity in question, which allows the body to become familiar with the behaviour. During the initial phases of learning a new skill, the bodily movements required to exercise it will typically feel alien and awkward, and one’s attempts to perform them are likely to be clumsy and involve greater effort. But as one practises, the movements will feel more and more familiar; one’s performance of them will become more graceful and fluid; and exercising the skill will require less effort. Imagine

Gestalt Perception and Seeing-As 101 learning to play a drum kit. At first, one will be unused to the bodily position one must take up to hold the drumsticks and to reach the pedal for the bass drum. One will find it difficult to keep time on the bass drum whilst playing a more complicated rhythm on the snare and cymbals. The movements required to strike the drums with the sticks will feel awkward, and one may hit them with varying amounts of force rather than playing evenly. But through practice, one’s body will start to find the activity familiar. The position and movements will feel less awkward. It will become easier to coordinate one’s limbs in the manner required to play the different parts of the kit. One’s playing will become fluent. Motor skills or habits do not just involve patterns of bodily movement; to acquire a motor habit is also to gain a new way of perceiving the world. To exercise any skill, one must be in the right sort of worldly place. I can only exercise my skill at playing volleyball, for example, if I am in an environment that contains a ball, a net, a suitable surface, and a partner. I cannot engage in this activity if I am seated in a theatre. Furthermore, in order to exercise a skill, I must also know how to move my body to perform the relevant actions. To return the volley ball over the net, I must know which movements are needed for my fists to make contact with the ball and strike it at the right angle and with sufficient force to send it back to my partner (there may be a number of different movements that will achieve this objective). It follows that, for any motor skill, someone who possesses it will have the ability to pick out environments where it can be exercised, and the capacity to select the bodily movements needed to exercise the skill in those environments. On Merleau-Ponty’s account, these abilities are manifest in the perception of the appropriate parts of the surrounding world as offering an opportunity to exercise a particular skill by executing a particular set of bodily movements. I see the volleyball coming towards me over the net, as requiring me to run to this particular spot, hold my arms at this angle, and hit it upwards with this amount of force and in this direction to return it over the net. On Merleau-Ponty’s account, the subject’s motor skills mean that her body is attuned to certain possibilities for action—those offered by environments in which she can exercise her skills. When the person enters such an environment, it ‘suggests’ to her body how she might interact with it. These ‘suggestions’ are then taken up in perception so that the subject perceives it as inviting her to act in those ways. Gestalt perception aims to faithfully track the contours of the world. For Merleau-Ponty, this means responding to the world’s ‘suggestions’ by picking out Gestalten in the world’s matter that present the perceiver with opportunities for action that she could really take up. However, the normal human perceiver can also pick out Gestalt forms in an imaginative or playful way, where this activity does not aim to reveal the world’s nature. Imagine, for example, gazing at a richly patterned carpet. A particular collection of swirling ornaments may take on, for the perceiver, the form of a horse. This

102 Komarine Romdenh-Romluc imaginative seeing is seeing-as. Although seeing-as does not aim to faithfully track the world’s nature, it nevertheless involves finding or picking out a Gestalt form in the world rather than simply creating one from nothing. Just as the large equine being in the field down the road ‘suggests’ a horse Gestalt to me (or more properly, to my body), so that I perceive him as a horse, so, too, the patterns in the carpet ‘suggest’ a horse-ish form to me so that I see a horse in the swirling ornaments. However, the ‘prompt’ from the world in seeing-as is more indeterminate and ambiguous than in Gestalt perception. Creativity is thus required to take up the world’s ‘suggestion’, and the resulting experience will have more to do with the subject and how he or she has responded than with the nature of the world.4 The subject’s motor skills also play a very similar role in seeing-as to the one they play in Gestalt perception. As we saw above, the subject’s skill at doing x attunes her body to places in the world where x can be done. When the person enters such an environment, the world ‘suggests’ to her body that this skill can be exercised there. The subject takes up this suggestion and perceives that environment as offering an opportunity to do x. In seeing-as, the world makes an ambiguous suggestion that the subject’s body finds somehow reminiscent of, or to resonate with, promptings connected with doing x. The subject—or her bodily self—responds by taking up this indeterminate suggestion to yield an experience of seeing-as x. For example, the subject’s skill at riding a motorbike sensitizes her to places in which this skill can be exercised—say, those containing a motorbike and an open road. The same skill also attunes her to the promptings from the world that are reminiscent of places that are appropriate for the exercise of that skill. The curve of an abstract doodle may be reminiscent of the curve of a motorbike wheel, and so she sees the doodle as a motorbike wheel, that is, part of a machine that is for-riding. The way in which the body finds the world’s promptings similar or alike cannot be further analysed. Indeed, it pertains to what Merleau-Ponty calls ‘the enigma of the body’ (Merleau-Ponty 1964c: 164). In both Gestalt perceptions and seeing-as experiences, the subject’s motor skills attune her to the world so that she is sensitive to its promptings. She responds to these by perceiving the world as offering her opportunities to exercise her skills. However, the two differ in an important respect: Gestalt perceptions are capable of being true or false, whereas seeing-as experiences are not. This is, of course, connected to the fact that Gestalt perceptions aim to track the contours of the world in terms of its value for one’s actions, whilst seeing-as experiences do not. This is not to say that seeing-as experiences are completely divorced from the nature of the world, since—as stated above—they are undergone in response to the world’s promptings. Where an experience aims to track the world’s nature in terms of its possibilities for action, the subject is committed to things being as they appear. Conversely, where an experience does not have this aim, the subject is not thus committed. It is because one is committed in

Gestalt Perception and Seeing-As 103 the case of Gestalt perception that one’s perceptual experience is capable of being veridical or illusory. In being committed to things being F, one has taken some sort of stance concerning the nature of the world, and one can be either right or wrong about its being that way. Conversely, the lack of commitment in cases of seeing-as explains why seeing-as is not capable of being either veridical or illusory. But whilst there is something right about this thought, cases of known illusion present a problem for this way of putting things. These are cases that we classify as illusions, which means they are capable of being true or false. Yet it does not seem that the perceiver is committed to the content of her experience’s being the case, because she knows that her experience is illusory. For example, the Müller-Lyer lines are classed as an optical illusion because one sees what are in fact equal lines as being unequal in length. Nevertheless, one may know that they are really equal, and so one is not committed to the lines being unequal lengths. However, Merleau-Ponty’s framework offers a way to deal with this problem. The objection above understands being committed to p as knowing that p. It is plausible to think that the sort of knowledge at stake is propositional knowledge, which is constituted by true beliefs (formed in the appropriate ways, or with adequate justification). The perceiver is thus committed to p in virtue of her mental states. Let us call this sort of commitment, ‘cognitive commitment’. Merleau-Ponty’s framework allows us to identify another form of commitment, which we can call ‘bodily commitment’. Bodily commitment to p consists in the body’s responding as if p were true. This can include emotional responses, bodily expectations about the sort of actions that can be performed that are constituted by the body’s readying itself for action, and actually acting in ways that are appropriate to p’s being the case. On Merleau-Ponty’s picture, our cognitive and bodily commitments can influence each other, and in many cases they are in line so that one acts, responds emotionally, readies oneself for action, and so on in ways that are consonant with how one believes the world to be. However, one’s cognitive and bodily commitments need not line up in every case. Certain experiments conducted by Rozin and colleagues (1986) nicely illustrate this point. They showed that subjects are reluctant to eat a piece of fudge shaped like dog faeces, even when they know it is fudge and therefore perfectly safe to eat. In this case, the subjects are cognitively committed to the item’s being fudge. But they are bodily committed to its being excrement. This bodily commitment manifests in disgust, shrinking away from the item as it approaches one’s mouth, shuddering as one places it near one’s lips, and so on. The notion of bodily commitment allows us to distinguish between Gestalt perception and seeing-as in terms of being committed. It explains why seeing-as cases are not illusory, whilst accommodating known illusions. We can say that a Gestalt perception involves bodily commitment to p, and it is this that means it is capable of being veridical or illusory. When I see my dog in front of me, I am bodily committed to his really being there. My

104 Komarine Romdenh-Romluc bodily responses constitute my taking a stance on the nature of the world in front of me, and this opens the possibility of my stance being either right or wrong. Seeing-as, in contrast, does not involve bodily commitment. The lack of bodily commitment means that the subject’s experience is not of the right sort to be either illusory or veridical. When I see the duck–rabbit as a duck, I am not bodily committed to its being a duck. Since I do not take a stance on this issue, there is no possibility of my stance being either right or wrong. My experience is thus incapable of being either veridical or illusory. In cases of known illusion, the perceiver’s bodily and cognitive commitments come apart. She knows that what she sees is an illusion, and so lacks cognitive commitment to it, whilst nevertheless being bodily committed to what she sees. In a case where the perceiver sees the Müller-Lyer lines as unequal whilst knowing that they are, in fact, the same length, the perceiver is not cognitively committed to their being unequal, but she is bodily committed to this state of affairs, and so her experience can be classed as an illusion. The difference in commitment between Gestalt perceptions and seeing-as experiences also shows up in their phenomenology. We have already noted that both experiences are similar insofar as they both have a Gestalt structure— the subject is presented with something (or some things) that have a particular form, rather than with a disorganized mass of impressions. Gestalten, in Merleau-Ponty’s view, refer to the perceiver’s capacities for action. As noted above, Merleau-Ponty takes each entity that one might encounter to correspond to a cluster of possibilities for action, which serve to define entities of that particular sort. To see something as having a particular Gestalt form is thus to see it as offering one (some of) the possibilities for action that are typical for such objects. This is true for both Gestalt perception and seeing-as experiences. However, these possibilities are experienced differently in each case. In Gestalt perception, one experiences the world as really having the form it takes in one’s perception of it. One experiences the possibilities for action it offers as being such that one could really take them up. But in seeing-as, one experiences the Gestalt form as ‘laid over’ what one perceives to be the real nature of the world. For example, when I see a horse standing in a field (Gestalt perception), my experience presents the world as really being such that there is an equine being in front of me. In contrast, when I see the carpet ornaments as a horse, I continue to perceive the carpet as really being a carpet (I continue to have a Gestalt perception of it as a carpet). We can understand this experience of a form being ‘laid over’ the world’s real nature as follows. She perceives a part of her environment as offering her conflicting possibilities for action. The subject’s experience presents her with opportunities to act that correspond with the real nature of the world. But it also presents her with conflicting opportunities that constitute the Gestalt form that is ‘laid over’ the world’s real nature. In our example, the carpet invites the perceiver to walk on it, to lie on it, to ruffle his or her toes

Gestalt Perception and Seeing-As 105 in its pile. At the same time, the horse that the subject imaginatively sees in its ornaments invites her to stroke its nose. The imaginatively seen possibilities for action (i.e. those that constitute the seeing-as experience) are far less numerous than those she perceives to be real. The subject experiences the horse imaginatively seen in the carpet’s ornaments as offering her far fewer possibilities for action than a real horse—she does not experience it as something that can be ridden or fed. In contrast, she experiences the carpet as offering her a sizeable number of the possibilities for action typically associated with carpets. An imaginatively seen entity thus lacks the fullness of an object presented in a Gestalt perception, which is why the subject is not bodily committed to its presence.5 In this way, imaginatively seen entities are presented as overlaying the world’s real nature, rather than being a real part of it. Merleau-Ponty writes that [t]hey are not there in the same way as [things presented in Gestalt perception]. But they are not elsewhere . . . I do not look at [an imaginatively seen entity] as I do at a thing; I do not fix it in its place. (Merleau-Ponty 1964c: 164) Finally, the capacity to see-as underlies, for Merleau-Ponty, the human ability for certain kinds of artistic expression. Some works of art present their audience with recognizable entities. These may be real—as in the case of a portrait of Merleau-Ponty. Or they may be merely imaginary—like a picture of a frog painted with no particular frog in mind. We can say that works of art like this are of things. They are images of them. The relation between a work of art and what it is of, where the artwork is an image of some entity or entities, is peculiarly intimate. Unlike the relation between, say, me and my name, which is largely arbitrary (I could have been given a great many other names), works of art that are of things are likenesses of them. Not any old likeness of an entity can be classed as a work of art or as an image of it. For one thing, art works must clearly be produced by humans or other similar creatures. An accidental likeness of something, such a patch of lichen that resembles a face, will not count as a work of art. Neither are images mere replicas or clones of the things they depict. One white coffee mug is a replica or clone of a second—qualitatively identical—mug. But it is not an image of it, much less an artwork that is of it. Instead, artworks that are of things somehow translate them into different media. For example, a still life of a French gîte translates the dwelling into paper and ink. Since this is so, an artwork cannot be a likeness of its object in virtue of sharing a number of its properties. Whilst the two will have some qualities in common, the fact that the former is a translation of the latter means that they will not share enough properties for this to be what it is for one to be a likeness of the other. The still life of the French gîte is flat, rendered in shades of grey and black, and made from paper and ink. But the gîte itself is a multi-hued home built from wood and brick. It is not flat but takes up a volume of space. It

106 Komarine Romdenh-Romluc has an inside where people may live and secret crevices where insects and small mammals may make their homes. Merleau-Ponty holds that those art works that are of things are likenesses of them in that perceivers will see them as being of their objects. The account of seeing-as presented above explains what this means. A perceiver is able to see an artwork as being of a particular object because she possesses motor skills that could be used to interact with objects of that sort. Those skills attune her body to ‘prompts’ from the world that it immediately finds reminiscent of ‘prompts’ from those situations in which the skills in question could be exercised. The ‘prompts’ are taken up in perception so that the perceiver sees the world as having a particular Gestalt form (one that is constituted by a particular cluster of possibilities for action). However, she does not see the world as really being that way. Instead, she sees the form as ‘overlaying’ the world’s real nature. Someone looking at the picture of the French gîte sees the form of a dwelling ‘laid over’ paper and ink. Merleau-Ponty also uses his analysis to explain what the artist does when she creates an image. The artist rearranges some part of the world so that others can see it as something. Sometimes, the artist will first have a seeing-as experience and then emphasize the relevant features of the world so that others can see it as she does. Arguably, this is the case for (certain aspects of) the cave paintings at Lascaux. Merleau-Ponty talks of them as ‘using the wall’s mass’ (1964c: 164), and looking at the paintings makes it clear what he is referring to here. Real features of the wall are incorporated into the designs so that a bulbous mass of rock becomes the clouds above the animals’ heads, and strata lines in the rock below the figures are emphasized to become the ground beneath their hooves. One can imagine the artists first seeing those elements as sky and ground, then emphasizing them with paint so that we can see them as such, too. In other cases, the artist’s work will not begin with a seeing-as experience. She will, instead, begin to alter the world from nothing, as it were, so that something that she and others can see as an object gradually takes shape. This is so when the artist produces a sketch, which begins as a blank sheet of paper, which she alters with pencil marks so that it can be seen as something.

Conclusion In this chapter, I examine two related phenomena—Gestalt perception and seeing-as—and explained how Merleau-Ponty analyses them. Both types of seeing involve the use of the perceiver’s motor skills, which she uses to respond to ‘prompts’ from the world. In the case of Gestalt perception, these ‘prompts’ are taken up in perception to yield an experience of the world as really having a certain form. The perceiver is committed to the world’s being the way it is presented. In other words, she—or more properly, her bodily self—takes a stance on things being a certain way, and so the experience may be veridical or illusory. Nevertheless, in both cases, the perceiver is still in

Gestalt Perception and Seeing-As 107 direct contact with the world. Merleau-Ponty accounts for this by holding that a Gestalt perception—whether veridical or illusory—is constituted by a stretch of the world, perceived in a certain way. The perceiver discerns or picks out a Gestalt form in the matter of the world. Seeing-as experiences, in contrast, do not present the world as really having the seen-as form. Instead, the perceiver experiences a Gestalt form as ‘laid over’ the world’s real nature. She is not committed to the world’s being this way, and so seeing-as experiences are not capable of being true or false. Merleau-Ponty takes the capacity to see-as to be what underlies the creation and appreciation of art works that are images of their objects. The artist alters the world so that others can see it as exhibiting a particular form.

Notes 1. See, for example, the discussion of Wittgenstein on seeing-as in (Hunter 1981). 2. I discuss Merleau-Ponty’s account of hallucination in Romdenh-Romluc (2009). 3. Travis himself does not make the connection with Grice’s work. But the second notion of meaning he identifies is clearly the one that Grice calls natural meaning. 4. There are other sorts of actions that are relevant here, for example, ‘mental’ actions such as categorizing the imaginatively seen form in certain ways alongside other horse-ish forms. I leave these aside here as there is insufficient space to discuss this adequately. 5. We might say that the imaginatively seen form is grasped by the body, but the body is not committed to its presence.

References Grice, P. (1989). Studies in the Way of Words. Cambridge, MA: Harvard University Press. Hunter, J. F. M. (1981). Wittgenstein on seeing and seeing as. Philosophical Investigations 4(2): 33–49. Jastrow, J. (1899). The mind’s eye. Popular Science Monthly 54: 299–312. Merleau-Ponty, M. (1963). Structure of Behaviour (trans. A. Fisher). Boston, MA: Beacon Press. ——— (1964a). The primacy of perception and its philosophical consequences (trans. J. E. Edie). In M. Merleau-Ponty (ed.), The Primacy of Perception. Evanston: Northwestern University Press, 12–42. ——— (1964b). Phenomenology and the sciences of man (trans. John Wild). In M. Merleau-Ponty (ed.), The Primacy of Perception. Evanston: Northwestern University Press, 43–95. ——— (1964c). Eye and mind (trans. Carleton Dallery). In M. Merleau-Ponty (ed.), The Primacy of Perception. Evanston: Northwestern University Press, 159–192. ——— (2013). Phenomenology of Perception (trans. D. Landes). London: Routledge. Romdenh-Romluc, K. (2009). Merleau-Ponty’s account of hallucination. European Journal of Philosophy 17(1): 76–90. Rozin, P., Millman, L., and Nemeroff, C. (1986). Operation of the laws of sympathetic magic in disgust and other domains. Journal of Personality and Social Psychology 50(4): 703–712. Travis, C. (2004). The silence of the senses. Mind 113: 57–94.

5

Aspect-Perception and the History of Mathematics Akihiro Kanamori

In broad strokes, mathematics is a vast yet multifarious edifice and mode of reasoning based on networks of conceptual constructions. With its richness, variety and complexity, any discussion of the nature of mathematics cannot but account for these networks through its evolution in history and practice. What is of most import is the emergence of knowledge, and the carriers of mathematical knowledge are proofs, more generally arguments and procedures, as embedded in larger contexts. One does not really get to know a proposition but, rather, a proof, the complex of argument taken together as a conceptual construction. Propositions, or rather their prose statements, gain or absorb their sense from the proofs made on their behalf, yet proofs can achieve an autonomous status beyond their initial contexts. Proofs are not merely stratagems or strategies; they, and thus their evolution, are what carry forth mathematical knowledge. Especially with this emphasis on proofs, aspect-perception—seeing an aspect, seeing something as something, seeing something in something— emerges as a schematic for or approach to what and how we know, and this for quite substantial mathematics. There are sometimes many proofs for a single statement, and a proof argument can cover many statements. Proofs can have a commonality, itself a proof; proofs can be seen as the same under a new light; and disparate proofs can be correlated, this correlation itself amounting to a proof. Less malleably, statements themselves can be seen as the same in one way and different in another, this bolstered by their proofs. With this, aspect-perception counsels the history of mathematics, taken in two neighboring senses. There is the history, the patient accounting of people and their mathematical accomplishments over time, and there is the mathematics, evolutionary analysis of results and proofs over various contexts. In both, there would seem to be the novel or the surprising. Whether or not there is creativity involved, according to one measure or another, analysis through aspects fosters understanding of the byways of mathematics. In what follows, the first section briefly describes and elaborates aspectperception with an eye to mathematics. Then in each of the succeeding two sections, substantial topics are presented that particularly draw out and show aspect-perception at work.

110 Akihiro Kanamori

1. Aspect-Perception Aspect-perception is a sort of meta-concept, one collecting a range of very different experiences mediating between seeing and thinking. Outwardly simple to instantiate, but inwardly of intrinsic difficulty, it defies easy reckoning, but, once seen, it invites extension, application, and articulation. For the discussion and scrutiny of mathematics, it serves to elaborate and to focus aspect-perception in certain directions and with certain emphases. And for this, it serves to proceed through a deliberate arrangement of some loci classici for aspect-perception in the writings of Ludwig Wittgenstein. Aspect-perception was a recurring motif for Wittgenstein in his discussions of perception, language, and mathematics. His later writings especially are filled with remarks, some ambitious and others elliptical, gnawing on a variety of phenomena of aspect-perception. It was already at work in his Notebooks 1914-16 and his 1921 Tractatus, with its Bildhaftigkeit, has at 5.5423 the Necker cube:

Figure 5.1

The figure can be seen in two ways as a cube, the left square in the front or the right square in the front. ‘For we really see two different facts’ (Wittgenstein 1921). Among the points that were made here: a symbol, serving to articulate a truth, involves a projection by us into a space of possibility, hence multifarious relationships, not merely the interpretation of a sign. This early juncture in Wittgenstein correlates with aspects of symbolization in mathematics: as explicit in succeeding sections, different modes of organization to be brought out for purposes of proof can be carried by one geometric diagram or one algebraic equation. In his 1934 Brown Book (cf. 1958: II,§16), Wittgenstein discusses ‘seeing it as a face’ in the ‘picture-face’, a circular figure with four dashes inside,

History of Mathematics 111 and in a picture puzzle, where ‘what at first sight appears as “mere dashes” later appears as a face’. ‘And in this way “seeing dashes as a face” does not involve a comparison between a group of dashes and a real human face.’ We are taking or interpreting the dashes as a face. This middle juncture in Wittgenstein, at an aspect more conceptual than visual, correlates with aspects of contextual imposition in mathematics: as exhibited by the ‘commonality’ (**) in §2, there can be a structure or a proof, logical yet lean, whose ‘physiognomy’ can be newly seen by being placed in a rich conceptual or historical context. In Wittgenstein’s mature Philosophical Investigations (1953) aspectperception comes to the fore in Part II, Section xi.1 ‘I contemplate a face, and then suddenly notice its likeness to another. I see that it has not changed; and yet I see it differently. I call this experience [Erfahrung] “noticing an aspect”’ (p. 193c). Wittgenstein works through an investigation of various schematic figures, particularly the Jastrow duck–rabbit (p. 194e), a figure which can be seen either as a duck with its beak to the left or a rabbit with its ears to the left. With this, he draws out the distinction between ‘the “continuous seeing” of an aspect’ (seeing, with immediacy; later, ‘regarding-as’), and ‘the “dawning” of an aspect’ (sudden recognition). With ‘continuous seeing’ he navigates a subtle middle road between being caused by the figure to perceive and the imposition of a subjective, private inner experience, undermining both as explanations of the phenomenon of seeing-as. Here follows a sequential arrangement of quotations, citing page and paragraph: 198c: The concept of a representation of what is seen, like that of a copy, is very elastic, and so together with it the concept of what is seen. The two are intimately connected. (Which is not to say that they are alike.) 199b: If you search in a figure (1) for another figure (2), and then find it, you see (1) in a new way. Not only can you give a new kind of description of it, but noticing the second figure was a new visual experience. 200f: When it looks as if there were no room for such a form between other ones you have to look for it in another dimension. If there is no room here, there is room in another dimension. [An example of imaginary numbers for the real numbers follows.] 201b: . . . the aspects in a change of aspects are those ones which the figure might sometimes have permanently in a picture. 203e: ‘The phenomenon is at first surprising, but a physiological explanation of it will certainly be found.’—Our problem is not a causal one but a conceptual one. 204g: Here it is difficult to see that what is at issue is the fixing of concepts. A concept forces itself on one. 208d: One kind of aspect might be called ‘aspects of organization’. When the aspect changes parts of the picture go together which before did not.

112 Akihiro Kanamori 212a: . . . what I perceive in the dawning of an aspect is not a property of the object, but an internal relation between it and other objects. For a visual context of some complexity pointing us towards the appreciation of aspects in mathematics, one can consider the Cubist paintings of Pablo Picasso and Georges Braque. These elicit aspects of aspect-perception emphasized by Wittgenstein like seeing faces and objects from various perspectives; continuous seeing; dawning of an aspect; aspects there to be seen; possible blindness to an aspect for a fully competent person and making sense of bringing such a person to see the aspect. Moreover, there is a shifting of aspects beyond complementary pairs, as several aspects can be kept in play at once, some superposing on others, some internal to others, some at an intersection of others, and, with the painters’ willful obstructionism, some petering out at borders and some incompatible with others in varying degrees. With an eye to mathematics, we can locate aspect-perception among broad philosophical abstractions in the following ways: 1. Aspect-perception is not (merely) psychological or empirical, but substantially logical, in working in spaces of logical possibility. 2. Aspect-perception is not in the service of conventionalism and is not (only) about language. 3. Aspect-perception, while having to do with fact and truth, is orthogonal to value. 4. Aspect-perception can figure as a mode of analysis of concepts and states of affairs. 5. Aspect-perception maintains objectivity, as aspects are there to be seen, but through a multifarious conception involving modality. What, then, is the place and import of aspect-perception in mathematics and its history? The above points situate aspect-perception between seeing and thinking as logical—and so having to do with truth and objectivity— and not about convention or value and as possibly participating in analysis. In these various ways, aspect-perception can be seen to be fitting and indeed inherent in mathematical activity. At the very least, aspect-perception provides language, and so a way of thinking, for discussing and analyzing concepts, proofs, and procedures—how they are different or the same and how they can be compared or correlated. More substantially, since mathematics is a multifarious edifice of conceptual constructions, attention to aspects itself promotes seeing, seeing anew, and gaining insights. This is particularly so in connection with how we gauge simplicity, how we account for surprise, and how we come to understand mathematics. On these last points, earlier remarks circa 1939 of Wittgenstein from Remarks on the Foundations of Mathematics have particular resonance. Part III starts with a discussion of proof, beginning: ‘“A mathematical proof

History of Mathematics 113 must be surveyable [übersichtlich].” Only a structure whose reproduction is an easy task is called a “proof”.’ Aspect-perception casts light here, since seeing an argument organized in a specific way, for example, through the projection of another or as figuring in a larger context, can lead to (the dawning of) a perception of simplicity and thereby newly found perspicuity. In this way, aspect-perception shows the limits of logic conceived to be (merely) a sequence of local implications. In Part I, Appendix II, Wittgenstein discusses the surprising in mathematics. The first specific situation he considered is when a long algebraic expression is seen shrunk into a compact form and where being surprised shows (§2) ‘a phenomenon of failure to command a clear view [übersehen] and of the change of aspect of a seen complex.’ For one surely has this surprise only when one does not yet know the way. Not when one has the whole of it before one’s eyes. . . . The surprise and the interest, then come, so to speak, from the outside. After the dawning of the aspect, there is no surprise, and what then remains of the surprise is the idea of seeing the logical space of possibilities. Wittgenstein subsequently wrote (§4), ‘There’s no mystery here!’—but then how can we have so much as believed that there was one?—Well, I have retraced the path over and over again and over and over again been surprised; and I never had the idea that here one can understand something.—So ‘There’s no mystery here!’ means ‘Just look about you!’ Though only elliptical, Wittgenstein here is suggesting that understanding, especially of novelty, as coming into play with the seeing and taking in of aspects. Aspect-perception and mathematics have further useful involvements. Aspect-perception is an intrinsically difficult meta-concept in and through which to find one’s way, and by going into the precise, structured setting of mathematics one can better gauge and reflect on its shades and shadows. One can draw out aspects and deploy them to make deliberate conceptual arrangements for communicating mathematics. And aspect-perception provides an opportunity to bring in large historical and mathematical issues of context and method and to widen the interpretive portal to ancient mathematics. In the succeeding sections, we show aspect-perception at work in mathematics by going successively through two topics, chosen in part for their differing features to illuminate the breadth of aspects. Section 2 takes up the classical and conceptual issue of the irrationality of square roots, bringing out aspects geometric and algebraic, ancient and modern. Section 3 sets out a circularity in the development of the calculus having to do with the

114 Akihiro Kanamori derivative of the sine function, retraces features of the concept in ancient mathematics, and considers possible ways out of the circularity, thus drawing out new aspects. A substantial point to keep in mind is that these topics are conceptually complex; aspect-perception can work and, indeed, is of considerable interest at higher levels of mathematics.2 In each of the sections is presented a ‘new’ result ‘found’ by the author, but one sees that creativity is belied to a substantial extent by context.

2. Irrationality of Square Roots For a whole number n which is not a square number (4  22, 9  32, 16  42, . . .), its square root n is irrational, that is, not a ratio of two whole numbers. This section attends to this irrationality; we shall see that its various aspects are far-ranging over time and mathematical context but, perhaps surprisingly, have a commonality. The irrationality can itself be viewed as an aspect of n separate from the seeing of it as or for its calculation as in Old Babylonian mathematics circa 1800 bce, an aspect embedded in conceptualizations about the nature of number and of mathematical proof as first seen in Greek mathematics. In particular, the irrationality of 2 was a pivotal result of Greek geometry established in the later 5th century bce. This result played an important role in expanding Greek concepts of quantity, and for contextually discussing 2 as well as the general n , it is worth setting out, however briefly, aspects of quantity as then and now conceived. For the Greeks, a number is a collection of units, what we denote today by 2, 3, 4, . . . with less a connotation of order than of cardinality. Numbers can be added and multiplied. A magnitude is a line, a (planar) region, a surface, a solid, or an angle. Magnitudes of the same kind can be added (e.g. region to region) and multiplied to get magnitudes of another kind (e.g. line to line to get a region). Respecting this understanding, we will deploy modern notation with its algebraic aspect, this itself partly to communicate further mathematical sense. For example, Proposition I 47 of Euclid’s Elements, the Pythagorean Theorem, states rhetorically that ‘[i]n right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle’ with a ‘square’ qua region. We will simply write the arithmetical a2 + b2  c2 where a and b are (the lengths of) the legs of a right triangle and c (the length of) the hypotenuse. A ratio is a comparison between two numbers or two magnitudes of the same kind (e.g. region to region). Having ratio 2 to 3 we might today write as a relation 2:3 or a quantity 23 , with the first being closer to the Greek concept. There is a careful historiographical tradition promoting the first, but we will nonetheless deliberately deploy the latter in what follows. There are several aspects to be understood here: the fractional notation itself can be read as the Greek ratio; it can be read as part of a modern numerical-algebraic construal; and finally, the two faces are assertively to be seen as coherent.

History of Mathematics 115 A proportion is an equality of two ratios. We deploy the modern  as if for numerical quantities, this again having several aspects to be understood: it can be read as the Greek proportion, it can be read as an identity of two numerical quantities, and finally, the two faces are assertively to be seen as coherent. In what follows, the notation itself is thus to convey a breadth of aspect as well as a change of aspect, something not always made explicit. Two magnitudes are incommensurable if there is no ‘unit’ magnitude of which both are multiples. While we today objectify n as a (real) number, that n is irrational is also to convey, in what follows, a Greek geometric sense: a square containing n square units has a side which is incommensurable with the unit. The pivotal result that 2 is irrational was for the Greeks that the side s and diagonal d of a square are incommensurable: d2  s2 + s2  2s2 by the Pythagorean Theorem, and the ratio ds ( 2) is not a ratio of numbers. To say that this result triggered a Grundlagenkrise would be an exaggeration, but it undoubtedly stimulated both the development of ratio and proportion for general magnitudes in geometry and a rigorization of the elements and means of proof. One of the compelling results of the broader context was just the generalization that n for non-square numbers n is irrational, sometimes called Theaetetus’s Theorem. Theaetetus (ca. 369 bce) is, of course, the great Platonic dialogue on epistemology. Socrates takes young Theaetetus (ca. 417–369 bce) on a journey from knowledge as perception, to knowledge as true judgement, to knowledge as true judgement with logos (an account), and, in a remarkable circle, returns to perception: How can there even be knowledge of the first syllable SO of “Socrates”—is it a simple or a complex? Early in the dialogue (147c–148d), Theaetetus suggested conceptual clarification vis-à-vis square roots. He first noted that the elder geometer Theodorus (of Cyrene, ca. 465–398 bce) had proved by diagrams the irrationality of n for non-square n up to 17. Then dividing the numbers into the ‘square’ and the ‘oblong’, he observed that they can be distinguished according to whether their square roots are numbers or irrational. In view of this and derivative commentary, Theaetetus has in varying degrees been credited with much of the content of the arithmetical Book VII of Euclid’s Elements and of Book X, the meditation on incommensurability and by far the longest book. The avenues and byways of the Theodorus result and the Theaetetus generalization have been much discussed from both the historical and mathematical perspectives.3 In what follows we point out aspects there to be seen that coordinate across time and technique, and to this purpose we first review proofs for the irrationality of 2 . The argument most often given today is algebraic, about 2 . Assume that 2  ba for (whole) numbers a and b so that, squaring, a2  2b2 · a2 is thus even and so, consequently, is a, say, a  2c. But then, 4c2  2b2, and so 2c2  b2 · b2 is thus even and so, consequently, is b, say, b  2d. But then, ba  dc . Now this reduction to a ratio of smaller numbers cannot be repeated forever

116 Akihiro Kanamori (infinite regress), or had we started with the least possibility for b, we would have a contradiction (reductio ad absurdum). Cast geometrically in terms of the side and the diagonal of squares within squares, this is plausibly the earliest proof, found by the “Pythagoreans” in the first deductive theory, the even versus the odd (even times even is even, odd times odd is odd, and so forth). The proof is diagrammatically suggested in Plato’s Meno 82b–85b, and, as an example of reasoning per impossible, in Aristotle’s Prior Analytics I 23. Another proof proceeds directly on a square, the features conveyable in the diagram showing half a square with side s and diagonal d.

Figure 5.2

On the diagonal, a length s is laid off, getting AB, and then a perpendicular BD is constructed. The three ss consequently indicate equal line segments, as can be seen using a series of what can be deduced to be isosceles triangles: the triangle ABE (formed by introducing line segment BE), the triangle BDE, and the triangle BCD. Now triangle BCD is also half of a square, with side s and diagonal d, given in terms s and d as above. So, if s and d were commensurable, then so would be s and d. But this reduction cannot be repeated forever (infinite regress), or had we started out with commensurability with s and d being the smallest possible multiples of a unit, we would have a contradiction (reductio). From the algebraic aspect, one sees that the components of a ratio have been made smaller: d d 2s  d , where 0  d  s  s.   s s ds

History of Mathematics 117 This proof correlates with the Greek process of anthyphairesis, the Euclidean algorithm for line segments, whereby one works towards a common unit for two magnitudes by iteratively subtracting off the smaller from the larger. Because of this, the proof or something similar has been thought by some historians to be the earliest proof of incommensurability.4 The proof appeared as a simple approach to irrationality in the secondary literature as early as in (Rademacher and Toeplitz 1930: 23) and, recently, with the simple diagram as shown above, in Apostol (2000). Proceeding to n , Knorr (1975, chap. VI) worked out various diagrammatic versions of the 2 even–odd proof as possible reconstructions for Theodorus’s n result n up to 17, and Fowler (1999: 10.3) provided various anthyphairetic proofs up to 19. The following proof of the general Theaetetus result appears to be new; at least it does not seem to appear put just so in the historical and mathematical literature. Assume that ba  n . Laying off copies of b on a, the anthyphairetic ‘division algorithm’, let a  qb + r in algebraic terms with ‘quotient’ q and ‘remainder’ r, where 0 < r < b (r  0 would imply that ba is a number and  n a square). Consider the following diagram generalizing the previous one for  2 .

Figure 5.3

On the hypotenuse of length a, a length qb has been laid off, and so the two es are, in fact, the same as in the 2 case. This time, we appeal to similarity; the two triangles are seen to have pairwise the same angles, and so we have the proportion a ce c 2  ce  , or a  . c r r There is a factor of b on both sides of the latter: a  b n ; c2  a2 − q2b2  b2n − q2b2 by the Pythagorean Theorem; and ce  qbr, since qbc  er again by similarity. Reducing by b,

118 Akihiro Kanamori n

bn  q2 b  qr , where 0  r  b, r

so that the ratio ba  n has been reduced to a ratio of smaller numbers. But as before, this reduction cannot be repeated forever (infinite regress) or had we started out with commensurability with smallest possible multiples, we would have a contradiction (reductio). Since r  a − qb and qa  q2b + qr, one sees again that from the algebraic aspect, the components of a ratio have been made smaller:

*

n

bn  qa , where 0  a  qb  b. a  qb

The author found this proof, and there is an initial sense of surprise in that through all the centuries there seems no record of a proof put just so. Is this creative? Novel? One should be loath to speculate in general for mathematics, but this is not an atypical episode in its progress. Perhaps there is surprise at first, but there quickly comes understanding by viewing aspects that are really there to be seen. With the 2 anthyphairetic proof given above as precedent, one is led to such a proof in order to account for the generality of Theaetetus’s Theorem and his having been alleged to have had a proof. It could, in fact, have been the original proof; its use of the division algorithm and ratios is within the resources that were presumably available already to the elder Theodorus. One notices an aspect of generality emerging in context, like a face out of a picture puzzle. Be that as it may, historians in their ruminations have attributed proofs to Theaetetus that can be drawn out from propositions in the arithmetical Books VII and VIII of Elements, the first having been attributed to Theaetetus himself in his efforts to rigorize his theorem.5 Scanning these books, there are several propositions that lead readily to Theaetetus’s Theorem. According to Book VII, ‘a number is a multitude composed of units’ (Definition 2); ‘a number is part of [divides] a number, the less of the greater, when it measures the greater’ (Definition 3); and ‘numbers relatively prime are those which are measured by a unit alone as a common measure’ (Definition  12). Assuming that ba  n so that a2  nb2, each of the following propositions about numbers readily implies that n is a square: 1. (VII 27) If r and s are relatively prime, then r2 and s2 are relatively prime. (Assume that a and b are the least possible so that they are relatively prime. As b2 divides a2, by the proposition b2 must be the unit. Hence, b must be the unit, and so n is a square.)

History of Mathematics 119 2. (VIII 14) If r2 divides s2, then r divides s. (It follows that b divides a, and so n is a square. This proposition is not used elsewhere in the Elements and seems earmarked for Theaetetus’s Theorem.) 3. (VIII 22) If r, s, t are in continued proportion (i.e. rs  ts ) and r is a a2 nb square, then t is a square. (Since nb  n and a2 is a square, n is a square.) Having allowed the positive conclusion that n, after all, could be a square, only the argument from VII 27 still depends on least choices for a and b. However, all the propositions depend on the much-used VII 20, which is about least choices: If

a c  b d

and a and b are least possibilities for this ratio, then a divides c and b divides d. The proof of VII 20 given in the Elements seems roundabout, and we give a sequentially direct proof, for example, that b must divide d: assume to the qa contrary that d  qb + r with the division algorithm, where 0 < r < b. ba  qb c  qa (VII 17), and this together with ba  dc implies ba  d  qb (VII 12). But this contradicts the leastness assumption as d – qb  r < b. VII 20 itself leads quickly to Theaetetus’s Theorem: Assume

a a n nb  n , so that   . b b a n

Then if b is the least possibility for this ratio, then b divides a, and so n is square. These proofs of Theaetetus’s Theorem drawn from the Elements are arithmetical and veer towards reductio formulations, while the ‘new’ proof given earlier is diagrammatic and more suggestive of infinite regress. Is there, after all, a commonality of aspect? Yes, it is there to be seen but somewhat hidden. It is seen through a simple algebraic proof of Theaetetus’s Theorem using a scaling factor, which is mysterious at first: If ba  n and there is a number q such that q  n  q  1 so that qb < a < (q + 1)b, we then have the algebraic reduction

**

a a  b b

 

  bn  qa , where 0  a  qb  b. a  qb n  q

n q

This q is just the q of the division algorithm a  qb + r of the diagrammatic proof, and () is a rendering of the () after that proof. As for the

120 Akihiro Kanamori arithmetical proof, the ratio reduction is there but only as part of the proof of VII 20 given above, at the use of VII 12. These aspects of various propositions and proofs are all there to be seen in a kind of whirl, the interconnections leading to understanding. Today, prime numbers and the Fundamental Theorem of Arithmetic, that every number has a unique factorization into prime numbers, are basic to number theory, and it is a simple exercise in counting prime factors to establish Theaetetus’s Theorem. However, for more than two millennia until Gauss the new simplicity afforded by the fundamental theorem was not readily attainable. There have recently been several accounts of the irrationality of k n ab initio that exhibit a minimum of resources though without conveying historical resonance, and these ultimately turn on (), what was there to be seen.6 Richard Dedekind in his 1872 Stetigkeit und irrationale Zahlen, the foundational essay in which he formulated the real numbers in terms of Dedekind cuts, provided (IV), what has been considered a short and interesting proof of the irrationality of n for non-square n: assume that ba  n with b the least possibility and q  n  q  1 . Then algebraically

***

 bn  qa 

2

 n  a  qb    q2  n  a 2  nb2  2

However, a2 − nb2 is zero by assumption and so is the left side, and hence, bn  qa  n a  qb

where 0  a  qb  b,

contrary to the leastness of b. Again the scaling ratio of () has emerged, but how had Dedekind gotten to it? During this time, Dedekind was steeped in algebraic number theory, particularly with his introduction of ideals. The ring Z  n  consists of x  y n , where x and y are integers and the ring has a norm given by N x  y n  x  y n x  y n  x2  ny 2 . The norm of a product is the product of the norms—Brahmagupta’s identity, first discovered by the 7th century ce Indian mathematician. In these terms, () above is just expressing



 





N q  n  a  b n





   N   q  n   N  a  b n   .

The appearance of the scaling factor n  q of () is motivated here in terms of norm reconstruing distance. Also, in this wider context of algebraic structures, it is well known that unique factorization into ‘prime’ elements

History of Mathematics 121 may not hold, and so there is a logical reason to favor the Dedekind approach to the irrationality. That () emerges as a commonality in proofs of Theaetetus’s Theorem is itself a notable aspect. Although indicating a proof on its own, () remains thin and mysterious in juxtaposition with the ostensible significance of the result, both historical and mathematical. One sees more sides and angles, whether about number, discovery or proof, in the other proofs—embedded as they are in larger ways of thinking—and these aspects garner a mathematical understanding of the proofs and related propositions.

3. Derivative of Sine In this section, a basic circularity in textbook developments of calculus is brought to the fore, and this logical node is related to the ancient determination of the area of the circle. How to progressively get past this node is considered, the several ways bringing out different aspects of analysis, parametrization, and conceptualization. There is quite a lot of mathematical and historical complexity here, but this is requisite for bringing out the subtleties of aspect-perception in this case, especially of seeing something as something and in something. The calculus of Newton and Leibniz revolutionized mathematics in the 17th century, with dramatically new methods and procedures that solved age-old problems and stimulated remarkable scientific advances. At the heart is a bifurcation into the differential calculus, which investigated instantaneous rates of change, like velocity and acceleration, and the integral calculus, which systematized total size or value, like areas and volumes. And the Fundamental Theorem of Calculus brought the two together as opposite sides of the same coin. In modern standardized accounts, the differential calculus is developed with the notion of limit. Functions working on the real numbers are differentiated, that is, corresponding functions, their derivatives, are determined that are to characterize their rates of change. The differentiation of the trigonometric functions is a consequential part of elementary differential calculus. The process can be reduced to determining that the derivative of the sine function is the cosine function, and this devolves, fortunately, to the determination of the derivative of the sine function evaluated at 0. This amounts to showing

*

lim  0

sin 



 1,

that the limit as θ approaches 0 of the ratio of sin θ (the sine of θ) to θ is 1. This is the first interesting limit presented in calculus courses, bringing together angles and lengths. How is it proved?

122 Akihiro Kanamori In all textbooks of calculus save for a vanishing few, a geometric argument is invoked with the following accompanying diagram:

Figure 5.4

Consider the arc AB subtended by (a small) angle θ on the unit circle, the circle of radius 1, with center O. The altitude dropped from A has length sin  θ, ‘opposite over hypotenuse’, for the angle θ; the length of the circular arc AB is θ, the (radian) measure of the angle (with 2π for one complete revolution); and the length of AB is tan θ (the tangent of θ), ‘opposite over adjacent’. Once

**

sin     tan 

is established, pursuing the algebraic aspect and dividing through by sin θ and then taking reciprocals yields

1

sin 





sin   cos  , tan 

and since cos θ (the cosine of θ), ‘adjacent over hypotenuse’, approaches 1 as θ approaches 0, () follows. In geometric aspect, the first inequality of () as a comparison of lengths is visually evident, but the second is less so. One can, however, proceed with areas: The area of triangle OAB (formed by introducing line segment AB) is 12 sin  , ‘half the base times the height’; the area of circular sector OAB is  2 , since the ratio of this area to the area π of the unit circle is proportional to the ratio of θ to the circumference 2π; and 12 tan  is the area of triangle OAB. With the figures subsumed one to the next, () follows by comparison of areas. But this is a circular argument! It relies on the area of the unit circle being π, where 2π is the circumference, but the proof of this would have to entail taking a limit like (), or at least the comparison of lengths () as in the

History of Mathematics 123 diagram. And underlying this, what after all is the length of a curve, like an arc? We can elaborate on, and better see, this issue by looking at the determination of the area of a circle in Greek mathematics.7 Archimedes in his treatise Measurement of a Circle famously established that the area of a circle of radius r is equal to the area of a right triangle with sides r and the circumference. With this latter area being 12  r  (2 r) , we today pursue the algebraic aspect and state the area as πr2. Archimedes briefly sketched the argument in Measurement in terms of the right triangle; it is more directly articulated through Propositions 3 through 6 of his On the Sphere and Cylinder I. The method was to inscribe regular n-gons (polygons with n equal sides) in the circle of radius r, to circumscribe with such, and then to take a limit as n gets larger and larger via the Eudoxan method of exhaustion. The following diagram taken from On the Sphere pictures a sliver of the argument:

Figure 5.5

The triangle OAC is one of the n triangles making up an inscribing n-gon, 1 the circular arc AC is n of the circumference, and the triangle OAC is one of the n triangles making up a circumscribing n-gon. This figure is just a coupling of the previous figure scaled to radius r with its mirror image, and so in (modern, radian) measure we would have: The angle AOC is 2n , and so with half of this as the angle   n of the previous figure, the line segment AC has length twice r sin n , or 2r sin n . Similarly, the line segment AC has length 2r tan n . Finally, the arc AC has length 2n r , based on the circumference being 2πr. Archimedes in effect used the version

2r sin

 n



 2 r  2r tan n n

124 Akihiro Kanamori of () to show, for the Eudoxan exhaustion—we would now say for the taking of a limit—that the perimeter of the inscribing and circumscribing polygons approximate the circumference from below and from above. This appeal to () is not itself justified in Measure, but the first two “assumptions” in the preface to On the Sphere serve: (1) the shortest distance between two points is that of the straight line connecting them (so sin θ < θ), and (2) for two curves convex in the same direction and joining the same points, the one that contains the other has the greater length (so8 θ < tan θ). With such assumptions, Archimedes had set out the conditions for how his early predecessors had constructed arc length. Archimedes’ work evidently built on a pre-Euclidean tradition of geometric constructions,9 in which an important motif had been how to rectify a curve, that is, render it as a straight line. Indeed, Archimedes stated and conceptualized his area theorem as one sublimating the circumference as a straight line, the side of a triangle; he could not in any case have stated the area as πr2, the Greek geometric multiplication having to do with areas of rectangles and not generally allowed for magnitudes. These aspects of area and length are illuminated by Euclid’s Elements XII 2: Circles are to one another as the squares on their diameters. This had been applied over a century before by Hippocrates of Chios in his ‘quadrature of the lunes’, and Euclid managed a proof in his system with a paradigmatic use of the method of exhaustion that borders on Archimedes’ later use. Commentators have pointed out how XII 2 falls short of getting to the actual ratio π, but in thinking through the aspects here, Euclid could not have gone further. In his rigorization over his predecessors, Euclid had famously restricted geometric constructions to straightedge and compass, and he had no way of rectifying a curve and so of formulating the ratio  π. For Euclid, and Greek theoretical mathematics, area was an essentially simpler concept, from the point of view of proof, than length (for curves); area could be worked through congruent figures, and there were no beginning, ‘common notions’ for length. Comparison of areas with one figure subsumed in another is simpler than a comparison of length (of curves), and XII 2 epitomizes how far one can go with the first. Archimedes went further to the actual ratio π, for 20 centuries called ‘Archimedes’ constant’, but this depended on his assumptions (1) and (2). Especially with this historical background uncovering aspects of length and area for the circle, one can arguably take () as logically immediate as part of the concept of length. We today have a mathematical concept of rectifiable curves, a concept based on small straight chords approximating small arcs so that polygonal paths approximate curves. Seeing the concept of length from this aspect, Archimedes’ assumptions (1) and (2) are more complicated in theory than () itself. Moreover, there is little

History of Mathematics 125 explanatory value in proving (), as it is presupposed in the definition of arc length. The logical question remains whether we can avoid the theft of assuming what we want and move forward to the derivative of the sine with honest toil. There are several ways, each illuminating further aspects of how we are to take analysis and definition, from arc length to the sine function itself. (a) Taking area as basic, define the measure of an angle itself in terms of the area of the subtended sector. To scale for radian measure, define the measure of an angle to be twice the area of the sector it subtends in the unit circle. Then by comparison of areas, 12 sin   12   12 tan  , and () is immediate. With this, one could deduce à la Archimedes that the area of the unit circle is π, where 2π is the circumference, so the measure for one complete revolution is 2π, confirming that we do, indeed, have the standard radian measure. Defining angle measure in this way may be a pedagogical or curricular shortcoming, but the shift in logical aspect is quickly re-coordinated and, moreover, resonates with how the conceptualization of area is simpler than that of length. G. H. Hardy’s classic A Course in Pure Mathematics (1908: §§158, 217) and Tom Apostol’s calculus textbook (1961: §1.38) are singular in pointing out the logical difficulty of defining the measure of an angle in terms of an unrigorized notion of arc length, and in advocating the definition of the measure of an angle in terms of area. Hardy (1908) advocated several approaches to the development of the trigonometric functions, two of which are conglomerated in (c) below. Apostol’s 1961 work is the rare calculus textbook of recent memory that does not proceed circularly; it develops the integral calculus prior to the differential calculus, defines area as a definite integral, and only later defines length for rectifiable curves.10 (b) First define the length of a rectifiable curve in the usual way as an integral. Then, get the derivative of sine using methods of calculus: Let x  sin θ and y  cos θ so that with Pythagorean relation y  1  x2 , dy x  . dx 1  x2 Anticipating the use of the length integral, note that 2

x 1  dy  1  .    1 1  x2  dx  1  x2 2

Since, according to the parametrization, θ is the length of the arc from 0 to x,

126 Akihiro Kanamori

 

x

0

1 1  t2

dt.

By the Fundamental Theorem of Calculus, d 1 ,  dx 1  x2 so that by the Inverse Function (or Chain) Rule, dx 1   1  x2 , d  d dx or in terms of θ, d sin   cos  . d One can similarly get dd cos   sin  . This approach underscores how length can be readily comprehended with infinitesimal analysis and how the derivative of the sine being cosine can be rigorously established by a judicious ordering of the development of calculus. Importantly, the approach depends on the Fundamental Theorem, which, in turn, depends on the conceptualization of area as a definite integral. In this logical aspect, too, area is thus to be conceptually subsumed first. There would be a pedagogical or curricular shortcoming here as well, this time with the derivative of the sine popping out somewhat mysteriously. The author found this non-circular proof that the derivative of the sine is the cosine and could not find this approach in the literature. Is this a new theorem? Is it creative? Novel? Here, a logical gap was filled with familiar methods. The proof can be given as a student exercise, once a direction is set and markers laid. The task set would be to outline, with an astute ordering of the topics, a logical development of the calculus through to the trigonometric functions. If the Fundamental Theorem and the length of a rectifiable curve as an integral are put first, then the above route becomes available to the derivatives of the trigonometric functions. This logical aspect of the derivative of sine was there to be found, emerging with enough structure. (c) Taking seriously the study of the trigonometric functions as part of mathematical analysis—the rigorous investigation of functions on the real and complex numbers through limits, differentiation, integration, and infinite series—define the sine function as an infinite series. One can follow a historical track as in the following.

History of Mathematics 127 Let f(x) by a function defined by the integral in (b): f  x 



x

0

1 1  t2

dt

Newton knew this to be, as in (b), the inverse of the sine function: f(x)  θ exactly when sin θ  x. He had come early on to the general binomial series, and so in his 1669 De analysi (cf. Newton 1968: 233ff.) he expanded the 1 integrand 1t 2 as an infinite series, integrated it term by term, and then, applying a key technique for inverting series term by term, determined the infinite series for sine:

S  x  x 

x 3 x 5 x7    ... 3! 5! 7!

One can, however, take this ab initio as simply a function to investigate. Term-by-term differentiation yields

C  x  1 

x 2 x 4 x6    ... 2! 4! 6!

Hence, by series manipulation, S2(x) + C2(x)  1, which sets the stage as the Pythagorean Theorem for the unit circle. Next, define π as a parameter, the least positive x, such that S(x)  0. With these, one works out that as x goes from 0 to π, the point (S(x), C(x)) in the coordinate plane traverses half the unit circle and, through the length formula, that the circumference of the unit circle is 2π. Hence, taking sin x  S(x) and cos x  C(x) retrieves the familiar trigonometric functions and their properties, as well as the derivative of the sine being cosine.11 This approach draws out how the trigonometric functions can be developed separately and autonomously in the framework of mathematical analysis. The coordination with the classical study of the circle and its measurement then has a considerable aspectual variance: one can take the geometry of the circle as the main motivation, one can bring out interactions between the geometric and the analytic, or one can even avoid geometric ‘intuitions’, a thematic feature of analysis into the 19th century. Lest the analytic approach to the trigonometric functions still seems arcane x 1 or historically Whiggish, consider the function g  x   0 1t 4 dt , where the ‘2’ of the previous integral has been replaced by ‘4’. This ‘lemniscatic integral’ arose as the length of the ‘lemniscate of Bernoulli’ at the end of the 17th century, and it was the first integral which defied the Leibnizian program of finding equivalent expressions in terms of ‘known’ functions (algebraic, trigonometric, or exponential functions and their inverses). At the end of the

128 Akihiro Kanamori 18th century, the young Gauss focused on the inverse of the function g(x) and found its crucial property of double periodicity. By 1827 the young Abel had also studied the inverse function, and in 1829 Jacobi wrote a treatise on the subject, from which such functions came to be known as elliptic functions, the integrals elliptic integrals, and the curves they parametrize elliptic curves. Thence, elliptic functions have played a large, unifying role in number theory, algebra, and geometry as they were extended into the complex plane. On that score, by 1857 Riemann had shown that the complex parametrizations are on a torus, a ‘doughnut’, with double periodicity an intrinsic feature. This is how the geometric figure of the torus came to be of central import in modern mathematics—the arc of discovery going in the opposite x 1 direction from the circle to the integral f  x   0 1t 2 dt . There is a broad matter of aspect to be reckoned with here, and, as a matter of fact, throughout this topic, as well as the previous. Taking mathematics as based on networks of conceptual constructions, one sees through aspects various historical and logical progressions. Simply seeing a certain ‘face’ on a topic in mathematics is to make connections with familiar contexts and modes of thinking, and this leads to the sometimes sudden dawning of logical connections. That aspects are logical thus has a further dimension in mathematics, that there are webs of logical connections. We can be provoked to seeing logical sequencings of results and themes, all there to be found. As we look and see, we can develop and reconstruct mathematics. In this, aspect-perception counsels the history of mathematics and draws forth an understanding of it, that is, its truths as embedded in its proofs. Stepping back further from the two topics presented in this chapter, one may surmise that many pieces of mathematics can be so presented as pictures at an exhibition of mathematics, with aspects and aspect-perception helping to get one about. Between seeing and thinking, aspect-perception is logical and can participate in the analysis of concepts. In the discussion of the irrationality of n , we saw conceptions of number themselves at play, the diagrammatic geometry of the Greeks stimulating a remarkable advance, various arithmetical manipulations of number domesticating the irrationality, and the play of recent systematizations reinforcing a commonality. In the discussion of the derivative of the sine function, we saw a basic limit issue of calculus swirling with the ancient determination of the area of a circle, the involvement of Greeks conceptualizations of area and length, different approaches to establishing a rigorous progression, and how older concepts can be transmuted in a broad new context. Venturing some generalizing remarks, across mathematics there are many angles, faces, and views and the noticing, continuous seeing, and dawning of many aspects. Especially in mathematics, however, aspect-perception is not just about conventions or language. Rather, aspects get at objectivity from a range of perspectives and, thus, collectively track and convey necessity, generality, and truth.

History of Mathematics 129

Notes Aspects of this chapter were presented at 2013 seminars at Carnegie Mellon University and at the University of Helsinki; many thanks to the organizers for having provided the opportunity. The chapter has greatly benefitted from discussions with Juliet Floyd. 1. Part II is renamed Philosophy of Psychology—a Fragment in the recent edition (2009) of Philosophical Investigations. 2. This belies objections at times lodged against Wittgenstein that he only raised philosophical issues of pertinence to very simple mathematics. 3. See Knorr (1975) and Fowler (1999) for extended historical reconstructions based on different approaches, and see Conway and Shipman (2013) for the most recent mathematical tour. 4. See Knorr (1975, chap. II, sect. II) for a critical analysis. Knorr (1998) late in his life maintained, however, that a specific diagrammatic rendition of the proof was the original one. 5. Cf. Knorr (1975, chap. VII). 6. See Beigel (1991) and references therein. Conway and Guy (1996: 185) conveys a proof in terms of fractional parts, which again amounts to (). 7. This circularity was pointed by Richman (1993) and, in the context of ancient mathematics, by Seidenberg (1972). Both pursue the trail in ancient mathematics at some length. 8. For this, one imagines in the first diagram a mirror image of the figure put atop it, say with a new point C corresponding to the old B. Comparing the arc CAB to the path CAB with (2), one gets 2θ < 2 tan θ. 9. Cf. Knorr (1986). 10. The calculus textbook by Spivak (2008, III.15) defines the sine and cosine functions as does Apostol (1961) and is not circular, but, on the other hand, it also (problem 27) gives the circular approach to the derivative of the sine as ‘traditional’. 11. See the classics Landau (1934, chap. 16) and Knopp (1921: §24) for details on this development.

References Apostol, Tom M. (1961). Calculus, Vol. 1. New York: Blaisdell. Apostol, Tom M. (2000). Irrationality of the square root of two—A geometric proof. The American Mathematical Monthly 107: 841–842. Beigel, Richard. (1991). Irrationality without number theory. The American Mathematical Monthly 98: 332–335. Conway, John H. and Guy, Richard K. (1996). The Book of Numbers. New York: Springer-Verlag. Conway, John H. and Shipman, Joseph. (2013). Extreme proofs I: The irrationality of 2 . Mathematical Intelligencer 35(3): 2–7. Fowler, David H. (1999). The Mathematics of Plato’s Academy: A New Reconstruction, 2nd edition. Oxford: Clarendon Press. Hardy, G. H. (1908). A Course in Pure Mathematics. Cambridge: Cambridge at the University Press. Knopp, Konrad. (1921). Theorie und Awendung der unendlichen Reihen. Berlin: Julius Springer. Translated as The Theory and Applications of Infinite Series. Dover Publications, 1990.

130 Akihiro Kanamori Knorr, Wilbur R. (1975). The Evolution of the Euclidean Elements. Dordrecht: D. Reidel Publishing Company. Knorr, Wilbur R. (1986). The Ancient Tradition of Geometric Problems. Boston: Birkhäuser. Knorr, Wilbur R. (1998). ‘Rational diameters’ and the discovery of incommensurability. American Mathematical Monthly 105(5): 421–429. Landau, Edmund. (1934). Einführung in die Differentialrechnung und Integralrechnung. Groningen: Nordhoff. Translated as Differential and Integral Calculus. New York: Chelsea Publishing Company, 1951. Newton, Isaac. (1968). De analysi per aequationes numero terminorum infinitas. In D. T. Whiteside (ed.), The Mathematical Papers of Isaac Newton, Volume II 1667–1670. Cambridge: Cambridge University Press, 206–276. Rademacher, Hans and Toeplitz, Otto. (1930). Von Zahlen und Figuren: Proben mathematischen Denkens für Liebhaber der Mathematik. Berlin: Julius Springer. Latest version in English translation The Enjoyment of Mathematics: Selections From Mathematics for the Amateur. Dover, 1990. Richman, Fred. (1993). A circular argument. The College Mathematics Journal 24: 160–162. Seidenberg, Abraham. (1972). On the area of a semi-circle. Archive for History of Exact Sciences 9: 171–211. Spivak, Michael. (2008). Calculus, 4th edition. Houston: Publish-or-Perish Press. Wittgenstein, Ludwig. (1953). Philosophical Investigations (trans. G. E. M. Anscombe). Oxford: Basil Blackwell. Wittgenstein, Ludwig. (1958). The Blue and Brown Books. Oxford: Basil Blackwell. Wittgenstein, Ludwig. (2009). Philosophical Investigations (trans. G. E. M. Anscombe, P. M. S. Hacker and Joachim Schulte). Oxford: Wiley-Blackwell.

6

Seeing-As and Mathematical Creativity Michael Beaney and Bob Clark

Introduction Over the last 20 years, aspect perception has been receiving increasing philosophical attention, inspired by the continuing influence of the remarks on seeing-as that form the first 20 pages or so of section xi of Part II of Wittgenstein’s Philosophical Investigations, first published in 1953. The initial influence of these remarks occurred mainly in aesthetics, Richard Wollheim’s Art and Its Objects (1968) being a prominent example. But it soon became a standard topic in philosophy of mind and philosophical psychology, and books on Wittgenstein’s later philosophy, in particular, would often contain a chapter on aspect perception.1 The topic tended to be seen, however, as somewhat self-contained, and its significance for areas outside aesthetics and philosophical psychology was rarely discussed. In 1990 Stephen Mulhall published On Being in the World, offering the first full-length account of aspect perception through a comparison of the views of Wittgenstein and Heidegger, which demonstrated its broader relevance. From the early 1990s, after his break with Peter Hacker, with whom he had collaborated on a series of books on Wittgenstein in the 1980s, Gordon Baker began to explore and stress the methodological importance of Wittgenstein’s thinking about aspect perception; and after his premature death, a collection of his papers was published in 2004, Wittgenstein’s Method: Neglected Aspects. Coupled with the emergence of ‘New Wittgensteinian’ readings led by Cora Diamond and James Conant, and greater attention to Wittgenstein’s last writings, as reflected in talk of the ‘third Wittgenstein’,2 these developments culminated in the first collection of essays devoted to aspect perception, Seeing Wittgenstein Anew, published in 2010. In introducing the essays in this collection, the editors William Day and Victor Krebs write that they contain the “recurring discovery . . . that there is something to be found in his remarks on aspect-seeing that is crucial to, yet all but overlooked in, the reception of the later Wittgenstein” (2010: 1). These remarks, they go on, far from being a detour in Wittgenstein’s “long and meandering journeyings” (as he put it in his preface to the Investigations), are “the expression of a theme whose figures and turns we might have been hearing, however faintly, all along” (2010:  5). They constitute

132 Michael Beaney and Bob Clark “Wittgenstein’s indirect meditation on the difficulties of receiving his (later) philosophical methods” (2010: 10). These claims are both bold and controversial. Peter Hacker, for example, has long held that the remarks on seeing-as are relatively unimportant. This is reflected, most recently, in the decision made by Hacker and his collaborator on the revised translation of the Investigations, Joachim Schulte, to rename Part  II of the Investigations ‘Philosophy of Psychology—A Fragment’, to emphasize not only its draft character but also its tangential connection to Part  I.3 The present essay is written in the conviction that Wittgenstein’s remarks on aspect perception are indeed of fundamental methodological importance, not just to Wittgenstein’s philosophy but to all areas of philosophy. We focus here on their relevance to philosophy of mathematics, for two reasons. First, this relevance has been almost entirely ignored,4 yet thinking through the questions raised by aspect perception sheds light on issues in philosophy of mathematics, especially in relation to mathematical creativity. Second, it is still the case that scholars underestimate how important the philosophy of mathematics was to Wittgenstein; earlier drafts of the Philosophical Investigations, for example, included material on the subject. His remarks on rule-following are now recognized to be as fundamental to his philosophy of mathematics as they are to his philosophy of mind. The same applies to his remarks on seeing-as, but there is still work to be done to show this. In this chapter we want to link Wittgenstein’s remarks on seeing-as with another aspect of his investigations in the philosophy of mathematics. This involves a contrast between two ways of seeing the development of mathematical concepts over time. Remarking on Cantor’s ‘diagonal proof’ of the uncountability of the real numbers, Wittgenstein points to what he describes as a “dangerous, deceptive thing” (see RFM, p. 131). Certain ways of looking at mathematical conceptual development, he suggests, can make us assume we are discovering a ‘fact of nature’, when actually what is going on may rather be thought of as ‘the determination of a concept’ (ibid.).5 This relates closely to Wittgenstein’s characterization of mathematics as ‘invented’ rather than ‘discovered’ (see e.g. RFM, p. 99). But there is another way of looking at this contrast, which we also want to bring out in what follows. Discussions of aspect perception often focus on the first-person singular—what I see in the duck–rabbit picture or how I see a change of aspect in the Necker cube. Consideration of conceptual development in mathematics and its relationship with aspect perception, however, reveals the importance of recognizing the role of first-person plural views. We—users of mathematics, members in a wide sense of the mathematical community— take certain aspects of mathematics to be thus-and-so rather than otherwise. This enables us to see what is going on in mathematical creativity, in its widest sense, not simply as a Gestalt switch in an individual’s phenomenology but as involving a change in aspect perception within a whole community. Recognition of the importance of first-person plural aspect perception, in

Seeing-As and Mathematical Creativity 133 other words, not only sheds light on the historical development of mathematical concepts but also deepens our appreciation of the role that the community of mathematics users plays in such development. With this in mind, then, we proceed as follows. In section 1 we consider one of the most important sources—perhaps the most important source—of philosophical methodology in Plato’s Meno. It is here that mathematical methodology first starts influencing philosophy, and, surprising as it may seem, we can see aspect perception as playing a pivotal role. In section 2 we turn from ancient Greek geometry to ancient Greek arithmetic, examining the conceptual development that occurred in the ‘discovery’ of irrational numbers. In section 3 we look at the emergence of non-Euclidean geometry; and in section 4 we switch back again to arithmetic and consider the ‘discovery’ of transfinite numbers. Recognizing the role that first-person plural aspect perception plays, a central theme in our discussion is the way that criteria for the relevant mathematical concepts come apart in these key cases, enabling different aspects of those concepts to be seen, which then allows something like a spontaneous choice to be made as to which aspect to take as primary. As we conclude in the final section, what is going on here is best described neither as ‘discovery’ nor as ‘invention’ of something entirely new. There are facts to be revealed and creativity to be exhibited, but what is crucial is the opening up of different aspects of something, the perception of which prompts a choice that sooner or later ‘catches on’ in a mathematical community and proves fruitful.

1. Meno’s Paradox and Geometry Philosophical methodology, and especially philosophical analysis (in its various forms), has its roots in ancient Greek geometry and Plato’s dialogues. The influence of Greek geometry on philosophy is first revealed in Plato’s Meno, the dialogue in which Socrates cross-examines a slave boy in an attempt to get him to ‘recollect’ the answer to a geometrical problem. The cross-examining is itself intended to answer what we now know as Meno’s paradox or the paradox of inquiry, which poses a dilemma that threatens the very possibility of searching for knowledge. Either we know something, in which case inquiry is pointless; or we do not, in which case we will not know what to inquire into. Either way, it would seem, there can be no genuine inquiry. Much has been written about this over the centuries, and new aspects have revealed themselves in many of these discussions.6 Here we want to explore a further aspect that, we feel, has not been appreciated: an aspect, indeed, in which aspect perception itself is seen as central to the solution to both the geometrical problem and Meno’s paradox. The geometrical problem that Socrates induces the slave boy to solve is that of constructing a square with twice the area of a given square. Socrates starts by asking the boy if he knows what a square is, a geometrical figure

134 Michael Beaney and Bob Clark with four equal sides and where two equal lines can be drawn through its centre from the midpoints of its sides. (Throughout the interrogation, we are to imagine Socrates drawing lines in the sand to illustrate what is being said; see Figure 6.1, where the initial square is represented as ABCD.) The boy says that he does, and when Socrates asks how big the square would be if its sides were two feet long, the boy comes to see that its area would be four square feet. He also recognizes that a figure twice as big would have an area of eight square feet. However, when Socrates asks him how long the sides of such a square would be in comparison with the sides of the original square, the boy immediately replies “twice the length”. By being shown a construction of a square with sides of this length, however, the boy realizes that its area is sixteen, not eight square feet (AEFG in Figure 6.1). Since the sides clearly need to be between two and four feet in length, the boy then suggests that they would be three feet long, but again he is brought to see his error by being shown that this would yield a square with an area of nine, not eight square feet. At this point, the boy is thoroughly perplexed, as Socrates breaks off to comment to Meno. (See Meno, 82b–84c.) It is here that a move of fundamental importance in Plato’s work—and in the history of philosophy generally—is made. With the boy now primed to ‘learn’, Socrates goes on to help him provide the right answer. Drawing lines

Figure 6.1

Seeing-As and Mathematical Creativity 135 from the corners of the original square and the other three squares that have the same area, Socrates gets the boy to recognize that the resulting figure is a square which does indeed have twice the area of the initial square (BHID in Figure 6.1). Telling the boy that the line that goes across two opposite corners of a square is called the ‘diagonal’, Socrates formulates the answer to the problem he set, to which the boy assents (Meno, 84d–85b). This answer can be stated as follows: (AN) A square with twice the area of a given square can be constructed from the diagonal of the given square (i.e. has sides whose length is the length of the diagonal of the given square). Knowledge of (AN), Socrates concludes to Meno, has been attained by the boy merely by using his own resources. As Socrates commented to Meno at the time (82e), the fact that the boy initially got the answer wrong shows that he was not being told what to believe. Even if Socrates suggested something, it was still the boy who had to accept or reject it, based on what he himself thought. So he must have had something within him, which he was ‘recovering’, that enabled him to solve the problem; and this process of ‘recovering’, Socrates suggests, is recollection (85c–d). This idea of ‘recollection’ has, of course, been hugely controversial. But the general shape of Socrates’ answer to Meno’s paradox is compelling. We cannot ‘fully’ know the answer to a question prior to inquiry, or else we would be impaled on the first horn of the dilemma; yet there must be something we know, if we are to avoid the second horn. The doctrine of ‘recollection’ is intended to provide some such position between the two horns. What the boy has within him is some geometrical and arithmetical knowledge, such as that a square has four equal sides and that 2 × 2  4. Socrates may need to give him some further terms with which to articulate the solution, as captured in (AN), such as the term ‘diagonal’, but this, too, presupposes that the boy has some knowledge. What is absolutely crucial, however, is that the boy himself must recognize the solution as the solution, and it is here that aspect perception is involved. For the turning point in the boy’s understanding occurs when he switches from seeing the original square as made up of four smaller squares to seeing it as made up of two triangles. Once this switch has occurred, the way is open to seeing the square on the diagonal as twice as big, and hence the solution to the geometrical problem. This can then be ‘proved’ by showing that the larger square is composed of four triangles, each of which has half the area of the original square; but the key step is the aspect switch. This switch is something that the boy must make for himself, and this, we suggest, is what lies behind Socrates’ claim about ‘recollection’—whatever other problems this notion may generate. The boy is capable of seeing this aspect (and perhaps has even seen it on other occasions), and no one can see it for him. It would not be inappropriate for the boy to say something along the lines of “Of course, I should have realized (‘recollected’) that it

136 Michael Beaney and Bob Clark can be seen as two triangles, which obviously solves the problem”. The use of such terms as ‘of course’ and ‘obviously’ suggest just how naturally it is for the boy to come to see the answer.7 As well as underlying the particular geometrical example in the Meno, aspect perception also provides a model for answering the general paradox of inquiry. For in aspect perception there is indeed something already grasped, in some form, for an aspect (or new aspect) to be recognized. In pursuing an inquiry, we must have enough prior knowledge to do so fruitfully and to be able to recognize an answer when it comes. Yet at the heart of what we ‘learn’ may be the perception of a new aspect (an aspect that we have not previously seen—or at any rate, not in that particular context of inquiry).

2. Incommensurability and Arithmetic Recognition of the diagonal of a square—and in particular, the ‘incommensurability’ between the diagonal and the side of a square—was to raise philosophically significant issues in arithmetic as well as in geometry. Here the ‘discovery’ of irrational numbers was a key stage in the development of the mathematical concept of a number, and lying at the core of this development was a move that essentially required a shift of conceptual aspect. We elucidate this in the present section. The problem of incommensurability is mentioned by Aristotle in the Prior Analytics. Giving an example of a reductio proof, Aristotle refers somewhat cryptically to a proof of incommensurability: All those who reach a conclusion through the impossible deduce the falsehood by a syllogism, but prove the initial thesis from a hypothesis, when something impossible results from the assumption of the contradictory. For example, one proves that the diagonal is incommensurable because odd numbers turn out to be equal to even ones if one assumes that it is commensurable. Now that odd numbers turn out to be equal to even ones is deduced by syllogism, but that the diagonal is incommensurable is proved from a hypothesis, since a falsehood results because of its contradictory. (Prior Analytics, I, 23, 41a23–9) This can be filled out as follows. Supposing the side of a square to be of unit length, we can use the result that Meno’s slave boy ‘recollected’ (and which is generalized in Pythagoras’s theorem) to deduce that the square formed on the diagonal has an area of 2 square units. Now, Aristotle claims, this diagonal is ‘incommensurable’. That is, the length of the diagonal of the square is not expressible as a ratio of the side of the square—there is no rational number equal to 2 . The proof, as Aristotle says, proceeds by contradiction.8 Here it is in present-day terminology:

Seeing-As and Mathematical Creativity 137 Suppose

2 is a rational number. That is,

2

a for some integers a b

and b. We can assume that a and b are not both even, for if they are we

a , repeating if necessary if the b a new numerator and denominator are both even. So, let 2  where b can divide them both by 2 leaving 2 

at most one of a and b is even. Now we argue as follows: 2

a  a  2b  a2  2b2 b

That is, a2 is even. So a must be even, since the square of an odd number is odd. Since at most one of a and b is even, it follows that b is odd. But now, since a is even, it must be that a  2p for some integer p. So a2  4p2. But from above, a2  2b2. So 2b2  4p2, or, dividing by 2, b2  2p2. That is, b2 is even and so b is even. There is our contradiction. From the supposition that the length of the diagonal of a square ( 2 units, where the side is 1 unit) is commensurable with the side (i.e. that

2

a for some integers a and b), we have b

proved that “odd numbers turn out to be equal to even ones”, as Aristotle puts it—we have proved that our integer b is both odd and even. So “that the diagonal is incommensurable [viz. that 2 is irrational] is proved from a hypothesis, since a falsehood results because of its contradictory [that the diagonal is commensurable, viz. that 2 is rational]”. What role did this proof of incommensurability play in the development of the concept of number? It might seem natural to us now to suppose that it forced mathematicians to recognize that numbers could be irrational as well as rational. But there is another way of seeing things that emerges once we appreciate two aspects that the ancient Greeks, prior to the discovery of this proof, took numbers to have. Numbers were viewed both as measures of lengths and as ratios. The proof of incommensurability shows that one cannot hold both views. The ancient Greeks continued to regard numbers as measures of lengths and hence came to reject the view that all numbers could be expressed as ratios, that is, are rational. But it would also have been possible to have held firm to the view that numbers are ratios and hence to have abandoned the view that all lengths can be numbered. Another way to express this would be to say that the ancient Greek concept of number, prior to the proof of incommensurability, had two key criteria: (1) numbers are measures of lengths, and (2) numbers are expressible as ratios. (1) was understood as an identity claim: every length can be assigned a number and every number can be represented by a length; in other words, there is a one—one correlation between numbers (in arithmetic) and lengths (in geometry), with a given length chosen as the unit length. (2) is simply the claim that all numbers are rational. In many cases (the ‘rational’ ones), these two criteria give the same result. But the proof of incommensurability showed that in some cases (the ‘irrational’ ones), such as in considering the

138 Michael Beaney and Bob Clark diagonal of a unit square, the two criteria come apart, forcing us to decide which of the two criteria is the more important. We know which way the decision went, of course. The second criterion was rejected in order to keep the first. 2 is a number in good standing, though irrational, and every length is (still) measurable numerically. The concept of number was determined accordingly: it was broadened to include irrational as well as rational numbers. We now take all this for granted, but if we go back to the origin of the determination, we can see that it was by no means necessary. At the core of this determination was a choice of conceptual aspect, and although we might find it hard now to see things in any other way, it is important to recognize that the choice was there and that our concept of number might have developed in another way. We should also note that while the choice between the different ways of seeing—of determining the concept—was, we might say, forced by mathematics itself (the proof above), the outcome of the choice was not so determined. The choice between criteria, whatever its motivation, does not answer uniquely to intra-mathematical considerations; mathematics itself, we might say, allows either choice, while eventually accepting the choice that is made. A rationale for the decision is not hard to find. Had mathematicians stuck to the view that numbers are ratios, then not only would they have had to conclude that some lengths cannot be numbered, but they would also have had to hold that while some expressions of the form ‘ n ’—that is, ‘the square root of n’—denote numbers (such as ‘ 25’, representing the length of the hypotenuse of a right-angle triangle the other two sides of which have lengths 3 and 4), other such expressions (such as ‘ 2’) do not. Of course, they came to hold instead that only some such expressions denote rational numbers; but that does not seem so radical a view as holding that some such expressions, despite their similarity of form, do not denote numbers at all. Before looking at some other examples involving seeing-as, let us briefly take stock, drawing out some implications for our understanding of mathematical creativity. We have suggested that both aspect perception and what we called aspect conception are involved in mathematics and play a role in mathematical creativity. The former was illustrated in recognizing that a square of twice the area of a given square can be constructed on the diagonal of that square, as shown (implicitly) by the slave boy in Plato’s Meno. This is aspect perception in a familiar sense. Just as one can switch from seeing the duck–rabbit as a duck to seeing it as a rabbit, so one can switch from seeing a square as made of four smaller squares to seeing it as made up of two triangles (although it may be that one has to ‘imagine’ lines in the diagram being considered; see Figure 6.1 above). Arguably, at the heart of the creativity involved in solving this mathematical problem is a switch in aspect perception. The point can be readily generalized: as anyone who has had experience of doing Euclidean geometry knows, solving problems of construction or proof involves looking for different ways of seeing figures, perhaps by adding appropriate auxiliary lines, to draw on the resources of theorems already proved.9

Seeing-As and Mathematical Creativity 139 Aspect conception was illustrated in the conceptual creativity involved in broadening the concept of number to include the irrationals as well as rationals. Here, it would seem, there is not so much aspect switch as aspect focus. Two aspects are already present, and the creativity arises from distinguishing them and taking one as primary and rejecting or subordinating the other. We might put the contrast in duck–rabbit terms. In the geometrical case, we see something as a duck but need to see it as a rabbit in order to solve our problem. In the arithmetical case, we see something simultaneously (somehow) as both a duck and a rabbit, but need to take it as just a rabbit, say, to make progress. In the geometrical case, too, however, both aspects are somehow already there (requiring mere ‘recollection’), and progress occurs by selecting and focusing on one; so the key difference does not lie here. What is different in the arithmetical case is the way that the aspect focusing results in the determination of a concept. One aspect is chosen as the defining criterion for the broader concept (number), with the other aspect relegated as a criterion for the subconcept (rational number). Where, previously, we had one concept and some unclarity as to its applicability, we now have more than one concept (number, rational number, irrational number) and greater clarity as to their applicability and logical interrelations. A final point may be made at this halfway stage. The creativity involved in each of these two examples itself has many aspects. What we have hoped to show is that one central (though not defining) aspect is the role played by aspect perception, and more specifically, by what we have called aspect focusing. We do not want to generalize and claim that all creativity involves aspect focusing; but this does seem to lie at the heart of the creativity involved in these particular mathematical examples. If that is right, then to understand the creativity involved, we need to ‘re-create’ in ourselves that aspect focusing, in just the same way that the slave boy had to recognize for himself the aspect switch required to solve the geometrical problem. We can only understand the creativity from ‘inside’, by attempting to recover the original situation in which the aspect perception and focusing occurred. This may take some of the ‘mystery’ out of creativity but only by enabling us to experience, first-hand, the wonderful mental operations that are involved in creative processes. What we do is to bring creativity back from its mysterious to its everyday use.

3. Straight Lines and Non-Euclidean Geometry We mentioned the move from first-person singular to first-person plural aspect perception in the introduction, and we can see this illustrated in the two cases we have just considered. The slave boy in the Meno may need to make the aspect switch for himself to appreciate the solution to the geometrical problem Socrates posed, and we may all have to do it for ourselves if we are also to appreciate the solution, but this possibility is rooted in our

140 Michael Beaney and Bob Clark shared capacities for aspect perception, and this solution only works because of these shared capacities. In the case of the concept of number coming to include both rational and irrational numbers, this is no less a matter of a new way of seeing things—here through aspect focus rather than aspect switch—‘catching on’ in determining a community’s joint conceptualization. We will see this illustrated further in the next two sections. We will look in the present section at the development of non-Euclidean geometry. As in the case of the concept of number just discussed, we can see this development as occurring through criteria for the application of concepts coming apart. We will focus here on two different but connected pairs of criteria: one general, concerning how we see geometry as a whole, and one specific, concerning how we conceptualize ‘straight line’. With regard to the first, what we have is a contrast between seeing geometry as an abstract axiomatic system, with associated apriority, certainty and necessity, and seeing it as descriptive of empirical physical space. From the time of the ancient Greeks to the middle of the nineteenth century, however, these two ways of seeing geometry seemed ineluctably joined. (Euclidean) geometry, it seemed, could be known a priori and yet was also applicable to physical space. In the late eighteenth century, Kant gave an account of how both could be maintained; but although this account was controversial from the beginning, it was only with the advent of non-Euclidean geometry that a diagnosis of the problems became possible. So how did these two ways of seeing geometry come apart? As in the case of number discussed earlier, certain intra-mathematical developments seem to have been required. Interestingly, the creative work of Bolyai and Lobachevsky was apparently not sufficient (contrary, perhaps, to conventional ways of telling the story); their work was at first received grudgingly (where it was recognized at all), and it took some time for non-Euclidean geometry to win acceptance.10 So let us sketch some of the key developments. Talk of non-Euclidean geometry, as is well known, directs us back to Euclid and his (in)famous fifth postulate. Here is how Euclid formulates it: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. (Elements, I, Post. 5) We will take up the question of what it is for a line to be straight shortly. Compared to the first four postulates, this fifth postulate seems relatively complex. So if it is true, might we not be able to prove it from the other axioms (definitions, postulates and common notions)? Proclus was just one influential commentator who thought so, writing that it should be struck from the postulates altogether since it is a theorem (Commentary, p. 150). Skipping millennia of mathematical investigation into this infamous postulate, however, we now know it to be independent of the other Euclidean

Seeing-As and Mathematical Creativity 141 axioms. In the 1820s, in one of those remarkable apparent coincidences in the history of ideas, János Bolyai and Nikolai Lobachevsky independently developed geometries that denied it. Each seems also to have thought of these alternative geometries as genuine rivals to Euclidean geometry, in the sense of being candidates for providing the true description of empirical space. At any rate, there no longer seemed any a priori reason for regarding one as the ‘true’ geometry.11 We see here the beginning of the idea that the two ways of seeing geometry— as an a priori, axiomatic system and as a true description of physical space— might come apart. But now matters become a little complicated, since Bolyai’s and Lobachevsky’s work did not immediately persuade the larger mathematical community to see matters in this light. There were notable exceptions. As far back as 1817, in a letter to Heinrich Olbers, Gauss had written, ‘I am becoming more and more convinced that the [physical] necessity of our [Euclidean] geometry cannot be proved, at least not by human reason nor for human reason’ (quoted in Kline 1972: 872). But Gauss did not go public with such thoughts, and overall the general consensus lagged behind. Why was this so? Let us just take the case of Lobachevsky. He was able to think creatively of a geometrical system denying the parallel postulate as a possible geometry of space, but his methods did not directly deal—or, at least, not obviously so—with traditional geometric objects existing in empirically experienced space. His trigonometric formulae did not seem to apply to what we might normally describe as straight lines, ordinary triangles, and the like. They might be true, as it appeared, only of curved lines, curvilinear triangles, and so on; and these curves, again as it appeared, might simply be embedded within Euclidean space. If so, then there would be no need to change any of the ideas underlying our understanding of space itself. So Lobachevskian ‘geometry’ could easily be relegated to the status of a curiosity, inapplicable to the world as a whole. There remained one final key piece of the jigsaw, which brings us to the second pair of criteria mentioned earlier, concerning how we conceptualize ‘straight line’. What is it for a line to be straight? Euclid defined a straight line as “a line which lies evenly with the points on itself” (Elements, I, Def. 4). But it is unclear how to understand this definition, without some kind of appeal to visual phenomenology.12 In 1868, however, Bernhard Riemann supplied a more rigorous answer, which can be summarized as follows. He first defined, analytically, the length of a line in terms of the limit of its (infinitesimal) increment. With length thus defined, a line joining two points can then be defined as ‘straight’ if all nearby lines joining the two points are longer (see Riemann 1868). A straight line, then, is the shortest distance between two points. This may come as no surprise, but importantly for our purposes, this works in any space—particularly in the curvilinear space of Lobachevsky. Furthermore, it then becomes possible to define the curvature of a space intrinsically. Gauss, earlier in the century, had specified how the curvature of a surface embedded in space was an intrinsic property of the

142 Michael Beaney and Bob Clark surface. Riemann, however, showed how any space could have its curvature defined intrinsically. This is the final piece of the jigsaw. Following Riemann, there can be seen to be many possibilities for geometries (in the plural). A non-Euclidean geometry might thus turn out to provide the ‘true’ description of empirical space; but whether it does or does not is not a question to be answered a priori. The contrast drawn earlier between geometry as descriptive of empirical space and geometry as an abstract axiomatic system is reflected, at the more specific level, in the contrast between two ways of seeing a straight line: as a line that looks straight (“which lies evenly with the points on itself”) and as what measures the shortest distance between two points (a so-called geodesic). Once again, we have two criteria, in this case for what is to count as a straight line, that come apart in conceptual development, in this case in giving rise to non-Euclidean geometry. As ever, there needs to be a mathematical background for such coming apart. But, importantly, the mathematics itself does not dictate which of the criteria we keep and which we drop, just as the background psychological prerequisites for seeing the duck–rabbit in two ways do not give us a reason for seeing it as one rather than the other. We may think we have good reasons for preferring one criterion over the other—in the case of non-Euclidean geometry, perhaps, concerning their applications in physics. But such reasons remain outside of mathematics itself, and we need to look at the wider context, and the way in which the decision to accord primacy to one of the criteria ‘caught on’, in understanding the relevant conceptual developments and hence the mathematical creativity involved.

4. Infinity and Transfinite Arithmetic It was in discussing the way in which ideas of numerical infinity developed that Wittgenstein drew a distinction that has informed our approach in this chapter, namely, between discovering matters of fact and determining concepts. In this section we explain what Wittgenstein had in mind here by taking precisely this case of the development of our mathematical concept of infinity. We have talked of how crucial the ‘catching on’ of a certain way of seeing things can be for conceptual development in a mathematical community. We consider here an actual historical case of an individually creative way of seeing that did not catch on in the mathematical community and contrast it with a way of determining the concept of infinity that did catch on, in spite of Wittgenstein’s strictures on it. What goes on in the development of mathematical concepts, we suggest, is neither wholly ‘discovery’ nor wholly ‘invention’; there is a distinctive role for creativity to play. We will draw out further implications in the concluding section that follows. The way of seeing infinity that did not catch on is due to the seventeenthcentury mathematician John Wallis (1616–1703). Wallis, it should be noted,

Seeing-As and Mathematical Creativity 143 was no outsider to the mathematical community of his time. He influenced Isaac Newton, and his tenure as Savilian Professor of Geometry at Oxford (from 1649 until his death) overlapped with Newton’s Lucasian Professorship at Cambridge (1669–1702). Nevertheless, the development we outline has been largely forgotten. In his Arithmetica Infinitorum of 1655, Wallis claimed that infinity (which he was first to denote using the symbol ‘’) was less than any negative number.13 He arrived at this surprising result by considerations to do with a certain kind of continuity of operation along the ‘number line’ (another invention of his, which did catch on). In brief explanation, suppose we divide a fixed positive number by an ever-decreasing positive variable; then the quotient will increase as this variable divisor diminishes. It made sense to Wallis to consider this procedure as being carried on continuously, decreasing the divisor to zero and then successively through decreasing negative quantities. Considering the increase in the quotient to continue seamlessly as this procedure is carried out, and noting that a positive dividend divided by a negative divisor gives a negative quotient, we will have, taking 1 as our dividend, for instance, ...

1 1 1 1 1 1 1 1 1 1          ...   ... 10 5 2 1 0.5 0.2 0.1 0.01 0.001 0 ...

1 1 1 1 1  ...      ... 0.1 0.2 0.5 1 0

1 1  1. So  1. That is,   1. In words, infinity is less than 1 0 negative one. Nowadays  does not count as a number at all, although 1 the notation survives, in talk of limits, for example. ‘ Lim     ’ is acceptx 0 x able contemporary notation. Overall, though, Wallis’s ‘’ is not a part of present-day mathematics. It did not catch on. Before looking further at Wallis’s ‘proof’ that negative numbers are larger than infinity, let us turn to another attempt at determining a concept of infinity, that of Georg Cantor (1845–1918). Cantor’s development of the concept did, in many ways, catch on, and his line of thinking is nowadays well known. Here is a criticism that Wittgenstein made of it, however, involving the distinction mentioned earlier:

But

The dangerous, deceptive thing about the idea: “The real numbers cannot be arranged in a series”, or again “The set . . . is not denumerable” is that it makes the determination of a concept—concept formation—look like a fact of nature. (RFM, II, 19; p. 131) An (infinite) set is said to be ‘denumerable’ if it can be put into one–one correspondence with the set N of natural (‘counting’) numbers, {1, 2, 3, . . .}.

144 Michael Beaney and Bob Clark This relies on the so-called Hume–Cantor Principle,14 according to which two sets have the same number of members if and only if their members can be one–one correlated. The set Cantor claims to be non-denumerable, a claim criticized by Wittgenstein for the way it makes concept formation look like a fact of nature, is the set of real numbers R (the rational as well as irrational numbers—all the numbers on Wallis’s number line, in other words). And the idea of the real numbers ‘being arranged in a series’ is part of Cantor’s ‘proof’15 of this set’s non-denumerability. For suppose R is denumerable. Then its members can be put into one—one correspondence with the members of N. Assume that this correspondence has been made explicit, with the real numbers listed in order, in their decimal expansion. Every real number then has a position on the list: it will be either the first, or the second, or the third, and so on. But now consider a number which differs from the first number on the list at the first place of decimals, from the second at the second place, from the third at the third place, and so on. It is clear both that there is such a number and that it is nowhere on the list. So the list does not, after all, contain every real number, contrary to the assumption. Hence no such list can be made and R is not denumerable. This is a version of Cantor’s so-called ‘diagonal argument’ (which first appeared in Cantor 1891). Cantor symbolized the cardinality (the number of members) of the infinite set N by ‘0’ (‘aleph-zero’) and the cardinality of the infinite set R by ‘1’ (‘aleph-one’), and called the series beginning with 0 and 1 the ‘transfinite cardinal numbers’. The details need not concern us here. Generally speaking, the notion of ‘infinity-as-0’ is nowadays accepted by mathematicians; Wallis’s notion of ‘infinity-as-’, on the other hand, is not accepted, as we saw. Should we accept current practice and go with Cantor? What about Wittgenstein’s complaints? Is there anything to be said at all for Wallis’s determination of the concept of infinity? We can clarify matters by considering these questions in the light of our account of the way that criteria for the use of concepts can come apart in conceptual development. Here the governing thought is very simple: two criteria for the use of a certain mathematical concept may coincide in finite cases, but when we turn to infinite cases, they do indeed come apart, forcing us to make a choice as to which to give primacy. Let us consider Wallis first. In the move from dividing by a very small positive number to dividing by a negative number, passing through (and assuming a continuity involved in) dividing by zero, the implication is that we move from small to smaller. But we end up with negative numbers larger than . It seems Wallis proves that 1 <  < −1 by using the fact that −1 < 0 < 1. This seems to be a contradiction. Consider, now, the following two criteria for the use of ‘