The History of Continua: Philosophical and Mathematical Perspectives [1 ed.] 0198809646, 9780198809647

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The History of Continua

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The History of Continua Philosophical and Mathematical Perspectives Edited by

S T EWA RT SHA P I R O A N D G E O F F R EY H E L L M A N

1

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3 Great Clarendon Street, Oxford OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © Stewart Shapiro and Geoffrey Hellman 2021 The moral rights of the authors have been asserted First Edition published in 2021 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2020938610 ISBN 978–0–19–880964–7 Printed and bound in Great Britain by Clays Ltd, Elcograf S.p.A. Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work.

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Contents List of Figures List of Contributors

Introduction Stewart Shapiro and Geoffrey Hellman 1. Divisibility or Indivisibility: The Notion of Continuity from the Presocratics to Aristotle Barbara M. Sattler 2. Contiguity, Continuity, and Continuous Change: Alexander of Aphrodisias on Aristotle Orna Harari 3. Infinity and Continuity: Thomas Bradwardine and His Contemporaries Edith Dudley Sylla 4. Continuous Extension and Indivisibles in Galileo Samuel Levey

vii ix

1

6

27

49 82

5. The Indivisibles of the Continuum: Seventeenth-Century Adventures in Infinitesimal Mathematics Douglas M. Jesseph

104

6. The Continuum, the Infinitely Small, and the Law of Continuity in Leibniz Samuel Levey

123

7. Continuity and Intuition in Eighteenth-Century Analysis and in Kant Daniel Sutherland

158

8. Bolzano on Continuity Paul Rusnock

187

9. Cantor and Continuity Akihiro Kanamori

219

10. Dedekind on Continuity Emmylou Haffner and Dirk Schlimm

255

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vi contents

11. What Is a Number? Continua, Magnitudes, Quantities Charles McCarty

283

12. Continuity in Intuitionism Charles McCarty

299

13. The Peircean Continuum Francisco Vargas and Matthew E. Moore

328

14. Points as Higher-Order Constructs: Whitehead’s Method of Extensive Abstraction Achille C. Varzi

347

15. The Predicative Conception of the Continuum Peter Koellner

379

16. Point-Free Continuum Giangiacomo Gerla

427

17. Intuitionistic/Constructive Accounts of the Continuum Today John L. Bell

476

18. Contemporary Infinitesimalist Theories of Continua and Their Late Nineteenth- and Early Twentieth-Century Forerunners Philip Ehrlich

502

Index

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List of Figures 1.1. Zeno’s paradox.

12

4.1. Galileo, Dialogo (1632), 229.

84

4.2. Cavalieri, Exercitationes Geometria Sex (1647), 4.

87

4.3. Galileo, Discorsi (1638), 32.

90

4.4. Galileo, Discorsi (1638), Prop. 1, Thm 1. 165.

98

4.5. Huygens, Horologium, Prop. V.

100

5.1. Cavalieri’s indivisibles.

105

5.2. Cavalieri’s integration of ‘cossic powers’.

107

5.3. Torricelli’s quadrature of the parabolic segment.

110

5.4. Torricelli’s ‘paradox of indivisibles’.

111

5.5. Torricelli’s curved indivisibles.

113

5.6. Roberval on the quadrature and tangent construction for the cycloid.

115

5.7. Wallis and the quadrature of the cubic parabola.

119

6.1. Sorites argument ‘transposed to continuous quantity’ in Leibniz’s Pacidius Philalethi (A 6.3.540).

128

6.2. Leibniz, Pacidius Philalethi (1676), A 6.3.550.

130

6.3. Leibniz, DQA (1676), Prop 6, A 7.6.528.

143

6.4. Figure from Knobloch (2002), 66, for DQA Proposition 6.

144

6.5. Leibniz, Cum prodiisset (c. 1701), 44.

151

14.1. An abstractive class of concentric squares converging to an ideal point.

354

14.2. An abstractive class of concentric rectangles converging to an ideal line segment.

354

14.3. An abstractive class of concentric squares and an abstractive class of concentric circles converging to the same point.

357

14.4. An abstractive class of concentric squares and an abstractive class of cotangential circles converging to the same point. The former class covers the latter, but not vice versa.

361

14.5. An abstractive class of cotangential circles and the abstractive class of circumscribed squares converging to the same point. The former class covers the latter, but not vice versa.

363

14.6. The intended distinction between two connected regions (left) and two disconnected ones (right).

366

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viii list of figures 14.7. A counterexample to the intended distinction.

366

16.1. An apparent point.

434

16.2. A and B are externally tangent.

444

16.3. A and B are internally tangent.

444

16.4. A and B are externally diametrically tangent to C.

444

16.5. A and B are internally diametrically tangent to C.

445

16.6. A is concentric with B.

445

16.7. The expressive power of our language.

447

16.8. The falsity of triangular inequality.

458

16.9. L is almost contained in S.

461

17.1. Function with two relative maxima.

487

17.2. Piecewise linear function.

488

17.3. Microintervals.

492

17.4. Microneighbourhoods.

495

17.5. Defining smooth natural numbers.

500

18.1. Comparative graphs of select real functions.

515

18.2. Early stages of the recursive unfolding of the surreal number tree.

528

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List of Contributors John L. Bell Professor Emeritus of Philosophy, University of Western Ontario Philip Ehrlich Professor Emeritus of Philosophy, Ohio University Giangiacomo Gerla Professor Emeritus of Logic and Foundation of Mathematics, University of Salerno, Italy Emmylou Haffner Institut de Mathématiques d’Orsay, Université Paris Saclay Orna Harari Associate Professor of Classics and Philosophy, Tel Aviv University Douglas M. Jesseph Professor of Philosophy, University of South Florida Akihiro Kanamori Professor of Mathematics, Boston University Peter Koellner Professor of Philosophy, Harvard University Samuel Levey Professor of Philosophy, Dartmouth College Charles McCarty Senior Visiting Fellow, Sidney M. Edelstein Center, The Hebrew University of Jerusalem Matthew E. Moore Professor of Philosophy, Brooklyn College Paul Rusnock Philosophy, University of Ottawa Barbara M. Sattler Professor of Ancient and Medieval Philosophy, Ruhr-Universität Bochum Dirk Schlimm Associate Professor of Philosophy, McGill University Daniel Sutherland Associate Professor of Philosophy, University of Illinois at Chicago Edith Dudley Sylla Professor Emerita of History, North Carolina State University Francisco Vargas Associate Professor of Mathematics, El Bosque University at Bogotá Achille C. Varzi Professor of Philosophy, Columbia University

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Introduction Stewart Shapiro and Geoffrey Hellman

The idea (or, perhaps better, the need) for this volume became clear to us when we were working on our monograph, Varieties of continua: from regions to points and back.1 We deveoped an interest in various contemporary accounts of continuity: the prevailing Dedekind–Cantor account, smooth infinitesimal analysis (or synthetic differential geometry), and intuitionisic analysis. Each of these theories sanctions some long-standing properties that have been attributed to the continuous, at the expense of other properties so attributed. The intuitionistic theories violate the intermediate value theorem, while the Dedekind–Cantor one gives up the thesis that continua are unified wholes, and cannot be divided cleanly. The slogan is that continua are viscous, or sticky. We were surprised to learn that many philosophers and even some mathematicians take the Dedekind–Cantor conception of continuity to be not only the right one, but the only one. Some were surprised to learn that there are any other notions. They may have heard something of the history, but many take it for granted that we now have the one correct account of continuity. The once longstanding “intuitions” that support the other accounts are not to be taken seriously (and perhaps never should have been). Our view is that there is no single, monolithic property of continuity. It is more of a cluster concept that can be sharpened, and developed rigorously, in mutually incompatible ways. Early on, we were led to the Aristotelian idea that continua are not composed of points, or indivisible parts. In terms of contemporary metaphysics, the theme is that continua are gunky: every part of a continous substance has a proper part. This, of course, is not sanctioned in the contemporary Dedekind–Cantor account nor, arguably, in the intuitionistic accounts either (depending on what counts as a “point” in those contexts). The bulk of our project in Varieties of continua was to develop various gunky, or point-free, accounts of continuity: one-dimensional and two- (and three- …) dimensional, Euclidean and non-Euclidean, with actual infinity and without actual infinity (another theme derived from Aristotle and maintained for two millennia).

1 Oxford, Oxford University Press, 2018. Stewart Shapiro and Geoffrey Hellman, Introduction In: The History of Continua: Philosophical and Mathematical Perspectives. Edited by: Stewart Shapiro and Geoffrey Hellman, Oxford University Press (2021). © Stewart Shapiro and Geoffrey Hellman. DOI: 10.1093/OSO/9780198809647.003.0001

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2 introduction We went on to compare these accounts with their contemporary counterparts on various mathematical, logical, and metaphysical grounds. Although it is obvious that mathematical and philosophical thought about the continuous developed considerably over the ages, we could not find any comprehensive treatment of this history.2 So the time is right for such a project. The two of us have a deep interest in the history of philosophy and mathematics, but we are not scholars in these areas. So an edited volume seemed the right course to follow. Peter Momtchiloff, from Oxford University Press, enthusiastically received our proposal, and was very generous concerning content and length. The present volume is the result. Our idea that a volume on this topic is most timely, and most needed, must have been correct, judging from the stellar group of scholars we were able to recruit. They are among the top researchers in each period that is covered. At one point, early on, someone suggested that we keep this project quiet for a time, so that we would not be “scooped” by someone else. Our reply was that we would, of course, welcome more work in this area, but we could not see anyone coming up with a lineup anywhere near as good as the one we have here. The papers here all speak for themselves, and the reader can determine the content of each from its title and its place in the volume. So we will rest content here with a very brief remark on each paper. We begin in ancient Greece. Barbara Sattler’s contribution concerns the metaphysical and natural philosophy that underlies the ancient discussions, arguing that the mathematics is a less central concern. The main focus is, of course, Aristotle, and his precursors, notably Parmenides and Zeno. A central theme of the paper is Aristotle’s response to Zeno’s paradoxes.3 Orna Harari covers the period in antiquity after Aristotle, focusing primarily on Alexander of Aphrodisias. The close look at one of Aristotle’s successors helps illuminate both accounts. Edith Dudley Sylla turns to the medieval period. Her main focus is a (relatively) recently discovered manuscript, by Thomas Bradwardine, and its relation to medieval views before, during, and after Bradwardine’s time. Many of the issues under debate today were prevalent then. Next is the so-called early modern period, when mathematicians developed the calculus and, with that, the rise of infinitesimal techniques. In effect, continuous magnitudes are treated as infinite sums of indivisible elements, each of which is infinitely small. The philosophical issues dominated thinking during that period. We are delighted to have two contributions by Samuel Levey. The first is on Galileo Galilei, whose entry to the theme of the continuum is the analysis of 2 There is Infinity and continuity in ancient and medieval thought, edited by N. Kretzmann (Ithaca, Cornell University Press, 1983). 3 We might add that Barbara, and her colleague Sarah Broadie, were most helpful to us in our work on Varieties of continua, aiding us in understanding the main Aristotelian themes. We eagerly await Barbara’s forthcoming book on Aristotle on continuity.

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introduction 3 continuous uniform and accelerated motion, a common concern of mathematicians, scientists, and philosophers during this most fascinating era. Levey’s second contribution is on Gottfried Wilhelm Leibniz, who was profoundly influenced by Galileo. Leibniz famously dubbed the continuum a “labyrinth”. The reason for this is, in large part, that “the discussion of continuity and of the indivisibles that appear to be its elements” requires “consideration of the infinite”. In between these two papers, there is one by Douglas Jesseph, who focuses attention on Bonaventura Cavalieri, whose mentor was Galileo. Cavalieri contributed greatly to the so-called “method of indivisibles” that formed the basis for the infinitesimal techniques developed by Leibniz, Newton, and others. Daniel Sutherland’s article turns the reader’s attention to Immanuel Kant, and the role and place of continuity and intuition in eighteenth-century analysis. It focuses on issues raised by continuity for the representation of the infinitely small and, in particular, on the status of geometrical and kinematic representations. Paul Rusnock covers Bernard Bolzano, a fascinating figure of the early nineteenth century. Bolzano was one of the first important mathematicians and philosophers to insist that continua are composed of points, and he made a heroic attempt to come to grips with the underlying issues concerning the infinite. Next up are the two figures most responsible for the contemporary hegemony. Akihiro Kanamori covers Georg Cantor. This article provides the rich mathematical and historical basis for Cantor’s initial work on limits and continuity and ascent from early conceptualizations to new ones, from interactive research to solo advance. Cantor proceeded to more and more specific results, just as he developed more and more set theory. Emmylou Haffner and Dirk Schlimm cover Richard Dedekind, providing a detailed view of both foundational and mathematical aspects. Dedekind, of course, characterized the property of continuity for the real numbers in terms of what are now called “Dedekind cuts” on the rational numbers—thus presupposing that continua are composed of points, or point-like elements. Haffner and Schlimm go on to consider some of Dedekind’s more mathematical treatments of continuity, notably the definition of the Riemann surface in his joint work with Heinrich Weber (1882). They show how Dedekind’s approaches became increasingly abstract, while at the same time retaining a common methodology. We have two outstanding contributions by Charles McCarty. The first uses a lucid analysis of the mathematician Paul du Bois-Reymond to argue for a constructive account of continuity, in opposition to the contemporary Dedekind– Cantor dominance. McCarty’s second paper, a nice companion to the first, treats Hermann Weyl and, more importantly, L. E. J. Brouwer. Francisco Vargas and Matthew E. Moore cover Charles Sanders Peirce, who once dubbed the notion of continuity “the master-key which …unlocks the arcana of philosophy”. Roughly, Peirce’s account has it that a continuous substance has a lot more points than a region of Dedekind–Cantor space—in effect there is no

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4 introduction set of all such parts. Vargas and Moore cover the defining features of Peirce’s mathematical theory of continuity, giving a model for that theory in Zermelo– Fraenkel set theory. They go on to summarize Peirce’s own attempts to put his conception into a rigorous form. Alfred North Whitehead is known for presenting a point-free, or gunky, account of continuity, and he showed how to recover points as a kind of “extensive abstraction”, a limit of sets or sequences of regions.⁴ Achille Varzi presents Whitehead’s various attempts along these lines. Each of the final four papers in the volume presents a more or less contemporary take on continuity that is outside the Dedekind–Cantor framework. Peter Koellner gives us an account based on predicativity—the rejection of impredicative definitions—derived from the work of Henri Poincaré, Bertrand Russell, Hermann Weyl, and, especially, Solomon Feferman. So far as we know, all other views take a continuous substance, like space or space-time, to be given as a whole, in its entirety. In contrast to the Dedekind–Cantor views, many theorists (Aristotle, most of the medieval writers, Leibniz, Kant, …) insist that the parts of a continuous substance constitute a potential infinity: our ability to carve out and describe parts of continua is only potential, in the sense that there is no completed totality of all such parts. A distinctive feature of the predicativist view is that it takes a continuum to be itself potential while—at least as presented rigorously by the leading proof-theorist Solomon Feferman—using classical logic (but not modal logic). This seems to be the result of accepting some aspects of the Dedekind– Cantor picture. Roughly, (1) we think of a line, say, as a collection of points; (2) we think of the points as real numbers (as in a number-line); and (3) we think of a real number as a set of natural numbers. But we then insist that all such sets must be defined in a predicative manner. So the continuum (or this continuum) is produced in stages: as we define some sets of natural numbers, we are then able to define more such sets, and there is no stage where all such sets have been, or can be, defined. Giangiacomo Gerla presents a survey of contemporary accounts (including Varieties of continua) that do not take continua to be composed of points. He notes that a central issue, in each case, is to recover the notion of being a point, typically via some sort of abstraction in the mould of Whitehead’s extensive abstraction. John Bell covers contemporary accounts of continuity that invoke intuitionistic logic. Most become inconsistent if a generalized version of excluded middle is imposed. A central concern is the extent to which each account sanctions a longstanding intuition that continua are wholes, and cannot be divided cleanly. As noted above, this “indecomposibility” is lost on the Dedekind–Cantor accounts.

⁴ Similar techniques are used in our Varieties of continua.

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introduction 5 Indeed, such accounts view a continuum as a set of points. With classical logic, each subset has a complement. Philip Ehrlich provides a rich presentation of theories that, unlike the Dedekind–Cantor accounts, accept the existence of infinitesimals. Such accounts thus violate the Archimedean principle, adopted in both Aristotle and Euclid, but the payoff is considerable. Ehrlich shows how such accounts derive from forerunners from the late nineteenth century and the early decades of the twentieth century.

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1 Divisibility or Indivisibility The Notion of Continuity from the Presocratics to Aristotle Barbara M. Sattler

1. Introduction While mathematical practice in ancient times provided some inspiration for the debate about continuity in early Greek thinking up to the time of Aristotle, mathematics is not where the main debate—as far as it has been handed down to us—happens. Rather, the discussion about continuity is a debate within metaphysics and natural philosophy. We will see that the main thinkers to contribute to the development of an understanding of continuity are Parmenides, Zeno, and Aristotle. And while a modern understanding of continuity may seem to be essentially anti-Aristotelian,1 Aristotle will prove to be the thinker who prepared many of the crucial features of a modern account of continuity.2 The main point of controversy about continuity in early Greek times is divisibility, as this chapter aims to show. All parties to this dispute agree that magnitudes which are continuous (suneches) are homogeneous and without any gaps.3 They disagree, however, on which inferences to draw from this for the possibility of divisibility—whether it implies divisibility or indivisibility. The first philosophically interesting and systematic usage of the notion of continuity we find in Parmenides poem. He understands being continuous as being completely homogeneous and without any differences. Parmenides infers from the lack of any differences that what is continuous is also indivisible, since what is completely homogeneous does not provide any (sufficient) reason for it to be divided anywhere; thus it is not divisible.

1 For a notable exception of a modern conception of the continuum that is inspired by Aristotle, and thus either assumes no points as basic constituents, or even no actual infinity; see Linnebo, Hellman, and Shapiro 2016 and Hellman and Shapiro 2018. 2 Even if notoriously he rejects the assumption of an actual infinity, see especially Physics book III. 3 Gaplessness is here used in an intuitive sense as being without any holes, interruptions, or sudden changes; not in the modern mathematical sense in terms of completeness according to which rational numbers do not form a continuum, while real numbers do. Barbara M. Sattler, Divisibility or Indivisibility: The Notion of Continuity from the Presocratics to Aristotle In: The History of Continua: Philosophical and Mathematical Perspectives. Edited by: Stewart Shapiro and Geoffrey Hellman, Oxford University Press (2021). © Stewart Shapiro and Geoffrey Hellman. DOI: 10.1093/OSO/9780198809647.003.0002

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parmenides’ account of continuity 7 Zeno will be shown to strengthen Parmenides’ understanding of continuity, by demonstrating that we would get into inconsistencies if we were to assume divisibility: given that there are no internal differences that could give rise to any division, if we assumed what is continuous to be divisible at any particular point, then it seems it could be divided everywhere. But if it can be divided everywhere, the parts we would thus derive cannot be thought of consistently according to Zeno, as we will see below. By contrast, Aristotle embraces the idea of continua as being divisible anywhere, which he takes up from the activity of the mathematicians of his time. While geometers of this time presuppose the magnitudes they deal with to be divisible anywhere, interestingly they do not discuss continuity in the mathematical texts handed down to us. We find, however, a full discussion of this notion as appropriated for natural philosophy in Aristotle’s Physics, and divisibility ad infinitum is a crucial feature. Aristotle reacts to the divisibility problem raised by Zeno’s paradoxes with a complex of logical tools: he shows that these problems can be avoided with the help of a new understanding of the part–whole relationship, a two-fold understanding of limits, a new understanding of the notion of infinity, and a careful distinction between actual division and potential divisibility.

2. Parmenides’ Account of Continuity The Greek term for being continuous, suneches, literally means ‘holding together’. We will see that this holding together can be understood in rather different ways— things are temporally uninterrupted, spatially connected, or ontologically holding together. In Greek literature before Parmenides, the word suneches refers mainly to uninterrupted activity⁴ and as such implies a certain temporal extension (two days or ten years) during which this activity takes place. With Parmenides, however, being continuous no longer refers to an activity, but rather to an ontological feature: being continuous is a characteristic of what truly or ultimately is (to eon, Being), which is the only thing that can be thought consistently.⁵ What truly is has nothing to do with any kind of activity; indeed, Parmenides explicitly claims that it is unmoved or unmovable.⁶ For Parmenides, being continuous implies complete homogeneity and, ultimately, indivisibility. Three passages from fragment 8 of his poem set out Parmenides’ notion of continuity in particular. ⁴ See, for example, Odyssey IX, lines 74–75 (“there for two nights and two days continuously we lay, eating our hearts for weariness and sorrow”), or Hesiod, Theogony 635–636 (“they, with bitter wrath, were fighting continually with one another at that time for ten full years”). For more details, cf. Sattler 2019, on which the Parmenides part of the paper draws. ⁵ For an account of how to understand what truly is according to Parmenides, see Sattler 2011. ⁶ The verbal adjective akinêton can indicate either a possibility (unmovable) or a passive resulting state (unmoved).

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8 continuity from the presocratics to aristotle

2.1 Being Continuous Excludes Any Temporal Differences In fragment 8, lines 5–6, Parmenides claims that “neither was it [what truly is] nor will it be, since it is now all together, one, continuous”. Being “now all together, one, continuous” is named as the reason why temporal differences that are captured as “was” and “will be” cannot be said of what truly is. “What was and will be” seem to be the things we deal with in our everyday world (which for Parmenides cannot be objects of knowledge but are what we ordinary mortals refer to in our opinions). These things are spread out temporally: they were there in (some part of) the past or will be there in (some part of) the future. By contrast, what truly is, is not subject to these temporal differences, because, according to Parmenides, it is altogether now, one, continuous. It is now—this has been understood either as indicating atemporality, being beyond time;⁷ or as indicating some present that we can never address as past or future.⁸ In both cases, ‘now’ cannot be temporally extended if it is to be strictly distinguished from was and will be; otherwise there will be a time when it would be right to say of it that it was or that it will be.⁹ So according to the first passage, Parmenides denies that what truly is is extended in time in the way everyday perceptible things are; it is continuous in the sense of not allowing for any temporal differences, like was and will be. Being “now all together”, “one”, and “continuous” thus prevents what truly is from being stretched out in time.

2.2 Being Continuous Implies Being Homogeneous, Full of Being, and No More or Less In lines 22–25 Parmenides makes it clear that being continuous excludes not only temporal differences, but also other kinds of differences: (1) And it is not divisible since it is all homogeneous.1⁰ (2) Nor is it more anywhere (or at any point), which would prevent it from being one continuous, nor less, but it is as a whole full of being.

⁷ See Owen 1966; similarly Mourelatos 2008, pp. 105–107. ⁸ See Coxon 2009, p. 196, who understands it as “total coexistence in the present”. And it needs to be a present that has neither come into being nor will pass away, since Parmenides argues against generation for what truly is. ⁹ Some scholars have read the ‘now’ as indicating eternal temporal duration, for example, Tarán 1965, p. 179, Gallop 1984, p. 15, and Palmer 2009; cf. also Schofield 1970. For a critique of this reading, see Sattler 2019. 1⁰ Oude diaireton could either mean ‘not divisible’ or ‘not divided’. Since Parmenides seems to deduce necessary features of what truly is in fragment 8, ‘not divisible’ seems to be the more appropriate translation.

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parmenides’ account of continuity 9 (3) Through that it is all continuous, for Being is in contact with Being (fr. 8, lines 22–25). The first step in this argument claims Parmenides’ Being to be all homogeneous; this implies being indivisible. The second step rules out that it is more or less—a condition that would prevent it from being continuous. Instead, it is as a whole full of being, which seems to mean equally full, neither more nor less. The last part of the third step, “Being is in contact with Being”, points out that all of Being is connected, and so, presumably, there is nothing in between anywhere that is not Being, which would undermine the homogeneity of what truly is. Let us clarify the sense in which the continuity referred to here should be understood and how it relates to being homogeneous. The first part of the third step, “through that it is all continuous”, reads as a conclusion—what precedes thus should explain why what truly is is all continuous.11 The features we are given in (1) and (2) that should guarantee continuity are that it is not divisible, it is all homogeneous, it is not more anywhere nor less, and it is as a whole full of being. What is important for us here is that being continuous includes all these features. Accordingly, being homogeneous (homoion) is a weaker notion than being continuous, since being continuous means being homogeneous plus fulfilling some further criteria. The Greek word homoion basically means ‘of the same kind’.12 Thus, being homoion here seems naturally understood as being homogeneous and one with respect to kind or genus—what truly is is not divisible into different kinds or genera. This would still leave open the possibility of other, internal differences, like quantitative or qualitative differences, which are at least in part excluded with the following lines that there is no ‘more or less’ that would prevent Being from being continuous. Being continuous has been understood in very different senses in this passage— in a temporal, spatial, ontological, and logical sense.13 For the time being, I will simply suggest that being continuous not only implies indivisibility in kind and genus, but also excludes some other kind of differences (be they temporal, spatial, ontological, or logical); the next passage will bring more clarity on this question.

11 The following clause, “Being is in contact with Being”, is either a summary or reformulation of (1)–(2), or, as is sometimes the case with Parmenides, an additional reason that is only provided after the conclusion. 12 See also, for example, Mourelatos 2008, pp. 113–114, 131 and Tarán 1965. 13 For example, Owen 1960, pp. 96–97 understands it temporally; Schofield 1970, pp. 132–134 takes it in a spatial sense; Tarán 1965, pp. 106–108 understands it ontologically; and Coxon 2009, pp. 325– 326 suggests a logical sense. Since the temporal differences we encountered in fragment 8, lines 5–6 were differences of tense and thus completely different from differences of ‘more nor less’, it would be strange if these temporal differences were now taken up by ‘more or less’.

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10 continuity from the presocratics to aristotle

2.3 Conditions That Would Prevent Continuity In lines 43–49 Parmenides spells out conditions under which what truly is would no longer be continuous: what would prevent it from being continuous could be either non-Being or an unequally distributed Being. The former would lead to it being larger or smaller here or there, the latter to more or less Being. But since there is no non-Being1⁴ and not more of Being here and less there,1⁵ the continuity of Being as a whole is granted. The explicit aim of these lines is to demonstrate the completeness of what truly is; but part of its completeness consists in its being continuous.1⁶ While “larger and smaller” suggest quantitative difference—it is what we would call quantities that are larger or smaller—“more or less” could also cover qualitative differences (for example, more or less hotness, blueness, etc.). Accordingly, it seems plausible that Parmenides wants to rule out what we would call quantitative as well as qualitative differences here—not any specific quantitative and qualitative differences, but rather quantitative and qualitative differences in general.1⁷ Summing up the discussion of all three passages, we can say that being continuous for Parmenides requires something to be homogeneous and to exclude any kinds of differences which can be further specified as follows: temporal differences, as well as what we may term qualitative and quantitative differences. The necessary result of continuity thus understood is indivisibility. For Parmenides seems to assume that a division is possible only where there is inhomogeneity and thus differences—if something is divisible then it must be divisible by virtue of a difference within itself such that one part of it can be separated from the part from which it differs. But since for Parmenides what is continuous is homogeneous in every respect and excludes differences, it is necessarily indivisible.1⁸

3. Zeno’s Paradoxes: Negative Consequences of Infinite Divisibility Parmenides argues for what is continuous to be indivisible. Zeno strengthens Parmenides’ understanding of continuity by showing the negative consequences the assumption of divisibility would have: it would undermine any strong notion of unity, and the parts of such a division could not be consistently thought.

1⁴ As shown in fragments 2, 6, and 7. 1⁵ As demonstrated in lines 22–25. 1⁶ While the word suneches is not mentioned, the discussion so far shows that this passage systematizes the conditions that would prevent something from being continuous. 1⁷ Of course, it is not obvious that Parmenides would have distinguished between quality and quantity in the way familiar to us at least since Aristotle; but he clearly excludes two distinct kinds of differences that we may capture as quantitative and qualitative. 1⁸ See Sattler 2019 for support of this claim.

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zeno’s paradoxes 11 Let us start with a brief look at the challenge divisibility seems to pose for an understanding of unity according to Zeno in fragments 1 and 2:1⁹ Fragment 1: And Themistius says that Zeno’s argument tries to prove that what is, is one, from its being continuous and indivisible. ‘For’ runs the argument, ‘if it were divided, it would not be one in the strict sense because of the infinite divisibility of bodies.’ Fragment 2:2⁰ For, he argues, if it were divisible, then suppose the process of dichotomy to have taken place: then either there will be left certain ultimate magnitudes, which are minima and indivisible, but infinite in number, and so the whole will be made up of minima but of an infinite number of them; or else it will vanish and will be divided away into nothing, and so be made up of parts that are nothing. Both of which conclusions are absurd. It cannot therefore be divided, but remains one. Further, since it is everywhere homogeneous, if it is divisible it will be divisible everywhere alike, and not divisible at one point and indivisible at another. Suppose it therefore is everywhere divided.21 Then it is clear again that nothing remains and it vanishes, and so that, if it is made up of parts, it is made up of parts that are nothing. For so long as any part having magnitude is left, the process of division is not complete. And so, he argues, it is obvious from these considerations that what is is indivisible, without parts, and one.

These paradoxes claim that if we assume some one thing to be divisible and thus to have parts, this one will not be one in a strict sense any longer. It rests on the background assumption that if it were divisible, it would have parts and thus it would not only be one (whole), but also at the same time many (parts). This is, however, impossible, since being one and many are mutually exclusive notions for the Eleatics, as Zeno makes clear in fragment 8.22 Hence assuming divisibility leads to the contradictory result that one is also many. Only if our one is continuous and thus indivisible will we be dealing with what is really one. Furthermore, the assumption that the one is divisible and divided everywhere also shows that we cannot conceive these parts in any consistent way (as he makes clear in the plurality paradox in fragment 2 just quoted). A variant of this problem can also be seen with one of Zeno’s probably more famous paradoxes, his first 1⁹ I am using H.D.P. Lee’s 1967 edition of Zeno’s fragments, since it is more encompassing than Diels/Krantz; I will also use his translations. Fragment 1 in Lee is found in Simplicius, Physics 139.19–22; fragment 2 in Simplicius, Physics 139.27ff. The discussion of Zeno draws from Sattler 2020b, ch. 3. 2⁰ Porphyry attributes fragment 2 to Parmenides, but Alexander and Simplicius think it more likely by Zeno; cf. also Lee 1967, p. 12. 21 This sentence demonstrates Zeno’s move from divisibility to being divided—a central point in Zeno’s paradoxes that will be attacked by Aristotle. 22 Lee fragment 8 = DK29 A21.

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12 continuity from the presocratics to aristotle paradox of motion. According to this paradox, if something moves over a certain distance, for example a runner wants to cover a certain finite track AB in a finite time FG, she first has to cover half of this distance AC (Figure 1.1). But before the runner can cover the distance AC, she must have covered already half of this distance, AD, and before that, half of this half, AE, etc. So she will have to pass an infinite number of spatial segments before reaching the end.23 A

E 3

D 2

C 1

F

B Distance G Time

Figure 1.1 Zeno’s paradox.

This seemingly simple paradox raises several problems, but the only one that concerns us here is that it seems to show that by covering a finite distance, a runner has to pass an infinite number of spatial segments, which, according to Zeno, cannot be done. Zeno’s paradoxes strengthen Parmenides’ understanding of continuity, by demonstrating that even if we were to leave Parmenides’ position to the side, divisibility still seems to get us into inconsistencies for the following reason: given that there are no internal differences that would account for any division, if we assume what is continuous to be divisible at any particular point then it seems to be divisible anywhere—as we just saw, the finite race track seems to be divisible ad infinitum. But if it is divisible anywhere, it seems it can be divided everywhere.2⁴ The parts we would thus derive are, however, problematic according to Zeno (as we saw in fragment 2). For either we assume that the process of division can go on without any restrictions, in which case the parts either (1) have to be divided until there is nothing left; then we will have to assume that these parts of nil extension make up an extended whole, which Zeno claims to be absurd. Or (2) these parts have some extension, but this only means that the division is not complete yet, since these parts would in principle be further divisible, and the parts are thus undetermined. Alternatively, the process of division cannot go on forever, but, at a certain point, reaches something like indivisible minima. Then (3) the one finite whole will be made up of infinitely many extended minima, which according to Zeno is not thinkable. We may be puzzled why he assumes infinitely many minima in this case. Fragment 11 suggests an answer by pointing out that if there is a plurality, it has to 23 In Aristotle’s Physics 263a4–11 this paradox is presented in two forms, which in accordance with the secondary literature can be called ‘progressive’ and ‘regressive’. The regressive variant is just the given state of affairs, while the progressive form assumes that after the runner has covered the first half, she then again has to cover the first half of the remaining distance and then again the first half of the still remaining distance, etc. However, in logical terms, both versions are equivalent. For a more detailed discussion of this paradox, see Sattler 2020b, ch. 3. 2⁴ The move from being divisible anywhere to being divided everywhere will be questioned by Aristotle.

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a mathematical conception of continuity 13 be both finite (the number they are) and infinite (for in order for there to be two separate things, there always needs to be some other third thing in between, and another fourth thing between the second and third, etc.). We know that in fact all three avenues considered by Zeno to be absurd have been pursued by the tradition to follow, the last one, in modified form, by the atomists, the second one by Aristotle, and the first one by modern mathematics.2⁵ For Zeno, however, assuming divisibility of what is continuous2⁶ leads to two major problems: such a continuous whole would only be one in a weak sense, as it would be many parts at the same time, and all possibilities of conceiving of such parts lead to absurdities. Accordingly, what is continuous has to be indivisible for the Eleatics.2⁷

4. A Mathematical Conception of Continuity While our contemporary understanding of continuity developed within the field of mathematics, so far we have seen that for the early Greeks the notion was discussed within the arena of metaphysics and natural philosophy.2⁸ Surprisingly, we do not find a discussion or explicit definition of continuity as such in the mathematical texts handed down to us from the time before or just after Aristotle. Also in Euclid, our best mathematical source close to the time of Aristotle, the term suneches is not defined and not very often used.2⁹ Nevertheless, we find clear indications that crucial features of the most prominent understanding of continuity later on can be found with the mathematicians, as the following points suggest. There are a couple of passages in the Elements in which Euclid uses the term suneches as being successive in the way continuous lines are.3⁰ In these passages, suneches seems to be understood as two-place—one straight line is continuous with another straight line. But there is also a one-place understanding of continua as being infinitely divisible, which is presupposed generally in geometrical constructions: geometers have to understand their geometrical objects—lines, 2⁵ Grünbaum 1968, for example, adopts this first route, that the extended whole would have to be made up of unextended parts, and points out that there is a way to allow for such part–whole relations in mathematics, where a line, and thus something extended, is understood to consist of extensionless parts. Since extension is simply a feature of the set making up the line, not of any of the individual members of this set, there is, according to Grünbaum, no paradox. And Grünbaum simply assumes that the same holds in the physical realm. It is not clear, however, that what holds true of mathematical things can also be said of physical things—that an extended physical thing can consist of unextended physical parts and that this is not just a mathematical description. 2⁶ We saw that for Zeno this means assuming unrestricted divisibility, since there is no more reason to assume a division here rather than there. 2⁷ There are a couple of philosophers who deal with some notion of suneches between Zeno and Aristotle, such as Philolaos, Gorgias, and Anaxagoras, but I do not have space to discuss them here. 2⁸ The following section, as well as sections 5 and 6, draw on Sattler 2020b, ch. 7. 2⁹ And most of the time the term refers to a continued proportion—to a continuous ratio as in Elements book VIII, proposition 8. 3⁰ As, for example, in book I, postulate 2, or book XI, proposition 1.

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14 continuity from the presocratics to aristotle surfaces, and solids—as magnitudes that are always further divisible and thus divisible without end.31 This becomes clear, for example, from the discussion of the anonymous treatise On Indivisible Lines, where the reply to the postulate of indivisible lines frequently relies on the assumption that mathematicians treat their geometrical objects as being always further divisible.32 While not explicitly discussed, this idea of magnitudes being divisible without end seems to have been taken so much for granted by mathematicians that they do not have to pay it any special attention. It is clear that when they assume crossing lines and similar constructions, there is no reflection of atomistic worries, such as that a line crossing another line would need to go between two atoms; rather infinite divisibility just seems to be presupposed. Finally, also Aristotle claims the mathematicians to understand magnitudes in the sense of being infinitely divisible in his discussion of the infinite in Physics book III.33 We see that the mathematical understanding of magnitudes is crucially different from the Eleatic notion. While the mathematicians seem to share the Eleatic assumption that being continuous implies being homogeneous—for the mathematicians this means that each possible part of a continuous magnitude is treated alike—the infinite divisibility of the mathematical continuum is the very opposite of the indivisibility Parmenides assumed.3⁴ The mathematical understanding of continuity in the sense of infinite divisibility is presupposed by any geometrical operation that involves the mathematical bisection of a line, surface, or body.3⁵ For his physics Aristotle can take up this implicit understanding of magnitudes from the mathematicians,3⁶ even if they do not prominently capture it with the term suneches. Aristotle’s account of continuity thus seems to combine Eleatic terminology with a reflection on Greek mathematics.3⁷ 31 From a contemporary point of view, we may ask whether this understanding of geometrical magnitudes as being always further divisible would map onto the real numbers, as the DedekindCantor continuum does, or only onto the rational numbers. So does the infinite divisibility simply mean that we could in principle make infinitely many cuts (which would be the case with the rational numbers), or does it mean that we can divide it exactly wherever we want, even at, let us say, 𝜋 (for example with the help of a circle that we roll along the line we want to cut)? 32 See De Lineis Insecabilibus 969b20ff. 33 203b17–18. And in Physics 200b18–20 Aristotle refers to existing accounts of being suneches as being infinitely divisible that are likely to be mathematical accounts, cf. Sattler 2020b, ch. 7. 3⁴ Zeno attacks the assumption of infinite divisibility in his paradoxes, but we do not have the textual evidence to say whether the mathematicians responded to this attack, or simply ignored Zeno, or perhaps did not feel attacked because Zeno showed infinite divisibility to lead into paradoxes for our basic conceptions concerning the physical realm (like motion and a plurality of physical things), not explicitly for the mathematical realm. 3⁵ See, for example, De Caelo 303a2 and also De Lineis Insecabilibus 969b29–70a5. 3⁶ Cf. also Waschkies 1977. 3⁷ Karasmanis 2009, p. 250 claims that Aristotle understood the continuum in a physical way and thus did not approach continuity (which he understands as Zenonian infinite divisibility) and magnitude from the phenomenon of incommensurability. Karasmanis claims that a “Zenonian infinite divisibility never cuts a line into incommensurable segments. However, we do not have any evidence that Aristotle had understood this problem.” I think Karasmanis is right in claiming that Aristotle did not approach continuity and magnitude from the problem of incommensurability. But this is not

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aristotle’s account of continuity 15

5. Aristotle’s Account of Continuity While Aristotle can take over the understanding of continua as being infinitely divisible from a mathematical context, he develops it in such a way as to be able to deal with problems specific to motion and to account for the specific form of unity required for physical but not mathematical continua. For example, in the physical realm there can be a difference between being continuous and being contiguous, as he points out in Physics V, 3, while this difference is not to be found in mathematics. For mathematicians, two lines in the same plane that touch are one,3⁸ as long as they do not simply intersect or form an angle; they are continuous. In a physical context, however, two things that are continuous in themselves and next to each other do not become one thing simply by touching, at least not if they are solid things. Accordingly, in Physics V, 3 Aristotle gives us two criteria for being continuous. Continuous are those things (a) whose limits touch—this is what makes for continuous things as they are in mathematics. Additionally, however, in the physical realm, (b) these limits have to be one, for otherwise we would be dealing only with neighbouring things.3⁹ In a physical context, we can distinguish merely touching limits from those that have become one—only where the limits are one do we have an object that moves as a whole. We should also bear in mind that in spite of some mathematical background, the paradigm continua for Aristotle are certain physical magnitudes, such as physical bodies, time, distance, and motion. Liquid continua, like water, and things like moulding dough may have in part different features—for example, the distinction between contiguity and continuity works with bodies, but does not seem to work with water.⁴⁰ Aristotle’s account of continuity is meant to fit the physical realm⁴1 because he did not understand this problem, but rather because a ‘Zenonian infinite divisibility’ can but need not cut a line into incommensurable segments, so this is a problem not essential for giving a basic account of continuity. As Aristotle himself points out in his Prior Analytics 65b16–20 (a passage Karasmanis explicitly refers to), infinite divisibility and incommensurability are two different problems. What this shows for our immediate investigation is that Aristotle does not necessarily take over all the problems that can be found in the vicinity of a mathematical notion. 3⁸ As are two surfaces or bodies. 3⁹ Accordingly, Aristotle sometimes talks about limits in the plural (so in Physics 231a22) and sometimes about limit in the singular (227a12). We may be worried about his use of the plural, since the limits of lines are points and Aristotle explicitly claims that points cannot touch (and if limits touch, would they not have to have limits themselves?). Aristotle may be talking loosely, when he talks about limits in the plural (as he does when mentioning points touching in 227a29), and actually think about a limit at which things touch; or he may think that with physical things we are never really dealing with points. ⁴⁰ Furthermore, once we physically actualize a part of Aristotle’s paradigmatic continua, we cannot simply get the whole back (we will need glue, etc.); this does, however, not hold for water. ⁴1 The distinction between physical and theoretical divisibility will become highly important in the continuum discussion in the thirteenth and fourteenth centuries with Thomas Aquinas’s ‘Minimatheorie’ in his commentary on Aristotle’s Physics and Duns Scotus’s reaction to that in his Opus Oxoniense. But it is not something Aristotle is concerned with as his understanding of continua is clearly tailored to the physical and not the mathematical realm (even though derived from a reflection on mathematics).

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16 continuity from the presocratics to aristotle insofar as it contains bodies and their motions and what is involved in these motions, like continua over which a locomotion will take place, such as a path on the ground, and the time in which this locomotion is performed. Aristotle’s two criteria for something being continuous, rather than merely contiguous, together form the first of two characterizations of continuity that we find in Aristotle’s Physics: (1) Continuous are those things whose limit, at which they touch, is one (book V, chapter 3). (2) Continuous is that which is divisible into what is always further divisible (book VI, chapter 2). We see that the first account of continuity is two-place, ‘A is continuous with B’, while the second is one-place, ‘A is continuous’—two different understandings of being continuous that Aristotle can take up from the mathematicians. Aristotle does not explain how he thinks the two accounts are connected, but as he moves without further ado between the two characterizations,⁴2 they are clearly meant to be closely related.⁴3 We can read them as two different approaches to the very same notion of a continuum: the first, two-place account tells us what a continuum is like if we begin our consideration with individual things and want to know how a continuum can be ‘made up’ from this starting point—metaphorically speaking, since strictly speaking continua cannot be understood as the result of adding individual single things. The second, one-place account, by contrast, provides an analysis of the continuum that takes it apart conceptually. Aristotle’s two-place account of continuity starts with the assumption of two different things getting so close as to become one—this seems to lead him to a weaker sense of a continuum than that which he employs later, when dealing with uninterrupted stretches of time and space.⁴⁴ The examples he gives us in Physics 227a10–17, in which things are connected by glue or a rivet, seem to be constructed so as to illustrate the way two things that touch can become one, but these examples have not necessarily been selected to demonstrate all the aspects of a continuum. The strong clinging together of the whole is well illustrated by the glue image, but the pieces glued together are not a uniform whole in every respect in the way Parmenides’ continuum seemed to be (the glued whole could still be more here and less there); similarly with two things being connected by ⁴2 So in Physics VI, 1–2. ⁴3 Scholars agree that the two accounts have to be seen as in some sense related to each other, but there is some disagreement on whether Aristotle does indeed provide us with two definitions in his Physics or whether one of them is the actual definition, while the other one is either preparatory or a further development. ⁴⁴ I talk of ‘space’ in this paper, but for our purposes it does not matter whether Aristotle only thinks of spatial distance, or does indeed have an account of space (this is usually doubted in the literature, but I argue for it in Sattler 2020a).

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aristotle’s account of continuity 17 a rivet. How then can Aristotle take being glued or being riveted to be examples of a continuum? If we glue together two pieces of wood to form one plank, then insofar as we regard this glued result as a plank, it is continuous. If we look at the plank, however, in light of the grain of the wood or its colour, the result might not be considered continuous—the end points of the original two pieces of wood have not literally merged. This shows that in order for something to be understood as being continuous, it need not necessarily be homogeneous in all respects; rather one respect in which it can be considered homogeneous has to be found. This account of continuity gives us a weaker notion than the one we saw Parmenides employing. But considering something as continuous in just one respect, as Aristotle seems to do in Physics V, 3, enables us to recognize differences that can serve as starting points for a division. For example, if we consider the length of a stone that has been painted with different colours in arbitrarily chosen sections, we could start cutting the stone at any of the points where the colour changes. Differences in colour here serve as a reason for dividing the stone, and thus for understanding the length of the stone as divisible, even though the length of the stone is continuous in its extension without any differences. Furthermore, for Aristotle, if the stone is divisible in its continuous length at a point where we find a difference in colour, then the length of the stone will be divisible at any other point too. For with respect to the continuous length, no possible partition is in any way more natural than any other. Thus, the stone is as divisible as one likes. Zeno also entertained the thought that once we start with divisions, no division is more natural than another. But this is exactly what he thought would lead us into conceptual problems because it would lead to the assumption of infinitely many parts which he took to be inconceivable, and thus Zeno rejected divisibility. Aristotle, by contrast, will show that if we follow his account of continuity, this feature will not be problematic.⁴⁵ Aristotle’s one-place account claims that continuous is “what is divisible into what is always further divisible” (232b24–25). For Aristotle, this implies that continua can be divided wherever we please, and we will always receive proper parts that are themselves continua and hence further divisible.⁴⁶ The parts that result from such divisions are of the same kind as the whole. For example, if we divide a length, each part will also be a length. And each part will also be a continuum, a ‘divisible’. Hence, the unity of (possible) parts of the whole will be of the same kind as the unity of the whole. This self-similarity is clearly expressed in 227a15–16: “And in whatever way that which holds together is one, so too will the whole be one.”

⁴⁵ The main point here is that Aristotle introduces a new account of infinity and distinguishes between divisibility and being divided; see below. ⁴⁶ See also 231a21–29 and 231b15–18.

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18 continuity from the presocratics to aristotle For Aristotle nothing indivisible can be a part of a continuum, because the limits of indivisibles can never become one.⁴⁷ For a limit must be different from what it limits, and so for the limits of something indivisible to become one would require that the indivisible thing is, after all, divisible to a certain degree, namely, into the limit and that which is limited. And if a continuum cannot be composed of indivisible parts, then neither can it be divided into indivisibles for Aristotle. Accordingly, possible limits can be set within a continuum as one pleases (we are not restricted by any indivisibles). This definition of continuity as what is always further divisible into divisibles is equivalent to infinite divisibility (we will clarify below how to conceive of the infinity employed in this account).⁴⁸ Aristotle shares the thought with Parmenides that a continuum is homogeneous; for Parmenides this means that a continuum cannot contain differences in any respect, for Aristotle it means that at least with respect to the dimension in question all possible sections of the whole have to be treated as equal. However, the consequences drawn by the two thinkers differ significantly: for Parmenides, the lack of any differences implies that a continuum is an indivisible whole without any possible parts. For Aristotle, by contrast, the lack of any internal differences (with respect to the dimension in question) means that a continuum can be divided wherever and however we please, ad infinitum, in the way mathematicians assume magnitudes to be divisible. As we saw above, Parmenides’ assumption seems to rest on the idea that divisions are possible only where there are differences in reality: if something is divisible then it must be divisible by virtue of differences within itself such that one part can be partitioned off from the part from which it differs. But what truly is does not allow for ontological differences in any respect, according to Parmenides. Hence, no logical or epistemic divisions are possible either, including the merely conceptual divisions in a measurement process. This rejection of divisibility is supported by Zeno’s paradoxes that seem to display problems with infinite divisibility. Aristotle, by contrast, allows for epistemic divisions independently of ontological differences: I can mark off a part of a continuum, even if this continuum itself does not display any differences. Such divisions seem to be completely arbitrary; however, they may provide additional information about the thing epistemically divided, allowing us, for example, to quantify it.⁴⁹ Moreover, in order to consider ⁴⁷ With something extended that is strictly indivisible physically, it is not possible to distinguish between its limits and what is limited in a way that the limits could become one (even though in the case of extended indivisibles one could still distinguish theoretically between the edges and the rest). This is in fact an argument already employed by Plato in his Sophist against Parmenides’ simple One; cf. also Furley 1967, pp. 89–90 and Sorabji 1983, ch. 24. ⁴⁸ Cf. also Physics III, 1, 200b18–20. ⁴⁹ Such a quantification takes place, for example, when we measure out a large table with a small ruler by establishing how many times the ruler fits into it. Here the arbitrariness can be seen from the fact that I can measure out and thus ‘divide’ the table with the help of different units, inches or centimetres,

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implications of aristotle’s concept of a continuum 19 something as a continuum in an Aristotelian framework, that thing is considered only in a certain respect, and needs to be homogeneous solely in this very respect. It is not required that the thing is homogeneous in every respect in order to qualify as being continuous. Thus, against Parmenides, Aristotle shows that the divisibility of a thing can be thought of without contradictions. Moreover, Aristotle develops a sophisticated apparatus to show how infinite divisibility can consistently be thought of— a challenge put forth by Zeno’s paradoxes. While contemporary mathematical treatments of Zeno’s paradoxes simply accept Zeno’s understanding of time, space, and motion and provide the mathematical apparatus to deal with what seemed to be paradoxical consequences in Zeno’s time, Aristotle shows that the first step to refute Zeno’s paradoxes is to scrutinize Zeno’s understanding of these concepts by giving a new account of continuity.⁵⁰ So let us look at the implications of Aristotle’s concept of continuity. It requires a new understanding of unity, which implies a different account of parts, infinity, and limits.⁵1

6. Implications of Aristotle’s Concept of a Continuum 6.1 A New Understanding of Parts Aristotelian continua can have parts—they are always further divisible into possible parts. They can, however, not be defined as the sum of such parts in the sense that the parts would be prior to the whole. While a mathematician nowadays may 1 1 1 understand a continuum, like a line of the length 1, as the sum of and and , 2 4 8 etc., this only works for mathematical continua, not for physical ones, at least not for space, time, and motion, the paradigm continua for Aristotle. This is, however, what Zeno’s paradoxes presuppose. In addition, Zeno also assumes a second part–whole model that is in fact not compatible with it. But since Zeno never clarifies the notion of a whole, both part–whole relations are employed at the same time. If we look at the runner paradox or the plurality paradox in fragment 2, we see that, on the one hand, the whole is presupposed as being prior to the parts, since it is only by dividing the whole that we gain the parts. On the other hand, the whole is meant to be constituted by the parts, since the puzzle is how non-extended parts can make up something extended or how infinitely many parts can make up some finite whole. We may think that this switch in the relation for example, without one being more natural than the other. In spite of this ‘arbitrariness’, I will still get new useful information from the measurement process. ⁵⁰ Cf. Sattler 2020b, chs. 3, 7, 8. ⁵1 For a more detailed discussion of these implications, see Sattler 2020b, ch. 7, with which this section in part overlaps.

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20 continuity from the presocratics to aristotle of whole and part is unproblematic. After all, if I take apart my model airplane into its parts, I should be able to put it together to form the whole as which I bought it. However, in the case of the model airplane, this works because the whole was originally built from discrete parts, each of which has its unique function (like the wings, and the motor). But if we look at a whole that is not originally made up out of (functionally) different parts, at continua, like a piece of wood, things look different: if we cut a continuum into parts, we may not in the same easy way get a whole out of it again (we will need some glue or rivets).⁵2 These parts do not constitute a whole in the way they do in the case of the model airplane, and thus the parts cannot necessarily be taken as the building blocks for the whole. With continuous things, parts are posterior to the whole, while the idea of parts making up a whole presupposes parts prior to the whole. Accordingly, Zeno can be shown to work implicitly with two different understandings of a whole: one kind is a whole because it has not been divided, the other because it has been put together from prior parts. Aristotle tries to show that continua have to be thought of as wholes that are prior to any possible parts. Parts of a continuum are not given but we can actualize them (i.e., ‘construct’ them by a division). However, such an actualization is restricted in several ways, according to Aristotle: (1) If such parts are physically actualized, we usually lose the original whole. Parts of a continuum can be actualized either conceptually or physically. We actualize a part conceptually, for example, when we measure out a plank by using a ruler several times to figure out how often it fits into the length of the plank. This conceptual division can be carried out either by simply conceiving a mark on the continuous plank or by physically marking off a part with a pencil without changing the continuum (we can later on remove the mark again). A part is physically actualized when the initial plank is cut into two pieces, in which case the two potential parts will have been transformed into two new continuous wholes. The parts therefore only stay parts as long as they are solely conceptually, and not physically, separate entities. (2) The actualization is also limited by the fact that not all parts that are possible theoretically can be actualized physically;⁵3 for example, only conceptually can a division go on without end. In the physical realm, a physical division can no longer be actualized after we have reached a certain smallness, since at a certain point it ⁵2 Continua like water or syrup will indeed give us back the original whole, but this fact seem to be specific to liquid continua and things like moulding dough, and does not necessarily apply to continua over which a locomotion will take place, like wooden planks, a stone, or hard asphalt. ⁵3 Aristotle does not discuss this second restriction explicitly, but there are traces that he clearly restricts some divisibility to theory and thus distinguishes it from physical divisibility, cf. Miller 1982, pp. 89–90. Furthermore, Physics VII, 5 shows that while a force moving a certain thing may in theory be always further divisible conceptually, this does not correspond to the actual physical divisions possible. For in the empirical realm there is a lower threshold up to which a force could still move a thing, while we do not have any reasons to assume such a lower limit in mere theory.

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implications of aristotle’s concept of a continuum 21 will not be possible to divide the plank any further with our saws and other means. Thus, not all parts that can be actualized in theory can become physically actual. Theoretically it seems such continua are infinitely divisible. However, there is a restriction to the actualization of possible divisions, and thus of having parts, also in theory, as we will see if we now turn to the understanding of infinity that Aristotle’s account of continuity presupposes.

6.2 A New Understanding of Infinity With Zeno’s first paradox of motion we encountered the problem how a finite whole can contain infinitely many parts (‘covering a finite distance a runner has to traverse an infinite number of spatial pieces’). The infinity of parts seems to be in tension with the fact that they belong to a finite whole. Aristotle deals with this problem by pointing out that (a) there is a clear distinction between infinity of addition and infinity of division, and (b) there are not actually infinitely many parts given, but rather an infinite potentiality of actualizing different parts, at least in theory. We have already seen above two restrictions for the actualization of the parts of a continuum. However, with respect to the theoretically infinite potentiality for actualizing parts, there is another restriction we should bear in mind: (3) Also theoretically, a continuous whole cannot be thought of as being divided all at once into all possible parts up to the smallest part. This restriction in theory contains in fact two aspects: on the one hand, not even in theory can all possible parts be brought about simultaneously, since some parts will overlap and so we have to decide for one or the other. Furthermore, infinite divisibility does not imply that something can be infinitely divided, as Zeno assumed. While continua are infinitely divisible, they cannot have been divided infinitely in such a way that we now have infinitely many parts.⁵⁴ This restriction can be seen from a passage in Physics book III, 206a16–25, in which Aristotle discusses the sense in which a magnitude can be said to be infinite. It is not infinite in extension, but by division, and for Aristotle this means infinite in potentiality. This potentiality is clearly set apart from the potentiality involved in the process of a statue coming into being, for there will be a time when the potentiality of some material to be turned into a statue has been fully actualized, but the potentiality of infinite division will never be fully actualized in the same way. Rather, its potentiality is essentially restricted in a twofold way: first, not all of the potential can be actualized at the very same time, and second, not all of the potential can be actualized over some period of time. Some bronze ⁵⁴ According to Aristotle, this would lead to sizeless parts (De gen. et corr. 316b19–27) and to an actual infinity, which according to Aristotle cannot be thought.

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22 continuity from the presocratics to aristotle can fully actualize its potential to become a statue at one time—at the time a sculptor creating a statue has finished her work, all of the potential of the bronze to be a statue is actualized. By contrast, a magnitude that is potentially divisible ad infinitum cannot have all of its potential actualized at the very same time. Rather, the actualization of such a potential requires time, analogous to the way temporal events, like a day or the Olympic Games, can only have part of their whole actualized at one time—only when one part is gone can another part be actualized. Not all of the possibly infinitely many parts of a continuum can be actualized simultaneously.⁵⁵ Furthermore, in contrast to temporal events like a day or the Olympic Games, infinite divisibility can also not be completely actualized over a period of time, as a day is once it is over.⁵⁶ The infinite divisibility of a magnitude is such that even if we actualize the potential, there will never be a finite period of time over which the actualization can be completed; the actualization of this potential always leaves some potentiality. Not every part that can potentially be conceived of can actually be there—neither at a particular time nor over some extended stretch of time—but any part can be actualized. In this way, infinitely many parts are possible, even though infinitely many parts can never be actual for Aristotle, not even in theory. One important consequence of this understanding of the potentiality in question is that the whole continuum cannot be thought of as the sum of its parts, if by ‘sum’ we understand that all parts are given prior to the sum. The threefold restriction of the potentiality of parts of continua discussed allows Aristotle to counter one of the assumptions we saw at work in Zeno’s paradoxes: that something’s being divided is directly inferred from something’s being divisible. By contrast, Aristotle shows that while any of the potential divisions can be actualized, this does not imply that they all can be actualized at the same time. According to Aristotle, the infinity rightly attributed to continua is an infinity of division, not of extension; it is only potential and can never be fully actualized. Thus, continua, being infinite, are incomplete with respect to division, since if we try to divide them, we will never reach an end.⁵⁷ But this kind of incompleteness of continua can be easily combined with the fact that continua are almost always finite things, in contrast to what the runner paradox seems to suggest. Continua may, however, seem to be indeterminate, given that their parts are not determined. But this would only be a problem if continua were understood as being constituted by their parts, which, as we saw, they are not. ⁵⁵ This is true for time qua continuum in the most restrictive sense. With spatial continua several parts can be actualized at the same time, though never all at once, as Aristotle states. ⁵⁶ I follow the understanding of Coope 2012 that infinite division is essentially only potential in that it has no complete occurrence. Such processes are occurring, but cannot occur, in the sense that the whole of it occurs. Infinite division is nevertheless actual in that it is fulfilling its potential; however, it is doing so incompletely. ⁵⁷ Or, as Coope 2012, p. 277, expresses it, “in the case of an infinite process, there is no such thing as the whole of it”.

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implications of aristotle’s concept of a continuum 23

6.3 A New Twofold Concept of a Limit Today we are used to the distinction between closed and open intervals, that is, between intervals which contain their end points and those that do not. And with open intervals, it is easy for us to distinguish between a point which belongs to this interval and an end point of this interval that is not contained by it. A predecessor of the first distinction we find for the first time in Aristotle when he analyses the end points of motion and rest. A predecessor to the second distinction can be found in the distinction of two kinds of limits that characterizes Aristotle’s notion of continuity: (a) ‘Outer limits’ are boundaries marking off a continuum from its surrounding, one whole thing from other things. Within the Aristotelian framework, these limits are normally given—either both the beginning and the end are given, as we find it, for example, with the beginning and the end of a race track; or the beginning is given and the end point is set as the aim or goal, for instance, the finishing condition at which a certain change is aimed; or the beginning is given and the end will eventually become clear, as in a stroll.⁵⁸ (b) ‘Inner limits’—what Aristotle calls division points or marks—are limits that mark off possible parts within a continuum.⁵⁹ These division points are usually not given, but they can be constructed for certain purposes: for example, when measuring a long beam with a small ruler we mark off parts of the beam with the help of such marks. Like outer limits, inner limits possess one dimension less than what they limit for Aristotle. For example, the inner limits of a line are without extension, they are point-like. Inner limits not only allow for divisions within a continuum, but also, as the flip side, guarantee the internal continuity. Thus while outer limits differentiate one whole from another and thus guarantee the unity of a continuum vis-àvis other continua, inner limits guarantee the internal unity of a continuum by showing that we can divide it wherever we please without ever finding a gap.⁶⁰ These two kinds of limits are clearly distinguished: inner limits can be set as we please within the continuum and may always stay potential, while outer limits cannot be set as we like and at least the limit marking the beginning is necessarily

⁵⁸ See, for example, Metaphysics 1022a4–13, meaning 1 and 3; and Physics 227a25. ⁵⁹ See, for example, Physics 262a23 and 29, and 262b7. ⁶⁰ In this sense inner limits may be compared to the nilsquare in Bell which is the ‘glue’ holding the continuum together. Since mathematical operations, like dividing a continuum wherever we please, are considered geometrically, rather than arithmetically by the Greeks of Aristotle’s time, it seems to me that their examples correspond to continua that we could also count as gapless, similar to our number line, even if arithmetically they could not ensure the exclusion of gaps.

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24 continuity from the presocratics to aristotle actualized. If inner limits are actualized, two different kinds of actualizations are possible. We can actualize an inner limit physically or merely conceptually. As we saw above, the second leads to parts that are still only potential parts—what we have actualized so far is only which of all the possible parts we are focusing on, for example, the three parts our 1-metre measuring rod gives us when we are measuring a beam that turns out to be 3 metres long. If, by contrast, we actualize these parts by actualizing the inner limits physically, we turn these inner limits into outer limits, as when the beam is cut into three pieces. The limits Aristotle is especially interested in are the limits of time, space, and motion. Whereas time as such does not have an outer limit for Aristotle, a certain time does have a beginning and an end. The time a motion takes will have outer limits, the original inner limits of infinite time that are treated like outer limits when time is perceived with respect to a certain motion. Space as such, as well as a particular space, is limited for Aristotle, and motion usually also has a beginning and an end.⁶1 And all three kinds of continua have inner limits as we please. The outer limit of most concern for Aristotle is the outer limit between a motion and the ensuing or preceding rest, or, in a change from being not-F to being F, the limit between the state of being not-F and being F. For there is no last point of motion nor a first point of rest according to Aristotle. We see that while Aristotle may think of continua in some respects quite differently from us, this idea that we do not find a first or last point within a continuous interval is something he is the first to prepare. For him, the limit between motion and rest is the only point within a motion or rest that does not separate motion from motion or rest from rest. It can be seen in analogy to the way we think of the limit of a continuous open interval nowadays: it is the point of an interval that in contrast to all other possible points does not have an 𝜀-surrounding. Although this limit is ‘in between’ motion and rest, Aristotle sometimes ascribes it to rest. In general, if we have a change from being not-F to being F, then the one point that we may characterize either as the end of one state or as the beginning of the other state for Aristotle belongs to the second state. It’s clear that it cannot fully belong to both, otherwise a contradiction would hold true at that point—something being F and not-F in the same respect. The reason why Aristotle ascribes it to the latter situation seems to be his assumption that while there is no last instant at which something is not the case, there may be a first instant at which it is the case.⁶2 For example, if something is turning black, there is a continuum of shades and thus no last shade before black, but there may be a first instance at which we say something is black.

⁶1 The exception being the infinite circular motion of the heavens.

⁶2 Cf. Strobach 1998, p. 57.

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references 25

7. Conclusion We have seen that the notion of continuity in ancient time underwent significant developments. While some form of homogeneity was central to all accounts of continuity (in all respects for Parmenides, in at least one respect for Aristotle), the question of divisibility was hotly disputed. For Parmenides something being continuous implies that there are no differences, but without differences there is nothing that would give rise to any division. Zeno supports this conclusion by showing that the assumption of divisibility would lead to infinite divisibility, which in turn seems to lead to fatal paradoxes. By contrast, Aristotle tries to show that continuity allows for infinite divisibility, even though it does not allow for something being infinitely divided. His account of continuity, which takes up mathematical practice, prepares the path for crucial features of our contemporary notions of continuity: it introduces the idea, which we take for granted, that divisibility ad infinitum is unproblematic.⁶3 Furthermore, it prepares a way of thinking about the limits of a continuum that makes it clear that there is no first nor last point, very much in the way we think of continuous intervals today. And indeed with Aristotle we find the first conceptual steps for the idea of an 𝜀-surrounding. While we may think of contemporary notions of continua as being anti-Aristotelian, if we look at the notion of continuity before Aristotle, we see that in fact we share a lot of common ground with his account, a ground that he was the first to prepare.

References Coope, Ursula (2012). “Aristotle on the Infinite”. In: Oxford Handbook of Aristotle. Ed. by Christopher Shields. Oxford: Oxford University Press, pp. 267–286. Coxon, A.H. (2009). Fragments of Parmenides: A Critical Text with Introduction and Translation, the Ancient Testimonia and a Commentary. Revised and expanded. Las Vegas: Parmenides Publishing. Furley, David (1967). Two Studies in Greek Atomists. Princeton: Princeton University Press. Gallop, David (1984). Parmenides of Elea, Fragments: A Text and Translation with an Introduction. Toronto: University of Toronto Press. Grünbaum, Adolf (1968). Modern Science and Zeno’s Paradoxes. London: Allen and Unwin. Hellman, Geoffrey and Stewart Shapiro (2018). Varieties of Continua. New York: Oxford University Press.

⁶3 Even if our account of why it is unproblematic is different.

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26 continuity from the presocratics to aristotle Karasmanis, Vassilis (2009). “Continuity and Incommensurability in Ancient Greek Philosophy and Mathematics”. In: Philosophical Inquiry 31, pp. 249–260. Lee, H.D.P., ed. (1967). Zeno of Elea: A Text, with Translation and Notes. Originally published 1936. Amsterdam: Hakert. Linnebo, Øystein, Geoffrey Hellman, and Stewart Shapiro (2016). “Aristotelian Continua”. In: Philosophia Mathematica. III, 24, pp. 214–246. Miller, Fred (1982). “Aristotle Against the Atomists”. In: Infinity and Continuity in Ancient and Medieval Thought. Ed. by Norman Kretzmann. Ithaca and London: Cornell University Press, pp. 87–111. Mourelatos, Alexander (2008). The Route of Parmenides. Revised and expanded. Las Vegas: Parmenides Publishing. Owen, G.L.E. (1960). “Eleatic Questions”. In: Classical Quarterly 10, pp. 84–102. Owen, G.L.E. (1966). “Plato and Parmenides on the Timeless Present”. In: The Monist 50, pp. 317–340. Palmer, John (2009). Parmenides and Presocratic Philosophy. Oxford: Oxford University Press. Sattler, Barbara (2011). “Parmenides’ System: The Logical Origins of his Monism”. In: Proceedings of the Boston Area Colloquium on Ancient Philosophy 2009/2010. Leiden and Boston: Brill, pp. 25–70. Sattler, Barbara (2019). “The Notion of Continuity in Parmenides”. In: Philosophical Inquiry 43, pp. 40–53. Sattler, Barbara (2020a). “Space in Ancient Times: From the Presocratics to Aristotle”. In: Space: A History. Oxford Philosophical Concepts. Ed. by Andrew Janiak. Oxford: Oxford University Press, pp. 11–51. Sattler, Barbara (2020b). The Concept of Motion in Ancient Greek Thought: Foundations in Logic, Method, and Mathematics. Cambridge: Cambridge University Press. Schofield, Malcolm (1970). “Did Parmenides Discover Eternity?” In: Archiv für Begriffsgeschichte der Philosophie 52, pp. 113–135. Sorabji, Richard (1983). Time, Creation and the Continuum: Theories in Antiquity and the Early Middle Ages. London: Duckworth. Strobach, Niko (1998). The Moment of Change: A Systematic History in the Philosophy of Space and Time. Dordrecht: Kluwer. Tarán, Leonardo (1965). Parmenides: A Text with Translation, Commentary, and Critical Essays. Princeton: Princeton University Press. Waschkies, Hans-Joachim (1977). Von Eudoxus zu Aristoteles: das Fortwirken der Eudoxischen Proportionstheorie in der Aristotelischen Lehre vom Kontinuum. Amsterdam: Grüner.

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2 Contiguity, Continuity, and Continuous Change Alexander of Aphrodisias on Aristotle Orna Harari

Aristotle’s account of continuity is mainly studied from the viewpoint of the contemporary notion of continuity.* Studies of this account either point to the difference between Aristotle’s and modern notions of continuity, analyse it with contemporary mathematical tools, or criticize it in light of the modern notion.1 This approach highlights the geometrical features of continuity, particularly infinite divisibility, and downplays its physical features, such as unity. This study, by contrast, draws attention to the latter features, by examining a Peripatetic interpretation of Aristotle’s account of continuity. It focuses on Alexander of Aphrodisias’ interpretation of Aristotle’s definitions of contact and contiguity, and shows that it results in a significant modification of the notion of continuity that Aristotle defines in Physics V.3. Specifically, I argue that (1) Alexander’s interpretation of the definitions of contact and contiguity is based on his assumption that continuity is equivalent to unity; (2) this assumption led him to distinguish three senses of continuity: the more general, the strict, and the strict and primary; (3) the latter sense that holds for continuous wholes whose motion is one is the sense that Alexander contrasts with contiguity in his interpretation of Physics V.3; (4) this sense is incompatible with Aristotle’s account of continuous motion but useful in its explaining the beginning of change and offering a criterion that determines the actual division of a continuum; finally (5) Alexander avoids the atomistic implications of this strong sense of continuity by grounding the actual divisions of a continuum in the efficacy of the cause of change.

* I am grateful to István Bodnár and Vincenzo de Risi for their valuable comments on an earlier version of this article. 1 For the first approach see, e.g., Wieland 1962; for the second, White 1992; and for the third, Bostock 1991. Orna Harari, Contiguity, Continuity, and Continuous Change: Alexander of Aphrodisias on Aristotle In: The History of Continua: Philosophical and Mathematical Perspectives. Edited by: Stewart Shapiro and Geoffrey Hellman, Oxford University Press (2021). © Stewart Shapiro and Geoffrey Hellman. DOI: 10.1093/OSO/9780198809647.003.0003

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28 contiguity, continuity, and continuous change

1. Alexander on Aristotle’s Definition of the Contiguous In Physics V.3 Aristotle defines the contiguous (τὸ ἐχόμενον) and the continuous (τὸ συνεχές) in terms of the relation between the extremities of their constituents. By these definitions, the constituents of both contiguous and continuous things are successive and in contact, while those of continuous things do not merely touch but are one and the same (ταὐτό καὶ ἕν) (227a6–12).2 Through these definitions Aristotle conveys the ideas that contiguous things are in contact and not merely adjacent, and that continuous things share a limit, but his exact formulation of this distinction is unclear. He says that the extremities of things in contact are ‘together’ (ἅμα) (226b23) and defines the expression ‘together in respect of place’ (ἅμα κατὰ τόπον) as holding for things that are in one primary place (226b21–22). A natural way of understanding this definition is an appeal to Physics IV.2, where the term ‘primary place’ is defined as the place that contains nothing but one thing (209b1). In another interpretation, in Physics V.3 Aristotle uses the term ‘primary place’ in a loose sense that allows two distinct things to be in one place, for example in the same room or in the same house.3 However, neither interpretation captures Aristotle’s notion of the contiguous. The former cannot account for the topological distinction between the extremities of contiguous and continuous things because a primary place contains nothing but one thing; the latter cannot account for the topological distinction between being adjacent and being in contact because the loose sense of primary place does not entail that there is nothing between the distinct extremities of contiguous things. In a fragment from his commentary on Physics V.3 Alexander offers an original solution to this difficulty.⁴ He argues that the definition of ‘together in respect of place’ is irrelevant to the definition of contact because here the term ‘together’ does not mean ‘together in place’ but ‘coincide’ (τὸ ἐφαρμόζειν).⁵ To support this interpretation he argues that in the definition of contact the term ‘extremities’ (ἄκρα) refers to the limits and not to the parts of contiguous things, and that since limits are in place only accidentally they are not together in place but together with one another, i.e. they coincide. Further, he argues that the distinction between these senses of ‘together’ has a basis in Aristotle’s phrasing. Alexander points out that in the definition of contact Aristotle uses the term ‘together’

2 In De caelo I.1 Aristotle defines continuity in terms of the monadic property, being divisible into divisible parts (268a6–7). In Physics VI.1 he argues that this property is a consequence of the relational definition of continuity as holding for things whose extremities are one and the same (231b13–18). For a more detailed discussion of these accounts see Barbara Sattler’s contribution to this volume. 3 For the latter interpretation see Simplicius In Phys. 869.8–29 Diels; cf. Ross 1936. ⁴ Unless otherwise noted, all fragments from and testimonies on Alexander’s lost commentary on the Physics are from Simplicius’ commentary on this work. ⁵ Cf. In Met. 232.8–9 Hayduck, where Alexander says that the limits of things in contact are one because they coincide.

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aristotle’s definition of the contiguous 29 without the qualification ‘in respect of place’ and concludes that the former term signifies a different sense of ‘together’, which he identifies with coincide (870.17–28 Diels).⁶ In his examination of the consequences of this interpretation, David Furley cites a fragment from Alexander’s commentary on Physics IV.4 concerning Aristotle’s claim that the extremities of discrete and touching bodies are equal because they are ‘in the same’ (ἐν τῷ αὐτῷ) (211a31–34). Here, in keeping with his interpretation of the definition of contact, Alexander does not understand the expression ‘in the same’ as elliptical for ‘in the same place’ but says that it stands for ‘together’ in the sense of coincide (569.39 Diels): For things that have nothing between them are also said to be in the same . . . For things that coincide with one another and neither make a volume nor have something between are in the same. For in the case of things that touch and overlap, the limits of both become one, while in the case of continuous things even the one is lost. For continuous things are those that have no intermediate limit in actuality.⁷ (570.1–7 Diels)

Here Alexander characterizes the contiguous in the terms that Aristotle uses in his definition of the continuous: that the touching limits of contiguous things become one (227a11–12), and consequently offers another account of the continuous: that it holds for things that have no intermediate limit in actuality. In his discussion of this fragment, Furley argues that the difference between Alexander’s and Aristotle’s accounts notwithstanding, Alexander’s notion of continuity is not significantly different from Aristotle’s. In so doing, he considers the possibility that Alexander’s account of the continuous implies that the mere mention of an intermediate point on a line turns its continuity into contact. But in the end he concludes that Alexander is faithful to Aristotle in holding, as Aristotle does in Physics VIII.8, that continuous things are potentially divisible (Furley 1982, p. 24 and White 1992, pp. 24, 27.). This conclusion is reasonable in the light of the evidence that Furley examined; but other evidence shows that Alexander’s interpretation of the distinction between the contiguous and the continuous is more complex than Furley assumes. Contrary to the impression made by the above fragment, in his commentary on Aristotle’s Metaphysics Alexander does not abandon Aristotle’s definition of the continuous in terms of a common limit. In his commentary on Metaphysics V.4 he rephrases Aristotle’s claim that in organic fusion (σύμφυσις) ‘there is something identical in both parts’ (1014b23–24), saying, ‘things that are organically fused are those which are continuous with the thing with which

⁶ Cf. Philoponus, In Phys. 528.13–14 Vitelli.

⁷ All translations are mine.

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30 contiguity, continuity, and continuous change they are organically fused, by having the same common limit present in both’ (358.21–23 Hayduck). Similarly, in his commentary on Metaphysics V.6 he adds to Aristotle’s characterizations of the continuous the definition found in Physics V.3: ‘things that have a common limit are continuous in the strict sense’ (362.18– 19 Hayduck). Moreover, a fragment from Alexander’s commentary on Physics VI.1 suggests that he does not strengthen Aristotle’s definition of the continuous, as the above fragment implies, but allows the term ‘continuous’ to hold in a more general sense for the contiguous. Here he says that when Aristotle rejects the possibility that continuous things are divisible into indivisible constituents on the ground that ‘the extremity of continuous things is one and in contact’ (231b17–18),⁸ he uses the term ‘continuous’ in a more general sense because ‘the extremities are one insofar as the extremities of continuous things are in contact’ (931.24–26 Diels). This evidence seems to support Furley’s view, in implying that Alexander’s interpretation of the definition of contact did not lead him to change Aristotle’s definition of the continuous. However, closer examination shows that the above fragment and evidence are manifestations of Alexander’s fundamentally different approach to Aristotle’s account of continuity, which led him to understand Aristotle’s definitions of the contiguous and the continuous in Physics V.3 from the viewpoint of Aristotle’s discussions of unity. This examination reconciles the apparent incompatibility between the above fragment and the other evidence on Alexander’s account of the continuous, and prepares the ground for showing that the notion of continuity in Alexander’s interpretation of Physics V.3 is not the notion that Aristotle defines there.

2. Continuity and Unity In Physics V.3 Aristotle draws the following conclusion from his definition of the continuous: This being defined, it is clear that the continuous is found in things from which a certain unity naturally (πέφυκε) comes to be in virtue of their mutual contact (σύναψις). And in whatever way that which holds them together is one, so the whole too becomes one, for example either by a rivet, by glue, by contact, or by natural adhesion (προσφύσις). (227a12–17)

⁸ Aristotle’s argument is based on the assumption that indivisible entities that have no parts cannot be together or one (i.e., contiguous or continuous respectively) because their extremities are not distinct from the other part of the entity (231a26–29).

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continuity and unity 31 This passage admits two interpretations. In one, Aristotle claims that things that are naturally unified are continuous, and then lists several ways by which things form a unity, not continuity. In the other interpretation, the first sentence of the above passage implies that continuity is equivalent to unity and in the second sentence Aristotle lists all the ways whereby things form unity, but also continuity. Obviously, the first interpretation is preferable, since the second is inconsistent with Aristotle’s distinction between the contiguous and the continuous. Specifically, it implies that unities formed by a rivet, by glue, and by contact are continuous and not contiguous, as Aristotle’s definition entails. Recently recovered evidence on Alexander’s interpretation of Physics V.3 found in Byzantine scholia to Aristotle’s Physics clearly indicates that he favours the second interpretation; he also reconciles the inconsistency between this interpretation and Aristotle’s definition of the contiguous through the view found in the fragment from his commentary on Physics VI.1 (Rashed 2011). That is, he regards unities formed by contact as continuous in a derivative sense: Continuous things are different and of many kinds (πολυειδῆ). Continuity in the strict sense [is found] in lines, surfaces, and bodies [understood] as natural but things become continuous also by a rivet and by glue and in such things [it is found] in a more derivative sense. (Scholium 261 in Rashed 2011)

This short scholium provides no information on Alexander’s justification of his interpretation of Physics V.3. But his longer discussion of a similar problem that arises in the context of Aristotle’s account of the different senses of the term ‘one’ in Metaphysics V.6 sheds light on this interpretation. The ambiguity between unity and continuity found in Physics V.3 is more emphasized in Metaphysics V.6. Here Aristotle says that in one sense, things are called ‘one’ in virtue of their own nature because they are continuous, like a bundle and glued pieces of wood (1015b36–1016a1). But shortly afterwards he says that things that are continuous by virtue of themselves are not those that become one by contact (1016a7). Like the account found in Physics V.3, this account can be interpreted in two ways: in one, the examples of a bundle and glued pieces of wood illustrate cases of unity but not of continuity, and in the other they are examples of continuous things but not of things that are continuous by virtue of themselves. Yet unlike in Physics V.3, here it is more difficult to decide between these interpretations. On the one hand, the first interpretation implies that Aristotle clarifies this sense of ‘one’ through examples that do not accord with its characterization in terms of continuity, and it appears to be at odds with the example immediately following of a line, about which Aristotle explicitly says that it is continuous (1016a2). On the other hand, the second interpretation is clearly incompatible with Aristotle’s explicit claim that pieces of wood that touch each other are not continuous at all (1016a9). In keeping with his interpretation

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32 contiguity, continuity, and continuous change of Physics V.3, in his commentary on this discussion Alexander prefers the second interpretation: He calls things that are held together (συνεχόμενα) by something in any way whatsoever ‘continuous’ in a more general sense, like a bundle that is held together by a band. Similarly, he calls continuous also things that are unified by glue, like books. But things [that are one] by having a common limit are strictly speaking continuous.⁹ In this sense, a line is one; for it is continuous even if it is not straight but if a certain part of it is bent. And bodily parts, for example leg and arm, are continuous in this way . . . Generally, he says that things that are one not by contact, like those that are [held together] by a band or glue, but because they are joined together by a common limit are continuous in themselves and strictly speaking; for pieces of wood that touch each other are called neither wood, nor one body strictly speaking, nor do they form a certain unity at all. (363.16–364.2 Hayduck)

Here Alexander addresses the difficulty in understanding Aristotle’s exact stance on the relation between unity and continuity through the view found in the fragment from his commentary on Physics VI.1 that things in contact are continuous in a more general sense. He thereby distinguishes two senses of continuity: the more general sense, which holds for bundles and glued materials, and the strict sense, which holds for things that share a limit, for instance straight or bent lines and bodily organs. Through this distinction Alexander avoids the difficulties of the first interpretation. He reconciles the examples of a bundle and glued pieces of wood with Aristotle’s account of unity in terms of continuity by viewing them as cases of continuity in the more general sense and understands Aristotle’s explicit use of the term ‘continuous’ for the example of a line as a case of the strict sense of continuity. Alexander encounters the difficulty that arises from the second interpretation in a less subtle way. In his paraphrase of Aristotle’s words he simply replaces Aristotle’s claim that pieces of wood that touch each other are not continuous at all with the claim that they do not form a unity at all, thereby allowing them to be continuous in the more general sense. This analysis brings to light Alexander’s exegetical approach to Physics V.3, by showing that it rests on the assumption that continuity is equivalent to unity. This assumption leads Alexander to understand Aristotle’s definitions of the contiguous and the continuous in light of Aristotle’s discussions of unity in Physics V.3 and Metaphysics V.6, which by a certain interpretation can imply that things in contact are continuous. This interpretation blurs the distinction between the contiguous and the continuous, and accordingly requires the distinction between continuity in the strict sense and continuity in the more general sense to account

⁹ I read κοινόν for κοινότερον with De Sepúlveda’s Latin translation.

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the definition of the continuous 33 for Aristotle’s distinction between the contiguous and the continuous. At first glance, this exegetical approach has no significant consequences for Alexander’s notion of continuity; after all, the strict sense of continuity is not different from the definition of the continuous found in Physics V.3. However, as scholium 261 suggests and the commentary on Metaphysics V.6 confirms, continuity, in Alexander’s view, has many senses. Therefore it is not straightforwardly evident that in his commentary on Physics V.3 Alexander understands Aristotle’s distinction between the contiguous and the continuous in terms of his distinction between the more general and the strict senses of continuity. To settle this matter, I return to Alexander’s commentary on Metaphysics V.6 and show that he understands Physics V.3 in the light of a third sense of continuity: the strict and primary.

3. Alexander’s Interpretation of the Definition of the Continuous in Physics V.3 In Metaphysics V.6 Aristotle says that things that in themselves have one motion and cannot move differently are continuous (1016a5–6). In his commentary on this chapter Alexander takes this claim to be a certain definition of the continuous and regards things whose motion is one as continuous in the strict and primary sense (363.25–29 Hayduck). According to Alexander’s discussion, this sense differs from the strict sense of continuity in its scope. As we saw above, the latter holds for bent lines and bodily organs, whereas the strict and primary sense holds for straight lines that have no bend (363.32; 364.3–4 Hayduck) and for parts of bodily organs, e.g., the shin and the thigh (364.4–5 Hayduck). The exact significance of this sense of continuity becomes clear from Alexander’s interpretation of Aristotle’s characterization of one motion as indivisible in respect of time (1016a6), as implying that it is impossible that one part of such continuous things moves and another rests (363.30–31 Hayduck). In clarifying this implication, Alexander first says that the shin and the thigh cannot have one part in motion and another at rest because then they would be bent (364.6–8 Hayduck). But later he offers a stronger explanation that clarifies why the parts of such continuous things necessarily move together: ‘A part of a straight line that has magnitude cannot at all either move in virtue of itself or be at rest in virtue of itself without the whole’ (364.14–16 Hayduck). This explanation is closely related to Aristotle’s account of place and the distinction between the contiguous and the continuous that it requires. Specifically, Aristotle’s definition of place as the inner surface of the surrounding body requires him to distinguish a surrounding body that is continuous with the body that it surrounds from a contiguous surrounding body whose inner surface is the place of another body. In Physics IV.4 Aristotle differentiates these cases through two distinctions: one, between being in something as a part in a whole and being in

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34 contiguity, continuity, and continuous change place (221a29–31); the other, between moving with something and moving in something (221a34–36). In his commentary on Physics IV.4 Alexander grounds the latter distinction in the former: Continuous parts are not in place if indeed that which is in place moves in it, whereas that which is in a whole (and a continuous part is suchlike) does not move in the whole but with it.1⁰ (570.25–28 Diels)

Alexander’s explanation in the commentary on Metaphysics V.6 of why the parts of things that are continuous in the strict and primary sense necessarily move together is based on this account. It regards these continuous things as parts of a whole and denies parts motion and rest on the ground that their motion and rest depend on the motion of the whole, i.e., they do not move or rest by virtue of themselves but with the whole that moves or rests by virtue of itself. The following examination of the fragments from Alexander’s commentary on Physics V.3 shows that this sense of continuity—the strict and primary that holds for continuous parts of a whole—is the sense of continuity that he contrasts with Aristotle’s definitions of contact and contiguity. One of the difficulties in Furley’s account is that Alexander’s interpretation of the definition of contact appears strained. In this account, Alexander holds that the definition of ‘together in place’ is irrelevant to the definition of contact immediately following in terms of ‘together’, but also that the former definition plays no role in Physics V.3. An examination of other fragments from Alexander’s commentary on Physics V.3 does better justice to his interpretation and shows that, in his view, the definition of ‘together in place’ serves in Aristotle’s account of the continuous. As we saw above, in propounding his interpretation of the definition of contact, Alexander argues that here the term ‘extremity’ means ‘limit’ and not ‘part’. In his description of Alexander’s argument for rejecting the latter meaning, Furley presents only one of the considerations that Alexander offers in support of his interpretation: that if the extremities of things in contact are parts and they are together in one primary place, these parts do not touch each other but coincide with one another (870.16 Diels; Furley 1982, 26). However, this is not Alexander’s main reason for rejecting the possibility that in the definition of contact Aristotle uses the term ‘extremity’ in the sense of ‘part’. Alexander’s full argument is the following: For if he says that [the extremities are together] as parts, these [sc. things in contact] are not together. For parts that are together in one primary place are precisely continuous. But the parts of discrete and touching bodies are not in one primary place (for they are neither continuous with each other nor does each 1⁰ Cf. scholium 46 in Rashed 2011.

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the definition of the continuous 35 of their proper parts coincide with one another), but the parts themselves touch each other. (870.12–17 Diels)

From this passage it is clear that Alexander’s argument in support of his interpretation of the definition of contact is primarily based on the assumption that things that are together in one primary place are continuous. Here he rejects the possibility that in the definition of contact ‘extremity’ means ‘part’ on the grounds that parts that are together in one primary place are continuous and that things in contact are not. So this passage confirms Simplicius’ claim that Alexander was forced to introduce another sense of ‘together’ in order to preserve his interpretation that things that are together are continuous (870.28–871.2 Diels). Another fragment from Alexander’s commentary on Physics V.3 where he clarifies the expression ‘one primary place’ illuminates the rationale behind this interpretation: As Alexander says, one place is that which is not divided but continuous because also things that are said to be together in place are continuous with one another, as the parts of the continuum are. For things are said to be together not because two bodies or more are in numerically one and the same place, so that they pass through each other (it has been proved in the previous book that this is impossible) but because those things that are neither divided nor surrounded by a proper surface have no proper place; for the limit of the surrounding body, which surrounds things that are unified in this way, is one and continuous. (868.26–869.4 Diels)

According to this passage, the problem that Alexander’s interpretation of the definition of contact addresses is not how parts or limits can be in place, as Furley argues, but how two or more things can be together in the same place (Furley 1982, pp. 22–3). Alexander’s solution is based on Physics IV.4’s distinction between being in place and being in a whole, which implies that continuous parts are not in place, or in Alexander’s words ‘have no proper place’ (Rashed 2011, p. 318). This solution highlights the significance of Alexander’s assumption that things that are together in place are continuous. It shows that the sense of continuity that Alexander contrasts with contiguity in his interpretation of Physics V.3 is the strict and primary sense that holds for continuous parts of wholes that are neither in place nor move by virtue of themselves.11 It also shows that being a part of Alexander’s interpretation of Aristotle’s distinction between to be in 11 Since we have no evidence as to Alexander’s interpretation of Aristotle’s definition of the continuous in 227a10–12, we cannot know whether he understood this definition as holding for the strict and primary or for the strict sense of continuity. The examples in scholium 261 give no definite answer to this question. Still, the available evidence clearly indicates that the sense of continuity that Alexander contrasts with the definitions of contact and contiguity is the strict and primary sense.

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36 contiguity, continuity, and continuous change place and to be in a whole, the continuous things that have no limit in actuality mentioned in the above fragment from Alexander’s commentary on Physics VI.4 are continuous in the strict and primary sense. In view of Furley’s interpretation of this fragment, this conclusion gives rise to two questions: one, whether this sense of continuity entails the unwelcome consequence that the mere mention of an intermediate point on a line results in a loss of continuity; two, whether in this fragment Alexander is faithful to Aristotle’s account of potential divisibility in Physics VIII.8. I address these questions in the following section.

4. Potential Divisibility The conclusion that things that are continuous in the strict and primary sense are wholes provides a context for understanding Alexander’s claim in the fragment from his commentary on Physics IV.4 that between continuous things there is no limit in actuality. Specifically, it suggests that Alexander’s appeal to the distinction between potentiality and actuality is based on his interpretation of Aristotle’s discussion of the term ‘whole’ in Metaphysics V.26. Here Aristotle says that bounded and continuous wholes are unities when a certain unity is formed from several parts, especially if they are present in potentiality (1023b32–34), but does not explain what ‘being present in potentiality’ means. In his commentary on this chapter Alexander understands this characterization as a clarification of the term ‘continuous’: ‘things whose parts are present in them in potentiality are continuous in the strict sense’ (425.27–30 Hayduck). Here Alexander does not use the expression ‘strict and primary’ but his description of these continuous things clearly accords with the examples of the strict and primary sense of continuity that he presents in his commentary on Metaphysics V.6. In bodily parts, such as the shin and the thigh, and in a straight line there is nothing in actuality, like the joint and the apex of an angle, which demarcates one part from another. Further, Alexander’s claim in the commentary on Metaphysics V.26 that things are continuous even if their parts are present in them in actuality (425.30–31 Hayduck) corresponds with the strict sense of continuity found in the commentary on Metaphysics V.6; in bodily organs such as a leg, and in bent lines, there is an actual point that distinguishes one part from another, i.e., the shin and the thigh or the parts of a line that extend in different directions. Understood in these contexts, the strict and primary sense of continuity is indeed stronger than the notion of continuity that Aristotle defines in Physics V.3, but it does not imply that the mere mention of an intermediate point on a continuous line turns its continuity into contact. Alexander’s understanding of continuity in terms of unity led him to distinguish different degrees of continuity. The strict and primary sense of continuity is the strongest sense, but continuous things that have a limit in actuality are not contiguous or, in Alexander’s terms,

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potential divisibility 37 continuous in the more general sense, but are continuous in the strict sense. Consequently, a joint or an angle, and still less the mere mention of an intermediate point, do not turn continuity into contact. Nevertheless, Alexander’s account of the strict and primary sense of continuity implies that he weakens Aristotle’s condition for actualizing an intermediate point in continuous entities in this sense. According to Alexander’s commentary on Metaphysics V.6, the defining characteristic of continuous entities in the strict and primary sense is that all their parts move together and that it is impossible that one of their parts moves and another rests. This characteristic implies that a continuous whole in the strict and primary sense that undergoes continuous change, i.e., part by part, is not continuous in this sense. Alexander does not explicitly draw this consequence but a brief comment found in his commentary on Metaphysics II.2 goes that way. There he explains Aristotle’s claim that the case of the infinite divisibility of a line is different from the case of attaining knowledge through infinitely many definitions because a person who traverses an infinitely divisible line does not count the divisions (994a24–25): For this reason a person who traverses an infinitely divisible continuous line does not traverse it in this way, i.e., by counting the divisions and by taking them in actuality to the extent that it is possible to divide them . . . but he traverses the parts at one go (κατ’ ἀθρόα) because the infinity is not in the line in actuality but in potentiality. (164.5–10 Hayduck)

Although, according to Physics VIII.8, counting the parts of a continuous line is a way of dividing it (263a25–26), the above account of how motion along a straight line divides it is significantly different from Aristotle’s. According to Aristotle, an intermediate point on a straight line is actualized when an object that moves along it divides the line by stopping at this point and restarting its motion (262a21–25); Alexander’s claim that a person who traverses a potentially divisible line does so at one go implies that his condition for actualizing an intermediate point on a line is weaker than Aristotle’s. Specifically, it implies that to actualize a point on a line, an object need not stop and restart its motion but its gradual passage along these points, rather than traversing the parts at one go, is enough. This passage has no exact parallel in Alexander’s extant writings or in evidence on his lost works, but a version of this view appears in several fragments from his commentary on Physics VI, where he argues that a part but not the whole can undergo qualitative change at one go. Before I discuss these fragments, I examine the following fragment from Alexander’s commentary on Physics VIII.8, which casts further doubt on Furley’s claim that Alexander is faithful to Aristotle’s account of potential divisibility: Through the words ‘a person who moves continuously [traverses infinity accidentally and not strictly speaking], Aristotle explains how it is possible to traverse

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38 contiguity, continuity, and continuous change potentially infinite [divisions]. Since a line along which the thing that has moved has moved has the accident of being infinitely divisible in potentiality and having infinite halves in potentiality, a thing that traversed the line has accidentally traversed the infinite halves in potentiality; that is to say it has traversed the line to which [the attribute] having infinite halves in potentiality belongs accidentally. For the essence of the line is not this, i.e., having infinite halves, since then a thing that has moved along the line would no longer traverse infinite halves accidentally but essentially, if indeed it has essentially moved along a line whose being is having infinite halves. But since this is not the being of a line but it is its accident, whereas the being of a line is length without breadth, a thing that traversed the line has essentially traversed a length without breadth and accidentally also all those accidents that hold for a line. For example, white if the line happens to be white and similarly each of the other accidents, one of which is having infinite halves in potentiality. (1291.34–1292.11 Diels)

As Simplicius says, this interpretation is not based on Aristotle’s exact words (1292.21–24 Diels).12 What Aristotle says in the lemma on which Alexander comments is that having an infinite number of halves is an accident of a line (263b7–8) and not that having infinite halves in potentiality is an accident of a line. As Simplicius further notes (1292.21–23 Diels), Alexander’s claim that infinite potential divisibility is an accident of a line is in conflict with Aristotle’s account of continuous magnitudes because their infinite divisibility or their divisibility into divisible parts is, according to Physics VI.1, a consequence of the above definition of the continuous (231b13–18) and, according to De caelo I.1 (268a6– 7), it is the definition of the continuous. These discrepancies aside, it is difficult to understand the motivation behind Alexander’s interpretation. The question how is it possible to traverse potentially infinite divisions is odd. The problem that Aristotle addresses in this context—Zeno’s dichotomy paradox—is how is it possible to traverse infinite divisions, and the answer is that it is possible to do so if the divisions are potential (263b5–6). Alexander’s question suggests that, in his view, potential divisibility does not suffice to explain how it is possible to traverse a line, therefore he assumes that the line’s divisions should be not only potential, as Aristotle says, but also accidental. Further, Alexander’s answer to this question is unclear. It gives rise to the questions why does Alexander use the perfect tense ‘has traversed’ (διελήλυθε) and ‘has moved’ (κεκίνηται) and what does it mean essentially to traverse a length without breadth and accidentally infinite halves in potentiality?13 A possible way of understanding this answer is to see it as an

12 According to Simplicius, Alexander presented Aristotle’s text with the addition of the word δυνάμει (in potentiality). Simplicius remarks that this addition is not found in the manuscripts that came down to him (1292.14–15 Diels). 13 Alexander refers to the definition of a line found in Euclid’s Elements I, def. 2.

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change at one go 39 attempt to address the question that Simplicius raises in this context: i.e., how can infinite divisibility be potential if this potentiality, unlike the potentiality of bronze to become a statue, is never actualized (1293.10–1294.2 Diels)?1⁴ By this understanding, Alexander’s use of the perfect tense is meant to indicate that an object that traverses a line can complete its motion; and his claim that it has accidentally traversed infinite potential divisions suggests that the actual divisions of the line are not determined by its potential infinite divisibility but by its essence. Admittedly, this is not a satisfactory interpretation of this difficult fragment but Alexander propounds a similar view in the fragments from his commentary on Physics VI, where he offers his account of change at one go. Next, I examine this account and argue that (1) Alexander offers a criterion for determining the actual divisions of a body that undergoes change by appealing not to the body’s essence, as the above passage suggests, but to the efficacy of the cause of change; (2) he does so in reply to the question how can motion begin; and (3) the strict and primary sense of continuity plays a role in this account.

5. Alexander’s Account of Change at One Go in Its Context In Physics VI.1 Aristotle establishes three theses: (1) continuous entities cannot be composed of indivisible parts (231a21–b6); (2) continuous entities are divisible into divisible parts (231b10–18); and (3) magnitude, motion, and time have the same underlying structure, i.e., either all three are composed of and divided into indivisible parts or all three are composed of and divided into divisible parts (231b18–20). Aristotle’s argument in support of the third thesis has in view his atomist opponents, therefore it focuses on the first horn of this thesis. It shows first, that the assumption of indivisible magnitudes implies indivisible motions, i.e., jumps (κινήματα), by which an object has moved along a magnitude without previously having been in the process of moving along it (231b21–232a4); and second, that jumps along indivisible magnitude imply indivisible instants, i.e., ‘nows’ (τὰ νῦν), because a body that moves by jumps cannot move along a part of this magnitude in a shorter time (232a18–22). This argument brings to light the counter intuitive consequences of the assumption of indivisible magnitudes but it does not establish a necessary connection between the underlying structure of magnitudes, motion, and time.1⁵ The two horns of the third thesis are not 1⁴ The distinction between the potentiality of bronze to become a statue and the never realized potentiality of infinite division is based on Aristotle’s Physics III.6, 206a18–25. Rashed 2011, scholium 748 indicates that Alexander appealed to this distinction in his commentary on Physics VIII.8. 1⁵ Aristotle’s argument does not lead to an outright rejection of atomism. As Simplicius reports, the Epicureans accepted the assumption that the underlying structures of magnitudes, motion, and time are isomorphic but held that like magnitudes, motion and time are composed of partless constituents and that over these constituents a thing does not move but has moved (934.23–30 Diels = fr. 277 Usener). On this problem see Sorabji 1982, pp. 55–59. For an analysis of Aristotle’s argument see Miller 1982, pp. 102–11.

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40 contiguity, continuity, and continuous change equivalent: whereas the assumption of indivisible magnitudes implies that an object cannot traverse a part of a magnitude before it traverses the whole magnitude, the assumption of divisible magnitudes does not exclude the possibility that an object traverses the whole divisible magnitude without previously passing through its parts.1⁶ Aristotle’s view on this matter is unclear. In Physics IV.11 he takes for granted that the continuity of motion derives from the continuity of magnitude and that the continuity of time derives from the continuity of motion (219a10– 13, 219b15–16; cf. Physics VI.2, 233a10–12), but in other contexts he allows motion at one go. In Physics VIII.3 he says that the divisibility of the body that undergoes alteration, growth, and diminution does not imply that these changes are necessarily divisible because they can be completed at one go (ἀθρόως), as in the case of freezing (253b21–26). Similarly, in De sensu 6 he distinguishes locomotion from alteration, arguing that in the latter a body can undergo change at one go, without a part of it being altered before another (446b28–447a3). The question whether the divisibility of magnitude necessarily implies the divisibility of motion was already raised by Aristotle’s immediate successor, Theophrastus. In his Paraphrase of Aristotle’s Physics Themistius reports that Theophrastus questioned Aristotle’s claim in Physics VI.4 that qualitative change from white to black necessarily takes place part by part (234b10–16) and appealed to the case of change from darkness to light as a possible counter example (191.30–192.2 Schenkl = fr. 155A Fortenbaugh et al.; 197.5–8 Schenkl = fr. 155B Fortenbaugh et al.). Similarly, in his commentary on Physics I.3 Simplicius reports that Theophrastus asked whether it is possible to say that alteration sometimes takes place at one go and that the change of a part does not always precede the change of the whole (107.12–16 Diels = fr. 155C Fortenbaugh et al.). Simplicius’ commentary on the Physics and the Byzantine scholia contain evidence that Alexander addressed Theophrastus’ question in a way that Marwan Rashed describes as original and paradoxical: he holds that the whole body undergoes continuous change but a part of it is altered at one go (Rashed 2011, p. 103).1⁷ The advantage of this view over the possibility that the whole undergoes alteration at one go becomes clear from the following examination of Alexander’s formulation of Theophrastus’ question.

1⁶ Richard Sorabji surveys the history of leaps or jumps over divisible magnitudes in Sorabji 1983, pp. 52–61 and 384–402. Alexander is not mentioned in that survey. 1⁷ Sharples dismisses this evidence found in Simplicius’ commentary on the grounds that it is incompatible with Themistius’ reports that Alexander holds that all change occurs in time. He argues further that in his commentary on De sensu 6, where Alexander propounds the same view, he is constrained by the need to comment on Aristotle’s text. See Sharples 1998, pp. 78–79 and Sharples 1994, p. 342. However, this evidence should be taken seriously for three reasons: (1) this view is found also in the Byzantine scholia, which are independent of Simplicius’ commentary on the Physics; (2) Alexander’s view is compatible with Themistius’ report; (3) the view that Alexander propounds in his commentary on De sensu 6 is not exactly Aristotle’s view but the view found in one of the fragments where he says that alteration begins at one go. I substantiate the two latter points below.

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change at one go 41 In his commentary on Physics VI.4 Simplicius reports that certain unnamed commentators asked how Aristotle’s claim that a changing thing has something in that from which it changes and something in that to which it changes (234b15) is compatible with his argument against Melissus in Physics I.3 where he rejects the latter’s assumption that what has come into being always has a beginning, saying ‘as if change does not take place at one go’ (186a15–16). He further reports that Alexander argued that this question should not be addressed in this context because in Physics VI.4 Aristotle discusses a changing body that has not yet completed its change. Regarding this body, Alexander argues, it is generally necessary that a part of it be in that from which it changes and a part of it be in that to which it changes. Still, Alexander does not dismiss this question altogether but says that it may be raised with regard to Aristotle’s claim that anything that has changed was previously in a process of change (Physics VI.6, 237a17–19). He answers this question in the negative on the grounds that in Physics VI.1 (232a10– 11) Aristotle shows that this account is absurd (966.26–967.4 Diels). From this report we can see that Alexander refines Theophrastus’ question. Whereas Theophrastus regards the claim that change takes place part by part, as well as the claim that a change of a part necessarily precedes the change of the whole, as incompatible with the possibility of change at one go, Alexander distinguishes these claims. He holds that the possibility of change at one go does not call into question Aristotle’s claim that change takes place part by part and rejects the possibility that a thing that has completed its change was not previously in a process of change. Another report on Alexander’s discussion of change at one go shows that through this distinction, Alexander allows change at one go and at the same time guarantees that anything that has completed its change was previously in a process of change. According to this report, Alexander distinguishes two cases of change at one go: (1) a change that does not occur in time, where the whole changes at one go without previously being in a process of change, for example, contact, and (2) a change where the whole undergoes change part by part but each of its parts changes at one go, as in the case of freezing (997.30–998.13 Diels). The example of contact clarifies how this view is compatible with Themistius’ report that Alexander holds that all change is in time (197.4 Schenkl). In his commentary on Aristotle’s De sensu Alexander says that, like contact, the activity (ἐνέργεια) of the senses does not take time because it occurs without a previous process of generation (οὐ διὰ γενέσεως) (125.12–24 Wendland).1⁸ Later in this commentary he mentions the propagation of light and sight as other cases of change that occur at one go and without time and claims that they are not cases of change at all. They come about without a process (οὐ διὰ κινήσεως) because, like being on the right of something, they occur due 1⁸ Cf. Aristotle, Metaphysics III.5, 1002a32–1002b5 and Alexander’s commentary ad loc., 232.2–16 Hayduck.

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42 contiguity, continuity, and continuous change to the relation of presence between the source of light and the illuminated air or between the visible object and the eye (131.21–132.16 Wendland).1⁹ From this account we can see that through his reformulation of Theophrastus’ question and the distinction between two types of change at one go, Alexander reconciles Aristotle’s claim that certain changes take place at one go with the characteristics of continuous motion and change. He secures Aristotle’s claim that anything that has completed its change was previously in a process of change, by denying that the change from darkness to light and other cases where a whole changes at one go are cases of change strictly speaking. Further, he treats other cases of change at one go, e.g., freezing, as changes that take place part by part and understands Aristotle’s claim that they are changes at one go as holding not for the whole but for the parts. In offering this answer, Alexander seemingly holds the rope at both ends. He regards the change of the whole as continuous and the change of its parts as taking place at one go. However, closer examination of Alexander’s account of this type of change shows that this answer is not a forced attempt to reconcile incompatible passages in the Aristotelian corpus but a solution to the question how change can begin in terms of the above account of accidental divisibility and the strict and primary sense of continuity.

6. Change at One Go and Accidental Divisibility The main fragment that informs us about Alexander’s account of motion at one go has three parts. In the first Alexander says that there is a process of change (κίνησις) even when the whole undergoes change at one go because between the qualities from which and into which it changes there are intermediate qualities that it gradually acquires before it completes its change (e.g., a surface that changes as a whole from white to black by gradually becoming darker and darker). In the second he says that this account requires further inquiry because Aristotle’s claim later in Physics VI.4 (235a17–18, cf. 235a35–36) that quality is accidentally divisible implies that quality is co-divided with the body that undergoes change, therefore qualitative change takes place part by part (e.g., a black colour spreads over a white surface). And in the third he concludes that it is better to hold that a part and not the whole of a changing body undergoes change at one go, e.g., when the surface of a body that turns in the direction of the sun is sunburnt before the other parts (968.5–29 Diels). From this fragment we see that Alexander’s account of change at one go is an alternative to the two accounts found in the first

1⁹ Alexander’s view that relational change is not change in the strict sense goes back to Aristotle’s Physics V.2, 225b11–13 and Metaphysics XIV.1, 1088a29–35. He expresses this view also in his De anima 43.9–11; Mantissa 143.6–145.16. On Alexander’s account of hearing and change at one go see Towey 1991, pp. 13–16.

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change at one go and accidental divisibility 43 two parts of this fragment. Further, Alexander’s appeal to Aristotle’s claim that quality is accidentally divisible suggests that he rejects the first account because Aristotle’s view that the number of qualities is finite (De sensu 6, 445b21–29, 446a19–20) excludes the possibility that a change from one shade to another takes place gradually by passing through intermediate shades. But it is unclear from this fragment why Alexander prefers the third account. A fragment from Alexander’s commentary on Physics VIII.3, where he again refers to Aristotle’s reply to Melissus in Physics I.3, clarifies this point. In this fragment Alexander draws the following conclusion from Aristotle’s claim in Physics VIII.3 that increase, decrease, and qualitative change can take place at one go: Through what he said now, Aristotle shows what he meant in the first book of this treatise when he said in reply to Melissus ‘as if change does not take place at one go’. For he did not say ‘at one go’ in the sense of instantaneous (ἄχρονος)—for this is false—but [he refers to] the beginning (ἀρχή) of that which changes at one go and not part by part toward changing.2⁰ (1199.16–20 Diels)

This fragment sheds fresh light on Alexander’s account of change at one go, in showing that it specifically holds for the beginning of change.21 This account also features in Alexander’s commentary on De sensu 6. Here, as in the above fragment, Alexander says that Aristotle’s claim that qualitative change can take place at one go is compatible with the view that change is in time because alteration begins at one go without a part of it being changed before the other; he adds: ‘if [the whole] began (ἤρξατο) its alteration and freezing at one go, it does not follow that it has already completed its process of freezing immediately upon beginning it’ (133.4–6 Wendland). Alexander’s stress on the beginning of change suggests that through his account of change at one go he addresses the question how change can begin, which arises from Zeno’s dichotomy paradox. Aristotle’s arguments against this paradox notwithstanding, it remained problematic in the Aristotelian tradition primarily because Aristotle’s accounts of the end of change and of the beginning of change

2⁰ Diels does not close the quotation marks but Rashed 2011, scholium 573 indicates that the last sentence is part of the quotation from Alexander. In his comment on this scholium Rashed argues that here Alexander disambiguates the adjective ἀθρόος, which may have the temporal meaning of ‘instantaneous’ and the spatial meaning of ‘at one go’ or ‘all at once’ (Rashed 2011, p. 516). However, Alexander’s claim that this adjective does not mean ‘instantaneous’ is better understood in light of Alexander’s distinction between the two types of change at one go: the propagation of light and sight, which are relational changes, hence not changes in the strict sense and freezing, where a part changes at one go but the whole changes part by part. 21 Cf. Croese 1998. Irma Croese does not cite the above fragment in support of her interpretation. Sextus Empiricus attributes a similar view to the Stoics (Against the Professors 10.121–142). On this see Long and Sedley 1987, pp. 303–304 and White 1992, pp. 314–326.

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44 contiguity, continuity, and continuous change are asymmetrical.22 In Physics VI.5 he argues that whereas there is a last stage at which something has completed its motion (253b30–33), there is no first stage at which something begins its motion because the first distance traversed can be infinitely divided into sub-distances converging on the starting point (236a35– b36).23 Aristotle’s distinction between potential and actual divisibility guarantees that change can begin, by implying that the number of actual divisions is finite, but it provides no definite answer to the question how many actual divisions there are. In another fragment Alexander presents an account of change at one go through which he addresses the question of the beginning of change, but also offers a criterion for determining the actual division of the magnitude that undergoes change. In his commentary on Physics VI.6 Simplicius reports that Alexander addressed the question of the correspondence between the divisibility of magnitudes and motion as follows: Alexander inquires again how qualitative change has its division in correspondence with the division of the magnitude that moves if indeed certain things change at one go, as Aristotle says in the first book of this treatise when he himself refers to Melissus. And here too Alexander notes that even if not the whole but a part of it changes at one go, neither motion nor time would be co-divided with the division of this part and decides the matter by saying ‘it is not co-divided with all its parts but with the parts of the thing that moves insofar as it moves’.2⁴ (978.35–979.7 Diels)

From this report we see that in his interpretation of Physics VI.6 Alexander does not follow Aristotle’s view found in Physics IV.11 and VI.2 that the divisibility of magnitude entails the divisibility of motion and time but prefers the view found in Physics VIII.3 and De sensu 6 that the divisibility of magnitude does not necessarily entail the divisibility of change. This report brings to light an important element in Alexander’s interpretation of the latter view. It shows that considering his assumption that the divisibility of motion does not necessarily follow from the divisibility of magnitude, Alexander identifies the factor that determines the divisibility of motion and time with a characteristic of the object that undergoes

22 As Themistius and Simplicius report, Theophrastus found this asymmetry surprising (Themistius, Paraphrase of Aristotle’s Physics, 195.8–21 Schenkl = fr. 156A Fortenbaugh et al.; Simplicius In Phys. 986.3–17 Diels = fr. 156B Fortenbaugh et al). Rashed 2011, scholium 362 indicates that Alexander addressed this problem through the assumption of change at one go. Here he says that even if quality is indivisible in nature, there is no first stage of change unless change takes place at one go. Alexander devotes Quaestio I.22 to this problem. Here he argues that since the parts of a continuous whole exist in potentiality, a thing that moves over a magnitude has previously traversed a potential infinity (36.5–9 Bruns). I show below that this view underlies Alexander’s account of change at one go. 23 For a lucid analysis of these arguments see White 1992, pp. 48–53. 2⁴ Cf. scholium 350.

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change at one go and accidental divisibility 45 change insofar as it moves. This view calls to mind Alexander’s account in the fragment from his commentary on Physics VIII.8 of the potential divisibility of a line. Similarly to this fragment, here the characteristic that determines the motion of or along a magnitude is not the underlying structure of continuous magnitudes, i.e., their divisibility into divisible parts, but another characteristic: the essence in the case of the line, and a characteristic of the moving body insofar as it moves in the present fragment.2⁵ The exact significance of this characteristic can be understood from the context in which appears Alexander’s account of the beginning of change in terms of change at one go. In Physics VIII.3 Aristotle argues against thinkers who hold that all things are always in motion though this fact escapes our perception, saying that there cannot be a continuous process of increase and decrease. In so doing, he says that this theory is similar to that which infers from the fact that a drop of water wears away a portion of a stone that half of this portion has previously been worn away in half the time on the basis of the wrong assumption that the divisibility of this portion entails that each of its parts was affected by the drop of water separately and not together (253b9–26). In his commentary on this discussion Simplicius reports that Alexander understood Aristotle’s claim that the parts of the portion of stone were affected by the drop of water together and not separately as holding also for the cause, i.e., the water: ‘because the parts of the mover are not in the whole in actuality, so as to be able to cause motion by virtue of themselves’ (1198.18–19 Diels). Alexander’s claims that motion begins at one go, and that the cause of motion acts at one go, explain how motion can begin and also give a criterion for determining the magnitude that the first stage of change covers, by regarding it as dependent on the efficacy of the cause. By this account, x drops of water wear x away a certain portion of a stone but drops do not affect the stone at all.2⁶ This 2 account is based on the assumption that the first stage of change is not subject to the principle that in continuous motion a thing that has completed its motion was previously in a process of change because there is a threshold below which a cause cannot initiate change. Further, this account implies that the first portion of stone that has been worn away could not previously have undergone a process of change part by part because there is a minimal magnitude that can affect change and a corresponding minimal magnitude that can undergo change. In his commentary 2⁵ Cf. Rashed 2011, pp. 379–380. 2⁶ In the passage from his commentary on De sensu 6 Alexander says that the parts contribute to the perceptibility of the whole. Richard Sorabji interprets Aristotle’s De sensu 6 in the same way and suggests that Aristotle can maintain that qualitative change is continuous, by holding that a body is changing to the next quality all the time and that this change will eventually lead to the next manifest quality (Sorabji 1983, p. 411). For a similar interpretation see Murphy 2008 esp. p. 196. The interpretation of Aristotle aside, nothing in Alexander’s account suggests that he interpreted Aristotle in this way, and it is unlikely that in his commentary on Physics VIII.3 he assumes latent gradual change because here Aristotle argues against the view that all things are always in motion but this fact escapes our perception.

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46 contiguity, continuity, and continuous change on De sensu 6 where Alexander discusses the causal efficacy of perceptible objects he explicitly posits this assumption: The motion [caused] by the perceptible object becomes perceptible in actuality not only by means of the quality but also by means of the quantity of the potentiality, not because each of the [parts] is perceptible individually but because through the composition [each part] contributes to the whole [that is made up] from them toward its being able to move perception in actuality. For being in the whole in this way they are perceptible in potentiality and not as being able ever to become in their own right [perceptible] in actuality, but as parts. (120.1–6 Wendland)

Apart from confirming that Alexander assumes that there is a minimal magnitude that can affect change, this passage explains the claim found in the fragment from his commentary on Physics VI.4 that motion and time are co-divided with the parts of the thing that moves insofar as it moves. It shows that the actual division of motion and time is a feature of the object that undergoes change because this division depends on the efficacy of the cause of change. This cause acts at one go and can affect no lesser a change than that which the object initially undergoes. Furthermore, Alexander’s explanation of why the cause of change acts at one go found in Simplicius’ report and in the above passage indicates that the strict and primary sense of continuity facilitates his solution to the question of the beginning of change. This explanation rests on the assumption that the cause is a whole, whose parts are not causally efficacious by virtue of themselves. In his commentary on De sensu 6 Alexander explicitly relates this explanation to continuous wholes: ‘Just as the parts of the continuum are in the whole in potentiality, so the affections of the parts, which are perceptible in potentiality, are perceptible, i.e., because they are parts in a whole’ (116.19–21 Wendland). This analysis shows that the strict and primary sense of continuity serves Alexander in addressing the question of the beginning of change and provides a criterion for the divisibility of motion and time. It may also suggest that by introducing the strict and primary sense of continuity, Alexander concedes to atomism. He allows discontinuous change and also assumes, much like the Epicureans, minimal quanta of motion. However, his view on these two matters is more subtle. Regarding motion, Alexander secures continuous motion by holding that certain apparent cases of change at one go, e.g., the propagation of light, are not cases of change strictly speaking and that other cases, e.g., freezing, begin at one go but the change of the whole takes place part by part.2⁷ Similarly, his explanation of the divisibility of motion in causal terms enables him to avoid the

2⁷ Cf. Rashed 2011, p. 105.

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references 47 atomist assumption of minimal quanta of motion. According to this explanation, the magnitude that can initiate change and the magnitude that can undergo change are not minimal by their own nature because the precise threshold of causing and undergoing change is determined by the relation between cause and effect and varies from case to case. The minimal amount of water required for initiating the process of eroding a stone is greater than the minimal amount of acid required for initiating this process and correspondingly the amount of stone that is initially eroded by these two causes differs in proportion to the efficacy of these causes. Indeed, in his commentary on De sensu 6 Alexander offers a similar argument against Diodorus Cronus, contending that there is no minimal perceptible by virtue of its own nature because imperceptibility is a consequence of perception’s weakness (122.18–23 Wendland). Thus Alexander is aware of an affinity between the strict and primary sense of continuity and atoms, employs it in his account of the beginning of change, but avoids its atomistic implications by appealing to the relation between cause and effect.

References Bostock, D. (1991). “Aristotle on Continuity in Physics VI”. In: Aristotle’s Physics: A Collection of Essays. Ed. by L. Judson. Oxford: Oxford University Press, pp. 179–212. Croese, I. M. (1998). Simplicius on Continuous and Instantaneous Change: Neoplatonic Elements in Simplicius’ Interpretation of Aristotelian Physics. Utrecht: Zeno Institute for Philosophy. Furley, D. (1982). “The Greek Commentators’ Treatment of Aristotle’s Theory of the Continuous”. In: Kertzmann 1982, pp. 17–36. Kertzmann, N., ed. (1982). Infinity and Continuity in Ancient and Medieval Thought. Oxford: Oxford University Press. Long, A. A. and D. N. Sedley (1987). The Hellenistic Philosophers. Vol. 1. Cambridge: Cambridge University Press. Miller, F. D., Jr. (1982). “Aristotle against the Atomists”. In: Kertzmann 1982, pp. 87–111. Murphy, D. (2008). “Alteration and Aristotle’s Theory of Change in Physics 6”. In: Oxford Studies in Ancient Philosophy. Vol. 34, pp. 185–218. Rashed, M., ed., trans., and comm. (2011). Alexandre d’Aphrodise: Commentair perdu à la Physique d’Aristote (Livres IV–VIII. Le scholies byzantines). Berlin and Boston: de Gruyter. Ross, W. D. (1936). Aristotle’s Physics: A Revised Text with Introduction and Commentary. Trans. and comm. by W. D. Ross. Oxford: Oxford University Press.

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48 contiguity, continuity, and continuous change Sharples, R. W. (1994). “Aristotle and Hellenstic Philosophy”. In: Phronesis 39.3, pp. 338–346. Sharples, R. W. (1998). Theophrastus of Eresus: Sources for His Life, Writings, Thought, and Influence. Leiden, Boston, and Köln: Brill. Sorabji, R. (1982). “Atoms and Time Atoms”. In: Kertzmann 1982, pp. 37–86. Sorabji, R. (1983). Time, Creation, and the Continuum: Theories in Antiquity and the Early Middle Ages. Ithaca: Cornell University Press. Towey, A. (1991). “Aristotle and Alexander on Hearing and Instantaneous Change: A Dilemma in Aristotle’s Account of Hearing”. In: The Second Sense: Studies on Hearing and Musical Judgement from Antiquity to the Seventeenth Century. Ed. by C. Burnett, M. Fend, and F. Gouk. London: The Warburg Institute, pp. 7–18. White, M. J. (1992). The Continuous and the Discrete: Ancient Physical Theories from a Contemporary Perspective. Oxford: Oxford University Press. Wieland, W. (1962). Die aristotelische Physik. Göttingen: Vandenhoeck & Ruprecht.

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3 Infinity and Continuity Thomas Bradwardine and His Contemporaries Edith Dudley Sylla

In about 1330 Thomas Bradwardine produced a work, De continuo or On the continuous, in which he argued against recent authors who had proposed indivisibilist or atomist conceptions of the cosmos, in particular Henry of Harclay, who had argued in favour of the composition of continua from infinitely many immediate indivisibles, Walter Chatton, who had argued for the composition of continua from finitely many indivisibles, and Robert Grosseteste, who, like Harclay, had argued for infinitely many indivisibles, but had supposed that they were mediate and not immediate.1 John Murdoch, who included an edition of Bradwardine’s De continuo in his 1957 Ph.D. dissertation, characterized these fourteenth-century atomistic theories as mathematical (Murdoch 1957). The indivisibles they proposed were not very small three-dimensional bodies like Democritus’ atoms, but rather like mathematical points. Also, the arguments that Bradwardine opposed to the atomistic theories were most importantly geometric. To the argument that in using geometrical arguments against the atomists he was assuming what he claimed to be proving, Bradwardine argued that this was not the case, because Euclid had not explicitly stated as a supposition that magnitudes are not composed of indivisibles, and because many of Euclid’s theorems would hold if continua such as lines were composed of infinitely many mediate indivisibles, even if finitely many indivisibles or immediate indivisibles would be inconsistent with various of Euclid’s demonstrations. There are three known manuscripts of De continuo (one extremely short) and even the longest text may be incomplete.2 Early in the work, Bradwardine assumes 1 By Bradwardine’s definitions, ‘For one thing mediately to be, to have been, or to be about to be, after another thing is the same as to be, to have been, or to be about to be with a mean between them’ and ‘For one thing immediately to be, to have been, or to be about to be, after another thing is the same as to be, to have been, or to be about to be without a mean between them.’ 2 The two fairly complete manuscripts are Torun, Poland, Gymnasial-Bibliotheck, Ksiaznica Miejska im Mikolaja Kopernica, MS R 4o 2, pp. 153–192, which is the most complete, and Erfurt, Wissenschaftliche Bibliothek der Stadt, MS Amploniana, 4o 385, fols. 17r–48r and 55v, which omits more than half of the discussion of the definitions of the two infinities, categorematic and syncategorematic, Edith Dudley Sylla, Infinity and Continuity: Thomas Bradwardine and His Contemporaries In: The History of Continua: Philosophical and Mathematical Perspectives. Edited by: Stewart Shapiro and Geoffrey Hellman, Oxford University Press (2021). © Stewart Shapiro and Geoffrey Hellman. DOI: 10.1093/OSO/9780198809647.003.0004

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50 thomas bradwardine and his contemporaries that all continua whether mathematical or natural are similarly structured or isomorphic, saying: Supposition 30. If one continuum has an infinite or finite number of immediate atoms, any [continuum] so has them. This is evident enough by means of supposition 3.3 Supposition 31. If one continuum is composed in a certain manner (secundum aliquem modum) out of indivisibles, any [continuum] is so composed, and if one is not composed of atoms, none is.

Only the Torun manuscript includes Parts VII and VIII, following after the refutation of the views of Harclay, Chatton, and Grosseteste. In Part VII, Bradwardine concludes: Part VII. Conclusion 138. No continuum is composed of infinitely many indivisibles. Conclusion 141. No continuum is integrated out of atoms. Whence this follows and is elicited: Every continuum is composed of infinitely many continua of the same species as it.

After conclusion 141, Bradwardine apostrophizes: For indeed, the foundation of nature is so composed, the pillar of mathematics [so] strengthened and the fabric of all physics [so] made firm.

Then in the next part Bradwardine concludes: Part VIII. Conclusion 143. It is possible per se for every substance to lack every accident. Conclusion 144. Everything which is neither part nor cause of another can be corrupted while that other thing is wholly saved. Whence (Conclusion 145): There can be a continuous and finite thing without some indivisible making [it] continuous or finite. Conclusion 151. There are no surfaces, lines or points at all (superficiem, lineam sive punctum omnino non esse). Whence [the following is] manifest: A continuum is not made continuous or finite by means of such [entities], but by means of itself.⁴

and ends in the middle of conclusion 134, where Bradwardine is arguing against composition out of infinitely many indivisibles. In addition there is Paris, BNF nouv. acq. lat. 625, f. 71v, which has little more than the definitions. See Murdoch 1957, 328–329n1. 3 Supposition 3 is ‘Where there is no cause of diversity or dissimilitude, it is considered to be the same’ (Ubi diversitatis vel dissimilitudinis nulla est causa, simile iudicatur). ⁴ Selections from De continuo are contained in Murdoch 1987. For this paper I use revised texts and English translations that John Murdoch made for a planned but unpublished three-volume work on

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thomas bradwardine and his contemporaries 51 I have here omitted the arguments for these conclusions, as well as conclusions themselves that are part of reduction to absurdity arguments. Given what is found in Parts I–VI of De continuo, Parts VII and VIII come as a shock. If Bradwardine ends by proving that geometric indivisibles do not exist, what was the point of including in De continuo suppositions 30 and 31, which are naturally understood to imply that geometric continua and natural continua are isomorphic or have similar structures? How might we explain this outcome? The view that most indivisibles are not things existing in the external world, but mental concepts existing in the minds of mathematicians was the view of many fourteenth-century nominalists.⁵ Bradwardine, however, is not usually considered to have been a nominalist. George Molland, who edited Bradwardine’s Geometria speculativa, lists among his conclusions that ‘Bradwardine’s attitude toward mathematics was realist rather than conceptualist’ (Molland 1978, 174). When did other fourteenth-century Aristotelians begin to consider that mathematics was in the mind or imagination? When and how did Bradwardine come to the conclusion that geometric indivisibles—surfaces, lines, and points—are not things (res) in the external world? Might Parts VII and VIII have been added to an incomplete manuscript by someone else? Or did Bradwardine himself undergo a conversion to the denial of the real existence of points, lines, surfaces, and other indivisibles between Parts VI and VII? Should we suppose that Bradwardine did not change his mind on indivisibles in the midst of composing De continuo, but rather adopted varying positions, as did some nominalists who wrote about mathematical indivisibles even though they said in the same works that indivisibles do not exist?⁶ Supporting the hypothesis that someone other than Bradwardine might have played a role in putting together the currently existing text of De continuo is the fact that that there are some twenty passages in De continuo in which the purpose of a given text is explained using the third-person singular (Murdoch 1957). On the other hand, the way in which De continuo promotes the importance of mathematics fits very well with what we know of Bradwardine from his other the context and significance of fourteenth-century atomism in the Latin West. I am grateful to John Murdoch for having given me copies of his editions and translations decades ago. He should not be held responsible for any errors I may have made in editing or interpreting the texts. I have not seen microfilms of the manuscripts, let alone the manuscripts themselves. The versions I have of Murdoch’s transcriptions and translations may not represent his most up to date versions. More of Murdoch’s many published papers on this topic will be cited in later footnotes. ⁵ For the appearance of the same position in Europe, I will call those who deny the existence of mathematical indivisibles ‘nominalists’ but I have in mind ontological minimalism together with assessing the truth of propositions by making use of the so-called personal suppositions of their terms, not some other aspects related to the problem of universals that might come to mind. Alternative labels might be Ockhamists, terminists, or fourteenth-century moderni. ⁶ See Rimini 1979, in 2 Sent. dist 2, q. 2, vol. 4, p. 291, ‘Quia tamen praeter hoc quodlibet etiam aliter deficit, respondeo ad secundum supponendo pro nunc puncta esse in magnitudine secundum quod multi imaginantur, et false iudicio meo. Hoc tamen supposito dico . . .’; cf. Sylla 2005, p. 262n37.

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52 thomas bradwardine and his contemporaries works. For instance, De continuo includes an over-the-top rhetorical passage extolling the power of mathematics in the pursuit of physical disciplines: mathematics . . . discerns more sharply than its sister disciplines, it shoots its arrow more directly, and it protects itself with a safer shield. Nobody can hope to carry away victory in a physical contest without the counsel, aid, and comfort of mathematics. Mathematics reveals every genuine truth, knows every hidden secret, and bears the key of all subtlety of letters. Anyone, therefore, who presumes to do physical science without mathematics will never enter the door of wisdom. (Murdoch 1987, pp. 133–134)

Beyond praise of mathematics, De continuo has been linked to Bradwardine also by its quasi-axiomatic format, similar to the formats of other works by him, with definitions, suppositions, and then conclusions deduced from these principles. Suppose that Bradwardine began De continuo with the goal of axiomatizing a combined discipline of mathematical and natural continua and then encountered a roadblock or anomaly. If we take De continuo as a whole, including Parts VII and VIII from the Torun manuscript, as the work of Bradwardine, we might interpret what we find as Bradwardine testing the small axiomatic or deductive discipline or science he has set up and then finding that the axiomatic system he has been testing is inconsistent or self-contradictory. No set of hypotheses about indivisibles works, neither the supposition that they are immediate, nor the supposition that they are mediate. Neither the supposition that they are finitely many, nor the supposition that they are infinitely many. Where is the flaw? The flaw seems to be the supposition that mathematical and natural indivisibles are isomorphic. Most obviously problematic are suppositions 30 and 31, quoted above, which, for instance, enable an argument about liquids to be used to support conclusions about geometric surfaces. At the start of Part VIII we find Bradwardine beginning the disproof of the existence of indivisibles (led by surfaces, lines, and points) with: Conclusion 142: Consequently, we must posit in advance that: no prime matter or substance or primary or secondary quality is corrupted in making liquid bodies continuous or discontinuous, and the same thing holds with respect to quantity and indivisibles of quantity.

Here the flaws in the suppositions of Bradwardine’s deductive scientific discipline are beginning to show themselves – quantities and indivisibles of quantity separate from matter are not thought to be subject to physical processes such as corruption in the way that liquid bodies are. In A History of Mathematics, Carl Boyer writes about the foundations of Euclidean geometry:

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thomas bradwardine and his contemporaries 53 Euclid’s Elements did have a deductive structure, to be sure, but it was replete with concealed assumptions, meaningless definitions, and logical inadequacies. Hilbert understood that not all terms in mathematics can be defined and, therefore, began his treatment of geometry with three undefined objects – point, line, and plane – and six undefined relations – being on, being in, being between, being congruent, being parallel, and being continuous. In place of Euclid’s five axioms (or common notions) and five postulates, Hilbert formulated for his geometry a set of twenty-one assumptions, since known as Hilbert’s axioms. Eight of these concern incidence and include Euclid’s first postulate, four are on order properties, five are on congruency, three are on continuity (assumptions not explicitly mentioned by Euclid), and one is a parallel postulate essentially equivalent to Euclid’s fifth postulate. (Boyer 1989, 609)

Along these lines, examining Bradwardine’s conclusions and his arguments against the assumption of indivisibles in continua, one could argue that what Bradwardine’s De continuo ultimately demonstrates (though he does not say so) is that his proposed axiomatic system is flawed and should be revised or rejected. It simply did not work to construe the properties of physical continua as isomorphic to the properties of geometric continua. Given that Bradwardine uses axiomatic or deductive approaches in several of his works, I would like to suggest that Bradwardine recognized that he had demonstrated that his set of suppositions was inconsistent and that, if he had continued to work on the continuum further, he might have tried to modify his set of suppositions or axioms. This may explain why the text in the Torun manuscript calls the end of De continuo as it exists the end of Book I (Murdoch 1987, 104). Of course, we need not assume that designing an axiomatic system was Bradwardine’s primary goal. Suppose his primary goal was to establish so-called mixed mathematical disciplines. Then he would need somehow to include mathematical and natural or physical concerns and would have to determine how the mathematical and the physical or natural would interact. His concept of an Aristotelian science would be that a scientific discipline should have principles and demonstrate conclusions. One would start with definitions and suppositions applicable to the particular subject matter. But does it make sense to have a single discipline for mathematical and physical continua? The outcome of Bradwardine’s attempt to follow that path seems to show that it was not a good idea. Better separate the discipline of geometry from the disciplines of physical or natural sciences. This could be considered the major contribution of Bradwardine to the history of concepts of continuity, i.e., the distinction between physical and more mathematical continuity, except that historians have yet to reveal the primary reason for the more general distinction between mathematics and physics in the fourteenth century (Sylla 2015). It could have resulted from the influence

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54 thomas bradwardine and his contemporaries of Averroes’s commentary on Aristotle’s Physics.⁷ It could have resulted from nominalism and the logica moderna. Given what we know about Bradwardine, I am sure that just because of the negative outcome of De continuo as far as it was completed, he would not have given up his goal of creating mixed mathematical disciplines, visible in his On the proportions of velocities in motions. Bradwardine had a preference for mathematical or demonstrative sciences, but he also functioned in a university system that emphasized disputations in which alternative points of view were pitted against each other. This would help to explain why he devoted so much of the space of De continuo to challenging the recently proposed atomist or indivisibilist theories of Henry of Harclay, Walter Chatton, Robert Grosseteste, and others. If De continuo is paired with Bradwardine’s De proportionibus velocitatum in motibus, written not long before, one might well assume that his purposes in the two works were similar, and that in both he aimed to produce a well-founded discipline applying mathematical tools to deal with scientific problems. If one looks at the size of Bradwardine’s later De causa Dei, contra Pelagium, et de virtute causarum, ad suos Mertonenses (876 large pages in print), one cannot miss the scope of Bradwardine’s intellectual ambition and energy. No wonder that he did not continue further with his project De continuo since he had moved on to greater things. At the end of this paper, I will describe briefly Bradwardine’s use of mathematics in De causa Dei to argue against the eternity of the world on the grounds of the problems with infinities that it would imply. What, then, might a reader of a book on philosophical concepts of the continuum find of interest in a chapter on Bradwardine’s De continuo? It might be of interest to see how fourteenth-century Aristotelians attempted to apply mathematics to continuity, a subject familiar to the fourteenth-century Aristotelians from Aristotle’s Physics and De caelo. Aristotle had proposed two definitions of the continuous. One was that two things are continuous if their limits are unified. This does not seem to offer much scope for application of mathematics. Aristotle’s other definition was that something continuous is infinitely divisible. That did seem to offer scope for mathematization, although it meant that the concepts of infinity and continuity had to be considered together (Murdoch 1982b). But then neither of the fundamental subdisciplines of mathematics common in the fourteenth century—that is, arithmetic and geometry—had much to offer with regard to mathematizing infinity. The numbers that arithmetic dealt with were limited to positive whole numbers. Infinity was not a number. Except for contrived cases such as the so-called gyrative line, most magnitudes dealt with by geometry were limited by the finitude of the Aristotelian cosmos.⁸ Before the continuum could be thought to be embodied in the real numbers, the number concept had to ⁷ For Averroes’s ‘divorce of mathematics and physics’ in his long commentary on Aristotle’s Physics, see Glasner 2009. ⁸ The gyrative line could be formed by mentally dividing a cylinder into proportional parts – the first half, the next quarter, the next eighth, the next sixteenth, ad infinitum, and then imagining a line to be wound around the cylinder once in every proportional part, ad infinitum. This would result in an

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thomas bradwardine and his contemporaries 55 be extended several times beyond positive whole numbers.⁹ Bradwardine applied Euclidean geometry to disprove theories of indivisibles, but perhaps another approach to magnitudes was required, such as later was provided by the calculus. When he wrote his dissertation in 1957 John Murdoch started with the assumption that to apply mathematics to science was a good thing. Many would agree with him. Was not the use of mathematics an important way in which early modern scientists surpassed their medieval predecessors? Like most historians of mathematics, Murdoch also assumed that Cantor’s mathematics of infinity and continuity took the correct approach. It was therefore disappointing to a medievalist that so many Aristotelians such as Robert Grosseteste and Henry of Harclay thought that infinities such as the infinity of points in a line could have ratios to each other. In the context of this book on historical concepts of the continuum, however, I now think that it is interesting to study how historical philosophers, mathematicians, or scientists worked without our always keeping in mind how they were right or wrong compared to the most recent science or mathematics. One step that fourteenth-century Aristotelians took had the effect of differentiating the claims of mathematics and physics about the continuum, even though this might not have been the original intent. Whether this was good or bad, it happened. While we may consider it to Bradwardine’s credit that he had a preferential option for mathematics, it is also interesting to see how Walter Burley (who will be discussed below, not claiming that he was the first to do this) used logic and the syncategorematic sense of ‘infinite’ to describe a model or structure of superimposed divisions of magnitudes. The model Burley describes could provide a basis for inquiries into infinity and continuity in the way that real numbers provided a model or structure for Dedekind and Cantor in their development of a mathematics of continuity and infinity in their time. At the end of his 1987 paper on De continuo Murdoch wrote: Even though one can take exception to the conclusiveness of some of Bradwardine’s refutatory claims and can remain somewhat unconvinced by some of the

infinitely long line in a finite space, such that the line never reaches the other end and there is never a last loop around the cylinder. ⁹ See Neal 2002, 1, ‘In the early modern period a crucial transformation occurred in the classical conception of number and magnitude. For the Greeks . . . numbers were merely collections of discrete units that measured some multiple. Magnitude, on the other hand, was usually described as being continuous, or being divisible into parts that are infinitely divisible. A distinction was made between arithmetic and geometry; arithmetic dealt with discrete or unextended quantity while geometry dealt with continuous or extended quantity. In the early modern period a transformation occurred in this classical conception of number and magnitude. For instance, Simon Stevin (1548–1620) insisted in his 1585 Arithmetique that the traditional Greek notion of numbers was wrong: he believed that numbers were continuous rather than discrete. Stevin also developed a system for indefinite decimal expansions of number that implicitly contained the idea of a numerical continuum.’

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56 thomas bradwardine and his contemporaries elements of his account of the relations between Euclidean geometry and the continuity of geometrical magnitudes, the Tractatus de continuo is an important document within the history of fourteenth-century natural philosophy. Not only does it provide us with an outstanding example of the application of mathematics that is increasingly apparent as one uncovers the history of later medieval natural philosophy, but it addresses the issue of the justification of applying the mathematics in question. That is a much rarer fourteenth-century occurrence, and Bradwardine deserves considerable credit for broaching the issue, whatever his success may have been in resolving it. (Murdoch 1987, 119)

Given the huge long-term achievements of mathematical physics, we can applaud Bradwardine’s efforts to promote the use of mathematics in physics, even if he was not always successful. With this understanding, let us now turn to a brief survey of De continuo.

1. Parts I and II of De Continuo Whether or not with an ambitious axiomatizing goal, Bradwardine began De continuo with definitions and suppositions, on the basis of which he proved affirmative conclusions. He then deployed what he had presented up to that point to argue in Parts IV–VI against various types of indivisibilism or atomism. Part I of De continuo opens with a comprehensive list of twenty-four definitions starting with the terms ‘continuum’, ‘permanent continuum’, ‘successive continuum’, ‘body’, ‘surface’, ‘line’, ‘indivisible’, ‘point’, ‘time’, ‘instant, ‘motion’, ‘motum esse’ (or indivisible completion of motion), ‘matter of motion’, ‘degree of motion’, ‘superposed’, and ‘imposed’. Then follow definitions of phrases rather than of single words: ‘mediately to be, to have been, or to be about to be after another thing’, ‘immediately to be, to have been, or to be about to be, after another thing’, ‘to begin to be through the affirmation of the present and the negation of the past’, ‘to begin to be through the negation of the present and the affirmation of the future’, ‘to cease to be through the negation of the present and the affirmation of the past’, ‘to cease to be through the affirmation of the present and the negation of the future’, ‘absolutely privative infinite’ (or infinite taken categorematically), and ‘relationally privative infinite’ (or infinite taken syncategorematically). Many of these terms and phrases are used in what exists of De continuo, but not the last two definitions of different senses of ‘infinite’. There is, however, in the Torun manuscript a long commentary on the last two definitions of senses of ‘infinite’, which might have been meant to substitute for further intended parts of De continuo that were never finished. Ten suppositions follow in Part II. Some are plausibly, as claimed, ‘self-evident to everyone’, such as the second, ‘If finite be added to finite, the total will be finite,’

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part iii: bradwardine’s affirmative conclusions 57 but some are not. For instance, Bradwardine supposes that ‘all sciences are true in which a continuum is not assumed to be composed of indivisibles’. This is not selfevident and it is so awkwardly stated that its sense is unclear. There is, however, a comment explaining that ‘he asserts this because sometimes he uses things proved in other sciences as if they were self-evident, since it would be far too lengthy a task to prove them all’. The ninth supposition states that ‘with whatever velocity or slowness one mobile may be moved or one space traversed, so can any’. It seems to me that this may presuppose that continua are not composed of indivisibles. Consider the example of a rod rotating around an end at rest together with the argument that velocities decrease continuously to zero towards the resting end of the rod. Would this be possible if the rod were composed of atoms?

2. Part III: Bradwardine’s Affirmative Conclusions After the definitions and suppositions, Bradwardine argues for positive conclusions, many of which are geometrical. For instance, he concludes: 10. It is impossible that one large part of [one] straight line be imposed to another straight line and another large part [of the same line] be superposed to, or laterally distant from, that [other line].1⁰ 11. A single straight line cannot have two points continued in another [straight line] and also have a large part superposed to or laterally distant from that [other straight line]. 12. It is clearly impossible that one large part of a straight line be superposed to a second straight line and another [large part of it] be laterally distant from that [second line]. 13. It is not possible for a single straight line to have two points superposed, or one [point] imposed, to another [straight line], and yet have another [point] superposed to and a large part laterally distant from that [other straight line]. 14. Any straight line intersecting [another] straight line cuts it in some point and not in more than one. 15. Straight lines meeting in some point have no other point intrinsic to them [both]. Whence a porism is inferred from this: The radii of a circle do not meet before the centre [of the circle], nor do the straight lines drawn from the base of a triangle to the angle opposite touch one another before that [angle].11 1⁰ Here, as Bradwardine defined his terms, ‘imposed’ means continued in the same direction from the end of another line and ‘superposed’ means adhering (to the side) without mean. 11 Unpublished draft translation by John Murdoch. Arguments for each of these conclusions omitted. See Murdoch 1964.

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58 thomas bradwardine and his contemporaries Bradwardine demonstrates these conclusions to prepare to argue geometrically against any atomist theory that might propose that indivisibles are immediate or that there are more indivisibles in a larger magnitude than in a smaller one of the same type.

3. Part IV: Arguments against Immediate Indivisibles In Part IV of De continuo, then, Bradwardine uses the definitions, suppositions, and the conclusions of Part III to prove that the proposal of immediately joined indivisibles in continua is self-contradictory. His first conclusions against the view that indivisibles are immediate are: 35. If immediate points be assumed in a continuum, then points immediate to the centre of a circle, of a square, or of any body equally correspond to the outer points of the circumference of the circle, the side of the square, and the surface of the body. 36. If this is so, infinitely many points are immediately conjoined to any given centre. Whence it is clear: no two points in a plane surface nor indivisibles in any continuum are joined to one another without a mean. 37. If this is so, infinitely many indivisibles fall between any two indivisibles of any continuum whatsoever.12

In sum, the assumption that indivisibles are immediate implies that they are not immediate because many indivisibles fall between any two indivisibles, an obvious self-contradiction. Most of Bradwardine’s conclusions in the refutatory parts of De continuo are in this conditional format, saying ‘if so, then such and such would follow’ and the conditions are generally counterfactual, the series of conclusions belonging to a reduction to absurdity argument. In Part IV of De continuo, the condition is ‘If immediate points be assumed in a continuum’ and, of course, Bradwardine is attempting to prove that to assume that there are immediate points in a continuum leads to self-contradictory results. Lest an unwary reader mistakenly assume that Bradwardine intends the consequents of his conclusions to be true, one needs only to look at Bradwardine’s next conclusions, which start: 38. If this is so, there are eight points, and no more, immediate to a point located in the middle of a plane surface. Whence this is manifest: there are twenty-six points, and no more, immediate to a point located in the middle of a body. 12 Latin in Murdoch 1987, 124. Here I have made my own translation to preserve word order and to replace Murdoch’s ‘infinite number’ with ‘infinitely many’, given that fourteenth-century authors do not in general take ‘infinite’ to be a number.

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part iv: arguments against immediate indivisibles 59 39. If this is so, no circular line has more than eight points, no finite straight line more than three, and no outermost surface of any body more than twenty-six.

Conclusions 38–39 are obviously inconsistent with conclusions 35–37. With the constant repetition of ‘si sic’ or ‘if so’, Part IV almost seems as if it might be the record of a disputation De obligationibus in which the respondent always has to keep in mind the positum of the given exercise, while accepting or rejecting other propositions posed to him.13 Given that the implications of an antecedent condition are tracked through several stages and then abruptly followed by another separate series of conclusions starting from the same antecedent but reaching an unconnected first conclusion, it is important to keep questioning the ultimate goal of the overall argument. This pattern is repeated in the transition from Part IV to Part V, in the latter of which Bradwardine argues against the supposition that there are finitely many indivisibles in a continuum (as found in Walter Chatton). Part V seems to start afresh, taking no account of the conclusions of Part IV. This could be reasonable if all the conclusions of Part IV are conditional on assuming that indivisibles are immediate to one another, whereas Part V does not make this assumption. But the last conclusion of Part IV is that in no continuum are atoms immediately joined, whereas the second conclusion of Part V is that if a continuum is composed of finitely many atoms, the atoms in that continuum will be immediately joined. On this basis it would seem to follow that all the following conclusions 59–114 of Part V are superfluous, since they assume or imply that indivisibles are immediate which, according to Part IV, is never the case. As far as I know, beyond the existence of three late fourteenth-century manuscripts containing all or part of De continuo, there is little to no evidence that any fourteenth-century scholar read or attempted to assess the demonstrative force of Bradwardine’s arguments against immediate indivisibles or against any other version of atomism. Even if this is the case, the fact that Bradwardine made a mathematical attempt to disprove several atomistic theories is worthy of note. On the other hand, given that at some time during the fourteenth century there was a strong move, maybe not only of nominalists, to assert that mathematical indivisibles are mental concepts and not real things, Bradwardine’s arguments may already have been out of the main stream when he made them. An important contribution that the fourteenth century offered to the project of applying mathematics to physical sciences was to open up the scope for creativity or originality in devising mathematical structures to match to nature. No longer were quantities always to be abstracted from quantities existing the natural world. One could make original hypotheses, say to explain observations by a continuist 13 On obligations see Spade and Yrjönsuuri 2017.

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60 thomas bradwardine and his contemporaries hypothesis or by an atomistic hypothesis, and one could elaborate hypotheses mathematically before comparing the results to experience. Whether or not a Galileo or some other early modern scientist benefited from the thinking of the fourteenth century need not control whether it is interesting to consider what was done in the fourteenth century in its own terms.

4. Atomism and Its Place in Medieval Philosophy and Theology In 2004 Christoph Grellard and Aurélien Robert organized a conference that led to a 2009 book with the title Atomism in Late Medieval Philosophy and Theology. According to Grellard and Robert: the principal tasks of this book are, first, to distinguish the singularity of fourteenth-century atomism, compared to other periods; second, to show that the understanding of the debates over this period is far more complicated than it is usually asserted in the old as in the recent historiography; and third, to ask whether fourteenth-century atomism is rather mathematical, physical or even metaphysical, as some of the contributors have tried to challenge the prevailing view about the mathematical nature of indivisibilism at that time. It is a difficult task to catch the essence of medieval atomism – if it exists . . . According to John E. Murdoch . . . fourteenth-century atomism presents some particular features that allow the historian of philosophy and science to isolate this period from other traditions . . . In this volume, we will focus on fourteenthcentury discussions of indivisibles and atoms, in order to take stock of the situation on recent historiography and above all to discuss Murdoch’s hypothesis, which is the prevailing one today. Indeed, all the chapters presented here try to respond, implicitly or explicitly, to the question: are debates on atomism in the fourteenth century purely mathematical and geometrical?1⁴

Murdoch did emphasize the mathematical nature of fourteenth-century discussions of continuity. But as he worked on the texts of the atomists that Bradwardine criticized, Murdoch also introduced the concept of ‘analytical languages’, by which 1⁴ Grellard and Robert 2009. Grellard and Robert may exaggerate Murdoch’s commitment to the purely mathematical and geometrical nature of fourteenth-century debates because of what he says in an early short paper: Murdoch 1974a, 29, ‘Bien que je n’aie fait jusqu’ici que résumer quelques-uns des problèmes qui sont au fond du débat sur l’atomisme au quatorzième siècle, j’espère que l’argument que j’ai présenté comme exemple peut donner une idée exacte de la substance de tout le débat. Le trait le plus frappant est le caractère mathématique et surtout géométrique de l’atomisme proposé et aussi bien des critiques contre cet atomisme. Il est évident que ce qui rendait nécessaire cet emploi continuel de la mathématique provenait de l’idée que l’atome médiévale n’avait pas d’extension.’

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atomism 61 he referred to frequently used techniques of analysis, both logical and mathematical, shared by fourteenth-century natural philosophers and theologians (Murdoch 1974b; Murdoch 1975). The existence of such common analytical techniques helps to explain the observation that seemingly prepackaged discussions of continuity and infinity often pop up in the midst of fourteenth-century theologians’ discussions of such topics as the motion of angels or the eternity of the world. Scholars had developed and continued to develop a common analytical language or technique for solving problems involving infinity and continuity, and this technique was ready for use when they participated in disputations, whether philosophical or theological. Authors were not necessarily more interested in continuity and infinity than in God or angels, as is sometimes suggested, but they were trained and practised in the moves to make in discussing questions related to the analytical languages, including the language of infinity and continuity. How, then, did infinity and continuity become subjects of disputation in the fourteenth century? And how did Bradwardine’s arguments against immediate indivisibles relate to Harclay’s advocacy of such immediate indivisibles? Harclay had addressed problems of continuity and infinity in his Ordinary Questions, especially in question XXIX, where he addressed the possible future eternity of the world. In a 1981 paper on Harclay’s two ordinary questions on the eternity of the world, Murdoch wrote: Taking both quaestiones into account . . . it is fair to say that Harclay makes but two basic claims: (1) That there can be unequal infinites, and (2) that continua are composed of an infinite number of indivisibles that are immediately next to one another. It is to these two contentions that he devotes the most substantial part of his efforts, and not to the problems of the possibility of a past or future eternal world as such, even though he clearly admits the possibility of both . . . (Murdoch 1982a)

When one looks at the recently published edition of Harclay’s Ordinary Questions, it’s hard to miss a sudden switch in his reasoning as he comes to the issue of continuity. In the midst of writing about the future eternity of the world, Harclay says: 55. Furthermore, another reply can be made specifically to that made on the authority of Grosseteste, that there are more points in a larger continuum than in a smaller one. All arguments that prove that a continuum cannot be composed of indivisibles are in opposition to this point of view [namely, opposed to the view that there may be more indivisibles in

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62 thomas bradwardine and his contemporaries one continuum than another], for also prove that there are not more points in one continuum than in another . . .1⁵

When John Murdoch focused on the sections of Harclay’s work that relate to continuity and infinity, it appeared to him that the thesis to which Harclay had the greatest attachment was that one infinite can be larger or smaller than another (Murdoch 1982a). Looking more broadly at the context, however, it appears that Harclay kept coming back to the issue whether one infinite may be greater than another of the same type partly because it was an integral part of a proposal that Robert Grosseteste had made that God can measure the extent of continua by the infinitely many indivisibles contained within them – which makes no sense if there are no infinities of indivisibles of different sizes in continua of different sizes. Let us look at Harclay’s question in more detail, then, and at what he says about Grosseteste. In his notes on Aristotle’s Physics, Robert Grosseteste had been concerned about measures of length (whether of time or distance) given the fact that an unknown length is measured by comparing it to a known length. What could be said about measuring length if there were only one length in the cosmos? His proposal was that although humans cannot measure lengths by the infinite numbers of indivisibles contained in them, God can intuit the indivisibles and can use them to measure lengths, a continuous line containing twice as many indivisibles as its half. Although humans cannot compare infinities of indivisibles, he said, God could do it.1⁶ Several passages from Harclay’s question XXIX are relevant here. The first reveals the question that Harclay was interested in before coming to his question on the continuum and the second contains Harclay’s main argument that indivisibles are immediate to each other in continua. Harclay’s question is ‘Could the world last eternally into the future?’ There is one principal argument: 1. That it can : a mobile will be eternal, therefore both time and motion could last eternally. Proof of the consequence: a mobile, as long as it is a mobile, can be in motion, for it would not be a mobile unless it were apt by nature to be in motion. Proof of the antecedent: celestial as well as elemental

1⁵ Harclay 2008. Here and in the following notes the numbers in boldface are the paragraphs of question XXIX. Henninger, the editor, thanks John Murdoch, who allowed him to use his transcription and translation of ‘Harclay’s difficult Question XXIX on the continuum and the infinite’, and notes that it was Murdoch who discovered the best manuscript of question XXIX (Tortosa, Catedral de Tortosa, 88), which is the question of interest here. 1⁶ In a sense Grosseteste belongs to a different context than the others Bradwardine addresses, because he was thinking in Platonic terms and had the idea, for instance, that light is the first corporeal form, which by its expansion causes definite extensions to exist within previously amorphous matter. See Panti 2014.

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atomism 63 bodies will last eternally according to the tenets of faith, and every body is mobile. Furthermore, a proof that time will be eternal: motion will be eternal, consequently so will time. Proof of the antecedent: for the bodies of the blessed shall undergo local motion in the highest heaven through their endowment of agility: The just will shine forth and shall run to and fro like sparks among the reeds. (Wisdom, three). Furthermore, even the bodies of the damned shall undergo a perpetual motion of alteration in hell: they shall pass from the waters of snow to extreme heat. But motion occurs in time, therefore, .

Unlike the norm, this principal argument is in favour of the conclusion that Harclay eventually advocates. Among the arguments that the world will not be eternal in the future is the argument: 8. Furthermore, if the world will last eternally, there will always be more revolutions of the moon than of the sun; yet both will be infinite in number, and hence one infinite will be greater than another.

To this Harclay responds that one infinite can be greater than another. In support of this conclusion he cites Robert Grosseteste: 32. Furthermore, the authority of Robert, the Bishop of Lincoln, speaks for this view in the chapter on time in four on the Physics. For he says that the primary measure of time by means of which Nature first measures a quantity of time cannot be any quantity known to us . . . Therefore, the primary measure, as he says, is a certain infinite multitude of instants contained in time. For there is a first and minimal element in numbers, although not in continuous things. And God, who knows this multitude, measures other time by means of its repetition. For one can find ratios of the same type among infinite numbers as among finite numbers, as for example of a double to a half, and so on. Therefore, since one time contains a more numerous infinite of instants than another, it will be greater than it. 33. This opinion of that most learned man is one of very great importance and true in my judgement. The reason for this is his understanding of a measure. According to Aristotle, in ten of the Metaphysics, the nature of a measure implies that it be a minimum and indivisible in so far as it is a measure. Now, the first essential element of the measure of any quantity is number; for a continuous quantity measures another only by means of number, as, for example, a proportion of the total or the number of times the one contains the other . . .

Harclay continues to talk about measures of continuous quantities in a way that is significant but too long to be quoted here. He concludes paragraph 33:

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64 thomas bradwardine and his contemporaries Consequently, a perfect measure of a continuous quantity exists only by virtue of an indivisible of discrete quantity, namely, by virtue of a unit, and similarly by means of an indivisible of continuous quantity, namely a point. Moreover, no quantity can be perfectly measured unless we know how many indivisible points it contains. Yet, since these are infinite in number, they cannot be known by a creature, but only by God, who has disposed all things in number, weight, and measure (Wisdom eleven). 34. This is the view of that man, and I hold it to be the truest . . .

Harclay then quotes in detail from Grosseteste’s comments on Aristotle’s Physics, where Grosseteste’s idea of God’s measurement by infinitely many indivisibles is applied to numbers, time, motion, and so forth. God comprehends infinities that are beyond human comprehension: 38. [continuing to quote Grosseteste’s Physics commentary]: ‘What is more, I am bold to say that to that very God of whose wisdom there is no number, every infinite number is finite, more so than two is finite to me. For finite to him is the infinite number composed of all even , the infinite number composed of all odd , and similarly all infinite numbers that can be infinitely divided. For, just as those things which are truly finite in themselves are infinite to us’ (that is, ) things that have no termination as far as we are concerned and are thus not terminated by an act in se do have a termination relative to God.

And then suddenly Harclay brings up, as an example, the infinite divisibility of a continuum: [38 cont.] for example, the division of a continuum has a termination relative to divine cognition, just as if were only finitely divisible, ‘so those things which are truly infinite in themselves are finite to him. He has created, moreover, all things in number, weight, and measure. He is the first and most certain Measurer; by means of infinite numbers, to him finite, he has measured these lines which he has created. He has measured and counted the line of a cubit by some infinite number to him both definite and finite, a two-cubit line by a doubly infinite number, and a half-cubit line by a number half as infinite. Moreover, there is one infinite number of points in all lines of one cubit, by which number he most exactly and determinately measures all lines of a single cubit. But then how is the first measured line measured and counted in the first place? I suppose that by the infinite number of points in that line, which number of points – finite, however, to the one measuring – is not found in any other greater , and a smaller in every smaller one. Only he to whom infinite numbers are finite, and to whom

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atomism 65 one infinite number is large and another small, can measure in this manner. Thus, if the infinite is finite to no created being, no created being can measure in this way. Moreover, it is certain that the infinite is not finite to our intelligence; just the converse is the case. Whence we cannot measure in the aforesaid manner . . . ’

Then Harclay continues in paragraph 39 to quote Grosseteste on the way in which God measured time by infinite indivisibles before the creation of the cosmos and the beginning of time as we know it. After having finished with his arguments based on the authority of Grosseteste, Harclay states and replies to arguments against the inequality of infinities (paragraphs 40–54), at the conclusion of which he switches the argument about inequality of infinites from time and motion to permanent continua and, in particular, to lines, ending with geometrical arguments that are often quoted from al-Ghazzali’s Metaphysics via John Duns Scotus, arguments which are at least partly responsible for John Murdoch’s characterization of fourteenth-century atomism as basically mathematical. Then he goes on to passage 55, which I have already quoted above. Passage 55 is obviously a sudden turning point in the argument. According to Harclay, the inequality of infinities and the composition of continua from indivisibles stand or fall together. Grosseteste’s doctrine that God (but not humans) can measure continua by infinitely many indivisibles implies that infinities can be of different sizes, so Harclay is led to argue that, yes, infinities can be of different sizes. But then this combination of doctrines implies that indivisibles can be immediate to each other, and so Harclay argues that, yes, indivisibles can be immediate to each other. The sudden switch here from discussion of the future eternity of the world to arguments about the relation of indivisibles to continua is worth further consideration. Were fourteenth-century Aristotelians really concentrating on the analysis of continuity or was their attention on theological problems? Several historians have suggested that fourteenth-century authors were more interested in infinity and continuity than they were interested in theology. When these authors elaborated analytical languages and then applied them in philosophy and theology alike, did they assume false equivalences? Bradwardine in De continuo repeatedly assumed that the relations of continua and invisibles in geometry were isomorphic to the relations of continua and indivisibles in natural philosophy. But what about the thought that De continuo as a whole could be said to show that natural and mathematical continua are not isomorphic? Let us return, then, to Harclay’s ordinary question XXIX. In passage 68 of his reply, Harclay says in defence of his view that indivisibles are immediate: 68. One of the arguments for this view is, it seems to me, inescapable. It is impious to say that God does not now actually perceive every point that can be designated in a continuum. Take, then, the first point of a line, inchoative of the

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66 thomas bradwardine and his contemporaries line; God perceives that point and any point other than this in this line up to the more immediate point [i.e., the point that God sees that is more immediate to the inchoative point] (Accipio igitur primum punctum in linea incoativum lineae; Deus videt illum punctum et quodlibet aliud punctum ab isto in hac linea usque ad illum punctum immediatiorem quem Deus videt). Does another line fall or not? If not, then God perceives this point to be immediate to the other one. If such a line does intercede, then since points can be assigned in the line , these mean points had not been perceived (non erant visa) by God. Proof of this consequence: by hypothesis (positum), a line falls between this first point and any other point that God perceives; thus, you are maintaining that God was not perceiving (Deus non videbat) the mean point just now discovered.1⁷

This passage or something like it as paraphrased by William of Alnwick has repeatedly been analysed as containing a logical error, but as it appears here the conclusion that points are immediate appears to be a direct consequence of God’s perception of every point (points and lines being discussed because they are assumed to be a simple case for the analysis of the relations of any continua and indivisibles). Bringing in what God sees or perceives makes a difference, because for God there are no stages in perception and everything is actual. For God it makes no sense to argue that when you have mediate points you can pick two points and then infer that there is a point between them. Where has this third point come from? God supposedly eternally unchangingly sees all the points of the line. Did he not initially see the third point? It follows that in God’s sight points are immediate—there are no points not seen by God. Or? Or it follows that to assume that God sees all the points in a line together with assuming that the points are mediate leads to a logical contradiction. One must conclude, then, either that God does not know all the points of a line (or know all the indivisibles in any continuum) or that the points are immediate. Some medieval scholars concluded indeed that God does not know particulars.1⁸ Near the end of question XXIX, paragraph 121, Harclay extols God’s ability to know

1⁷ I have modified the translation slightly. For God’s relation to questions of indivisibles and continua, see Sylla 1997; Sylla 1998a; Sylla 1998b. 1⁸ Cf. Richardson 2015, §3.4 ‘Avicenna considers God’s immutability to entail that his knowledge, volition, and action are also immutable. He denies that God knows particular things except in a universal way.’ This, of course, can have far-reaching consequences. Richardson goes on, ‘So, according to Avicenna, God cannot act otherwise than he does; furthermore he cannot plan or execute any particular event. So God cannot cause a miracle.’ I have quoted Richardson on Avicenna here simply because it was ready to hand, but there were fourteenth-century Latin Aristotelians who held similar views, for instance, Pierre Aureoli. See Sylla 2005 (note 8), p. 267.

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gerard of odo on infinity and continuity 67 minuscule quantities, raising issues that might be related to possible conceptions of continuity and infinity: Moreover, God knows a quantity so small that, were we to subtract it ad infinitum, we could not exhaust a quantity of one foot. Proof : a one-foot quantity can be infinitely divided into equal parts, as has been said above, just into parts of the same ratio, and God knows all such possible . Consequently, just as, were we to go on dividing ad infinitum, there would be no end that arrives at a quantity as small as that which God actually perceives, so, in the same way, if God were to present us with that small quantity and we were to take it away an infinite number of times relative to us, it would never exhaust a quantity of one foot. Nor, if we that given quantity an infinite number of times relative to us, and join and collect what we have taken together, we would never reach a one-foot quantity as a sum.

So Harclay’s investigation into what God knows of the universe leads him to think of different orders of magnitude for infinities or infinitesimals, and although it does not lead him to a new fully articulated theory of infinities, it raises some possibilities. As a historical generalization, then, it might be said that fourteenthcentury disputations concerning infinity and continuity raised many issues and contained some insights, even if they did not lead to any sort of revolutionary change of consensus. Most historians have thought that when late medieval scholars thought that one infinity could be larger or smaller than another infinity in cases like the infinitely many points in a whole line vs. the infinitely many points in half the line they were failing to understand the correct mathematics of continuity and infinity that Georg Cantor would later propose. But rather than focusing on what the scholars of the fourteenth century did not achieve, let us look further into what they did achieve, using, as was their wont, logic more than mathematics, or perhaps one might say using mathematical logic.

5. Gerard of Odo on Infinity and Continuity The papers included in Atomism in Late Medieval Philosophy and Theology, mentioned above, show that there were at least some fourteenth-century individuals who asserted the real existence of physical as opposed to mathematical atoms. Historians are currently inquiring further into the works of these ‘atomists’. In more than one case an atomist was condemned and his writings burned—for instance this happened to Nicholas of Autrecourt at Paris and it happened to John Wyclif at Oxford. While the complaints against these authors may not have primarily targeted their atomism, the burning of their writings, as well as the accompanying controversy surrounding them, has slowed the efforts of historians

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68 thomas bradwardine and his contemporaries to study their positions on atomism and continuity in detail. Progress, however, is currently being made. One atomistic author copies of whose works have been identified is Gerard of Odo, a Continental atomist not named by Bradwardine, but whose work Bradwardine might have known. Odo argued that the division of continua comes to an end with extended indivisibles that cannot be divided further.1⁹ Sander de Boer argues that Gerard of Odo was not simply thinking in terms of Aristotle’s Physics, Book VI, but was instead committed to indivisibles for other reasons. When Odo describes the means by which such indivisibles can combine to cause something extended he uses concepts inconsistent with Euclidean geometry. In 1328 Gerard of Odo lectured as the Franciscan bachelor of the Sentences at Paris, but he had lectured on the Sentences in the Franciscan house of studies at Toulouse already in the 1310s (Sylla 2011). In preparation for these public disputations, he likely studied and made use of various logical and mathematical tools—John Murdoch’s ‘analytical languages’—including techniques of analysis related to infinity and continuity. Extant manuscripts contain disputations by Gerard of Odo on the question whether a continuum is divisible in infinitum (Utrum continuum sit divisibile in infinitum; see de Boer 2009; de Boer 2012). When Odo lectured and disputed on the Sentences at Paris and then wrote up the results, he used his analysis of the question whether a continuum is divisible in infinitum once in connection with Book I, distinction 37 and another time in connection with Book II, distinction 44. As Sander de Boer reports: In both distinctions, the immediate context for discussing the structure of continua is the question whether God is omnipresent in the universe by means of his presence (praesentia), his essence (essentia) and his power (potentia). The position that Odo defends in these two texts is the same, namely that all continua are composed of a finite number of indivisibles . . . (de Boer 2012)

Was what Odo had to say on the question whether all continua are composed of finitely many indivisibles shaped by the questions from the Sentences within which it was later included or was his first writing on the continuum independent of his questions on the Sentences? In his commentary on the Sentences, Book I, Distinction 37, Odo asks, ‘Whether God is [sit] throughout every part of space by [his] presence, essence and power?’ The first principal argument is: The first [of these questions] is to be answered in the negative. For any being that could simultaneously exist in all parts of space could know how many are all the parts of space and of a continuum by the intuition of his presence, that is, of [his] knowledge, could designate each and every [part] through the permeation of his 1⁹ de Boer 2009; see also de Boer 2012. Some of what Odo writes may lie behind Bradwardine’s use of superposition and other geometrical concepts in arguing against the atomists.

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gerard of odo on infinity and continuity 69 essence, and could separate all of them from one another by the employment or exercise of his power. But neither God nor any being is seen to be able to accomplish this. For all these known, designated and separated parts are either divisible or indivisible. If they are divisible, then there remain some parts that are neither known, designated nor separated [from one another]; consequently, not all [parts] are known, designated or separated . . . If, on the other hand, these parts are indivisible, then they will not be parts; for a continuum is not composed of indivisibles, as is proved in [Book] VI of the Physics.2⁰

Odo closes this part of his commentary by comparing Aristotle’s opinion to his own. Two of the comparisons are: [5] The fifth thing implied by Aristotle’s view is that the possible divisions of a whole continuum are no more numerous than the possible divisions of one part of it, indeed, that [for example] there are no more instants in a hundred thousand years than in one hour or no more parts in a hundred thousand leagues or in the whole world than in a millet seed, since, [in all these, the totals] on both sides are infinite in number, and no multitude or plurality can be conceived to be greater than an infinite multitude. The other view [Odo’s view] implies the opposite. Yet the first alternative seems completely irrational, the latter, however, quite consonant with reason. [6] The sixth inconsistency which follows from Aristotle’s view is that God, even with his omnipotence, cannot know exactly how many parts he has placed in a continuum. This is apparent from the following: If the Divine Intellect is to know how many parts He has placed in a continuum, He must also discern the resolution and division of the whole continuum into all of its parts. However, given Aristotle’s view, this is impossible for him to accomplish, since the division would have to be terminated at the ultimate parts of the whole for this to be possible. Therefore, if, [as Aristotle maintains], every part is further divisible, the division is not yet terminated, and thus one could still not have in hand how many parts there are, since any part includes again an infinity of parts and any of that infinity yet another infinity. The opposite [Odo’s] view implies that God knows a definite number of parts that He has placed in a continuum. However, this seems more possible than the first alternative. For it appears most irrational that God should be able to have placed the totality of those parts in existence and that He neither has, nor can have knowledge of a definite number of them.21

2⁰ John Murdoch’s unpublished draft translation, slightly edited. Based on his unpublished text from MSS Valencia, Cated. 139, ff. 120v–125v and Naples. Bib. Naz. VII, B25, ff. 234v–244v. 21 John Murdoch, unpublished translation.

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70 thomas bradwardine and his contemporaries Odo then replies to the principal argument at the start of this section: One must reply . . . that God does know all parts of a continuum by the intuition of his knowledge, that He can designate each and every one of them by the permeation of his essence and that He can separate them from one another by the employment of his power. Consequently, the minor premise of the argument is false. Moreover, when it is proved that these parts would be either divisible or indivisible, I say that they are indivisible and that there is no inconsistency in the parts of a continuum being such.22

In his commentary on the Sentences, Book II, last question, Gerard of Odo repeats this conclusion about God’s presence in every part of the universe. So Odo, like Harclay, is concerned to understand the relations of God to the cosmos. The science or analytical technique of infinity and continuity is elaborated to address such an issue, but it also had an independent existence.

6. Gerard of Odo and Walter Burley There is another context in which Gerard of Odo interacted with Walter Burley, like Bradwardine an early Oxford Calculator. While at Oxford, Burley had written a commentary with questions on Aristotle’s Physics, but he revised it after teaching the Physics at Paris. As well as likely interacting at Paris, Burley and Odo probably interacted in disputations at Toulouse before 1320. In Burley’s revised, Parisian Expositio of the Physics, Gerard of Odo is named in the margins of some manuscripts (Maier 1964, p. 462). In this interchange there is another way to apply mathematics and logic to continuity, one which is worthy of our attention. In this case mathematics and logic are applied to framing a rigorous way, or to ‘model’, if this is not too loaded a term, Aristotle’s second definition of what it means to be continuous, namely that the continuous is infinitely divisible. No matter how many times a continuum has been divided the resulting parts are themselves still divisible. For a long time scholars had proposed that a continuum can be divided into proportional parts in infinitum. It was commonly said that a continuum cannot be divided into infinitely many parts of the same quantity, because then the whole continuum would be infinite, but it can be divided into infinitely many proportional parts that together add up to no more than the original quantity. This consideration was pushed further, however, by distinguishing between parts that are physically separated from each other and parts that are only conceived of, but not physically separated. One has the two halves of a whole which are distinguished in thought. Then each of these halves can be divided in two, 22 John Murdoch, unpublished translation.

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gerard of odo and walter burley 71 so that one conceives of four quarters of the original whole. Dividing each part in half again one has eight eighths, and so forth. As William of Ockham argued, these are actual parts, even though they are not physically separated—are not the two halves of a plank of wood actual?23 Are not the four quarters actual? Then the infinitely many parts that could be produced by dividing in half ad infinitum are actual, but there is no ultimate division terminating the infinitely repeated division. Although the human mind cannot conceive clearly the result of an infinitely repeated division, it can conceive how the infinite division is carried out according to this description. Logic plays a role in this argument. The infinity under examination here is said to be not the infinity of one thing, which would be a categorematic infinite, but an infinity of many things, which in relation to each other are said to be syncategorematically infinite.2⁴ As already mentioned, in Bradwardine’s De continuo, categorematic and syncategorematic infinities are defined and, in the Torun manuscript, commented on at length even though they are not used in what follows. As Bradwardine defines what he calls a relationally privative (or syncategorematic) infinite, a syncategorematic infinite in the plural in discrete quantities is some finites and finites more than these, and finites more than the more, and so forth without ultimate terminating end, and these are some and more and not so many but that more (non tot quin maius).2⁵ As fourteenth-century logicians understood the situation, the effect of syncategorematic terms is to change the supposition (reference) of the categorematic terms in the same proposition such that one not technically correct proposition is expounded and said to be equivalent to two or more technically correct propositions in which the term at issue has different types of supposition (reference) in different propositions. In the case of infinite divisibility of a continuum what is syncategorematically infinite is not the continuum, but the infinite possible divisions that can be made producing infinitely many parts. The divisions and the parts can be considered as discrete, though the continuum that is divided is, of course, not discrete. What is taken in a syncategorematic sense in the proposition is not the term ‘infinite’ itself, but the term ‘without end’ included within the meaning of the word ‘infinite’ which 23 For the existence in act of the parts even though they remain unmoved, see Murdoch 1982c. 2⁴ In his Sophismata, f. 86r, William Heytesbury writes, ‘Et est terminum teneri sincathegorematice sic pro omnibus suis suppositis supponere ut per illam suppositionem non contingat illum terminum predicari de aliquo vel aliquibus suis suppositis, sicut cum infinita sunt finita, notum est quod iste terminus infinita supponit pro finitis quia pro duobus tribus et quatuor et sic in infinitum et tamen propter illam suppositionem non contingit illum terminum infinitum predicari de aliquibus illorum, quia non sequitur infinita sunt finita ergo ista finita sunt infinita vel ista finita sunt finita.’ Sylla 1981, p. 636n29. 2⁵ My translation from Murdoch’s unpublished edition: ‘Tamen infinita privative et secundum quid in plurali et in quantitate discreta debet intellegi diffiniri hoc modo: Aliqua finita, et finita plura istis, et finita plura illis pluribus, et sic sine fine ultimo terminante; et hec sunt aliqua et plura, et non tot quin plura.’

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72 thomas bradwardine and his contemporaries is equivalent to ‘many without end finite’. So the proposition ‘infinitely divisible is a continuum’ should be understood as equivalent to two propositions: ‘1) A continuum is divisible some number of times or into so many parts; and 2) it is not divisible so many times or into so many parts (tot) without (quin) being divisible more times or into more parts.’2⁶ Here the repeated division into proportional parts, as described above, provides the infinitely many parts for which the terms in the definition of the syncategorematic infinite can have supposition. As may be apparent, exposition of the proposition ‘infinitely divisible is a continuum’ in the syncategorematic sense is taken to be analogous to the exposition of the proposition ‘infinite are the numbers’. In both cases there are one or more examples of the given property and then an unending iterative method for identifying more. So there is some low number (numbers were often understood to begin with 2) and for every number there is a successor number which can be obtained by adding 1. Given that one has 2, then one has its successor, namely 3, and so forth, and the whole series of numbers is infinite or has no end because there is always a successor to each number. In fact it should be noted that medieval mathematicians customarily grounded the infinity of the integers on the infinite divisibility of the continuum, so that however many parts of the continuum may have been counted, a greater number could be obtained by dividing further and counting further smaller parts of the continuum. This seems to suggest that were it not for the infinite divisibility of the continuum, there might have been only finitely many things to count in the universe. In the mind of the mathematician (or for God who is outside of time) all these parts or cuts are simultaneous. In Book VI, text comment 10, of Burley’s final (Parisian) Expositio of Aristotle’s Physics, the third question asked is whether the continuum is divisible into always divisibles. Burley begins his answer: And it seems that not, because in whatever has a minimum, division does not proceed in infinitum in those. But in natural continuous things there is a minimum. Therefore in them division does not proceed in infinitum and consequently a natural continuum is not divisible in always divisibles.2⁷ 2⁶ For the effect of the word quin in the definition of a syncategorematic infinite, see Spain 1992. Quin is said to connect in consequence two negatives. Thus one may say that because all humans are animals, there is not a human if there is not an animal. For the syncategorematic infinite, one says that if there is a syncategorematic infinite, there are so many (tot) (e.g., parts), and there are not so many (non tot) if there are not more (quin maius). Here quin does not signify a consecution (consecutionem) in time, but an inference (illationem). Peter of Spain explains, ‘Everything that is inferred, insofar as it is inferred, follows, and everything that implies, insofar as it implies, is antecedent . . . That is why the word ‘quin’ signifies consecution via an inference or in an inference.’ 2⁷ Burley 1501, 177rb. In the 1972 reprint by Georg Olms Verlag, this edition has been given the title (not appearing in the 1501 edition) In Physicam Aristotelis Expositio et Quaestiones, a title so widely repeated that I have been unable to uproot its use. (The earlier Oxford commentary extant in only one manuscript is said to be an expositio and quaestiones.) See Sylla 2002.

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gerard of odo and walter burley 73 Aristotle, however, disagrees. Some (identified in the margins of some manuscripts as Gerard of Odo) say that no finite continuum is divisible into always divisibles. And they prove this as follows: if some continuum were divisible into always divisibles, at some time (aliquando) it would be divisible in infinitum, which is false. The first argument that Odo gives for this is: if a magnitude were divided into infinitely many parts, it would be infinite, but no magnitude is infinite. This sets off a give and take. First one says that the argument holds if the parts are actual, but not if they are only potential. Against this it is replied that the parts are actual, not potential. Next it is argued that if there were infinitely many parts of the same size, then taken together they would make an infinite magnitude, but this does not follow if there are proportional parts rather than parts of the same size. Against this it is argued that if in a continuum there may be infinite parts of the same proportion, then there may be in the continuum infinite parts of the same quantity. By the time of Burley and Odo, to make use of proportional parts was a common move in scholastic disputations, but the standard move is to take the first half, and then half of what remains, and half of what remains after that, so that the parts get smaller and smaller and when the division is continued ad infinitum, the sum of all the parts is the original quantity. Here, however, Odo proposes to divide the original extension into halves and then divide each of these halves into halves, producing four quarters; then to divide each of the four quarters producing eight eighths; then to divide the eight eighths producing sixteen sixteenths, ad infinitum. At each step all the newly noticed parts such as quarters or sixteenths are equal to each other, and so forth, and the parts of different sizes are superimposed on each other. Here the text in the 1501 printing of Burley’s Expositio appears to be faulty, because it says that all these parts taken together will be infinite in act. But Burley’s reply to Odo takes the necessary next step, which is to say that all these parts are not actually separated from each other but all remain in the same place. One thinks of the two halves and then one thinks how these two halves together contain four quarters, and eight eighths, ad infinitum, all in the same place. The heavenly spheres are divisible in this way, Burley says, even though their parts cannot be separated and moved away from each other. All continua are divisible into divisibles in infinitum in the way that the heavens are divisible. Therefore every continuum is divisible into always divisibles, understanding by ‘to be divisible’ to have part outside part. Further in reply to principal arguments, Burley concedes that a continuum is divisible in infinitum, and that in this case the continuum would have infinitely many parts. It will not follow that the continuum is infinite in act, however, because it is not composed of infinitely many parts of a certain given quantity or of parts which are totally outside each other. If something were composed of infinitely many parts of the same given quantity of which any one was totally outside the

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74 thomas bradwardine and his contemporaries other, it would be infinitely extended, but this is not so given the mode of division he has described. This follows, he says, from the definition of the syncategorematic infinite: But neither designating nor determining the quantity, I say that in any continuum there are infinite parts of the same quantity of which any is totally outside the other, because there are not so many parts of the same quantity in any continuum but that (quin) there are more parts of the same quantity [to each other] in the same continuum . . . It is true that if a continuum is divisible in infinitum in parts of the same proportion, it will also be divisible in as many parts of the same quantity, and this is true because it is not divisible in so many parts but that it is divisible in more (verum est quia non in tot quin in plures) . . . Whence it is not divisible into many parts of the same quantity, unless it is divisible into twice as many parts of the same quantity, because in the halves of those parts, and again in halves of the halves and so forth in infinitum. And if it were said that what is divisible in infinitum in parts of the same quantity of which any is outside the other is infinite, I say that it is not necessary that every divisible in infinitum in parts of the same quantity is infinite in act, but every divisible in infinitum into parts of such and such a quantity (in partes tante vel tante quantitatis) of which any is totally outside the other is infinite in act.

To understand this argument one has to differentiate between different groups of parts of the same quantity. The parts that are of the same quantity are the parts that are all the same fraction of the original, such as quarters or sixteenths. On the other hand, there is no maximum number of parts of the same quantity, because given any number of parts that are the same fraction of the original quantity such as sixteenths, one can have twice as many parts at a new level by conceptually dividing in half the parts previously considered. So there are sixteen sixteenths of the same quantity, but more than this there are thirty-two parts that are 1/32 of the original. There is no final level of division, the finer and finer divisions into smaller and smaller parts continue without end. One might imagine zooming into the model, continually magnifying the scale, so that finer and finer divisions come into view, all with the same structure. It appears to me that the different levels of finer and finer division might be considered as an additional dimension. As further division is considered the parts on a single level become a larger and larger multitude, while at the same time the levels of division keep increasing ad infinitum.2⁸ This model of infinite divisibility using the concept of the syncategorematic infinite is ingenious, but even more is achieved because it is a model in the imagination of mathematicians, not something in the external world (what is in

2⁸ Perhaps this is something like fractals, where the situation appears the same at all levels.

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infinity and imagined cases 75 the world might be the original whole). The increasingly many divisions and parts are not achieved by a process over time of making one division after another, but rather all the infinitely many divisions are brought about at once conceptually, by definition, or by positing that there are some conceptual divisions and resulting parts and not so many divisions or parts but that more. As Bradwardine says, the definition of the syncategorematic infinite is not the definition of a thing (res), but the definition of the meaning of a term (diffinitio quid nominis).2⁹ There is more in this text than can be described here. For instance, there is a second question whether something divisible in infinitum could be completely divided in act (potest esse actu ultimate divisum). Burley concludes that a continuum cannot be divided according to every way in which it is divisible (divisum secundum quodlibet secundum quod est divisibile). Not every possibility can be reduced to act in the same instant. Not even all possibilities having no order among themselves can be reduced to act in the same instant. To say otherwise is not even probable. Many things are possible in the divided sense which are not possible together or at the same instant (Burley 1501, f. 178rb). I think this means that although the syncategorematic infinite is not one thing, it refers to some things that, because of their relations to each other, are actually infinitely many.

7. Infinity and Imagined Cases in Bradwardine’s De Causa Dei Although I am attempting not to see fourteenth-century speculation about infinity and continuity through the eyes of Cantor or other modern conceptions of infinity and continuity, there is one last example of fourteenth-century work on infinity and continuity that is too much fun to resist, although I will paraphrase for the sake of economy. In his huge theological work De causa Dei, Thomas Bradwardine made use of quasi-mathematical imaginary cases when he argued against the eternity of the world on the grounds, for instance, that if the world were eternal one could match one to one the infinitely many souls of all the popes or other leaders who had lived in an eternal world to the (presumably larger?)3⁰ infinitely many bodies of all the 2⁹ ‘Et hic videmus mirabile, quia diffinitio de suo diffinito minime predicatur, diffinitum de seipso, nec diffinitio de seipsa, quia si infinitum privative secundum quid est quantum finitum et maius isto, et cetera, tunc aliquod quantum est maius se. Alia duo simili modo patent. Et hoc est quia est diffinitio quid nominis et non rei, et illud nomen non est signum alicuius rei determinate, sed est consignum. Sicud etiam diffinitio chymere non predicatur de suo diffinito, et cetera’. John Murdoch unpublished text. 3⁰ Bradwardine 1618 Book I, Cap. I, Coroll., Part 40, pp. 119–123. ‘Contra Aristotelem astruentem mundum non habuisse principium temporale, & non fuisse creatum, nec generationem hominium terminandam, neque mundum seu statum mundi presentem ullo tempore finiendum. Contra Anaximandrum quoque Platonem, Anaxagoram, Empedoclem, & Maurorum vaniloquos circa idem . . .’ p. 121C, ff. Compare Lukács 2013: Lukács includes the beginning of the text paraphrased here, but breaks before the introduction of heaps.

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76 thomas bradwardine and his contemporaries humans who had ever lived. He begins by saying that he will disprove the eternity of the world by more or less mathematical arguments (videtur quod ista sententia posset per alias rationes quasi-mathematicas quodammodo mathematice reprobare). If there are infinite souls, he says, they would have had, would have, or will have infinitely many individual bodies (corpora propria infinita). Take, therefore, those infinitely many bodies, or some infinitely many of them (illa corpora infinita, vel aliqua infinita ex eis) and put them in a series such that each body has a body preceding it and also a body following it next in order with no body in between. Let that series of bodies be ordered in a real or imagined straight line beginning from the centre of the horizon and extending infinitely to the west. Or let them be arranged along a gyrative line31 from the centre through proportional parts of the semidiameter (i.e., radius) of the horizon to the circumference, carried around infinitely many times. Or they could be arranged according to unequal proportional parts along some finite straight line, for instance, the radius of the horizon to the direction of the sunset. Let no one be disturbed by the corpulence of these bodies. They are posited truly or imaginarily in that infinite straight line, true or imaginary, or in that gyrative line, or finite straight line, however you wish to situate them. And let the later bodies always be smaller as is necessary, or in their place put surfaces, lines, or points (which, however are called bodies), in the aforesaid way. This argument, he explains, solely concerns multitude and cares nothing about size. These things thus arranged, whether by the omnipotence of God truly or by the fictional imagination (His itaque per Dei omnipotentiam vere vel per imaginationem ficte ita disposita), then arrange also the souls against the bodies, each soul corresponding entirely to its body. Let A be the multitude of all the souls and let B be the whole multitude of bodies. Therefore multitude A of single units, and B of single units and all to all and vice versa correspond equally. For any soul has its single body and any body has its one proper soul . . . So that this is clearly demonstrated, and with the souls and bodies arranged as supposed, whether by God’s omnipotence or by imagination, let it be done in this way, the first soul is allocated to the first body, the second to the second, and so forth. When this distribution is completed, any soul will have one body and any body will have a unique soul . . . To make the argument clearer, suppose that the world and the human species are eternal in the past and that eternally there is a single ruler of all, let’s say an emperor or Pope – all of whom, for the sake of this example, will be called Popes. Assume according to the hypothesis that infinitely many humans have preceded and similarly infinitely many Popes, which means that now there are, only considering Popes, infinitely many Papal souls, which, if they are distributed

31 See the description of a gyrative line in note 8.

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infinity and imagined cases 77 to the bodies in the previous way will suffice to animate each and every body. Indeed they would suffice to give to each body a thousand souls, and however many souls are granted to each. This can be demonstrated most clearly as before. A similar result could have been found above if the second soul was given to the first body, the fourth soul was given to the second body, the sixth soul to the third body, and so forth. If this was done, in the end every body would have a soul, but there would be left infinitely many [unmatched] souls. Who could bear such repugnant and unseemly superfluity of souls for animating so many bodies when it would be entirely sufficient to have fewer? (Sed quis inconvenientias et repugnantias tantas feret? Ut quid etiam, quaeso, tanta et tam excessiva superfluitas animarum pro tot corporibus seu hominibus animandis, quando plene sufficiunt pauciores.) Even Papal souls would suffice or as many fewer as you wish. Or [if you proceeded inversely], who could bear also such an excess of superfluous bodies for receiving so many spirits? This is not becoming to the most wise God (or even to nature, as will be shown). Then Bradwardine proposes another case: Suppose all the bodies are piled in one heap without order (Adhuc autem positis corporibus B secundum ordinem praetaxatum, vel sine ordine ad cumulum unam confusam), and let the souls A come and each assume one of these bodies, and never two. And all will occupy as many bodies as is necessary and not more. When this has been done there will be some infinite multitude of animated bodies and none (quo facto erit aliqua multitudo infinita corporum animata, et nulla). That there will be some animated bodies follows evidently. That there will be none is proved, because no reason can be given why this one will be more than that. Why would the whole multitude B be occupied by these souls when fewer would suffice, suppose for instance the Papal bodies, call them C, and even fewer while still infinite, call them D, and so forth for other infinities? Why would B be occupied rather than C, or C rather than D, and so forth? For it does not necessarily follow that they occupy all of B, nor all of C, nor all of D, and so forth. Indeed if the souls do not take more bodies than are necessary, they will occupy no infinite multitude, because given any infinite multitude, a smaller will suffice for them. And if falsigraphus wishes to quibble, it may be said that that is said to be more than another which contains the whole and something more or some quantum beyond or extra . . . Take away all the old bodies and in their place let there be created as many new ones, etc., as before. Or let there be put as many coins (denarii) in a heap, or according to an ordered series, from which any soul or any man, when he has been animated, takes one for himself, and not more, etc., as above. But how is this superfluity to be accounted (reputanda)? For how can the souls take fewer of these bodies than each taking one, a single soul a single body, neither some of them, nor all together, taking anything beyond . . . I break off my paraphrase here. And so Bradwardine continues, trying out alternative scenarios by which the arithmetic of infinite collections seems to lead

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78 thomas bradwardine and his contemporaries to paradoxes. Results that seem to follow when the souls are arranged in a line and the bodies are arranged in a parallel line, making obvious how to pair souls with bodies, fall apart when the bodies are left in a disordered heap. I take it that the heap of coins is introduced as if they will somehow enable an orderly accounting.

8. Conclusion How should we reckon reasoning about infinities such as this? Minimally, we can say that, at least part of the time, this is what fourteenth-century grappling with continuity and infinity looks like. On the positive side, I would say that by their fourteenth-century modern or nominalistic view that the entities of mathematics exist in the mind and not in the outside world, thinkers began to open up possibilities for creativity in constructing new mathematical subdisciplines. As far as infinity is concerned, this view of mathematics made it possible to bypass the repeated argument that the potential infinite requires a process and time for its gradual emergence. It also allowed mathematicians to consider what might be constructed in their own minds or imaginations, rather than always tying their thoughts to what they speculated that God might see or know – not to say that some scholastics did not continue to refer to God in their thinking about the continuum—using God and the imagination as alternative avenues for reasoning.32 Some fourteenth-century thinkers advocated actual infinities as opposed to potential infinities. The fourteenth century did not come up with the view that the infinity of the continuum may be of a different order than the infinity of the integers, but perhaps, first, mathematicians had to expand the concept of numbers from positive whole numbers to real numbers and so forth, before they had a model or structure with the help of which such a step could be taken. In a certain sense the parts in the geometric model for parts in a divided continuum as described by Odo and Burley are doubly infinite because there are infinite levels of

32 William of Alnwick writes that God and humans can conceive of things in the same way. See Alnwick 1937. Quodlibet, Question IX, Utrum Deus cognoscat infinita. Doubt 4, response to third reponse to Aquinas’s opinion (=B) (Ad 3 Ad B). ‘Ad tertium argumentum dicendum quod non est contra me. Concedo enim quod infinitum secundum modum infiniti sit cognoscibile tam ab intellectu divino quam ab intellectu nostro, quia tunc unumquodque cognoscitur secundum modum sui quando cognoscitur secundum rationem suam definitivam sed ratio definitiva infiniti in rebus est quod eius quantitate accepta semper restat aliquid ulterius accipere, ex III Physicorum. Secundum hanc rationem a nobis cognoscitur, alioquin a nobis non definiretur. Deus etiam cognoscit eodem modo . . . Item infinitum prout est in rebus in actu admixto cum potentia, est positivum; sed omne positivum sub ratione positivi est cognoscibile ab intellectu divino et etiam creato; ergo infinitum in potentia, prout infinitum est in potentia admixta cum actu, est cognoscibile a Deo et a nobis. Unde si mundus in perpetuum duraret, intellectus cognosceret tempus et motum caeli sub ratione qua infinita sunt, quia cognosceret tempus sub ratione qua tempus et motum sub ratione qua motus et, ut sic, infinita sunt secundum successionem, ideo intellectus noster cognoscit infinitum successivum per modum infiniti successivi.’ On Gregory of Rimini, see Sylla 2005, pp. 264–268.

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references 79 division and, as the levels of division increase in infinitum, the numbers of parts at single successive levels also increase in infinitum. If I am right in my hypothesis that in De continuo Bradwardine attempted to construct an axiomatic discipline covering both geometric and natural indivisibles and that he ended up showing that his hypotheses were inconsistent, this does not mean that the effort was not worth making. I do not mean to suggest that what was done in the fourteenth century was still influential in the nineteenth or twentieth centuries. It is said that most mathematicians today consider themselves Platonists. I think that Bradwardine and his contemporaries showed that there can be intense work to develop mathematical sciences by people who are Aristotelians, even Aristotelians of the nominalistic type. The history as it occured was contingent, and it might have gone in a different direction (Mancosu 2009).

References Alnwick, William of (1937). Quaestiones Disputatae de esse intelligibili et de Quodlibet. Ed. by Athanasias Ledoux. London: Florence Ad Claras Aquas, ex typographia Collegii s. Bonaventurae. Boyer, Carl B. (1989). A History of Mathematics. 2nd ed. revised by Uta C. Merzbach. New York: John Wiley and Sons. Bradwardine, Thomas (1618). De causa Dei contra Pelagium et de virtute Causarum, ad suos Mertonenses, libri Tres. London. Reprint Frankfurt am Main, 1964. Burley, Walter (1501). Super octo libros Physicorum. Venice. Reprinted by Georg Olms Verlag (Hildesheim and New York), 1972. Glasner, Ruth (2009). Averroes’ Physics. Oxford: Oxford University Press. Grellard, Christophe and Aurélien Robert, eds. (2009). Atomism in Late Medieval Philosophy and Theology. Leiden and Boston: Brill. Harclay, Henry of (2008). Ordinary Questions I–XXIX. Ed. by Mark G. Henninger (SJ). Trans. by Raymond Edwards and Mark G. Henninger. Oxford and New York: Oxford University Press. Lukács, Anna, ed. (2013). Thomas Bradwardine: De causa Dei contra Pelagium et de virtute causarum. Göttingen: V & R unipress. Maier, Anneliese (1964). Ausgehendes Mittelalter: Gesammelte Aufsätze zur Geistesgeschichte des 14. Jahrhunderts. Vol. 1. Rome: Edizioni de Storia e Letteratura. Mancosu, Paolo (2009). “Measuring the Size of Infinite Collections: Was Cantor’s Theory of Infinite Number Inevitable?”. In: Review of Symbolic Logic 2.4, pp. 612–646. Molland, George (1978). “An Examination of Bradwardine’s Geometry”. In: Archive for History of Exact Sciences 19. Reprinted in Mathematics and the Medieval Ancestry of Physics, edited by George Molland, Routledge, 1995.

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80 thomas bradwardine and his contemporaries Murdoch, John E. (1957). “Geometry and the Continuum in the Fourteenth Century: A Philosophical Analysis of Thomas Bradwardine’s Tractatus de Continuo”. Available on ProQuest. Ph.D. University of Wisconsin. Murdoch, John E. (1964). “Superposition, Congruence and Continuity in the Middle Ages”. In: Mélanges Alexandre Koyré. Vol. 1. Paris, pp. 416–441. Murdoch, John E. (1974a). “Naissance et déeveloppement de l’atomisme au bas moyen âge latin”. In: Cahier d’études mediévales, II. La science de la nature: théories et pratiques, pp. 1–30. Murdoch, John E. (1974b). “Philosophy and the Enterprise of Science in the Later Middle Ages”. In: The Interaction between Science and Philosophy. Ed. by Yehuda Elkana. Atlantic Highlands, NJ: Humanities Press, pp. 51–74. Murdoch, John E. (1975). “From Social to Intellectual Factorrs: An Aspect of the Unitary Character of Late Medieval Learning”. In: The Cultural Context of Medieval Learning. Ed. by John E. Murdoch and Edith Sylla. Dordrecht: Reidel, pp. 271–339. Murdoch, John E. (1982a). “Henry of Harclay and the Infinite”. In: Studi sul XIV secolo in memoria de Anneliese Maier. Ed. by A. Maieru and A. Paravicini-Bagliani. Rome: Edizioni di Storia e Letteratura, pp. 219–261. Murdoch, John E. (1982b). “Infinity and Continuity”. In: The Cambridge History of Later Medieval Philosophy. Ed. by Norman Kretzmann, Anthony Kenny, and Jan Pinborg. Cambridge: Cambridge University Press, pp. 564–591. Murdoch, John E. (1982c). “William of Ockham and the Logic of Infinity and Continuity”. In: Infinity and Continuity in Ancient and Medieval Thought. Ed. by N. Kretzmann. Ithaca, NY: Cornell University Press, pp. 165–206. Murdoch, John E. (1987). “Thomas Bradwardine: Mathematics and Continuity in the Fourteenth Century”. In: Mathematics and Its Applications to Science and Natural Philosophy in the Middle Ages: Essays in Honor of Marshall Clagett. Ed. by Edward Grant and John E. Murdoch. Cambridge: Cambridge University Press, pp. 103–261. Neal, Katherine (2002). From Discrete to Continuous: The Broadening of Number Concepts in Early Modern England. Boston: Kluwer Academic Publishers. Panti, Cecilia (2014). “Natural Continuity and the Mathematical Proofs against Indivisibilism in Roger Bacon’s De Celestibus (Communia Naturalium, II)”. In: Roger Bacon’s Communia Naturalium. A 13th-Century Philosopher’s Workshop. Micrologus’ Library. Florence: SISMEL Edizioni del Galluzzo. Peter of Spain (1992). Syncategoreumata. Ed. by L.M. de Rijk. Trans. by Joke Spruyt. Oxford, New York, and Köln: E.J. Brill. Richardson, Kara (2015). “Causation in Arabic and Islamic Thought”. In: The Stanford Encyclopedia of Philosophy. Ed. by Edward N. Zalta. Winter 2015. Metaphysics Research Lab, Stanford University. Rimini, Gregory of (1979). Gregorii Ariminensis OESA Lectura super primum et secundum Sententiarum.

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references 81 Spade, Paul Vincent and Mikko Yrjönsuuri (2017). “Medieval Theories of Obligationes”. In: The Stanford Encyclopedia of Philosophy. Ed. by Edward N. Zalta. Winter 2017. Metaphysics Research Lab, Stanford University. Sylla, Edith D. (1981). “William Heytesbury on the Sophism “Infinita sunt finita” ”. In: Sprache und Erkenntnis im Mittelalter. Ed. by Albert Zimmerman. Vol. 13/2. Miscellanea Mediaevalia. Berlin and New York: De Gruyter, pp. 628–636. Sylla, Edith D. (1997). “Thomas Bradwardine’s De continuo and the Structure of Fourteenth-Century Learning”. In: Texts and Contexts in Ancient and Medieval Science: Studies on the Occasion of John E. Murdoch’s Seventieth Birthday. Ed. by Edith Sylla and Michael McVaugh. Leiden: Brill, pp. 148–186. Sylla, Edith D. (1998a). “God and the Continuum in the Later Middle Ages: The Relations of Philosophy to Theology, Logic, and Mathematics”. In: Was ist Philosophie im Mittelalter? Ed. by Jan A. Aertsen and Andreas Speer. Vol. 26. Miscellanea Mediaevalia. Berlin and New York: De Gruyter, pp. 628–636. Sylla, Edith D. (1998b). “God, Indivisibles, and Logic in the Later Middle Ages: Adam Wodeham’s Response to Henry of Harclay”. In: Medieval Philosophy and Theology 7, pp. 69–87. Sylla, Edith D. (2002). “Walter Burley’s Practice as a Commentator on Aristotle’s Physics”. In: Medioeva 27, pp. 301–307. Sylla, Edith D. (2005). “Swester Katrei and Gregory of Rimini: Angels, God, and Mathematics in the Fourteenth Century”. In: Mathematics and the Divine. Ed. by Teun Koetsier and Luc Bergmans. Amsterdam: Elsevier, pp. 249–271. Sylla, Edith D. (2011). “Disputationes Collativae: Walter Burley’s Tractatus Primus and Gregory of Rimini’s Lectura super primum et secundum Sententiarum”. In: Documenti e studi sulla tradizione filosofica medievale 22, pp. 383–464. Sylla, Edith D. (2015). “Averroes and Fourteenth-Century Theories of Alteration: Minima Naturalia and the Distinction between Mathematics and Physics”. In: Averroes’ Natural Philosophy and Its Reception in the Latin West. Ed. by Paul J.J.M. Bakker. Leuven: Leuven University Press, pp. 141–192.

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4 Continuous Extension and Indivisibles in Galileo Samuel Levey

The revolution in seventeenth-century mathematics that culminated in the differential and integral calculus is understood to arise from two striking developments of the era: analytic geometry, which represents geometrical magnitudes in algebraic terms; and various infinitary methods in geometry that render curves, planes, or solids as comprising infinitely many infinitesimal or indivisible elements. The first was due to work by Viète and especially Descartes, which furnished an algebraic framework for geometry that could exactly represent curves by polynomial expressions built around the fundamental operations of addition, subtraction, multiplication, division, and powers or the extraction of roots. Descartes’s La Géométrie, published in 1637, would be the pivotal text. The second began to emerge piecemeal in different forms in the work of many authors, including Fermat, Descartes, Kepler, Pascal, and others. The use of infinitary methods was not altogether new to the period and was regarded as a revival of secret ancient methods; Leibniz speculated that Archimedes and Conon relied on infinitesimals to make their own discoveries. Still, the consolidation of infinitary techniques into a commonly studied method would take root decisively with Cavalieri’s ‘method of indivisibles’, first published in his Geometria indivisibilibus continuorum nova quadam ratione promota of 1635, which represents plane or solid figures as containing infinite collections of ‘lines’ or ‘planes’, respectively, and studies relations between those lines or planes in order to establish theorems about the quadratures or cubatures of the figures themselves. It is in this context of mathematical innovation that the most notable seventeenth-century discussions of continuity and the composition of the continuum take place, in the writings of Galileo and Leibniz. Our subject in this chapter is Galileo, who was a mentor to Cavalieri and whose works profoundly influenced Leibniz. Galileo’s entry to the theme of the continuum and indivisibles is the analysis of continuous uniform and accelerated motion. He presents this first in his Dialogo sopra i due massimi sistemi del mondo of 1632, where he takes bold mathematical steps using infinite collections of indivisibles. He subsequently

Samuel Levey, Continuous Extension and Indivisibles in Galileo In: The History of Continua: Philosophical and Mathematical Perspectives. Edited by: Stewart Shapiro and Geoffrey Hellman, Oxford University Press (2021). © Stewart Shapiro and Geoffrey Hellman. DOI: 10.1093/OSO/9780198809647.003.0005

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galileo on continuous motion 83 revisits the topic in Discorsi e dimonstrazioni matematiche, intorno à due nuove scienze of 1638, where he explores the presuppositions and paradoxes associated with such techniques and appears to step back from the method of indivisibles, issuing a philosophical warning about the infinite in mathematics.

1. Galileo on Continuous Motion and Acceleration in the Dialogo The question of continuity in Galileo is not how to give a precise definition or to say just what continuity is. He takes an intuitive conception of continuity for granted, roughly the Aristotelian idea of a magnitude’s being ever divisible into lesser magnitudes of the same kind. But Galileo’s studies of continuous phenomena such as motion and extension drive him to innovate in geometry, and specifically to analyse continuous magnitudes in terms of infinitely many indivisible elements. The extraction of infinity from continuity of course dates back to ancient times; Galileo’s steadfast use of infinity in the form of infinite given multitudes of indivisibles to underpin arithmetical handling of the magnitudes is the signature of the seventeenth-century break with tradition. A clear example comes in Galileo’s Dialogo, when he gives a demonstration for his theorem that a ‘falling body moving uniformly for an equal time with the degree of velocity acquired would pass over double the space passed during its accelerated motion’ (Dia. 160/D67 225).1 The interlocutor Sagredo explains: You meant that commencing from rest and progressively increasing the velocity by equal additions, which are those of the successive integers beginning with 1, or rather with 0 (which represents the state of rest), and arranging these thus and taking consecutively as many as you please, so that the minimum degree is 0 and the maximum is 5, for example, then all these degrees of speed with which the body moves make a sum of 15. And if the body were moved at this maximum degree for the same number as there are of these, the total of all these speeds would be double the above; that is, 30. Hence if the body moved for the same time with a uniform speed of this maximum degree of 5, it would have to pass through double the space which it passed during the time in which it was accelerated and started from the state of rest. (Dia. 161–162/D67 228)

This preliminary analysis presents motion as if it were a sequence of discrete changes. Galileo’s spokesman Salviati then adapts the example to continuous uniform and uniformly accelerated motion, ‘for the increases in the accelerated 1 Abbreviations of primary texts and translations used in this chapter are listed in the bibliography. In addition, I use the following two: Dia. = Dialogo and Dis. = Discorsi, where page references are to volumes 7 and 8, respectively, in EN. Translations of Galileo generally follow those of Stillman Drake (D67 and D74) and translations of Huygens follow those of Richard Blackwell (B), as noted in the texts, though I have sometimes made minor modifications without comment.

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84 continuous extension and indivisibles in galileo M

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Figure 4.1 Galileo, Dialogo (1632), 229.

motion being continuous, one cannot divide the ever-increasing degrees of speed into any determinate number; changing from moment to moment, they are always infinite’ (Dia. 162/D67 229). Here Galileo expresses the degrees of speed not as integers but as line segments, and the interval of motion as a line AC divided into six equal subintervals (Figure 4.1): Hence we may better exemplify our meaning by imagining a triangle, which shall be this one, ABC. Taking in the side AC any number of equal parts AD, DE, EF, and FG, and drawing through the points D, E, F, and G straight lines parallel to the base BC, I want you to imagine the sections marked along the side AC to be equal times. Then the parallels drawn through the points D, E, F, and G are to represent the degrees of speed, accelerated and increasing equally in equal times. Now A represents the state of rest from which the moving body, departing, has acquired in the time AD the velocity DH, and in the next period the speed will have increased from the degree DH to the degree EI, and will progressively become greater in the succeeding times, according to the growth of the lines FK, GL, etc. But since the acceleration is made continuously from moment to moment, and not discretely (intercisamente) from one time to another, and the point A is assumed as the instant of minimum speed (that is, the state of rest and the first instant of the subsequent time AD), it is obvious that before the degree of speed DH was acquired in the time AD, infinite others of lesser and lesser degree have been passed through. These were achieved during the infinite instants that there are in the time DA corresponding to the infinite points on the line DA. Therefore to represent the infinite degrees of speed which come before the degree DH, there must be understood to be infinite lines, always shorter and shorter, drawn through the infinity of points of the line DA, parallel to DH. (Dia. 162/D67 228–229)

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galileo on continuous motion 85 Continuity requires using ‘infinite lines’, that is, infinitely many horizontal line segments, to represent all the ‘infinite degrees of speed’ in the accelerated motion. It also requires that the individual lines have no thickness; otherwise, the lines would be rectangles and their differences would represent discrete changes of velocity, ‘steps’ in the diagonal AB, contrary to the hypothesis of continuous uniform acceleration. Galileo leaves this unstated, but it is clear he intends the lines to be ‘indivisibles’. Now he introduces a striking mathematical step of representing this infinity of lines as a surface of a plane figure: This infinity of lines is ultimately represented (rappresenta) here by the surface of the triangle AHD. Thus we may understand that whatever space is traversed by the moving body with a motion which begins from rest and continues uniformly accelerating, it has consumed and made use of infinite degrees of increasing speed corresponding to the infinite lines which, starting from the point A, are understood as drawn parallel to the line HD and to IE, KF, LG, and BC, the motion being continued as long as you please. (Dia. 162/D67 229)

The area of the triangle AHD thus represents the sum of all the infinitely many increasing speeds of the body in accelerated motion. Galileo then compares this to the case of uniform motion by construction of a parallelogram whose lines are all of the same length and equal to the highest degree of speed reached at time B, namely, length BC. Now let us complete the parallelogram AMBC and extend to its side BM not only the parallels marked in the triangle, but the infinity of those which are assumed to be produced from all the points on the side AC. And just as BC was the maximum of all the infinitude in the triangle, representing the highest degree of speed acquired by the moving body in its accelerated motion, while the whole surface of the triangle was the sum total of all the speeds (la massa e somma di tutta la velocita) with which such a distance was traversed in the time AC, so the parallelogram becomes the total and aggregate of just as many degrees of speed (il parallelogrammo viene ad esser una massa ed aggregato di altrettanti gradi di velocità) but with each one of them equal to the maximum BC. This total of speeds is double that of the total of the increasing speeds in the triangle, just as the parallelogram is double the triangle. And therefore if the falling body makes use of the accelerated degrees of speed conforming to the triangle ABC and has passed over a certain space in a certain time, it is indeed reasonable and probable that by making use of the uniform velocities corresponding to the parallelogram it would pass with uniform motion during the same time through double the space which it passed with the accelerated motion. (Dia. 163/ D67 229)

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86 continuous extension and indivisibles in galileo Galileo’s mathematical technique here is an early use of what will come to be called ‘the method of indivisibles’, later codified in Cavalieri’s Geometria indivisibilibus of 1635 and Exercitationes Geometria Sex of 1647. Galileo leaves implicit in his demonstration the idea, later stressed by Cavalieri, that one can infer from a comparison of lines to a comparison of figures. But Galileo’s treatment of all the ‘infinite lines’ parallel in a figure as jointly having a magnitude representable as the area of the figure is precisely the philosophical and mathematical step Cavalieri develops in his account with the concept of ‘all the lines’ (omnes linea) of any given figure as a new category of magnitude. On Cavalieri’s approach, ‘all the lines’ of a given plane figure (or omnia plana for a given solid figure) are defined by the continuous passage of a plane through it between opposite tangents parallel to an assigned reference line (or plane) which he calls the regula (GIC 99). Cavalieri holds that when ‘all the lines’ or ‘all the planes’ of one figure F can be shown to stand in a definite ratio to those of another figure G, a proportion can then be formed on the basis of which it can be inferred that the areas of F and G themselves stand in the same ratio. Cavalieri sees this principle of comparison as ‘the great foundation of this new geometry of mine’ (GIC 115). As he is well aware, his principle requires that ‘all the lines … of any plane figures, and all the planes of any solids, are magnitudes having ratio to one another’ (GIC 108). But given the continuity of the plane figures and solids themselves, and the passage of the plane through them that determines their indivisibles, an obvious worry arises; as Cavalieri wrote to Galileo in 1621, ‘it seems that all the lines of a given figure are infinite and hence not covered by the definition of magnitudes which have ratios’ (EN 13.81).2 Cavalieri conceived of two ways that the indivisibles of figures can be compared: either, first, collectively, so that ‘all the lines’ of a given figure (say, EFGH in the diagram below) are compared at once to all those of another figure (ABCD); or, second, distributively, so that individual lines matched by the same moment of the passage of the moving plane (e.g., FH and BD) are compared. The distributive method is devised in the late-added seventh book of the Geometria, then expanded in the Exercitationes, and it appears to be crafted to satisfy mathematicians sceptical of the use of the infinite in demonstrations. It invokes the same fundamental principle that if two figures F and G enclosed by ‘parallel tangents’ such that the lines lf of F and the lines lg of G stand in a ratio such that lf : lg = a:b, then F:G = a:b. (Similarly for solids and their plane sections.) But whereas in the collective method the lines lf and lg in question were ‘all the lines’ and so infinitely many, in the distributive method Cavalieri’s principle is interpreted in terms of each pair of individual lines. (Cavalieri seeks to use Euclid V.5 to establish 2 Guldin would later famously object to Cavalieri’s method precisely on the ground that ‘there is no proportion or ratio of an infinite to an infinite’ (1635–41, 4.341).

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the composition of the continuum? 87 I

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Figure 4.2 Cavalieri, Exercitationes Geometria Sex (1647), 4.

the validity of the implication from ratios of pairs of individual lines to ratios of the figures to which they belong.3) The distributive method is less general than the collective method, since it requires compared figures to have precisely the same ‘altitude’ in order to pair their respective lines individually; but, if successful, it would avoid the charge levelled against the collective method of using infinities in ratios and proportions.

2. The Composition of the Continuum? Galileo’s and Cavalieri’s use of indivisibles in the geometry of continuous magnitudes suggests a view of the continuum as composed of indivisibles: the plane figures are aggregates of the lines, solids are aggregates of planes. In a passage from his Exercitationes, Cavalieri provides a famous image of a figure as a piece of cloth woven of parallel threads deprived of their thickness: It is evident that we can think of plane figures as like pieces of cloth woven out of parallel threads, and solids as like books made up of parallel folios. But whereas the threads in a piece of cloth and the folios in a book are always finite in number and in fact have some thickness, in our method the lines in plane figures, and planes in solids, are to be supposed indefinite in number and without any thickness. (EGS 3–4)⁴

3 For details, see Andersen 1985, 350ff. ⁴ Cavalieri’s ‘indefinite’ here aims to sidestep controversy about the use of infinity in mathematics, though, as in note 2 above, Guldin was not assuaged. Nor, apparently, was Galileo; see below. See also Andersen 1985, and Jesseph in this volume.

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88 continuous extension and indivisibles in galileo The idea that Cavalieri’s method of indivisibles expresses an underlying view of the composition of the continuum is attractive, and his readers often assumed as much. For example, writing in 1700, J. C. Sturm would describe the method of indivisibles this way: The Method of indivisibles … goes to work after a way which seems to be more natural than any other, by supposing plane Figures to consist of innumerable lines, and solid of innumerable Plans (called their indivisible Parts or Elements because the Lines are conceived without latitude and the Plans without any thickness), and relying on this self-evident Axiom, That if all the Indivisibles of one Magnitude collectively taken, be equal or proportional to all the corresponding indivisibles of another, or taken separately each to each, then also those Magnitudes will be equal or proportional among themselves. (Sturm 1700, Preface section IX)

Yet it must be noted that in fact Cavalieri is officially neutral about whether the indivisibles of a given figure compose the figure as its parts or instead stand in some non-compositional relation to it. As he writes to Galileo in a letter of 2 October 1634: ‘I do not declare to compose the continuum from indivisibles’ (EN 18.67). And again on 28 June 1639: ‘I have not dared to say that the continuum was composed of these [indivisibles]’ (EN 16.138). All that his mathematics strictly requires is the consistency of assigning magnitude to omnes linea and omnia plana, and Cavalieri declines to venture into the metaphysics of the continuum in defence of his mathematics. In that respect, Galileo stands in contrast to his student. In the 1632 Dialogo, Galileo does not pursue the question of the nature of the lines and their relation to the figures in which they occur: whether they are parts, whether they are magnitudes, whether their aggregate is properly speaking a sum, etc. But in 1638, three years after the publication of Cavalieri’s Geometria, Galileo returns to the topic of continuous magnitudes in his Discorsi e dimonstrazioni matematiche, intorno à due nuove scienze. There Galileo makes dexterous use of ancient paradoxes—notably the Rota Aristotelica—to argue for the hypothesis that any continuous magnitude contains infinitely many indivisibles that are, in his term, parti non quanti or ‘parts without quantity’ (Dis. 71–72). He explicitly equates a circle with an infinilateral polygon whose sides are lati non quanti, and further holds that ‘What is thus said of simple lines is to be understood also of surfaces and solid bodies, considering those as composed of infinitely many unquantifiable atoms [infiniti atomi non quanti]’ and that the ‘highest and ultimate resolution’ of surfaces and solid bodies is ‘into the prime components unquantifiable and infinitely many [componenti non quanti ed infiniti]’ (Dis. 71–72). Thus Galileo avows exactly what Cavalieri had not dared to say: the continuum is composed from indivisibles.

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galileo on the continuum and indivisibles 89

3. Galileo on the Continuum and Indivisibles in the Discorsi In the Discorsi’s First Day, while the interlocutors puzzle over the resistance and cohesion of bodies, Galileo’s spokesman Salviati proposes to explain the integrity of ‘the most minute particles’ by the presence of tiny void points within matter. Salviati advances the idea ‘not as the true solution, but as a kind of fantasy full of undigested things’ (Dis. 66/D74 27). A minute particle might contain any number of tiny voids holding it together. Could it contain an infinity of voids? Well, since paradoxes are at hand, let us see how it might be demonstrated that in a finite continuous extension it is not impossible for infinitely many voids to be found. At the same time, we shall see, if nothing else, at least a solution of the most admirable problem put by Aristotle among those he himself called admirable. (Dis. 68/D74 28–29)

The most admirable problem is the Rota Aristotelica, or ‘Aristotle’s wheel’ paradox, and Galileo uses it to argue for infinitely many indivisibles in the continuum. The paradox describes the motion of a wheel made of a pair of concentric circles as it rolls along a line. Suppose the outer circle (say, radius 𝛼) rolls forward for a single revolution. In doing so it will cover a distance equal to its circumference (2𝜋𝛼). The inner circle (radius 𝛽) fixed to it also revolves exactly once but traverses the same distance—thus appearing to roll a distance greater than its own circumference (2𝜋𝛼 > 2𝜋𝛽) in a single revolution. (Inverting the case, if instead the interior circle ‘drives’ the motion by rolling forward one revolution, it will travel a distance equal to its own circumference; but then the outer circle in a single revolution traverses a distance smaller than its own circumference.) How could this be? Galileo approaches the problem by first developing an analysis of the motion of concentric regular polygons—he starts with hexagons—and then extending it to the case of the circles, taking the circles as infinilateral polygons. Consider the case where a larger hexagon ABCDEF drives the motion by rolling along its perimeter (Figure 4.3). The motion of the larger hexagon measures out a line equal to its perimeter, each side stamping down a part of the line in succession as it rolls. The smaller hexagon HIKLMN traverses a parallel line of similar length, but its sides only touch some parts of this line, alternately touching down and then skipping over segments as it is lifted away by the tumbling transit of the larger hexagon ABCDEF. For the hexagons, and indeed for any polygons of n sides, the length of the line AS ‘passed over and measured’ by the larger will be equal to the sum of the n segments of line HT touched by the smaller’s sides plus the lengths of the n void spaces in HT that the smaller polygon skipped over. Approximately equal: to be precise, the line (here HT) composed of the smaller polygon’s n equal sides plus n equal void spaces exceeds the length of the larger polygon’s perimeter (here given

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by AS) by exactly the difference between the length of the larger polygon’s side and the length of an equal void space skipped by the smaller polygon.⁵ As the number n of sides increases, this excess amount shrinks. Galileo’s Salviati describes the case for polygons of 100,000 sides, and then moves to the decisive point: I return to the consideration of the polygons discussed earlier, the effect of which is intelligible and already understood. I say that in polygons of one hundred thousand sides, the line passed over and measured by the perimeter of the larger—that is, by the hundred thousand sides extended continuously—is equal to that measured by the hundred thousand sides of the smaller, but with the interposition of one hundred thousand void spaces. And just so, I shall say, in the circles (which are polygons of infinitely many sides), the line passed over by the infinitely many sides of the larger circle, arranged continuously , is equal in length to the line passed over by the infinitely many sides of the smaller, but in the latter case with the interposition of just as many voids [d’altrettanti vacui] between them. And just as the sides are not quantified [lati non son quanti], but are infinitely many [ma bene infiniti], so too the interposed voids are not quantified [vacui non son quanti], but are infinitely many; that is, for the former infinitely many points all filled, and for the latter , infinitely many points, part of them filled points and part of them voids. (Dis. 71/D74 32–33)

⁵ See Drake (1974), 33 fn. 14.

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galileo on the continuum and indivisibles 91 The step to the limit case of the circle as a polygon with infinitely many ‘not quantified sides’—lati non quanti—is striking, as is the image of a finite continuous quantity as composed of infinitely many non quanti points, ‘part of them filled and part of them void’. These are the indivisibles that appear to be ultimate constituents of the continuum. The distinction between quanti and non quanti is crucial in Galileo’s discussion. Quanti parts have finite measure and are further divisible. Non quanti parts, by contrast, have no measure and are indivisible. They are not infinitely small quantities; they are not quantities at all. In calling the indivisibles non quanti (‘unquantified’ in Drake’s translation), Galileo is removing them from the domain of mathematical measurement proper. As Eberhard Knobloch nicely observes,⁶ Galileo shifts from having a little earlier spoken of the lines as ‘measured’ (misurata) by the finitely many sides of the finite polygons to saying only that they are ‘passed over’ (passata) by the infinitely many sides of the circles. The sides are no longer strictly fit to measure the lines they touch: they are lati non quanti, i.e., without quantity, and so lacking the basic mathematical character required for measurement. The introduction of non quanti indivisibles into the composition of continuous quantities allows two claims in the analysis of Aristotle’s wheel. First, in the limit case of the circles, the two lines traversed by the larger and smaller circles are exactly equal. There is no excess difference left over, for the difference between the length of the void spaces in HT and the length of the ‘sides’ of the larger circle has vanished along with the ‘quantity’ of the sides and subsegments marked out on the line traversed. Secondly, the interposition of void points in HT that are ‘skipped’ by the rotation of the smaller circle makes it possible to suppose that the length of HT, which itself contains both touched and skipped points, is not constrained to be equal to the circumference of the smaller circle. Rather, although HT is indeed traversed by the motion of the smaller circle, nonetheless HT is ‘expanded’ relative to the smaller circle’s circumference by the presence in it of void points so that HT is equal to AS. The paradox of the wheel is thus solved—if not necessarily truly, then at least by ‘a kind of fantasy filled with undigested things’.⁷

⁶ Knobloch (1999), 92. ⁷ Later mathematics would solve Aristotle’s wheel differently. The non-driving inner circle does not ‘roll’, that is, individual points on its circumference do not follow a cycloid path. The path of a circumferential point on the inner circle is instead a curtate cycloid. (In the inverse case where it is the inner circle that drives the motion, a point on the non-driving outer circle follows a prolate cycloid.) Only the driving circle follows a true cycloid. Further, the straight line HT determined by the motion of the smaller circle is both perfectly continuous and not peppered with void points. HT is in effect the ‘tangent line’ to the aggregate of all the curtate cycloids traced out by the circumferential points of the inner circle. Each curtate cycloid intersects HT at only a single point, so there is no ‘slipping’. And each and every point on HT is touched by some point’s path, so none are ‘skipped’. The tangent line to an aggregate of true cycloids of circumferential points on a given circle will be equal in length to the circle’s circumference, as in the case of AS. So a rolling circle’s path for one revolution is equal in length to its circumference. But the tangent lines corresponding to the aggregate of curtate or prolate cycloids will

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92 continuous extension and indivisibles in galileo But the analysis of Aristotle’s wheel is only an entry to Galileo’s larger theme concerning the composition of the continuum. He elaborates the idea of composing a finite quantity from an infinity of non quanti parts in order to show how the doctrine of indivisible voids can be deployed to make sense of the possibility of the expansion (and presumably contraction) of lines or solids into spaces of different sizes. Salviati presses forward: Here I want you to note how, if a line is resolved and divided into parts that are quantified [in parti quante], and consequently numbered [numerate], we cannot then arrange these into a greater extension than that which they occupied when they were continuous and joined, without the interposition of just as many [altrettanti] void spaces. But imagining the line resolved into unquantifiable parts [parti non quante]—that is, into infinitely many indivisibles—we can conceive it immensely [immenso] expanded without the interposition of any quantified void spaces, though not without infinitely many indivisible voids. What is thus said of simple lines is to be understood also of surfaces and solid bodies, considering those as composed of infinitely many unquantifiable atoms [infiniti atomi non quanti]; for when we wish to divide them into quantifiable parts [parti quante], doubtless we cannot arrange those in a larger space than that originally occupied by the solid unless quantified voids [quanti vacui] are interposed—void, I mean, at least of the material of the solid. But if we take the highest and ultimate resolution into the prime components unquantifiable and infinitely many [componenti non quanti ed infiniti], then we can conceive such components as being expanded into immense space [in spazio immenso] without the interposition of any quantified void spaces, but only of infinitely many unquantifiable voids [vacui infiniti non quanti]. In this way there would be no contradiction in expanding, for instance, a little globe of gold into a very great space without introducing quantifiable void spaces [spazii vacui quanti]—provided, however, that gold is assumed to be composed of infinitely many indivisibles. (Dis. 71–72/D74 33–34)

Let us focus on just a few points. With the conception of lines and solids as composed of infinitely many non quanti indivisibles in mind—a familiar precursor to contemporary point-set analysis of the continuum—Galileo is observing, correctly, how the metrical properties of collections are not directly determined by those of their elements if the elements are allowed to be both infinitely many and to have, individually, no positive measure. The same infinite collection of non quanti points might constitute a line of any finite length, or a globe of any finite size,

not be equal to the circumference of the smaller or larger circles whose circumferential points follow those non-rolling paths. Losing track of the subtle differences between these types of rotating motions yields the illusion that the non-driving circle paradoxically rolls a distance unequal to its circumference.

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why infinitely many unquantifiable parts? 93 depending on how the points are arranged. Or more carefully: any assignment of measure might be consistent with a collection of infinitely many non quanti points; there is no contradiction in assigning different sizes to such collections.⁸ Still, it should be noted that Galileo’s appeal to the presence of non quanti voids in the lines or solids does not obviously explain the differences in measure for the different arrangements. For example, it’s not clear why non quanti voids should expand things any more than non quanti atoms would on their own. The appeal to voids seems to serve as a placeholder for whatever it is that makes the difference in the ‘arrangement’ of the non quanti atoms to yield different measures. Nonetheless, an account of the composition of continuous magnitudes from non quanti indivisibles at least makes room for the possibility that the very same measureless elements might be able to be combined in different ways so as to compose quantities of different magnitudes. In the case of material quantities, the indivisibles may compose bodies of different sizes. In the case of void quantities, the composition of indivisibles may yield different measures of resistance, thus allowing for a model of how minute particles might hold together with great tenacity.

4. Why Infinitely Many Unquantifiable Parts? Galileo’s audience may wonder whether the postulation of infinitely many non quanti indivisibles in the analysis of Aristotle’s wheel is satisfactorily justified. Salviati moves seamlessly from the case of finite polygons to the case of the circle as a polygon with infinitely many sides. Is it so easy? Even granting the traditional understanding of the continuum that holds it to be infinitely divisible—as Galileo attributes it to the Peripatetics: ‘divisible into ever-divisible parts’ (Dis. 92/D74 54)—what underwrites the further idea that a continuous extension contains infinitely many elements? Here Galileo has an independent line of argument to offer. Infinite divisibility presupposes the existence of infinitely many indivisibles: Salv. Now let us pass to another consideration, which is that the line, and every continuum, being divisible into ever divisibles, I do not see how to escape their composition from infinitely many indivisibles; for division and subdivision that can be carried on forever assumes that the parts are infinitely many. Otherwise subdivision would come to an end. And the existence of infinitely many parts has a consequence that they are unquantifiable, since infinitely many quantified

⁸ Classic contemporary point-set analysis allows unions of infinitely many zero-dimensional points (or singletons) to have any positive measure, though with the proviso that the cardinality of the union be uncountable; countably infinite unions of points would still have measure zero. For discussion, see Grünbaum (1952) and Skyrms (1983).

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94 continuous extension and indivisibles in galileo [parts] make up an infinite extension. And thus we have a continuum composed of infinitely many indivisibles. (Dis. 80/D74 42)

Two assumptions worth notice arise to view.⁹ First, divisibility requires parts, so divisibility into ever divisibles—infinite divisibility—requires infinitely many parts. Second, for parts to be unquantifiable, non quanti, they must be indivisible; so divisible parts are always quantified parts. Either assumption might be challenged, but let us accept the second for the sake of argument and consider only the first. What is required for a continuum to be divisible into ever divisibles? If divisibility requires parts, then clearly no given finite number of parts will suffice. For any such number n you like, there must be at least n parts in a continuum. Whether from those facts it follows that there are infinitely many parts depends to some extent on what one understands by the term ‘infinite’. Galileo himself notes that he regards this formula—for any number n, the continuum contains n parts—as picking out something intermediate between ‘finite’ and ‘infinite’. Speaking of discrete quantity it appears to me that there is a third, or middle, term; it is that of answering to any [ogni] designated number. Thus in the present case, if asked whether the quantified parts in the continuum are finite or infinitely many [infinite], the most suitable reply is ‘neither finite nor infinitely many, but so many as [ma tante che] to correspond to every specified number’. To do that, it is necessary that these be not included within a limited number, because then they would not answer to a greater ; yet it is not necessary that they be infinitely many, since no specified number is infinite [infinito]. And thus at the choice of the questioner we may cut a given line into a hundred quantified parts, into a thousand, and into a hundred thousand, according to whatever number he likes, but not into infinitely many . (Dis. 81/D74 43–44)

This is an eminently reasonable position. But it appears to be at odds with an earlier reply by Salviati to a question from Simplicio that had initiated the whole discussion of how many quantified parts there can be in the continuum: Simp. But if we can continue forever the division into quantified parts, what need have we, in this respect, to introduce the unquantifiable? Salv. The very ability to continue forever the division into quantified parts implies the necessity of composition from infinitely many unquantifiables. For, getting

⁹ A third assumption is that infinitely many quantified parts would make up an infinite extension; Galileo here ignores the case of infinitely many quantified parts of unequal sizes forming a convergent series. If equal quantified parts are assumed, the inference is correct.

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why infinitely many unquantifiable parts? 95 down to the real trouble, I ask you to tell me boldly whether in your opinion the quantified parts of the continuum are finite, or infinitely many? (Dis. 80/D74 42)

Salviati insists that there will have to be infinitely many parts. But now with the ‘middle term’ between finite and infinite available, his argument is undercut. For it seems that, strictly speaking, there is not a necessity of composition from infinitely many parts, but only composition from ‘so many’ parts ‘as to correspond to any designated number’. And this seems like a perfectly good way to suppose that a division into quantified parts can ‘continue forever’. More to the point: without an inference to a necessity for infinitely many parts in a continuum, the further claim that there must be unquantified indivisible parts falls through. For that claim rested on the idea that infinitely many quantified parts taken together will compose an infinite extension. But as Salviati himself has explained, it is consistent with an ‘intermediate’ division of a continuum of finite extension that the parts are always only quantified parts. Perhaps a further argument could be given for holding that an intermediate division must presuppose infinitely many parts. It’s tempting to consider, for example, Cantor’s argument that the potential infinity associated with a variable presupposes an actual infinity for its range of values.1⁰ Maybe in order to explain how it’s possible for a continuum to allow a division into n parts no matter which value for n one chooses, we must assume that it contains infinitely many parts. And from there Galileo’s argument for the necessity of unquantified parts in the continuum could be reinstated. But as yet Galileo has not supplied premises that would serve as a bridge from an intermediate division to the necessity of infinitely many parts. So the hypothesis of infinitely many unquantified indivisibles in the continuum appears not to be adequately justified by this particular line of argument even on its own terms. A related point arises in the discussion of infinite division. The interlocutors are troubled by the question of whether a division of a continuum by stages could reduce it to its indivisible components. Evidently there is no last division imposed, ‘for there always remains another’ (Dis. 92/D74 54). At any stage, there are only finitely many parts, each of them quantified. How then could a division into infinitely unquantified indivisible parts be understood as effected? ‘And yet’, claims Salviati, there indeed exists a last and highest, and it is that which resolves the line into infinitely many indivisibles. I admit that one will never arrive at this by successively dividing into a greater and greater multitude of parts. But by employing the method I propose, that of distinguishing and resolving the 1⁰ See Cantor (1886, 9), Hallett (1986, 24ff.).

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96 continuous extension and indivisibles in galileo whole infinitude at one fell swoop—an artifice that should not be denied to me— I believe that they should be satisfied, and should allow this composition of the continuum out of absolutely indivisible atoms. (Dis. 92/D74 54)

The method Galileo describes is that of bending a straight line into a circle, i.e., an infinilateral polygon. Just as bending a line at angles into a square or an octagon marks out four sides or eight sides, and so on, bending it into a circle yields ‘at one fell swoop’ infinitely many sides. That such a resolution is made into its infinitely many points cannot be denied, any more than that into four parts in forming a square, or into its thousand in forming a milligon, inasmuch as none of the conditions are lacking here that are found in the polygon of one thousand or one hundred thousand sides. (Dis. 92/D74 53–54).

The weight of the argument is carried by its identification of a circle with an infinilateral polygon. If this is accepted at face value, then Salviati’s claim that the line is resolved into infinitely many components by this method is a plausible one. And it need not be a circle; any continuous curve would do.

5. Galileo and Huygens: Infinity, Indivisibles, and Geometrical Rigour One of Galileo’s most celebrated conclusions in the Discorsi’s discussion of the paradoxes is that ‘one infinity cannot be said to be greater or less than or equal to another’ (EN 8.78/D74 40). Galileo takes the natural numbers as his example of an infinity and argues as follows. Since the natural numbers include both the square numbers and non-square numbers, there are more naturals than squares. Yet there are just as many squares as there are roots, since every root has its own square and every square its own root; and there are just as many naturals as roots, since every natural is a root and every root is a natural. It follows that there are just as many squares as naturals. We thus appear to have contradictory results: the natural numbers are both greater than and equal to the square numbers, which is absurd. (See Dis. 78–79.)11 Galileo, speaking through Salviati, concludes: I don’t see how any other decision can be reached than to say that all the numbers are infinitely many [infiniti]; all the squares infinitely many; all their 11 This is now sometimes called ‘Galileo’s Paradox’ in contemporary literature. See Levey (2015) for discussion.

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galileo and huygens 97 roots infinitely many; that the multitude [moltitudine] of squares is not less than that of all numbers, nor is the latter greater than the former. And in the final conclusion, the attributes of equal, greater and less have no place in infinite, but only in bounded quantity [quantità terminate]. (Dis. 79/D74 41)

That conclusion is Galileo’s signature thesis about the infinite, solving the paradox about the natural numbers by excluding mathematical comparisons between ‘infinites’ that yielded the contradiction. It can be tempting to think that it also delivers a warning about mathematical methods such as Cavalieri’s. If the basic vocabulary of ratios has no place in the infinite, then comparisons among ‘all the lines’ of figures is called into question and with it a crucial element of the method of indivisibles. Yet there is room to suppose that Galileo’s restriction of comparisons to ‘bounded quantity’ will accommodate Cavalieri’s method. For the infinite collections of lines or plane sections are in an obvious way bounded by the figures to which they belong. Cavalieri grants magnitude to infinite collections of indivisibles, making them eligible for ratio, but it is always finite magnitude. (By contrast, the natural numbers collectively ‘summed’ would be an infinite, unbounded magnitude.) And Galileo himself shows no qualms about relying on one-to-one correspondences to establish that there are ‘just as many’ elements in one collection as in another in mathematical proofs in his treatise ‘On Naturally Accelerated Motion’, embedded in the Discorsi on its Third Day. So at least some form of equality among infinites—among infinite multitudes at any rate—is acceptable by his lights. Nonetheless, in the Discorsi, as compared to the earlier Dialogo, Galileo appears to distance himself from Cavalierian techniques in a way that may signal growing reluctance about the use of infinity in his calculations. In ‘On Naturally Accelerated Motion’, Galileo revisits his theorem comparing uniform and uniformly accelerated motion from the Dialogo, providing a new proof whose differences from the original one, though subtle, merit attention (Figure 4.4). PROPOSITION I. THEOREM I The time in which a certain space is traversed by a moveable in uniformly accelerated movement from rest is equal to the time in which the same space would be traversed by the same moveable carried in uniform motion whose degree of speed is one-half the maximum and final degree of speed of the previous, uniformly accelerated, motion. Let line AB represent the time in which the space CD is traversed by a moveable in uniformly accelerated movement from rest at C. Let EB, drawn in any way upon AB, represent the maximum and final degree of speed increased in the instants of the time AB. All the lines reaching AE from single points of the line AB and drawn parallel to BE will represent the increasing degrees of speed after

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Figure 4.4 Galileo, Discorsi (1638), Prop. 1, Thm 1. 165. the instant A. Next, I bisect BE at F, and I draw FG and AG parallel to BA and BF; the parallelogram AGFB will [thus] be constructed, equal to the triangle AEB, its side GF bisecting AE at I. Now if the parallels in triangle AEB are extended as far as IG, we shall have the aggregate of all parallels contained in the quadrilateral equal to the aggregate of those included in triangle AEB. For those in triangle IEF are matched by those contained in triangle GIA, while those which are in the trapezium AIFB are common. Since each instant and all instants of time AB correspond to each point and all points of line AB, from which points the parallels drawn and included within triangle AEB represent increasing degrees of the increased speed, while the parallels contained within the parallelogram represent in the same way just as many degrees of speed not increased but equable, it appears that there are just as many momenta of speed consumed in the accelerated motion according to the increasing parallels of triangle AEB, as in the equable motion according to the parallels of the parallelogram GB. For the deficit of momenta in the first half of the accelerated motion (the momenta represented by the parallels in triangle AGI falling short) is made up by the momenta represented by the parallels of triangle IEF. It is therefore evident that equal spaces will be run through in the same time by two moveables, of which one is moved with a motion uniformly accelerated from rest, and the other with equable motion having a momentum one-half the momentum of maximum speed of the accelerated motion; which was [the proposition] intended. (Dis. 208–209/D74 165–166)

As in the earlier demonstration from the Dialogo, Galileo moves effortlessly between facts about the relations among the parallels in the figures and facts

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galileo and huygens 99 about relations between the figures themselves, implicitly relying on proportions between them. For example, the similarity of triangles GIA and IEF is used to argue that the parallels of GIA contain just as many ‘momenta’ of speed as the parallels of IEF. Galileo’s language here in the Discorsi is careful: the mention of ‘sums’ of lines from the Dialogo has now been dropped in favour of ‘aggregates’. Most notably, there is no mention of infinity: neither the infinity of instants, points, or parallels, nor the infinity of degrees of speed. This is left implicit, along with the unstated assumption of the continuity of the motion that would appear to require it. Galileo operates instead with a half-acknowledged principle that the one-to-one correspondence between instants, points, parallels, and degrees of speed implies that there are ‘just as many’ of each in their respective aggregates, however many that should be—including even infinitely many. But the infinity is not optional for the argument. Without the infinity of the parallels, the inferences between facts about aggregates of parallels and facts about areas of figures would fall through. For it is the continuity of the figures that stands in for the continuity of the motions, and without allowing the parallels to be infinite, the continuity of the figures will be lost in their representation by the parallels they contain. The geometry of Galileo’s argument could be reconstructed to avoid even this implicit reliance on an infinity of parallels in the figures, letting the continuity of the figures carry the work entirely, as in the traditional method of exhaustion. But the replacement proof would then lose the advantage of directness. Interestingly, Huygens, in his Horologium oscillatorium sive de motu pendulorum ad horologia aptato demonstrationes geometricae (1673), provides precisely such a proof of this theorem, still ‘using Galileo’s method’ but writing down ‘more accurately the demonstration which he gave in a less perfect form’ (Hor. 28–29/B 40). Huygens had in 1659 performed a series of mathematical studies using a version of the method of indivisibles, which he clearly prized for its role in discovery while remaining wary of its apparent lack of rigour. In the Horologium, he proves again many of his results but opts now for traditional geometrical arguments. His recasting of Galileo’s demonstration in more perfect (i.e., rigorous Euclidean) form, at Propositio V, is an exemplar of the double-reductio method of exhaustion; it’s worth quoting in full (Figure 4.5): The distance crossed in a certain time by a body which begins its fall from rest is half the distance it would cross in an equal time with uniform motion having the velocity acquired at the last moment of fall. Let AH be the total time of the fall. Let H be the total time of fall. In that time the moving body crosses a distance whose quantity is designated by the plane P. Draw HL, of any length, perpendicular to AH, and let it represent the velocity acquired at the end of the fall. Next complete the rectangle AHLM, and let it represent the amount of distance which would be crossed in the time AH with the velocity

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Figure 4.5 Huygens, Horologium, Prop. V. HL. What must be shown is that the plane P is half of the rectangle MH, or, after drawing the diagonal AL, that the plane P equals the triangle AHL. If the plane P is not equal to the triangle AHL, then it will be either smaller or larger. First assume that the plane P is smaller than the triangle AHL, if that would be possible. Let AH be divided into a number of equal parts AC, CE, EG, etc. And on the triangle AHL let the figure be circumscribed which is composed of rectangles whose altitude equals each part of this division of AH, namely, the rectangles BC, DE, FG. Also within the same triangle let there be inscribed another figure composed of rectangles of the same altitude, namely, KE, OG, etc. Let all this be done in such a way that the excess of the former figure over the latter figure is less than the excess of the triangle AHL over the plane P. It is clear that this can be done since the entire excess of the circumscribed figure over the inscribed figure equals the lowest rectangle which has HL as a base. Therefore the whole excess of the triangle AHL over the inscribed figure will be less than its excess over the plane P, and thus the figure inscribed in the triangle is larger than the plane P. Further, since the straight line AH represents the entire time of fall, its equal parts AC, CE, EG represent equal parts of that time. And since the velocities of falling bodies increase in the same proportion as the times of fall, and since AHL is the velocity acquired at the end of the whole time, CK will be the velocity acquired at the end of the first part of the time, for AH is to AC as HL is to CK. Likewise EO will be the velocity acquired at the end of the second part of the time, and so forth. But it is clear that in the first time AC some distance larger than zero is crossed by the moving body. And in the second time CE the distance crossed is greater than KE since the distance KE would have been crossed in the time CE by a uniform motion with the velocity CK. For the distances crossed by uniform motion have a ratio composed of the ratios of the times and of the velocities. And hence, since in the time AH the distance MH would have been crossed with

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galileo and huygens 101 a uniform velocity HL, it follows that in the time CE the distance KE would be crossed with a velocity CK, since the ratio of the rectangle MH to the rectangle KE is composed of the ratios of AH to CE and of HL to CK. Now, as was said, the distance KE would be crossed in the time CE with the uniform velocity CK, but the body is moved in the time CE with an accelerated motion which already has the velocity CK at the beginning of this time. Hence it is clear that, in this accelerated motion in the time CE, a greater distance than KE is crossed. For the same reason, in the third period of time EG, a greater distance than OG is crossed since this is what would be crossed in that time EG with the uniform velocity EO. And so successively, and each part of the time AH, the body crosses a greater distance than the rectangles inscribed in the figure adjacent to each part of the time. Hence the whole distance crossed by an accelerated motion will be greater than the inscribed figure. But that distance was assumed to be equal to the plane P. Thus the inscribed figure will be smaller than the distance P, which is absurd because this was shown to be larger than that distance. Thus the plane P is not smaller than the triangle AHL. Next it will be shown that it is also not larger. Assume that it is larger, if that be possible. Let AHB be divided into equal parts, and using these parts as altitudes, let figures composed of rectangles be inscribed and circumscribed again on the triangle AHL, as before. Let this be done in such a way that the one exceeds the other by an amount less than the amount by which the plane P exceeds the triangle AHL. Hence the circumscribed figure will necessarily be less than the plane P. Now it is clear that in the first part of the time AC the distance crossed by the body is less than BC, since this would be crossed in the same time with the uniform velocity CK which the body requires only at the end of the time AC. Likewise in the second part of the time CE the distance crossed by an accelerated motion is less than DE since this would be crossed in the same time CE with the uniform velocity EO which it acquires only at the end of the time CE. And in like fashion, in each part of the time AH, less distance is crossed by the moving body than is represented by the rectangles circumscribed on the figure adjacent to each part of the time. Hence the whole distance crossed by an accelerated motion will be less than this circumscribed figure. But that distance was assumed to be equal to the plane P. Thus the plane P will be less than the circumscribed figure, which is absurd since this figure was shown to be less than the plane P. Therefore the plane P is not larger than the triangle AHL. But it was also shown that it is not less than the triangle. Therefore it must be equal. QED. (Hor. 29–31/B 40–42)

Huygens’s exquisite proof involves no reference implicit or explicit to infinity. It relies on the continuity of the line AH to ensure that the excess of the step figures—represented by the ‘lowest rectangle’ that has HL as its base—can be made,

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102 continuous extension and indivisibles in galileo respectively, smaller than the difference between P and the inscribed figure (in the first reductio ad absurdum) and smaller than the difference between P and the circumscribed figure (in the second reductio). Continuity guarantees that the height of any given rectangle can be made as small as one wishes, and since the base is fixed to be a given finite length HL, the area of the lowest rectangle can thereby be taken as small as one wishes. Unlike in Galileo’s proof, this continuity is never cashed out as an aggregate of parallels in the triangle AHL and rectangle AHLM. Dubious ratios between infinites are thus avoided, as is the suggestion that a surface might consist of indivisible lines. The tradeoff is that Huygens’s proof must follow the doubly indirect and correspondingly elaborate method of exhaustion. Compared to the ‘less perfect form’ of Galileo’s direct demonstration using parallels, it is a cumbersome affair. If only there were a technique as effective for discovery and calculation as the new direct infinitary methods and as rigorous as the traditional indirect ones to make for a consilience of ancient and modern ideas. As it happens, Huygens gave a personal copy of his new Horologium oscillatorium to his premier student, who inscribed on its title page that he had come to possess this work ‘ex dono authoris’ (A 7.4.30). The student was Leibniz.

References Cavalieri, Bonaventure. 1635. Geometria indivisibilibus continuorum nova quadam ratione promota. Bologna. [GIC] Cavalieri, Bonaventure. 1647. Exercitationes Geometricae Sex. Bologna. [EGS] Galilei, Galileo. 1890–1909. Le Opere di Galileo Galilei. Edizione Nationale. 20 vols. Edited by Antonio Favaro. Florence: Barbera. [EN. Cited by volume and page.] Galilei, Galileo. 1967. Dialogue Concerning the Two Chief World Systems, 2nd ed. Translated and edited by Stillman Drake. Berkeley and Los Angeles: University of California Press. [D67] Galilei, Galileo. 1974. Two New Sciences, Including Centers of Gravity and force of Percussion, 2nd ed. Translated and edited with commentary by Stillman Drake. Toronto: Wall and Emerson, Inc. [D74] Guldin, Paul. 1635–41. Centrobaryca seu centro gravitatis trium specierum quantitatis continue. 4 vols. Vienna. Huygens, Christiaan. 1673. Horologium oscillatorium sive de motu pendulorum ad horologia aptato demonstrationes geometricae. Paris: F. Muguet. [Hor.] Huygens, Christiaan. 1986. The Pendulum Clock or Geometrical Demonstrations Concerning the Motion of Pendula as Applied to Clocks. Translated by Richard J. Blackwell. Ames: The Iowa State Press. [B]

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references 103 Leibniz. 1923–. Samtliche Schriften un Briefe. Philosophische Schriften. Berlin: Akademie-Verlag. [A. Cited by series, volume, and page.] Sturm, J. C. 1700. Mathesis enucleata: Or the Elements of Mathematicks. London.

Secondary Literature Andersen, Kirsti. 1985. “Cavalieri’s Method of Indivisibles.” Archive for the History of Exact Sciences, 31: 291–367. Blay, Michael. 1993. Reasoning with the Infinite: From the Closed World to the Mathematical Universe. Chicago: Chicago University Press. Cantor, George. 1886. “Über die verschiedenen Ansichten in Bezug auf die actualunendlichenen Zahlen.” Bihang Till Koniglen Svenska Vetenskaps Akademiens Handligar 11: 1–10. Drabkin, I. E. 1950. “Aristotle’s Wheel: Notes on the History of a Paradox.” Osiris 9: 126–198. Grünbaum, Adolph. 1952. “A Consistent Conception of the Extended Linear Continuum as an Aggregate of Unextended Elements.” Philosophy of Science 19: 288–306. Hallet, Michael. 1986. Cantorian Set Theory and Limitation of Size. Oxford: Oxford University Press. Holden, Thomas. 2004. The Architecture of Matter: Galileo to Kant. New York: Oxford University Press. Jesseph, Douglas. 2021. “The Indivisibles of the Continuum: Seventeenth-Century Adventures in Infinitesimal Mathematics.” This volume. Knobloch, Eberhard. 1999. “Galileo and Leibniz: Different Approaches to Infinity.” Archive for History of Exact Sciences 54: 87–99. Knobloch, Eberhard. 2011. “Galileo and German Thinkers: Leibniz.” In L. Pepe, ed., Galileo e la scuola galileiana nelle Università del Seicento (Bologna: Cooperativa Libraria Universitaria Bologna), 127–139. Levey, Samuel. 2015. “Comparability of Infinities and Infinite Multitude in Galileo and Leibniz.” In N. Goethe, P. Beeley, and D. Rabouin, eds., G. W. Leibniz: Interrelations between Mathematics and Philosophy, Archimedes Series 41 (Dordrecht: SpringerVerlag), 157–187. Mancosu, Paulo. 1996. Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century. New York: Oxford University Press. Skyrms, Brian. 1983. “Zeno’s Paradox of Measure.” In R. S. Cohen and L. Lauden, eds., Physics, Philosophy and Psychoanalysis (Dordrecht: D. Reidel Publishing Company), 223–254.

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5 The Indivisibles of the Continuum Seventeenth-Century Adventures in Infinitesimal Mathematics Douglas M. Jesseph

The seventeenth century was a period of profound change in European mathematics. At the beginning of the century, the prominent mathematician’s standard employment was the preparation of editions of ancient authors with extensive commentary. By 1700 the calculus was well established and a vast collection of new results had been obtained by the employment of new methods. The new techniques solved such problems as finding areas of curvilinear figures, determining arc lengths, drawing tangents, and investigating minima and maxima with essentially algorithmic procedures that achieved an immense number of new results. Two elements are of particular significance in the development of the calculus: the advent of analytic geometry that represents geometric magnitudes algebraically, and the rise of infinitesimal techniques that permits continuous magnitudes to be treated as infinite sums of infinitely small elements. My concern is with the second of these, and specifically with the growth of the ‘method of indivisibles’ that formed the basis for the infinitesimal techniques developed by Leibniz, Newton, and others.

1. Cavalieri and the Method of Indivisibles Classical Greek mathematics drew a firm distinction between two species of quantity: the continuous magnitudes of geometry and the discrete multitudes of arithmetic. Geometric magnitudes such as lines, angles, surfaces, and solids are all divisible into lesser magnitudes of the same kind (lines divide into lines, angles into angles, etc.). In contrast, the discrete multitudes of arithmetic are only finitely divisible and are ultimately composed of indivisible units. Viewed in this way, geometry and arithmetic are two fundamentally distinct sciences with radically different objects. This explains, for instance, why the theory of ratios is developed

Douglas M. Jesseph, The Indivisibles of the Continuum: Seventeenth-Century Adventures in Infinitesimal Mathematics In: The History of Continua: Philosophical and Mathematical Perspectives. Edited by: Stewart Shapiro and Geoffrey Hellman, Oxford University Press (2021). © Stewart Shapiro and Geoffrey Hellman. DOI: 10.1093/OSO/9780198809647.003.0006

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cavalieri and the method of indivisibles 105 twice in the Euclidean Elements, first in Book V with regard to continuous magnitudes and again in Book VII for discrete multitudes. A significant part of the reason for this distinction is that it avoids difficult issues concerning the composition of continua: if every continuous magnitude has parts that are also continuous, there is no puzzle about how, for instance, a line segment might be composed out of dimensionless, indivisible points. Although a line can truly be said to ‘contain’ points, these are not, strictly speaking, parts of the line: it cannot be resolved into them, nor can it be composed from them. The classical approach runs against the contemporary orthodoxy, namely that a line is composed of uncountably many points. A significant step towards the contemporary point of view came about in the seventeenth century with the development of the method of indivisibles. The first exposition of the method was in Bonaventura Cavalieri’s Geometria indivisibilibus continuorum nova quadam ratione promota (Cavalieri 1635). The method takes its point of departure from the fact that we can establish a correspondence between the lines contained in a plane figure. Consider the circle ABCD and the oblong figure of equal height EFGH (Figure 5.1). By passing the line LM (which Cavalieri termed the regula) upwards to position IK, the successive intersection of the regula and the figures will be a collection of lines. Cavalieri spoke of the transit of the regula as producing ‘all the lines’ or ‘the indivisibles’ of the figure.1 Cavalieri insisted that he did not seek to compose the continuum from indivisibles. Instead, he offered indivisibles as a new species of magnitude that could be incorporated into the general theory of ratios and magnitudes presented in Book V of the Euclidean Elements. Thus, a central result in Cavalieri’s Geometria is Theorem III of Book II: ‘All the lines . . . of any plane figures, and all the planes of any solids, are magnitudes having a ratio to one another’ (Cavalieri 1635, 108). To show that indivisibles behave as classical magnitudes, Cavalieri argued that E

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Figure 5.1 Cavalieri’s indivisibles. 1 In an analogous way, the transit of a plane regula through a solid would produce ‘all the planes’ of the solid or the indivisibles of the solid. I will consider only the two-dimensional case.

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106 17th-century adventures in infinitesimal mathematics collections of indivisibles stand in well-defined ordering relations. For instance, he postulated that the indivisibles of congruent figures are equal; that if figure A is a proper part of B, that the indivisibles of A are less than those of B; and that if figure A can be decomposed into two figures B and C, then the indivisibles of A are equal to the sum of the indivisibles of B and C.2 Cavalieri employed his method of indivisibles almost exclusively in pursuit of the goal of determining the areas of figures or the volumes of solids. In the twodimensional case, the basic procedure he employed was to establish a ratio between the indivisibles of two figures, and then conclude that the areas of the figures stand in the same ratio as their indivisibles. As he put the matter: It is clear from this that when we want to find what ratio two plane figures or two solids have to one another, it is sufficient for us to find what ratio all of the lines of the figure stand in (and in the case of solids, what ratio holds between all of the planes), relative to a given regula, which I lay as the great foundation of my new geometry. (Cavalieri 1635, 115)

This procedure leads to what has become known as ‘Cavalieri’s principle’, namely that if two figures having equal altitudes and sections made by lines parallel to the bases at equal distances are always in a given ratio, then the areas of the figures stand in that same ratio. The strongly finitistic character of Greek geometry and traditional concerns about the composition of the continuum make it natural to worry whether this method is, indeed, completely rigorous. To forestall concerns about the intelligibility of ‘all the lines’ as a metric concept, Cavalieri argued that it was no more conceptually problematic than the notion of an area. In his words: I should indicate that when I consider all the lines or all the planes of some figure, I do not compare their number, which we do not know, but only the magnitude, which is equal to the area occupied by these same lines and is congruent with them. Because this area is contained in boundaries, so therefore the magnitude of all the lines is contained in the same boundaries, and therefore they can be added or subtracted, although we do not know their number. I say that this is enough to make them comparable to one another, for otherwise neither could the same areas of the figures be comparable to one another. (Cavalieri 1635, 111)

In a further effort to vindicate the rigour of reasoning from indivisibles, Cavalieri introduced an alternative formulation of his method in Book VII of the Geometria. Following Andersen (1985, section IX) we can call this the ‘distributive’ 2 These postulates appear at the beginning of Book II of the Geometria. See Giusti (1980, chapter 3) and Andersen (1985, sections V and VI) for detailed accounts of this aspect of Cavalieri’s programme.

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cavalieri and the method of indivisibles 107 method of indivisibles, in contrast to the ‘collective’ method that is employed throughout the first six books of the Geometria. The difference between these two approaches lies in the kind of comparison made between the indivisibles of two figures. The collective method considers the ratio between the whole collection of lines (such as ‘all the lines’ in figure EFGH and ABCD of Figure 5.1). In contrast, the distributive method compares only lines to one another (such as RS to NO, FH to BD, or TV to PQ). Cavalieri recognized that some would worry that his collective method appears to compose the continuum out of indivisibles,3 but he took the distributive method to face fewer concerns about its rigour, since it requires only that pairs of lines contained within figures be compared, with no suggestion that continuous magnitudes are literally composed of indivisibles. Although the distributive method played a secondary role in Cavalieri’s Geometria, his Exercitationes Geometricae Sex of 1647 featured it more prominently. The principal result in that work is the investigations of areas determined by ‘cossic powers’, or curves of the form axn . Indeed, Cavalieri proved a geometric 1 a equivalent of the theorem ∫0 xn dx = an+1 . His formulation of the result reads (n+1)

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Figure 5.2 Cavalieri’s integration of ‘cossic powers’.

3 In the Preface to Book VII of the Geometria, where the collective method is set out, Cavalieri declared, ‘It is by no means unknown to me that many things concerning the composition of the continuum and also concerning the infinite are disputed by philosophers, and this will perhaps seem to many to be prejudicial to my principles; in consequence of which they will surely be hesitant, because it might appear as if the concepts of all the lines or all the planes are chimerical, as if they were more obscure than inscrutable darkness’ (Cavalieri 1635, 482).

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108 17th-century adventures in infinitesimal mathematics In any parallelogram such as BD, with the base CD as regula, if any parallel to CD such as EF is taken, and if the diameter AC is drawn, which cuts the line EF in G, then as DA is to AF, so CD or EF will be to FG. And let AC be called the first diagonal. And again, as DA2 is to AF2 , let EF be to FH, and let this be understood in all the parallels to CD, so that all of these homologous lines HF terminate in the curve AHC. Similarly, as DA3 is to AF3 , let also EF be to FI, and likewise in the remaining parallels, to describe the curve CIA. And as AD4 is to AF4 let EF be to FL, and likewise in the remaining parallels to describe the curve CLA. Which procedure can be supposed continued in the other cases. Then CHA is called the second diagonal, CIA the third diagonal, CLA the fourth diagonal, and so forth. Similarly, the triangle AGCD is called the first diagonal space of the parallelogram, the trilinear figure AHCD is the second space, AICD the third, ALCD the fourth, and so on. I say therefore that the parallelogram BD is twice the first space, triple the second space, quadruple the third space, quintuple the fourth space, and so forth. (Cavalieri 1647, 279)

The geometric style of presentation of the result is somewhat confusing to the modern reader, but the idea is simple enough: if the line CA is straight, the ratio between the two triangles CBA and CDA is 1:1; if the line is a parabola, increasing as the square of the distance covered along CD, then the ratio of the trilinear figure CBAHC to the figure CDAHC is 2:1, and so forth for higher powers. In his proof Cavalieri introduced the notion of ‘all the squares’ of a figure (which arise from squaring the lines contained within a figure), as well as ‘all the cubes’ and ‘all the square-squares’ for higher powers. The details of Cavalieri’s proof need not detain us,⁴ but the main idea should be tolerably clear.

2. Torricelli and the Extension of Indivisibles In the course of events, Cavalieri was correct in thinking that the method of indivisibles would face resistance. Opposition was far from universal, as the method gained considerable currency in the 1650s. Broadly speaking, the objections raised to the method of indivisibles fall into two categories, which we can term ‘conceptual’ and ‘consequential’. The conceptual criticisms focus on the difficulty of reconciling the method with traditional strictures against the comprehensibility of the infinite or the notorious problems of composing continua from indivisible parts. The consequential criticisms aim to show that the method of indivisibles yields false results and must therefore be rejected as unrigorous.

⁴ They can be found in Palmieri (2009).

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torricelli and the extension of indivisibles 109 These two kinds of criticisms are not entirely unrelated, since a method whose foundations are murky might well be expected to deliver false consequences. Still, there is a tolerably clear division between these sorts of criticisms, and authors who objected to Cavalieri’s method tended to prefer one sort of objection to the other.⁵ For instance in the fourth volume of his Centrobaryca (which appeared in 1640), the Jesuit Paul Guldin attacked the method of indivisibles as conceptually unsound. To the first theorem of Book II of the Geometria (which asserts that the indivisibles of a figure are magnitudes in the Euclidean sense), Guldin objected: This proposition is completely false, and I oppose this argument to it: All the lines and all the planes of one figure and another are both infinite. But there is no proportion or ratio of a infinite to an infinite. Therefore, etc. Both the major and minor premises of the argument are clear to all geometers, and so do not need many words. Therefore, the conclusion of Cavalieri’s proposition is false. (Guldin 1635–41, 4: 343)

A number of consequential criticisms were raised by André Tacquet, in his 1651 treatise Cylindricorum et annularium libri IV.⁶ A representative one can be phrased thus: consider a right cone whose base has radius r and whose height is h. Now take the lateral surface of the cone as an aggregate of the peripheries of circles with diminishing radii, starting at the base with radius r and ending in a point at the vertex, with radius 0. In addition to forming the lateral surface of the cone, the aggregate of these circular peripheries is also equal to the area of the circle that forms the base of the cone. Therefore, the method of indivisibles entails the false consequence that the lateral area of the conic surface is equal to the area of the base. The contributions of Evangelista Torricelli to the theory of indivisibles are best understood against the background of the consequential criticisms of the method. Torricelli established his reputation with his Opera geometrica of 1644, in which he proved a number of important results. He made his commitment to Cavalieri’s methods explicit, declaring ‘it is certain that this geometry is a remarkably brief means of discovery, and confirms innumerable nearly inscrutable theorems by brief, direct, and affirmative demonstrations which could never be done by the doctrine of the ancients’, from which he concluded that in geometry ‘this is the true royal road, which was first opened and levelled for the public good by Cavalieri’ (Torricelli 1644, 3: 56). Torricelli often used a variety of proofs to derive a single result, notably proofs using classical ‘exhaustion’ arguments in the style of Archimedes as well as proofs by means of indivisibles. This approach is on

⁵ See Festa (1992) for an account of seventeenth-century objections to the method. ⁶ On Tacquet’s objections, which are rooted in the idea that ‘heterogeneous’ species of magnitude cannot be compared in ratios, see Descotes (2015a, 254–267).

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110 17th-century adventures in infinitesimal mathematics display in the Quadratura parabolae, which is one of the works published as part of the Opera geometrica. Torricelli provided ten different classical proofs showing how to determine the area enclosed by a parabolic segment, to which he added eleven proofs using the method of indivisibles.⁷ A clear part of the motivation for this procedure was the thought that the new method could be confirmed by showing that it generated the same results as classical techniques, whose rigour was unchallenged.⁸ Moreover, the relative ease and simplicity of proofs using indivisibles contrast with the complexity of traditional ‘exhaustion’ techniques, thus offering an additional motivation for the new method.⁹ Torricelli’s approach to the method of indivisibles can be seen in one of his proofs for the quadrature of the parabola. Taking the parabolic segment ABC with tangent CD and diameter AD (Figure 5.3), he sought the ratio between the D

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Figure 5.3 Torricelli’s quadrature of the parabolic segment.

⁷ He observed: ‘Thus far what has been said of the dimension of the parabola has been in the style of the ancients. It remains that we investigate the same dimension of the parabola by a certain new yet remarkable reasoning; that is by way of the geometry of indivisibles, and this in different ways’ (Torricelli 1644, 3: 55). ⁸ In this, Torricelli followed Cavalieri. In the second book of the Geometria he declared: ‘It seems worthwhile to observe, in confirmation of what we have assumed, how many things that have been shown by Euclid, Archimedes, and others are equally demonstrated by me, and my conclusions agree completely with theirs, which can be an evident sign that I have assumed the truth in my principles’ (Cavalieri 1635, 112). ⁹ De Gandt concludes that ‘[by] this disposition of the material, the reader is first invited to observe how difficult and unnatural the demonstrations are without indivisibles. Then comes the geometry of indivisibles, this new and admirable way, that demonstrates innumerable theorems ‘by brief, direct, and affirmative demonstrations,’ and in comparison with which ancient geometry makes a ‘pitiful’ figure’ (De Gandt, 1987, 152–153).

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torricelli and the extension of indivisibles 111 parallelogram ACED and the trilinear figure ABCD. He first demonstrated that, for any given diameter GF, the ratio FG∶IB is the same as the ratio between the areas of circles with diameters FG and IG. Since the line GF was taken arbitrarily, this result generalizes to hold for any diameter parallel to AD. Thus, Torricelli concluded that all the lines FG are to all the lines IB as all the circles with diameter FG are to all circles with diameter IG. From these considerations, he concluded that All of the first lines taken together (that is the parallelogram AE) will be to all of the second lines taken together (that is the trilinear figure ABCD) as all of the third circles together (that is the cylinder AE) to all of the fourth circles together (that is, the cone ACD). And thus the parallelogram is triple the area of the trilinear space. (Torricelli 1644, 3: 57)

The most interesting feature of this argument is the fact that where Cavalieri had taken pains to avoid the suggestion that a collection of indivisibles could compose a continuous magnitude of higher dimension, Torricelli did not hesitate to claim that a collection of lines can compose a surface or a collection of circles could compose a cone or cylinder. Although he did not baulk at asserting that an infinite collection of indivisible lines can compose a surface, Torricelli was well aware of the objections that had been raised against indivisibles, and he went to some trouble to overcome them. As a result, he ultimately developed an understanding of indivisibles that significantly modified Cavalieri’s initial formulation of the method. The best source for these objections and Torricelli’s response is a manuscript that bears the title De indivisibilium doctrina perperam usurpata, or ‘On the doctrine of indivisibles wrongly applied’ (Torricelli 1919–44, 1: 417–432).1⁰ Torricelli’s account of the improper use of indivisibles opens with a simple example that shows how the method can seem to lead to paradox. We begin (as in Figure 5.4) with a rectangle, whose side AB is twice the length of the side AD. Having drawn the diagonal DB, we can consider two orthogonal line segments, EF and EG, that meet in the diagonal. By construction, EF is twice EG, and this holds A

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Figure 5.4 Torricelli’s ‘paradox of indivisibles’. 1⁰ The manuscript and its connection with Torricelli’s treatment of indivisibles have been discussed more extensively in the literature than my account here (De Gandt, 1987 and Bascelli, 2015).

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112 17th-century adventures in infinitesimal mathematics for every pair of orthogonal segments that meet in DB. We therefore seem led to the conclusion that ‘all the lines’ of the triangle ABD stand in the ratio 1:2 to ‘all the lines’ in the triangle CDB. Consequently, the method of indivisibles (whether understood collectively or distributively) seems to require the conclusion that the area of ADB is twice the area of CDB. This is an unwelcome result, since the two triangles are equal by construction.11 The central issue raised by this example is ‘whether a property that holds for each indivisible also holds for their aggregate’ (Bascelli 2015, 119). Torricelli focused on three important facts connected with this seeming paradox. First, the areas of the triangles ABD and CDB are equal; second, the paired indivisibles such as EF and EG are unequal; and third, the areas are aggregates of these indivisibles. From these considerations, he concluded that the indivisibles must differ in ‘thickness’—so that EF is a (relatively) long and thin line, while EG is (comparatively) shorter and thicker. This way of handling the inconsistency also requires that the points along the line DC are ‘larger’ than the corresponding points along the shorter line DA. More generally, Torricelli’s approach requires that indivisibles not be one dimension less than the geometric objects from which they are generated. That is to say, the indivisibles of a one-dimensional line are not zero-dimensional points, but tiny ‘linelets’ with extension; the indivisibles of a twodimensional surface are not one-dimensional lines, but rectangles of minuscule breadth; and the indivisibles of a three-dimensional solid are not two-dimensional plane surfaces, but parallelepipeds of negligible height. In Bascelli’s words, the paradox ‘disappears when we imagine points and lines as minuscule geometrical entities with two or three dimensions’ (2015, 125). In addition to being extended (whether in one, two, or three dimensions), Torricelli’s indivisibles have internal structure, most notably curvature. The intuition at the root of this doctrine is simple enough. If we consider two concentric circles C1 and C2 with a common radius r, the points of intersection between the circles and r can be designated p1 and p2 (Figure 5.5). It is clear that ‘all the points’ on the periphery of C1 stand in one-to-one correspondence with ‘all the points’ on the periphery of C2 , so Torricelli requires that the indivisibles on C1 are smaller than those on C2 in order to avoid the paradoxical consequence that the peripheries are equal. He further noted that the curvature of C1 at every point on the periphery is greater than that of C2 , and he attributed curvature to his indivisibles, so that the indivisible at p1 is not only smaller than that at p2 , but more tightly curved. This innovation allowed Torricelli to extend the method of indivisibles to the consideration of arc lengths and the calculation of solids of revolution.12 11 It should be noted that Cavalieri’s formulation of the method of indivisibles requires that the indivisibles of two figures be compared with respect to a common regula. In the case outlined here (and in other versions of the same sort of paradox), this requirement is ignored. Thus, strictly speaking, Cavalieri’s methods can avoid the objection. Nevertheless, the pairing of ‘all the lines’ contained in two figures clearly invites this sort of critique, and Cavalieri’s restriction seems rather ad hoc. 12 Bascelli (2015, 125–131) surveys the novelty of Torricelli’s indivisibles with respect to this issue.

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torricelli and the extension of indivisibles 113

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Figure 5.5 Torricelli’s curved indivisibles.

The most remarkable result enabled by this innovation was Torricelli’s proof that the volume of the solid of revolution generated from a branch of the rectangular hyperbola about its axis has infinite length but finite volume. Extolling the virtues of his method of curved indivisibles, Torricelli declared: Our method, which we will apply in the proof of the aforementioned theorem, proceeds by curved indivisibles, without following the example of anyone else, though not without the previous approval of geometers. We will consider all the cylindrical surfaces describable about a common axis in our acute hyperbolic solid, but since Cavalieri himself offered no part of this subject in his Geometria, we have judged that our method of reasoning should be confirmed by some examples . . . Curved indivisibles adequate for these demonstrations are peripheries of circles in plane figures, and spherical, cylindrical, and conic surfaces in solid figures. They have the advantage of fitting perfectly and having, so to speak, a thickness which is always equal and uniform. (Torricelli 1644, 3: 94–95)

Torricelli’s main result was regarded alternately as an example of the power of his methods and as an expression of the inherently paradoxical nature of infinitesimal reasoning, and it established his reputation as one of Europe’s leading mathematicians.13 We can credit Torricelli with significant innovation in the theory of indivisibles, and mathematicians who employed the method were generally eager to adopt his conception of indivisibles as having extension and internal structure. It is worth noting, however, that his success in overcoming the consequential objections to indivisibles comes at the expense of ignoring the conceptual objections. The method in its Torricellian guise may avoid the charge that it delivers false results, but accepting it requires making sense of the notion that two points can differ in magnitude, or one line segment can be thicker than another. As we will see, these 13 See Mancosu (1996, 129–149) on this result and its philosophical reception.

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114 17th-century adventures in infinitesimal mathematics sorts of questions persisted, but proponents of the method of indivisibles were generally happy to ignore them.

3. Roberval’s Method of Indivisibles The contributions of Gilles Personne de Roberval to the method of indivisibles are very significant, but their origins, chronology, and impact are difficult to assess with any great degree of confidence. Roberval held the chair of mathematics at the Collège Royal in Paris from 1633 until his death in 1675. The chair was awarded on the basis of triennial competitions, with challenge problems set by the chair’s incumbent. This arrangement gave Roberval ample incentive to keep his methods to himself, and he published relatively little pure mathematics in the course of his career.1⁴ He did, however, correspond widely (often through the mediation of Marin Mersenne) and his results were generally communicated by way of correspondence. Roberval’s most important single work on indivisibles is a compendium known as the Traité des Indivisibles. This remained unpublished during his lifetime but appeared in 1693 as part of the Divers Ouvrages de M. De Roberval edited and published by the Académie Royale des Sciences (Roberval, 1693, 247–360). Roberval’s procedures are similar in many respects to those of Torricelli, which is to say that he considered plane surfaces as composed out of infinitely many parts and he attributed dimension and internal structure to indivisibles. Roberval compared his approach to indivisibles with those of Cavalieri in a 1647 letter to Torricelli, writing: We consider our infinities or indivisibles in this way: a line is made up, as it were, of an infinite or indefinite number of lines, a surface from an infinite or indefinite number of surfaces, a solid of solids, an angle of angles, an indefinite number of indefinite units; and indeed, we conceive of a square-square to be composed of square-squares, and so forth. Each one of these categories has its own properties, Moreover, by breaking each figure down into its own infinite parts, we almost always observe a certain equality, or at least a relation, between the height or width of those parts. (Roberval 1693, 286)

Roberval also employed a method of ‘composition of motions’ which involved analysing a curve as produced by a point in motion, and he took the motion at any instant as a composition of two motions with different directions. In essence, the method decomposes a continuous motion at any instant into indivisible

1⁴ See Julien (2015b) for a detailed account of Roberval’s use of the method of indivisibles.

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roberval’s method of indivisibles 115 ‘tendencies’ that jointly determine its magnitude and direction. This method was not a complete innovation on Roberval’s part. It has classical antecedents in the definitions of certain curves by the compounded motions of geometric magnitudes, as well as in Galileo’s construction of the parabola from uniform rectilinear motion along one axis and uniformly accelerated motion along another.1⁵ By exploiting both infinitesimal techniques and the composition of motions Roberval produced quite a number of results which extended well beyond what his predecessors had achieved. Roberval’s methods are best illustrated in his study of the cycloid—the curve traced by the point on the periphery of a circle as the circle rolls across a straight line.1⁶ In Figure 5.6, the circle with diameter AB moves to the right, rotating without slipping as it moves until point A arrives at D, making the line AC equal to half of the circumference of the circle. The cycloid is the curve A 8 9 10 11 12 13 14 D traced by the composition of rectilinear motion parallel to the line AB and rotational motion in the circumference AGB. Using an argument based on the division of the relevant geometric magnitudes into infinitely many infinitesimal parts, Roberval determined the path of the cycloid and the area of the space enclosed by it and the straight lines AC and CD. The first step in his

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Figure 5.6 Roberval on the quadrature and tangent construction for the cycloid.

1⁵ For details on Galileo’s use of indivisibles in the context of his analysis of motion, see Julien (2015a). 1⁶ Torricelli had also examined the cycloid after Marin Mersenne had invited Italian mathematicians (notably Galileo and Cavalieri) to investigate the curve and its properties. This ultimately led to a priority dispute between Torricelli and Roberval. See Jesseph (2007, 417–418) on this issue.

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116 17th-century adventures in infinitesimal mathematics argument is to notice that as the point A rotates in the circumference to point E, it will be carried up to a height A1, while the rectilinear motion of the circle will carry its centre through the interval AM. Similarly, rotation to point F is paired with rectilinear motion through AN, and so forth for all the points in the periphery of the circle. This allows the construction of two curves: the cycloid itself and the companion curve A 1 2 3 4 5 6 7 D, which is formed by pairing the points M, N, O, P, Q, R, S, T, with the heights 1, 2, 3, etc. The lines E1, F2, G3, etc. are the sines of the angles of rotation of the point, and the difference between the cycloid and its companion curve is always equal to the sine of the angle of rotation. To find the area of the space enclosed by the curve, Roberval argued that [T]wo curves enclose a space, being separated from one another by all the sines and joining together again at the two extremities A, D. Now, each part contained between these two curves is equal to each part of the area of the circle AEB contained in the circumference and diameter, for both are composed of equal lines, namely of the heights A1, A2, etc, and of the sines E1, F2, etc., which are the same as these diameters M, N, O, etc., and thus the figure A 4 D 12 is equal to the semicircle AHB. Now the line A 1 2 3 D divides the parallelogram ABCD in two equal parts, because the lines of one half are equal to the lines of the other half, and the line AC is equal to the line BD, and consequently (according to Archimedes), half of the parallelogram is equal to the circle. Then adding the semicircle (i.e., the space between the two curves), we will have one and a half circles equal to the space A 8 9 D C; and doing the same for the other half of the figure, the whole figure of the cycloid will be three times the circle. (Roberval, 1693, p. 192)

Roberval determined the tangent to the cycloid by an argument from composition of motions. To find the tangent at an arbitrary point (for instance, point 13 in Figure 5.6), he first took the tangent to the circle at that point; because the cycloid is described by a combination of circular and rectilinear motions, this will determine the tangent to the circular component of the complex motion. Taking the longitudinal velocity as represented by the diameter of the circle, Roberval then constructed an equilateral parallelogram whose sides are parallel to the circular tangent and the diameter. The equality of the sides of the parallelogram is dictated by the supposition that the circular and rectilinear motions are equal, and the diagonal of the parallelogram will be the tangent to the curve. As Roberval explained: To find the tangent to the figure at a given point, I draw a tangent to the circle which passes through that point, since each point of the circle moves along the tangent to along the tangent of the circle. I then consider the motion we have given to our point in being carried by the diameter moving parallel to

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john wallis and the arithmetic of infinities 117 itself. Drawing from the same point the line of this movement, if I complete the parallelogram (which must always have its four sides equal since the path of the point A through the circumference is equal to the path of the diameter AB along the line AC), and if from the same point I draw the diagonal, I will have the tangent to the figure which has had these two motions for its composition, namely the circular and the rectilinear. This is how we proceed when we suppose the movements to be equal. But if we had supposed that instead of being equal the movements were in some other ratio . . . the parallelogram would have to have been constructed with its sides in that ratio. (Roberval, 1693, pp. 192–193)

Roberval’s approach to indivisibles clearly has much in common with Torricelli’s. By taking indivisibles as infinitely small magnitudes of the same dimension as the lines, figures, or solids they compose, he could avoid the paradoxes that seemed to threaten Cavalieri’s methods, just as Torricelli had done. Likewise, Roberval’s use of the composition of motions is closely connected with Torricelli’s notion that indivisibles have curvature and internal structure: a continuous motion has indivisible ‘tendencies’ in different directions that answer to Torricelli’s notion of an indivisible point having curvature. The most salient difference between the two is on the question of the rigour of the method: where Torricelli spent considerable time and effort to establish the reliability of arguments from indivisibles (such as proving the same result multiple times, with both classical exhaustion techniques and indivisibles), Roberval seems to have been content to leave questions of rigour to others.

4. John Wallis and the Arithmetic of Infinities In 1656 John Wallis (Oxford’s Savilian Professor of Geometry) published a collection entitled Operum Mathematicorum Pars Altera (Wallis 1656). This included two works that employed the method of indivisibles: De Sectionibus Conicis Nova Methodo Expositis Tractatus (‘A Treatise of Conic Sections Set Forth by a New Method’) and Arithmetica Infinitorum (‘The Arithmetic of Infinities’). In explaining the basis of his new approach to the classical theory of conic sections, Wallis announced, ‘I suppose, to begin with (according to the Geometry of Indivisibles of Bonaventura Cavalieri) any plane to be made up (so to speak) out of an infinity of parallel lines; or (which I prefer) from an infinity of parallelograms of the same altitude. Let the altitude of any one of them be 1/∞ of the whole . . . and the altitude of all together being equal to the altitude of the figure’ (Wallis, 1656, 2: 4). Wallis then distinguished an infinitely narrow parallelogram from an indivisible line by characterizing the former as ‘dilatable, or having so much thickness that by infinite multiplication it can acquire a certain altitude or latitude, namely as much as is in the altitude of a figure’ (Wallis 1656, 2: 4).

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118 17th-century adventures in infinitesimal mathematics Although Wallis’s use of indivisibles in the theory of conic sections yielded nothing that was not already obtainable with ancient methods, the Arithmetica Infinitorum produced important new results. The principal result is the ‘Wal𝜋 2×2×4×4×6×6 . . . lis Product’ = . Wallis credited Torricelli with sparking his in2 3×3×5×5×7×7 . . . terest in the new techniques, writing in the letter of dedication to the Arithmetica Infinitorum that ‘at the end of 1650 I came across Torricelli’s mathematical writings . . . where among other things he expounds Cavalieri’s method of indivisibles. I did not have Cavalieri’s work itself at hand and sought it several times in vain at the booksellers, but his method (as expounded by Torricelli) pleased me all the more because, from the time I first encountered mathematics, I did not know that something of the kind had ever been considered’ (Wallis 1656, 3: 3). In Wallis’s procedure geometric problems are represented analytically by equations and solved by using infinite series summations to determine the area of a figure, taking the figure to be an infinite sum of infinitely small elements. As an example we can consider his approach to the quadrature of the cubic parabola. He commenced his reasoning with arithmetical results, observing that: 1 1 1 0+1 = = + 1+1 2 4 4 0+1+8 9 3 1 1 = = = + 8+8+8 24 8 4 8 0 + 1 + 8 + 27 36 4 1 1 = = = + 27 + 27 + 27 + 27 108 12 4 8 100 5 1 1 0 + 1 + 8 + 27 + 64 = = = + 64 + 64 + 64 + 64 + 64 320 16 4 16 From these initial cases, Wallis inferred ‘by induction’ that as the number of terms in the sums increases, the ratio approaches arbitrarily near to the ratio 1:4. From this he concluded that, in the infinite case, the precise result 1:4 will obtain. Given this result, Wallis then turned to the quadrature of the cubic parabola, treating it as an infinite sum of lines forming a series of cubic quantities (as in Figure 5.7): And indeed let AOT (with diameter AT, and corresponding ordinates TO, TO, &c.) be the complement of the cubic semiparabola AOD (with diameter AD and corresponding ordinates DO, DO, &c.). Therefore, (by Proposition 45 of the Treatise of Conic Sections) the straight lines DO, DO, &c. or their equals AT, AT, &c. are in subtriplicate ratio of the straight lines AD, AD &c. or their equals TO, TO, &c. And conversely these TO, TO, &c. are in triplicate ratio of the straight

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john wallis and the arithmetic of infinities 119 A

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Figure 5.7 Wallis and the quadrature of the cubic parabola.

lines AT, AT, &c. Therefore the whole figure AOT (consisting of the infinity of straight lines TO, TO, &c. in triplicate ratio of the arithmetically proportional straight lines AT, AT, &c.) will be to the parallelogram TD (consisting of just as many lines all equal to TO) as one to four. Which was to be shown. And consequently the semiparabola AOD (the residuum of the parallelogram) is to the parallelogram itself as one to four. (Wallis, 1693–9, vol. 1, p. 383)

Like Torricelli and Roberval, Wallis took a two-dimensional figure as literally composed of an infinite collection of lines, but he understood lines as infinitely narrow two-dimensional figures rather than one-dimensional objects. Wallis’s method is also noteworthy for its use of arithmetical principles involving ratios of infinite series to the solution of geometric problems. Despite his departures from traditional geometry, Wallis held that the method of indivisibles was essentially equivalent to the classical technique of exhaustion, although he admitted that it must be used with care in order to avoid paradox. As he put the matter at the beginning of the fourth chapter of his 1670 treatise Mechanica: ‘this doctrine of indivisibles (now everywhere accepted, and after Cavalieri, approved by the most celebrated mathematicians) replaces the continued circumscription of figures of the ancients; for it is shorter, nor is it less demonstrative, if it is applied with due caution’ (Wallis, 1670, 112).

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120 17th-century adventures in infinitesimal mathematics

5. Conclusion By the time Wallis published his Artithmetica Infinitorum the method of indivisibles had become generally accepted as a tool for investigating the properties of curves, surfaces, and solids. When Blaise Pascal published his essays on the cycloid in 1658, he unhesitatingly employed the method.1⁷ Pascal insisted that ‘Everything that is demonstrated by the true rules of indivisibles will also be demonstrable rigorously and in the manner of the ancients; and thus the one method differs from the other only in the form of expression’ (Pascal 1658, 10). In his Geometrical Lectures delivered at Cambridge in the mid 1660s Isaac Barrow could term it ‘the most expeditious method of all for investigating the dimensions of surfaces and solids’, and one that ‘when rightly applied it is no less certain and infallible’.1⁸ Barrow’s admission that the method must be ‘rightly applied’ acknowledges that objections had been raised to the rigour and reliability of indivisible techniques, and Pascal’s insistence on the equivalence of indivisibles and the classical method of exhaustion repeats a theme we have seen in Torricelli and Wallis. For all that the method of indivisibles had become commonly accepted in the two decades after the publication of Cavalieri’s Geometria, it is clear that its foundations had been altered significantly. Authors continued to write of surfaces being composed of lines or solids being composed of planes, but they had adopted an entirely different understanding of terms such as ‘point’, ‘line’, and ‘plane’ than the classical notions that Cavalieri had employed. Indeed, the ‘indivisibles’ of the 1660s were understood to be divisible but infinitely small. Following Torricelli’s lead, mathematicians of the era took the ‘lines’ that compose surfaces to be infinitely narrow parallelograms or the ‘planes’ composing solids to be infinitely thin parallelepipeds. This move avoided concerns about how a continuous magnitude of a given dimension could be composed out of collections of objects of a lower dimension, and the strategy helped overcome consequential objections to the method. Nevertheless, the reformulated version of the method of indivisibles raised important questions about whether a mathematical theory committed to infinites1⁷ Pascal set a series of challenge problems in 1658, seeking the area, arc length, and centre of gravity for the cycloid, as well as the volume and centre of gravity for its centre of revolution. He was apparently unaware of Roberval’s research into the cycloid, in which all the planar problems had been solved. The contest was then reduced to the solution of the three-dimensional problems, with Roberval agreeing not to enter the competition. The contest produced its share of acrimony when Pascal declared that none of the submitted solutions (including efforts by Wallis) were adequate. He then published his Lettre de A. Dettonville á Monsieur de Carcavy (Pascal, 1658). This pseudonymous tract contained a series of essays that set out Pascal’s use of the method of indivisibles and solved all of the stated problems. For details on the cycloid challenge problems, see Costabel (1964) and Descotes (2015b). 1⁸ This assessment appears in the second of Barrow’s Lectiones Geometricae, which were delivered at Cambridge in 1664 and 1666, first published in 1669. Cf. Barrow (1860, 2: 183).

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references 121 imals could be rigorously formulated. On this point, the general attitude of mathematicians in the 1660s was one of casual indifference. When the question of rigour was raised, the typical response was to claim that the method of indivisibles was equivalent to classical exhaustion techniques, so that the language of indivisibles amounted to nothing more than a notational variant of exhaustion arguments, whose rigour was unchallenged. The successes of the method of indivisibles were soon to be significantly augmented by the advent of the calculus, in the form of Isaac Newton’s method of fluxions and the calculus differentialis of Gottfried Wilhelm Leibniz. The calculus extended the earlier method by providing a powerful algorithmic technique for finding tangents, quadratures, and arc lengths. The calculus retained the earlier methods’ reliance upon infinitesimal magnitudes, and the development of the calculus was a continuation of the fundamental ideas involved in the method of indivisibles. As might be expected, questions about the foundations of infinitesimal methods remained unresolved well into the eighteenth century and were the focus of philosophical and methodological controversy.

References Andersen, Kirsti: 1985, Cavalieri’s Method of Indivisibles. Archive for History of the Exact Sciences 24: 292–367. Barrow, Isaac: 1860, The Mathematical Works, 2 vols. bound as one, ed. W. Whewell. Cambridge: Cambridge University Press. Bascelli, Tiziana: 2015, Torricelli’s Indivisibles, in Seventeenth-Century Indivisibles Revisited, ed. Vincent Jullien. Science Networks Historical Studies, vol. 49. Heidelberg, New York, Dordrecht, and London: Springer, 105–136. Cavalieri, Bonaventura: 1635, Geometria indivisibilibus continuorum nova quadam ratione promota. Bologna. Cavalieri, Bonaventura: 1647, Exercitationes Geometricae Sex. Bologna. Costabel, Pierre: 1964, Essai sur les secrets des traités de la roulette, in L’œvre scientifique de Pascal, ed. Pierre Costabel. Paris: PUF, 169–206. De Gandt, François: 1987, Les indivisibles de Torricelli, in L’Œuvre de Torricelli: science galiléene et nouvelle géométrie, ed. François De Gandt. Nice: Les Belles Lettres, 147–206. Descotes, Dominique: 2015a, Pascal’s Indivisibles, in Seventeenth-Century Indivisibles Revisited, ed. Vincent Jullien. Science Networks Historical Studies, vol. 49. Heidelberg, New York, Dordrecht, and London: Springer, 249–273. Descotes, Dominique: 2015b, Two Jesuits against the Indivisibles, in SeventeenthCentury Indivisibles Revisited, ed. Vincent Jullien. Science Networks Historical Studies, vol. 49. Heidelberg, New York, Dordrecht, and London: Springer, 211–249.

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122 17th-century adventures in infinitesimal mathematics Festa, Egidio: 1992, Queles aspectes de la controverse sur les indivisibles, in Geometria e Atomismo nella Scuola Galileiana, ed. Massimo Bucciantini and Maurizio Torrini. Florence: S. Olschiki, 193–207. Giusti, Enrico: 1980, Bonaventura Cavalieri and the Theory of Indivisbiles. Milan: Edizioni Cremonese. Guldin, Paul: 1635–41, Centrobaryca, seu de centro gravitatis trium specierum quantitatis continue, 4 vols. Vienna. Jesseph, Douglas: 2007, Descartes, Pascal, and the Epistemology of Mathematics: The Case of the Cycloid. Perspectives on Science 15: 410–433. Julien, Vincent: 2015a, Indivisibles in the Work of Galileo, in Seventeenth-Century Indivisibles Revisited, ed. Vincent Jullien. Science Networks Historical Studies, vol. 49. Heidelberg, New York, Dordrecht, and London: Springer, 87–103. Julien, Vincent: 2015b, Roberval’s Indivisibles, in Seventeenth-Century Indivisibles Revisited, ed. Vincent Jullien. Science Networks Historical Studies, vol. 49. Heidelberg, New York, Dordrecht, and London: Springer, 177–210. Mancosu, Paolo: 1996: Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century. Oxford and New York: Oxford University Press. Palmieri, Paolo: 2009, Superposition: On Cavalieri’s Practice of Mathematics, Archive for History of the Exact Sciences 63: 471–495. Pascal, Blaise: 1658: Lettre de A. Dettonville á Monsieur de Carcavy. Paris. Torricelli, Evangelista: 1644, Opera geometrica, 5 vols. bound as one. Florence. Torricelli, Evangelista: 1919–44, Opere di Evangelista Torricelli, ed. Gino Loria and Giuseppe Vassura, 4 vols. Faenza: Montanari. Wallis, John: 1656, Operum Mathematicorum Pars Altera. Oxford. Wallis, John: 1670, Mechanica: Sive, De Motu Tractatus Geometricus. London.

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6 The Continuum, the Infinitely Small, and the Law of Continuity in Leibniz Samuel Levey

Leibniz’s Labyrinth and Early Modern Geometry Continuity is not one topic but many, and Leibniz’s studies under that name range widely across mathematics, physics, and metaphysics. Leibniz describes the problem of the composition of the continuum as a ‘labyrinth’ that ‘consists in the discussion of continuity and of the indivisibles that appear to be its elements, and where consideration of the infinite must enter in’ (GP 4.29).1 Here we shall limit our focus to a few points in what we might call his philosophy of geometry. The core of Leibniz’s thought about the composition of the continuum and the role of the infinitely small in his new calculus can be seen in a pair of writings dating to 1676. One is his dialogue on motion and the continuum, Pacidius Philalethi: Prima de Motu Philosophia. The other is his early masterwork in infinitesimal geometry, De quadratura arithmetica circuli ellipseos et hyperbolae corollarium est trigonometria sine tabulis (DQA). In addition, Leibniz has remarkable insights to offer on continuity as defined for continuous magnitudes and as defined for functions; and his so-called law of continuity plays a distinctive role in his justification of the calculus. These can be seen across a handful of later documents, notably, Lettre sur un principe général (GP 3.51–55, 1687), Principium quoddam generale (GM 5.129–135, 1688), Specimen geometriae luciferae (GM 7.260–299, c.1695), Cum prodiisset (H&O, 1701), and In Euclidis PROTA (GM 5.188–211, c.1712). Here we shall give Pacidius Philalethi and DQA the lion’s share of attention, with forays into the later texts where they clarify Leibniz’s views or indicate new frontiers emerging in his thought. Before forging ahead into Leibniz, however, it will be important to recall the historical context of seventeenth-century geometry and the legacy of Greek mathematics within it.

1 Abbreviations used in this chapter are noted in the bibliography. Translations generally follow existing sources, where available, as noted in the texts, though I have sometimes made minor modifications without comment. I am especially indebted to the translations of Pacidius Philalethi by Richard Arthur (LOC), and of Cum prodiisset by J.M. Child (EMM). My thanks additionally to Richard Arthur for help with various passages in De quadratura arithmetica. Samuel Levey, The Continuum, the Infinitely Small, and the Law of Continuity in Leibniz In: The History of Continua: Philosophical and Mathematical Perspectives. Edited by: Stewart Shapiro and Geoffrey Hellman, Oxford University Press (2021). © Stewart Shapiro and Geoffrey Hellman. DOI: 10.1093/OSO/9780198809647.003.0007

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124 the continuum in leibniz Euclid’s style of demonstration in the Elements remained the standard of rigour for seventeenth-century mathematicians, for its “logical” aspect of deductive proof from axioms and definitions, of course, but also for its strictures concerning (i) the concept of magnitude and (ii) permitted forms of construction. Consider those briefly in turn. Book V of Euclid’s Elements sets out the Eudoxian theory of ratio and proportion, providing much of the framework for a general mathematical theory of magnitude. Euclid tells us (Vd3) that ratios are relations of comparative size (as we would write: a > b, or a = b, or a < b) between magnitudes of the same kind. And further (Vd5), pairs of magnitudes stand in the same ratio (a:b = c:d) when the comparative size of the first pair (a > b, or etc.) is the same as the comparative size of the second under all equimultiple transformations (so that for natural numbers m and n, if ma > nb, then mc > nd, and so on for the rest). Pairs of magnitudes having the same ratio are then called proportional (Vd6) no matter what kind of magnitude (lines, planes, or solids) is being compared within each ratio. The theory of proportions built along these lines allows for a general concept of magnitude available for arithmetical treatment to be abstracted from the taxonomy of particular geometrical kinds. Euclid also lays down a notable restriction. Ratios are to hold among magnitudes satisfying a requirement of finite comparability, enshrined in the so-called axiom of Archimedes: Magnitudes are said to have ratio to one another which, being multiplied, are capable of exceeding one another. (Vd4)

Eligibility to stand in ratio with given magnitudes of the same kind is a limiting condition for a proposed magnitude to be a fit object for mathematical analysis. Archimedes’s axiom dictates that magnitudes must be finitely comparable to one another if they are to stand in ratio; ‘infinite multiplication’ is not countenanced. No multiple of a finite magnitude can exceed an infinite magnitude and no multiple of the infinitely small can exceed one that is finite, however, so there can be no ratio of a finite magnitude to something infinite or infinitely small with respect to it. Such quantities ‘incomparable’ to ordinary finite magnitudes—i.e., magnitudes given to us in perception—are thus ruled out. This finitism is widely recognized in ancient and early modern mathematics as a point of rigour in demonstration, although use of infinity or infinitesimals in the less exacting contexts of experiment and discovery was also familiar. Today Archimedes’ axiom is often written as the principle that for any x, y > 0 such that x > y, there is a natural number n such that ny > x. Quantities that do not satisfy it are called non-Archimedean.2

2 For discussion of non-Archimedean fields, see Erlich in this volume.

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leibniz’s labyrinth and early modern geometry 125 For constructions, Euclidean geometry allows only the use of the ruler and compass, an idealized ‘stretched cord’ that may be rotated around a fixed end. This limits the domain of curves and shapes that can be handled in geometry, and with it as well the range of solutions that can be considered to given problems. It was appreciated that various curves not constructible by Euclidean means could bear on the solution of outstanding problems—as, for example, Archimedes’ spiral might be employed to ‘square the circle’—but curves obtainable only by such alternative methods of construction were deemed not truly geometric. One property presupposed by the Elements but never defined or stated is continuity. For example, Proposition 1 of Book I is ‘to construct an equilateral triangle on a given finite straight line’, which Euclid accomplishes by drawing circles from the line’s endpoints, with the line as the radius, and taking a point of intersection of the two circles as a third vertex. The proof requires some guarantee that the circles have such a point of intersection. In the Greek context in which geometry is a study of figures and their construction, this guarantee may have been implicit in assumptions about curve tracing.3 But to early modern eyes, in which geometry is evolving into a more abstract study not just of curves and figures but of the space in which they are constructed, it seems that a postulate of continuity would need to be added in order to ensure the existence of such points of intersection. Ancient geometers could and did work quite effectively with an informal, intuitive conception of continuity and without an explicit definition. Also, principles descending from Aristotle gave widely accepted shape to the understanding of continuity: in continuous bodies (i.e., adjacent bodies that are continuous with one another) the neighbouring boundaries are one and the same and contained in each other (Physics, 227a12), and continuous magnitudes are infinitely divisible into further parts but never into indivisible elements (De Caelo, 268a7–8; Physics, 231a1–b, 237a33–34).⁴ Aristotle’s scepticism about the infinite and its place in mathematics (Physics, 207b27–34) was of course also influential. Still, by the time of the early modern period the absence of a definition of continuity left it unclear, or at least open for dispute, which sorts of technical innovations in the analysis of continuous quantities could be accepted as rigorously correct and which could not. Archimedes and Conon were suspected by Leibniz of having already experimented with ‘secret methods’ for integration that involved infinities and infinitesimals as a means of discovery. But fully rigorous proof required replacing those devices with a strictly finitistic method, exemplified in the classical, indirect

3 See De Risi (2019b). For discussion of geometry as a study of space, see De Risi (2015, 40ff., and Forthcoming). ⁴ For discussion of Aristotle on continuity and related notions, see especially White (1992); in this volume, see the chapters by Sattler and by Harari.

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126 the continuum in leibniz double-reductio method of exhaustion.⁵ Seventeenth-century mathematicians, in experimenting anew with infinitary techniques, confronted again the question of rigour. Most notably, Cavalieri argued, in his Geometria indivisibilibus infinitorum nova quadam ratione promota of 1635, that the category of magnitude should be expanded to include omnes linea or omnia plana (‘all the lines’ or ‘all the planes’) contained in any given plane or solid figure—collections of what he called indivisibles—which could then be put directly into ratios and proportions.⁶ Since ‘all the lines’ or ‘all the planes’ of a figure in Cavalieri’s method were widely regarded to be infinite collections, this threatened to clash with the Archimedean axiom. Further, Cavalieri’s picture suggests that the continuum is composed from infinitely many indivisible elements, a view taken explicitly by Galileo, which defied the tradition handed down from Aristotle and codified in Euclid. Pressure gradually rose as well against the classical restriction of geometry to simple ruler and compass constructions. Work by Viète and especially Descartes furnished an analytic framework for geometry that could exactly represent curves in a letter algebra by polynomial expressions built around the fundamental operations of addition, subtraction, multiplication, division, and powers or the extraction of roots. In La Géométrie, published in 1637, Descartes accepts an ancient distinction between ‘geometric’ curves, which can be traced by ruler and compass constructions, and ‘mechanical’ curves, which cannot. But he relocates the boundary between the two by expanding the range of constructions that are to be accepted as exact, arguing for the admission of ‘new compasses’ beyond the straight line with a fixed end of the Euclidean tradition.⁷ Descartes regards this not as a replacement of the original science but as a correction to the ancients with respect to the existing content of geometry, clarifying the requirements of exactness already implicit in their own position. Now to treat all the curves that I mean to introduce here, only one additional assumption is necessary, namely, two or more lines can be moved, one upon the other, determining by their intersection other curves. (Géométrie Book II, 316/G 44.)

⁵ The classical method of exhaustion for demonstrating quadratures—proving a given figure to be equal in size to a certain square (hence ‘quadrature’)—involves ‘squeezing’ a figure between sequences of inscribed or circumscribed polygons. One reductio uses inscribed polygons to show that the area of the given figure cannot be smaller than the assigned square; the other reductio uses circumscribed polygons to show that the area of the figure cannot be greater than the assigned square. (In each case, if one assumes a difference of a certain amount, that supposition can be reduced to absurdity.) From this it follows that the area of the figure and the assigned square must be equal. For a full-dress example from the early modern period, see Huygen’s demonstration of his Propositio V of the Horologium oscillatorium (1673), reproduced in my chapter on Galileo in this volume. ⁶ See in this volume Jesseph, ‘The Indivisibles of the Continuum: Seventeenth-Century Adventures in Infinitesimal Mathematics’ and my ‘Continuous Extension and Indivisibles in Galileo’. ⁷ See Bos (2001).

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leibniz on the continuum 127 This ‘one additional assumption’ yields a reclassification of various historically important curves. The conchoid and cissoid, for example, ‘should be accepted’ as geometric curves, contrary to the ancient view, although the spiral and the quadratrix ‘really do belong only to mechanics’ (G 44). Descartes’s proposal also embodies a great general mathematical achievement. For, as he argues, the curves constructible under his definition are precisely those that can be expressed by polynomial equations of finite degree—what would come to be called algebraic curves—thereby yielding an extraordinary harmony between analytical and constructive methods in the new ‘algebraization’ of geometry. Nonetheless, it leaves many curves outside the scope of geometry, ones that Leibniz would eventually call transcendental curves.⁸ Leibniz steps into this moment with eyes on the ancient ideas and on the burgeoning early modern innovations. And he wants it all: rigorous demonstration, infinitary analysis of continuous curvature or instantaneous change, a letter algebra for mathematical reasoning (a ‘calculus’), and the widest possible scope for geometry to include both algebraic and transcendental curves. In this context, for our limited purposes, we can divide Leibniz’s discussion of continuity into two main parts: (1) his analysis of the continuum, both his early treatment of the composition of the continuum and his later new definition of a continuous magnitude; and (2) his use of the infinitely small in representing the properties of continuous curves and magnitudes in his calculus and the role of his ‘law of continuity’ in the justification of the calculus, with its corollary definition of continuity for the newly emerging concept of function.

1. Leibniz on the Continuum 1.1 Composition of the Continuum in Pacidius Philalethi, 1676 Leibniz’s dialogue Pacidius Philalethi is addressed to the study of motion, but its underlying topic is the composition of continuum (A 6.3.548).⁹ Motion is defined initially as a ‘change of place’ (A 6.3.534) and the analysis of change itself finds that there can be no such thing as a momentary ‘state of change’ but rather change turns out to be an aggregate of something’s being in two opposite states at two immediately neighbouring moments (A 6.3.541, 558). This topology of change holds equally for change of qualities and change of place, and so too for natural transitions in all cases: distinguished intervals of motion, space, time, and matter are separated by pairs of boundary points. Adjacent intervals touch at their boundaries but do not coincide at them and so are not, in Leibniz’s view, strictly ⁸ On transcendental curves in Leibniz, see Blasjo (2017). ⁹ A full discussion is in Levey (2003).

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128 the continuum in leibniz continuous. Here Leibniz follows Aristotle’s discussion at Physics V.3,227a10–b2 of the continuous and the contiguous: I recall that Aristotle, too, distinguishes the contiguous from the continuous in such a way that those things are continuous whose extrema are one, and contiguous whose extrema are together. (A 6.3.537/LOC 149)

Adjacent boundaries of contiguous magnitudes touch in the strong sense of leaving no gap at all between them. Leibniz says they are ‘indistant’ (A 6.3.557), and the implication appears to be that while such boundary points are in topologically distinct positions, they are not thereby in metrically different locations. Continuous magnitudes, by contrast, do not merely touch but are connected by a common boundary: they literally ‘hold together’ (con-tenere, sun-echein). On the Aristotelian definition, continuity and contiguity are thus both understood as relations, ways neighbouring magnitudes are (or can be) together. Explicit too in Leibniz’s treatment is the idea that the boundaries between neighbouring intervals are indeed ‘minima’: not extended parts or subintervals, but indivisible elements. Like Galileo before him, Leibniz in Pacidius Philalethi invokes paradoxes from antiquity to argue for the existence of indivisible elements in the continuum. The dialogue’s interlocutors argue for this by a deft handling of the classical Sorites paradox. Leibniz’s principal spokesman, Pacidius, first gives the argument in the context of ‘discrete quantity’, or quantity with pre-assigned units of measure, as wealth is measured ultimately by the penny, extracting the key conclusion from the young interlocutor Charinus: pa.: Do you admit, then, that either nobody ever becomes rich or poor, or one can become so by the gain or loss of one penny? ch.: I am forced to admit this. (A 6.3.539/LOC 155) Leibniz then has Pacidius ‘transpose the argument from discrete to continuous quantity’ (A 6.3.540). This time the aim is to establish the existence of minimal elements in space, time, and motion. The interlocutors consider the motion of a movable point a that approaches a fixed point H (Figure 6.1).

[not near]

[near]

a

H F

C

[D] E B

Figure 6.1 Sorites argument ‘transposed to continuous quantity’ in Leibniz’s Pacidius Philalethi (A 6.3.540).

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leibniz on the continuum 129 At some point, say at B, the motion of a carries it from being not near to near to H. The interlocutors conclude that a’s motion can accomplish this only by virtue of containing a ‘last part’, just as it is finally by the gain or loss of a single penny that someone becomes rich or poor. Since motion is a continuous quantity, however, this last part has to be a minimum; ‘for if it were not a minimum, then something could be removed from it, leaving intact whatever produces the nearness’, and hence ‘there are minima in reality’ (A 6.3.540/LOC 156).1⁰ Transposed to the continuous in this way the Sorites has taken on the form of another ancient paradox, namely, Zeno’s dichotomy. The moveable point a may as well be Achilles and its passage toward the nearpoint to H his transit of the mile. Here the dichotomy is used not to prove motion to be impossible; motion’s reality is taken for granted. Instead, the paradox is used to show that in order for anything to succeed in traversing a given interval it must contain a minimal element by virtue of which it actually reaches to the end of the interval. The minima of space and time are points and moments. Lines and planes will also be counted as minima, the indivisible boundaries or sections of continua of two and three dimensions, respectively. Leibniz holds that there are minima in matter as well, for every surface, edge, or vertex of any part of matter is a minimum (see A 6.3.552, 555, 565f.). With those several ideas in place, the resulting account of change holds that change consists of an aggregate of two adjacent minimal elements containing opposite states. But while this tells us what happens at the boundaries of a given change, it does not yet reveal how to understand the structure of the extended intervals those boundaries delimit. This is especially urgent in the case of motion, of course, since it seems that motion itself is precisely a quantity that stretches across intervals, like the transit of moveable point a to H or Achilles’ motion across the mile. If a moving point x now located at A traverses continuously to some space to a distant location C, in a following moment it must occupy a different location. If it is at A now, what is its position at the next moment? Leibniz says its next position is either immediately after A or else ‘mediately’, i.e., at a distant point such that there are other positions intermediate between it and A. If mediately, Leibniz identifies two options, both unacceptable. First, the moving point x somehow leaps instantaneously to the distant point without passing through the intermediate points at all, an ‘excruciating’ idea (A 6.3.556) and contrary to reason, even if we imagined the leaped-over space to be actually infinitely small (A 6.3.560). Alternatively, x is at all the intermediate points at once, and so expands to occupy ‘the whole place’, which is declared ‘absurd’ (A 6.3.557). Consider instead that x’s next location is a point immediately next to A, say, at B. Following suit all the way to C, it then appears that x’s motion across an interval will

1⁰ See Levey (2002) for further discussion of the Sorites in Leibniz.

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130 the continuum in leibniz L

N

7

7

2

b c

0 d 3

4

g

5

6 9

M

P

Figure 6.2 Leibniz, Pacidius Philalethi (1676), A 6.3.550.

be composed of minimal elements of motion and the line it traverses is likewise composed of points; that is, points would be the ultimate parts of a continuum. But, says Leibniz, it is impossible for a space to be ‘an aggregate of nothing but points’ or for time to be ‘an aggregate of nothing but moments’ (A 6.3.548/LOC 173). Focusing on the line as a model case, Leibniz argues as follows (A 6.3.548ff.). Suppose a line were composed entirely of points. This must then be from either a finite or an infinite number of points. If it is composed of a finite number of points, then the line could not be divided into any given number of equal parts, contrary to established principles of geometry. If it is composed of an infinite number of points, then there follows a violation of Euclid’s part–whole axiom. Take a rectangular parallelogram LNPM and construct its diagonal NM (Figure 6.2). The number of points on the opposite sides LM and NP will be the same, as can be established by connecting them via parallel lines. But these parallels also intersect the diagonal NM, placing them also into one-to-one correspondence with the points on the sides. This implies the equality of LM, NP, and NM. But of course, as in any parallelogram, the diagonal NM is greater than its side, each of which is provably equal to only part of the diagonal. Thus the whole of the diagonal would be equal to a part, contradicting Euclid’s principle (in Book I, common notion 5) that the whole is greater than the part. ‘Whence it is established that lines are not composed of points’ (A 6.3.550/LOC 177). The elements of this ‘diagonal paradox’ are of course already on display in Galileo’s Discorsi of 1638, both in discussion of the paradox of natural numbers— which Leibniz expressly notes (A 6.3.550)—and in the geometrical demonstration of Proposition 1, Theorem 1 of ‘On Falling Bodies’. Pacidius Philalethi is clearly written with Galileo’s Discorsi in mind, and Leibniz’s analysis here provides a direct critique of Galileo’s account of a continuous extension as ultimately composed

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leibniz on the continuum 131 from points, i.e., non quanti indivisible parts. The two disagree about the scope of Euclid’s part–whole principle, with Galileo holding that the terms ‘greater’, ‘less’, and ‘equal’ do not apply in the infinite, whereas Leibniz writes, ‘I believe it to be no less true in the infinite than in the finite that the part is less than the whole’ (A 6.3.551/LOC 179; see also A 2.1.351–2, 2.2b .814). Leibniz’s own lesson from the paradox is to affirm the ancient doctrine that points are not parts of lines but only modes of them and are not to be understood as pre-existing or ontologically independent elements in their own right. Likewise for all minima in reality: vertices, edges, surfaces are all ‘extrema’, and none exist except those ‘made by an act of dividing’ (A 6.3.553). Leibniz continues: Nor are there any parts in the continuum before they are produced by a division. But not all the divisions that can be made are ever in fact made. (A 6.3.553/LOC 181)

Like Galileo, Leibniz accepts that any actual continuous magnitude will contain infinitely many indivisible elements. But Leibniz finds the idea that a continuum is a mere aggregate of minima or indivisibles to be impossible and fundamentally mistaken about the way in which the continuum is related to the indivisibles within it. The resolution of an ever-divisible quantity into a powder of points is not inevitable, as Galileo seemed to think, but contradictory. How Leibniz proposes to avoid the same conclusion we shall see shortly below. Continuous motion across an interval now appears utterly perplexing. A moving body x cannot in the next moment be in an immediately neighbouring location, on pain of reducing the continuum into an aggregate of points; but neither can it be in the whole of an extended interval, or leap over intervening spaces. It seems there is no coherent account of its location after the initial moment. Perhaps then motion cannot be continuous after all but must instead be interrupted by interspersed intervals of rest. Yet this is no good either, for in between the rests there must be motion, else motion will be reduced entirely to rests and nothing will move at all.11 How then to escape from this labyrinth? Leibniz proposes an answer. The key to it lies in rejecting what he sees as a tacit assumption about motion, namely, an assumption of its uniformity. In first introducing the problem about composing a continuum entirely from points, Pacidius had said: ‘If the present motion is an aggregate of two existences, it will be continued out of more existences, for we assumed it to be continuous and uniform’ (A 6.4.547). Leibniz does not explain his notion of uniformity in Pacidius Philalethi, though in other writings on geometry he appears to have in mind a ‘situational’ property of the distribution of points (and presumably of boundaries

11 Leibniz has Charinus provide a good summary of the enquiry and its difficulties at A 6.3.562.

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132 the continuum in leibniz more generally) in a space—roughly, being the same everywhere, or ‘isotropy’— something properly studied in an analysis situs of the sort he will develop in later writings.12 Continuity for magnitudes such as motion, space, or time is still treated here on an Aristotelian model as a relation among parts; by contrast, uniformity is a structural property of a space understood as having a system of internal relations among the elements it contains. So there is a glimmer here of a profound, new alternative approach to the questions being asked about the structure of motion in the Pacidius, though it has not yet flowered. In the context of the dialogue, Leibniz’s proposal appears to be that if continuous motion is perfectly uniform across an interval, there is no consistent way to assign a position to a moving body after the initial moment of motion. But if it is instead broken by changes—say, by the action of accelerative forces—into subintervals, then definite locations can be assigned to the moving body at the boundaries between adjacent subintervals, their ends and beginnings. Here Leibniz has Charinus target the concept of uniformity to escape the labyrinth; his example is again of a ‘moving point’ traversing a line AC, which he considers as cut into neighbouring parts AB and DC, so that B and D are immediately next to each other. ch.: Since uniformity cannot be denied in place and time considered in themselves, it therefore remains for it to be denied in motion itself. And in particular it must be denied that another point can be assumed immediately next to the point D in the same way that the point D was assumed immediately next to point B. pa.: But by what right do you deny this, since there is no prerogative in a continuous uniform line for one point over another? ch.: But our discussion is not about a continuous uniform line, in which two such points B and D immediately next to each other could not even be assumed, but about the line AC which has already been cut into parts by nature; because we suppose change to happen in such a way that at one moment the moving point will exist at the endpoint B of one of its parts AB, and at another moment at the endpoint D of the other part DC. [ . . . ] I deny, therefore, that another point could be assumed in the line DC immediately next to D for I believe that no point should be admitted in the nature of things unless it is the endpoint of something extended. (A 6.3.563–4/LOC 205, with omissions) 12 For example, Leibniz distinguishes continuity from uniformity in In Euclidis PROTA (GM 5.206). For a text contemporaneous with Pacidius Philalethi, see Leibniz’s paper from 1676 Generatio quidem rectae et circuli (in La charactéristique géométrique, 66); see also Leibniz’s essay Uniformis Locus from 1692 (in De Risi (2007, 582–585)) and his reading notes to Arnauld’s Nouveaux Elémens de géométrie from 1695 (in De Risi (2015, 146–151)). For discussion of Leibniz’s analysis situs, see De Risi (2007, 2018).

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leibniz on the continuum 133 In this way Leibniz ties his ontology of points as modes to the idea of nonuniformity: there can be no third point assumed immediately after B and D, lest the continuum be composed from minima, and so the line cannot have points assigned in it uniformly in that fashion. The question then becomes why points are assigned non-uniformly in the line as Charinus proposes. Since it is motion that divides the continuum and actually assigns points to the line, motion itself must be non-uniform. And here comes to light the central thesis in Leibniz’s theory of motion—indeed the very centrepiece of his solution to the paradoxes of the continuum. ch.: Then what if we say that the motion of a moving thing is actually divided into an infinity of other motions, each different from the other, and that it does not persist the same and uniform for any stretch of time? pa.: Absolutely right, and you yourself see that this is the only thing left for us to say. But it is also consistent with reason, for there is no body which is not acted upon by those around it at every single moment. (A 6.3.564–5/LOC 209, with omissions) With the hypothesis of non-uniformity that was instituted to solve the paradoxes now explained dynamically as the structure belonging to motion infinitely divided by actions, Charinus announces the new analysis of motion: ch.: So now we have the cause of the division and the non-uniformity, and can explain how it is that the division is arranged and the points assigned in this way rather than that. The whole thing therefore reduces to this: at any moment which is actually assigned we will say that the moving thing is at a new point. And although the moments and points that are assigned are indeed infinite, there are never more than two immediately next to each other in the same line, for indivisibles are nothing but bounds. (A 6.3.565/LOC 209) Motion can neither be a purely discrete series of points, as if it were just so many grains of sand, nor can it be purely continuous like an undivided geometrical line. Charinus’s proposal tries to find a middle path between pure discreteness and pure continuity. Motion across an interval is divided, not into points, but rather into finer subintervals of motion, and so on ad infinitum. Pairs of immediately neighbouring points, such as B and D, lying at the boundary between extended subintervals are densely ordered throughout the full interval. But motion is not thereby resolved into ‘nothing but an aggregate of points’. Leibniz illustrates the new proposed structure of motion, and more generally of the continuum as it exists in nature, with an image that contrasts strikingly with Cavalieri’s earlier depiction of geometrical figures as pieces of cloth woven from

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134 the continuum in leibniz indivisible threads deprived of thickness (see EGS, 3–4). Leibniz reverses the order of priority here between the continuum and its indivisibles: the cloth is not built up as an aggregate of indivisible elements—as if from threads or grains of sand—into which it could in principle be decomposed. Rather, the indivisibles arise as modes of the fabric, its folds: pa.: Accordingly the division of the continuum must not be considered to be like the division of sand into grains, but like that of a sheet of paper or tunic into folds. And so although there occur some folds smaller than others infinite in number, a body is never thereby dissolved into points or, what is the same, minima. On the contrary . . . although it is torn into parts, not all the parts of the parts are so torn in their turn; instead at any time they merely take shape, and are transformed; and yet in this way there is no dissolution all the way down into points, even though any point is distinguished from any other by motion. It is just as if we suppose a tunic to be scored with folds multiplied to infinity in such a way that there is no fold so small that it is not subdivided by a new fold . . . And the tunic cannot be said to be resolved all the way down into points; instead, although some folds are smaller than others to infinity, bodies are always extended and points never become parts, but always remain mere extrema. (A 6.3.555/LOC 185–7) Leibniz’s analysis of the division of the continuum grants to quantities such as matter and motion a fractal-like structure13 of parts folded within further parts ad infinitum, with no resolution into ultimate parts as first elements. It is his great alternative both to the Peripatetic account of the continuum as divisible into a potential infinity of ever divisibles and to Galileo’s picture of the continuum as ultimately resolvable into an infinity of non quanti parts. The division of the continuum into folds within folds is actual and infinite, but there is never a resolution into unquantifiable, indivisible parts. In a later document, In Euclidis PROTA (1712), Leibniz even more drastically questions the basic topological presuppositions of the Galilean and Cavalierian picture of solids composed of planes, plane figures of lines, and lines of points, by proposing that Euclid’s classification of extended objects into solids, planes, and lines is itself impoverished, an ‘imperfect doctrine’ (GM 5.187). He imagines magnitudes of intermediate dimension between solids and planes, and likewise between planes and lines, calling them ‘ascending’ and ‘descending’ surfaces: Also, from this it is clear that surface is not sufficiently defined, for indeed, as I said, it is not settled that everything which has width and lacks depth is of the same dimension. In fact, if descending surfaces should be defined, then there is 13 See Levey (2003) and Bouquiaux (1994, 241).

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leibniz on the continuum 135 a magnitude whose section cannot have a section that is in turn a magnitude (assuming that every section of a surface is either a line or a point). Accordingly, it may be that an ascending dimension exists intermediately between a surface and a solid, whose section is a surface, which would not now be the highest dimension but rather it would be possible in turn to assign another section. (GM 5.187–8)

Leibniz apparently does not further develop these ideas challenging the classical Euclidean taxonomy of dimension—which will later get their due in fractal geometry—but they can nonetheless be seen as comprehensively filling out his alternative to Cavalieri and Galileo. Their conception of the continuum turns out, in his view, to belong to a caricature of the extended world that hasn’t begun to understand the depth of its structure. The view of the physical continuum as folded into parts within parts ad infinitum but never resolved into indivisibles becomes Leibniz’s signature view of the topology of matter and motion, one he upholds for the rest of his career. Not long after he develops this view Leibniz will articulate a further layer of theory that posits indivisible ‘unities’—unitates, and, later, monades—in the metaphysical foundations of the natural world to ground its part–whole scaffolding. His early analysis of the continuum, having ‘passed through that labyrinth’, then gives rise to a new, ‘true conception of substance and matter’ (A 6.3.449). His distinctive account of the division of the continuum itself remains intact, providing a critical through-line that connects his mathematical and metaphysical enquiries in natural philosophy.

1.2 The Continuum as Ideal and a New Definition of Continuity for Magnitudes Crucial to Leibniz’s full philosophical account of the continuum is a distinction between the actual quantities of nature and the ideal quantities of geometry.1⁴ Strictly, continuity itself belongs not to actual things such as matter and motion but only to their ideal counterparts as conceived in mathematical and geometrical mental construction. Geometry prescinds from the actual divisions of things in nature and treats its ideal subjects—in effect, space and time and its possible contents abstractly represented—as indefinitely divisible, in all possible ways, into parts. As he will later write to De Volder: A mathematical line is like the arithmetical unit: for both the parts are only possible and completely indefinite. But in real things, namely, in bodies, the

1⁴ See, for example, Hartz and Cover (1988), Hartz (1992), Levey (1999).

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136 the continuum in leibniz parts are not indefinite (as they are in space, a mental thing), but are actually assigned a certain way as nature has actually instituted divisions and subdivisions in accordance with the varieties of motion. (GP 2.268)

Perfectly straight lines or continuous curves, exact spheres or triangles, and so on, belong only to the realm of reason, not to the world of actual bodies and motions, which are absolutely speaking discrete quantities. ‘Matter is a discrete being, not a continuous one’ Leibniz writes in 1676, ‘it is only contiguous and united by motion’ (A 6.3.474). He further associates the distinction between the discrete and the continuous with contrasting orders of priority between whole and part: in discrete quantity, the parts are prior and the whole is the result of the parts, whereas in continuous quantity the opposite is true, and the whole is prior to the parts into which it may be divided (A 6.3.502, 520; see also GP 4.491f, GP 3.622, GP 7.562). Thus for Leibniz the distinction between discrete and continuous is not just topological but metaphysical. The contiguous world of matter cleaves apart in a fundamental way from the continuous constructions imagined in geometry. Still, Leibniz maintains, the approximation of the discrete to the continuous is so close as to allow the mathematical and geometrical sciences to provide accuracy of analysis to any desired degree: Even if no straight lines or circles can exist in nature, it is nonetheless sufficient that shapes can exist that differ from straight lines and circles so little that the error is less than any given error—which is sufficient in order to demonstrate certainty as well as usage. (A 6.4.159)

For the purposes of calculation and mathematical reasoning, it is as if the natural world were ideally continuous. As is clear, then, Leibniz departs from Aristotle on the metaphysics of the continuum by holding that continuity belongs only to ideal magnitudes and not to actual physical ones. But Leibniz also comes to differ from Aristotle on the topology of the continuum, even taken as an ideal quantity, and in a profound way that turns geometry away from its ancient form of a study of curves and figures and toward its future as a study of space. Whereas Leibniz initially follows Aristotle in saying that those things are continuous whose boundaries are one, he later develops a subtly different account, on display in a number of important texts. A good example comes in Specimen geometriae luciferae (c.1695), where Leibniz writes: A continuum is a whole any two of whose co-integrating [cointegrantes] parts (that is, parts which, taken together, coincide with the whole) have something in common, and if indeed the parts are not redundant, that is, they have no common part, or if the sum of their magnitudes is equal to the total magnitude, then they have at least a boundary in common. (GM 7.284)

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leibniz on the continuum 137 As noted already, for Aristotle continuity is a relation of natural ‘connectness’ between given pairs of adjacent parts: continuous things don’t merely touch, they share a common boundary. Leibniz transforms this into a constellation of precisely defined ideas, a mereological analysis recognizably akin to set-theoretic topology. A whole has many possible ways of being divided into co-integrating parts (many possible ‘covers’, as one might say today). Co-integrating parts may overlap in a common part, or in a boundary, or not at all. On Leibniz’s new definition, in a continuum, any possible pair of co-integrating parts must overlap at least in a boundary. Thus for Leibniz continuity now comes out not as a relation between given adjacent parts but as a general structural property of a whole that reflects all possible ways of dividing it into parts. Leibniz offers similar definitions of continuity in a number of texts and develops his analysis in order to apply it to mathematical problems.1⁵ Notably, Leibniz trains his attention on Euclid’s proof at Elements I.1 and sees a gap: Euclid fails to demonstrate that the two circles actually intersect. Leibniz writes: In Elements I.1, where he [Euclid] teaches how to construct an equilateral triangle on a given basis, he assumes that two circles drawn from the endpoints of the straight line (taken as the base) and with radius equal to the straight line, cut each other. This is not evident at all. (LH 35, I, 2, Bl. 6–7, transcribed in Echeverría 1980, vol. 2, 43–51; quoted in De Risi (2019a))

To make this evident, Leibniz says, Euclid would need to show two things not yet established: First: that the circumference of each circle is partly inside and partly outside the other circle (for if this is established, the circumference would necessarily intersect the other circumference, given the definition of section). Second: that the radius of the circle, extended, will intersect the circumference in two points only. (Ibid.)

Similarly, In Euclidis PROTA offers the following reasoning to prove the existence of a point of intersection for a line that crosses a co-planar circle, based on continuity: It follows from the nature of continuity that every continuum which is partly inside and partly outside a figure falls on its boundary. In fact, any two parts of a continuum that together make the whole have something in common even 1⁵ See De Risi (2019a).

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138 the continuum in leibniz though they have no part in common. Let there be, then, two parts of a straight line, one inside and the other outside the circle. They have a common point. This point is also common to the circle, for it is in the part of the plane falling inside the circle and also in the part of the plane in which the straight line falling outside the circle lies. But anything that is common to the two parts of the plane, is in their common section, that is, the circumference. (GM 5.196)

This at least satisfies the first of the two steps he had earlier identified as necessary for completing Euclid’s proof. Leibniz even constructs a symbolic proof of this axiom that the line intersects the circle using the tools of what he calls his Calculo situs (GM 5.197; see De Risi (2019a)). And he notes that the result will generalize, applying not only to a straight line crossing a circle but to ‘any continuous curve whatsoever’ partly inside and partly outside any surface, and likewise for any surface crossing any solid, there will be an intersection somewhere at the boundary (GM 7.284). In taking up the topic of Elements I.1 and trying to furnish an argument to guarantee the existence of a point of intersection, Leibniz belongs to a lineage of mathematicians who sought to close the purported gap by introducing some axiom of intersection. What sets Leibniz apart is his effort to derive such an axiom from first principles about continuity. It makes sense that Leibniz would try to do so. As De Risi notes, more limited ad hoc principles of intersection might close the gap in Elements I.1, and even all the other ‘gaps’ in the Elements where rulerand-compass constructions assume, without proof, the existence of their points of intersection.1⁶ One can stipulate that for all crossing curves so constructible, the underlying plane contains corresponding points of intersection; this does not require a theory of continuity of the plane. Likewise, taking the further step with Descartes to expand constructions to all algebraic curves, one can posit that the plane contains all points corresponding to algebraic numbers, and still not have to fall back on a general theory of continuity. But if all transcendental curves are to be included under an axiom of intersection, a stronger guarantee of completeness for the class of points in the plane is required. For Leibniz, whose calculus was expressly intended to do for Archimedes what Viète and Descartes did for Euclid and Apollonius and to expand geometry proper to encompass transcendental curves,1⁷ it should be no surprise to find him advancing the frontiers of the study of continuity as well.

1⁶ De Risi (Forthcoming). 1⁷ See, for example, A 3.1.358, 3.5.239, 3.8.103, 7.4.594, 7.6.88.

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leibniz on the infinitely small 139

2. Leibniz on the Infinitely Small 2.1 The Infinitely Small and the Archimedean Principle of Equality in DQA 1676 Until recently Leibniz was commonly taken to have developed his differential calculus without concern for rigour or conceptual clarity in the foundations. It’s not hard to understand the criticism. Consider the notion of a differential as an infinitely small difference in the first textbook of the Leibnizian differential calculus, Analyse des infinitements petits (1696), by Guillaume L’Hôpital, a follower of Leibniz: Definition I. Those quantities are called variable which increase or decrease continually, as opposed to constant quantities that remain the same while others change. Definition II. The infinitely small portion by which a variable quantity increases or decreases is called the Differential. (Analyse, 1–2, with omissions)

L’Hôpital in short order states the two fundamental postulates of the work: Postulate I. We suppose that two quantities that differ by an infinitely small quantity may be used interchangeably, or (what amounts to the same thing) that a quantity which is increased or decreased by another quantity that is infinitely smaller than it is, may be considered as remaining the same. Postulate II. We suppose that a curved line may be considered as an assemblage of infinitely many straight lines, each one being infinitely small, or (what amounts to the same thing) as a polygon with an infinite number of sides, each being infinitely small, which determine the curvature of the line by the angles formed among themselves. (Analyse, 2–3, with omissions)

Focus, for example, on a puzzling implication of Definition II and Postulate I. It seems that a variable quantity ‘increases or decreases continually’ by infinitely small differences, yet a quantity so changed may be considered as ‘remaining the same’. How can that be? The trouble is not simply a matter of loose explication. The basic rules of the calculus seem to require the differential to be both something and nothing, behaving as a non-zero value and as zero. Take the simple case of calculating the derivative of y = x2 , that is, the rate of change of the value y with respect to the variable x as x increases by an infinitely small increment to become x + dx. The value of the differential dy with respect to this change in x is calculated

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140 the continuum in leibniz as the difference between the value y takes on when x becomes x + dx and the original value of y. So for y = x2 we have: dy ((x + dx)2 − y) = dx dx Then by the rules of the calculus we work out the right-hand side as follows: (x + dx)2 − x2 dx x2 + 2xdx + (dx)2 − x2 = dx 2xdx + (dx)2 = dx = 2x + dx =

= 2x

(1) (2) (3) (4) (5)

The final two steps of the calculation appear to point up an inconsistency. Discarding dx from 2x + dx to reach 2x in the last step, at (5), would seem to require dx to be zero. Yet dividing through by dx to obtain 2x + dx from (2xdx + (dx)2 )/dx in the next to last step, at (4), requires dx to be non-zero, since division by zero is not allowed. And there are many similar cases. (How is one to understand the ratio dy/dx in cases in which the values of dy and dx appear to go to zero?) The infinitesimal calculus appears vexing, or even inconsistent. Popular history has it that Leibniz’s focus on obtaining results rather than dealing with issues of foundations, plus his gift for effective notation, proved to be great advantages to his followers in continental Europe over their English counterparts, whose struggle with difficult foundational matters slowed their progress. It was left to the nineteenth century’s ‘rigorous reformulation’ of the calculus in finitist terms to free the calculus from the conceptual obscurity introduced by Leibniz’s use of infinitesimals. Analysis needed to rid itself of the idea of quantities that defy the requirement of finitely comparability enshrined in Archimedes’s axiom. The ‘incomparability’ of infinitely small quantities meant that they failed to meet accepted standards of rigour, notwithstanding Leibniz’s late claims, under pressure from critics, that infinitesimals were only fictions. This history, however, has its facts almost upside down with respect to Leibniz. Leibniz’s calculus was developed with scrupulous care for foundational matters and was never wedded to the idea of infinitesimals as fixed quantities greater than zero but less than any finite value. His great 1676 work on quadratures of conic sections, De Quadratura Arithmetica (DQA), articulating the central ideas behind the calculus, is finitist at its heart and explicitly formulates its basic tenets to accord with classical standards. In DQA, Leibniz employs infinitesimals as a

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leibniz on the infinitely small 141 convenient shorthand or abbreviation: a fiction that allows one to reason as if curves and figures were built up from infinitesimal linelets or rectangles, while the notation for infinitesimals, rigorously taken, represents relations among variable finite quantities that can be taken as small as one wishes. Leibniz’s particular canons of rigour differ somewhat from now contemporary standards—both in what principles he accepts and what assumptions he is willing to leave implicit— but not by ignoring the stricture imposed by Archimedes’ axiom. To the contrary, Leibniz’s approach relies directly on the fundamental idea in Archimedes’ axiom in the demonstrations of crucial ‘foundations’ theorems in DQA. Leibniz’s DQA is best understood as a work in infinitesimal geometry rather than a proper presentation of the calculus itself whose key procedures are not distilled in DQA into a small algebra of rules, as one finds in Nova Methodus of 1684 with its ‘Algorithm’ for differentials of constants, sums, differences, products, quotients, powers, and roots.1⁸ A key achievement of the DQA is, as the title implies, an ‘arithmetical quadrature of the circle,’ that is, an expression of the ratio of the area of the circle to that of a square constructed on its diameter in arithmetical terms using rational numbers, given by Leibniz’s series: 𝜋/4 = 1/1 − 1/3 + 1/5 − 1/7 + 1/9 − 1/11, etc. ad infinitum (A 7.6.674). Leibniz’s claim is that the series is not merely an approximation of the area of the circle—as one might achieve through exhaustion methods—but an exact expression of it: For, when I say that the circle is equal to this infinite series 1/1−1/3+1/5−1/7+1/9−1/11 etc., supposing that the circumscribed square is 1, I certainly say something more than if I exhibit it in the manner of an approximation. Every infinite series indeed contains an approximation as exact as is desired; but also, a true and exact equality. (A 7.6.439/Quintana (2018, 71))

The use of infinite series allows the geometrical method of exhaustion to be transformed into an arithmetical technique in which a sequence of step-space approximations is replaced by a sequence of rational numbers, which can then be summed. Of course this is an extension of classical arithmetic into the newly emerging theory of infinite series or, as Leibniz calls them, progressions.1⁹ The aim is revolutionary, to transform Archimedes’ method of quadrature in the same way that Descartes and Viète transformed the geometry of Euclid and Apollonius (see A 3.5.239). Leibniz says that ‘just as rectilinear problems are reduced to a calculus and to numbers by means of equations, in the same way the difficulty of curvilinear [problems] is transferred from geometry to arithmetic by means of progressions’ (A 7.6.88–89; also, A 7.6.600–601) 1⁸ Knobloch (2002) shows, however, how Leibniz tacitly operates in DQA with a precise set of rules that form an ‘arithmetic of the infinite’. 1⁹ In an earlier text he distinguished between arithmetica pura with finitely many terms and the new arithmetic continuorum with infinitely many (see A 7.4.263, 265).

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142 the continuum in leibniz DQA offers a clear window onto Leibniz’s fundamental philosophical concerns and the rigour of his approach. Our focus will fall not on Leibniz’s ‘arithmetizing’ geometry through infinite series, but on the fundamental role of the concepts of equality and infinity in DQA. Leibniz claims to ‘lay the foundations of the whole method of indivisibles in the firmest possible way’ (A 7.6.521) at Proposition 6 (more below). The reference to indivisibles of course alludes to Cavalieri, and it sets the stage for a method of direct calculation of curved spaces by means of infinitely many rectilinear elements overlaid on them. As Leibniz notes, Cavalieri’s method comes out as a limited, special case of his own, so in a way Cavalieri receives a new foundation in Leibniz’s scheme. But Leibniz does not mean that his new method is to be taken as a refinement of Cavalieri; in fact, he sees it as a great departure with vaulting ambitions. When I speak of the Geometry of indivisibles, I envisage something much wider than that of Cavalieri, which seems to me to be a trivial portion of that of Archimedes. (A 7.6.498)

For Leibniz, the indivisibles of his own method are understood as infinitesimals: that is, not as lines or threads deprived of width but as magnitudes of the same dimensional order as the extended objects they make up, only infinitely small in comparison. A plane figure, for example, is represented not as an aggregate of ‘all the lines’ but as a sum composed of infinitely small rectangles. (Leibniz argues that Cavalieri’s method would lead to paradox in certain instances and that his own new approach will avoid the same fate.2⁰) Particularly critical for grasping the conceptual foundation of his method, however, is the fact that for Leibniz the idea of ‘infinitely small’ is itself understood in terms of a variable finite quantity that can be made arbitrarily small. The idea of a non-Archimedean infinitesimal is only a fiction in the account, given for heuristic value and economy of expression. Indeed, in the first seven propositions of DQA, where Leibniz lays down his foundations, the discussion is conducted entirely in terms of finite quantities and plainly resembles, while also innovating upon, the Archimedean method of exhaustion. Infinitesimals are introduced subsequently (starting in Prop. 11) as a shorthand to speed proof and allow direct calculations in place of the more elaborate double-reductio pattern of the method of exhaustion and even Leibniz’s single-reductio reworking of it in the early parts of DQA. The key to Leibniz’s conceptual foundations for his new geometrical method comes in the demonstration of Proposition 6, where he shows that the quadrature

2⁰ See Leibniz’s analysis of the rectangular hyperbola at A 7.4.438 and the scholium of DQA Prop. 22 (A 7.6.583–584). See also Knobloch (1999, 2002), Mancosu (1996, 128–129), and Arthur (2013).

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leibniz on the infinitely small 143 K A 1B 2B

1T 1M 2T1G 2M 1N 1D

2G 3T

3G 3M

4T

1C

1F 2D

1E 1P

Q

2N

2C

γ

2F

ν

2E 3B

2P

φ ψ

3C 3N 3D

R 3F

5D

H 4B

1L

2L

3L

4C

β

S

3E 3P

4D

6D

δ

(μ) μ

ξ

λ

Figure 6.3 Leibniz, DQA (1676), Prop 6, A 7.6.528.

of any of a very broad class of curves can be taken by constructing step spaces built up of rectangles (Figure 6.3).21 The demonstration is intricate; Leibniz calls it spinosissima, ‘most thorny’, and it involves elaborate preparatory work for the subsequent Proposition 7. It is also mathematically prescient; the technique Leibniz develops here, in 1675–6, is Riemannian integration. We’ll observe just a few elements to draw out a central philosophical point. Consider the mixtilinear space under the curve A1 D2 D3 D4 D between ordinates 1 L and 4 D; call that space Q (for ‘the whole Quadrilineal’ (A 7.6.531); Leibniz introduces ‘Q’ for the counterpart in Proposition 7 (A 7.6.536)). Leibniz proceeds to express Q as composed of unequal ‘elementary rectangles’ fitted to the curve, forming a step space (spatium gradiformis), call it A, whose area is the sum of those elementary rectangles. It may be helpful to think of the step space A as constructed in stages. At any stage, each elementary rectangle in A is finite and its end fits approximately to the curve, leaving some difference from its sector of Q. Leibniz then constructs ‘complementary rectangles’ to bound this difference: for each elementary rectangle, the difference between it and its sector of Q will be smaller than the corresponding complementary rectangle (‘e.r.’ and ‘c.r.’ in Figure 6.4). 21 For discussion, see Knobloch (2002, 2018), Arthur (2008, 2013), Levey (2008), Rabouin (2015), and Blasjo (2017a, 2017b); the last two offer a sceptical critique of some implications the others draw from Proposition 6 of DQA.

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144 the continuum in leibniz 1N

1B

e.r.

e.r.

2N

2B 1P

e.r.

3B

e.r.

2P

3N

Figure 6.4 Figure from Knobloch (2002), 66, for DQA Proposition 6.

The total difference between the step space A and the mixtilinear space Q, |A – Q|, then, is smaller than the sum of all the complementary rectangles. That sum itself can be no greater than the product of the sum of the bases of complementary rectangles, given by the line 4 D1 L, with the maximum height 𝜓4 D of any complementary (or elementary) rectangle, hence |A – Q| < 𝜓 4 D1 L. As Leibniz notes, the maximum height 𝜓 4 D of any rectangle can be made arbitrarily small, and so likewise 𝜓 4 D1 L can be made arbitrarily small. (Think of subsequent stages of construction of the step space being formed by taking 𝜓4 D smaller and smaller, increasing the number of rectangles and making the fit more precise.) Therefore, for any value E that one might assign as the ‘error’ in the calculation of Q’s area by A, given by the difference |A – Q|, it can be shown that, by taking 𝜓 4 D sufficiently small, |A – Q| < 𝜓 4 D1 L < E. More briefly: for any value E, it can be shown that |A – Q| < E. Leibniz concludes: Now with these established, the proof is thus finished: The difference between the whole Quadrilineal and the step space is smaller than the rectangle 𝜓 4 D1 L, by article 6. And points on the curve can be taken in so small an interval and in so great a number that the rectangle 𝜓4 D1 L is smaller than a given space, by article. 7. By the same work, the difference between this Quadrilineal (which is the subject of this Proposition) and the step space can therefore be made smaller than a given quantity. Q.E.D. (A 7.6.532)

Note that Leibniz’s argument assumes that the curve is uniformly continuous: that in any interval on the curve, however small, points can always be taken ‘in so great a number’ to allow 𝜓 4 D1 L to be made smaller than a given space. Continuity is thus presupposed to license the inference to the last conclusion that the step space can be made smaller than any given quantity. And now that conclusion takes

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leibniz on the infinitely small 145 on a profound meaning for Leibniz, in light of Archimedes’ axiom that for any magnitudes x, y > 0 such that x > y, there is a natural number n such that ny > x. For a corollary of the axiom is what we might call the Archimedean principle of equality: for any quantities x, y, If, for any natural number n, the difference |x – y| < 1/n, then x = y.

Quantities that differ by no finitely assignable amount are equal. In the present case, since for any value 1/n no matter how small, the difference |A – Q| can be made still smaller, it follows that difference is null, that is, A = Q. Put otherwise: if A and Q are not equal, there must be some finite value E by which they differ (or which is smaller than their difference). But by Leibniz’s construction of the step space, it can always be shown, by taking the maximum height of elementary rectangles 𝜓 4 D small enough, that for any proposed ‘error’ E the difference must be less than E. Hence, A and Q are, in fact, equal. Leibniz does not make explicit the inference from the difference being able to be made smaller than any given quantity to equality in Proposition 6 of DQA. But it comes through quite clearly in his subsequent discussion of Proposition 7 (Leibniz’s ‘transmutation’ theorem) whose proof involves the same logic and concludes: ‘Therefore anyone contradicting our assertion [that the sum of the rectangles is the same as the area of the quadrilineal] could easily be convinced by showing that the error is smaller than any that can be assigned, and therefore null’ (A 7.6.542): errorem esse. . .nullum, there is no error.22 As noted, Leibniz’s analysis in Proposition 6 of DQA is conducted entirely in terms of finite quantities; to be sure, variable finite quantities in some cases, such as 𝜓4 D and quantities related to it such as 𝜓 4 D1 L. But there is no appeal to fixed infinite or infinitely small values. Now, of course, the smaller 𝜓4 D is taken, the greater the number of elementary rectangles will be that are overlaid on Q. And for no given finite height 𝜓 4 D and number n of rectangles will the corresponding ‘stage’ An (so to speak) in the construction of the step space be exactly equal to Q, though the smaller 𝜓 4 D is taken and the greater the number of rectangles, the smaller the difference becomes. So it is inviting to describe matters as follows: when 𝜓4 D is infinitely small, there is a final step space A? that consists of infinitely many infinitesimally thin rectangles that fill the mixtilinear space Q so precisely that the difference |A – Q| has itself become infinitely small and, thus, effectively null or negligible. This image of the space Q as equivalent to a step space built up from infinitely many infinitesimal rectangles is a powerful heuristic, and the device of infinitely small quantities both aids the imagination and abbreviates

22 The Archimedean principle of equality at work here is the counterpart of his claim that the arithmetical expression of the quadrature of the circle by the infinite series 𝜋/4 = 1/1 − 1/3 + 1/5 − 1/7 + 1/9 − 1/11 etc. is an exact equality and not merely an approximation.

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146 the continuum in leibniz the reasoning. But it is only a fiction—there is no terminal step space reached in the demonstration—and proofs employing it can be recast in more prolix form to make explicit the underlying relations among variable finite quantities. When Leibniz finally starts using the device of the infinitely small in DQA, at Proposition 11, it is taken in the same way: we reason as if curves and figures were built up from infinitesimal parts, though strictly to be infinitely small is just to be able to be made less than any given quantity. Leibniz’s aim in the DQA, of course, is not to eliminate infinitesimals but to promote their use in geometry through his new method of indivisibles, illustrated in the rest of the treatise by finding arithmetical quadratures of conic sections and various related theorems. He regards his demonstrations of Proposition 6 and 7 as having laid down rigorous foundations in Archimedean style that will support the subsequent uses of infinitesimals from Proposition 11 onward. In a scholium to Proposition 23, Leibniz notes: The things we have said up to now about infinite and infinitely small quantities will appear obscure to some, as does anything new; nevertheless, with a little reflection they will be easily comprehended by everyone, and whoever comprehends them will recognize their fruitfulness. Nor does it matter whether there are such quantities in the nature of things, for it suffices that they be introduced by a fiction, since they allow economies of speech and thought, and therefore economies of discovery as well as of demonstration, so that it is not always necessary to use inscribed or circumscribed figures, and to infer by reductio ad absurdum, and to show that the error is smaller than any assignable; although what we have said in Props. 6, 7 & 8 establishes that it can easily be done by those means.23 (A 7.6.585)

Leibniz’s view in DQA of the infinitely small and its correlative idea of the infinitely large as useful fictions, whose underlying truth is understood in terms of relations among variable finite quantities that could be spelled out painstakingly if necessary, will be his longstanding position. As he writes thirty years later to Bartholomew des Bosses: Speaking philosophically, I maintain that there are no more infinitely small magnitudes than there are infinitely large ones, that is, no more infinitesimals than infinituples. For I hold both to be fictions of the mind through an abbreviated way of speaking, adapted to calculation, as imaginary roots in algebra are too. Meanwhile I have demonstrated that these expressions are very useful for abbreviating thought and thus for discovery, and cannot lead to error, since it

23 Translation thanks to Richard Arthur, personal communication.

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leibniz on the infinitely small 147 suffices to substitute for the infinitely small something as small as one wishes, so that the error is smaller than any given, whence it follows that there can be no error. (1706. GP II 305)

The links between the Archimedean principle of equality and the content of Leibniz’s view of infinite and infinitely small quantities in his mathematics are indicated across his writings. An especially clear statement occurs in a 1695 response to criticisms of the calculus by Bernard Nieuwentijt, where Leibniz explains that his infinitely small differential (say, dx in x + dx) is not to be taken to be a fixed small quantity: Such an increment cannot be exhibited by construction. Certainly, I agree with Euclid bk. 5, defin. 5, that only those homogeneous quantities are comparable of which one when multiplied by a number, that is, a finite number, can exceed the other. And I hold that any entities whose difference is not such a quantity are equal. . . . This is precisely what is meant by saying that the difference is smaller than any given. (GM 5.322)

It needs to be observed that for Leibniz the terms ‘infinite’ and ‘infinitely small’ in mathematics are syncategorematic: rather than denoting special categories of non-finite magnitudes (as they would if they were ‘categorematic’ terms), they make their semantic contributions by way of imposing systems of logical and mathematical relations on the other terms in the statements in which they occur. To say that a difference is ‘infinitely small’ is not to imply the existence of a fixed infinitesimal quantity, but to say that the difference can be made less than any given finite value. Care is required here, however, to note also that Leibniz embraces the infinite in multitude: there are infinitely many bodies in the world, infinitely many parts in every portion of matter, infinitely many natural numbers, and so on, where ‘infinite’—still taken syncategorematically—means ‘more than can be specified by any finite number’.2⁴ Such a multitude is infinite, Leibniz says (echoing Cavalieri), ‘distributively’ across its many elements, not ‘collectively’ as a single whole or magnitude (see GP 2.315). For Leibniz there is no genuine infinite magnitude, great or small: no infinite line, no infinite quantity, no infinite number. Both sides of his stance on the infinite are on display here as he writes in the New Essays: It is perfectly correct to say that there is an infinity of things, i.e., that there are always more of them than one can specify. But it is easy to demonstrate that there is no infinite number, nor any infinite line or other infinite quantity, if these are taken to be genuine wholes. The Scholastics were taking that view, or should have been doing so, when they allowed a ‘syncategorematic’ infinite, as they called it, but not a ‘categorematic’ one. (NE 157) 2⁴ For example, see A 6.4.1393; GP 1.416.

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148 the continuum in leibniz The idea that the language of the infinite and infinitely small is syncategorematic couples nicely with Leibniz’s view of infinites and infinitesimals as fictional quantities. The combination suggests that the fiction is intended by Leibniz to be ‘logical’ in character: infinite and infinitely small quantities can be written out of the mathematics altogether via a syncategorematic analysis in favour of expressions referring only to finite quantities and their relations.2⁵ In any case it is clear that Leibniz regards the fictions as dispensable. They can be replaced by proofs given in Leibniz’s updated style of Archimedes if full rigour is wanted, and the mathematics in which they figure is not committed to the existence of ‘actual’ infinitesimals in nature for its justification.

2.2 The Archimedean Principle of Equality and the Law of Continuity We saw earlier that Leibniz’s appeal to the Archimedean principle of equality in Propositions 6 and 7 in DQA presupposes the continuity of the curve whose quadrature was to be demonstrated. To justify the claim that the difference can always be made smaller than any given difference, Leibniz must hold, as he explicitly does, that in any interval of the curve, however small, points can be taken in ‘so great a number’ that the maximum height 𝜓4 D of any rectangle in the step space can be made smaller than any given line. In general the viability of the principle of equality will always rely on some background postulate of continuity strong enough to imply that there can be no least difference—no error that cannot be made smaller—short of equality itself. Leibniz’s reasoning about equality in DQA is thus entwined with a more general assumption of continuity. In subsequent documents the significance of this assumption becomes more salient, as Leibniz declares his mathematical method to fall under a ‘principle of general order’ that he first publishes as such in 1687 and will later call his law of continuity:2⁶ When the difference between two cases, in what is given or presupposed, can be made less than any given quantity, it is necessary likewise that the difference in what is sought, or in the consequences that result from what is presupposed, is made less than any given quantity. Or to speak more commonly: when the cases (or givens) continually approach one another, so that one finally passes over into the other, the same must happen in the corresponding consequences or results (or what is sought). (A 6.4.2032)

2⁵ See Ishiguro (1990), Arthur (2013), Levey (2008), Knobloch (1994, 2002). 2⁶ Leibniz asserts the principle of continuity already as early as 1678 in De Corporum Concursu (LH 35.9.23). See also A 2.1.470.

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leibniz on the infinitely small 149 Leibniz is formulating a quite general and abstract idea of continuity: not continuity as it applies to an extended magnitude such as space (or time or motion), but continuity as it applies to the concept of a function. In particular, it provides essentially the same definition of a continuous function familiar from Weierstrass. Letting values x and x0 be the ‘two cases’ or ‘what is given’, and f (x) and f (x0 ) be ‘what results’, Leibniz’s definition says that as |x – x0 | < 1/n for any n > 0, then | f (x) – f (x0 )| is less than 1/m for any m > 0. The concept of function itself is only just emerging at this time (Leibniz used the term ‘function’ for a quantity related to points on a curve in 1673; in 1698 Johann Bernoulli began to use it for a single variable, with Leibniz’s approval), and it will need to be completely clarified before Leibniz’s ideas can have their full mathematical import. But the key system of inequalities central to the historic epsilon-delta definition of continuity are already well in place here in 1687. In the wider context for this particular passage it’s evident as well that Leibniz means to present his ‘principle of general order’ as useful not only to mathematics but also to physics and enquiry into the natural laws more broadly. That is, he sees himself as delivering an insight from mathematical reasoning to the rest of natural philosophy. The law of continuity seems to emerge for Leibniz from his reflection on the principle of equality and the Archimedean pattern of argument underlying his use of the infinite and infinitely small in mathematics. It is the general precondition for their validity, and for Leibniz it comes to be a metatheoretical principle for enquiry across multiple domains. In Cum prodiisset (c.1701), Leibniz defends his use of the infinite and infinitely small in his calculus explicitly by appeal to his ‘Lege continuitatis’ (H&O 40), which he states as the following ‘postulate’: In any proposed continuous transition that ends in a certain limiting case [terminus], it is permissible to formulate a general reasoning [ratiocinationem communem] in which that final limiting case [ultimus terminus] is included. (H&O 40)

He illustrates this with a series of examples: we include equality under a common reasoning with differences, rest with motions, parallel lines with converging lines, the parabola with ellipses (H&O 40–41). Leibniz acknowledges that ‘from this postulate arise certain expressions which are generally used for the sake of convenience, but seem to contain an absurdity’—that equality is a case of difference, rest is a case of motion, parallel lines meet, etc.—though the absurdity ‘ceases when its meaning is substituted’ (41–42). On analogy with imaginary roots, we speak of a straight line BP that, after rotation, ultimately becomes parallel with another fixed line VA but still ‘converges towards it or makes an angle with it, only that the angle is then infinitely small’ (42). Likewise, ‘when one straight line is equal to another, it is said to be unequal to it, but that the difference is infinitely small’ (42). In truth,

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150 the continuum in leibniz of course, things that are absolutely equal have no difference at all, parallel lines never meet, a parabola is not an ellipse at all, etc. (42). Leibniz says we are free to ‘feign’ (fingere) a ‘transitional state’ (status transitus) of infinitely small difference that lies between inequalities and equality, or infinitely small velocity between motions and rest, an ellipse with an infinitely distant focus between ellipses and the parabola, lines that intersect at infinity between converging lines and parallels, to transact the passage to the final limiting case so as to include it in the series (42). Such ideas can serve as aids to the mind in discovery, as Leibniz suspects they did for Archimedes and Conon. Whether the feigned quantities are even possible is open to dispute; but, in any case, geometry itself does not depend on the metaphysics (43). The fictions can be bypassed: all that is needed is to fill in the meaning of the apparently absurd expressions and the absurdity ceases. Leibniz then directly draws the lesson of these cases for the language of infinite and infinitely small quantities: It will be sufficient if, when we speak of infinitely great (or more strictly, infinite) or infinitely small quantities (that is, the infinitesimal quantities of our studies), it is understood that we mean quantities that are indefinitely great or indefinitely small, i.e., as great as you please or as small as you please, so that the error that anyone may assign is less than that which he himself assigned [ut error quem aliquis assignat, sit minor quam quem ipse assignavit]. Also, since in general it will appear that, when any small error is assigned, it can be shown that it should be still smaller, it follows that there is no error at all. (H&O 43)

The DQA roots are still clearly evident here in Cum prodiisset: substituting the Archimedean meaning eliminates the seeming absurdity. Leibniz further notes the connection with what we might call the range of values for the variable magnitudes underlying the talk of the infinitely great or small. Thus, by indefinitely small and infinitely great, we understand something arbitrarily great or arbitrarily small, so that each conducts itself as a sort of class (genus), and not as an ultimate instance of this class. (H&O 43/EMM 150)

This syncategorematic way of understanding the language of infinite and infinitely small appears to be Leibniz’s most favoured account. Even so, it is not therefore mandatory in his view. In a particularly concessive instrumentalist remark, he suggests that one who wishes to take those expressions at face value can do so, and even then without accepting the dubious metaphysics. The feigned status transitus can be left as an unreduced fiction. For the utility of the expressions and the fact

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leibniz on the infinitely small 151 that the reasoning in which they figure can be verified by rigorous Archimedean demonstrations suffices to justify their use: If anyone should understand something ultimate, or anyhow rigorously infinite, he can do this, and even without falling back on a controversy about the reality of extensions, or of infinite continuums in general, or of the infinitely small, indeed, even if he thinks that such things are utterly impossible; for it will suffice to employ them usefully in calculation, just as algebraists retain imaginary roots with great profit. For they contain abbreviations of reasoning, which can manifestly be verified in every case in a rigorous manner by the method already stated. (H&O 43/EMM 150)

Having sketched his informal philosophical rationale for the use of expressions for infinite and infinitely small quantities, Leibniz turns to the more exacting work of justifying the individual rules of the differential calculus on the basis of the law of continuity. But it seems right to show little more clearly that the Algorithm (as they call it) of our differential calculus, set forth by me in the year 1684, may be proved to be perfectly correct [verissimus esse comprobetur]. (H&O 44/EMM 150–151)

We won’t review the details of Leibniz’s subsequent justifications of the rules for sums, differences, products, quotients, and powers (46–48). But it is worth pausing to note the way Leibniz initially sets the stage by giving his ‘proper meaning’ for the expression ‘dy is the element of y’, and correspondingly for his famous ‘dy/dx’. Leibniz explains his expressions by reference to a sample curve, a parabola AY of the equation x2 = ay, or as he writes it, xx = ay (Figure 6.5): A T 1X

2X

dx 1Y

2Y

D

Figure 6.5 Leibniz, Cum prodiisset (c. 1701), 44.

Leibniz sets dx to be the difference between the abscissas A1 X and A2 X, that is, dx = |A1 X – A2 X|, and dy as the difference |1 X1 Y – 2 X2 Y| of the ordinates 1 X1 Y and 2 X2 Y. (Note that dx and dy are given as finite differences.) Solving now to obtain dy : dx, he writes:

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152 the continuum in leibniz Since y = xx : a, by the same law we have, y + dy = xx + 2xdx + dxdx, : a; and taking away y from the one side and xx : a from the other, we have left dy : dx = 2x + dx : a ;2⁷ and this is the general rule expressing the ratio of the difference of the ordinates to the difference of the abscissas, or if the chord 1 Y2 Y is produced until it meets the axis in T, then the ratio of the ordinate 1 X1 Y to T1 X, the part of the axis intercepted between the point of intersection and the ordinate, will be as 2x + dx to a. (H&O 44/EMM 151)

Having obtained the general rule for this ratio, Leibniz invokes his law of continuity to apply the general rule to the limiting case in which the ordinate 2 X2 Y has been moved into exact coincidence with 1 X1 Y, i.e., to include the ultimate case in which the difference dx = 0 together with the ordinary cases of finite differences in which dx > 0. Now, since by our postulate it is permissible to include in a general reasoning also the case where the ordinate 2 X2 Y, having been moved closer and closer to the fixed ordinate 1 X1 Y until it finally coincides with it, it is clear that in this case dx will be equal to zero [nihilo] or should be discarded, and so it is clear that, since in this case T1 Y is the tangent, 1 X1 Y to T1 X is as 2x to a.

So the difference dx in this limiting case in which it is equal to zero may be discarded from the calculation on the right-hand side of dy : dx = 2x + dx : a, while yet retaining its meaning in the ratio dy : dx on the left-hand side, thereby giving the slope of the tangent T at the point 1 X. Thus the term dx behaves in the calculation as if it were an infinitely small quantity, a fictional quantity in status transitus between something and nothing. But in fact dx stands for a variable finite quantity, and its behaviour reflects precisely the fact—ensured by the continuity of the curve AY—that the difference between the abscissas can be taken as small as one wishes, all the way to zero. Leibniz goes on: Hence it may be seen that in all our differential calculus there is no need to call equal those things that have an infinitely small difference, but those things are taken as equal that have no difference at all, provided that the calculation is supposed to have been rendered general, applying equally to the case where the difference is something and to where it is zero [nullum]; and only when the calculation has been purged as far as possible through legitimate omissions and ratios of non-vanishing quantities until at last application is made to the ultimate case, is the difference assumed to be zero. (H&O 44–45/EMM 151–152) 2⁷ There’s an obvious slip where the equation in Gerhardt’s text reads ‘dy ∶ dx = 2x + dy ∶ a’.

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leibniz on the infinitely small 153 Richard Arthur nicely observes here: It can be appreciated, I think, how close this is to a modern justification of differentiation in terms of limits. Leibniz has here assumed that the difference dx is a finite difference (we would write Δx), and when an expression for Δy/Δx has been found in which x has as far as possible been cancelled, he effectively takes the limit as Δx → 0 to arrive at an expression for dy/dx. ... What this application of the Law of Continuity legitimates, then, is proceeding as if dx and dy are infinitely small quantities that can be neglected in the last step of the calculation; whereas what in fact they stand for are finite differences that are assumed to vary in accordance with the Archimedean Axiom, so that they can be made arbitrarily small. (Arthur 2013, 564, with omission)

So, more broadly, the law of continuity supports a certain mathematical fiction: the idea that in the case of continuous transitions, the ultimate case falling under the general reasoning belongs to the same kind as the predecessor cases. A circle is a polygon, an infinite polygon; a parabola is an ellipse, one with an infinitely distant focus; equality is a difference, just infinitely small, etc. The fiction is useful, for it allows us to move directly, via a sort of shorthand, to apply the general reasoning to the limiting case, as the law guarantees. But it is always dispensable in favour of the indirect method of Archimedean demonstration. As Leibniz writes in a famous letter on the calculus to Varignon, composed near the same time as Cum prodiisset, ‘anyone who is not satisfied with this can be shown in the manner of Archimedes that the error is less than any assignable quantity and cannot be given by any construction’ (L 546). Leibniz’s law of continuity is a methodological precept and correspondingly vague, despite his supplying clear corollaries of it in the context of specific applications. Also, Leibniz offers rather little to justify the law itself, apart from broad appeals to a principle of sufficient reason. As he writes to Varignon, giving a rather different expression to the law: ‘the rules of the finite are found to succeed in the infinite’ and ‘the rules of the infinite apply in the finite’, and ‘this is because everything is governed by reason; otherwise there could be no science and no rule, and this would not at all conform with the nature of the sovereign principle’ (L 544). It is not hard to think of counterexamples to the law so stated, however, and at a minimum the law of continuity requires additional sharpening and guidance to be used correctly in specific cases.2⁸ Leibniz himself says that we can safely use the language of infinites and infinitesimals only when we have an accompanying demonstration as security, even if that demonstration is a general one, such as those provided by Propositions 6 and 7 in DQA, proving the soundness of a method for a class of similar cases. (As early as 1673, commenting 2⁸ See Jesseph (2015).

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154 the continuum in leibniz on Galileo’s paradox of the natural numbers, Leibniz says, ‘one ought not to say anything about the infinite as a whole except when there is a demonstration of it’ (A 6.3.168/LOC 7–8).) Thus the final character and soundness of this justification from the law of continuity for the use of the infinite and infinitely small in Leibniz’s mathematics remains open to question.

References Cavalieri, Bonaventure. 1635. Geometria indivisibilibus continuorum nova quadam ratione promota. Bologna. [GIC] Cavalieri, Bonaventure. 1647. Exercitationes Geometricae Sex. Bologna. [EGS] Descartes, René. 1637. La Géométrie. Leyde: Jean Maire. [Géométrie] Descartes, René. 1925. The Geometry of René Descartes. Translated by David E. Smith and Marica L. Latham. New York: Dover Publications, Inc. [G] Galilei, Galileo. 1898. Opere, Vols. 1–8. Ed. Antonio Favaro. Florence: Edizione Nazionale. [EN] Galilei, Galileo. 1974. Two New Sciences, Including Centers of Gravity and force of Percussion, 2nd ed. Translated and edited with commentary by Stillman Drake. Toronto: Wall and Emerson, Inc. [D74] Leibniz, G.W. 1846. Historia et origo calculi differentialis, ed. C.I. Gerhardt. Hannover: Hahn. [H&O] Leibniz, G.W. 1849–1863. Mathematische Schriften von Gottfried Wilhelm Leibniz, Vols. 1–7. Ed. C.I. Gerhardt. Berlin: A. Asher; Halle: H.W. Schmidt. [GM. Cited by volume and page.] Leibniz, G.W. 1875–1890. Die Philosophischen Schriften, Vols. 1–7. Ed. C.I. Gerhardt. Berlin: Weidmannsche Buchhandlung. [GP. Cited by volume and page.] Leibniz, G.W. 1920. The Early Mathematical Manuscripts of Leibniz. Edited and translated by J.M. Child. Chicago and London: Open Court Publishing. [EMM] Leibniz, G.W. 1923–. Sämtliche Schriften un Briefe. Berlin: Akademie-Verlag. [A. Cited by series, volume, page.] Leibniz, G.W. 1969. Philosophical Papers and Letters. Edited and translated by Leroy Loemker. Second edition. Dordrecht and Boston: Reidel. [L] Leibniz, G.W. 2001. Labyrinth of the Continuum: writings on the continuum problem, 1672–1686. Translated and edited with commentary by Richard T. W. Arthur. New Haven: Yale University Press. [LOC] L’Hôpital, Guillame. 1696. Analyse des Infiniment Petits pour l’Intelligence des Lignes Courbes. Paris. [Analyse]

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references 155 L’Hôpital, Guillame. 2015. L’Hôpital’s Analyse des Infiniment Petits: An Annotated Translation with Source Material by Johann Bernoulli. Edited and translated by Robert E. Bradley, Salvatore J. Petrilli, and C. Edward Sandifer. Basil: Birkhäuser.

Secondary Literature Arthur, Richard. 2008. “Leery Bedfellows: Leibniz and Newton on the Status of Infinitesimals.” In Infinitesimals Differences: Controversies between Leibniz and his Contemporaries, ed. U. Goldenbaum and D. Jesseph, 7–30. Berlin: Walter de Gruyter. Arthur, Richard. 2013. “Leibniz’s Syncategorematic Infinitesimals.” Archive for History of Exact Sciences 67: 553–593. Blasjo, Viktor. 2017. Transcendental Curves in the Leibnizian Calculus (Studies in the History of Mathematical Inquiry). Duxford, UK: Woodhead Publishing. Blasjo, Viktor. 2017a. “On What Has Been Called Leibniz’s Rigorous Foundation of Infinitesimal Geometry by means of Riemannian Sums.” Historia Mathematica 44: 134–149. Blasjo, Viktor. 2017b. “Reply to Knobloch.” Historia Mathematica 44: 420–422. Bos, Henk. 1974. “Differentials, Higher-Order Differentials, and the Derivatives in the Leibnizian Calculus.” Archive for History of Exact Sciences 14: 1–90. Bos, Henk. 2001. Redefining Geometrical Exactness: Descartes’ Transformation of the Early Modern Concept of Construction. New York: Springer. Bouquiaux, Laurence. 1994. L’harmonie et le chaos: le rationalism leibnizien et la ‘nouvelle science’. Louvain-la-Neuve: Editions de l’Insitut supérieur de philosophie: Edition Peeters. De Risi, Vincenzo. 2007. Geometry and Monadology: Leibniz’s Analysis Situs and Philosophy of Space. Basel/Boston: Birkhäuser. De Risi, Vincenzo. 2015. Leibniz on the Parallel Postulate and the Foundations of Geometry. Basel/Boston: Birkhäuser. De Risi, Vincenzo. 2018. ‘Analysis Situs, the Foundations of Mathematics and a Geometry of Space.’ In The Oxford Handbook of Leibniz, ed. M.R. Antognazza, 247–258. Oxford: Oxford University Press. De Risi, Vincenzo. 2019a. “Leibniz on the Continuity of Space.” In Leibniz and the Structure of Sciences: Modern and New Essays on Logic, Mathematics, Epistemology, ed. V. De Risi, Boston Studies in Philosophy and History of Science, 111–169. Berlin: Springer. De Risi, Vincenzo. 2019b. “Gapless Lines and Gapless Proofs: Intersections and Continuity in Euclidean Geometry.” Apeiron, 2019 (https://doi.org/10.1515/apeiron2019-0012)

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156 the continuum in leibniz De Risi, Vincenzo. Forthcoming. “Has Euclid Proven Elements I, 1? The Early Modern Debate on Intersections and Continuity.” In Reading Mathematics in Early Modern Europe, ed. P. Beeley, Y. Nasifoglu, and B. Wardhaugh. London: Routledge. Echeverría, Javier. 1980. La caractéristique géométrique de Leibniz en 1679. (PhD diss.) Paris: Sorbonne. Goldenbaum, Ursula, and D. Jesseph (eds.). 2008. Infinitesimals Differences: Controversies between Leibniz and his Contemporaries. Berlin and New York: Walter de Gruyter. Hartz, Glenn. 1992. “Leibniz’s Phenomenalisms.” The Philosophical Review 101: 511–549. Hartz, Glenn, and Cover, J.A.C. 1988. “Space and Time in the Leibnizian Metaphysic.” Noûs 22: 493–519. Ishiguro, Hidé. 1990. Leibniz’s Philosophy of Logic and Language, 2nd ed. Cambridge: Cambridge University Press. Jesseph, Douglas. 2015. “Leibniz on the Elimination of Infinitesimals.” In G.W. Leibniz, Interrelations between Mathematics and Philosophy, ed. P. Beeley, N. Goethe, and D. Rabouin, 189–205. Dordrecht: Springer. Knobloch, Eberhard. 1994. “The Infinite in Leibniz’s Mathematics: The Historiographical Method of Comprehension in Context.” In Trends in the Historiography of Science, ed. Kostas Gavroglu et al., 265–278. Dordrecht: Kluwer. Knobloch, Eberhard. 1999. “Galileo and Leibniz: Different Approaches to Infinity.” Archive for History of Exact Sciences 54: 87–99. Knobloch, Eberhard. 2002. ‘Leibniz’s Rigorous Foundation of Infinitesimal Geometry by Means of Riemannian Sums.’ Synthese 133 (1–2): 59–73. Knobloch, Eberhard. 2011. “Galileo and German Thinkers: Leibniz.” In Galileo e la scuola galileiana nelle Università del Seicento, ed. L. Pepe, 127–139. Bologna: Cooperativa Libraria Universitaria Bologna. Knobloch, Eberhard. 2017. “Letter to the Editors.” Historia Mathematica 44: 280–282. Levey, Samuel. 1998. “Leibniz on Mathematics and the Actually Infinite Division of Matter.” The Philosophical Review, 107: 49–96. Levey, Samuel. 1999. “Matter and Two Concepts of Continuity in Leibniz.” Philosophical Studies 94: 81–118. Levey, Samuel. 2002. “Leibniz and the Sorites.” The Leibniz Review 12: 25–49. Levey, Samuel. 2003. “The Interval of Motion in Leibniz’s Pacidius Philalethi.” Noûs 37: 371–416. Levey, Samuel. 2008. “Archimedes, Infinitesimals, and the Law of Continuity: On Leibniz’s Fictionalism.” In Infinitesimals Differences: Controversies between Leibniz and his Contemporaries, ed. U. Goldenbaum and D. Jesseph, 107–134. Berlin and New York: Walter de Gruyter. Levey, Samuel. 2015. ‘Comparability of Infinities and Infinite Multitude in Galileo and Leibniz.’ In G.W. Leibniz: Interrelations between Mathematics and Philosophy, ed. N. Goethe, P. Beeley, and D. Rabouin, 157–187. Dordrecht: Springer-Verlag.

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references 157 Mancosu, Paulo. 1996. Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century. New York: Oxford University Press. Quintana, Federico Raffo. 2018. “Leibniz on the Requisites of an Exact Arithmetical Quadrature.” Studies in History and Philosophy of Science 67: 65–73. Rabouin, David. 2015. “Leibniz’s Rigorous Foundations of the Method of Indivisibles”. In Seventeenth-Century Indivisibles Revisited, ed. Vincent Jullien, 347–364. Basil: Birkhäuser. White, Michael J. 1992. The Continuous and the Discrete: Ancient Physical Theories from a Contemporary Perspective. Oxford: Clarendon Press.

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7 Continuity and Intuition in Eighteenth-Century Analysis and in Kant Daniel Sutherland

1. Introduction Any attempt to give a complete account of the understanding of continuity in the eighteenth century and in Kant’s philosophy would require an examination of its role in metaphysical debates, in issues that arose in pure geometry, and in the foundations of analysis. This paper will focus on the last of these areas and the issues raised by continuity for the representation of the infinitely small. It will focus in particular on the status of geometrical and kinematic representations, which Kant classifies as intuitive representations, in the foundations of eighteenth century analysis and in Kant’s understanding of those foundations. There is a widely disseminated history of eighteenth-century analysis that is quite unflattering to the role of intuition. The British tradition descending from Newton employed his theory of fluxions and fluents in order to address worries about the foundations of analysis, including issues concerning continuity, and that theory rested on geometric and kinematic representations. In contrast, Continental mathematicians and natural philosophers, under the influence of Leibniz, were much less concerned with foundations and ‘metaphysical’ issues concerning continuity, and in particular did not appeal to geometrical and kinematic representations. Despite Newton’s enormous contributions, eighteenth-century British mathematical physics stagnated, while Continental mathematical physics made tremendous advances. The well-established narrative in the history of mathematics lays blame for this difference largely on the British tradition’s insistence on intuitive representations. Against this backdrop, Kant’s philosophy of mathematics does not fare well. Kant’s sympathies with Newton and the British tradition, his insistence on intuitive representation, and his references to Newton’s theory of fluxions place him squarely on the wrong side of this divide. Furthermore, Kant was writing in the midst of the remarkable progress on the Continent a full century after Newton’s Principia, and was in a position to know better, so that his views seem antiquated and naïve. Finally, the history of the fate of intuition in accounting for continuity in eighteenth-century analysis plays a role in a larger narrative about Daniel Sutherland, Continuity and Intuition in Eighteenth-Century Analysis and in Kant In: The History of Continua: Philosophical and Mathematical Perspectives. Edited by: Stewart Shapiro and Geoffrey Hellman, Oxford University Press (2021). © Stewart Shapiro and Geoffrey Hellman. DOI: 10.1093/OSO/9780198809647.003.0008

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seventeenth- and eighteenth-century analysis 159 the expulsion of intuition from mathematics, first in the foundations of analysis at the end of the nineteenth century and then in arithmetic at the end of the nineteenth and beginning of the twentieth century. Kant fares poorly in this larger narrative as well.1 The purpose of this article is twofold. First, relying on relatively recent reassessments of the history of analysis, it will attempt to bring forward a more accurate account of intuitive representation in eighteenth-century analysis and the relation between British and Continental mathematics. Second, it will give a better account of Kant’s place in that history. The result shows that although Kant did no better at navigating the labyrinth of the continuum than his contemporaries, he had a more interesting and reasonable account of the foundations of analysis than an easy reading of either Kant or that history provides. It also permits a more accurate and interesting account of how and when a conception of foundations of analysis without intuitive representations emerged, and how that paved the way for Bolzano and Cauchy. Section 2 of the paper reviews the widely held history of seventeenth- and eighteenth-century analysis, while Section 3 introduces Kant’s views and their relation to that history. Section 4 provides an improved historical narrative concerning the role of intuition in the foundations of eighteenth-century analysis. Section 5 provides a quick survey of the views of several of Kant’s immediate predecessors in Germany that likely influenced him. Section 6 then considers Kant’s views more closely with the benefit of the improved understanding of the historical context. It argues that his understanding of fluxions, infinitesimals, and limits was in step with widely held views at the time. Section 7 describes the emergence of a new understanding of the foundations of analysis in the middle of the eighteenth century and its relation to the larger arc of the history of analysis.

2. A Brief History of Seventeenth- and Eighteenth-Century Analysis There is a tidy story about treatments of continuity in the history of analysis after Newton and Leibniz that culminates with the work of Cauchy and Weierstrass, one that is unflattering to seventeenth- and eighteenth-century British mathematics and anyone sympathetic to it, including Kant. A long tradition of scholarship backs up this history, and what follows is necessarily a radically condensed version whose purpose is only to convey its main moral. According to this narrative, British mathematicians were far too beholden to the Greek mathematical tradition, first, 1 See, for example, Kitcher (1975), pp. 40–1. Coffa (1993), Chapters 1 to 3, provide a partiularly unsympathetic account of Kant along these lines.

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160 continuity in 18th-century analysis and in kant in giving priority to geometry over algebra and the related developing techniques of analysis, and second, in honouring the distinction between analytic and synthetic method. That is, they held that in whatever way algebra might be useful for the discovery of geometrical truths, the results were not considered adequately proved and certain until they were demonstrated from geometrical axioms and definitions, derivations that essentially employed geometrical representations. This attitude persisted through the development of analysis and worries about the legitimacy of appealing to infinitesimals, and spurred an appeal to the continuous motion of a point in generating geometrical curves. The appeal to geometrical and kinematic representations ultimately turned out to be a mistake. Isaac Barrow was especially important in establishing this general attitude and approach in Britain. His Geometrical Lectures of 1760, in particular, was influential, and is founded on the generation of geometrical magnitudes through the motion of points and lines. Barrow was Newton’s teacher and strongly influenced him. While Newton made free use of the techniques of analysis he developed in discovering the results presented in the Principia, he employed geometrical demonstrations in the Principia itself. Influenced by the Greek mathematical tradition and Barrow, he used this mode of demonstration, both because it secured reference and hence the meaningfulness for the words and symbols employed, and because it conferred a certainty on the results; he wanted above all to convince his readers with the certainty of these intuitive representations. The techniques of analysis Newton used were based on his method of fluxions, which used a ‘dot notation’ to represent the continuous rate of change of some quantity, a fluent. The history of Newton’s development of the theory of fluxions and its relation to infinitesimals is complex, but at least later in his career, Newton saw the continuous change of fluents and fluxions, and in particular the continuous motion of a point in generating lines, as a way of accounting for the continuity of physical quantities while sidestepping concerns about the seemingly contradictory properties of infinitesimals. Newton’s geometrical and kinematic understanding of fluxions and fluents were made manifest in the synthetic demonstrations of the Principia. To allay lingering potential worries about this appeal to fluxions and fluents and to provide it with further justification—and in keeping with both his veneration of Greek mathematics and his belief in the need for demonstrations according to the synthetic method—Newton developed his account of prime and ultimate ratios. This was a precursor to our modern notion of limit that explains the values of a limit through the representation of ratios of ever-decreasing finite quantities, above all the representation of ever-decreasing lines and areas over ever-decreasing intervals of time in the representation of instantaneous velocities. Newton’s prime and ultimate ratios still appealed to geometric and kinematic representations, while providing a more precise representation of the ratio between magnitudes as the magnitudes became ever smaller. Among British mathemati-

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seventeenth- and eighteenth-century analysis 161 cians, Newton’s account of prime and ultimate ratios was thought in a general way to correspond to Archimedes’ method of exhaustion employed in Euclid’s Elements, which connected Newton’s demonstrations to the Greek mathematical tradition. Newton’s emphasis on Greek mathematics and on the need for kinematic and geometrical representations to provide adequate and certain demonstrations impressed itself on British mathematics for over a century. The British approach to mathematics culminates with Colin Maclaurin’s Treatise of Fluxions, published in 1742, which was prompted in part by Berkeley’s trenchant attack on the coherence of appeals to either infinitesimals or fluxions. The first volume of Maclaurin’s treatise showed, in laborious detail, how Newton’s fluxional calculus could be grounded in geometrical and kinematic representations and vindicated the idea that it corresponds to the Archimedean method of exhaustion. British mathematics during this period contrasted sharply with that on the Continent, in particular in France, Switzerland, and Germany. The latter mathematicians were of course concerned to get correct results, but following Leibniz’s example, they were less focused on the certainty of their demonstrations and more interested in employing Leibnizian tools to achieve new results in mathematical physics. Complex phenomena of the natural world provided a standard against which to judge the success of those results, which replaced synthetic demonstrations in pure mathematics and the certainty they promised. These mathematicians were less interested in foundations and hence less worried about employing the notion of infinitesimals, and more interested in making new discoveries, whose correctness was validated by their successful description and prediction of complex features of the natural world. Leibniz’s notation was more perspicuous and easier to manipulate than Newton’s dot notation, which facilitated those discoveries. The lack of concern with foundations was part of a shift towards a more abstract view of mathematics based on symbolic representations and algebraic manipulations rather than geometrical or kinematic representations. And the French approach proved far more fruitful; eighteenthcentury advances in analysis were driven by work on the Continent by mathematicians such as D’Alembert, L’Huilier, the Bernoullis, Euler, Lagrange, and Laplace, who adopted an increasingly abstract treatment of mathematics—abstract in the sense that it made less and less contact with concrete spatial and temporal representations, relying instead on the algebraic manipulation of symbolic representations. In comparison, British mathematics in the eighteenth century stagnated. According to the history we are recounting, the British were handicapped by their concern for foundations and with their admiration for Greek standards of certainty and insistence on synthetic method; their emphasis on demonstrations employing geometrical and kinematic representations was the primary factor keeping them from making progress.

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162 continuity in 18th-century analysis and in kant There were external factors as well. The Newton–Leibniz priority dispute, in particular, was divisive, and led those on either side of the Channel to prefer their own approach. The effect on the British was especially damaging, however, since it prevented them from appreciating the tremendous advances being made on the Continent and the shift in attitude about the foundations of analysis. According to the history of analysis we are recounting, mathematics was evolving towards a more abstract understanding of its subject matter and methods, and away from the perceived need to ground it in spatial or temporal representations. Mathematics was undergoing a deeply important change, and British mathematicians simply missed the boat. It was only at the end of the eighteenth century, after ignoring what was happening on the Continent for too long, that they realized their mistake, acknowledged the superiority of the Continental approach, and began to catch up, a process that took until the 1830s. This history of eighteenth-century analysis fits into a larger narrative about the evolution of mathematics through the end of the nineteenth century and into the twentieth century—the increasingly abstract understanding of mathematics, as well as the rigorization and arithmetization of mathematics. These developments are marked by more rigorous expression of basic properties of continuity and of numbers, respectively, with an explicit rejection of any appeal to geometrical, kinematic, or other ‘intuitive’ forms of representation. This broader history is supported by and also confirms the view that eighteenth-century British mathematics suffered precisely because of its insistence on synthetic proofs and geometrical and kinematic representations in the foundations of analysis to overcome worries about continuity.

3. Kant’s Views and Their Place in This History Kant’s place in this history of mathematics is especially unflattering. It is often noted that, unlike Descartes and Leibniz, Kant was not a philosophermathematician making revolutionary contributions to mathematics, nor a highly influential mathematician such as D’Alembert, much less an Euler. He also did not write mathematics texts as did Christian Wolff, or make relatively modest contributions while writing textbooks, such as Kant’s contemporaries Abraham Kästner, Johann Segner, and Wenceslaus Karsten. Kant was not, nor represented himself as, a mathematician; he was a philosopher, on the outside looking in. He was concerned with the foundations of mathematics, with a particular emphasis on explaining mathematical cognition and its implications for human cognition in general. It would therefore be possible to think that Kant’s knowledge of mathematics was relatively unsophisticated, especially in light of the tremendous advances of the late eighteenth century.

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kant’s views and their place in this history 163 This initial judgement needs to be moderated, however, for Kant was no mathematical rube. Kant’s Metaphysical Foundations of Natural Science shows a deep engagement with Newton’s Principia and a sophisticated understanding of the mathematics required for it. While to his credit, this might give the impression that he was more interested in Newton than eighteenth-century developments and that he was a full century behind. That, too, is incorrect. While Kant was not in a position to offer mathematical solutions, he understood and grappled with issues raised by continuum mechanics in his day, as well as subtle issues concerning the relation of universal gravitation to the oblateness of the earth.2 Nevertheless, when it comes to the foundations of analysis, important passages in Kant suggest that he uncritically adopted Newton’s theory of fluxions as a solution to worries about continuity, and that he took geometrical and kinematic representations to be essential to the mathematics required for natural science. In the Anticipations of Perception of the Critique of Pure Reason, for example, Kant argues that space and time are quanta continua, and then states: Magnitudes of this sort can also be called flowing [fließende], since the synthesis (of the productive imagination) in their generation is a progress in time, the continuity of which is customarily designated by the expression ‘flowing’ (‘elapsing’) [Fließens (Verfließens)]. (A170/B211–12]

Kant’s identification of continuity with ‘flowing’ magnitudes is a clear reference to Newton’s theory of fluents and fluxions; ‘fließende Größe [flowing magnitudes]’ was the German translation of fluent. Kant does more than endorse Newtonian fluents and fluxions, and the appeal to motion on which it rests, for the foundations of analysis. He makes the strong claim that we can only represent any line whatsoever as generated by the motion of a point. In §24 of the Transcendental Deduction, he states: We cannot think of a line without drawing it in thought, we cannot think of a circle without describing it . . . (B154)

Kant repeats this claim elsewhere in the Critique: . . . in order to cognize something in space, e.g., a line, I must draw it, and thus synthetically bring about a determinate combination of the given manifold, so that the unity of this action is at the same time the unity of consciousness (in the concept of a line), and thereby is an object (a determinate space) first cognized. (B137–8)

2 See Friedman (2013), p. 142 on continuum mechanics, pp. 483ff. on the oblateness of the earth.

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164 continuity in 18th-century analysis and in kant I cannot represent to myself any line, no matter how small it may be, without drawing it in thought, i.e., successively generating all its parts from one point, and thereby first delineating this intuition. (A162/B203)

While Kant holds that motion is required for the representation of any sort of line, whether straight, curved, or wildly wiggling, Kant’s discussion of the drawing of a line and of a circle is a clear reference to Euclid’s construction postulates, a connection he also makes elsewhere.3 Kant showed the same veneration for Greek mathematics as Newton and the British mathematicians; he held up Euclid as a model of human knowledge and as a paradigm of certain synthetic a priori knowledge, beyond the meddling of philosophically motivated reforms. Kant also thought that motion was essential to Euclidean geometry, repeatedly drawing attention to the role of motion in the construction of geometrical figures. There were additional reasons for introducing motion into Euclidean geometry: the rigid motion of entire figures is presupposed by Euclid’s appeal to congruence from the earliest propositions of the Elements.⁴ Having admitted the motion of figures into Euclidean geometry, it was natural to connect that motion to a more general kinematic approach to magnitudes and to analysis in particular. Significantly, Newton himself made this connection in his mathematical papers: The geometry of the ancients had, of course, primarily to do with magnitudes, but propositions about magnitudes were from time to time demonstrated by means of local motion: as for instance, when the equality of triangles in Proposition 4 of Book I of Euclid’s Elements were demonstrated by transporting either one of the triangles into the other’s place. Also, the genesis of magnitudes through continuous motion was received in geometry: when for instance, a straight line was drawn along a straight line to generate an area, and an area was drawn along a straight line to generate a solid. If the straight line which is drawn into another be of a given length, there will be generated a parallelogram area. If its length be continuously changed according to some fixed law, a curvilinear area will be generated . . . If times, forces, motions and speeds of motion be expressed by means of lines, areas, solids or angles, then these quantities too can be treated in geometry. Quantities increasing by continuous flow we call fluents, the speed of flowing we call fluxions, and the momentary increments we call moments.⁵

3 Kant also discusses drawing a straight line and describing a circle in a document sent to his friend and defender, Johann Friedrich Schultz (11:184ff.), who published it in the Allgemeine LiteraturZeitung under the title ‘On Kästner’s Treatises’. See Kant (2014). Friedman (1992, 2000, Forthcoming) emphasizes the role of motion in translations and rotations corresponding to Euclidean constructions. ⁴ See Sutherland (2005) for further discussion of the reliance on congruence as it had been historically understood and motion. ⁵ Newton, Mathematical Papers, 8:455. Quoted and discussed in Guicciardini (1999), p. 103.

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kant’s views and their place in this history 165 Newton thus identifies the motion required for Euclidean geometry with the motion underlying his theory of fluxions. Kant agrees with Newton on this point. Furthermore, Kant’s explicit connection of flowing magnitudes to continuity strongly suggests that he thinks that the theory of fluxions and fluents accounts for the continuity required for mathematical physics. Kant does not, however, appeal to motion only to explain the continuity required for analysis, as a precondition for Euclidean congruence and Euclid’s construction postulates, and the representation of any geometrical figure. Kant incorporates the idea that a line is generated by the motion of a point into the heart of his philosophy. In §24 of the Transcendental Deduction, Kant argues that the faculty of understanding is responsible for producing the representation of any combination of any manifold, including a manifold in the form of inner sense, that is, in time. He argues that any combination of the manifold in the form of inner sense requires a ‘synthetic influence of the understanding on inner sense’, under the name of a ‘figurative synthesis’. Kant claims that we can see this in ourselves, and it is at this juncture that Kant makes the claim at B154, quoted above, concerning the drawing of a line and the describing of a circle. Kant then continues by making a further claim that the drawing of a line is a condition of representing time itself, which is bound up with representing succession: . . . we cannot even represent time without, in drawing a straight line (which is to be the external figurative representation of time), attending merely to the action of the synthesis of the manifold through which we successively determine the inner sense, and thereby attending to the succession of this determination in inner sense. (B154)

Motion, he claims, is what ‘first produces the concept of succession at all’. The motion Kant has in mind here is ‘motion, as action of the subject (not as determination of an object)’. Kant clarifies the distinction between these two notions in a footnote: Motion of an object in space does not belong in a pure science, thus also not in geometry; for that something is moveable cannot be cognized a priori but only through experience. But motion, as description of a space, is a pure act of the successive synthesis of the manifold in outer intuition in general through productive imagination, and belongs not only to geometry but even to transcendental philosophy. (B154–5n)

Kant thereby places motion of a point not only at the basis of geometrical representation and analysis, but at the core of his transcendental philosophy.⁶ It is difficult ⁶ Michael Friedman highlights this footnote and emphasizes the centrality of motion in Kant’s transcendental philosophy more broadly. See Friedman (1992) and (2000).

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166 continuity in 18th-century analysis and in kant to escape the impression that Kant’s transcendental philosophy was influenced not just by the role of motion in Euclidean geometry, but by its role in the theory of fluxions.⁷ Newton’s view that geometrical and kinematic representations are required for the certainty and meaningfulness of mathematics also very likely contributed to Kant’s conviction that intuition is required for all human cognition. As early as 1763, when Kant wrote his Prize Essay, Kant holds that the distinctive certainty of which mathematics is capable depends on intuition. Kant had not yet developed his distinction between intuitive and conceptual representations and the faculties to which they belong; he also had not yet developed his theory of the construction of mathematical concepts. Nevertheless, Kant argues that the certainty of mathematics is grounded in the representation of concrete particulars in intuition, a sensible representation that allows things to be known ‘with the degree of assurance characteristic of seeing something with one’s own eyes’ (2:291). At the same time Kant states, in his discussion of the arithmetic of numbers and general arithmetic, i.e., algebra, that one manipulates signs and that, temporarily, ‘the things themselves are completely forgotten’, but that after drawing a conclusion, ‘the meaning of the symbolic conclusion is deciphered’ (2:278). The meaning of numbers and algebraic symbols, as well as the signs of geometry, is ultimately grounded in their ‘representation of things themselves’ (2:279), a reflection of the traditional view of the meaningfulness of mathematics with roots in Greek mathematics and endorsed by British mathematicians. Although Kant does not directly discuss the foundations of analysis in this book, he does speak admiringly of Newton in the Introduction, whose method he says ‘transformed the chaos of physical hypotheses into a secure procedure based on experience and geometry’ (2:275). The reference to geometry can only be to the synthetic method Newton employed in the Principia. In sum, it is overwhelmingly likely that Kant’s view that intuitive representations play a role in all mathematical cognition was supported and reinforced by his understanding of Newton’s approach to the foundations of analysis, one that emphasizes the role of geometrical and kinematic representations. Kant’s conviction that intuition is required for all mathematical cognition had a tremendous influence on the development of his critical philosophy. It provided Kant’s strongest case for the existence of synthetic a priori knowledge, a key element of his argument for his critical philosophy. It also provided a model for the role of intuition in human cognition more generally. Moreover, Kant held that mathematics is a science of magnitude, that the representation of magnitudes is made possible by intuition, and that objects of experience are extensive magnitudes whose real properties all have intensive magnitudes. His view that intuition

⁷ See Kitcher (1975), pp. 40–1. Friedman (1992), pp. 75ff. provides an extended discussion of Newton’s theory of fluents and fluxions and their relation to Kant’s views.

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kant’s views and their place in this history 167 is required for mathematics is incorporated into his understanding of all human experience and influenced his entire theoretical philosophy. The main moral of the history of analysis we’ve been recounting makes Kant look particularly bad. He, like Barrow, Newton, and other British mathematicians, was seduced by Greek mathematics into a false view of mathematics and what it requires, which led him to emphasize the role of intuition in mathematical cognition. In particular, he erroneously appealed to Newton’s theory of fluxions and geometrical and kinematic representations to overcome serious issues concerning continuity. Kant’s philosophy of mathematics, especially his insistence on the role of intuitive representations in all mathematical cognition, was simply mistaken. But the implications of this mistake extend well beyond his philosophy of mathematics, since it deeply influenced his understanding of the nature of all human cognition and shaped his critical philosophy. What is particularly damning, however, is that Kant fell into this error while writing on the Continent a full century after Newton’s Principia. Kant seems to have missed the remarkable advances in eighteenth-century analysis on the Continent that set aside worries about foundations, used Leibniz’s notation of differentials, and appealed to infinitesimals. While some excuse might be made for Newton and his seventeenth-century followers, Kant was in a position to have known better. Kant’s passing references to the Newtonian theory of fluxions underscores the impression that, at least when it came to the foundations of analysis, his views were fairly unsophisticated, despite their impact on his philosophy of mathematics and critical philosophy more generally. Finally, as noted in the introduction, the history of analysis we have been recounting supports and is itself reinforced by a larger history of mathematics, in which intuition is definitively expelled from the foundations of analysis at the end of the nineteenth century, followed by the elimination of any role for intuition in the foundations of arithmetic. Intuition is even removed from the foundations of geometry by the advent of axiomatic foundations and the distinction between, on the one hand, the logical deduction of propositions of geometry and, on the other, empirical questions of applicability. These advances finally and definitively extinguished Kant’s claim that intuition is required for mathematics. The history of analysis we have reviewed, however, is Whiggish. In the next two sections, we will see that the history of seventeenth- and eighteenth-century analysis is not as straightforward as the version we have been considering, and that in light of a corrected history and a closer look at his views, Kant’s understanding of the foundations of analysis are more sophisticated than one might have thought. The corrected history also reveals that a new understanding of the foundations of analysis first emerged in the second half of the eighteenth century, one that for the first time took foundations seriously while setting aside appeal to geometrical and kinematic representations.

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168 continuity in 18th-century analysis and in kant

4. Toward a Less Whiggish History of Analysis To make room for a more accurate history of eighteenth-century analysis, it is helpful to first consider the source of the Whiggish history outlined in Section 1.⁸ Its origins are found in British mathematicians at the end of the eighteenth century and beginning of the nineteenth. Playfair called attention to the fact that British mathematicians were lagging behind their counterparts on the Continent, Woodhouse introduced Leibnizian notation to Britain, and Babbage, Herschel, and Peacock founded the Analytical Society to advocate for Leibnizian notions and analytic and algebraic techniques; by the 1830s, Britain was recovering from its mistake. These nineteenth-century British mathematicians blamed Newton’s inferior dot notation and British chauvinism stoked by the Newton–Leibniz controversy, which led to a self-imposed isolation that weakened British mathematics by reducing their resources for innovative ideas. Above all, however, they laid blame on the demand for geometrical and kinematic representations in analysis. Even the chauvinism was a problem precisely because it reinforced the view that Newton’s synthetic method was the proper approach to mathematics, with its appeal to geometrical and kinematic representations. The views of these early reformers were then reflected in later histories of analysis, such as the work of Rupert Hall, Morris Klein, and others.⁹ And the view that British mathematicians of the eighteenth century were held back by Newton’s emphasis on geometrical and kinematic methods and had little or no influence on the developments in analysis on the Continent also featured prominently in later histories.1⁰ This account of the source of stagnation is too simplistic, however. First, there were also significant external sociological forces within Britain that further weakened British mathematics during this time. In the eighteenth century, Cambridge incorporated mathematics, above all geometry, into the curriculum, but the content of the curriculum treated mathematics as an exercise in careful reasoning, and focused on past mathematical results rather than on innovation. Mathematics did not have a high rank in the hierarchy of disciplines, and the pursuit of pure mathematics for its own sake was not particularly encouraged.11 This attitude extended into natural philosophy as well; while Newton was still president of the Royal Society, there was resistance to the mathematical approach to nature he endorsed. Soon after his death, there were disputes between those who endorsed a Newtonian mathematical approach to nature and those who pursued non-mathematical natural history. The struggle favoured the natural historians ⁸ What follows in this section is indebted above all to the work of Judith Grabiner and Nicolò Guicciardini. See the references that follow. ⁹ Guicciardini (2004), pp. 219–20. 1⁰ Grabiner (1997), pp. 393–4. 11 Guicciardini (2004), p. 252.

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toward a less whiggish history of analysis 169 over the so-called ‘philomaths’, and under the presidency of Joseph Banks, who strongly supported the former, eventually resulted in the resignation, in 1784, of an important number of the philomaths.12 Pure mathematics as both a field for advancement and a mathematical approach to nature found neither status nor support in Britain during the eighteenth century. This contrasted sharply with the high regard for mathematics on the Continent, in particular France, during the same period. Those who focused on mathematics and mathematical physics were hired into university positions, and their contributions were highly valued. Thus, not all responsibility for the state of eighteenthcentury British mathematics can be attributed to their insistence on foundations and intuitive representations. Even if there were such external factors holding British mathematics back, however, it could still be the case that the British emphasis on synthetic method and geometrical and kinematic representation hindered British progress. But here, too, the actual history is more complex in ways that are important for understanding shifting attitudes towards mathematics in Britain and on the Continent. While Newton did promote a veneration of Greek mathematics and thought that synthetic proofs were required to place results on secure foundations, Newton relied heavily on symbolic notation and algebraic manipulations. For example, Newton made important contributions to ‘common analysis’, that is, the use of finite equations to study ‘geometrical’ (i.e., algebraic) curves, which were published in his Arithmetica Universalis.13 He also relied on symbolic representation and algebraic manipulations in his analytic method of infinite series and fluxions. Newton’s appeal to fluxions and fluents was first and foremost a method, one that used his symbolic dot notation. At least early in his researches, Newton refers to infinitesimals in conjunction with this method, though he later came to emphasize the motion underlying fluents as an alternative to the problematic notion of infinitesimals, and in light of continuing concerns about the foundations of his results, developed the theory of prime and ultimate ratios. The relations among these ideas in the development of Newton’s views is complex and we won’t attempt to reconstruct them, but Newton later distinguished between the analytic and the synthetic fluxional calculus; the first corresponded to the use of symbol manipulation in his dot notation to derive results, while the latter referred to the synthetic method of geometrical proofs found in the Principia. Significantly, different British mathematicians responded to Newton in different ways. One strand of thought indeed focused on grounding analysis in geometrical and kinematic representations and looked to Greek mathematics as a model. But there was another strand that focused on Newton’s analytic methods and the use of infinite series, which included mathematicians such as Brook Taylor, 12 Guicciardini (2004), pp. 250–2. 13 Guicciardini (2004), pp. 225, p. 231.

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170 continuity in 18th-century analysis and in kant James Stirling, Roger Cotes, and Abraham de Moivre.1⁴ Maclaurin was also deeply influenced by the analytic methods inspired by Newton. Perhaps the most famous achievement of these followers of Newton is the Taylor series—the representation of a function as an infinite sum of terms determined by the derivatives of the function at a single point. As Guicciardini states: … a group of British mathematicians active in the first decades of the eighteenth century successfully extended research lines presented in highly algorithmic works left by Newton on common and new analysis. All these early followers of Newton, contrary to the received view, were proudly aware of the fact that they were contributing to a new, highly symbolical, kind of mathematics. (2004, p. 253)

In short, British concerns about foundations did not prevent productive advances in analysis. This brings us to Maclaurin’s monumental Treatise of Fluxions of 1742. Maclaurin had two major aims in this work corresponding to its two volumes. Volume I was dedicated to clarifying the foundations of the method of fluxions. As noted above, British mathematicians thought that Newton’s theory of prime and ultimate ratios corresponded to the Archimedean method of exhaustion found in Book X of the Elements, which provided a secure geometrical foundation for the reasoning underlying the theory of fluxions. Nevertheless, the correspondence had not been backed up with demonstration, and general concerns about foundations were fanned by Berkeley’s attack on the methods of analysis in The Analyst of 1734. Berkeley aimed his criticisms not just at infinitesimals but the theory of fluxions and prime and ultimate ratios. And his attack was not just polemical; his criticisms were deeply incisive and recognized as such. At least part of Maclaurin’s intention in the first volume was to provide a definitive response to Berkeley and to finally lay to rest lingering worries about fluxions. He did so by showing how the theory of fluxions and prime and ultimate ratios could indeed be grounded in the Archimedean method of exhaustion. It took Maclaurin almost 500 pages to do so. Yet this was only the first volume. Having vindicated Newton’s approach, Maclaurin used the second volume, which was more than 350 additional pages, to introduce, defend, and develop the analytic fluxional calculus, with its emphasis on the algebraic manipulation of symbols—Newton’s dot notation. He was explicit about his understanding of the relation between the two volumes; because the foundations of the method of fluxions were secured by appeal to geometrical and 1⁴ Guicciardini (2004), pp. 232.

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toward a less whiggish history of analysis 171 kinematic representation in the first volume, one could dispense with those representations and get on with algorithmic analytic methods in the second. Maclaurin thereby unified the analytic and synthetic strands of Newton’s influence.1⁵ The importance of Maclaurin’s accomplishments in Volume II should not be underestimated; they directly contributed to the development of analysis. According to the Whiggish history of analysis, Continental mathematicians made rapid advances on their own, ignoring their plodding counterparts in Britain. In fact, however, they were paying attention. Clairaut, a French mathematician, promoted Newton’s work on the Continent, and corresponded with Maclaurin. Euler was in correspondence with James Stirling, one of the ‘analytic’ Newtonians noted above, as well as Clairaut. Maclaurin’s work was known, read, and respected on the Continent, and he was even awarded two prizes from the Acadèmie des Science in Paris before publishing his Treatise on Fluxions. Before the latter appeared, Euler wrote to Clairaut asking him to keep him informed of Maclaurin’s anticipated work on analysis; once he read it, he praised it in a letter to Goldbach. D’Alembert read Maclaurin’s Treatise and commended it for its rigour. In fact, D’Alembert was influenced by Volume I as well as Volume II; D’Alembert endorsed the method of limits over infinitesimals in part because of Maclaurin’s work. Lagrange also credited Maclaurin.1⁶ What is particularly important for our purposes is that the eighteenth century saw the emergence of more nuanced attitudes towards foundations than an insistence on geometrical and kinematic representations in Britain and a disregard for foundational issues on the Continent, reinforced by self-imposed isolation on each side. On the one hand, many of those who were concerned with foundations, especially but not only in Britain, came to think that geometric and kinematic representations could provide them, as Maclaurin had shown, and that once those foundations had been legitimated, algebraic methods applied to symbolic formulae were perfectly acceptable. On the other hand, many of those who were not concerned with foundations, especially but not only those on the Continent, came to think that Newton’s theory of fluxions and Leibniz’s infinitesimals were two roughly equivalent ways of getting at the same thing, and that Newtonian and Leibnizian notation could be translated one into the other. Furthermore, many converged in thinking that the method of limits was the best way to represent the properties required for carrying out derivations in analysis. D’Alembert and Maclaurin nicely exemplify the more nuanced attitudes towards foundations that emerged by the middle of the eighteenth century. Following Leibniz, the analysts on the Continent in the first half of the eighteenth century 1⁵ See Grabiner (1997) for a thorough defence of these claims. 1⁶ Grabiner (1997), p. 397.

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172 continuity in 18th-century analysis and in kant were simply not as concerned by foundational worries. What was important was that the analytic techniques based on Leibniz’s notation worked; as long as they did, the analysts were happy to employ Leibniz’s differentials, and, influenced by the success of his approach as well as by the Newton–Leibniz controversy, viewed them as superior to Newton’s appeal to fluxions. Following Leibniz, they largely set aside the puzzles surrounding continuity or the status of infinitesimals, which they viewed as ‘metaphysical’ questions that were unnecessary to answer. Despite a preference for differentials, their relaxed attitude towards foundations also led them to see little important difference between appeal to infinitesimals or to fluxions or to prime and ultimate ratios. Eventually, by the time D’Alembert wrote his important Encyclopédie articles on mathematics, he endorsed thinking of analysis in terms of limits over infinitesimals, and, as noted, he did so at least partially under the influence of Maclaurin. Meanwhile, on the other side of the Channel, the concern with foundations did not so much stifle the pursuit of analytic techniques as live alongside it. This is seen at first in different British mathematicians and then in one: Maclaurin. His attitude towards the relation of the two volumes of the Treatise of Fluxions demonstrates this. The concern with foundations did not prevent the British from developing new algorithms and analytic techniques or making contributions to analysis. In summary, the tidy history of analysis outlined in the previous section overemphasizes the influence of the synthetic method and geometrical approach in Britain, while ignoring the analytic fluxional calculus and advances in algorithmic approaches and treatments of infinite series. It also oversimplifies the divide between British mathematicians and their counterparts on the Continent. Of course, it remains true that most of the tremendous advances in eighteenthcentury analysis occurred on the Continent and in particular in France, and there is no doubt that those advances were in part made possible by setting aside foundational worries and using physical phenomena as a confirmation of results developed in the context of mathematical physics. But part of the responsibility also falls on external factors, that is, the difference in attitude towards and the level of support for mathematics in Britain and France. Part of the responsibility also falls on the clumsiness of the fluxional dot notation in comparison to Leibniz’s notation. While there is no doubt that there were some British mathematicians who were much more preoccupied with grounding the techniques of analysis than Continental mathematicians, the degree to which that impeded the progress of mathematics and blocked them from contributing to its development, both at home and abroad, as well as the isolation and stagnation of eighteenth-century British mathematics, has been exaggerated. The lack of quick progress in Britain relative to the Continent can at best be partially attributed to the British concern for foundations and their appeal to geometrical and kinematic representations in accounts of foundations.

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5. Foundations of Analysis in Kant’s German Predecessors Before considering Kant’s views in relation to eighteenth-century analysis, it is worthwhile to at least very briefly consider the views of a few German contributors that influenced Kant: Christian Wolff, Euler, and Abraham Kästner. Wolff was an ardent supporter of Leibniz and reflected the Continental prejudices against Newton and the theory of fluxions. In his 1716 Mathematisches Lexikon, Wolff defines infinite analysis as a science of discovering from known truths others that are unknown, which includes algebra as well as Leibniz’s differential calculus (p. 53); he defines infinite analysis as a science that finds from given infinitely small magnitudes other magnitudes that are unknown and finite; he credits John Wallis, Bernard Nieuwentüts, and especially ‘Herr von Leibniz’s incomparable differential and integral calculus’ (p. 54). Writing at the height of the Newton–Leibniz priority dispute, he does not mention Newton at all in this entry. There is no entry for differential calculus, but the entry for ‘differential magnitude’ states that it is what Leibniz called the infinitely small difference between two magnitudes. He adds that a differential magnitude is sometimes called an infinitesimal, sometimes an infinitely small quantity, and that ‘the English, following Mr. Newton, call it by the clumsy name fluxiones’. Note that although hostile to Newton, Wolff nevertheless treats the theory of fluxions as another, albeit clumsy, way of talking about the same thing. Euler embodied the more relaxed Continental attitude towards foundations; he was unimpressed with the sorts of worries that Berkeley raised in The Analyst and that plagued those who were concerned with foundations. His Introduction to Analysis of the Infinite first appeared in 1748 and became the standard textbook on the subject. It contains what we would now describe as ‘pre-calculus’.1⁷ He states in the Preface that he introduces the material in such a way that ‘the reader gradually and almost imperceptibly becomes acquainted with the idea of the infinite’ (p. v). Beginning in Chapter VII, Euler freely refers to the infinitely small, but does so without explaining the notion. Euler does say more, however, in his Foundations of Differential Calculus, the first edition of which was published in 1755.1⁸ In the Preface, he repeatedly emphasizes that vanishing increments are of no interest; it is only the ratios among them that are. He defines the differential calculus as a method for ‘determining the ratio of the vanishing increments that any functions take on when the variable, of which they are functions, is given a vanishing increment’ (p. vii). But Euler also insists that the vanishing increment, which he also calls a differential and which he represents as omega, ‘completely vanishes’. Using the

1⁷ See John D. Blanton’s Translator’s Introduction to Euler (2000), p. xiii. 1⁸ Kant apparently owned this work, though the edition he had was published in 1790, well after Kant developed his critical philosophy. See Warda (1992).

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174 continuity in 18th-century analysis and in kant function x2 as an example, he states that the ratio of the function value to the vanishing increment does not arrive at the ratio 2x before the increment itself, omega, completely vanishes (p. vii). Euler repeatedly makes it clear that the rigour of differential calculus requires that the vanishing increments must be ‘completely annihilated [plane annihilarentur]’ and that they be ‘absolutely nothing’ [omnino nulla] and absolutely equal to zero (pp. vii–viii passim). He holds this view despite the fact that the increment stands in a finite non-zero ratio, the problem which gave rise to the notion of an infinitesimal in the first place. He explicitly rejects the notion of an infinitesimal, that is, a quantity less than any other yet not zero: To many who have discussed the rules of differential calculus, it has seemed that there is a distinction between absolutely nothing and a special order of quantities infinitely small, which do not quite vanish completely but retain a certain quantity that is indeed less than any assignable quantity. Concerning these, it is correctly objected that geometric rigour has been neglected . . . if these infinitely small quantities which are neglected in calculus are not quite nothing, then necessarily an error must result that will be the greater the more these quantities are heaped up . . . Those quantities that shall be neglected must surely be held to be absolutely nothing. (p. viii)

Euler directly addresses the worry that motivated the notion of an infinitesimal in the first place: . . . It is difficult to say what possible advantage might be hoped for in distinguishing the infinitely small from absolutely nothing. Perhaps they fear that if they vanish completely, then their ratio will be taken away, to which they feel this whole business leads. It is avowed that it is impossible to conceive how two absolutely nothings can be compared. They are forced to admit that this magnitude is so small that it is seen as if it were nothing and can be neglected in calculations without error. Neither do they dare to assign any certain and definite magnitude, even though incomprehensibly small. Even if they were assumed to be two or three times smaller, the comparisons are always made in the same way. From this it is clear that this magnitude gives nothing necessary for undertaking a comparison, and so the comparison is not taken away even though that magnitude vanishes completely. (p. viii)

In short, Euler simply bites the bullet, and sees no contradiction in maintaining both that the vanishing increment is zero, and that it also stands in a non-zero ratio.1⁹ The concern for the rigour of differential calculus trumps any worries 1⁹ See Grabiner (2005), p. 35. Using h for the vanishing increment, she describes Euler’s position thus: ‘… although h is zero when added to finite quantities, it is not zero when considered in ratios’.

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kant’s german predecessors 175 about the properties of vanishing increments. Note, however, that in Euler’s view, ‘geometrical rigour’ does not mean that the quantity can in principle be represented by a finite extended magnitude, since the magnitude is taken to be zero. Instead, geometrical rigour for Euler means an exactness of the ratio, which is only achieved at the limit. Abraham Kästner, only five years older than Kant, was a highly respected mathematician and poet at the University of Göttingen. His Foundations of Mathematics, which first appeared in 1758, went into six editions and was widely read. He worked on proofs of the parallel postulate, and he also wrote History of Mathematics, published in 1796, which was unusually well sourced and careful for its time. While Kant had philosophical differences with him later in his career, he praised Kästner as a mathematician, and owned his Foundations of Mathematics, including the third volume, Foundations of Analysis of the Infinite.2⁰ Kästner likely influenced Kant’s understanding of mathematics, and analysis in particular. Kästner exemplifies the more tolerant attitude that emerged on the Continent in the eighteenth century. In his Foundations of Analysis of the Infinite,21 Kästner shows a willingness to accept different accounts of the infinitely small, including Euler’s. He states that ‘One can view infinitely small magnitudes as something actual, as a fiction, or with Mr. Euler (Inst. Calc. Diff. Chap. II), view it as 0; the results will retain their exactness.’ Note that Kästner is not bothered by the fact that Euler does not resolve the problem of a magnitude whose quantity is equal to zero standing in non-zero ratios; nor does he side with Euler in holding that one must take the infinitely small to be absolutely zero in order to preserve the rigour of analysis. Kästner was also quite open to the theory of fluxions, and in fact, endorses it. Under the heading ‘Foundations of the Differential Calculus,’ he introduces differentials, but also devotes fourteen sections (eleven pages) to explaining the concepts of the fluxional calculus. He follows this explanation with a long section discussing the priority dispute, in which he concludes that Newton and Leibniz developed their methods independently after reflecting on the work of previous mathematicians (p. 49). He later reveals that he has taught the method of fluxions, adding that it is important to the lover of mathematics to be able to use the teachers on the other side of the sea (p. 55)—hardly a hostile attitude towards Newton or eighteenth-century British mathematics. In the preface to the first edition, Kästner explains that he has mostly used the method of prime and ultimate ratios, which Newton had used in his Principia (though in §45 he says that Newton did not provide a method of reckoning for them). Kästner, like others in his time, thought the method of prime and ultimate ratios was the best means for reasoning about infinitely small magnitudes.

2⁰ Warda (1992). 21 I am referring to the third ‘greatly expanded’ edition published in 1789. Kant owned the first edition published in 1761, which did not include all of what follows.

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176 continuity in 18th-century analysis and in kant In §45 Kästner returns to the theory of fluxions and discusses Maclaurin, who Kästner says proved the method of fluxions with the utmost clarity (albeit awkwardly and at too great length). He goes on to defend the theory of fluxions against the claim that motion, and that which is connected to it—space, time, and speed—does not belong to geometry, but to mechanics. Like Newton, he argues that motion is required to extend a straight line and to draw a circle, as is required for Euclid’s Elements. Mechanics takes into account forces, but geometry, he says, only considers motion phoronomically, which he says is a distinction Kant noted in his Metaphysical Foundations of Natural Science.22 Kästner did not, however, think that the theory of fluxions was wholly interchangeable with the theory of differentials: One understands the British badly if, at the instigation of many introductions to differential calculus, one imagines that their fluxions and our differentials are identical. Fluxions expand our concept of the differential calculus, and show how one can think of the ratio of speeds rather than infinitely small changes. (p. 55)

He clearly thought there was an important place for the theory of fluxions. Kästner had remarkably high regard for Maclaurin’s foundations of the theory of fluxions and fluents. In fact, Kästner singles out Maclaurin for extraordinary praise in his preface to the first edition of this work in 1761. He begins the preface with the phrase ‘Mysteries must indeed be something very delightful to the human understanding,’ giving the following examples from analysis: Magnitudes that are different and yet are viewed as the same; infinitely large things that in comparison to others are nothing; and infinitely small things that in comparison to others are infinitely large … (p. iii)

This opening is followed by various references to the mysteries of the infinite and to the labyrinth mentioned by various authors. This introduction to the topic makes it all the more remarkable that Kästner goes on to state that ‘Maclaurin, in his Treatise of Fluxions, had dispelled all the imagined mysteries of the infinite’ (p. xiii) We have seen that Kästner refers to both differentials and to fluxions, and is at pains to defend the latter, believes it is worth understanding it in order to learn from the British, and that he has utmost regard for Maclaurin’s Treatise on Fluxions. He also thinks that the method of prime and ultimate ratios is the clearest and best way to provide a treatment of the analysis of the infinite. Kästner’s views reflect the widespread attitude towards the foundations of analysis on the continent by the mid-eighteenth century. With this backdrop in place, we turn to Kant’s views. 22 Kant’s Metaphysical Foundations appeared in 1786, so this passage must not have appeared until after the third edition of Kästner’s work.

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6. Foundations of Analysis in Kant’s Metaphysical Foundations of Natural Science 6.1 General Remarks I will not provide a detailed reconstruction and interpretation of Kant’s views on the foundations of analysis here; in keeping with the broad scope of this paper, I will summarize Kant’s views to place them in the context of eighteenth-century foundations. We saw in section 3 that in the Critique of Pure Reason Kant embraces the idea that motion is essential to geometry, and even places motion at the heart of his account of transcendental philosophy; furthermore, he uses the expression ‘flowing magnitudes’ in his discussions of continuity, a clear reference to Newton’s theory of fluents and fluxions. But the best place to look for Kant’s views on the continuity required for analysis is his Metaphysical Foundations of Natural Science, published in 1786, a year before the second edition of the Critique appeared. Kant’s aim in this work is to provide the metaphysical foundations required for Newton’s mathematical approach to natural philosophy. Kant holds that there is only as much science in natural philosophy as there is mathematics. Since analysis is absolutely essential to Newton’s natural science, one can expect that any foundations of analysis required for that science would be explicitly addressed in the Metaphysical Foundations, perhaps with its own chapter. Surprisingly, however, nowhere in the Metaphysical Foundations does Kant focus explicitly on the foundations of analysis and the puzzles arising from treatments of continuity. Perhaps he thought that the foundations of analysis should appear in its own work, and that issues concerning the infinite, and in particular the infinitely small, should appear there. In a reflection of uncertain date, Kant jotted the following note, which looks like topics to be covered in such a treatise: Metaphysics of the doctrine of magnitude, or the metaphysical foundations of mathematics. Of the magnitude through degree, unity, and multiplicity [Menge]. Of multiplicity that is greater than all number. Of continuous magnitudes, the infinite (the unmeasurable), infinitely small. (14:196)

Unfortunately, Kant did not write a work on the metaphysical foundations of mathematics. But even if Kant thought the proper place for a full discussion of the infinitely small belonged elsewhere, it is surprising that he does not address the issue head on in the Metaphysical Foundations. Perhaps Kant thought, like others in Britain and on the Continent influenced by the Newtonian tradition, that foundational worries had been adequately addressed by Maclaurin, but in the writings we have, Kant nowhere refers to Maclaurin. More importantly, foundational issues are not the sort that Kant skims over. We are left wanting a

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178 continuity in 18th-century analysis and in kant fuller account of Kant’s views on fluxions and fluents and his understanding of the foundations of analysis. Although he does not directly raise and address worries about the foundations, Kant does discuss continuity and infinitely small magnitudes at various points in the text as they arise. This allows us to piece together his views. It also brings us to the second important and especially surprising point: Kant does not even mention fluxions or fluents in any of these passages. What is more, Kant refers repeatedly to infinitely small magnitudes throughout the Metaphysical Foundations, especially in the chapters on dynamics and mechanics. His appeal to infinitely small magnitudes seems out of step with Kant’s general treatment of the infinite, for example in the Antinomies of the Critique, in which Kant argues against the notion of the mathematically infinite as a completed totality and for the view that it is to be understood as a never completed synthesis, as a progress to infinity; that is, he argues against understanding the mathematically infinite as an actual infinity and instead argues for understanding it as an infinity In potentia. The same view is found in Kant’s understanding of the continuity of space and time; they are infinitely divisible, not infinitely divided; smaller parts of space and time can be determined ad infinitum. His appeal to infinitely small magnitudes in the Metaphysical Foundations thus seems out of step with Kant’s critical philosophy. What, exactly, is Kant’s view?

6.2 Kant on the Infinitely Small and Limits Kant’s discussions of issues concerning the infinitely small are scattered through the Metaphysical Foundations. What he says about the infinitely small makes it clear that this phrase is not simply shorthand for a magnitude that is diminishing towards zero. Nor does it mean a magnitude or quantity that is simply absolute zero, as Euler insists. Kant attributes to infinitely small magnitudes both a quantity that is smaller than any other and is also non-zero. In short, he attributes to them the characteristic properties of an infinitesimal. In Explication 3 of the Phoronomy, for example, he explains how one can appropriately attribute rest to point B, the turnaround point of an object continuously decelerated and then accelerated, such as a ball thrown up in the air. We do so based on the representation of a sequence of ever-decreasing speeds over smaller and smaller intervals with the turnaround point as an endpoint. In other words, Kant is appealing to an ultimate ratio or limit of distances over times. He then claims that we represent the ‘motion of the object as uniformly slowed down so that the speed at point B is not completely diminished, but only to a degree that is smaller than any given speed’, despite the fact that the speed at point B is zero, so that if it continued at that speed, it would ‘in no way change its place … in all eternity’ (4:486).

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Kant’s most revealing comments on the infinitely small occur in Propositions 4 and 8 of the Dynamics. In Proposition 4, Kant explains how it is possible to mathematically represent the repulsive forces of the parts of an elastic matter, which increases as it is compressed and decreases as it expands. The mathematical representation of the repulsive forces of these parts can only be represented in proportion to the distances of the parts from one another. He then gives as an example the repulsion of the ‘smallest parts’ of air, which repel one another in inverse ratio to their distances from each other. This model works in a straightforward way if there are smallest parts that are separated from each other by some distance, but does not work for continuous matter that is represented as completely filling the space it occupies. In that case, there is no actual distance between the parts and they are all in contact, because the parts always constitute a continuum. But representing the distance as zero would not allow one to represent the force as a function of the distance of the parts. Hence the mathematical procedure for ‘constructing the concept’ of the force in proportion to the distance of the parts must represent the contact among the parts as an infinitely small yet non-zero distance. Here is a clear case of Kant appealing to infinitesimals in the mathematical representation of repulsive forces. The way in which Kant describes the mathematical construction is particularly revealing. He distinguishes between what necessarily belongs to the procedure of constructing a concept and what belongs to the concept in the object itself, and warns against attributing the former to the latter, which ‘completely misses [the mathematicians’] meaning, and misinterprets their language’ (4:505). The mistake is to attribute to the parts of a continuous matter an ‘actual [wirkliche]’ distance of the parts, by which he means a finite distance between the parts.23 By calling a finite distance an actual distance, Kant may be emphasizing not only their finiteness, but the fact that what is under consideration is the distances between actual parts of matter and the repulsive forces that are a function of those distances. Kant’s formulation suggests that infinitely small distances are, in some sense, less than actual. Kant puts his point as follows: So even in the case of something divisible to infinity, no actual distance of the parts—which always constitute a continuity with all expansion [Erweiterung] of the space of the whole—can be assumed, although the possibility of this expansion can only be made intuitive under the idea of an infinitely small distance. (4:505)2⁴ 23 Friedman (2013), p. 168 fn. 104, points out that this is the meaning of ‘actual’ in this context. 2⁴ Kant uses the term ‘Erweiterung’, which in most contexts refers to an expansion in the senses of an enlarging of a space, and Friedman translates this term as ‘Enlargement’. But the German term also strongly connotes the expanse occupied by something, and to emphasize that, I have chosen to translate it as ‘expansion’. See footnote 26.

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180 continuity in 18th-century analysis and in kant Kant’s terminology here is quite significant: an idea is a pure concept that has its sole origin in the understanding (not in a pure image of sensibility) and goes beyond the possibility of experience; it is a concept that belongs to the faculty of reason (A3320/B376). What is important for our present purposes is that although no object can be given for an idea of reason, it can nevertheless play an important role in guiding and structuring our cognition. Kant’s view is that the representation of a sequence of ever-smaller finite spaces makes those distances intuitively representable, and thereby makes the forces over those distances mathematically representable, while the sequence of these finite but ever-decreasing distances are thought ‘under’ the idea of an infinitely small distance in our taking them to approach an ultimate ratio or limit; that is, the idea of an infinitely small line is employed to think about the sequence in the limit. What is doing the work of allowing us to intuitively represent, and hence construct, and hence mathematically represent, the ratio of force over distance are the decreasing finite distances.2⁵ And this corresponds to the broadly held view, found in the Newtonian tradition and promoted on the Continent after D’Alembert, that the best way to mathematically represent either fluxions or infinitesimals is by means of the notion of limits. Kant reiterates this view in Proposition 8, where he says that the mere idea of space is ‘thought simply for determining the ratio of given spaces, but is not in fact a space’ (4:521). By the ‘idea of space’ he means an infinitely small space: . . . since the adjacent parts of a continuous matter are in contact with one another, whether it is further expanded or compressed, one then thinks these distances as infinitely small, and this infinitely small space as filled by its repulsive force to a greater or lesser degree. But the infinitely small intervening space is not at all different from contact. Hence it is only the idea of space, which serves to make intuitive the expansion2⁶ of a matter as a continuous magnitude, although it cannot, in fact, be actually conceived [begriffen] in this way. (4:521–2)

Note that Kant says here that the idea of space makes the enlargement of matter as a continuous magnitude intuitive, but then immediately adds that they cannot be actually conceived in this way. Kant’s view is that the representation of everdiminishing distances as approaching a limit under the idea of an infinitely small distance allows us to make the filling of an expanse by continuous matter intuitive, while the idea itself cannot be actually conceived. His appeal to the representation of finite quanta in the theory of limits fits well with his view that intuition is

2⁵ For detailed argument that Kant appeals to finite magnitudes to represent limits in the case of motion, see Sutherland (2014). 2⁶ Just as in Proposition 4, Remark 1, Kant uses the term ‘Erweiterung’, which I have translated ‘expansion’ to emphasize the connotation of filling an expanse. See footnote 24.

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changing notions 181 required for mathematical cognition, while his treatment of the concept of the infinitely small fits well with this theory of the regulative use of ideas of reason. But where do fluxions and fluents fit in? Why does he mention them in the Critique, but not in the Metaphysical Foundations? The context in which the theory of fluxions and fluents arises in the Critique provides an answer: it is part of an account of the conditions of human cognition, and in particular of an account of the successive synthesis of productive imagination. The theory of fluxions and fluents plays an important role at this level of explanation, for it is the continuity of this synthesis—its flowing—that allows us to cognize continuous quanta, that is, quanta for which no part is the smallest. Thus, the Newtonian theory enters at a deeper level of explaining the possibility of mathematical cognition, one that is not required for an account of the metaphysical foundations of natural science, where limits and the idea of an infinitely small magnitude suffice. In summary, Kant’s account reflects a widespread attitude towards foundations in the eighteenth century that looked to the theory of limits for a mathematical representation and was relatively open-minded about the interchangeability of infinitesimals and fluents. Kant’s views are not, after all, an unreconstructed and anachronistic recapitulation of Newton’s theory of fluxions and fluents. One might object that Kant does not provide an adequate account of the infinitely small— neither the apparently contradictory properties he attributes to them, nor the laws governing derivations involving them. But to expect Kant to have done so is tantamount to expecting him to have found a way out of the labyrinth of the continuum. Kant did not accomplish that. What he did accomplish, however, was an account of limits and the notion of an infinitesimal that made room for the best understanding of infinite analysis as it was carried out in his time.

7. Changing Notions of Foundations and the Role of Intuition in Eighteenth-Century Analysis As we have noted, foundations of analysis had not been a dominant concern on the Continent from the time of Leibniz. Suppressing worries about foundations and using physical phenomena described by analysis as a touchstone of results aided their rapid advances, which in turn led to less concern about the meaningfulness of equations at each step of a derivation. At the same time, the concept of a function, further developed and exploited by Euler, led to a focus on the properties of functions themselves, apart from an ability to directly translate what they expressed into physical terms.2⁷ These were harbingers of a new way of thinking about mathematics. But despite this relative lack of concern for foundations

2⁷ See Guicciardini (1999), Chapter 9.

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182 continuity in 18th-century analysis and in kant backing up the advances in mathematics, an interest in foundations emerged on the Continent in the second half of the eighteenth century. There were various external factors for this. Some thought that the rapid developments in analysis since Newton and Leibniz had played themselves out, which prompted a period of consolidation of advances and reflection on the nature of analysis. Many mathematicians on the Continent had also taken up university positions, which required them to teach mathematics, and that in turn led them both to think more carefully about the fundamental assumptions of analysis and to write textbooks that began by addressing the topic.2⁸ But one person on the Continent in particular, Lagrange, played a particularly important role in spurring interest in the foundations of analysis.2⁹ Until at least 1760, Lagrange followed others on the Continent with a relatively relaxed attitude about foundations and endorsed Newton’s and D’Alembert’s use of limits. In 1772, however, he published an article inspired by Euler’s work in which he maintained that any function could be given a power-series expansion and that derivatives and integrals of functions could be algebraically derived from those power-series functions, all without any need for differentials or infinitesimals or limits. This was the beginning of a very new approach to foundations of analysis, one that he began to employ in his Méchanique analytique, which first appeared in 1788. Nevertheless, it was not until 1799, when he published his ‘Discours sur l’object de la théorie des fonctions analytiques’, that he articulated reasons for rejecting appeal to infinitesimals. The very title of this work states that it would provide . . . the principles of the differential calculus, freed from all consideration of the infinitely small or vanishing quantities, of limits or fluxions, and reduced to the algebraic analysis of finite quantities.3⁰

This does not mean, however, that Lagrange had entirely eliminated the employment of the infinitely small in his work. In the Preface to the second edition of the Méchanique analytique, published in 1811, he states that: The ordinary notation of the differential calculus has been preserved, because it corresponds to the system of the infinitely small, which is adopted in this Treatise. When we have grasped the spirit of this system, and are convinced of the exactness of its results either by the geometric method of prime and ultimate ratios, or by the analytical method of derived functions, we can employ 2⁸ See Grabiner (1983), pp. 188–9. 2⁹ What follows is particularly indebted to the excellent Grabiner (2005), especially Chapter 2. Coffa (1993), pp. 25–6, citing Grabiner, notes the particularly important role that Lagrange played leading up to the work of Bolzano. 3⁰ Lagrange (1797), quoted in Grabiner (2005), p. 43.

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changing notions 183 the infinitely small as an instrument and as a convenience to abbreviate and simplify the demonstrations. This is how the demonstrations of the ancients are abbreviated by the method of indivisibles.31

Lagrange still employs differentials and the notion of the infinitely small, and while he asserts that we can be convinced of the legitimacy of their use by his algebraic approach, he also allows that we can also be convinced by the geometric method of prime and ultimate ratios. In summary, it took until the end of the eighteenth century and the beginning of the nineteenth for Lagrange to fully express and argue for an algebraic foundation for analysis. Nevertheless, the significance of his new approach cannot be overstated. This was not a return to or acknowledgement of the importance of foundations in anything like the Newtonian sense. Its aim was to avoid appeal to infinitesimals, fluxions, and fluents, and even prime and ultimate ratios. As a consequence, it avoided appeal to geometrical or kinematic representations in the foundations of analysis. Lagrange had a new vision of foundations which took algebra as its starting point. Significantly, he was not concerned with providing a foundation for algebra itself by supplying it with a geometrical or kinematic interpretation; he treated algebra as sufficiently clear and well established that it required no such grounding. It is only at this point in the history of analysis that geometrical and kinematic representations grounded in intuition were no longer regarded as necessary to secure foundations by those who were nonetheless concerned about foundations. Lagrange was a catalyst for this change. We just saw that in the second edition Preface of the Méchanique analytique published in 1811, Lagrange allowed for the use of differentials and establishing their legitimacy either by the ‘geometric method of prime and ultimate ratios’ or by his analytic method of derived functions. But he also states: One will not find figures in this work. The methods that I expound require neither constructions, nor geometrical or mechanical arguments, but only algebraic operations, subject to a regular and uniform course.32

This marks the real turning point for the history of geometrical and kinematic representations in analysis. The more careful history we have been examining in this article shows that the fate of intuition in eighteenth-century analysis is not what one might have thought. A general lack of concern for foundations was one factor contributing to rapid advances in analysis on the Continent. This was not, however, what led to the expulsion of geometric and kinematic representations from the foundations of 31 Lagrange (1811), p. ii. 32 Lagrange (1811), p.i.

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184 continuity in 18th-century analysis and in kant analysis. By the mid-eighteenth century, as we saw, Continental mathematicians were not hostile to the British approach to analysis, and following D’Alembert, Continental mathematicians were receptive to a theory of limits based on the geometrical method of prime and ultimate ratios. It was actually the emergence of a concern with foundations on the Continent, coupled with Lagrange’s radically new understanding of what constituted foundations, that paved the way for the eventual banishment of geometric and kinematic representations from the foundations of analysis. In fact, Lagrange’s algebraic foundations both inspired Bolzano and prepared the path for Cauchy’s employment of the algebra of inequalities later in the nineteenth century. The corrected history tells a more interesting story of the fate of intuition in the history of analysis, one that pinpoints a crucial change in the conception of what is required for foundations introduced by Lagrange. He introduced his new understanding of foundations, which was developed just as Kant was publishing his critical philosophy. At the same time the corrected history also helps us understand Kant’s insistence on the need for geometrical and kinematic representations in mathematical cognition, including in the foundations of analysis, and places it in a more favorable light. Moreover, Kant was not simply falling back on Newton’s theory of fluents and fluxions a century after he should have known better. It is only in specifically philosophical contexts in which he is accounting for our cognition of continuity that Kant alludes to the Newtonian theory. Kant’s views on the foundations of analysis, revealed in his Metaphysical Foundations, reflected the views of the best mathematicians of the day.

References Barrow, I. (1916). The Geometrical Lectures of Isaac Barrow, J. M. Child, ed. London: Open Court. Coffa, J. A. (1993). The Semantic Tradition from Kant to Carnap: To the Vienna Station. Cambridge: Cambridge University Press. Euler, L. (1988). Introduction to Analysis of the Infinite. Book I and Book II. John D. Blanton, transl. New York: Springer Verlag. Euler, L. (2000). Foundations of Differential Calculus. John D. Blanton, transl. New York: Springer Verlag. Friedman, M. (1992). Kant and the Exact Sciences. Cambridge, MA: Harvard University Press. Friedman, M. (2000). “Geometry, Construction and Intuition in Kant and His Successors,” in Between Logic and Intuition: Essays in Honor of Charles Parsons, Gila Sher and Richard Tiezten, eds. Cambridge: Cambridge University Press. Friedman, M. (2013). Kant’s Construction of Nature. Cambridge: Cambridge University Press.

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references 185 Friedman, M. (Forthcoming). “Space and Geometry in the B-Deduction,” in: Kant’s Philosophy of Mathematics, Vol. I: The Critical Philosophy and Its Background, C. Posey and O. Rechter, eds. Cambridge: Cambridge University Press. Grabiner, J. (1997). “Was Newton’s Calculus a Dead End? The Continental Influence of Maclaurin’s Treatise of Fluxions,” The American Mathematical Monthly, Vol. 104, No. 5 (May, 1997), 393–410. Grabiner, J. (2005). The Origins of Cauchy’s Rigorous Calculus. Mineola NY: Dover Publications. Guicciardini, N. (1999). Rereading the Principia: The Debate on Newton’s Mathematical Methods for Natural Philosophy from 1687 to 1736. Cambridge: Cambridge University Press. Guicciardini, N. (2004). “Dot-Age: Newton’s Mathematical Legacy in the 18th Century,” Early Science and Medicine, Vol. 9, No. 3, 218–256. Guicciardini, N. (2009). Newton on Mathematical Certainty and Method. Cambridge, MA: MIT Press. Kant, I. (1902–). Gesammelte Schriften. 29 vols. Berlin: G. Reimer, subsequently Walter de Gruyter & Co. Kant, I. (1992). Theoretical Philosophy, 1755–1770. D. Walford, trans. and ed. New York: Cambridge University Press. Kant, I. (1997). Critique of Pure Reason. Paul Guyer and Allen Wood, transl. New York: Cambridge University Press. Kant, I. (2004). Metaphysical Foundations of Natural Science. Michael Friedman, transl. New York: Cambridge University Press. Kant, I. (2014). “On Kästner’s Treatises,” Christian Onof and Dennis Schulting, transl. Kantian Review, Vol. 19, No. 2, 305–313. Kästner, A. (1761). Der mathematischen Anfangsgründe Dritter Theil, Zweyte Abt., Anfangsgründe der Analysis des Unendlichen. Göttingen: Rosenbusch. Kästner, A. (1796). Geschichte der Mathematik. Göttingen: Rosenbusch. Kitcher, P. (1975). “Kant and the Foundations of Mathematics,” The Philosophical Review, Vol. 84, No. 1, 23–50. Lagrange, J. L. (1799). “Discours sur l’object de la théorie des fonctions analytiques,” Journal de L’Ecole Polytechnique, Cahier 6, 2. Lagrange, J. L. (1811). Méchanique Analytique, 2nd ed. 2 vols. Paris: Mme. Ve. Courcier. Maclaurin, C. (1742). A Treatise of Fluxions in Two Books. Edinburgh: Ruddimans. Newton, Isaac. (1769). Universal arithmetick, or, A treatise of arithmetical composition and resolution. J. Raphson, transl. and S. Cunn, ed. London: W. Johnston. Newton, Isaac. (1962) The Principia: Mathematical Principles of Natural Philosophy. I. Bernard Cohen and Anne Whitman, transl. Berkeley: University of California Press. Newton, Isaac. (1967–81). The Mathematical Papers of Isaac Newton. 8 vols., D. T. Whiteside et al., eds. Cambridge: Cambridge University Press.

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186 continuity in 18th-century analysis and in kant Sutherland, D. (2005). “Kant on Fundamental Geometrical Relations,” Archiv für Geschichte der Philosophie 87. Bd., S. 117–158. Sutherland, D. (2014). “Kant on the Construction and Composition of Motion in the Phoronomy,” Canadian Journal of Philosophy, Vol. 44, Nos. 5–6, 686–718. Warda, A. (1922). Immanuel Kant’s Bücher. Berlin: Martin Breslauer. Wolff, Christian. (1965). Mathematisches Lexicon. Hildesheim: Georg Olms Verlag.

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8 Bolzano on Continuity Paul Rusnock

1. Introduction Bernard Bolzano (1781–1848) was a philosophical mathematician, especially interested in foundations and the analysis of important mathematical concepts. His work is prescient in many respects, sometimes astonishingly so, employing concepts and methods often thought to have been developed only considerably later. It also displays a high level of philosophical and methodological sophistication. At the same time, his mathematical technique had its limitations, and his work was not immune from error. Professor of the Science of Religion at the Charles University in Prague beginning in 1805, Bolzano was removed from his post at the end of 1819 for political reasons. His troubles with the authorities deprived him of contact with students and made it difficult for him to publish his work. As a result, most of his important contributions remained unknown well into the twentieth century, and his influence, though still significant, was not proportionate to his merit. The notion of continuity was a subject of sustained reflection throughout Bolzano’s life. He deals with the notion in many settings: the theory of space (geometry), the theory of time (chronometry), the theory of functions (analysis), physics (continuous processes, matter), and numerical continuity (the theory of measurable numbers). One can also distinguish earlier and later versions of most of his work in these areas. This paper aims to provide an overview of his thoughts on these matters, along with some indications of the historical and philosophical context of his work.1

1 In preparing this paper, I have been very fortunate to be have been able to draw on the excellent work of previous commentators, notably Gilles-Gaston Granger, Dale M. Johnson, Lukas Kraus, Jan Šebestík, and Petr Simon. I am also grateful to Jan Šebestík for communicating the contents of Simon’s article to me, and for his comments on an earlier draft of this paper; and to Aki Kanamori, Dirk Schlimm, and Stewart Shapiro for their helpful comments on a later version.

Paul Rusnock, Bolzano on Continuity In: The History of Continua: Philosophical and Mathematical Perspectives. Edited by: Stewart Shapiro and Geoffrey Hellman, Oxford University Press (2021). © Stewart Shapiro and Geoffrey Hellman. DOI: 10.1093/OSO/9780198809647.003.0009

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188 bolzano on continuity

2. Philosophical Context 2.1 Methodology One of Bolzano’s lifelong ambitions was to develop and present a comprehensive system of mathematics based upon a unified foundation, following the pattern established by earlier authors such as Wolff and Kästner. His writings contain two separate attempts to realize such a project, one dating from the first two decades of the nineteenth century, the other from the 1830s and 1840s. As the conceptual and methodological framework plays an important role in his work on continuity, I will give a brief summary of the relevant parts of his early and late views.

2.1.1 Early Views: The Contributions and Related Writings In 1810, Bolzano published the first issue of the Contributions to a Better-Grounded Presentation of Mathematics.2 In it, he set out his plans for a series of publications which would present his mathematical system, along with a definition of mathematics, a classification of the principal branches of the science, and a short treatise on mathematical method, or logic. He chose the form of a publication series, he explained, because it provided the opportunity to receive and respond to feedback from other scholars. His hopes for intelligent discussion of his ideas were, however, disappointed. Finding that almost no notice had been taken of his work, he decided not to publish any of the further planned instalments, some of which were already in a fairly advanced state of preparation.3 Instead, he published three samples of his more advanced work that he hoped would make the significance of his methodological innovations clear to a wider audience: a paper on power series,⁴ another presenting his famous proof of a special case of the intermediate value theorem,⁵ and a third dealing with the problems of calculating lengths, areas, and volumes.⁶ The third paper also includes fragments of his work in what we would today call point-set topology, including his early definitions of the concepts of line, surface, and solid (discussed below, section 3.1). The Contributions begins with a discussion of the concept of mathematics. After considering a number of proposed definitions, Bolzano endorses one according to which mathematics is the science that deals with the ‘general laws (forms)

2 BD = Beyträge zu einer begründeteren Darstellung der Mathematik, Erste Lieferung (Prague, 1810). For a more detailed discussion of this work, see P. Rusnock and J. Šebestík, ‘The Beyträge at 200: Bolzano’s quiet revolution in the philosophy of mathematics,’ Journal for the History of Analytic Philosophy, Vol. 1, no. 8 (2013). 3 Surviving drafts have been published in the collected works: BBGA = Bernard BolzanoGesamtausgabe, Series 2A, Vol. 5. ⁴ Der binomische Lehrsatz usw. (Prague, 1816). ⁵ RB = Rein analytischer Beweis usw. (Prague, 1817). ⁶ DP = Die Drey Probleme der Rectification, Complanation, und Cubirung usw. (Leipzig, 1817).

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philosophical context 189 to which things must conform in their existence’.⁷ Examples of such forms are the part–whole relation, the possibility of being ordered in various ways (e.g., being counted), time, and space. Some of these forms apply to all objects without exception, while others apply only to certain kinds—only actual objects, for example, are in space and time, but all objects may be counted. This gives rise to a hierarchy of mathematical disciplines, based upon the range of applicability of the forms they study. At its summit is universal mathematics, whose laws apply to all objects without exception. Among the subordinate disciplines are geometry (the science of space), chronometry (the science of time), and pure natural science. Bolzano tells us that arithmetic, algebra, analysis, and combinatorics belong to universal mathematics.⁸ Only in the second, unfinished and unpublished, instalment of the Contributions, however, does it become clear that the central concept of universal mathematics is that of a system or whole composed of parts (in his later writings, Bolzano uses the term ‘Inbegriff ’, or ‘collection’). In line with this approach, space as well as particular spatial objects such as lines and surfaces are conceived as structured collections of points, and time as a structured collection of instants. Moreover, space, time, and indeed continuous quantities in general have a common metrical structure, which is studied in universal mathematics. The Contributions, as noted above, also contain a part on mathematical method, or logic. Several features of this logic are important for our purposes. To begin with, Bolzano claims that, in addition to the subjective and contingent interrelations of the elements that comprise our knowledge, there is an independent, objective order among truths: This much . . . seems to me to be certain: in the realm of truths, i.e., in the collection of all true judgments, there is an objective connection, independent of our subjective recognition of it; and that, as a consequence, some of these judgments are the grounds for others, and the latter the consequences of the former.⁹

Several consequences are drawn from this: the reasons we have for accepting a given truth, to begin with, may not constitute the objective ground of that truth. Accordingly, we need to distinguish between proofs that merely convince us that something is true and those that indicate its objective grounds. In addition, there are two notions of principles (basic truths, axioms): a subjective one (an axiom is a truth that we accept immediately, without inferring it from others), and an objective one (a truth that has no objective ground, i.e., is an initial element of the deductive hierarchy). Accordingly, he maintains that obvious truths may not be objective axioms, and that axioms need not be obvious. Similarly, we ⁷ BD, I, §8 [MW= The Mathematical Works of Bernard Bolzano, p. 94]. ⁸ BD, I, §11 [MW, p. 96]; RB, Preface, I [MW, p. 254]. ⁹ BD, II, §2 [MW, p. 103].

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190 bolzano on continuity need to distinguish two sorts of definitions: definitions in the strict sense, which indicate the parts of a given concept, and orientations that merely permit us to determine the extension of the concept more or less accurately. In line with this, he points out that concepts clearly understood in themselves may be complex, and thus definable, while simple or primitive concepts, which are not objectively definable, may not be clearly understood in and of themselves. The logical and the epistemological orders are thus clearly separated. At this stage, Bolzano had some very strong hunches about the objective order; in particular, he seems to have thought that it must be unique: for every complex concept, there is one objective definition, for every truth that is not an axiom, exactly one objective proof, and in general there is essentially only one way of presenting the entirety of mathematics in accordance with its objective structure. His aim was to discover, and display, this structure.1⁰

2.1.2 Later Refinements The first instalment of the Contributions was, as Bolzano himself reminds us, just a first, partial sketch of both logic and the foundations of mathematics. As early as 1812, he resolved to write another treatment of logic and, throughout the decade of the 1820s, he worked on his masterwork on the subject, the Theory of Science.11 Once this work was done, he turned again to mathematics, working on a massive treatise to be called the Theory of Quantities [Größenlehre]. The introductory matter contains another discussion of the concept of mathematics, a brief presentation of his mature logical system, called ‘On the Mathematical Method’, and a detailed development of the theory of collections (still obviously the central notion of his mathematical system), among other things. For our purposes, the most important change from his earlier methodology is the clear recognition of what he calls ‘propositions in themselves’. Similar to Leibniz’s possible thoughts and Frege’s thoughts, propositions in themselves are abstract objects that are either true or false, the possible contents of judgements, and the meanings of sentences. Propositions, moreover, are complex entities, and their parts, provided that they are not themselves complete propositions, are called ideas in themselves. Actual thoughts in individual minds are called subjective propositions (or judgements) and ideas, while expressed propositions and ideas are linguistic items that are associated both with the subjective propositions and ideas 1⁰ He found support for these views in Aristotle and the Scholastics, but especially in Leibniz. Cf. his later remarks and references in WL, §198, note. Similar thoughts clearly motivated later thinkers such as Frege, and appear to have adherents even today. 11 WL = Wissenschaftslehre (Sulzbach, 1837). This work was essentially complete in 1830. The long delay before publication was mainly attributable to Bolzano’s troubles with the authorities, as he was forbidden to publish in the Hapsburg dominions following his dismissal. The ban was partially lifted after the death of the Emperor Francis, and Bolzano was permitted to publish several scientific papers under his name in the 1840s.

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philosophical context 191 they give voice to and with the objective propositions and ideas that are the matter [Stoff ] of the latter.12 Whereas previously Bolzano had spoken only of thoughts and their verbal expressions, he now has an additional degree of freedom. This is especially important in the case of defining terms in common (everyday or scientific) use, as, for example, ‘continuity’ or ‘extended spatial object’. When used by a particular person, these expressions will correspond to subjective ideas, which in turn will correspond to ideas in themselves. In this way, individuals associate ideas in themselves with linguistic expressions. In scientific contexts, the ideal we seek is to have universal agreement in such matters, i.e., to have everyone associate one and the same idea in itself with a given expression. In life, of course, different people may associate different ideas with a given expression, and one and the same person may do so as well at different times. These ideas may not even be coextensive. Often, however, there will at least be a number of cases where all or almost all agree that the term definitely applies to a given object, and a number of cases where it is agreed that the term definitely does not apply. Normally, there will be a third class of disputed cases, and a fourth class of cases that no one has even considered. There will also be claims stated using a given term, some of which will be false if the term is defined one way and true if it is defined in another, even though the evidence provided by previous usage is insufficient to decide which way we should go. For example, some of Bolzano’s contemporaries thought that every continuous function was differentiable at all but an isolated set of values, while others thought that the continuity of a function was equivalent to the intermediate value property. Given the narrower notion of function that these authors were working with, there was some justification for these claims. But if, with Bolzano, Dirichlet, and others, one broadens the concept of function, decisions impose themselves: should we seek a definition of continuous function that preserves these claims or not? Both options appear to be open. All the same, some choices will appear more fitting than others. The task is not unlike that of extending a function defined on the natural numbers to one defined on the reals or even the complex numbers. Even if, for example, there are infinitely many smooth functions that agree with the factorial function on the natural numbers, Euler’s Gamma function seems to provide a ‘natural’ extension. Given this, there is a pragmatic, not to say political, dimension to definition. One can of course follow Humpty Dumpty and propose whatever definition one likes for a given term. To have a definition widely or even universally adopted, however, usually requires at a minimum that one respect usage to the extent of preserving the clear cases of application, or at least giving a satisfactory explanation 12 Cf. W. Künne, ‘Propositions in Bolzano and Frege,’ Grazer philosophische Studien 53 (1997) 203–240.

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192 bolzano on continuity when any of these are rejected, and similarly for claims stated using the term. These explanations will usually point to systematic considerations, showing why the proposed definition is appropriate given the needs of science. As he puts it in an unfinished essay on the concepts of line, surface, and solid: It is a good thing that the question upon whose correct answer so much depends is not ‘of which parts does a certain idea as presently found in our consciousness, or even as it will be in the future, consist’ but rather ‘which parts must we ascribe to it so that it will best serve the ends for which we wish to use it?’13

Thus, on Bolzano’s mature view, in developing definitions for terms in common use, our task is not simply to analyse what we think or others think (insofar as we can determine this from what they say). Rather, we are to use the information obtained in this way, along with other systematic considerations, to arrive at a concept which best serves the purposes of science. This approach is most clearly stated in a late essay on the concept of beauty: [I]n most cases when we put forward a concept in a scientific context, it matters little whether the definition we propose for it actually presents in all its parts the concept we have hitherto connected with the word chosen to designate it, provided only that it is an appropriate concept, one that merits consideration at the place where we present it, and also to serve as the basis of the theses we are to present concerning its object.1⁴

Clearly, there are close similarities between this understanding and Carnap’s notion of explication.1⁵ Finally, Bolzano changes his view on the concept of mathematics and the classification of mathematical disciplines.1⁶ Though the new account differs significantly from the one given in 1810, it shares with it the one feature that is significant for my discussion, namely, that the sciences of quantity and order as such are more general and abstract than chronometry, geometry, and natural science, where they are applied.

13 ‘Versuch einer Erklärung der Begriffe von Linie, Fläche und Körper,’ BBGA 2A.11/1, p. 131. 1⁴ Über den Begriff des Schönen, §1, no. 4, p. 12 [BBGA 1.18, p. 104]. 1⁵ See, e.g., Meaning and Necessity, 2nd ed. (University of Chicago Press, 1956), §2, pp. 7–8: ‘The task of making more exact a vague or not quite exact concept used in everyday life or in an earlier stage of scientific or logical development, or rather of replacing it by a newly constructed, more exact concept, belongs among the most important tasks of logical analysis and logical construction. We call this the task of explicating, or of giving an explication for, the earlier concept . . . ’ 1⁶ ‘Einleitung zur Größenlehre,’ I, §§1–3 [BBGA 2A.7, pp. 25–45].

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philosophical context 193

2.2 Metaphysics Bolzano can sometimes be easily mistaken for someone writing much later than he actually did. Certainly, many parts of his logical and mathematical work leave such an impression. In the case of metaphysics, however, he is clearly very much a child of the eighteenth century, and continues to work within the Leibnizian– Wolffian tradition. He too would elaborate a monadology, or atomic theory, in which the atoms are simple substances. Unlike Leibniz, however, he would maintain that the atoms are spatial, each occupying a distinct point at any given instant. The point-masses of rational mechanics, the three-dimensional pointcontinuum studied in Bolzano’s geometry, and the one-dimensional timepointcontinuum which is the object of his chronometry are thus anything but idealizations for him: they are rather taken to be elements of an exact description of reality.1⁷ The commitment to simple substances, points, and instants, and the consequent obligation to explain how collections of unextended elements can constitute continuous, extended objects, stem from an ontological principle according to which there can be no complexity without simple entities: Every object, even the most complex, must have parts that are not themselves complex, but altogether simple. If the number of parts of which a whole consists is finite, then the truth of this claim is evident. For, in this case we must come to indivisible, i.e., simple, parts after a finite number of divisions, e.g., bisections. However, there may be wholes which contain an infinite number of parts, as we find, for example, in any spatial extension, any line, surface or solid. In the case of such objects, no division, if it generates only a finite number of parts, like a bisection or trisection, etc., will yield simple parts, no matter how often we repeat it. This creates the illusion that such an object does not even consist of simple parts. I claim, nevertheless, that such a whole, too, must have parts which are simple. Complexity is an attribute which obviously cannot exist without there being parts that produce it (i.e., parts that contain the reason or condition for it). If these parts are themselves complex, then they merely explain a complexity of a certain kind but not the complexity as such of the whole. In order to explain the latter sufficiently, there must be parts that are no longer complex, but simple. This is a condition for complexity that does not require any further condition. Hence, e.g., lines, surfaces and bodies contain parts that are not further divisible, 1⁷ For a concise introduction to Bolzano’s atom theory, see Peter Simons, ‘Bolzano’s monadology,’ British Journal for the History of Philosophy 23 (2015) 1074–1084. A more detailed exposition is given by Andrej Krause, Bolzanos Metaphysik (Freiburg and Munich: Alber Verlag, 2004). See also Lukas Kraus, Ontologie der Grenzen ausgedehnter Gegenstände (Berlin and Boston: De Gruyter, 2016).

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194 bolzano on continuity but simple, i.e., points, which, however, are not of the same kind as the wholes that they generate, since an infinite number of them is required. Thus geometers, who take the word ‘part’ in the narrower sense of ‘part of the same kind’, do not usually call them parts.1⁸

The development of an approach to geometry rooted in the theory of point-sets thus emerges quite naturally from Bolzano’s metaphysics.

3. Continuous Extension 3.1 Early Definitions of the Concepts of Line, Surface, and Solid As Dale Johnson has thoroughly documented, Bolzano’s reflections on continuity began quite early in his life, and were connected to debates in eighteenth-century academic metaphysics.1⁹ In an autobiographical fragment, he tells us that one of the first philosophical books he read, at age sixteen, was Baumgarten’s Metaphysics.2⁰ In it he discovered, he says, a number of errors, among them the claim that lines, surfaces, and solids could be composed of a finite number of points.21 He did not reject everything Baumgarten had written, however. Extension, the latter had claimed, belonged only to composite things, and only extended things have shape. Monads, as simple beings, thus have neither shape nor extension, though wholes composed of them can have both.22 So too in geometry, where multitudes of unextended points can constitute extended wholes.23 Though disagreeing with Baumgarten on the details, Bolzano would embrace and defend all of these claims.2⁴ In maintaining that a continuous extension could be composed of unextended points, Bolzano disagreed with another favourite author of his youth, A. G. Kästner, author of the massive treatise Anfangsgründe der Mathematik, which Bolzano had studied at university.2⁵ Kästner writes:

1⁸ WL, §61 [I.263–264]. See also §315, no. 7. 1⁹ ‘Prelude to dimension theory,’ section 2. 2⁰ ‘Zur Lebensbeschreibung,’ BBGA 2A.12/1, p. 67. 21 A. G. Baumgarten, Metaphysik (Halle, 1783), §196 et seq. 22 Baumgarten, Metaphysik, §196. 23 A line, for example, is said by Baumgarten (Metaphysik, §202) to be a series of points between separated points which are uninterruptedly next to one another. [Eine Linie ist eine Reihe Puncte, welche zwischen von einander entfernten Puncten ununterbrochen neben einander sind.] He also tells us that points are simple parts of extended things (§202), and that the parts of a continuous [stetig, ununterbrochen] thing are not separated by any distance (§201). 2⁴ See also ‘Aphorismen zur Physik,’ §5 [BBGA 2A.12/3, p. 120], where Bolzano states that atoms are unextended and shapeless. 2⁵ Cf. Bolzano, Lebensbeschreibung, p. 19.

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continuous extension 195 A point is the boundary of a line; and thus of all extension. Consequently it has neither extension nor parts, and a multitude of points arrayed next to one another [eine Menge von auseinandergesetzten Puncten] does not constitute a line.2⁶

In line with this view, Kästner did not seek to define lines, surfaces, and solids in terms of points, but instead proceeded in the opposite direction, defining a surface as a boundary or limit (Grenze) of a solid, a line as the boundary of a surface, and, as we have seen, a point as the boundary of a line.2⁷ In his first published work, Considerations on Some Objects of Elementary Geometry (1804), Bolzano takes issue with these definitions: As a genuine definition must contain only those characteristics of the concept to be defined which constitute its essence, and without which it cannot even be thought, the definitions of solid, surface, line, and point that come from the Schoolman Ockham, according to which a solid is the sort of extension that cannot be the boundary of any other, a surface the boundary of a solid, etc.— must be considered quite spurious, in that they . . . always require the idea of a solid in order for us to think merely of a point or a line, while it is obvious that we can very well think of a surface, line or point without thinking of a solid that they bound.2⁸

It would not, he continues, be such a terrible idea if someone were to proceed in the opposite direction, defining lines in terms of points, surfaces in terms of lines, etc.2⁹ Though he does not pursue this line of thought in the Considerations, it is abundantly clear in that work that he is already thinking of geometry as a theory of systems or collections of points. Also apparent are some of the basic notions he would use thereafter in his geometrical investigations: point, direction, distance, system (or collection). About a decade later, he followed up on his own suggestion, proposing definitions of the concepts of line, surface, and solid as notions of certain kinds of point-sets in a short treatise on the rectification of curves and related problems in

2⁶ A. G. Kästner, Anfangsgründe der Arithmetik, Geometrie, ebenen und sphärischen Trigonometrie und Perspectiv (6th edition Göttingen, 1800), Geometrie, Definition 5 (p. 178). 2⁷ Geometry, Definitions 4 and 5 (pp. 177–178). These definitions have obvious antecedents in ancient times. See, e.g., Aristotle, Metaphysics, 1060b 12–15; Euclid, Elements, Definitions 3, 6, 13, 14. 2⁸ Betrachtungen über einige Gegenstände der Elementarmathematik (Prague, 1804), II, §4 [MW, p. 69]. Cf. WL, §79 [I.369], where Bolzano adds a practical challenge—if we must always think first of a surface in order to think of a line (as the boundary of the former), and of a solid of which the surface is the limit, then it should be fairly easy to indicate the corresponding surface and solid for any given line. Go ahead and try it, he says, with a line of double curvature (e.g., a helix). 2⁹ Betrachtungen über einige Gegenstände der Elementarmathematik (Prague, 1804), II, §4 [MW, p. 69].

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196 bolzano on continuity 1817.3⁰ The genus is spatial object, systems of points, which may be either finite or infinite in number.31 We then have the following definition: A spatial object, at every point of which, beginning at a certain distance and for all smaller distances, there is at least one and at most only a finite multitude of points as neighbours, is called a line in general.32

(Clearly, by ‘neighbour’, Bolzano here means any other point belonging to the spatial object.33) We note that, according to this definition, some spatial objects that consist of several disjoint parts are classified as lines. This was no oversight: Bolzano himself gives as an example a hyperbola with its two branches, and follows up the above definition with a second: A spatial object of which every part which can be viewed as a line according to the definition just given, has at least one point in common with the remaining part which then likewise must be viewed as a line, is called an absolutely connected line.3⁴

To judge from this, he appears to assume that the points of division will belong to both parts.3⁵ Later sections extend this approach to higher dimensional spatial objects: A spatial object at each point of which, beginning from a certain distance and for all smaller distances, there is at least one and at most only a finite multitude of separate lines full of points is called a surface in general.3⁶ A spatial object at each point of which, starting from a certain distance and for all smaller distances, there exists at least one absolutely connected surface full of points, is called a solid in general.3⁷

The accompanying diagrams (along with others in his notebooks) indicate that Bolzano had cast his net fairly wide in looking for potential counterexamples. We 3⁰ Preliminary versions of these definitions may be found in one of Bolzano’s notebooks from 1813– 14: BBGA 2B.3/2, pp. 120–121. 31 DP = Die Drey Probleme der Rectification, der Complanation und der Cubirung usw. (Leipzig, 1817), §11 [MW, p. 301]. 32 DP, §11, Defn. 1 [MW, p. 301]. 33 Cf. ‘Über Haltung, Richtung, usw,’ §2 [Geometrické Práce (Prague, 1948), p. 143]. 3⁴ DP, §11, Defn. 2 [MW, p. 302]. 3⁵ Cf. Johnson, ‘Prelude to Dimension Theory,’ p. 277. 3⁶ DP, §35, Defn. 1 [MW, p. 322]. 3⁷ DP, §52, Defn. 1 [MW, p. 334].

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continuous extension 197 can also clearly see the recursive structure of these definitions, which could easily be extended to cover higher dimensions. Bolzano’s primary interest at this point is clearly the concept of dimension rather than extension or continuity as such. Nonetheless, he also identifies a general concept that is superordinate to those of line, surface, and solid: Obviously, at each point, e.g., m of every line there is no next point, i.e., no point that is so near to it that another could not be said to be even nearer to it. Rather, at every point m a certain distance can be found, e.g., mr, for which and for all smaller distances, e.g., mr′ , mr′ , mr″ , . . . it can be asserted that there are points in the line which have this distance from m. However, this property of lines also belongs to every surface and every solid, and is therefore, as it were, the higher concept (genus proximum) which comprehends all these three kinds of extension.3⁸

This higher concept, it seems clear, is that of an extended spatial object.3⁹ As we have seen, such objects might or might not consist of a single, connected piece. We thus have two potential candidates for an early definition of a spatial continuum. Bolzano says nothing about this in the 1817 paper, however, nor does he touch upon the questions of non-spatial continua, or continua in general. Another question that is neither asked nor answered is whether there are extended spatial objects that are neither lines, surfaces, solids, nor (finite) unions of objects of these kinds. Further comments address some potential objections: yes, he insists, an extended spatial object can be composed of unextended points. However, a merely finite set of points cannot constitute an extension, and not every infinite set does. One must, he points out, think of spatial objects as structured collections, and not as mere sums of points (where the arrangement of the parts is a matter of indifference).⁴⁰ Furthermore, if, as he claims, the parts of a line are points, we must abandon characterizations of extension according to which the parts of a continuous extension are immediately next to one another.⁴1 In battling these and similar misconceptions, he is clearly aware of the obligation to render his theory intuitive that was articulated so clearly by Russell a century later:

3⁸ DP, §12 [MW, p. 305]. 3⁹ As Bolzano would later point out, the first, informal, characterization is not equivalent to the second—it is entirely possible for there to be no ‘next point’ from a given point when the second condition is not satisfied (at some distance, and for all smaller distances, there is at least one point belonging to the spatial object)—consider, e.g., the point A within the set of points on a line AB whose distance from A is a rational multiple of d(A, B). See below, section 3.2.2. ⁴⁰ In his later theory of collections, Bolzano would use the term ‘Menge’ (in modern German ‘set’; often translated as ‘multitude’ in his writings) for collections for which the arrangement of parts is a matter of indifference and which therefore retain their identity under all rearrangements of their parts. ⁴1 DP, §12 [MW, p. 305].

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198 bolzano on continuity [T]here remains a feeling—of the kind that led Zeno to the contention that the arrow in its flight is at rest—which suggests that points and instants, even if they are infinitely numerous, can only give a jerky motion, not the smooth transitions with which the senses have made us familiar. This feeling is due, I believe, to a failure to realize imaginatively, as well as abstractly, the nature of continuous series as they appear in mathematics. When a theory has been apprehended logically, there is often a long and serious labour still required in order to feel it: it is necessary to dwell upon it, to thrust out from the mind, one by one, the misleading suggestions of false but more familiar theories, to acquire the kind of intimacy which, in the case of a foreign language, would enable us to think and dream in it, not merely to construct laborious sentences by the help of grammar and dictionary. It is, I believe, the absence of this kind of intimacy which makes many philosophers regard the mathematical doctrine of continuity as an inadequate explanation of the continuity which we experience in the world of sense.⁴2

3.2 Later Definitions of the Concepts of Extended Objects and Continua 3.2.1 Sources Writings from Bolzano’s later period contain three substantive treatments of the questions of extension and continua. The first text is an unfinished essay entitled ‘Attempted Definitions of the Concepts of a Line, Surface, and Solid’.⁴3 The manuscript that has survived contains a presentation of some key concepts of Bolzano’s mature logic and methodology, followed by a definition of the concept of an extended spatial object, and a justification of this definition. He also planned to give and justify definitions of the concepts of line, surface, and solid, and to present a literature review, but apparently did not finish these parts.⁴⁴ The manuscript is not dated.⁴⁵

⁴2 Bertrand Russell, Our Knowledge of the External World (1914; reprint Routledge Classics, 2009), p. 105. ⁴3 ‘Versuch einer Erklärung der Begriffe von Linie, Fläche,’ Jan Berg, ed. BBGA 2A.11/1, pp. 113– 150. ⁴⁴ BBGA 2A.11/1, p. 137. Bolzano followed a similar plan in his essay ‘Über den Begriff des Schönen.’ ⁴⁵ In the MS, Bolzano says that he had examined his definitions for eleven years; since he does not get far enough to present definitions of the concepts of line, surface, and solid, it is possible that he was referring to his earlier efforts, which would be compatible with a date as early as 1824 or 1825 (as we saw, the early definitions date from no later than 1813–14). A reference to §28 of the Logic, by which he most likely meant the Theory of Science (p. 140), however, was taken by both Dale Johnson and Jan Berg, who edited the manuscript, to support a date of around 1830 (BBGA 2A.11/1, p. 22; ‘Prelude to dimension theory,’ p. 281). On the other end, our next text dates from the early 1840s; assuming that Bolzano would not simply have set aside this manuscript if he were still writing about the problem, this

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continuous extension 199 The second text bears the title: ‘On the bearing, direction, curvature and scrolling of lines as well as surfaces, along with other related concepts’.⁴⁶ Although it too was never completed,⁴⁷ parts of it were read at several meetings of the Bohemian Royal Society in 1843 and 1844.⁴⁸ The manuscript was not published until 1948.⁴⁹ Our third source is the Paradoxes of the Infinite, written in 1847–8, edited by his friend František Příhonský, and published posthumously in 1851.⁵⁰ As the only text among our three sources published before 1948, the Paradoxes was the only presentation of Bolzano’s mature ideas available to nineteenth- and early twentieth-century authors such as Cantor, Dedekind, and Russell. Finally, Bolzano briefly mentions his views on these matters in the Theory of Science,⁵1 and in the introductory part of the unfinished Theory of Quantities.⁵2

3.2.2 The Concept of Extension in Bolzano’s Later Writings: 1) Versuch einer Erklärung Bolzano begins the geometrical part of the first essay by trying to identify a suitable genus of which lines, surfaces, and solids are species. He notes that many authors choose extended object for this purpose, and that some think it necessary to add continuous [stetig] to obtain their generic concept.⁵3 He objects to this addition because he thinks that those who propose it believe that there can be extended objects that are not continuous, e.g., a finite collection of points. In his opinion, this would make the notion of an extended object so wide that it would be equivalent to that of a system or collection of points, contrary to accepted usage.⁵⁴ He then adds: [I]t should be noted that the concept that one must connect with the word continuous [stetig] if all lines, surfaces and solids are supposed to be called continuous is in no way that which we elsewhere usually designate with the word connected (continuum). For not every object that the geometer with good would provide us with an upper limit of 1843, with a fair likelihood that it was written at least several years earlier than that. ⁴⁶ ‘Über Haltung, Richtung, usw,’ Foreword [Geometrické Práce, p. 142]. ⁴⁷ Bolzano, who suffered throughout his life from debilitating lung disease, tells us that he did not have the strength to finish it. See ‘Über Haltung usw.,’ Foreword; also B’s letters to M. J. Fesl of 30 June 1843 and 1 September 1843 in E. Winter, ed., Wissenschaft und Religion im Vormärz: Der Briefwechsel Bernard Bolzanos mit Michael Josef Fesl (Berlin, 1965), p. 329, pp. 331–332. ⁴⁸ Abh. d. kön.-böhm. Ges. d. Wiss., Series 5, Vol. 3, pp. 10, 11, 28. ⁴⁹ ‘Über Haltung, Richtung, Krümmung und Schnörkelung bei Linien sowohl als Flächen sammt einigen verwandten Begriffe,’ ed. J. Vojtěch, pp. 139–207 in Geometrické práce–Geometrische Arbeiten, Vol. 5 of Spisy Bernarda Bolzana–Bernard Bolzano’s Schriften (Prague, 1948). ⁵⁰ PdU = Paradoxien des Unendlichen (Leipzig, 1851). ⁵1 WL, §315, no. 7. ⁵2 BBGA 2A.7, p. 71–72 [MM-EX, p. 61]. ⁵3 ‘Versuch einer Erklärung,’ II, §2 [BBGA 2A.11/1, p. 138]. ⁵⁴ His remarks also suggest that he thought every extended object to be either a line, surface, or solid, or else a hybrid composed of parts of these kinds.

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200 bolzano on continuity reason considers to be a single extension is such a connected whole (line, surface, or solid). Thus it is known that the line expressed through the equation y = √(1 − x)(2 − x)(3 − x) consists of two entirely separate ovals;⁵⁵ and yet it is generally and rightly called a single line, namely, insofar as it is a spatial object determined by a single equation. If, however, we were to assign such a broad meaning to the word continuous [stetig] that even a whole consisting of several unconnected parts can be called a continuous whole, we would not only violate accepted usage, but also it would be superfluous to combine the words continuous and extended.⁵⁶

A couple of points are noteworthy here. Again, we see that Bolzano believes that a single extended object can consist of several disjoint parts, and that some objects of this sort are rightly called single lines (or surfaces, or solids), namely, when they are determined by a single equation.⁵⁷ We note, moreover, that for Bolzano the word ‘stetig’ (continuous), as used in such contexts, indicates that the object in question consists of a single, connected piece, and that the Latin ‘continuum’ refers to this property of connectedness (which is presumably to be met with only in extended objects). The genus, extended object, however, is still too broad for Bolzano. For, he observes, time also contains objects that can reasonably be called extended, e.g., temporal intervals. It is therefore, he thinks, justifiable to take extendedness to belong to other kinds of objects apart from the spatial. With this in mind, he settles on the notion of an extended spatial object as determining the genus he requires. Bolzano next motivates his definition of an extended object with a few observations.⁵⁸ To begin with, he expects that all will agree that extension is a property of a complex whole, or collection [Inbegriff ], since everyone recognizes parts within every extended object. As before, however, these collections cannot be mere sums or multitudes [Mengen], since the order of the parts matters. At the same time, mere complexity is obviously insufficient for extension, as shown by the example of a hundred points, each separated by an inch from the previous one, or a system of several instants, each separated by a second. Even infinite complexity is insufficient, as we shall presently see.⁵⁹ ⁵⁵ There is probably a transcription error here, and we should have something like: y = √(1 − x2 )(2 − x)(3 − x); clearly, too, Bolzano intends us to include both the negative and the positive roots, i.e., he wants us to consider the graph determined by the condition y2 = (1 − x2 )(2 − x)(3 − x). ⁵⁶ ‘Versuch einer Erklärung,’ II, §2 [BBGA 2A.11/1, p. 139]. ⁵⁷ Here we touch upon another terminological muddle, for, in the eighteenth century, Euler had applied the term ‘continuous’ to curves ‘whose nature is expressed by one determinate function of x’ (Introductio in analysin infinitorum (Lausanne, 1748), Vol. 2, Chapter 1, no. 9). We can see how, according to Euler’s definition, a hyperbola with two branches could be considered continuous or, as Bolzano says, a single line. ⁵⁸ ‘Versuch einer Erklärung,’ II, §6 [BBGA 2A.11/1, pp. 141–143]. ⁵⁹ Cf. II, §9 [BBGA 2A.11/1, p. 144].

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continuous extension 201 Why are these objects not extended? Bolzano suggests that the elements (points or instants) are isolated or unconnected. In the two cited examples, he remarks, for each point there is a next element, i.e., one at a distance such that no other lies at a smaller distance, while in extended objects this is not the case. If an object is to be extended, then, it cannot have any elements of this kind. Yet, he continues, this is not sufficient for extendedness. For consider two points a, b, along with a point c in the middle between them, and points d, e in the middle, respectively, between a, c and c, b, and so on ad infinitum. None of the points belonging to this system, he notes, has a closest neighbour; yet no geometer, he claims, would claim that these points are connected with each other and form an extension. For even though each point has neighbours that are arbitrarily close to it, for each there are also arbitrarily small distances at which it has no neighbour.⁶⁰ The property today called density-in-itself is thus not a sufficient condition for extendedness according to Bolzano.⁶1 In line with these reflections, he calls an element p of a spatial object connected [verbunden] iff for some positive number d, there is at least one element of the spatial object (neighbour) lying at the distance h from p, for all h ≤ d. An element that is not connected is called isolated.⁶2 A spatial object is extended, finally, if all of its points are connected. Clearly, this is the same generic concept we encountered in Three Problems. Having set out this definition, Bolzano attempts to justify it in the following sections. Of particular interest are his arguments in support of the view that extended objects can be collections of unextended parts.⁶3 This claim is conceded, he argues, by every geometer who says, e.g., that the equation y2 = ax represents a curve. For what else does this mean, he asks, if not that the collection of points satisfying this equation (relative to a coordinate system) constitutes a curve? It will not do, he continues, to say that although curves contain points, they are not composed of them. For to say that a curve is composed [zusammengesetzt] of points in no way requires a successive placing [setzen] by us of one point after the other, which admittedly would only ever result in a finite, unextended assemblage. Finally, §11 revisits Bolzano’s views on the definition of terms such as ‘extension’. It is difficult to decide, he says, whether a proposed definition is correct, especially because it may be that different people associate different ideas with a given term. His should be accepted, he argues, not because it tells you what you have been thinking all along (though he often optimistically assumes that others will have ⁶⁰ II, §6 [BBGA 2A.11/1, p. 143.] ⁶1 Cf. Kraus, Der Begriff des Kontinuums bei Bernard Bolzano, section 3.1; as Kraus notes, densityin-itself is clearly a necessary condition for extendedness in Bolzano’s sense. ⁶2 Bolzano clearly does not use the term ‘isolated’ in the sense it is most often used today, even though he just as clearly had formed that concept. Though this creates some difficulties for his readers, he can hardly be faulted for not conforming to terminological choices made by the mathematical community long after his death. ⁶3 II, §9 [BBGA 2A.11/1, p. 145].

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202 bolzano on continuity had similar thoughts), but rather because it best meets the logical requirements of definitions and best serves the needs of mathematics: . . . if the question is not what we involuntarily think by the word extension but rather what we should consider to be an essential constituent of the concept belonging to it, then it can be shown with sufficient certainty that this concept should be determined precisely as above. Namely, it can be shown that our concept is composed of the smallest number of constituents among those with the same extension, and that no other concept is more useful for deriving the most noteworthy attributes we wish to demonstrate of the objects considered in geometry.⁶⁴

After a handful of further remarks, the essay breaks off before even discussing the concepts of a line, surface, or solid.

3.2.3 The Concept of Extension in Bolzano’s Later Writings: 2) Über Haltung, Richtung usw. The second essay begins by covering much of the same ground: we have definitions of spatial objects as collections of points (§1), of a neighbour (§2), of isolated and non-isolated points (§3), and of an extended object (§4). A terminological difference should be noted, however: Bolzano now also calls extended objects continuous [continuirlich], even if they are not connected. Afterwards, he gives new definitions of the concepts of line, surface, and solid. The first runs as follows: An extension, each point of which for each sufficiently small distance has only so many neighbours that their collection at each of these distances does not, considered by itself, constitute an extension, I call a spatial object of single or simple extension, or a line.⁶⁵

A surface is then defined as a spatial object for which, at a certain and at all smaller distances, the collection of neighbours constitutes a line, and similarly for solids.⁶⁶ Once again, we observe the recursive structure of his definitions, and the ease with which they could be generalized to cover higher dimensions.

⁶⁴ II, §11, [BBGA 2A.11/1, pp. 146–147]. ⁶⁵ §7 [Geometrické Práce, p. 144]. In the previous section (§6), Bolzano had clearly explained how he would use the expression ‘for each sufficiently small distance’; in the present case, he means that there is some distance such that, for it and all smaller distances, the collection of neighbours at that distance does not constitute an extension. ⁶⁶ §7 [Geometrické Práce, p. 144].

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continuous extension 203 Without explaining why, Bolzano claims that the new definitions are ‘essentially better’ than those given in Three Problems.⁶⁷ In light of later developments, we can agree with this, seeing in them an attempt to make the concept of an object of dimension zero the base case for his recursive definitions.⁶⁸

3.2.4 The Concept of Extension in Bolzano’s Later Writings: 3) The Paradoxes of the Infinite The technical aspects of Bolzano’s discussion of extension in the Paradoxes are essentially the same as in the two earlier sources.⁶⁹ This being said, there are some new features. First, even more clearly than before, Bolzano recognizes that extendedness (or continuity) is an attribute that can belong to different kinds of objects: in addition to those previously recognized (spatial and temporal), he now adds the material, by which he means a collection of simple substances (atoms) that are arranged so as to constitute an extended object.⁷⁰ The property of extendedness or extension as such, along with that of connected extension, or continuity, are thus worthy of study in their own right, belonging to universal mathematics according to his early classification, and somewhere higher in the hierarchy than chronometry, geometry, and pure natural science according to his later one. Turning to the applied parts of the theory of quantities, we encounter the first paradoxes in the Theory of Time in the concept of time itself, especially insofar as it is supposed to be a continuous extension [stetige Ausdehnung]. However, the apparent contradictions, famous from ancient times, that people have claimed to discover in the concept of a continuous extension of a continuum [eine stetigen Ausdehnung eines Kontinuums] bear equally on the temporal, the spatial, and even the material; hence we shall consider them all together.⁷1

As Gilles-Gaston Granger observed, this was an important conceptual advance: . . . in the end, the notion of continuity is considered as a common structure ultimately depending . . . upon an abstract and general theory of quantity.⁷2

⁶⁷ §9 [Geometrické Práce, p. 145]. Cf. Johnson, ‘Prelude to dimension theory,’ p. 287. ⁶⁸ As my concern in this paper is simply continuity, I will not discuss the problem of dimension in detail. On this topic, see Johnson, Simon, and Sebestik (Log. et math. chez BB, Ch. II, no. 2). ⁶⁹ PdU, §§38–41. The definition of an extended object is in §38, and the definitions of line, surface, and solid in a note to §40. ⁷⁰ PdU, §38. ⁷1 PdU, §38. ⁷2 ‘Le concept de continu chez Aristote et Bolzano,’ Les études philosophiques (1969) 513–523, p. 515.

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204 bolzano on continuity Even if Bolzano himself may have been in no position to take it, it seems to me that we are but a short step from a fully abstract treatment of the question in terms of metric spaces. Some terminological divergences with the earlier texts should also be noted: Bolzano now speaks not simply of extended objects, but of continuous extension and continua, using the words ‘stetig’ and ‘Kontinuum’ as equivalent to ‘ausgedehnt’ and ‘Ausdehnung’, respectively. This usage is in direct opposition to his earlier remarks, where both ‘stetig’ and the Latin ‘continuum’ were thought to indicate not extendedness in Bolzano’s sense, but rather connectedness. As we saw, a Bolzanian continuum may consist of several disjoint pieces. In the Paradoxes, Bolzano gives an example that takes us further, showing that a continuum in his sense need not even consist of a finite number of connected, extended parts. Consider, he says, a straight line az. Let b be the midpoint of az, c the midpoint of bz, d the midpoint of cz, and so on ad infinitum. If we remove the points b, c, d, . . . along with z, we will, according to his definition, still have a line, which consists of infinitely many disjoint parts.⁷3 Perhaps the best-known part of the Paradoxes is §20, where Bolzano describes a surprising property of point-continua (in this case, line segments), namely, that the points contained in any line segment ac can be placed into one-to-one correspondence with the points contained in a segment ab which is a proper part of ac. This was not the first time this phenomenon had been noticed, as Bolzano was well aware.⁷⁴ In the Theory of Science,⁷⁵ Bolzano had stated that the existence of such a oneto-one correspondence was sufficient to establish that two collections had equal numbers of elements. Already in a notebook entry from 1813–14, we find him examining this principle, and asking whether it applies to infinite collections. At first, he writes that the principle is universally applicable only in the finite case, noting that if it also applied to infinite collections, we could not define the length of a line segment in terms of the number of points it contains. He then has a second thought: I can’t see anything that could be said against the following proof, which shows the falsity of that theorem. Because for every individual in A there is a corresponding individual in B, the multitude [Menge] of individuals in B cannot be smaller than that in A . . . , one can also state that B = A + p, where p means either 0 or some

⁷3 PdU, §41, no. 3. Bolzano notes that if the point z were reinstated, we would no longer have a line, since z would be isolated. ⁷⁴ PdU, §42. For the historical background, see J. Sebestik, ‘Le paradoxe de la réflexivité des ensembles infinis: Leibniz, Goldbach, Bolzano,’ pp. 175–194 in F. Monnoyeur, ed., Infini des mathématiciens, infini des philosophes (Paris: Belin, 1992); also Logique et mathématique chez Bernard Bolzano, pp. 452 et seq. ⁷⁵ WL, §87.

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continuous extension 205 positive number (or indeed ∞). So too, A cannot be smaller . . . than B; thus A = B + q, where q means some positive number or else is = 0. Consequently B = B + p + q, which is not possible unless p = 0 and q = 0. Note. Thus the multitudes of points in all lines are equal. And the multitude of points in a part [of a line] is just as large as that in the whole abc.⁷⁶

In the Paradoxes, he notes that Johann Carl Fischer had come to the same conclusion in 1807.⁷⁷ But in that later work, as in the earlier notebook entry, he rejects it, based upon the Euclidean principle that the whole is greater than the part.⁷⁸ Although Paolo Mancosu has made a case for the claim that Bolzano was not entirely misguided in taking this path,⁷⁹ I think it fair to say that this was not Bolzano’s finest hour, and that he should at least have pursued these thoughts further. For one thing, he had already acknowledged exceptions to the Euclidean rule: the length of a closed line segment ab, for example, is according to him the same as the length of the open segment, even though the latter is a proper part of the former.⁸⁰ Moreover, according to his official doctrine, number is an attribute of collections that retain their identity under any rearrangements of their parts.⁸1 And, as he points out elsewhere, one and the same collection of atoms can occupy all the points on a line segment ac as well as all the points contained in a line segment ab that is a proper part of the former.⁸2 If we accept the reasonable claim that the collection of atoms is just as numerous as the points it occupies, and the Euclidean principle that if two things are each equal to a third, then they are equal to each other,⁸3 it follows immediately that the line segments contain equal numbers of points. Georg Cantor read the Paradoxes and was aware of Bolzano’s definitions. In a series of papers entitled ‘On Infinite, Linear Point-Manifolds’ he praises Bolzano for his forthright defence of the actual infinite and of the proposition that a continuum can be composed of points.⁸⁴ He is also happy to find Bolzano speaking

⁷⁶ BBGA 2B.3/2, pp. 79–80. ⁷⁷ J. C. Fischer, Grundriß der gesammten höhern Mathematik usw., 3 vols. (Leipzig, 1807–9), Vol. 2, §51, note. ⁷⁸ Elements, Common Notion 5. ⁷⁹ P. Mancosu, ‘Measuring the sizes of infinite collections of natural numbers: was Cantor’s theory of infinite number inevitable?’ The Review of Symbolic Logic 2 (2009) 612–646. ⁸⁰ PdU, §41, no. 1. ⁸1 PdU, §4–11. ⁸2 He merely alludes to the argument in PdU, §59. It is presented in detail in §16 of the paper ‘Aphorismen zur Physik,’ which was presented at the Bohemian Royal Society [BBGA 2A.12/3, pp. 124–125], and in Příhonský’s Atomenlehre des sel. Bolzano, pp. 13–14. ⁸3 Elements, Common Notion 1. ⁸⁴ ‘Über unendliche lineare Punktmannigfaltigkeiten,’ no. 5 (1883), §7. Reprinted in E. Zermelo, ed., Georg Cantor: Gesammelte Abhandlungen mathematischen und philosophischen Inhalts (Berlin, 1932; reprint Hildesheim: Olms, 1962), pp. 179ff.

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206 bolzano on continuity in support of actually infinite numbers, though, for understandable reasons, he finds it impossible to agree with him on the details: The author lacks the general concept of a power as well as the precise concept of number [Anzahlbegriff ] needed for a genuine conceptual understanding of determinate infinite numbers. Both indeed appear in embryonic form as special cases in isolated passages, yet it seems to me that he does not work them out in full clarity and definiteness, which explains many of the inconsistencies and even errors of this valuable work.⁸⁵

This assessment seems exactly right to me. Cantor also discusses Bolzano’s definition of a continuum from §38 of the Paradoxes. Here, too, his remarks are critical: The Bolzanian definition of a continuum is certainly incorrect; it merely onesidedly expresses one property of the continuum, which, however, is satisfied by sets produced from Gn by removing an arbitrary ‘isolated’ point-set . . . ;⁸⁶ it is also satisfied by sets consisting of several separated continua. Obviously in such cases there is no continuum, even though according to Bolzano this is so. We see here a violation of the rule: ‘ad essentiam alicujus rei pertinet id, quo dato res necessario ponitur et quo sublato res necessario tollitur: vel id, sine quo res, et vice versa quod sine re nec esse nec concipi potest.’⁸⁷

In this case, Cantor seems to me to be on shakier ground. He seems clearly to be working with a version of Bolzano’s youthful views, according to which a definition grasps a unique essence that we are all already acquainted with. As we saw, however, Bolzano later moved beyond such a view of definition.⁸⁸ He would, I believe, have rightly deemed the criticism that his definition missed the essence of continuity misguided if not meaningless,⁸⁹ and recognized only three legitimate sorts of criticisms of his definitions, namely:

⁸⁵ ‘Über unendliche lineare Punktmannigfaltigkeiten,’ no. 5, §7; Gesammelte Abhandlungen, p. 180. ⁸⁶ Cantor uses ‘isolated’ in the modern sense, not in Bolzano’s sense. ⁸⁷ ‘Über unendliche lineare Punktmannigfaltigkeiten,’ no. 5, §10; Gesammelte Abhandlungen, p. 194. The quote is from Spinoza’s Ethics (Part 2, Definition 2): ‘That belongs to the essence of a thing, which, being given, the thing is also necessarily given, and, which being removed, the thing is also necessarily removed; in other words, that without which the thing, and which itself without the thing, can neither be nor be conceived.’ ⁸⁸ This point is one of my few disagreements with the findings of Johnson’s excellent article (see, e.g., ‘Prelude to dimension theory,’ p. 275, p. 291). ⁸⁹ Cantor’s definition, for example, would classify closed line segments as continua, while denying this name to open and half-open segments, but is it really clear that this agrees with the ‘essence’ of continuity?

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continuous extension 207 1. Logical—e.g., if it could be shown that the concept presented in the definition contained redundant elements (e.g., an equilateral triangle is a triangle with three equal sides and three equal angles), or that the definition was circular (e.g., an extended object is one composed of extended parts), etc. 2. Unjustified departure from accepted usage: even if departure from accepted usage can sometimes be justified (e.g., when the concept of a function was broadened to include relations not expressible via the customary kinds of formulas), we should try to respect it whenever possible, and failure to do so without good reason is a defect. 3. Systematic considerations (the most important point): we have an example, albeit one that Bolzano did not have the concepts to appreciate, in a criticism raised by Petr Simon, namely, that if, following Bolzano, we consider only spherical neighbourhoods in our definition of dimension, dimension will not be a topological invariant.⁹⁰ A definition of dimension according to which it is a topological invariant would thus arguably better serve the needs of mathematics. Only on the second point, it seems to me, would Cantor have any real basis for criticism (in any case, he raises no objections of the first or third kinds). Here, Bolzano would have had to concede that Cantor had a point, since he himself had remarked that words like ‘stetig’ and ‘Kontinuum’ were commonly taken to indicate not merely extension but also connectedness. At the very least, then, his choice of terms in the Paradoxes would require justification. Clearly, however, Bolzano also had the concept of a connected and extended spatial object, so we can see that at bottom this is a disagreement concerning terminology alone.

3.3 Final Remarks Although Bolzano’s understanding of dimension changed over time, his concept of spatial extension remained essentially the same throughout his life. A point p belonging to a spatial object is connected within the object, namely, iff there exists a distance d such that for it, and for all smaller distances, there exists a point belonging to the spatial object at that distance from p. And a spatial object is extended iff all of its points are connected. We find this concept in the early and late writings, and a special case of the notion is clearly expressed in his notebooks from 1803–8.⁹1 Throughout, we also have two candidates for the concept of a

⁹⁰ P. Simon, ‘Bolzano a teorie dimenze,’ as reported by Sebestik, Logique et mathématique chez Bernard Bolzano, p. 63. ⁹1 See the passage with the heading ‘Definition einer continuirlichen Linie,’ BBGA 2B.2/1, p. 28.

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208 bolzano on continuity spatial continuum, namely: (a) an extended spatial object and (b) an extended and connected spatial object. One important change that we do see is Bolzano’s increasingly clear understanding that spatial continuity or extension is but one manifestation of a general phenomenon. Given his early views on universal mathematics, this is not surprising, nor is the central role played by the general theory of quantities in his account. As Granger observed, this conceptual approach passes the buck, in that the structure of the real (or, as Bolzano called them, measurable) numbers becomes the ultimate basis of the continuity of spatial, temporal, and material objects. Granger appears to regret this, looking forward to the later separation of topological and metrical properties, and to purely topological definitions of connectedness and compactness and hence of a continuum.⁹2 Staying with Bolzano and his age, I prefer to highlight another aspect of his way of approaching continuity, to wit: it demands an account of the real numbers that is independent of the theories of space, time, and any of the other objects where the theory of real numbers can be applied. Not only did Bolzano understand this, he sought to provide what was necessary, with a fair measure of success.

4. The Numerical Continuum In Bolzano’s early and later classifications of the sub-disciplines of mathematics, the general theory of quantities occupies a more central place than the theories of time and space, and pure natural science; as we saw, it belongs to a more general part of mathematics which is applied in these sub-disciplines. As such, it requires a development that is independent of chronometry, geometry, etc.; accordingly, an approach similar to Newton’s,⁹3 where number is defined as a ratio of a given quantity to another arbitrarily designated as a unit, is ruled out. For from Bolzano’s standpoint, one cannot presuppose the existence of spatial, temporal, or other kinds of quantities when developing the general theory of quantity; on the contrary, the existence of the latter depends upon the prior existence of general concepts of quantity. Accordingly, in his Theory of Quantities, Bolzano presents a purely arithmetical theory of quantities (or rather quantity concepts), based on the notion of a natural number (Bolzano uses the term ‘actual’, ‘wirklich’) and the operations of addition, subtraction, multiplication, and division.⁹⁴ Concepts of this sort involving finitely

⁹2 Granger, ‘Le concept de continu chez Aristote et Bolzano,’ pp. 522–523. ⁹3 Universal Arithmetick (London, 1769), Article 1 (p. 2). ⁹⁴ For more detailed discussion of his theory, see B. van Rootselaar, ‘Bolzano’s theory of real numbers,’ Archive for History of Exact Sciences 2 (1963) 168–180; D. Laugwitz, ‘Bemerkungen zu Bolzanos Größenlehre,’ Archive for History of Exact Sciences 2 (1965) 398– 409. J. Sebestik, Logique

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the numerical continuum 209 many operations will, he notes, yield rational numbers. To obtain the full range of quantity-concepts required for science, therefore, we must also consider infinite quantity-concepts, those in which infinitely many operations are represented.⁹⁵ For example: 1 + 2 + 3 + 4 + ⋯ ad inf. 1 2

−

1 4

+

1 8

−

1 16

+ ⋯ ad inf.

1

1

1

1

2

4

8

16

(1 − ) (1 − ) (1 − ) (1 − a+

b 1+1+1+1+⋯ ad inf.,

) ⋯ ad inf.

where a, b ∈ ℕ

A number concept S is then said to be measurable if it satisfies the following condition: For every positive natural number q, there is an integer p such that: S=

p p+1 + P1 = − P2 q q

where P1 is either equal to zero or else ‘purely positive’, and P2 is purely positive.⁹⁶ It is not clear precisely how we are supposed to understand these infinite number concepts, nor what Bolzano means by a ‘purely positive’ number concept.⁹⁷ Bolzano also encounters some technical difficulties carrying out his plan, which I will not go into here.⁹⁸ All the same, the guiding idea is clear: a measurable number concept S will determine a set of ‘measuring fractions’, allowing us to say that, for each positive natural number q, there is a unique integer p such that: p p+1 ≤S< q q

et mathématique chez Bernard Bolzano, Ch. 4; Rusnock, Bolzano’s Philosophy and the Emergence of Modern Mathematics (Amsterdam, 2000), Chapter 5 Section 3; A. Behboud, Bolzanos Beiträge zur Mathematik und ihrer Philosophie (Bern, 2000), Part II, Ch. 4. ⁹⁵ BBGA 2A.8, p. 100; MW, p. 357. ⁹⁶ BBGA 2A.8, pp. 102–104; MW, pp. 358–361. ⁹⁷ A number of commentators, myself included, have considered the theory that results when his infinite concepts are identified with certain sequences of rational numbers (e.g., the partial sums of series like the first two examples above) and purely positive expressions with sequences all of whose terms after a certain one are greater than or equal to zero. ⁹⁸ For discussion of the problems, and possible fixes, see the studies mentioned in note 94.

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210 bolzano on continuity It can happen that different quantity concepts determine the same set of measuring fractions; for example: 4+

1 1 + 2 + 3 + ad inf.

4+

1 1 + 1 + 1 + ad inf.

and

In this case, they are said to be ‘equal in the process of measurement’.⁹⁹ Equality of measurable number concepts is accordingly defined in terms of measuring fractions, as are the order relations greater than and less than. A representation theorem then shows that there is an infinite number concept for every set of measuring fractions. Complete and consistent sets of measuring fractions thus emerge as principal objects of interest. The affinities between Bolzano’s approach and the ratio theory of Elements, Book V, as well as Dedekind’s cuts, are evident. Although there are unclarities and technical deficiencies in the most basic parts of his theory, the middle parts are fairly solid. In addition to propositions establishing the measurability of rational numbers, the closure of the measurable numbers under addition, subtraction, etc., they include proofs of two theorems establishing the completeness of the measurable numbers (a property that distinguishes them from the rationals in a way that makes them more suitable for providing a basis for continuity). The first is the Bolzano–Cauchy Theorem, stating that a Cauchy sequence of measurable numbers has a measurable limit;1⁰⁰ while the second, which is closely related to the least-upper bound property, runs as follows: If we know merely that a certain attribute B does not belong to all values of a variable measurable quantity X that are greater (less) than a certain U, but does belong to all that are smaller (greater) than U, then we may claim with certainty that there is a measurable number A which is the greatest (smallest) of those of which it may be said that all smaller (greater) X have the attribute B . . . 1⁰1

Both of these theorems occur already in the 1817 paper, Purely Analytic Theorem . . . , even though Bolzano had not developed a theory of real or measurable numbers at that time.1⁰2

⁹⁹ ‘Unendliche Größenbegriffe’, §54 [BBGA 2A.18, pp. 129–130; MW, p. 391]. 1⁰⁰ FL= Functionenlehre, §107 [BBGA 2A.8, p. 151; MW, pp. 412–413]. 1⁰1 ‘Unendliche Größenbegriffe,’ §109 [BBGA 2A.8, p. 156; MW, p. 416]. 1⁰2 RB, §7, §12. The Bolzano–Cauchy Theorem also appears in a slightly earlier paper on the Binomial Theorem (1816).

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the numerical continuum 211 A third noteworthy proposition is the Bolzano–Weierstrass Theorem,1⁰3 which Bolzano states roughly as follows: If the infinitely many measurable numbers x1 , x2 , x3 , x4 , . . . all lie within the interval (a, b), then there is a measurable number c within [a, b] such that, for every 𝜔 > 0 infinitely many of the xn lie within c ± 𝜔.1⁰⁴

His reference to this theorem suggests that he provided a proof for it in the theory of measurable numbers,1⁰⁵ but this proof has not been found among his papers.1⁰⁶ In addition to the standard measurable numbers, Bolzano also makes room for infinitely small ones,1⁰⁷ e.g.: 1 1 + 1 + 1 + 1 + ⋯ ad inf. Although such number concepts are equal to zero in the process of measurement, they are not identical to zero. Unsurprisingly in light of this, he also recognizes infinitely large numbers, which, however, are not measurable.1⁰⁸ For example: 1 + 2 + 3 + 3 + ⋯ ad inf. The Bolzanian numerical continuum thus extends, both in the large and the small, beyond the numbers identifiable through the process of measurement. This additional richness, however, does not carry over to temporal, spatial, or material continua. While acknowledging that the entirety of space and the entirety of time are infinite, he denies that the duration between two instants can be infinitely greater or infinitely smaller than the duration between two others, and similarly for distances separating points in space. Similarly, in physics, he maintains that any two forces will stand in a finite ratio.1⁰⁹ He argues for these claims based upon physical determinism, stating that the instantaneous state of the universe (locations and forces of the atoms) at any given instant completely determines the state of the universe at any later instant (barring any divine interventions in the

1⁰3 On the origin and use of this title, see the appendix at the end of this paper. 1⁰⁴ See, e.g., FL, §56 [BBGA, 2A.10/1, pp. 47–48; MW, p. 459]. 1⁰⁵ See BBGA 2A.10/1, p. 47, note d—where the editor reproduces a marginal note (which is not included in MW), which reads: ‘The §§ referred to here are proved in the theory on the measurability of numbers.’ 1⁰⁶ See, however, the manuscript entitled ‘Verbesserungen und Zusätze zur Functionenlehre,’ §21 [BBGA 2A.10/1, p. 180; MW, p. 582], where the key idea of such a proof is clearly present. 1⁰⁷ §§21–22 [BBGA 2A.8, pp. 112–113; MW, pp. 370–371]. 1⁰⁸ §27 [BBGA 2A.8, pp. 114–115; MW, p. 373]. 1⁰⁹ PdU, §27.

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212 bolzano on continuity interval), and that such determinism is incompatible with infinite ratios between durations, distances, or forces.11⁰

5. Continuous Functions As is well known, Bolzano was also a pioneer in the investigation of the continuity of functions. Although the concept of continuity he used was by no means original with him,111 he was among the first, if not the first, to understand how the definition could and should be used in proofs of theorems of analysis. In 1817, Bolzano wrote: According to a correct definition, the expression that a function fx varies according to the law of continuity for all values of x inside or outside certain limits means only that, if x is any such value, the difference f(x + 𝜔) − fx can be made smaller than any given quantity, provided 𝜔 can be taken as small as we please.112

The later work Theory of Functions is even more precise, and also distinguishes left, right, and two-sided continuity: If a function Fx . . . is so constituted that the variation it undergoes when . . . its variable passes from a determinate value x to the different value x+Δx diminishes ad infinitum as Δx diminishes ad infinitum—if, that is, Fx and F(x+Δx) (the latter of these at least from a certain value of the increment Δx and all smaller values) are measurable, and the absolute value of the difference F(x + Δx) − Fx becomes and 1 remains less than any given fraction if one takes Δx small enough (and however N smaller one may let it become): then I say that the function Fx is continuous for the value x, and this for a positive increment or in the positive direction, when that which has just been said occurs for a positive value of Δx; for a negative increment or in the negative direction, on the other hand, when that which has been said holds for a negative value of Δx; if, finally, the stated condition holds for a positive as well as a negative increment of x, I say, simply, that Fx is continuous at the value x.113

11⁰ PdU, §27. 111 Antecedents may be found as early as Leibniz (cf. Sebestik, Logique et mathématique chez Bernard Bolzano, pp. 73–74), and Bolzano himself credits Lagrange with using the term ‘continuous’ in his sense (FL, §39 [BBGA 2A.10/1, p. 38; MW, p. 450]). 112 RB, Preface, II (a) [MW, p. 256]. 113 FL, §38 [BBGA 2A.10/1, pp. 35–36; MW, pp. 448–449].

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continuous functions 213 These definitions are accompanied by a series of examples that illustrate important features of continuous functions and distinguish this property from others such as monotonicity and differentiability. In particular, he constructs: • a function that is not continuous anywhere;11⁴ • a function that is continuous at only one point;11⁵ • a function which, although continuous, changes from increasing to decreasing infinitely often in a finite interval;11⁶ • a function which, although continuous, changes its sign infinitely often in a finite interval;11⁷ and • a continuous function that neither increases nor decreases continually on any interval and, moreover, does not have a derivative at any point.11⁸ We also have an important collection of theorems, including the following: • A function continuous on a closed interval is bounded on the interval.11⁹ • If a function is continuous on a closed interval and takes on values that approach a given value arbitrarily closely, then the function takes on that value on the interval.12⁰ • A function continuous on a closed interval attains maximum and minimum values on the interval.121 • The Intermediate Value Theorem.122 Finally, Bolzano distinguishes between pointwise and uniform continuity, shows with an example that a function that is pointwise continuous on an open interval need not be uniformly continuous on that interval, and claims that continuity on a closed interval is sufficient for uniform continuity.123 There is

11⁴ FL, §37 [BBGA 2A.10/1, p. 35; MW, p. 448]. 11⁵ FL, §46 [BBGA 2A.10/1, p. 41; MW, pp. 453–454]. 11⁶ FL, §101 [BBGA 2A.10/1, pp. 72 f.; MW, pp. 480–481]. 11⁷ FL, §101 [BBGA 2A.10/1, pp. 72 f.; MW, pp. 480–481]. 11⁸ FL, §§111, 135 [BBGA 2A.10/1, pp. 79ff, p. 103; MW, pp. 487ff.; pp. 507–508]. Bolzano himself only claims and attempts to prove that his function is non-differentiable on a set of points that is dense in the interval. 11⁹ FL, §57 [BBGA 2A.10/1, p. 48; MW, p. 459]. 12⁰ FL, §58 [BBGA 2A.10/1, p. 48; MW, p. 460]. 121 FL, §60 [BBGA 2A.10/1, pp. 49–50; MW, pp. 460–461]. 122 FL, §65 [BBGA 2A.10/1, pp. 52–53; MW, pp. 463–464]. Bolzano proved a generalization of this theorem in his Purely Analytic Proof (1817). 123 FL, §49 [BBGA 2A.10/1, pp. 43–44; MW, p. 456]; ‘Verbesserungen und Zusätze zur Functionenlehre,’ §6 [BBGA 2A.10/1, pp. 171ff; MW, pp. 575 ff.]. See also P. Rusnock and A. Kerr-Lawson, ‘Bolzano and uniform continuity,’ Historia Mathematica 32 (2005) 303–311.

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214 bolzano on continuity also a useful fragment of a proof of this last result, today usually called Heine’s Theorem.12⁴ Several of the theorems mentioned above are proved using the Bolzano– Weierstrass Theorem. Thus, though they are stated for functions continuous on closed intervals, they can be easily be converted into proofs of the corresponding claims for functions continuous on compact sets—the key condition required for all the proofs being that a certain limit point of an infinite subset of a given set must itself be a member of the set.12⁵ We saw above that Bolzano made room for both infinitely large and infinitely small quantities in his theory of numbers, but thought that they had little if any application in the actual world. We find something similar in his treatment of functions. Having seen and partially mapped the vast territory of the theory of functions of classical real analysis, he narrows his focus to a special kind which he calls determinable [bestimmbar], which are not only continuous, but also have derivatives of all orders, and are representable by their Taylor series either for all values or for all but an isolated set of values.12⁶ What Bolzano calls determinable spatial extensions, in turn, are representable by means of determinable functions (thus establishing a link between the continuity of functions and continuous extension).12⁷ Given Bolzano’s views on determinism, this class of functions understandably also plays a key role in his physics.12⁸

6. Conclusion Bolzano covered a surprising amount of ground in his studies of classical pointcontinua, arriving at a number of concepts and techniques that are often thought to belong to a later age. Particularly noteworthy are his claim that extension can result from the appropriate arrangement of unextended elements, the consequent acceptance of the actual infinite, the situation of spatial, temporal, and material continuity within a more general and abstract framework, the clear understanding of the role played by the real (measurable) numbers in that framework and the consequent need for an independent account of the reals, the substantial materials he provided for such an account, his profound understanding of continuity in real 12⁴ I discuss this attempted proof, and show how to construct a correct proof based on similar ideas, in ‘Bolzano’s contributions to real analysis,’ pp. 99–116 in E. Morscher, ed., Bernard Bolzanos Leistungen in Logik, Mathematik, und Physik (St Augustin: Akademia, 2003). 12⁵ Cf. ‘Bolzano’s contributions to real analysis.’ 12⁶ PdU, §37, §45. ‘Über Haltung Richtung usw.,’ §9 [Geometrické Práce, p. 146]; cf. Sebestik, Logique et mathématique chez Bernard Bolzano, pp. 428 et seq. 12⁷ See, e.g., ‘Über Haltung, Richtung, usw.,’ §9 [Geometrické Práce, p. 146]. 12⁸ PdU, §27.

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appendix: a note on the bolzano–weierstrass theorem 215 function theory, and a fine collection of theorems about real numbers and realvalued functions. Not everything is done perfectly, and not everything is done the way we might like, but the breadth and depth of his achievements are undeniable.

7. Appendix: A Note on the Bolzano–Weierstrass Theorem As noted above, Bolzano has a clear claim to priority concerning the Bolzano– Weierstrass Theorem: he was the first to state it, and to use it appropriately in proofs of theorems of real analysis. He also tells us he had a proof of the theorem, a claim I see no reason to doubt, even though his surviving papers apparently do not contain one. On the other hand, though it is not impossible,12⁹ I think it highly unlikely that Weierstrass and his students had any knowledge of Bolzano’s later, unpublished work. Weierstrass should accordingly, I believe, be given credit for the independent discovery of the theorem (and of how to use it in analysis), as well as for producing the earliest surviving proof of it. The name Bolzano–Weierstrass is thus eminently fitting. Bolzano’s Theory of Functions first appeared in print in 1930, however, long after the theorem in question had come to be named after Bolzano and Weierstrass.13⁰ Given this, how did the name come to be? More or less by accident, it seems. Bolzano’s Purely Analytic Proof was well known to Weierstrass and his students, and Bolzano’s contributions were apparently generously acknowledged by Weierstrass in his lectures.131 In particular, Bolzano’s proof of the proposition stated in §12 of the Purely Analytic Proof seems to have inspired the Weierstrassian technique of repeatedly bisecting intervals, which was widely used in his circle, and in particular to prove the Bolzano–Weierstrass Theorem. Most likely, it was this acknowledged influence that led others to name the theorem after both Bolzano and Weierstrass, a choice that was more appropriate than any of them most likely knew.

12⁹ Bolzano did share some of his manuscripts with others, and, in 1846, gave three talks on his theory of functions at the Bohemian Royal Society in Prague. 13⁰ G. H. Moore thought that Schoenfliess, in 1899, was the first to call the theorem by this name. See his ‘Historians and philosophers of logic: are they compatible? The Bolzano–Weierstrass Theorem as a case study,’ History and Philosophy of Logic 20 (1999) 169–180. Surprisingly, Moore seems to have been unaware of Bolzano’s priority in this case. See, e.g., p. 171: ‘the theorem was actually found by Weierstrass alone, with concepts not known to Bolzano’. 131 See e.g., H. A. Schwartz’s letter to Cantor of 1 April 1870: ‘I share with you the opinion that Herr Weierstrass expressed in his courses, namely, that we could not have succeeded in numerous investigations without the method of proof that Herr Weierstrass constructed upon the principles of Bolzano.’ Quoted after H. Meschkowski, Probleme des Unendlichen: Werk und Leben Georg Cantors (Brunschweig, Vieweg & Sohn, 1967), p. 228.

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216 bolzano on continuity

References Works by Bolzano: (1) Abbreviations BBGA: Bernard Bolzano-Gesamtausgabe, series editors: Eduard Winter, Jan Berg, Friedrich Kambartel, Jaromír Loužil, Edgar Morscher, Bob van Rootselaar (Stuttgart-Bad Cannstatt, 1969–). MW: The Mathematical Works of Bernard Bolzano, ed. and tr. by S. B. Russ (Oxford University Press, 2004). BD: Beyträge zu einer begründeteren Darstellung der Mathematik. Erste Lieferung (Prague, 1810). Reprinted in facsimile Darmstadt, 1974. English translation in MW. RB: Rein analytischer Beweis des Lehrsatzes, daß zwischen je zwey Werthen, die ein entgegengesetztes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege (Prague, 1817). Also published in the Abh. d. kön. böhm. Ges. d. Wiss. 1817. Reprinted in 1894 and 1905 (with notes by P. E. B. Jourdain). English translation in MW. DP: Die drey Probleme der Rectification, der Complanation und der Cubirung, ohne Betrachtung des unendlich Kleinen, ohne die Annahmen des Archimedes, und ohne irgend eine nicht streng erweisliche Voraussetzung gelöst; zugleich als Probe einer gänzlichen Umstaltung der Raumwissenschaft, allen Mathematikern zur Prüfung vorgelegt (Leipzig, 1817). English translation in MW. WL: Wissenschaftslehre (Sulzbach, 1837); English translation: Theory of Science (Oxford University Press, 2014). PdU: Paradoxien des Unendlichen, ed. F. Příhonský (Leipzig, 1851). There are three later editions with notes by Hans Hahn (Leipzig, 1920), Bob van Rootselaar (Hamburg, 1975), and Christian Tapp (Hamburg, 2012). English translation in MW. FL: Functionenlehre, manuscript. BBGA, Series 2A, Vol. 10/1. English translation in MW.

(2) Other Works Betrachtungen über einige Gegenstände der Elementargeometrie (Prague, 1804). English translation in MW. Der binomische Lehrsatz und als Folgerung aus ihm der polynomische, und die Reihen, die zur Berechnung der Logarithmen und Exponentialgrössen dienen, genauer als bisher erwiesen (Prague, 1816). English translation in MW. “Versuch einer Erklärung der Begriffe von Linie, Fläche und Körper,” manuscript. BBGA 2A.11/1, pp. 113–150. “Unendliche Größenbegriffe,” manuscript. BBGA 2A.8, pp. 100–168. English translation in MW. “Aphorismen zur Physik,” manuscript. BBGA 2A.12/3, pp. 105–148.

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references 217 “Uiber Haltung, Richtung, Krümmung und Schnörkelung bei Linien sowohl als Flächen samt einigen verwandten Begriffen,” manuscript. Spisy Bernarda Bolzana– Bernard Bolzano’s Schriften, published by the Royal Bohemian Society of Sciences, 5 vols. (Prague 1931–48). Vol. 5: Geometrické práce–Geometrische Arbeiten, ed. J. Vojtěch (Prague, 1948), pp. 139–207. Wissenschaft und Religion im Vormärz: Der Briefwechsel Bernard Bolzanos mit Michael Josef Fesl, ed. E. Winter (Berlin, 1965).

Works by Other Authors Baumgarten, Alexander Gottlieb, Metaphysik (Halle, 1766). Behboud, Ali, Bolzanos Beiträge zur Mathematik und ihrer Philosophie (Bern Studies in the History and Philosophy of Science, 2000). Cantor, Georg, Georg Cantor: Gesammelte Abhandlungen mathematischen und philosophischen Inhalts, ed. E. Zermelo (Berlin, 1932; reprint Hildesheim: Olms, 1962). Fischer, Johann Carl, Grundriß der gesammten höhern Mathematik usw., 3 vols. (Leipzig, 1807–9). Gilles-Gaston Granger, “Le concept de continu chez Aristote et Bolzano,” Les Etudes philosophiques (1969) 513–523. Johnson, Dale M., “Prelude to dimension theory: the geometrical investigations of Bernard Bolzano,” Archive for History of Exact Sciences 17 (1977) 261–295. Kästner, Abraham Gotthelf, Anfangsgründe der Arithmetik, Geometrie, ebenen und sphärischen Trigonometrie und Perspectiv (6th edition Göttingen, 1800). Kraus, Lukas, Der Begriff des Kontinuums bei Bernard Bolzano (St Augustin: Academia, 2014). Kraus, Lukas, Ontologie der Grenzen ausgedehnter Gegenstände (Berlin and Boston: De Gruyter, 2016). Krause, Andrej, Bolzanos Metaphysik (Freiburg and Munich: Alber Verlag, 2004). Künne, Wolfgang, “Propositions in Bolzano and Frege,” Grazer philosophische Studien 53 (1997) 203–240. Laugwitz, Detlef, “Bemerkungen zu Bolzanos Grössenlehre,” Archive for History of Exact Sciences 2 (1965) 398–409. Mancosu, Paolo, “Measuring the sizes of infinite collections of natural numbers: was Cantor’s theory of infinite number inevitable?” The Review of Symbolic Logic 2 (2009) 612–646. Meschkowski, Herbert, Probleme des Unendlichen: Werk und Leben Georg Cantors (Brunschweig: Vieweg & Sohn, 1967).

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218 bolzano on continuity Moore, Gregory H., “Historians and philosophers of logic: are they compatible? The Bolzano–Weierstrass Theorem as a case study,” History and Philosophy of Logic 20 (1999) 169–180. Newton, Isaac, Universal Arithmetick (London, 1769). Příhonský, František, Atomenlehre des sel. Bolzano (Budissin, 1857). Reprinted in F. Příhonský, Neuer Anti-Kant, ed. E. Morscher and Ch. Thiel (St Augustin: Academia, 2003). Rootselaar, Bob van, “Bolzano’s theory of real numbers,” Archive for History of Exact Sciences 2 (1963) 168–180. Rusnock, Paul, Bolzano’s Philosophy and the Emergence of Modern Mathematics (Amsterdam: Rodopi, 2000). Rusnock, Paul, “Bolzano’s contributions to real analysis,” pp. 99–116 in E. Morscher, ed., Bernard Bolzanos Leistungen in Logik, Mathematik, und Physik (St Augustin: Academia, 2003). Rusnock, Paul and Angus Kerr-Lawson, “Bolzano and uniform continuity,” Historia Mathematica 32 (2005) 303–311. Rusnock, Paul and Jan Šebestík, “The Beyträge at 200: Bolzano’s quiet revolution in the philosophy of mathematics,” Journal for the History of Analytic Philosophy, Vol. 1, no. 8 (2013). Russell, Bertrand, Our Knowledge of the External World (La Salle: Open Court, 1914; reprint London: Routledge, 2009). Sebestik, Jan, Logique et mathématique chez Bernard Bolzano (Paris: Vrin, 1992). Sebestik, Jan, “Le paradoxe de la réflexivité des ensembles infinis: Leibniz, Goldbach, Bolzano,” pp. 175–194 in F. Monnoyeur, ed., Infini des mathématiciens, infini des philosophes (Paris: Belin, 1992). Simon, Petr, “Bernard Bolzano a teorie dimenze,” Pokroky mat., fys., astron. 26 (1981) 248–258. Simons, Peter, “Bolzano’s monadology,” British Journal for the History of Philosophy 23 (2015) 1074–1084.

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9 Cantor and Continuity Akihiro Kanamori

Georg Cantor (1845–1919), with his seminal work on sets and number, brought forth a new field of enquiry, set theory, and ushered in a way of proceeding in mathematics, one at base infinitary, topological, and combinatorial. While this was the thrust, his work at the beginning was embedded in issues and concerns of real analysis and contributed fundamentally to its nineteenth-century rigorization, a development turning on limits and continuity. And a continuing engagement with limits and continuity would be very much part of Cantor’s mathematical journey, even as dramatically new conceptualizations emerged. Evolutionary accounts of Cantor’s work mostly underscore his progressive ascent through set-theoretic constructs to transfinite number, this as the storied beginnings of set theory. In this article, we consider Cantor’s work with a steady focus on continuity, putting it first into the context of rigorization and then pursuing the increasingly set-theoretic constructs leading to its further elucidations. Beyond providing a narrative through the historical record about Cantor’s progress, we will bring out three aspectual motifs bearing on the history and nature of mathematics. First, with Cantor the first mathematician to be engaged with limits and continuity through progressive activity over many years, one can see how incipiently metaphysical conceptualizations can become systematically transmuted through mathematical formulations and results so that one can chart progress on the understanding of concepts. Second, with counterweight put on Cantor’s early career, one can see the drive of mathematical necessity pressing through Cantor’s work towards extensional mathematics, the increasing objectification of concepts compelled, and compelled only by, his mathematical investigation of aspects of continuity and culminating in the transfinite numbers and set theory. And third, while Cantor’s constructions and formulations may seem simple, even jejune, to us now with our familiarity with set theory and topology, one has to strive, for a hermeneutic interpretation, to see how difficult ⁰ Many thanks to Paul Rusnock and Dirk Schlimm for comments that have led to improvements in this paper. The papers in this volume by Rusnock on Bolzano and by Haffner and Schlimm on Dedekind have some overlap with this paper and are well worth reading for coordination.

Akihiro Kanamori, Cantor and Continuity In: The History of Continua: Philosophical and Mathematical Perspectives. Edited by: Stewart Shapiro and Geoffrey Hellman, Oxford University Press (2021). © Stewart Shapiro and Geoffrey Hellman. DOI: 10.1093/OSO/9780198809647.003.0010

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220 cantor and continuity it once would have been to achieve basic, especially founding, conceptualizations and results. This article has a pyramidal structure which exhibits first a mathematical and historical basis for Cantor’s initial work on limits and continuity and then his narrowing ascent from early conceptualizations to new ones, from interactive research to solo advance. There is successive tapering since continuity has wide ambience out of which Cantor proceeds to more and more specific results, just as he is developing more and more set theory. Section 1 has as a central pivot Cantor’s construction of the real numbers. Leading up to it, we draw in the relevant aspects of real analysis, much having to do with continuity and convergence, and following it, we set out aspects and consequences of constructions of the real numbers that establish a larger ground for mathematics in set theory and topology. Section 2 gets to Cantor’s work on uncountability and dimension, seminal for set theory while also of broad significance—and this is the emphasis here—for the topological investigation of continuity and continua. Finally, section 3 finishes up with point-sets and in particular perfect sets, which is a culmination of sorts for Cantor’s work on the integrated front of continuity and set theory. In several ways we follow well-trodden paths; it is through a particular arrangement and emphasis that we bring out the import and significance of Cantor’s work on continuity. The books [Ferreirós, 2007] and [Dauben, 1979] proved to be particularly valuable for information and orientation.

1. The Real Numbers In his earliest researches that anticipated his development of set theory and the transfinite, Cantor provided a construction—or theory—of the real numbers out of the rational numbers. Seen in terms of his overall accomplishments, this construction can be said to be basic and straightforward, an initial stone laid presaging remarkable advances. Nonetheless, it is worth dwelling on the construction and its role in Cantor’s research, especially as they have a larger significance when set in a broad context of ponderings about continua and continuity. This section is much longer than the others, being given over to establishing and working that context. Aristotle, in Physics, famously argued (III.5) that ‘infinity cannot exist as an actual thing’ but only has a ‘potential existence’, and, related to this, maintained (VI.1) that ‘anything continuous’ cannot be made up of ‘indivisibles’, e.g., ‘a line cannot be made up of points’. Cantor, with others like Riemann and Dedekind who developed continua, decidedly opted for the actual infinite in mathematics, and, as he formulated the real numbers, he identified them with points, conceiving the continuum as consisting extensionally of points. Cantor’s construction, together with Weierstrass’s and Dedekind’s, completed the ‘arithmetization’ of

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the real numbers 221 real analysis. No longer would number be the account of quantity, reckoning, and measuring; number becomes inherent and autonomous, given by arithmetic and order relations and completed by extension. This arithmetization was in the wake of an incisive nineteenth-century mathematical enquiry about continuity and convergence of series of functions. Of this, we give a brief, if necessarily potted, history. What has been called the nineteenth-century ‘rigorization’ of real analysis could fairly be said to have been initiated by Augustin-Louis Cauchy’s classic 1821 text Cours d’analyse, in which he set out formulations of function, limit, and continuity and encouraged the careful investigation of series (infinite sums) and convergence.1 He established [note III] inter alia a ‘pure existence’ proposition, the Intermediate Value Theorem. Cauchy’s initiative promoted norms and procedures for working with continuous functions, but also a Leibnizian ‘ideal of continuity’ whereby properties are to persist through limits. Fourier’s remarkable 1822 Théorie analytique de la chaleur brought ‘Fourier series’—certain series of sines and cosines—to the fore, and, with some leading to discontinuous functions, would exert conceptual pressure on the new initiative. Indeed, Abel, in an incisive 1826 paper on the binomial series, specified at one point that an appeal to the ‘Cauchy sum theorem’—that a (convergent, infinite) sum of continuous functions is again a continuous function—would have been unwarranted as it ‘suffers exceptions’, one being a simple sine series. It was left to Dirichlet in a penetrating 1829 paper to provide broad sufficient conditions, the ‘Dirichlet conditions’, for a (possibly discontinuous) function to be representable as a Fourier series. In the face of such developments, various mathematicians in the 1840s reaffirmed a new ‘ideal of continuity’ by formulating the appropriately articulated concept of uniform convergence, so that e.g., a uniformly convergent sequence of continuous functions does converge to a continuous function. In 1861 lectures at Berlin, Weierstrass carefully set out continuity and convergence in terms of the now familiar 𝜀–𝛿 language, the ‘epsilontics’. A plateau was reached by Riemann in his 1854 Habilitation. In 1868, after his untimely death, his colleague Dedekind published the lecture [Riemann, 1868b] and the dissertation [Riemann, 1868a]. From the first emanated Riemann’s farreaching concept of a continuous manifold, cast in terms of extensional, settheoretic conceptualizations and now fundamental to differential geometry. From the second we have the now familiar Riemann integral for the assimilation of arbitrary continuous functions, and, with it, Riemann’s magisterial extension of Dirichlet’s 1829 work to general trigonometric series—arbitrary series of sines

1 It is true that Bernard Bolzano in his 1817 text Rein analytischer Beweis [Bolzano, 1817] carefully attended to such matters—see the paper by Rusnock on Bolzano in this volume—but his work in general did not have much influence in the mainstream of mathematical development.

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222 cantor and continuity and cosines—arriving at necessary and sufficient conditions for a function to be representable by such a series. However one may impart significance to Cantor’s construction of the real numbers, it is best seen in light of this past as prologue, from both mathematical and conceptual perspectives. Cantor too worked on trigonometric series, getting to the next stage, the uniqueness of representation, and, for the articulation, it became necessary to have a construction of the real numbers in hand for conceptual grounding. Historically, uniform convergence had been developed with a similar motivation in mind, to better articulate results on representability by Fourier series. A side question might be raised here as to why constructions of the real numbers, being conceptually simple to modern eyes, appeared so relatively late. First, there was still a tradition persisting, going back to the Greek notion of magnitude (megethos), which based number on quantity. Occam’s razor is a hallmark in the development of mathematics, with mathematicians proceeding steadily with the fewest ontological assumptions, and it seems that only by Cantor’s time did having a construction of the real numbers as such become necessary, to have a ground for defining collections of real numbers based on taking arbitrary limits. Second, as for conceptual simplicity, basic conceptualizations simply formulated lend themselves to generalization, and we today in axiomatizations of complete metric spaces and the like tend to take what Cantor and others did with the constructions of the real numbers as jejune, underestimating the initial difficulty of carrying out a regressive analysis. In what follows, we briefly describe in section 1.1 Cantor’s early work on trigonometric series, set in its historical context, and describe in section 1.2 the [Cantor, 1872] construction of the real numbers and move into limit points and point-sets. In that year, there also appeared constructions of the real numbers in [Heine, 1872] and in [Dedekind, 1872]. The construction in [Heine, 1872] is Cantorian, with acknowledgement, and it is deployed there to establish governing results on functions and continuity. Section 1.3 sets this out and, embedding it into our narrative, serves to draw out the relevance of the Cantorian construction for functions and continuity. [Dedekind, 1872] provided a thematically different construction of the real numbers which would gain comparable standing. Section 1.4 makes comparisons, and serves to bring the Cantorian construction into sharper relief as well as raise issues about extensionalism.

1.1 Uniqueness of Trigonometric Series After his studies and some teaching at Berlin, in 1869 Cantor took up a position to teach as Privatdozent at the university at Halle, and upon arrival presented the faculty with a Habilitationsschrift in number theory. Eduard Heine was an

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the real numbers 223 elder colleague there who decades before had been a student of Dirichlet at Berlin. Working with him, Cantor soon made a consequential change of research direction, to real analysis and the study of trigonometric series. In his [1870], Heine had pointed out the role of uniform convergence in the work of Abel and Dirichlet and how the significance attributed to the representability of a function as a trigonometric series depended in large part on the uniqueness of the representation. He then built on Dirichlet’s 1829 work to establish the following, where by ‘generally’ he meant except at finitely many points. Theorem 1 [Heine, 1870, p. 355] ‘A generally continuous but not necessarily finite function f(x) can be expanded as a trigonometric series of the form 1

f(x) = a0 + Σ(an sin nx + bn cos nx) 2

in at most one way, if the series is subject to the condition that it is generally uniformly convergent. The series generally represents the function from −𝜋 to 𝜋.’ Heine mentioned that it was not known that a trigonometric series representing a continuous function must be uniformly convergent. Cantor set out to eliminate the ‘uniformly’ from the theorem, in itself a significant move since uniform convergence had become so woven into the representability by trigonometric series. Working as Heine had done with Riemann’s [1868a] key function F(x), the formal double integration of the trigonometric series, Cantor was able to establish: Theorem 2 [Cantor, 1870, p. 142] ‘When a function f(x) of a real variable x is given by a trigonometric series convergent for every value of x, then there is no other series of the same form which likewise converges for every value of x and represents the function f(x).’ The next step forward, to a further generalization, would be momentous. Heine [1870, p. 355] had acknowledged Cantor for proposing for uniqueness of representation that, as in Dirichlet’s work on Fourier series, there could be a finite number of exceptional points at which the convergence of the trigonometric series fails—thus the ‘generally’ in Theorem 1. Cantor, having reduced uniform convergence to simple convergence with Theorem 2, worked the nice properties of Riemann’s F(x) to allow finitely many exceptional points, effectively incorporating ‘generally’ into Theorem 2. Cantor then realized that a further elaboration with F(x) involving limits would allow infinitely many exceptional points in a systematic way. With this central insight about possibility, Cantor developed dramatically new conceptualizations to accommodate his arguments and results, particularly ‘point-sets’ and ‘derived’ such sets in a hierarchy of ‘kinds’. In these terms, he

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224 cantor and continuity generalized Theorem 2 to the following uniqueness theorem (the theorem is about the zero function, but of course, uniqueness of representation ensues by subtraction of two possible representations of the same function). Theorem 3 [Cantor, 1872, p. 130] ‘If an equation is of the form 0 = C0 + C1 + C2 + . . . + Cn + . . . , 1

where C0 = d0 , Cn = cn sin nx + dn cos nx for all values of x with the exception 2 of those corresponding to the points of a given point-set P of the 𝜈th kind in the interval (0, 2𝜋), where 𝜈 denotes an integer, then d0 = 0 and cn = dn = 0.’

1.2 Cantor [1872] It is at the beginning of his [1872] that Cantor presented his construction of the real numbers. The necessity for Cantor was that for Theorem 3 he had newly developed the topological notion ‘point-set of 𝜈th kind’ with respect to the linear continuum, and, heading towards a rigorously presented proof about a real function, he had to be able to apprehend the real numbers ‘corresponding to the points’ in such sets. On this, one can say, again, that the late construction of the real numbers had to do with its only becoming incumbent for proceeding further, and this, appropriately enough, at a next stage in the investigation of trigonometric series, a subject that was interwoven with conceptualizations of continuity and convergence. Cantor’s construction of the real numbers was not sui generis. Weierstrass, in Berlin lectures from the early 1860s, based his theory of analytic functions on a construction of the real numbers.2 He began with the natural numbers as collections of units—much like the Greek arithmos—and fractions as collections of aliquot parts 1/n—as did the Egyptians—to develop real numbers as series (infinite sums)—his Zahlengrößen. Thus, in a rather prolix way, the actual infinite could be said to have surfaced here, as well as elemental set-theoretic conceptualizations. There is evidence that Cantor lectured on his own construction of the real numbers in 1870,3 his second year at the university at Halle. What distinguishes Cantor’s construction is its simplicity at a higher level and how it was deployed in the development of new mathematics.

2 A general reference here is [Dugac, 1973]. 3 [Purkert and Ilgauds, 1987, p. 37].

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the real numbers 225 Starting with the rational numbers as given, Cantor [1872] specified (pp. 123f): When I speak of a numerical magnitude in a further sense [Zahlengröße im weiteren Sinne], it happens above all in the case that there is present an infinite series [Reihe], given by means of a law, of rational numbers (1)

a1 , a2 , a3 , . . .

which has the property that the difference an+m − an becomes infinitely small with increasing n, whatever the positive integer m may be; or in other words, that given an arbitrary (positive, rational) 𝜀 one can find an integer nl such that |an+m − an | < 𝜀, if n ≥ nl and m is an arbitrary positive integer. This property of the series (1) I will express by means of the words ‘The series (1) has a definite limit [bestimmte Grenze] b.’

‘Series’ here is evidently used in the sense we now specify by ‘sequence’; we refer to a sequence as above, as Cantor subsequently did, as a fundamental sequence.⁴ Cantor went on to emphasize that ‘has a definite limit b’ is to have no further sense than as set out, with b a ‘symbol [Zeichen]’ and different symbols b, b′ , b′′ , . . . to be associated with different sequences. He then defined, in terms of associated sequences, b = b′ if for any positive rational 𝜀, |an − a′n | < 𝜀 for n sufficiently large, and similarly, b > b′ and b < b′ . Finally, he stipulated that b ∗ b′ = b′′ for ∗ any of +, −, ×, / according to lim(an ∗ a′n − a′′n ) = 0 in the expected sense, that for any positive rational 𝜀 the value for sufficiently large n is within 𝜀 of 0. These definitions conform to the Leibnizian ‘ideal of continuity’ of properties persisting through limits. These order relations and arithmetical operations as defined can be regarded as extending those for the rational numbers, if e.g., we construe a rational number a as the definite limit of the constant sequence of a’s. In particular, that a sequence a1 , a2 , a3 , . . . ‘has a definite limit b’ has an a posteriori justification in lim(b−an ) = 0. Cantor considered that the domain A of rational numbers has been extended, by the introduction of definite limits, to a domain B. However circumspect Cantor had initially been about ‘definite limit’, he thence referred to the members of B as Zahlengrößen—and we have a construction of the real numbers. Cantor then entertained the extension of the domain B to a domain C by analogously introducing as ‘definite limits’ fundamental sequences of members of B. However, he pointed out that whereas there are members of B that do

⁴ See [Cantor, 1883a]. Such a sequence came to be known as a ‘Cauchy sequence’ in the twentieth century, there being a precedent in [Cauchy, 1821, p. 125].

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226 cantor and continuity not correspond to any rational number, every member of C ‘can be set equal [gleichgesetzt werden kann]’ to a member of B. Nonetheless (p. 126): it is . . . essential to maintain the conceptual distinction between the domains B and C, just as the identification of two numerical magnitudes b, b′ from B does not include their identity, but only expresses a certain relation which takes place between the series to which they refer.

This remark is revealing about Cantor at this juncture vis-à-vis number and identity: with process paramount, C is to be regarded as conceptually different from B; B itself is not quite the domain of real numbers as b = b′ does not entail their identity; and yet, there is commitment to number as given by ratio and order relations. Cantor is interestingly at a cusp of the intensional vs. extensional distinction here; he insists on a meaningful distinction between members of C and B, yet subscribes to how they ‘can be set equal’ on the way to identification. Maintaining the conceptual distinction and regarding B as consisting of Zahlengrößen ‘of the first kind’, Cantor proceeded to iterate the process of going from B to C to get from C to a domain D and so on, getting generally to Zahlengrößen of the 𝜈th kind. With respect to the intensional vs. the extensional distinction, this iteration as carried out with general collections of numbers (see below) would foster an increasingly extensional approach, at the very least because of the need to have simplicity through making identifications—and one has the naissance of Cantor’s extensional set theory. Cantor next went about correlating his Zahlengrößen with points on the straight line—so yes, he was inherently committed to the continuum as consisting extensionally of points. Once an origin o and a unit distance have been specified, the rational numbers correlate to points according to ratio. Then, any point is approached arbitrarily closely by a sequence of points corresponding to rational numbers in a fundamental sequence a1 , a2 , a3 , . . . . So (p. 127), ‘The distance of the point to be determined from o is equal to b, where b is the numerical magnitude [Zahlengröße] corresponding to the sequence.’ How about the converse? Cantor astutely saw (p. 128) the need to postulate an axiom to complete the correlation: . . . to every numerical magnitude [Zahlengröße] there corresponds a definite point of the line, whose coordinate is equal to that numerical magnitude. I call this proposition an axiom, since it is in the nature of this statement that it cannot be proven. Through it the numerical magnitudes also gain a certain objectivity, from which they are, however, quite independent.

We today so readily identify real numbers with points on the straight line, that Cantor’s initial identification may seem jejune or at best a pragmatic correlation.

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the real numbers 227 However, one can try to approach hermeneutic interpretation by seeing how Cantor in his day is taking the straight line qua linear continuum in a prior sense, one through which his Zahlengrößen are to gain ‘a certain objectivity’. A plausible way of thinking is that Cantor’s axiom is analogous to Church’s Thesis, correlating an informal notion with a formal one. Cantor would continue to be invested in the investigation of the continuum, with what he would increasingly call ‘arithmetic’ means. With Zahlengrößen identified with points on the line and collections of points being ‘point-sets [Punktmengen]’, Cantor formulated (p. 129) some concepts that would become basic for topology as well as crucial for his uniqueness theorem: By a limit point of a point-set P I understand a point of the line whose position is such that in any neighbourhood [Umgebung], infinitely many points of P are found, whereby it can happen that the same point itself also belongs to the set. By a neighbourhood of a point one should understand here any interval which contains the point in its interior. Accordingly, it is easy to prove that a point-set consisting of an infinite number of points always has at least one limit point.

This last proposition, with the presumption of the point-set being bounded, is recognizably the Bolzano–Weierstrass theorem, and it is indeed easy to prove given Cantor’s context of sets and real numbers. Weierstrass, in his lectures at Berlin, had the concepts of neighbourhood and limit point more basically and locally put. With Cantor’s synthetic approach involving actually infinite point-sets, there is a higher-order picture, one which will provide the basis for his development of set theory and topology. For any infinite point-set P, considering that ‘limit point of P’ is a welldetermined concept, Cantor took the limit points of P to form a new point-set P′ . Thus, for the first time, an operation on infinite sets was devised. P′ is the derived set of P, and if P′ is again infinite, it too has a (non-empty) derived set P′′ , and so on. Either this process can be iterated to get for each 𝜈 the 𝜈th derived set P(𝜈) of P, or else there is a least 𝜈 when P(𝜈) is finite. In the latter case Cantor stipulated P to be of the 𝜈th kind, and those P being of the 𝜈th kind for some (finite) 𝜈 as derived sets of the first species. With an evident correlation between Zahlengrößen of the 𝜈th kind and derived sets of the 𝜈th kind, Cantor pointed out that if one takes a single Zahlengröße of the 𝜈th kind and traces back through the fundamental sequences all the way back to the rational numbers, the resulting point-set Q of rational numbers is of 𝜈th kind—Q(𝜈) in fact consists of a single point. While the correlation with Zahlengrößen may have stimulated such analysis, the coming to the fore of derived sets as a systematization of the construction of the real numbers promoted a picture of the 𝜈th kinds not as different types but as of the same extensional domain.

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228 cantor and continuity Finally, with all this structure in place, Cantor established (pp. 130f.) the new uniqueness theorem, Theorem 3 above, which allows infinitely many exceptional points. One sees, through his proof idea of a further elaboration involving Riemann’s F(x) with attention to nested intervals, how he had come to entertain his derived point-sets of the first species.

1.3 Heine [1872] With [Heine, 1870] having been a motivation for [Cantor, 1872], it will be à propos to discuss Heine’s [1872], which followed up on an issue—uniform continuity— from his [1870] and relied on the Cantorian construction of the real numbers. Through the discussion one can bring out the operative efficacy of the construction for real analysis. Heine began [1872] with a lament that function theory as promulgated by Weierstrass in his Berlin lectures had not appeared in ‘authentic’ form, and suggested that in any case its truth rests on a ‘not fully definite [nicht völlig feststehenden] definition’ of the irrational numbers. Thanking Cantor for his number conceptualization with sequences, Heine would rigorously set out in his paper the ‘elements of function theory’ as per the title. Heine began with ‘number series [Zahlenreihe]’, fundamental sequences of rational numbers, like Cantor [1872]. He observed that if a1 , a2 , a3 , . . . and b1 , b2 , b3 , . . . are two such, then so are a1 ∗b1 , a2 ∗b2 , a3 ∗b3 , . . . for ∗ any of +, −, ×, / (taking care not to divide by 0). Heine then associated to each sequence a1 , a2 , a3 , . . . a ‘number symbol [Zahlzeichen]’, [a1 , a2 , a3 , . . . ]. Evidently writing ‘A = [a1 , a2 , a3 , . . . ]’ etc. to express abbreviation, he then formulated relations between symbols A = B, A > B, A + B = C, AB = C etc., each given in terms of associated sequences. Finally, Heine defined ‘limit’ for symbols, first taking a = [a, a, a, . . . ] for rational numbers a and, working with such, establishing criteria for general symbols. Thus, unlike Cantor [1872] who initially introduced ‘definite limit’ as an expression and, developing symbols, later justified its use, Heine developed symbols first and only later brought in ‘limit’ as a concept.⁵ Like Cantor, Heine next considered fundamental sequences consisting of ‘number symbols’ and so forth, getting to ‘irrationals of higher orders’. Cantor had pointed out that such an irrational number ‘can be set equal’ to one of first order, but insisted on maintaining the conceptual distinction for his account of pointsets of higher kind. Heine, on the other hand, merely sketched that ‘the irrationals of higher order are not new, agreeing with those of first order’, and proceeded

⁵ Years later, making his only reference to [Heine, 1872], Cantor [1883a, §9.8] describes how with a fundamental sequence (a𝜈 ) he correlates a number b, ‘for which one can expediently use the symbol (a𝜈 ) itself (as Heine, after many conversations with me on the subject, has proposed)’.

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the real numbers 229 to use [x1 , x2 , x3 , . . .] where the xi ’s could be irrational numbers. Thus taking an extensional view of the real numbers at the outset, Heine could be said to have proved the completeness of the real numbers under the taking of limits. With the real numbers thus formulated, Heine forthwith developed his function theory ab initio (pp. 180f.). ‘A single-valued function of a real variable x is an expression which is uniquely defined for every rational or irrational value of x.’ Proceeding to continuity (pp.182f.), A function f(x) is called continuous for a given individual value x = X if for any positive 𝜀 however small, there exists a positive 𝜂0 such that for no positive 𝜂 smaller than 𝜂0 does the value of f(X ± 𝜂) − f(X) exceed 𝜀.

This is essentially Weierstrass’s ‘epsilontics’ formulation of continuity at X. Heine then presented a characterization that he credited to Cantor: a function f(x) is continuous at x = X if and only if whenever X = [x1 , x2 , x3 , . . . ], f(x1 ), f(x2 ), f(x3 ), . . . is a fundamental sequence such that: f(X) = [f(x1 ), f(x2 ), f(x3 ), . . .].⁶ Heine next formulated pointwise and uniform continuity (p. 184): A function f(x) is called continuous from x = a to x = b if it is continuous for each individual value x = X between x = a and x = b, including the values a and b; it is called uniformly continuous from x = a to x = b if for each positive 𝜀 however small, there exists a positive 𝜂0 such that for all positive 𝜂 smaller than 𝜂0 , the value of f(x ± 𝜂) − f(x) remains below 𝜀.

As Heine emphasized, this last is to be so for all values of x and x ± 𝜂 between a and b. Working with Cantor’s characterization, Heine then, in quick order, established the following theorems, now seen as basic to continuity. Weierstrass had attended to these theorems in his lectures with his own, more involved construction of the real numbers. • Intermediate Value Theorem (pp. 185f.): a continuous function from x = a to x = b, with values at a and b of opposite sign, achieves 0 in the interval. • Greatest Lower Bound Theorem (pp. 186f.): a continuous function from x = a to x = b, which is never negative yet becomes arbitrarily small in the interval, achieves 0 in the interval. • Extreme Value Theorem (p. 188): a continuous function from x = a to x = b achieves both a maximum and a minimum in the interval.

⁶ It is notable that from a logical point of view, the only-if direction made the first, unavoidable use of the Countable Axiom of Choice in mathematics. See [Moore, 2015a, pp. 15f.].

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230 cantor and continuity In [1870, p. 361], Heine had pointed out the importance of uniform continuity in the work of Dirichlet and Abel. To conclude, he showed that, in the recurrent situation of intervals including the endpoints, uniform continuity and continuity coincide: Theorem 4 [Heine, 1872, p. 188] ‘A continuous function f(x) from x = a to x = b (for all individual values) is also uniformly continuous.’ This theorem has come to be called the Cantor–Heine Theorem. It requires a higher level of argumentation than for the basic continuity theorems, and can be seen as the thematic climax of Heine’s paper. Heine’s proof, turning on Cantor’s characterization, is sketched as follows: Suppose that a positive 3𝜀 is given. Let x1 be the largest y ≤ b such that a ≤ x ≤ y entails | f (x) − f (a)| ≤ 𝜀. (x1 is the greatest lower bound of those x such that | f (x)−f (a)| > 𝜀.) If x1 < b, note that | f (x1 )−f (a)| = 𝜀 (by a continuity argument). In that case, let x2 be the largest y ≤ b such that x1 ≤ x ≤ y entails |f (x)−f (x1 )| ≤ 𝜀. If x2 < b, note that | f (x2 ) − f (x1 )| = 𝜀. In this way, proceed for as long as possible to get a < x1 < x2 < x3 < . . . < b. If this sequence is finite, then the result is established. (In detail, if M is taken to be half the minimum of the |xn+1 − xn |’s, then a straightforward ‘triangle inequality’ argument shows that for any z1 and z2 between a and b, |z2 − z1 | < M entails |f(z2 ) − f(z1 )| ≤ 3𝜀.) If this sequence is infinite, then x1 , x2 , x3 , . . . is a fundamental sequence and f is continuous at [x1 , x2 , x3 , . . .], yet f(x1 ), f(x2 ), f(x3 ), . . . is not a fundamental sequence—which contradicts Cantor’s characterization. With the Cantorian construction of the real numbers playing a significant role throughout [Heine, 1872], it is worth describing how its arguments sit among those in the historical ‘rigorization’ of real analysis. In a historical context where the Intermediate Value Theorem qua principle was presupposed and applied as part of the sense of continuity, the Bohemian philosopher Bernard Bolzano [1817] in his conceptual approach and Cauchy in his expository text Cours d’analyse [1821, note III] enunciated and established it qua theorem by ‘purely analytic’ means.⁷ Bolzano first ‘proved’ (§7) that a fundamental sequence of partial sums converges; the argument is circular, in that the convergence cannot be proved except on some basis equivalent to it. Bolzano then proceeded (§12) to establish the Greatest Lower Bound Theorem, and with that, (§15) the Intermediate Value Theorem. Cauchy proved (pp. 460–462) the Intermediate Value Theorem by numerical approximation, constructing two fundamental sequences, one approaching the ⁷ For Bolzano, see also the paper by Rusnock in this volume.

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the real numbers 231 intermediate value from above and the other from below, their convergence then taken for granted. These argumentations are quite creditable as early gestures in the rigorization of real analysis. With them, one sees specifically how making explicit beforehand the Cantorian construction of the real numbers—so objectifying limits of fundamental sequences—renders argumentation for the Intermediate Value Theorem rigorous and routine, as in [Heine, 1872]. With respect to the Cantor–Heine Theorem on uniform continuity, it is first of all a notable historical happenstance that Bolzano in his Functionenlehre—written in the 1830s but only published a century later as [Bolzano, 1930]—had engaged with the concept of uniform continuity.⁸ In improvements written for his work, Bolzano stated (see [Russ, 2004, §6, pp. 575ff.]) the Cantor–Heine Theorem and made an unsuccessful attempt at proof. In his 1854 Berlin lectures on the definite integral, Heine’s teacher Dirichlet (see [1854, pp. 3–8]), with uniform continuity on closed intervals a needed refinement, discursively established the Cantor–Heine Theorem. Dirichlet’s argument and Heine’s proof, given above, proceed along the same lines, and it can be specified exactly where the latter has the sufficient buttress: where the Greatest Lower Bound Theorem provides for the increasing sequence of xi ’s and where Cantor’s characterization is applied to deny their infinitude.

1.4 Dedekind [1872] Richard Dedekind, in his essay Stetigkeit und irrationale Zahlen [1872], provided his now well-known construction of the real numbers.⁹ While Cantor was invested in his as part and parcel of his research, Dedekind had arrived at his as a matter of the conceptual analysis of continuity. In his preface, he recounts that he had done so already in 1858, and that it was his receipt of [Heine, 1872] that prompted publication. He also mentioned that he was just in receipt of [Cantor, 1872] going into press, and specifically pointed to Cantor’s axiom as correlated with his own ‘essence of continuity’. Dedekind initially set out (§1) the rational numbers and their ordering and reviewed (§2) how they can be correlated with points on a straight line, once an origin o and a unit distance have been specified. Presupposing the line to consist extensionally of points, Dedekind recalled how any point partitions the line into two parts, those points to the left and those points to the right, and formulated (§3) the ‘essence of continuity’ to be the converse, the following principle: ⁸ See [Rusnock and Kerr-Lawson, 2005] for this and what follows. Bolzano pointed out (see [Russ, 1 2004, §49, p. 456]) that the function f(x) = , while continuous, is not uniformly continuous in an 1−x open interval (i.e., excluding endpoints) around x = 1. ⁹ See the paper by Haffner and Schlimm in this volume for more on Dedekind’s essay and his work in general.

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232 cantor and continuity If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions.

Dedekind thus fixed on the straight line for his analysis and principle—or axiom— whereas Cantor worked up his Zahlengrößen first and, latterly correlating with the straight line, posited his axiom, through which his Zahlengrößen gain ‘a certain objectivity’. Pursuant of his principle, Dedekind formulated (§4) his now well-known cut [Schnitt] as any pair (A1 , A2 ) of non-empty classes that constitute a partition of the rational numbers, with any member of the first less than every member of the second. A rational number produces two cuts—one with that number as the maximum of the left set and the other with it as the minimum of the right set— these cuts to be regarded as essentially the same. Those cuts (A1 , A2 ) with no maximum nor minimum ‘create a new, irrational number 𝛼 which we regard as completely defined by this cut (A1 , A2 )’. In this, 𝛼 is like Cantor’s ‘definite limit’ in having no further sense than as given, though Dedekind’s principle objectifies the number as corresponding to ‘one and only one point’ on the straight line. After setting out the order relations between his real numbers according to set-inclusion of corresponding cuts, Dedekind established (§5) the thematically central result, that the real numbers—autonomous and no longer correlated with the straight line—satisfy his principle of continuity: given a cut of real numbers— a partition of the real numbers into two non-empty classes with any real number in the first less than every real number in the second—there is one and only one real number that produces the cut. This corresponds to Cantor’s observation that any member of his domain C, the result of taking limits of fundamental sequences from his domain B of Zahlengrößen, ‘can be set equal’ to a member of B. Rhetorically, having focused on order and continuity, Dedekind latterly attended (§6) to the formulation of the arithmetical operations for real numbers. He detailed only the addition of cuts: for cuts (A1 , A2 ) and (B1 , B2 ), define C1 to consist of those rational numbers c such that for some a in A1 and b in B1 , c ≤ a + b; then taking C2 to be the complement of C1 , (C1 , C2 ) is a cut that appropriately serves as the sum. Actually, the multiplication of cuts cannot be analogously defined because of the law of signs (−a)(−b) = ab, and a proper definition would have to involve intervals of rationals of differing signs. Dedekind suggested introducing the ideas of ‘variable magnitudes, functions, limiting values, and it would be best to base the definition of even the simplest arithmetic operations upon these ideas’— thus approaching Cantor’s definitions of the arithmetical operations. Dedekind concluded (§7) his essay by proving two ‘fundamental theorems of infinitesimal analysis’, each of which he noted is equivalent to his principle of continuity: ‘If a magnitude x grows continually but not beyond all limits it

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the real numbers 233 approaches a limiting value,’ and ‘If in the variation of a magnitude x we can, for every given positive magnitude 𝛿, assign a corresponding position from and after which x changes by less than 𝛿, then x approaches a limiting value.’ Were these stated in terms of sequences, then the first would assert that an increasing sequence bounded above has a limit, and the second, that a fundamental sequence has a limit. Stepping back, one sees that, however Dedekind actually arrived at his construction of the real numbers, his account and Cantor’s proceed in diametrically opposite directions. Dedekind began with the straight line and his principle of continuity and got to the existence of limits in real analysis; Cantor started with fundamental sequences having ‘definite limits’ and made his way to an axiom for correlating numbers with the linear continuum. With its explicit, existence postulation about the straight line, Dedekind’s principle has been deployed as an axiom to rigorize Euclidean geometry.1⁰ Once the real numbers have been defined and are in place, Dedekind’s principle is seen to be equivalent to fundamental sequences having limits, as well as to each of the Intermediate Value Theorem, the Greatest Lower Bound Theorem, and the Extreme Value Theorem (cf. [Heine, 1872]). As for constructions as definitions of the real numbers, it is informative to consider how Cantor himself in his later, 1883 Grundlagen (§9) saw and compared them. Cantor weighed three definitions of the real numbers, those of Weierstrass from his Berlin lectures, [Cantor, 1872], and [Dedekind, 1872]. After loosely describing the first, Cantor pointed out how Weierstrass was the first to avoid the ‘logical error’ of assuming a finished number exists to which a defining process aspires. Cantor here was bringing out the motivating point of genetic construction for rigorization. Then briefly sketching Dedekind’s definition, Cantor asserted that it ‘has the great disadvantage that the numbers of analysis never occur as ‘cuts’, but must be brought into this form with a great deal of artificiality and effort’. Cantor here speaks as the researcher in real analysis who finds Dedekind’s conceptualization distant from utilizability; recall Dedekind’s last efforts in his §6 and §7. Cantor subsequently settled into an extensive account of his own definition, more detailed than in [Cantor, 1872] and more in the style of [Heine, 1872]. In the middle of this comparative account, Cantor wrote (para. 7): The disadvantage in the [Weierstrass] and [Cantor] definitions is that the same (i.e., equal) numbers occur infinitely often, and that accordingly an unambiguous

1⁰ See [Greenberg, 2008, pp. 134ff.] and [Heath, 1956, pp. 236ff.]. From it, one can prove that if a circle has one point inside and one point outside another circle, then the two circles intersect in two points; with this, one can fill a well-known lacuna in the proof of the very first proposition of Euclid’s Elements.

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234 cantor and continuity overview of all real numbers is not immediately obtainable. This disadvantage can be overcome with the greatest ease by a specialization of the underlying sets (a𝜈 ) using one of the well-known unambiguous systems, such as, for instance, the decimal system or the simple development in continued fractions.

While Dedekind’s cuts can themselves serve as (stand for, be identified with) the real numbers, with the Weierstrass and the Cantor definitions one real number corresponds to infinitely many Zahlengrößen that are pairwise equal according to a derivative notion of equality. This falls short of the ideal of extensionalism, in that real numbers are not fixed as well-defined by construction. There are two ways of rectifying this, one by way of equivalence classes and the other, as Cantor mentions, by way of specializing fundamental sequences to correspond to decimal expansions or to continued fractions. The mode of equivalence classes actually has antecedence in the work of Dedekind. In [1857], Dedekind had proceeded in ℤ[x], the ring of polynomials in x with integer coefficients, by taking as unitary objects infinite collections of polynomials pairwise equivalent modulo a prime p. One can arguably date the entry of the actual infinite into mathematics here, in the sense of infinite totalities serving as unitary objects within an infinite mathematical system. Had Dedekind come to the genetic construction of the real numbers ‘from the other end’— Cantor’s approach—he might well have taken equivalence classes of fundamental sequences as the real numbers. As for the mode of specializing fundamental sequences, Cantor had engaged with it in the course of his research in June 1877, as brought out in an exchange of letters then with Dedekind.11 For specializing via decimal expansions, one would consider fundamental sequences only of the form: a1 is an integer, and an+1 = an + dn ⋅ 10−n for some integer dn satisfying 0 ≤ dn ≤ 9. With the happenstance that e.g., .3000 . . . = .2999 . . . , one has to further restrict consideration to those sequences without a tail of 0’s. For specializing via continued fractions, one would first call on the known fact that every number r in the interval (0, 1) would have a unique representation as a continued fraction 1

r= 𝛼1 +

1 𝛼2 + . . . +

11 See [Ewald, 1996, pp. 853ff.].

1 𝛼𝜈 + . . .

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uncountability and dimension 235 where each 𝛼i is a positive integer. Exactly when r is a rational number, there is a last 𝛼𝜈 as denominator, and we denote this by r = [𝛼1 , 𝛼2 , . . . , 𝛼v ]. With this, one would consider fundamental sequences only of the form: a1 is an integer, and an+1 is either: an , or else a1 + [a1 , . . . , an−1 , an ]—this last possibility only in the case that an has the form a1 + [a1 , . . . , an−1 ]. Cantor’s ‘specializing’ of fundamental sequences to those corresponding either to decimal expansions or to continued fractions does achieve the ideal of extensionalism in that there is a one-to-one correlation between sequences and real numbers. This however comes with the sacrifice of the ideals of simplicity and perspicuity as one incorporates a posteriori understandings, so much so that one might even say that one is looking at different constructions of the real numbers, not aspects of the same. Cantor’s [1872] construction of the real numbers was integrated with his research and has a basic relevance and applicability, as brought out in [Heine, 1872]. Continued fractions too were brought into his research, this for working the theme of one-to-one correlation, as he advanced into transfinite set theory.12 Cantor and Dedekind began a correspondence in late 1873 that would last, on and off, for decades, a correspondence that was stimulating for Cantor and is informative to us about his thinking and progress. We mentioned an exchange of letters in June 1877 above, and in an exchange a month earlier, Cantor and Dedekind discussed aspects of [Dedekind, 1872]. It is through this correspondence that we learn much about Cantor’s next advances, those squarely in transfinite set theory yet much having to do with continuity.

2. Uncountability and Dimension With his formulation of the real numbers in play, Cantor, in the initiating correspondence with Dedekind in late 1873, pursued a question about one-to-one correlation and the real numbers, a question that he had apparently considered several years earlier. The result was a compelling new mathematical understanding of cardinality as a concept applied to the real numbers, one that would stimulate Cantor to the development of transfinite numbers and set theory. Cantor then, in a letter to Dedekind of 5 January 1874, followed up with the question of whether there could be a one-to-one correlation between a line and a surface. In 1877, Cantor also compellingly settled this issue, stimulating the initial work on the invariance of dimension. These two results on one-to-one correlation would be the bulwark for setting up the concept of infinite cardinality. Both proofs still being embedded in real analysis

12 See section 2.2.

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236 cantor and continuity and talk of correlation, transfinite set theory would emerge with the consideration of arbitrary correspondence, in the form of the study of transfinite cardinality. In what follows, we describe these developments one by one, the main points of which are well known in the history of set theory as Cantor’s initial accomplishments, with emphasis put on the specificities of research activity and the underlying involvement of continuity.

2.1 Uncountability Cantor in a letter of 29 November 1873 to Dedekind posed the question:13 Take the totality [Inbegriff ] of all positive whole-numbered individuals n and designate it by (n). And imagine say the totality of all positive real numerical magnitudes [Zahlengrößen] x and designate it by (x). The question is simply, can (n) be correlated to (x) in such a way that to each individual of the one totality there corresponds one and only one of the other?

(‘Totality’ here is a deliberate translation of ‘Inbegriff ’.1⁴) Note the tentativeness of setting out in uncharted waters of totalities and correlation. Today, this primordial question is put: are the real numbers countable? Cantor opined that the answer would be no, that the explanation may be ‘very easy’. He did point out that it is not difficult to correlate one-to-one the totality of positive integers with the totality of rational numbers, and indeed with the totality of finite tuples of positive integers. Dedekind answered by return post that he could not answer the question. However, bringing in his algebraic experience, Dedekind included a full proof that even the totality of algebraic numbers, roots of polynomials, can be correlated oneto-one with the totality of positive integers.1⁵ Cantor in his responding letter of 2 December, encouraged, wrote that he had wondered about the question ‘already several years ago’; that he agreed with Dedekind that it has ‘no special practical interest’ and ‘for this reason does not deserve much effort’; but that it would be good to answer the question—a negative answer would provide a new proof, in light of the algebraic number correlation, of Liouville’s theorem that there are transcendental (i.e., non-algebraic) real numbers.1⁶

13 See [Ewald, 1996, p. 844]. 1⁴ What may first come to mind today for ‘Inbegriff ’ may be ‘essence’, ‘embodiment’, or ‘paradigm’. However, Cantor likely meant ‘totality’, with a precedent for this in [Bolzano, 1851], who used ‘Inbegriff ’ in proximity to ‘Ganzes’. ‘Totality’ conveys an appropriately incipient extensionalism, soon to become more substantive in Cantor’s work. 1⁵ See [Ewald, 1996, p. 848]. 1⁶ See [Ewald, 1996, pp. 844f.].

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uncountability and dimension 237 Presumably stimulated to success, Cantor in his letter of 7 December wrote that ‘only today do I believe myself to have finished’ and included for appraisal an argument that the totality of real numbers cannot be correlated one-to-one with the totality of positive integers.1⁷ On that day, transfinite set theory was born. Again Dedekind answered by return post, with ‘congratulations for the fine success’ and a much simplified version of the proof.1⁸ Cantor in his letter of 9 December announced that he had already simplified his proof, that it shows that for any sequence and any interval of real numbers, there is a real number in the interval not in the sequence.1⁹ In his notes, Dedekind remarked that their letters must have crossed.2⁰ In a letter of 25 December, Cantor wrote that, with the encouragement of Weierstrass at Berlin, he had written and submitted a short paper, this to become ‘On a property of the totality of real algebraic numbers’ [1874].21 Although the paper is where one points to for the birth of transfinite set theory, it is restricted in purpose, as Cantor pointed out in a letter of 27 December, because of ‘local circumstances’—these presumably being Weierstrass’s restrictive focus on the algebraic numbers.22 In the paper, Cantor established that the totality of the algebraic numbers is countable and that the totality of the real numbers is not. In his notes, Dedekind recorded that both proofs were taken ‘almost word for word’ from his letters.23 From the correspondence, it can fairly be said that the first result is actually Dedekind’s. As for the uncountability of the totality of the real numbers, the [Cantor, 1874] proof is schematically as follows: Suppose that a sequence and an interval of real numbers is given. The goal is to find a real number in the interval but not in the sequence. Let 𝛼1 and 𝛽1 be the earliest two members of the sequence, if any, in the interval, say with 𝛼1 < 𝛽1 . Generally, given 𝛼n , and 𝛽n , let 𝛼n+1 and 𝛽n+1 be the next two least members of the sequence, if any, in the interval (𝛼n , 𝛽n ), say with 𝛼n+1 < 𝛽n+1 . If ever this process terminates at a finite stage, then we are done, as the interval at that stage will have a real number not in the sequence. Assume then that this process is infinite, and let 𝛼 ∞ be the upper limit of the 𝛼n ’s, and let 𝛽 ∞ be the lower limit of the 𝛽n ’s. Then any real number 𝜂 such that 𝛼∞ ≤ 𝜂 ≤ 𝛽 ∞ cannot be in the sequence. This proof has an evident involvement of completeness, viz. the fundamental sequences of Cantor’s construction of the real numbers. But also, as Cantor

1⁷ See [Ewald, 1996, pp. 845f.]. 1⁸ See [Ewald, 1996, p. 849]. 1⁹ See [Ewald, 1996, pp. 846f.]. 2⁰ See [Ewald, 1996, p. 849]. 21 See [Ewald, 1996, p. 847]. 22 See [Ewald, 1996, pp. 847f.]. 23 See [Ewald, 1996, pp. 848f.]. This may have contributed to Dedekind not responding to Cantor’s letters for quite some time.

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238 cantor and continuity pointed out in a prescient footnote, the rth member of the sequence is not in the particular interval (𝛼r , 𝛽r ), thereby correlating the indexing of the sequence with the enumeration of the nested intervals. Almost two decades later, Cantor in [1891] would present his famous diagonal proof, abstract and no longer involving continuity, of a vast generalization of uncountability: For any set whatsoever, there is no one-to-one correlation of that set and the collection of functions from that set into a fixed two-element set.

2.2 Dimension Cantor already in a letter of 5 January 1874 to Dedekind raised a new question pursuant of the motif of one-to-one correlation:2⁴ Can a surface (say a square including its boundary) be one-to-one correlated to a line (say a straight line including its endpoints) so that to every point of the surface there corresponds a point of the line, and conversely to every point of the line there corresponds a point of the surface?

He opined that the answer is ‘very difficult’, that as with the previous question ‘one is so impelled to say no that one would like to hold the proof to be almost superfluous’. Again a primordial question, one for which an answer of no does look difficult to establish, but this time the pathways of proof would lead the other way, working against the initial surmise. Note, importantly, that Cantor could only hope to answer such a question—indeed, even pose it—with his construction of the real numbers in place to work combinatorial possibilities for one-to-one correlation. Dedekind did not respond, nor when Cantor brought up the question again in a letter of 15 May 1874. Fully three years would pass before there was again an exchange of letters, this initially about aspects of Dedekind’s [1872]. During this time, Cantor had evidently developed a new conceptualization. In a letter of 20 June 1877 to Dedekind, Cantor now set out his question:2⁵ The problem is to show that surfaces, bodies, indeed even continuous structures of 𝜌 dimensions can be correlated one-to-one with continuous lines, i.e., with structures of only one dimension—so that surfaces, bodies, indeed even continuous structures of 𝜌 dimensions have the same power [Mächtigkeit] as curves.

2⁴ See [Ewald, 1996, p. 850]. 2⁵ See [Ewald, 1996, pp. 853ff.].

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uncountability and dimension 239 Note the ‘continuous’, ‘dimension’, and especially ‘power’. Two totalities have the same power if there is a one-to-one correlation between them, and with this notion of cardinality Cantor had begun his ascent into transfinite set theory. In the letter, specifically addressing whether the 𝜌-tuples of real numbers in the closed unit interval [0,1] can be correlated one-to-one with the real numbers in [0, 1], Cantor now answered yes. Writing each such real number in infinite decimal expansion, he simply interlaced 𝜌 expansions into one. In the 𝜌 = 2 case, given .𝛼1,1 𝛼1,2 𝛼1,3 . . . and .𝛼2,1 𝛼2,2 𝛼2,3 . . . the result would be .𝛼1,1 𝛼2,1 𝛼1,2 𝛼2,2 𝛼1,3 𝛼2,3 . . . . By return post Dedekind pointed out the problem that, presuming in the cases e.g., of .3000 . . . = .2999 . . . one would choose the latter representation, those real numbers with expansion consisting of a tail of alternating 0’s would never be the result of an interlacing.2⁶ By postcard, Cantor acknowledged this, and noted the reduction of the issue to finding a one-to-one correlation between the interlaced real numbers with all the real numbers in [0, 1].2⁷ In a pivotal, long letter to Dedekind of 25 June 1877,2⁸ Cantor accepted that the ‘subject demands more complicated treatment’ and set out to prove a theorem that he now stated in Riemannian terms, (A): ‘A continuous manifold [Mannigfaltigkeit] extended in e dimensions can be correlated one-to-one with a continuous manifold in one dimension.’ His proof, ‘found even earlier than the other’, established that the 𝜌-tuples of real numbers in the closed unit interval [0, 1] can be correlated one-to-one with the real numbers in [0, 1]. Cantor first took an irrational number in [0, 1] to be represented as an infinite continued fraction,2⁹ and correlated one-to-one the 𝜌-tuples of irrational numbers in [0, 1] with the irrational numbers in [0, 1] by interlacing the fraction entries—in analogy to what he had tried with decimal expansions. This time, the known fact that continued fraction representations are unique ensures one-to-one correlation. What must now be established is (B): The irrational numbers in [0, 1] can be correlated one-to-one with all the numbers in [0, 1]. For this, he first enumerated the rational numbers in [0, 1] (recall his first, 1873 letter to Dedekind!) and then correlated them one-to-one with certain 0 < 𝜀i < 1 with 𝜀1 < 𝜀2 < 𝜀3 < . . . having limit 1. The proof then devolves to proving (C): The numbers in [0, 1] except for the 𝜀i ’s can be correlated one-to-one with all the numbers in [0, 1]. Since this in turn would follow if each half-open interval (𝜀i , 𝜀i+1 ] can be correlated one-toone with the closed interval [𝜀i , 𝜀i+1 ], one is left to proving the paradigmatic (D): The half-open interval (0, 1] can be correlated one-to-one with the closed interval [0, 1]. And this he establishes with a step function of line segments, providing a detailed diagram. 2⁶ See [Ewald, 1996, pp. 855f.]. See, several paragraphs below, how with an adjustment this problem can be avoided. 2⁷ See [Ewald, 1996, p. 856]. 2⁸ See [Ewald, 1996, pp. 856ff.]. 2⁹ cf. end of section 1.4.

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240 cantor and continuity Note how this proof proceeds by successive reduction of the problem, each step having to do with composing one-to-one correlations of various domains. One can say that Cantor was driven, almost by necessity, from analytic thinking about correlations to set-theoretic, combinatorial thinking about manipulating them. In modern terms, the initial continued-fraction correlation of the irrationals is continuous, while the rest is gradually worked down combinatorially to (D), which cannot be carried out with a continuous function. As here and generally in his work, as one-to-one correspondence comes to the fore, continuity recedes, this seen in the new ways that one must entertain arbitrary functions. Today, (D) is a simple exercise in set theory texts. Cantor’s overall result, as he later pointed out in his mature Beiträge [Cantor, 1895, §4], could be derived with ‘a few strokes of a pen’ in his cardinal arithmetic: 2ℵ0 ⋅ 2ℵ0 = 2ℵ0 +ℵ0 = 2ℵ0 . Finally, if one insists on working with infinite expansions, then it is straightforward, just working with infinite decimal expansions, to take all non-zero real numbers as successive blocks each consisting of a sequence of zeros followed by a non-zero digit—e.g., .|08|1|3|2|001|03|1|2| . . . —and, through interlacing according to blocks, one-to-one correlate pairs of real numbers in (0, 1] with all the real numbers in (0, 1]. What then remains, if [0, 1] is desired, is to apply (D). Cantor concluded the 25 June 1877 letter with remarks that positioned his result in a larger tradition. For years he had followed with interest the efforts of Gauss, Riemann, Helmholtz, and others at clarification of the foundations of geometry. The important investigations in this field proceed from a presupposition that Cantor too had held to be correct, though he alone had thought that it was a theorem in need of a proof. Attending the Gauss-Jubiläum,3⁰ he had aired (A) above as a question. There was acknowledgement that a proof was needed, to show the answer to be no. But ‘very recently’ he had arrived at the conviction that the answer is an unqualified yes, and thus he had found the proof presented in the letter. All deductions that depend on the erroneous presupposition are now inadmissible. ‘Rather, the difference that obtains between structures of different dimension-number must be sought in aspects completely different from the number of independent coordinates, which is taken to be characteristic.’ Dedekind in a substantive reply of 2 July 1877 first off avouched that Cantor’s proof is correct and congratulated him on the result.31 Dedekind however took issue with Cantor’s last remarks on the dissolution of dimension. He maintained that the ‘dimension-number of a continuous manifold remains its first and most important invariant’, though he would gladly concede that this invariance is in 3⁰ 30 April 1877 at Göttingen, on the centenary of his birth. 31 See [Ewald, 1996, pp. 863f.].

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uncountability and dimension 241 need of a proof. All authors have made the ‘completely natural presupposition’ that transformations of continuous manifolds via coordinates should also be via continuous functions. Thus, he believes the following theorem: If it is possible to establish a reciprocal, one-to-one, and complete correspondence between the points of a continuous manifold A of a dimensions and the points of a continuous manifold B of b dimensions, then this correspondence itself, if a and b are unequal, must be utterly discontinuous.

Dedekind here, with his acute sense of mathematical structure and consequent emphasis on structure-preserving mappings, had swung the pendulum back towards continuity. Riemann, and others, had put forward an informal theory of continuous manifolds with an implied concept of dimension based on the number of coordinates. Dedekind forthwith asserted the invariance of dimension of continuous manifolds under homeomorphisms, i.e., one-to-one correspondences of their points which are continuous in both directions. Framed as a question, his proposition can be fairly said to have stimulated the study of topological invariants. Cantor in his letter of 4 July 1877 responded that he had not intended to give the appearance of opposing the concept of 𝜌-fold continuous manifolds, but rather ‘to clarify it and to put it on the correct footing’.32 He agreed with Dedekind that if the correspondence is to be continuous, then only structures of the same dimension can be correlated one-to-one. He suggested that, if so, difficulties might arise in ‘limiting the concept of continuous correspondence in general’. Indeed, in the decades to come, how to frame continuity for mappings between continuous manifolds so as to establish Dedekind’s proposition would itself become a substantial issue. Cantor published his one-to-one correlation result in his ‘A contribution to the theory of manifolds’ [1878], a paper he pitched be to promulgating his concept of power [Mächtigkeit]. Having made the initial breach in [1874] with a negative result about the lack of a one-to-one correlation, he worked to secure the new ground by setting out the possibilities for having such correlations. With ‘manifold’ evidently meant in a broad sense, two manifolds have the same power if there is a one-to-one correlation between their elements. ‘If two manifolds M and N are not of the same power, then M either with a part [Bestandteil] of N or N with a part of M has the same power; in the first case we call the power of M smaller, and in the second we call it greater, than the power of N.’33 The class [Klasse] of manifolds of the power of the positive whole numbers is ‘particularly rich and extensive’,

32 See [Ewald, 1996, pp. 864f.]. 33 Note the locution ‘power of M’. Already here, at the incipience of Cantor’s theory of cardinality, we have the assertion of the trichotomy of cardinals. As set theory became axiomatized, it was seen that the trichotomy of cardinals is equivalent to the Axiom of Choice; see [Moore, 2015a, p. 10].

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242 cantor and continuity consisting of the algebraic numbers, the point-sets of the 𝜈th kind from [Cantor, 1872], the n-tuples of rational numbers, and so forth. If M is in this class, so also is any infinite part [Bestandteil] of M; and if M′ , M′′ , M′′′ . . . are all in this class, so is their union [Zusammenfassung].3⁴ Proceeding to n-fold continuous manifolds, Cantor first elaborated on how invariance of dimension under continuous correspondence had always been presupposed but should be demonstrated, and then offered up his theorem as to what becomes possible when no assumptions are made about the kind of correspondence.3⁵ Most of the paper is given over to a proof of his one-to-one correlation result, almost verbatim as given in his 25 June 1877 letter to Dedekind save for some notational refinements made in succeeding letters. At the end of the paper, having reduced considerations of power to linear manifolds, Cantor opined: . . . the question arises how the different parts of a continuous straight line, i.e., the different infinite manifolds of points that can be conceived in it, are related with respect to their powers. Let us divest this problem of its geometric guise, and understand (as has already been explained in §3) by a linear manifold of real numbers any conceivable totality [Inbegriff ] of infinitely many, distinct real numbers. Then the question arises, into how many and which classes [Klassen] do the linear manifolds fall, if manifolds of the same power are placed in the same class, and manifolds of different power into different classes? By an inductive procedure, whose more exact presentation will not be given here, the theorem is suggested that the number of classes of linear manifolds that this principle of sorting gives rise to is finite, and indeed, equals two. Thus the linear manifolds would consist of two classes, of which the first includes all manifolds that can be given the form of a function of 𝜈 (where 𝜈 ranges over the positive whole numbers), while the second class takes on all those manifolds that are reducible to the form of a function of x (where x can assume all real values ≥ 0 and ≤ 1). Corresponding to these two classes, therefore, would be only two powers of infinite linear manifolds; the exact study of this question we put off for another occasion.

Note that the ‘geometric guise’ can be divested through Cantor’s construction of the real numbers; how a linear manifold is any ‘conceivable totality’ of real numbers; how ‘classes’ consisting of these are being entertained; and how the initial ‘having the same power’ has become being ‘of a power’—an equivalence 3⁴ Note the set-theoretic delving. Logically speaking, this last assertion, put in axiomatic set theory as ‘the countable union of countable sets is countable’, requires the Countable Axiom of Choice. See [Moore, 2015a, pp. 9, 32]. 3⁵ Thus, Cantor set out the sequential thinking on the topic in reverse order relative to how it had been in his correspondence with Dedekind.

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uncountability and dimension 243 relation has become equivalence classes. We see entering line by line the more settheoretic posing and thinking. The ‘inductive procedures’ are presumably what evolved into the transfinite numbers in his coming papers. In suggesting the existence of only two power classes of linear manifolds, this passage has Cantor’s first statement of the Continuum Hypothesis, a primordial, dichotomous assertion that he would wrestle with ‘on another occasion’—the rest of his life—and set theory still wrestles with to the present day. Cantor’s ascent into set theory would be by himself, but the issue raised of the invariance of dimension, with its foregrounding of continuity, elicited quick reaction. Within a year, five publications appeared that offered proofs of the invariance of dimension in various formulations, by Jacob Lüroth, Johannes Thomae, Enno Jurgens, Eugen Netto, and soon after, by Cantor [1879a] himself.3⁶ The arguments were complex and would later be deemed as only partially successful, revealing a lack of command of the relevant topological notions at the time. In a letter of 29 December 1878 to Dedekind, Cantor wrote that he had seen the papers of the four others on the invariance of dimension, but that ‘the matter does not seem to me to be fully resolved’.3⁷ In a letter of 17 January 1879, Cantor claimed to have settled the question, sketching an argument that turned on contradicting the Intermediate Value Theorem were there a continuous oneto-one correlation between continuous manifolds of different dimensions.3⁸ As on previous occasions, Dedekind in a reply of 19 January helpfully responded with issues, though this time he saw a ‘real difficulty’.3⁹ In a postcard of 21 January, Cantor wrote, acknowledging the difficulty, that he would consider publishing ‘only in case I should succeed in settling the point’.⁴⁰ Cantor must have done so at least to his own satisfaction, for his [1879a] appeared shortly after with his proof. Cantor’s solution, his last work dealing directly with continuous correspondences, was thought to have settled the matter for decades. However, Guiseppe Peano’s [1890] ‘space filling curve’, a continuous mapping from the unit interval [0, 1] onto the unit square [0, 1]×[0, 1], was latterly seen to be a counterexample to Cantor’s formulation. As topological notions were developed, the stress brought on by the lack of firm ground led the young L. E. J. Brouwer to definitively establish the invariance of dimension in a paper [1911] that was seminal for algebraic topology. In retrospect, it is to his considerable credit that Dedekind made explicit the invariance of dimension as an issue in his 1877 correspondence with Cantor. Cantor [1878] publicized it, and he [1879a] pursued this new angle on continuity for a while, but he soon reverted to earlier conceptualizations to be followed up into

3⁶ 3⁷ 3⁸ 3⁹ ⁴⁰

See [Dauben, 1979, pp. 70ff.] for details and references. See [Ewald, 1996, pp. 866f.]. See [Ewald, 1996, pp. 867ff.]. See [Ewald, 1996, pp. 869f.]. See [Ewald, 1996, pp. 870f.].

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244 cantor and continuity the transfinite. The renewed pursuit of invariance of dimension in the twentieth century led to the new field of algebraic topology.

3. Point-Sets and Perfect Sets Through the early 1880s Cantor carried out what would be his major work, work which would be of basic significance for the subject he created, transfinite set theory. It featured continuing engagement with immanent topological notions, and through them, what would become his mature results having to do with continuity, involving continua and perfect sets. Through sustained effort, in large part driven by the urge to establish the Continuum Hypothesis, Cantor vindicated his early construction of the real numbers and his derived sets by iterating the derived set operation through limits and establishing a hierarchical structure of continuity. During this period, Cantor published a series of six papers under the title ‘On infinite, linear point-manifolds’ which documents his progress. Pursuing them in sequence one by one, we can see an overall forward logic in the progress of discovery. Section 3.1 describes the progress through the first four papers in the series, through (in modern terms): dense sets, derived sets of infinite order, the countable chain condition, and countability along the iteratively defined sequences of derived sets. Section 3.2 then describes the plateau reached, derived sets of uncountable sets, continua, and closed and perfect sets. These sections, describing Cantor’s major advances having to do with continuity, are comparatively short for several reasons. First, they describe work by one individual working on his own with novel conceptualizations and methods. Second, the new set-theoretic context thus established transcends continuity, which was to command our main focus. And third, the work builds on earlier, seminal formulations and results about continuity for which we have already provided comparatively elaborate historical and mathematical detail.

3.1 Point-Sets The first paper [Cantor, 1879b] in Cantor’s ‘linear point-manifolds’ series established a base camp for his further ascent through infinitary processes. Containing no new results, it framed ab initio Cantor’s earlier, basic work on limit points and power and cast it anew systematically. In doing so, it brought out aspects of simplicity and directness to constructs and results that had initially emerged in an encumbered way out of Cantor’s construction of the real numbers and considerations of one-to-one correlation.

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point-sets and perfect sets 245 In the grip of his recent initiatives Cantor had titled his intended series with the ponderous ‘linear point-manifolds [Punktmannigfaltigkeiten]’, but he quickly reverted in [1879b] to his earlier ‘point-set [Punktmenge]’ for a collection of real numbers. Recalling the [1872] operation of taking for an infinite point-set P the derived set P′ consisting of its limit points, he again set out its iteration P(𝜈) through 𝜈 stages, and deemed P to be of the first species if P(𝜈) is finite for some 𝜈. He then officially stipulated that P is of the second species if the series of P(𝜈) continues ad infinitum. Proceeding, Cantor brought to the fore what his [1874] uncountability result had turned on: a point-set P is everywhere-dense—for us, just dense—in an interval [𝛼, 𝛽] if every sub-interval contains a point of P.⁴1 He observed that if P is everywhere-dense in [𝛼, 𝛽], then [𝛼, 𝛽] is included in P′ and P is of the second species. Lastly, bringing in the concept of power Cantor reviewed its basics and focused on point-sets of two powers. Point-sets of the power of the natural numbers are now simply the countable [abzählbaren] point-sets. Any infinite point-set of the first species is countable, and so also are the rational and the algebraic numbers, which are of the second species. Point-sets of the power of the real numbers include any interval, and any interval from which a countable set is excluded—this recalling the [1878] dimension proof. Cantor concluded the paper by giving his [1874] uncountability result in the new terms, separating the two powers. [Cantor, 1880], though quite brief, is remarkable for a palpable extensionalism as set out in set-theoretic notation and terminology and indexed by symbols of infinity. Cantor deployed P ≡ Q for extensional equality; P ≡ O for P ‘does not exist [nicht vorhanden ist]’; {P1 , P2 , P3 . . . } for disjoint union; and, for inclusion P ⊆ Q, P is a ‘divisor’ of Q or Q is a ‘multiple’ of P. 𝔐(P1 , P2 , P3 , . . . ) is union, ‘the least common multiple’, and 𝔇(P1 , P2 , P3 , . . . ) is intersection, ‘the greatest common divisor’.⁴2 In these terms, Cantor considered point-sets of the second species. He observed, explicitly for the very first time, that the successive P′ , P′′ , P′′′ , . . . satisfy the set-theoretic P′ ⊇ P′′ ⊇ P′′′ . . . . Cantor could then move to the overall intersection 𝔇(P′ , P′′ , P′′′ , . . . ), which he denoted by the ‘symbol [Zeichen]’ P(∞) . With type distinctions collapsed, Cantor could further pursue the uniformity of construction through set inclusion and intersection. As long as one has infinite point-sets, one can continue with P(∞+n) , the nth derived set of P(∞) , and get to their intersection P(2∞) . The intersection of P(∞) ⊇ P(2∞) ⊇ P(3∞) . . . would be ⁴1 We write the now-standard [𝛼, 𝛽] for Cantor’s (𝛼, 𝛽). Cantor specifies that his ‘intervals’ contain their endpoints. ⁴2 [Ferreirós, 2007, p. 204] pointed out that this number-theoretic terminology agrees with Dedekind’s in his [Dedekind, 1871]. Dedekind there was famously developing his theory of ideals as a generalization of divisibility in number theory, and the terminology, used for ideals, is analogously appropriate. These set-theoretic relations and operations are now commonplace, but one must remember that at the time, working with them, especially with actually infinite totalities, was still quite novel. Both Cantor and Dedekind should be credited with domesticating set-theoretic operations on infinite totalities in the course of their work.

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246 cantor and continuity 2

2

3

denoted P(∞ ) . The intersection of P(∞) ⊇ P(∞ ) ⊇ P(∞ ) . . . would be denoted ∞ ∞ ∞ P(∞ ) . One gets to P(∞ ) , etc., and only the notation is running out. ‘We see here a dialectical generation of concepts, which leads on and on, free from any arbitrariness, in itself necessary and consequent.’ In a footnote, he mentioned that ‘ten years ago’ he had come to such concepts as a proper extension of the concept of number. Whatever is the case, unlike Cantor’s [1872] Zahlengrößen which gained a certain objectivity according to historical antecedence and their relevance to the straight line, these symbols of infinity emerged sui generis in the study of pointsets and the iteration of the derived set operation, as necessary instruments for indexing. The third paper [Cantor, 1882] broadened the context with new domains and issues, and in so doing established two new, significant results that mark the initial ascent into the elucidation of concepts of power and continuity. Recalling his [1878] work with manifolds consisting of n-tuples of real numbers, Cantor set out ‘limit point’, ‘derived,’ and ‘dense’ for these, now simply called ‘n-dimensional domains’, or again, ‘point-sets’. With this generality attained, Cantor emphasized the necessity of having ‘well-defined’ ‘manifolds (totalities, sets)’ for deploying the concept of power, ‘internally determined’ on the basis of definition and according to the law of the excluded middle. (With set-theoretic constructs in place, the next stipulation is the logical definability of prospective sets.⁴3) With this, ‘the theory of manifolds as conceived embraces arithmetic, function theory and geometry’. (Nascent set theory is beginning to be seen as foundational.) Bringing to the fore power as ‘unifying concept’, Cantor established: In an n-dimensional, infinite, continuous space A let an infinite number of ndimensional, continuous sub-domains (a) be [well-]defined, disjoint and contiguous at most on their boundaries; then the manifold (a) of such sub-domains is always countable.

In modern terms, Cantor had affirmed that ℝn , the space of n-tuples of real numbers, satisfies the countable chain condition: any collection of pairwise-disjoint open sets is countable. This follows directly from ℝn being separable—another modern topological notion—that is, having a countable dense subset; as Cantor had noted, the n-tuples of rational numbers are countable and are dense in ℝn . However, Cantor’s proof was much more roundabout, this indicative of his forging a new path through new, basic topological notions. That the real numbers satisfy the countable chain condition would soon become of crucial import. Cantor subsequently observed that if A is a continuous domain and M is a countable point-set consisting of points in A, then for any two points N, N′ in the ⁴3 Zermelo, in his axiomatization of set theory, famously included the Aussonderungsaxiom for just this purpose.

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point-sets and perfect sets 247 domain 𝔄 consisting of the points of A except those in M, there is a continuous, ‘analytically defined’ line connecting N and N′ and lying entirely in 𝔄.⁴⁴ His argument draws out his assumptions. Let L be a line in A connecting N and N′ . Then L can be segmented into finitely many lines NN1 , N1 N2 , . . ., Nn N with N1 , N2 , . . .Nn not in M. Each line can now be replaced by a circular arc avoiding M completely. The result is an ‘analytically defined’ line connecting N and N′ and avoiding M. With this observation, Cantor speculated about the possibilities of continuous motion in a discontinuous space; about how the hypothesis of the continuity of space may not actually conform to the reality of phenomenological space; and about how a revised mechanics might be investigated for spaces like 𝔄. With the broader context established by the first three, Cantor in his fourth paper [1883b] made headway by incorporating countability into the sequences of derived sets indexed with symbols of infinity. Notably, Cantor introduced further set-theoretic notation to new purpose: P + Q for disjoint union; P − Q for set difference when Q ⊆ P; and P1 + P2 + P3 . . . for infinite disjoint union. With this, he stipulated that a point-set Q is isolated if it contains none of its limit points, i.e., 𝔇(Q, Q′ ) ≡ O. For any point-set P, P − 𝔇(P, P′ ) is isolated. Crucially, countability enters the fray here: Every isolated point-set is countable. (Every point in an isolated point-set has a neighbourhood disjoint from the point-set, and by the [Cantor, 1882] countable chain condition, there are only countably many such neighbourhoods.) With this, one has: For any point-set P, if P′ is countable, then so is P. (P = (P − 𝔇(P, P′ )) + 𝔇(P, P′ ), and both point-sets are countable.) Through a series of extensions Cantor proceeded to establish the following, where 𝛼 is meant to refer to any of the symbols loosely indicated in [Cantor, 1880]: If P is a point-set such that P(𝛼) is countable, so is P. Recalling that P′ ⊇ P′′ ⊇ P′′′ . . . , P′ ≡ (P′ − P′′ ) + (P′′ − P′′′ ) + . . . + P(𝛼) is a disjoint union of countable sets. So P′ is countable, and hence P is as well.

3.2 Perfect Sets The fifth paper [1883c] in Cantor’s ‘linear point-manifolds’ series was conspicuously longer and magisterial, and Cantor published it separately, with a preface and footnotes, as an essay [1883a], his Grundlagen. In it Cantor presented his new conceptualization of number, the transfinite numbers [Anzahlen] couched in a carefully wrought philosophy of the infinite. Cantor’s Grundlagen, together with his mature presentation Beiträge [1895, 1897] of his theory of sets and number, ⁴⁴ Cantor initially made this observation in a letter of 7 April 1882 to Dedekind. See [Ewald, 1996, pp. 871f.].

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248 cantor and continuity would become the definitive publications from which transfinite set theory would emanate. In what follows, we persist with our scheme of going through Cantor’s publications, but now in a decidedly skewed fashion. We assume some familiarity with the transfinite (ordinal) numbers, and continuing our focus on continuity, emphasize Cantor’s historical progress, setting out the details of his culminating work on limit points and derived sets, the analysis with perfect sets. In keeping with the expository thrust of the Grundlagen Cantor briefly described (end of §3) his work on the iterations P(𝛼) of the derived set operation, indexed by the transfinite numbers, and (§9) his construction of the real numbers, making comparison with other treatments.⁴⁵ In the midst of §10, Cantor formulated a concept central to the last paper of the ‘linear point-manifolds’ series: a perfect set is a (non-empty) set P such that P′ = P. With this, Cantor proceeded to a resolution, in the Grundlagen spirit of coming to terms with number, of what is, or ought to be, a continuum. Cantor began his §10 discussion of the concept of the ‘continuum’ by recalling an age-old debate between partisans of Aristotle and of Epicurus, this leading to the regrettable impasse that the continuum is not analysable. He then opined that the ‘concept of time’ or the ‘intuition of time’ is not the way to proceed, nor is any appeal to the ‘form of intuition of space’. What is left then is to take the continuum to consist of points and to start with the concept of real number, this as arithmetically and mostly felicitously given by his limit construction. Taking as a foundation the ‘n-dimensional arithmetical space’, essentially ℝn , endowed with the usual distance, || ′ ′ ′ 2 2 2| |√(x1 − x1 ) + (x2 − x2 ) + . . . + (xn − xn ) || , Cantor specified that a point-set P ⊆ ℝn , to be a continuum, ought to be perfect, i.e., satisfy P′ = P. However, perfect sets are not necessarily dense. (To emphasize this, Cantor gave in an endnote the now famous ‘Cantor ternary set’, the totality of all real numbers given by z=

c𝜈 c c1 + 22 + . . . + 𝜈 + . . . 3 3 3

where the c𝜈 can be 0 or 2 and the series can be finite or infinite. This set is perfect yet nowhere dense, i.e., not dense in any interval, and serves today as a paradigmatic example of a set of real numbers with many distinctive properties, e.g., it is the prototype of a fractal.) So, Cantor came up with a second condition for a continuum. A point-set T is connected ‘if, for any two of its points t and t′

⁴⁵ We discussed Cantor’s comparisons already in section 1.4.

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point-sets and perfect sets 249 and for any arbitrarily small number 𝜀 there always exist [finitely many] points t1 , t2 , . . . , t𝜈 , such that the distances tt1 , t1 t2 , t2 t3 , . . . , t𝜈 t′ are all less than 𝜀’. Putting these concepts together, Cantor defined a ‘point-continuum inside [ℝn ] to be a perfect-connected set’. ‘Here “perfect” and “connected” are not merely words but completely general predicates of the continuum; they have been conceptually characterized in the sharpest way by the foregoing definitions.’ Thus, as with his construction of the real numbers, Cantor has formulated, through mathematical precisification in topological terms, the concept of continuum. Cantor concluded §10 forthwith with rhetorical remarks vis-à-vis Bolzano and Dedekind. Contra Bolzano, Cantor criticized Bolzano’s definition of a continuum in Paradoxien des Unendlichen [Bolzano, 1851]⁴⁶ with respect to connectedness: ‘Bolzano’s definition of the continuum is certainly not right; it expresses onesidedly just one property of the continuum, which is also satisfied by sets which result from [ℝn ] when one imagines an “isolated” point-set at a distance from [ℝn ]; in the same way it is also satisfied by sets which are made up of several separated continuua; obviously in such cases no continuum exists, although according to Bolzano this would be the case.’ This brings out how Cantor was not characterizing some one categorical continuum but rather entertaining a range of possibilities, marshalling them through his point-set theory. As things would go, the concept of manifold would gain ascendancy following Riemann’s articulation of n-dimensional manifolds, and connectedness would be built into modern formulations. In his classic Topology [Kuratowski, 1968, vol. II, chap. 5] Kuratowski investigated connected spaces according to a general topological definition and defined a continuum to be a compact, connected Hausdorff space. He then observed (p. 167) that a compact metric space is a continuum if and only if it is connected in Cantor’s sense, acknowledging [Cantor, 1883c]. As for Dedekind, Cantor opined that in [Dedekind, 1872], ‘only another property of the continuum has been one-sidedly emphasized, namely, that property which is in common with all “perfect” sets’. This remark is somewhat opaque, but some light is cast on it by their correspondence. In a letter of 15 September 1882 to Dedekind, Cantor initially raised the question of ‘what we are to understand by a continuum’, and wrote: ‘An attempt to generalize your concept of cut and to use it for the general definition of the continuum did not succeed.’ The sixth and last paper [Cantor, 1884] in Cantor’s ‘linear point-manifolds’ series was of comparable length to the previous, the Grundlagen, and continued its paragraph numbering with §§15–19. As counterpart to the Grundlagen, which was expansive in conceptualizations and philosophical underpinnings, Cantor in [1884] set out his mature mathematical work integrating continuity and cardinality, centering on perfect sets. ⁴⁶ This, though the only reference to Bolzano in Cantor’s works, shows that he was aware of it. Paradoxien is the first text that explicitly espoused the actual infinite.

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250 cantor and continuity After advancing some involved technicalities in §15 to serve as lemmas, in the major paragraph §16 Cantor detailed his characterizing results about the transfinite iterations P(𝛼) and perfect sets. In §17, he re-articulated his results in terms of a now basic topological notion, one to which he thus gave rise. A point-set P is closed [abgeschlossen] if it contains its limit points, i.e., P′ ⊆ P. Cantor observed that a point-set is closed exactly when it is of the form Q′ for some point-set Q. As set out in the Grundlagen, a set is of the first power if it is countable; a transfinite (ordinal) number is of the first number class if it is finite; and it is of the second number class if it is infinite and countable. The second number class is uncountable, and in fact its power is the least after that of the first number class. Systematically, Cantor now allowed P(𝛼) ≡ O. If P(𝛼) is finite for some 𝛼, then P(𝛼+1) ≡ O, and if P(𝛽) ≡ O for some 𝛽, then for 𝛾 ≥ 𝛽, P(𝛾) ≡ O. Theorem 5 [Cantor, 1884, p. 471] ‘If P is a closed point-set of the first power, there is always a smallest number of the first or second number classes, say 𝛼, so that P(𝛼) ≡ O, or what is said, such sets are reducible.’ Theorem 6 [Cantor, 1884, p. 471] ‘If P is a closed point-set of higher than the first power, then P is divided into a perfect set S and a set of the first power R, so that P ≡ R + S.’ Theorem 5 is a forward-direction version of the prominent [Cantor, 1883b] result for countable derived sets, afforded by the focal Grundlagen result that the second number class is uncountable. For closed P, one has P ⊇ P(1) ⊇ P(2) ⊇ . . . , and so P ≡ (P − P(1) ) + (P(1) − P(2) ) + . . . . The isolated sets P(𝛽) − P(𝛽+1) are each countable, so if P itself were countable, then there must be a countable 𝛼 such that P(𝛼) ≡ O—else there would be the contradiction that the second number class is countable. Theorem 6 is the crucial structure result for uncountable point-sets. With the §15 lemmas, Cantor established the theorem through an involved argument that was indicative of his forging a new conceptual path. In terms of now standard topological notions, we can render his argument perspicuously. A point-set Q ⊆ ℝn is open if ℝn − Q is closed. A collection B of open sets of ℝn is a basis if every open set is a union of members of B. Cantor in effect devised a countable basis for ℝn by taking the collection of n-spheres with rational radius and centre an n-tuple of rational numbers. To establish the theorem for an uncountable closed point-set P ⊆ ℝn , first fix a countable basis B for ℝn . The successive P ⊇ P(1) ⊇ P(2) ⊇ . . . ⊇ P(𝛼) ⊇ . . . are all closed, so for each 𝛼 let B𝛼 ⊆ B be such that ℝn − P(𝛼) = ⋃ B𝛼 . Then if P(𝛼+1) is a proper subset of P(𝛼) , B𝛼 is a proper subset of B𝛼+1 . Consequently, there must be a countable 𝛼 such that P(𝛼+1) ≡ P(𝛼) —else there would be the contradiction

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point-sets and perfect sets 251 that B is uncountable. Specifying 𝛼 to be the least such and setting S ≡ P(𝛼) , note that P ≡ (P − P(1) ) + (P(1) − P(2) ) + . . . + S , where each P(𝛼) − P(𝛼+1) is isolated and hence countable. Taking R to be their union, one has P ≡ R + S, where R is countable and S is perfect. Cantor had asserted, in summarizing remarks in the Grundlagen (§10), that the set R of Theorem 6 is reducible in the sense of Theorem 5, i.e., there is a countable 𝛼 such that R(𝛼) ≡ O. The Swedish mathematician Ivar Bendixson, in a letter to Cantor of May 1883, pointed out that R is not necessarily reducible. In a careful analysis that Cantor included in [Cantor, 1884, Theorem G], Bendixson showed that there is a countable 𝛼 such that, instead, R ∩ R(𝛼) ≡ O. Theorems 5 and 6 are nowadays often called the Cantor–Bendixson analysis, and the least 𝛼 such that P(𝛼) ≡ O in the first and P(𝛼+1) ≡ P(𝛼) in the second the Cantor–Bendixson rank. However, this eponymy hardly does justice to Cantor to whom the entire development of the iterations P(𝛼) leading to perfect sets ought to be credited. In §18, Cantor with his perfect sets in hand took the time to develop a theory of ‘content [Inhalt]’—his word—pursuing a subject with which he had been dialectically engaged in the earlier [Cantor, 1883b]. He essentially showed that, according to his formulation, the content of an arbitrary set is equal to that of a perfect set. In the last section, §19, Cantor affirmed the central role of perfect sets, as incipiently seen in his definition of the continuum (e.g., [1884, §10]) and later in his theory of content, and focusing on the subsets of the real numbers, proceeded to establish: Theorem 7 [Cantor, 1884, p. 485] Linear perfect sets have the power of the linear continuum [0, 1]. The proof, indicative of how far he had journeyed, marshalled the accumulated store of topological concepts and results to establish the requisite one-to-one correlation. Hence, ‘closed sets satisfy the Continuum Hypothesis’, in that either they are countable or have the power (cf. Theorem 6) of the real numbers. Cantor concluded optimistically, In future paragraphs it will be proven that this remarkable theorem has a further validity even for linear point-sets which are not closed, and just as much validity for all n-dimensional point-sets. From these future paragraphs . . . it will be concluded that the linear continuum has the power of the second number class (II).

That is, Cantor would establish the Continuum Hypothesis.

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252 cantor and continuity Here at last is revealed what must have been a driving motivation for Cantor’s research. Having suggested that there are only two classes of infinite sets of real numbers according to power (cf. end of [Cantor, 1878]), Cantor had persisted with an analysis of sets through the iteration of the derived set operation—this soon proceeding into the new terrain of transfinite numbers. Along the way, he had developed now basic topological notions to handle limits and continuity, this resulting in the perfect ‘kernel’ of uncountable sets. Perfect sets have the power of the continuum, as seen through accumulated experience, and the next stage would have been to extend the closed set result to all sets of real numbers, thereby establishing the Continuum Hypothesis. Cantor would fail to do this, and, as we now know, it could not be done in the setting that he was working in. Stepping back, we see that Cantor in his remarkable ascent developed, in the mathematical articulation of continuity, the basic topology of point-sets. And we also become aware, here at the end, that as a matter of mathematical practice—as with his construction of the real numbers—Cantor thus made lasting conceptual advances concerning continuity in the course of establishing necessary ground for the resolution of a problem. These aspects of Cantor’s work bring into sharper relief what we have focused on as Cantor’s steady engagement with continuity as he ascended into set theory.

References Bernard Bolzano. Rein analytischer Beweis des Lehrsatzes, daß zwischen je zwey Werthen, die ein entgegengesetztes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege. Gottlieb Hasse, Prague, 1817. New edition in Bernard BolzanoGesamtausgabe, Stuttgart–Bad Cannstatt, 1969–, Series 2A, Volume 10/1. Translated by Steve Russ in [Ewald, 1996], pages 225–248 and in [Russ, 2004], pages 253–277. Bernard Bolzano. Paradoxien des Unendlichen herausgegeben aus dem schriftlichen Nachlasse des Verfassers von Dr. Fr. Příhonský. Reclam, Leipzig, 1851. Translated in [Russ, 2004], pages 595–678. Bernard Bolzano. Functionenlehre. Herausgegenben und mit Anmerkungen versehen von Dr. Karel Rychlík. Königlich böhmische Gesellschaft der Wissenschaften, Prague, 1930. L. E. J. Brouwer. Beweis der Invarianz der Dimensionenzahl. Mathematische Annalen, 70:161–165, 1911. Georg Cantor. Beweis, dass eine für jeden reellen Wert von x durch eine trigonometrische Reihe gegebene Funktion f(x) sich nur auf eine einzige Weise in dieser Form darstellen lässt. Journal für die reine und angewandte Mathematik, 72:139–142, 1870. Reprinted in [Cantor, 1932], pages 80–83.

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references 253 Georg Cantor. Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen. Mathematische Annalen, 5:123–132, 1872. Reprinted in [Cantor, 1932], pages 92–102. Georg Cantor. Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen. Journal für die reine und angewandte Mathematik, 77:258–262, 1874. Reprinted in [Cantor, 1932], pages 115–118. Georg Cantor. Ein Beitrag zur Mannigfaltigkeitslehre. Journal für die reine und angewandte Mathematik, 84:242–258, 1878. Reprinted in [Cantor, 1932], pages 119–133. Georg Cantor. Über einen Satz aus der Theorie der stetigen Mannigfaltigkeiten. Nachrichten von der Königlichen Gesellschaft der Wissenschaften und der GeorgAugusts-Universität zu Göttingen, pages 127–135, 1879. Reprinted in [Cantor, 1932], pages 134–138. Georg Cantor. Ueber unendliche, lineare Punktmannigfaltigkeiten, 1. Mathematische Annalen, 15:139–145, 1879. Reprinted in [Cantor, 1932], pages 139–145. Georg Cantor. Ueber unendliche, lineare Punktmannigfaltigkeiten, 2. Mathematische Annalen, 17:355–358, 1880. Reprinted in [Cantor, 1932], pages 145–148. Georg Cantor. Ueber unendliche, lineare Punktmannigfaltigkeiten, 3. Mathematische Annalen, 19:113–121, 1882. Reprinted in [Cantor, 1932], pages 149–157. Georg Cantor. Grundlagen einer allgemeinen Mannigfaltigkeitslehre. Ein mathematischphilosophischer Versuch in der Lehre des Unendlichen. B. G. Teubner, Leipzig, 1883. Separate printing of [Cantor, 1883c]. A translation in [Ewald, 1996], pages 878–920. Georg Cantor. Ueber unendliche, lineare Punktmannigfaltigkeiten, 4. Mathematische Annalen, 21:51–58, 1883. Reprinted in [Cantor, 1932], pages 157–164. Georg Cantor. Ueber unendliche, lineare Punktmannigfaltigkeiten, 5. Mathematische Annalen, 21:545–586, 1883. Reprinted in [Cantor, 1932], pages 165–208. Separately printed as [Cantor, 1883a]. Georg Cantor. Ueber unendliche, lineare Punktmannigfaltigkeiten, 6. Mathematische Annalen, 23:453–488, 1884. Reprinted in [Cantor, 1932], pages 210–244. Georg Cantor. Über eine elementare Frage der Mannigfaltigkeitslehre. Jahresbericht der Deutschen Mathematiker-Vereinigung, 1:75–78, 1891. Reprinted in [Cantor, 1932], pages 278–280. Translated in [Ewald, 1996], pages 920–922. Georg Cantor. Beträge zur Begründung der transfiniten Mengenlehre, I. Mathematische Annalen, 46:481–512, 1895. Reprinted in [Cantor, 1932], pages 282–311. Georg Cantor. Beträge zur Begründung der transfiniten Mengenlehre, II. Mathematische Annalen, 49:207–246, 1897. Reprinted in [Cantor, 1932], pages 312–351. Augustin-Louis Cauchy. Cours d’analyse de l’École royale polytechnique. Imprimérie royale, Paris, 1821. Joseph W. Dauben. Georg Cantor: His Mathematics and Philosophy of the Infinite. Harvard University Press, Cambridge, 1979.

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254 cantor and continuity Richard Dedekind. Abriss einer Theorie der höheren Kongruenzen in bezug auf einen reellen Primzahl-Modulus. Journal für die reine und angewandte Mathematik, 54: 1–26, 1857. Richard Dedekind. Über die Komposition der binären quadratischen Formen. Supplement X of P. G. Lejeune Dirichlet, Vorlesungen über Zahlentheorie, second edition. Vieweg, Braunschweig, 1871. Richard Dedekind. Stetigkeit und irrationale Zahlen. F. Vieweg, Braunschweig, 1872. Translated in [Ewald, 1996], pages 765–779. Peter G. Lejeune Dirichlet. G. Lejeune-Dirichlets Vorlesungen über die Lehre von den einfachen und mehrfachen bestimmten Integralen. Vieweg, Braunschweig, 1854. 1854 lectures at the University of Berlin. Edited by Gustav Arendt. Pierre Dugac. Eléments d’analyse de Karl Weierstrass. Archive for History of Exact Sciences, 10:41–176, 1973. William Ewald. From Kant to Hilbert. Clarendon Press, Oxford, 1996. José Ferreirós. Labyrinth of Thought: A History of Set Theory and its Role in Modern Mathematics. Birkhäuser, Basel, 2007. Second, revised edition. Marvin J. Greenberg. Euclidean and Non-Euclidean Geometries. Freeman, New York, 2008. Fourth edition. Thomas L. Heath. The Thirteen Books of the Elements, volume I. Dover Publications, New York, 1956. Eduard Heine. Über trigonometrische Reihen. Journal für die reine und angewandte Mathematik, 71:353–365, 1870. Eduard Heine. Die Elemente der Functionenlehre. Journal für die reine und angewandte Mathematik, 74:172–188, 1872. Kazimierz Kuratowski. Topology. PWN—Polish Scientific Publishers, Warszawa, 1968. Second edition, English translation of the first, 1958 edition. Gregory H. Moore. Zermelo’s Axiom of Choice: Its Origins, Development, and Influence. Springer-Verlag, New York, 1982. Giuseppe Peano. Sur une courbe, qui remplit toute une aire plane. Mathematische Annalen, 36:157–160, 1890. Walter Purkert and Hans J. Ilgauds. Georg Cantor, 1845–1918. Birkhäuser, Basel, 1987. Bernhard Riemann. Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe. Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 13:87–131, 1868. Bernhard Riemann. Über die Hypothesen, welche der Geometrie zu Grunde liegen. Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 13: 133–152, 1868. Translated in [Ewald, 1996], pages 652–661. Paul Rusnock and Angus Kerr-Lawson. Bolzano and uniform continuity. Historia Mathematica, 32:303–311, 2005. Steve Russ. The Mathematical Works of Bernard Bolzano. Oxford University Press, Oxford, 2004.

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10 Dedekind on Continuity Emmylou Haffner and Dirk Schlimm

In this chapter, we will provide an overview of Richard Dedekind’s work on continuity, both foundational and mathematical. His seminal contribution to the foundations of analysis is the well-known 1872 booklet Stetigkeit und irrationale Zahlen (Continuity and irrational numbers), which is based on Dedekind’s insight into the essence of continuity that he arrived at in the fall of 1858. After analysing the intuitive understanding of the continuity of the geometric line, Dedekind characterized the property of continuity for the real numbers in terms of what are nowadays called ‘Dedekind cuts’ on the rational numbers. This treatment, which can be characterized as being ‘arithmetical’ as well as ‘axiomatic’, will be presented in detail. To better position Dedekind’s contributions in their historical context, we will also consider some of his more mathematical treatments of continuity in addition to his foundational work. Of particular interest is the definition of the Riemann surface in his joint work with Heinrich Weber (1882). Moreover, Dedekind’s reflections on space and continuity in his unpublished papers ‘Allgemeine Sätze über Räume’ (General theorems about spaces; before 1870) and ‘Beweis und Anwendungen eines allgemeinen Satzes über mehrfach ausgedehnte stetige Gebiete’ (Proof and applications of a general theorem about multiply extended continuous domains; 1892) illustrate the wide range and general coherency of his thoughts. By discussing Dedekind’s works in which the notion of continuity plays a central role, we will show how Dedekind’s approaches became increasingly abstract, while at the same time retaining a common methodology. I never imagined that my conception of irrational numbers would be of particular value, otherwise I would not have kept it to myself for almost fourteen years. Richard Dedekind Letter to Lipschitz, 10 June 1876 (Dedekind 1932, III, 470)

Emmylou Haffner and Dirk Schlimm, Dedekind on Continuity In: The History of Continua: Philosophical and Mathematical Perspectives. Edited by: Stewart Shapiro and Geoffrey Hellman, Oxford University Press (2021). © Stewart Shapiro and Geoffrey Hellman. DOI: 10.1093/OSO/9780198809647.003.0011

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256 dedekind on continuity

1. Introduction Richard Dedekind was born into a well-off family in Braunschweig (in the Duchy of Brunswick, now Germany) on 6 October 1831 and died in his hometown on 12 February 1916 at the age of eighty-four.1 He studied mathematics in Braunschweig and Göttingen, where he received his doctorate in 1852 as the last student supervised by Gauss. He received his Habilitation in Göttingen in 1854. Four years later, he became professor in Zürich, but returned to Braunschweig in 1862, where he retired in 1894 after thirty-two years at the Technische Hochschule. During his time in Göttingen, Dedekind was influenced by his teachers C. F. Gauss (1777– 1855) and Gauss’s successor P. G. Lejeune Dirichlet (1805–59), as well as by his fellow student and friend Bernhard Riemann (1826–66). Later, Heinrich Weber (1842–1913) became a close friend and collaborator. The influence of Dedekind on the development of modern mathematics can hardly be overestimated. In addition to his definition of continuity in terms of cuts, he made several other seminal contributions. In his work on the foundations of arithmetic, Was sind und was sollen die Zahlen? (1888), he gave an axiomatic characterization of the natural numbers and was the first to prove the Recursion Theorem; in his work on algebraic number theory Dedekind introduced the algebraic notions of fields and ideals; and together with Cantor he developed the theory of sets. In addition, Dedekind devoted much of his time to editing important mathematical works, such as four editions of Dirichlet’s Vorlesungen über Zahlentheorie and Riemann’s collected works and scientific Nachlass (Riemann 1876). The birth of Dedekind’s conception of continuity can be dated very precisely. It was on 24 November 1858 that he conceived of the idea of giving a definition of a continuous domain in terms of cuts (‘Schnitte’). He relates this information in the preface of his booklet Stetigkeit und irrationale Zahlen (Continuity and irrational numbers; in the following abbreviated by SZ), which was published in 1872. Given that Dedekind kept meticulous diaries about his daily activities, we have no reason to doubt his account.2 Dedekind had defended his Habilitation on 30 June 1854 in Göttingen with a talk on the introduction of new functions in mathematics (Dedekind 1854) and accepted a position as professor of mathematics at the Eidgenössisches Polytechnikum in Zürich in 1858.3 In autumn 1858 Dedekind was assigned to teach an introduction to analysis and he ‘felt more keenly than

1 For more information about Dedekind’s biography, see Biermann (1971) and (Scheel and Sonar 2014, 19–22). 2 Only one original booklet seems to have been preserved, from the years 1900–1, held at the archive of the library (Archiv der Bibliothek) of the Technische Universität Braunschweig. 3 See (Ewald 1996, 753–754) for some background on Dedekind and (Ewald 1996, 765–766) for some background on SZ.

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dedekind’s definition of continuity 257 ever before the lack of truly scientific foundation for arithmetic’ (Dedekind 1872, 767).⁴ To provide such a foundation was the goal of his text. In the following, we first give an exposition of his account of the continuity of the real numbers in section 2, including a brief look at the historical and methodological context of Dedekind’s work. In order to give a fuller picture of Dedekind’s mathematical work, we consider how Dedekind treated other continuous domains in the second part of the paper (section 3).

2. Dedekind’s Definition of Continuity 2.1 Background The nineteenth century was a period of great transformations, both within mathematics and regarding philosophical views about mathematics (Gray 1992, 2008). Dedekind played a crucial role in these developments. Since antiquity, geometry was considered to be the study of two- and three-dimensional (idealized) objects in the world and arithmetic the study of properties of collections of units. Later, analysis emerged to deal with numbers that represent quantities such as lengths and weights. For Kant, geometric and arithmetical knowledge was a priori, based on our (inner) intuition of space and time.⁵ However, with the emergence and growing acceptance of non-Euclidean geometries, mathematicians were forced to reconsider the status of geometry. Dedekind’s doctoral supervisor Gauss, for example, still viewed arithmetic as a source of a priori mathematical knowledge, but considered geometry to be a posteriori (Goldstein et al. 2007). To regard arithmetic as the basis on which to ground mathematics is an idea that Dedekind inherited from Gauss and Dirichlet, whom he paraphrases as having said that ‘every theorem of algebra and higher analysis, no matter how remote, can be expressed as a theorem about natural numbers’ (Dedekind 1888, 792).⁶ Dedekind reports in the Preface to SZ that in his 1858 course on analysis he had to appeal to geometric intuition for the justification of the monotone convergence theorem, and while he considers such an approach didactically useful, he does not consider it to be rigorous: In discussing the concept of the approach of a variable magnitude to a fixed limiting value . . . I took refuge in geometrical evidence. Even now I regard such invocation of geometric intuition [Anschauung] in a first presentation of the differential calculus as exceedingly useful from a pedagogic standpoint, and

⁴ The page references to (Dedekind 1872) are to the English translation in (Ewald 1996). ⁵ On Kant’s notion of continuity, see also the paper by Sutherland, in this volume. ⁶ The page references to (Dedekind 1888) are to the English translation in (Ewald 1996).

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258 dedekind on continuity indeed it is indispensable, if one does not wish to lose too much time. But no one will deny that this form of introduction into the differential calculus can make no claim to being scientific. For myself this feeling of dissatisfaction was so overpowering that I resolved to meditate on the question until I should find a purely arithmetical and perfectly rigorous foundation [Begründung] for the principles of infinitesimal analysis. (Dedekind 1872, 767)

Dedekind’s attitude is also expressed forcefully in a comment on du BoisReymond’s book on function theory (1882), contained in Dedekind’s Nachlass: ‘Apparently, B. has still not realized that every theorem of analysis is ultimately always a theorem about whole numbers’ (Dugac 1976, 199). Dedekind’s approach fits well into the so-called movement of arithmetization of analysis, of the same period. The phrase ‘arithmetization of mathematics’ was popularized by Felix Klein (1895), who borrowed it from Kronecker.⁷ It was used to describe a reworking of mathematics, in particular of analysis, such that mathematical developments that are based on intuition are only accepted after they were brought into a rigorous logical, arithmetical form. Although this was an essential part of the project for Kronecker, Klein dismisses the ‘mere putting of the argument into arithmetical form’. Dedekind also uses the verb ‘to arithmetize’ in this sense. For him, putting a definition or an argument into an arithmetical form is, indeed, a way to avoid resorting to intuition in mathematics. In addition, he refers to arithmetic as a means to clarify and rigorize Riemannian function theory, in a letter to Weber (11 November 1876; in Scheel and Sonar 2014, 50). Related to these developments is also the growing reluctance to consider analysis as being based on the notion of quantity. Epple (2003) characterizes the work on the foundations of analysis of Weierstrass, Cantor, Dedekind, and others in the period from 1860 to 1910 as contributing to ‘the end of the science of quantity’. That Dedekind felt very strongly about this issue can be seen from his comment on du Bois-Reymond’s approach that considers ‘the concept of quantity as the only and necessary key for understanding the remaining basic concepts of analysis’ (du Bois-Reymond 1882, 42). Referring to this passage and to SZ, Dedekind remarks: ‘This is the extreme opposite of my conception of arithmetic and analysis’ (Dugac 1976, 199). In the Introduction to his work on natural numbers (1888), Dedekind returns to this topic and declares: ‘I wholly reject the intrusion of measurable quantities’ in the treatment of irrational numbers (Dedekind 1888, 793). After the conception of the principle of continuity on 24 November 1858, Dedekind discussed it on and off with colleagues and students. For instance: in the Preface to SZ, he writes about having a conversation with his Zürich colleague Heinrich Durège on 30 November 1858, mentioning it occasionally at the beginning of his lectures, and giving a talk at the Wissenschaftlicher Verein in

⁷ For a detailed discussion, see Petri and Schappacher (2007).

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dedekind’s definition of continuity 259 Braunschweig (11 January 1864).⁸ Dedekind must also have communicated his plan to publish his theory of continuity already in 1870, as this is mentioned in a letter from Adolf Dauber written on 20 June 1871 (Dugac 1976, 194). He then considered the topic for a publication on the occasion of his father’s fiftieth anniversary in office (26 April 1872). The final push for the choice of the topic came from the fact that Eduard Heine had just published his account (Heine 1872). Dedekind fully agrees with it in substance, but thinks that his own account is ‘simpler and [brings] out the vital point more clearly’ (Dedekind 1872, 767). Dedekind’s booklet was written independently of Cantor’s definition, as he received Cantor’s paper in which continuity is defined in terms of fundamental sequences (Cantor 1872) only while he was writing the Preface of SZ.⁹

2.2 Stetigkeit und irrationale Zahlen (1872) Dedekind’s booklet Stetigkeit und irrationale Zahlen is only thirty-one pages long. It consists, after a short Preface, of the following seven sections, which show very nicely the didactic character of Dedekind’s presentation: 1. 2. 3. 4. 5. 6. 7.

Properties of rational numbers Comparison of the rational numbers with the points of a straight line Continuity of the straight line Creation of irrational numbers Continuity of the domain of real numbers Operations with real numbers Infinitesimal analysis

We shall follow the structure of Dedekind’s presentation in the present section. The original text is easy to follow also for a modern reader. Dedekind presupposes the system of rational numbers, which he calls R, and notes that it satisfies the conditions for being a field, which he had introduced in Supplement X of (Dirichlet 1871, 424): they are closed under the arithmetic operations of addition, subtraction, multiplication, and division. In addition to these properties, Dedekind points out that one can define the familiar order relations (‘’) on the rational numbers, such that they satisfy three conditions: (I) The relation is transitive. (II) If a ≠ c, then there are infinitely many rational numbers b, such that a < b and b < c. (III) Any definite rational number a separates the entire system R into two infinite classes A1 and A2 , such that all elements of A1 are less than a and a is less than every element of A2 . The number a itself can be assigned to either of the two ⁸ A brief biography of Durège can be found in (Knus and Scharlau 1985, 348–349). ⁹ In fact, Cantor is not mentioned in the draft of SZ that is held in Dedekind’s Nachlass (Dugac 1976, 203–209). See also Kanamori’s paper in this volume.

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260 dedekind on continuity classes, such that ‘every number of the first class A1 is less than every number of the second class A2 ’ (Dedekind 1872, 769). Next, Dedekind considers a geometric line L and notes that we can introduce the relation lies-to-the-right-of between two points that is analogous to the >relation on the rational numbers. Moreover, by picking a point on the line as the zero point o and choosing a definite unit of length, every rational number can be made to correspond to a definite point on the line, such that for any two numbers a and b with a > b, the corresponding points p and q are such that p lies-tothe-right-of q. In this way, the ‘analogy between rational numbers and the points of the straight line becomes a real correspondence [wirklicher Zusammenhang]’ (1872, 770), Dedekind writes, and the above three laws are also satisfied by the corresponding relation and points on a line. However, while the mapping between rational numbers and points on a line is injective, Dedekind notes that it is not surjective, because some cuts, e.g., (A1 , A2 ) such that A2 = {x ∈ R ∣ x2 > 2} and A1 = R\A2 , are not produced by a rational number.1⁰ To obtain a domain of numbers that has the same continuity as the straight line, additional numbers are necessary, which have to be ‘created’ according to the following principle that expresses ‘the essence of continuity’: If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions. (Dedekind 1872, 771)

Although Dedekind uses our geometric intuition of the straight line to motivate this principle, he emphasizes that ‘[t]he assumption of this property of the line is nothing else than an axiom by which we attribute to the line its continuity, by which we think continuity in the line’ (Dedekind 1872, 771–772). In the correspondence with Lipschitz in 1876, Dedekind remarks that he can indeed imagine space to be non-continuous (Dedekind 1932, III, 478). The epistemic priority of numbers over space and time is repeated again in the Introduction to (Dedekind 1888). To complete the domain R in such a way that the resulting domain is continuous, Dedekind introduces the notion of a cut of rational numbers: If now any separation of the system R into two classes A1 , A2 , is given which possesses only this characteristic property that every number a1 in A1 is less than every number a2 in A2 , then for brevity we shall call such a separation a cut [Schnitt] and we shall designate it by (A1 , A2 ). (Dedekind 1872, 772; emphasis in original)

1⁰ The mathematical terminology and symbolism used in this sentence are anachronistic.

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dedekind’s definition of continuity 261 If in a cut (A1 , A2 ) of rational numbers, there is either a greatest rational number in A1 or a least rational number in A2 , then Dedekind says that the cut is ‘produced’ (hervorgebracht) by that number. Later presentations of cuts simplify this by adding a convention that the rational number that produces the cut is always in A1 .11 However, he shows that not all cuts of rational numbers are actually produced by a rational number, which illustrates the discontinuity of R. For those cuts that are not produced by a rational number Dedekind introduces new numbers that are said to produce the cuts: Thus, whenever we have to do with a cut (A1 , A2 ) produced by no rational number, we create a new number, an irrational number 𝛼, which we regard as completely defined by this cut (A1 , A2 ); we shall say that the number 𝛼 corresponds to this cut, or that it produces this cut. (Dedekind 1872, 773; emphasis in original)

Notice that here Dedekind does not identify the real numbers with the system of cuts of rational numbers, but instead ‘creates’ irrational numbers that correspond to those cuts that are not produced by any rational number. Thus, Dedekind does not create the entire system of real numbers that corresponds to the system of cuts, but only the irrational numbers, which complement the rational numbers to form the real numbers. Dedekind himself did not put much weight on the newly created numbers; he wrote in a letter to Lipschitz on 10 June 1876 that ‘if one does not want to introduce new numbers, I have nothing against this’ and went on to reformulate his result without reference to numbers (Dedekind 1932, III, 471). The nature of Dedekind’s notion of creation, as it is used here, has been the topic of many scholarly discussions.12 For the system ℜ of real numbers, which consists of both rational and irrational numbers, Dedekind defines the less-than and greater-than relations and proves (in section 5) that it is continuous, i.e., that for any cut of ℜ there is a real number that produces it. He also shows how to define addition of real numbers in terms of their corresponding cuts of rational numbers (section 6). To illustrate Dedekind’s treatment of arithmetic operations on real numbers in terms of cuts, let us present his definition of addition of two real numbers 𝛼 and 𝛽 (Dedekind 1872, 776–777). To these numbers correspond two cuts, say (A1 , A2 ) and (B1 , B2 ), and the task is to define a cut (C1 , C2 ) that corresponds to (or is produced by, in Dedekind’s parlance) the real number 𝛾 (= 𝛼 + 𝛽). Dedekind separates all rational numbers c into the classes C1 and C2 , such that c is in C1 if

11 See, e.g., (Müller-Stach 2017, 134). 12 See, e.g., Tait (1996); Ferreirós (1999); Reck (2003); Sieg and Schlimm (2005, 2017). On the relation between Dedekind and Eudoxos’ theory of proportion, see Stein (1990) and the correspondence with Lipschitz from June and July 1876 in (Scharlau 1986, 56–80).

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262 dedekind on continuity there are a1 ∈ A1 and b1 ∈ B1 with a1 + b1 ≥ c; otherwise, c is in C2 . (We can also think of C1 as consisting of the elements a1 + b1 , for all a1 ∈ A1 and b1 ∈ B1 .) Because every element of C1 is less than every element of C2 , this separation indeed yields a cut (C1 , C2 ). Next, Dedekind shows that the addition of rational numbers is preserved, if we reason in terms of their corresponding cuts and the above definition of the sum of two cuts. (a) By the definition of the cuts corresponding to two rational numbers 𝛼 and 𝛽, we have that a1 ≤ 𝛼 and b1 ≤ 𝛽, so that for every number c1 ∈ C1 , we have c1 ≤ 𝛼 + 𝛽. Thus, a1 + b1 ≤ 𝛼 + 𝛽. (b) Assume now, for the sake of contradiction, that there is a c2 ∈ C2 with c2 < 𝛼 + 𝛽, i.e., there is some positive rational number 1 1 p, such that 𝛼 + 𝛽 = c2 + p. In this case, c2 = (𝛼 − p) + (𝛽 − p). However, this 1

2

1

2

contradicts the definition of c2 , because 𝛼 − p ∈ A1 and 𝛽 − p ∈ B1 . Therefore, 2 2 we have for every element c2 ∈ C2 that c2 ≥ 𝛼 + 𝛽. It follows from (a) and (b) that the cut (C1 , C2 ) is produced by the rational number 𝛼 + 𝛽. In other words, the definition of the sum of two cuts that correspond to two rational numbers 𝛼 and 𝛽 yields exactly the cut that corresponds to the rational number 𝛼 + 𝛽. After this argument, Dedekind proceeds by claiming that other arithmetic operations, including roots and logarithms, can be defined analogously.13 In this way, Dedekind proudly announces, ‘we arrive at real proofs of theorems (as, e.g., √2 ⋅ √3 = √6), which to the best of my knowledge have never been established before’ (Dedekind 1872, 777).1⁴ Finally, Dedekind connects his investigation to two well-known theorems of analysis by showing that both the monotone convergence theorem and the Cauchy condition of continuity are equivalent to his principle of continuity. In addition to these two theorems, Dedekind mentions two others in his notes on du Bois-Reymond’s (1882) book, namely the least-upperbound (and greatest-lower-bound) property and the analytic intermediate value theorem (Bolzano theorem).1⁵ The four theorems that Dedekind proves in these notes using his principle of continuity are, according to Dedekind, ‘just those for which I undertook the entire investigation (in the fall of 1858) in the first place’. He also remarks that his proofs are ‘almost always or always much shorter’ than the ones found in du Bois-Reymond’s book (Dugac 1976, 201).

13 As Kanamori points out on p. 232 in this volume, the law of signs renders the definition of multiplication of cuts less straightforward than that of addition. 1⁴ This claim is discussed in the correspondence with Lipschitz in 1876 (Dedekind 1932, III, 471 and 474) and is also mentioned more than ten years later in the Introduction to Was sind und was sollen die Zahlen? (Dedekind 1888, 794). 1⁵ For other equivalent theorems, see (Benis-Sinaceur 1994, 194).

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dedekind’s definition of continuity 263

2.3 Methodological Reflections 2.3.1 On Definitions and Ideals Dedekind’s definition of irrational numbers became for him a paradigmatic example of the right way of introducing mathematical domains. He refers to it in the introduction to his ‘Sur la théorie des nombres entiers algébriques’ (1877), in connection with his justification of the definition of ideals. Ideals were introduced by Dedekind in (Dirichlet 1871) as a way to generalize E. E. Kummer’s ideal numbers, which themselves were invented to ‘save’ the existence of a unique decomposition in primes in certain domains of complex integers (Kummer 1856).1⁶ Dedekind criticizes Kummer’s introduction of ideal numbers, because they were defined only indirectly through their property of divisibility, with direct references to particular numbers (e.g., cyclotomic numbers). This definition cannot be effectively generalized to any domain of complex numbers, and can ‘lead to hasty conclusions and incomplete proofs’. Dedekind demands instead ‘a precise definition covering all the ideal numbers . . . and at the same time a general definition of their multiplication’. Such a definition can be achieved by establishing ‘once and for all the common characteristic of the properties A, B, C, . . . that serve to introduce the particular ideal numbers and then to indicate how one can derive, from properties A, B corresponding to individual numbers, the property C corresponding to their product’. At this point in the text Dedekind adds a long footnote containing three methodological demands for the introduction of the real numbers, which is then followed by a summary of the main points of Stetigkeit und irrationale Zahlen: Assuming that the arithmetic of rational numbers is soundly based, the question is how one should introduce the irrational numbers and define the operations of addition, subtraction, multiplication and division on them. My first demand is that arithmetic remain free from intermixture with extraneous elements, and for this reason I reject the definition of real number as the ratio of two quantities of the same kind. On the contrary, the definition or creation of irrational number ought to be based on phenomena one can already define clearly in the domain R of rational numbers. Secondly, one should demand that all real irrational numbers be engendered simultaneously by a common definition, and not successively as roots of equations, as logarithms, etc. Thirdly, the definition should be of a kind which also permits a perfectly clear definition of the calculations (addition, etc.) one needs to make on the new numbers. (Dedekind 1877, 284–285)1⁷ 1⁶ Dedekind (1877) is a rewritten, reworked version of his 1871 work. 1⁷ The English translations in this section were taken from (Dedekind 1996, 57–58). Similar remarks can also be found in Dedekind’s letter to Lipschitz from 10 June 1876 (Dedekind 1932, III, 470–471).

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264 dedekind on continuity We see here again very clearly that Dedekind aimed at an introduction of the irrational numbers based only on the system of rational numbers, his rejection of the reliance on ‘extraneous’ magnitudes, his rejection of piecemeal definitions, and the insistence that a definition of new objects must also allow for the definition of operations on these objects. The strategy chosen by Dedekind for the definition of ideals from a number field1⁸ is analogous. Rather than considering a factor p (ideal or not), he proposes to consider the set of all the (actually existing) numbers divisible by p. This collection is called an ideal, and it is defined as the collection 𝔞 of numbers satisfying the following two necessary and sufficient conditions: I. The sum and difference of every pair of numbers of 𝔞 are again numbers of 𝔞. II. The product of any number in 𝔞 by any integer of the field is again a number of 𝔞. Dedekind defines a notion of divisibility for ideals by reinterpreting the relation of inclusion: an ideal 𝔞 is divisible by an ideal 𝔟, if 𝔞 ⊂ 𝔟. It is also possible to prove that if 𝔠 is an ideal divisible by an ideal 𝔞, then there exists one and only one ideal 𝔟 that satisfies the condition 𝔞𝔟 = 𝔠. By doing so, Dedekind translates to the domain of collections some of the arithmetical relationships between elements of the said collections, and opens the possibility of defining new arithmetical operations, and of calculating with them. In ideal theory, this possibility of moving computations to the ‘higher level’ is ensured by the equivalence between the laws of divisibility of numbers and of ideals.1⁹ The relation between rational numbers and cuts of rational numbers is analogous.2⁰

2.3.2 On Concepts, Sets, and Structures Dedekind’s general approach in mathematics has often been called ‘conceptual’, mainly on the basis of his own characterization. For example, in a letter to Lipschitz from 6 October 1876, Dedekind contrasts his approach to one that is based on forms of representation, like formulas: My efforts in number theory have been directed towards basing the work not on arbitrary representations or expressions but on simple foundational concepts and thereby—although the comparison may sound a bit grandiose—to achieve in number theory something analogous to what Riemann achieved in function theory . . . (Dedekind 1932, III, 468–469; transl. in Edwards 1983, 11)21

1⁸ There is no notion of ring in Dedekind’s introduction of ideals. 1⁹ Indeed, it is easy to show that when 𝔞 is divisible by 𝔟, then for all 𝛼 ∈ 𝔞 and 𝛽 ∈ 𝔟, 𝛽 divides 𝛼. 2⁰ See also (Ferreirós 1999, 134) on the relation between ideals and Dedekind’s definition of irrational numbers. 21 See Avigad (2006) for a discussion of this quote and the next in the context of Dedekind’s theory of ideals.

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dedekind’s definition of continuity 265 Around the same time he also contrasts the use of calculations to his own approach that is based on ‘fundamental characteristics’: [it] is preferable, as in the modern theory of functions, to seek proofs based immediately on fundamental characteristics, rather than on calculation, and indeed to construct the theory in such a way that it is able to predict the results of calculation . . . (Dedekind 1877, 92; quoted from Dedekind 1996, 102)

Although Dedekind did not shy away from making extensive calculations in his mathematical work and also paid careful attention to his choice of notation— as is evidenced from the manuscripts in his Nachlass22—the above quotes show his methodological stance for the presentation of his theories. The fundamental characteristics of concepts like natural numbers and ideals, as well as his ‘principle of continuity’, can be understood from a modern perspective as axioms that define the structures in question.23 In the years following the publication of SZ, Dedekind developed further his conception of sets, which he called ‘systems’, and mappings, which he considered to be fundamental to mathematics and thinking in general. Dedekind himself understood these notions to belong to logic, which has earned him the reputation of being a ‘logicist’ (though in a different sense than Frege and contemporary neo-logicists).2⁴ Because these set-theoretic notions were developed only after 1872, Dedekind did not yet have the mathematical tools to address isomorphisms between complete ordered fields and SZ does not have the modern set-theoretic flavour of Dedekind’s introduction of natural numbers (Dedekind 1888). Nevertheless, the system of cuts of rational numbers corresponds to a subset of the power-set of the rational numbers, and therefore Dedekind’s introduction of real numbers also relies on set-theoretic principles.2⁵ Because of his abstract approach with an emphasis on the structural relations between numbers instead of the properties of each individual number, Dedekind has rightly been described as a ‘methodological structuralist’ (Reck 2003). On the ontological status of Dedekind’s structures, there is an ongoing debate among scholars.2⁶

22 These are discussed by Haffner in (2018) and discussed further in a forthcoming paper on the notion of duality in Dedekind’s works. 23 See the account of ‘structural definitions’ in (Sieg and Schlimm 2017, 4–7); on Dedekind, see also Sieg and Schlimm (2005); Sieg and Morris (2018), Ferreirós (2017), and, for a more general account, Schlimm (2013). 2⁴ See Dedekind’s remarks in the Introduction to (Dedekind 1888). For a discussion, see BenisSinaceur (2015), Detlefsen (2011), Klev (2017), and Ferreirós (Forthcoming). 2⁵ For a discussion of the predicativity of Dedekind’s definition of cuts, see Yap (2009). 2⁶ See, e.g., Tait (1996), Corry (1996), Ferreirós (1999), Reck (2003, 2013), Yap (2009), Sieg and Morris (2018).

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266 dedekind on continuity

2.4 The Reception of Dedekind’s Account of Continuity Of the lectures on the differential and integral calculus that Dedekind held in Zürich, only the student notes of Heinrich Bechtold, who attended these lectures in the winter semester of 1861/2 have been published posthumously (Knus and Scharlau 1985). It might strike us as curious that in these lectures, which were attended mainly by engineers, Dedekind did not present a definition of continuity, but relied, when necessary, on the intermediate value theorem instead, which he could expect his audience to be familiar with (Knus and Scharlau 1985, 12). However, this is in accord with his own account, given in the Preface of SZ, where he writes that an appeal to geometric intuition is pedagogically useful and indispensable to make quick progress (see the quotation in section 2.1 above). The trade-off between rigorous foundations and the presentation of mathematics in teaching is also taken up in the correspondence with Lipschitz (Dedekind 1932, III, 471). How Dedekind approached continuous domains in his later research is presented in section 3. Among German mathematicians, Dedekind’s treatment of irrational numbers seems to have been accepted relatively quickly. Dedekind must have sent his booklet to Cantor, who thanked him for that in the first letter of the published correspondence, written on 28 April 1872 in Halle.2⁷ Herein Cantor remarks that his own point of view corresponds in substance with Dedekind’s and that the only difference lies in the conceptual introduction of the numbers. Cantor writes: ‘I fully agree that the essence of continuity consists in that which you point out as such’ (Noether and Cavaillès 1937, 12).2⁸ Similarly, Weber writes to Dedekind that of the foundational investigations of this kind, Dedekind’s ‘satisfied [him] most’ (letter from 18 May 1876; Scheel and Sonar 2014, 140). Two years later, he declares himself a complete convert to Dedekind’s approach and informs Dedekind that he presented it in his lectures on function theory.2⁹ Indeed, Weber introduces irrational numbers in terms of cuts in his influential Lehrbuch der Algebra (Weber 1895, 5). Mathematicians with a different philosophical outlook from Dedekind’s also readily adopted his definition of irrational numbers. The empiricist geometer Pasch, for example, used it in his introduction to analysis (Pasch 1882), which is based on lectures from 1878/9. He writes in the Preface: ‘With regard to [the 2⁷ Other colleagues who sent thank-you notes for the receipt of Dedekind’s book are Karl Hattendorff and Adolf Dauer (letters from 2 and 14 May 1872; in Dugac 1976, 163 and 194). 2⁸ The publication of SZ thus appears to have been the occasion for the start of fruitful and at times difficult correspondence between Dedekind and Cantor. Fraenkel (1932, 456) reports that Cantor and Dedekind met for the first time in 1872 by chance while vacationing in Gersau, Switzerland. See Ferreirós (1993) for a discussion of the relationship between Dedekind and Cantor. Dedekind’s definition of irrational numbers is discussed in (Cantor 1883, 566–567). 2⁹ Letter from 2 November 1878, in (Scheel and Sonar 2014, 207). See also the continuation of this exchange in (Scheel and Sonar 2014, 210, 212, 214–215).

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dedekind’s definition of continuity 267 concept of irrational numbers], I have followed the point of view of Dedekind (with minor modification), which corresponds most to the nature of the issue’ (Pasch 1882, IV). Pasch considers only the first halves of the cuts and calls them ‘number segments’ (Zahlenstrecken) (Pasch 1882, 3). In Pringsheim’s 1898 article on irrational numbers in the influential Encyclopedia of Mathematical Sciences Dedekind’s contribution is discussed alongside those of Cantor and Weierstrass. The need to postulate the continuity of the geometric line is here referred to as the ‘Cantor–Dedekind axiom’ (Pringsheim 1898). Not only professional mathematicians, but also high school teachers took an interest in Dedekind’s work. For example, the three ways of introducing irrational numbers by Weierstrass, Cantor, and Dedekind were discussed at a meeting of the Association of Teachers of Mathematics at Swiss High Schools (Vereinigung von Mathematiklehrern an schweizerischen Mittelschulen) by Conrad Brandenberger in 1902. Regarding their differences, he points out that Dedekind’s definition does not rest on arithmetical expressions and formal representations as do the others. He also notes that, on the one hand, while Cantor’s and Weierstrass’s definitions rely only on a subset of the rational numbers, Dedekind’s makes use of the set of all rational numbers. On the other hand, a real number is represented only by a single cut in Dedekind’s theory, but by more than one sum or fundamental sequence in the other theories. Brandenberger also remarks that ‘[t]he overview of the set of real numbers, provided that they are defined by cuts, leaves nothing to be desired’ (Brandenberger 1903, 206). Accordingly, Landau’s influential Grundlagen der Analysis (1930) presents the theory of real numbers based on Dedekind’s approach. In addition, Dedekind’s treatment of irrational numbers was adopted also by mathematicians in other European countries. For example, the Italian mathematician Ulisse Dini introduces irrational numbers using cuts of rational numbers in his textbook on the foundations of functions with real variables (Dini 1878, 6),3⁰ and his compatriot Federigo Enriques introduces a ‘postulate of continuity (by Dedekind)’ in his book on projective geometry (Enriques 1898, 78). In France, Jules Tannery mentions in his 1886 lectures on arithmetic that he learned about Dedekind’s SZ from Cantor, but remarks that he hasn’t seen the work yet. A few years later, however, Tannery presents irrational numbers on the basis of Dedekind’s work and writes: ‘I have, however, adopted here the fundamental idea of Mr. Dedekind, which seems to me to profoundly illuminate the nature of irrational number’ (Tannery 1894, 379). Without explicit reference to Dedekind, Camille Jordan uses cuts to introduce irrational numbers in his Cours d’Analyse de l’École Polytechnique (Jordan 1893, 2–8).31

3⁰ This is mentioned by Dedekind in the Preface of (Dedekind 1888). 31 For a discussion of Dedekind in the context of nineteenth- and early twentieth-century developments, see Bell (2005).

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268 dedekind on continuity In retrospect, Müller-Stach reports that Dedekind’s cuts ‘have been part of introductory lectures in analysis for many decades and still are to a certain extent’. However, he also remarks that foundational issues seem to have been disappearing from introductory lectures (Müller-Stach 2017, 30).

3. Other Treatments of Continuous Domains We have seen that Dedekind showed an early interest in giving an arithmetical foundation of the linear continuum, but his interest in rigorous, arithmetical definitions permeates his entire work and is clearly reflected in his mathematical practice. To illustrate this practice, we present some of Dedekind’s later work on continuous domains, in particular in relation to the work by Riemann.

3.1 Dedekind’s Early Interest in Riemann’s Work In an 1879 letter to Cantor, referring to, among other things, Riemann’s manifolds, Dedekind states that it would be ‘meritorious, if this whole ‘theory of domains’32 were presented ab ovo, without resorting to geometrical intuition’ and that one should ‘for example, define exactly and clearly the concept of a continuous line traced from a point a to a point b inside a domain G’.33 He then recalls an attempt, written in the late 1860s, related to the desire to provide a rigorous foundation to the so-called ‘Dirichlet principle’.3⁴ The text Dedekind refers to, entitled ‘Allgemeine Sätze über Räume’ (Dedekind 186?), which is only two pages long, can be found in his Nachlass and deals with what we would consider to be topological notions.3⁵ It illustrates how Dedekind’s attempts to rigorize, or arithmetize, parts of analysis related to continuity were not restricted to giving an arithmetical definition of irrational numbers, but that his concerns extend well beyond. In (Dedekind 186?), Dedekind begins by defining a ‘body’ (Körper3⁶) as a ‘system of points p, p′ , . . . ’, such that for each point p there exists a length 𝛿 with the property that all the points whose distance to p is less 32 Dedekind suggested using the shorter term ‘domain’, instead of ‘the more clumsy word “manifold” ’. 33 The passage of the letter to Cantor quoted here is reproduced in Noether’s editorial note to (Dedekind 186?) in (Dedekind 1932, II, 355). 3⁴ For a discussion of this principle, see (Bottazzini and Gray 2013, 271–275). 3⁵ This text, as well as Beweis und Anwendungen eines allgemeinen Satzes über mehrfach ausgedehnte stetige Gebiete (discussed in section 3.3 below), were published for the first time in 1931 in the second volume of Dedekind’s Gesammelte Werke. 3⁶ Here, the term ‘Körper’ is clearly used with a different meaning than in algebra and number theory (in which ‘Körper’ designates a field). It is a good indication for dating the manuscript to the late 1860s (see also Ferreirós 1999, 138), when Dedekind had not finished writing the Supplement X of Dirichlet (1871), in which the algebraic notion of field is introduced.

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other treatments of continuous domains 269 than 𝛿 also belong to the system (Dedekind 1932, II, 353). He then introduces the notion of a part of a body, and gives a theorem, which includes the definition of a ‘ball’ (Kugel): ‘All the points whose distance to a fixed point p is smaller than a given length 𝛿 form a body. It is named a ball, p is the centre and 𝛿 is the diameter’ (Dedekind 1932, II, 354). After defining the outside and inside of the body, as well as the notion of limit point and boundary, the text ends with a result stating that a system of points that are at the same time the boundary points of a body cannot themselves form a body. At the end of the 1860s, Dedekind was not only thinking about publishing Dirichlet’s lectures on potential theory (which were never published), but he was also already involved in the edition of Riemann’s Gesammelte Werke, a task that required a profound study of Riemann’s works and Nachlass. It is significant that the Dirichlet principle, at stake in the early work described above, plays an important role in Riemann’s works in function theory, which Dedekind also wished to reinterpret in a more rigorous manner. Dedekind wrote to his coeditor Heinrich Weber that he would only truly understand Riemann’s works if he could ‘overcome in [Dedekind’s] way, with the rigour that is customary in number theory, a whole series of obscurities’ in them (letter to Weber, 11 November 1874; Scheel and Sonar 2014, 50). The in-depth study of Riemann’s works involved in the edition of his Gesammelte Werke seems to have left a significant imprint on both Dedekind and Weber, who continued to discuss Riemannian topics in their letters, to the point of working together on the elaboration of a new foundation for Riemannian function theory. This discussion lasted two years3⁷ and resulted in Theorie der algebraischen Funktionen einer Veränderlichen (Theory of algebraic functions of one variable) (1882). How Dedekind’s idea of arithmetization is employed in this text is discussed next.

3.2 Arithmetical Treatment of Riemann Surfaces 3.2.1 Motivations for Arithmetizing Riemann Surfaces In Dedekind and Weber (1882), the authors propose an entirely new definition for the notion of the Riemann surface, based on an ‘analogy’ with number theory that allows them to transfer Dedekind’s concepts of field, module, and ideal to function theory. The transfer of Dedekind’s number-theoretical insights and methods gives them the tools to reformulate the fundamental notions of Riemannian function theory within a new arithmetical framework. The core concept of Riemann’s function theory is what we call today the ‘Riemann surface’. It offers a characterization of the algebraic functions in terms of their behaviour

3⁷ See (Scheel and Sonar 2014, 220–271).

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270 dedekind on continuity at singular points and at the boundary of the surface. Riemann’s theory draws an intimate link between analytical and geometrical conceptions of functions. The concept of a Riemann surface allows us to understand the multi-valuedness of algebraic functions in geometric terms. Rather than defining a function in the complex (Argand–Cauchy) plane, Riemann proposed to consider a surface with ‘piled’ sheets, each corresponding to a branch of the multi-valued function. The arithmetical rewriting of Riemann’s theory by Dedekind and Weber (1882) thus implies a new arithmetical definition of a continuous object—and indeed one whose singularities are crucial and must be characterized accurately.3⁸ In this process, the definition of a point of a Riemann surface is explicitly highlighted as being essential in the introduction of their paper. Dedekind and Weber explain that in order to propose new bases for Riemannian function theory, one important task presents itself as essential: to supply mathematicians with a completely precise and rigorous definition of the ‘point of a Riemann surface’ that can also serve as a basis for the investigation of continuity and related questions. (Dedekind and Weber 1882, 241)3⁹

3.2.2 Analogy with Number Theory Dedekind and Weber rely on an analogy between the theories of algebraic numbers and of algebraic functions, which allows them to define the notion of field of functions and set up the framework for what Weber calls the ‘theory of ideals of algebraic functions’ (letter from Weber, 2 February 1879; Scheel and Sonar 2014, 220). This is justified by the fact that a field of algebraic functions ‘coincides completely’ with a Riemann surface. To work out this coincidence, they take as basis a generalization of the theory of rational functions of one variable, in particular of the theorem according to which every polynomial of one variable admits a decomposition into linear factors. (Dedekind and Weber 1882, 238, transl. altered)

An algebraic function 𝜃 of z (z a complex variable) is defined as satisfying an irreducible polynomial equation a0 𝜃 n + a1 𝜃 n−1 + . . . + an−1 𝜃 + an = 0,

3⁸ See Haffner (2017) for a more thorough presentation of their work in relation to Dedekind’s arithmetization and conception of arithmetic. 3⁹ The English translation comes from (Stillwell 2012, 42–43). All other translations from Dedekind and Weber (1882) are also from Stillwell, with slight modifications when noted.

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other treatments of continuous domains 271 in which the coefficients a0 , a1 , . . . , an are polynomials in z with rational coefficients and without common divisors. It is thus a generalization of the concept of a rational function, a quotient of two polynomials with numerical coefficients. Dedekind and Weber’s aim is to treat the general case, that is, to provide a treatment valid for any genus p.⁴⁰ This general case is related to the surfaces of genus 0 ‘in the same way that general algebraic numbers are related to rational numbers’.⁴1 This comparison guides the ‘transfer’ (Übertragung) of methods from number theory to function theory. In order to develop Dedekind’s theory of ideals in function fields, Dedekind and Weber define for functions counterparts of the notions used in number theory as follows: Number theory

Function theory

rational integers rational numbers algebraic numbers algebraic integers

polynomials rational functions algebraic functions integral algebraic functions

The integral algebraic function, which corresponds to the notion of integer, is defined analogously to the way in which the algebraic integers are defined.⁴2 To treat the general case, Dedekind and Weber transfer and adapt to functions of one complex variable the number-theoretical methods ‘derived from Kummer’s creation of ideal numbers’, which proved to be ‘most successful’ in Dedekind’s own works in number theory (Dirichlet 1871; Dedekind 1877, 1879). To do so, the first step is the definition of a field of algebraic functions: In an analogy with number theory, a field of algebraic functions is understood to be a system of such functions with the property that application of the four fundamental operations of arithmetic to the functions of this system always leads to functions of the same system. (Dedekind and Weber 1882, 239, transl. slightly altered)

3.2.3 The Theory of Ideals of Functions Fields of algebraic functions set up the framework for the development of ideal theory, the most important tool in Dedekind and Weber’s work. An ideal of functions is defined with the same two necessary and sufficient conditions as an ⁴⁰ The genus of a connected, orientable surface is an integer representing, roughly speaking, the number of ‘holes’ it has. For example, the sphere has genus 0, and the torus has genus 1. ⁴1 Recall the definition of algebraic numbers: a number 𝜃 is an algebraic number if it satisfies an equation 𝜃 n + a1 𝜃 n−1 + . . . + an−1 𝜃 + an = 0 of finite degree n, whose coefficients ai are rational numbers. It is an algebraic integer if the coefficients ai are integers. ⁴2 See (Geyer 1981, 116) for a modern presentation of Dedekind and Weber’s concept of integral algebraic function.

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272 dedekind on continuity ideal of numbers (see section 2.3.1 above): a collection 𝔞 of algebraic functions is an ideal if and only if I. The sum and difference of every pair of functions of 𝔞 are again functions of 𝔞. II. The product of any function in 𝔞 by any integral function of the field is again a function of 𝔞.⁴3 The same notion of divisibility and product for ideals is defined as before, and the study of divisibility laws for functions is transposed to the study of divisibility laws for collections of functions. Thus, the theory of ideals of functions is developed in a strikingly similar fashion to what was done in number theory in (Dedekind 1877, 1879). Definitions, properties, and theorems are often similar word for word, with ‘function’ in place of ‘number’. Dedekind and Weber’s aim, in introducing ideals in function theory, is strictly to be able to analyse and unfold the arithmetical properties and laws of function fields, as they are the properties exploited in defining and studying the Riemann surface, once the correspondence on which they are relying has been established. Indeed, their arithmetical rewriting of Riemann’s theory is organized in two parts. First, they develop an analysis of a field of algebraic functions of one complex variable, which they consider as strictly formal because it does not consider numerical values for the functions. Next, they define the Riemann surface and related notions (e.g., the genus), and reformulate (and prove) several known theorems of the theory (e.g., the Riemann– Roch theorem). In the first ‘formal’ part, Dedekind and Weber study the arithmetical laws and theorems verified by ideals, following the path opened by the definition of a new notion of divisibility. Once this definition is introduced, the study of these laws and properties can follow the lines of elementary number theory. A prime ideal is ‘an ideal 𝔭 different from 𝔬’ such that ‘no other ideal than 𝔭 and 𝔬 divides 𝔭’ (Dedekind and Weber 1882, 266). The arithmetical properties obtained thus appear of a striking simplicity and familiarity, for example: Each ideal different from 𝔬 is either a prime ideal or else uniquely expressible as the product of prime (and only prime) ideals. (Dedekind and Weber 1882, 271)

It is important to note that the methods of proof are also similar to the ones used in number theory. Dedekind and Weber thus set up an arithmetical framework, in which they can study the ideals of functions using the methods and ways of writing customary in number theory. Their arithmetical approach, and especially ⁴3 See (Dedekind and Weber 1882, 264).

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other treatments of continuous domains 273 the reformulation of inclusion of ideals as a divisibility relation, is not simply an ad hoc introduction of familiar terminology, but it reflects the reshaping of the whole research project in an arithmetical form.

3.2.4 The New Definition of the Riemann Surface It is important to underline that the study of the arithmetical structure of the field of functions (which is completed with algebraic considerations of norms, basis, and discriminant) is not Dedekind and Weber’s final goal. Instead, they proceed to define the notion of the Riemann surface. The first step for the new definition of the Riemann surface is the definition of a point. A point is defined as a mapping between the field of functions studied and what we call the Riemann sphere (i.e., the numerical constants and ∞) (Dedekind and Weber 1882, 293–294). Dedekind and Weber explicitly state that a geometrical or spatial representation of the point should be dismissed, and that the point must only be understood as the attribution of numerical values to functions. The crucial property is that points are in a one-to-one correspondence with the prime ideals of the field. A point is said to generate (erzeugen) the corresponding prime ideal. With this one-to-one correspondence, Dedekind and Weber can use their ‘formal’ results to pursue their rewriting (and proofs) of the properties of the Riemann surface. Dedekind and Weber indicate how it is possible to obtain ‘all the existing points exactly once’, thus forming a collection of all the points. This collection, which contains all the points exactly once, is called a ‘simple totality’ by Dedekind and Weber. However, this collection does not possess the structure of a Riemann surface: it does not yet describe the singularities of the functions, nor the multiplicity and ramification of the surface. In order to complete the definition, Dedekind and Weber define complexes of points, which they call ‘polygons’:⁴⁴ We give the name polygons to complexes of points, which may contain the same point more than once and denote them by 𝔄, 𝔅, ℭ, . . . We also let 𝔄𝔅 denote the polygon obtained from the points of the polygons 𝔄 and 𝔅 put together, in such a way that a point 𝔓 that appears r-tuply in 𝔄 and s-tuply in 𝔅, appears (r + s)-tuply in 𝔄𝔅. (Dedekind and Weber 1882, 299, transl. altered)

Polygons are extremely useful in the proofs. From the definition of the product of polygons one can deduce the ‘meaning of the power of a point 𝔓r ’: it is a point repeated r times on itself. Here, one starts to see the ramification of the surface. Besides, it is possible to decompose a polygon into the product of all its points,

⁴⁴ Polygons correspond to what are today called ‘positive divisors’.

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274 dedekind on continuity each with its own power (i.e., the number of times it appears in the polygon). This provides a decomposition of the polygon into prime elements, from which is deduced a correspondence between polygons and composed (non-prime) ideals. The polygon r

r

𝔄 = 𝔓r 𝔓11 𝔓22 . . . corresponds to the ideal r

r

𝔞 = 𝔭r 𝔭11 𝔭22 . . . , whose prime factors correspond to the points of the polygon.⁴⁵ Thereby the oneto-one correspondence between ideals and polygons is established, and: the laws of divisibility of polygons agree completely with the laws of divisibility of integers and of ideals. Points play the role of prime factors. (Dedekind and Weber 1882, 299, transl. slightly altered)

The one-to-one correspondence justifies the ‘complete coincidence’ between a field of algebraic functions and a Riemann surface for this class of functions. It is also the justification for the possibility of using, for Riemann surfaces, the results obtained for ideals of functions. For example, define the greatest common divisor and the least common multiple of two polygons, with a definition modelled on the definition of the GCD and LCM of two numbers in terms of powers of their prime factors. With the above definitions in place, Dedekind and Weber have everything they need to define the Riemann surface with its structure. The Riemann surface thus defined is called the ‘absolute Riemann surface’. It is defined as the product of all the polygons ‘moving through’ ℂ ∪ {∞}, that is, taking successively all the values of the Riemann sphere. In this way, all the points with their (possible) multiplicities are obtained. This product of polygons is ∏ 𝔄 = Tℨz in which T is the simple totality of all the points once, and ℨz is a finite polygon called a ‘ramification polygon’ of T in z (Dedekind and Weber 1882, 301). The ramification polygon provides a characterization of the ramification of the surface. Its points are the ramification points. This polygon corresponds to a determinate ideal, which Dedekind and Weber defined in the first part of the paper, and which is called the ramification ideal.⁴⁶

⁴⁵ The decomposition in prime elements for ideals is featured on the postal stamp of the German Democratic Republic dedicated to Dedekind on his 150th anniversary. ⁴⁶ See (Stillwell 2012, 99–103) and (Haffner 2017).

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other treatments of continuous domains 275 In the last step of the definition of the Riemann surface, the characterization that was, until then, a discrete characterization becomes a continuous one. And the correspondence between the surface and the field implies that all singularities and properties of the surface can be expressed (and studied) in terms of ideals and, in fact, of divisibility of ideals, thus providing a thoroughly arithmetical treatment of the Riemann surface. Indeed, the change of framework made by Dedekind and Weber’s rewriting of Riemann’s theory implies a reformulation of all the notions of Riemann’s theory: genus, ramification number, number of double points, etc. are all rewritten in the algebraico-arithmetical framework using polygons, as are properties related to the differentials of the functions, which are intimately related to continuity. Analogously to how the desire to provide a rigorous foundation for the differential calculus prompted the writing of SZ and a new and (from Dedekind’s viewpoint) improved definition of the real numbers, the desire to provide a rigorous foundation of Riemannian function theory, a large part of which relies on differential calculus on the surface, prompted the introduction of a new and (from Dedekind and Weber’s viewpoint) improved definition of the Riemann surface. These works show how Dedekind’s foundational and methodological worries and reflections are deeply intertwined with his mathematical practice. The resort to arithmetical methods and tools, and indeed to a thorough arithmetization, is characteristic of Dedekind’s approach. Dedekind and Weber’s paper had a sparse and delayed reception. Dedekind himself saw the risk that his contemporaries might not react as well as he hoped to their paper (Scheel and Sonar 2014, 271). In his Lehrbuch der Algebra (1895), Weber himself departed considerably from the approach they used in 1882. Rather than using modules and ideals, Weber used the notion of ‘Functionale’, which is closer to Kronecker’s approach. Hilbert used some of their ideas for his Nullstellensatz. In Brill and Noether (1892), the authors explicitly exclude Dedekind and Weber’s theory (as well as Kronecker’s) from their report on algebraic functions. They consider Dedekind and Weber’s theory mainly as an application of number theory to function theory. In general, the adoption of ideal theory as a basis for a treatment of Riemannian function theory did not stir up much enthusiasm. A first significant response can be found in Hensel and Landsberg’s 1902 book dedicated to Dedekind (Hensel and Landsberg 1902). As emphasized by Dieudonné, the algebraic tools necessary to develop Dedekind and Weber’s approach in dimensions higher than one were only developed ‘very progressively between 1890 and 1950, not systematically, and often by algebraists whose concern was only superficially related to algebraic geometry’ (Dieudonné 1974, 124). In particular, it required the development of the theory of commutative fields (founded by Steinitz in 1910) and developments of commutative ring theory (by Noether and Krull from 1927

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276 dedekind on continuity onwards). A generalization of Dedekind and Weber’s ‘abstract Riemann surface’ (now known as the Zariski–Riemann surface) was then given by Zariski in 1940.⁴⁷

3.3 Set-Theoretical Treatments of More General Continuous Domains Dedekind’s aim of developing rigorous foundations to the study of continuity and related questions is also pursued in Beweis und Anwendungen eines allgemeinen Satzes über mehrfach ausgedehnte stetige Gebiete (1892). In this unpublished paper, he proposes an application of ‘his method of proof ’ to ground analysis for functions of several real variables. He uses the conceptual apparatus developed in Dedekind (1888) (i.e., sets, mappings, etc.) to develop an approach similar to cuts, yet more general. Briefly recalling his definition of real numbers in terms of cuts, as well as other possible definitions such as Cantor’s, Dedekind explains that the preference for a certain method of proof or definition is ‘often just a matter of habit or taste’. He takes this remark as an opportunity to expose ‘as an example of [his] own proof method’ a few results which can be ‘applied to known results’ in analysis. The reference to (Dedekind 1888) serves as background to introduce a number of set-theoretical notions which are used as a basis for his research. As in his previous foundational works, but also in (Dedekind and Weber 1882), Dedekind starts by setting up the foundations of a more general, here set-theoretic, conceptual arsenal on which he relies for the definition of new tools and the proof of known theorems—the ‘many applications’ he mentions in his introduction. In addition to the basic first definitions of (elementary) set theory, he introduces the notion of ‘partial cut’ (Teilschnitt): As a partial cut 𝜑 of a system S, I understand a division of all its parts into two kinds, namely in pure [rein] and impure [unrein] parts, which satisfy the three following conditions: 1. Each part of S is either pure or impure, but never both at the same time. 2. Each part of a pure part is pure. 3. The sum [i.e., union] of two pure parts is pure. (Dedekind 1892, 358)

The last two conditions can be expressed as: ‘The sum of two parts is impure if and only if at least one of them is impure.’ If one interprets 𝜑(A) as = 0 if A is impure and = 1 if A is pure, then this result can be stated as 𝜑(A + B) = 𝜑(A)𝜑(B). We find, here again, the idea of a morphism. This notion of Teilschnitt allows for a generalization of the principle of continuity given in Dedekind (1872). Dedekind introduces several notions that correspond to notions developed by Hausdorff

⁴⁷ For a more detailed account, see Dieudonné (1974).

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conclusion 277 (e.g., his Berührpunkt, abgeschlossene Menge),⁴⁸ and then defines the notions of ‘deepest’ and ‘highest’ points, which are used to state and prove, by the end of his fifteen-page paper, the theorem: If a continuous function in T takes positive as well as negative values, then there exists a deepest point in which the function vanishes. (Dedekind 1892, 368)

In this theorem, one recognizes a more general version of the intermediate value theorem.

4. Conclusion We have presented above Dedekind’s seminal treatment of continuity in terms of cuts that he developed in 1858. Twenty years after the publication of SZ (1872), he confidently maintained that he had realized his Principle of Continuity to be a secure (sicher) and at the same time simple foundation for infinitesimal analysis and for the study of all continuous domains and [he] always applied it to all of the most important questions relating to them with the desired success. (Dedekind 1932, II, 356)

The fact that his definition was quickly adopted by many mathematicians, despite the fact that alternative equivalent definitions were also given around the same time by Cantor and Weierstrass, confirms his evaluation. Moreover, we have argued that Dedekind’s definition was not the result of an isolated mathematical effort, but that it was deeply rooted in his rigorous and arithmetical approach to mathematics. After the further development of the notions of sets and mappings in Was sind und was sollen die Zahlen? (1888), Dedekind incorporated these notions into his general methodology, thus making his approach appear more abstract. The consideration of sets of objects as mathematical objects themselves is an integral part of his definitions of cuts and ideals. At the same time, Dedekind’s insistence on arithmetization (understood in a much broader sense than just the reduction of mathematical notions to the positive integers, as Kronecker would have it) and the rejection of geometrical intuition as the basis for mathematical arguments, were guiding principles for his mathematical practice. We have illustrated this in Dedekind’s reshaping of Riemann’s work that does away with geometrical intuitions and that allows for a treatment of continuous domains that is analogous, both with respect to definitions and to methods of proof, to his work in number theory.

⁴⁸ See the note by Noether (Dedekind 1892, 369–370) for links to Hausdorff ’s set-theoretic topology.

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278 dedekind on continuity

Acknowledgements The authors would like to thank the participants of the conference In Memoriam: Richard Dedekind (1831–1916), held in Braunschweig, Germany on 6–8 October 2016 for many discussions that helped to shape this paper. We also would like to thank Sorin Bangu, Pascal Bertin, Viviane Fairbank, José Ferreirós, Akihiro Kanamori, Ansten Klev, Daniel Lovsted, Daniel B. Miller, Paul Rusnock, Stewart Shapiro, Hourya Sinaceur, and Tristan Tondino for comments on an earlier version of this paper.

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references 281 Klev, A. (2017). Dedekind’s logicism. Philosophia Mathematica, 25(3):341–368. Knus, M.-A. and Scharlau, W., editors (1985). Richard Dedekind. Vorlesung über Differential- und Integralrechung 1861/62, volume 1 of Dokumente zur Geschichte der Mathematik. Deutsche Mathematiker Vereinigung, Vieweg, Braunschweig, Wiesbaden. Krull, W. (1932). Allgemeine Bewertungstheorie. Journal für die reine und angewandte Mathematik, 167:160–196. Kummer, E. E. (1856). Theorie der idealen Primfactoren der complexen Zahlen, welche aus den Wurzeln der Gleichung von = 1 gebildet sind, wenn n eine zusammengesetzte Zahl ist. Math. Abh. köngl. Akad. Wiss., 1–47. Repr. in Collected Papers, vol. 1, Contributions to Number Theory, 583–629. Ed. A. Weil. Springer, 1975. Landau, E. (1930). Grundlagen der Analysis. Akademische Verlagsgesellschaft, Leipzig. Müller-Stach, S., editor (2017). Richard Dedekind: Was sind und was sollen die Zahlen? Stetigkeit und Irrationale Zahlen. Springer Spektrum, Berlin. Noether, E. (1927). Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern. Mathematische Annalen, 96:26–61. Reprinted as No. 30 in Gesammelte Abhandlungen, pp. 493–528. Noether, E. and Cavaillès, J., editors (1937). Briefwechsel Cantor-Dedekind. Number 518 in Actualités Scientifiques et Industrielles. Hermann & Cie, Paris. Pasch, M. (1882). Einleitung in die Differential- und Integralrechnung. B. G. Teubner, Leipzig. Petri, B. and Schappacher, N. (2007). On arithmetization. In Goldstein, C., Schappacher, N., and Schwermer, J., editors, The Shaping of Arithmetic after C. F. Gauss’s Disquisitiones Arithmeticae, pages 343–374. Springer, Berlin, Heidelberg. Pringsheim, A. (1898). Irrationalzahlen und Konvergenz unendlicher Prozesse. In Meyer, W. F., editor, Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, volume 1: Arithmetik und Algebra, Part 1, chapter I A 3, pages 49–146. B. G. Teubner, Leipzig. Reck, E. (2003). Dedekind’s structuralism: An interpretation and partial defense. Synthese, 137:369–419. Reck, E. (2013). Frege or Dedekind? Towards a reevaluation of their legacies. In Reck, E., editor, The Historical Turn in Analytic Philosophy, pages 139–170. Palgrave Macmillan, London. Riemann, B. (1876). Gesammelte mathematische Werke und wissenschaftlicher Nachlass. Teubner, Leipzig. Edited by Heinrich Weber, with the assistance of Richard Dedekind. 2nd revised edition 1892, with a supplement added in 1902, repr. by Dover, New York, 1953. Scharlau, W. (1986). Rudolf Lipschitz: Briefwechsel mit Cantor, Dedekind, Helmholtz, Kronecker, Weierstrass und anderen, volume 2 of Dokumente zur Geschichte der Mathematik. Deutsche Mathematiker Vereinigung, Vieweg, Braunschweig, Wiesbaden.

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282 dedekind on continuity Scheel, K. and Sonar, T. (2014). Der Briefwechsel Richard Dedekind: Heinrich Weber. W. de Gruyter, Berlin. Schlimm, D. (2013). Axioms in mathematical practice. Philosophia Mathematica, 21(1):37–92. Sieg, W. and Morris, R. (2018). Dedekind’s structuralism: Creating concepts and making derivations. In Reck, E., editor, Logic, Philosophy of Mathematics, and their History: Essays in Honor of W. W. Tait. College Publications, London. Sieg, W. and Schlimm, D. (2005). Dedekind’s analysis of number: Systems and axioms. Synthese, 147(1):121–170. Sieg, W. and Schlimm, D. (2017). Dedekind’s abstract concepts: Models and mappings. Philosophia Mathematica, 25(3):292–317. Stein, H. (1990). Eudoxos and Dedekind: On the ancient Greek theory of ratios and its relation to modern mathematics. Synthese, 84:163–211. Steinitz, E. (1910). Algebraische Theorie der Körper. Journal für die reine und angewandte Mathematik, 137:167–309. Stillwell, J. (2012). Theory of Algebraic Functions of One Variable. Translation of Dedekind and Weber (1882). History of Mathematics. American Mathematical Society, Providence. Tait, W. W. (1996). Frege versus Cantor and Dedekind: On the concept of number. In Schirn, M., editor, Frege: Importance and Legacy, pages 70–113. DeGruyter, Berlin. Tannery, J. (1886). Introduction à la théorie des fonctions d’une variable. Hermann, Paris. Tannery, J. (1894). Leçons d’arithmétique théorique et pratique. Armand Colin & Cie, Paris. Weber, H. (1895). Lehrbuch der Algebra, volume 1. Vieweg, Braunschweig. Yap, A. (2009). Predicativity and structuralism in Dedekind’s construction of the reals. Erkenntnis, 71(2):157–173. Zariski, O. (1940). Local uniformization of algebraic varieties. Annals of Mathematics, 41:852–896.

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11 What Is a Number? Continua, Magnitudes, Quantities Charles McCarty

For Father Adam Olszewski As well as to continuous water there corresponds a continuous wetness, so to a continuous magnitude there corresponds a continuous number. [Stevin 1585 3] First, I set out four propositions. Then, I return to consider each individually. 1. I know that Herman Weyl called his book Das Kontinuum, The Continuum. Even so, there is no such thing as the continuum. In reality, there exists a multiplicity of continuous phenomena without any single, readily discernible, analytical substructure both common and exclusive to all. 2. The most successful effort hitherto to close off much-needed discussion on the natures of continua, the Cantor–Dedekind Axiom, is—as its earliest proponents claimed—unprovable, but not because, as they thought, the axiom is a judgement from veridical intuition. Rather, the axiom is disprovable. 3. An important theoretical undertaking, one straddling the divide between a priori and a posteriori, has largely been omitted from foundational studies of continua. I refer to axiomatizations of the notion ‘continuous magnitude’, die axiomatische Grössenlehre. 4. Otto Hölder’s Theorem [1901] on measurement establishes a reciprocity between axioms for magnitudes and the extents of constructed numerical domains by which the former constrain the latter, a constraint that can be exploited to test proposed arithmetizations empirically.

Part 1 I begin with a brief survey of efforts to define the term ‘continuum’. I maintain that this survey—with its lack of unanimity—could be extended indefinitely.

Charles McCarty, What Is a Number?: Continua, Magnitudes, Quantities In: The History of Continua: Philosophical and Mathematical Perspectives. Edited by: Stewart Shapiro and Geoffrey Hellman, Oxford University Press (2021). © Stewart Shapiro and Geoffrey Hellman. DOI: 10.1093/OSO/9780198809647.003.0012

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284 what is a number? continua, magnitudes, quantities (a). In his textbook of 1907—highly influential in its day—E.W. Hobson defined ‘continuum’ as follows. We give the name continuum to an aggregate which possesses the two properties of being connex [i.e., dense] and being perfect [Cauchy complete]. This is in the first instance taken to be the definition of the meaning of the word continuum as it is used in Analysis. [Hobson 1907 50]

Hobson’s term ‘continuum’ expressed an order-theoretic concept, applying to any system of points, lines, numbers, or functions endowed with a suitable order having the requisite properties. Even Brouwer’s intuitionistic real numbers manifest both density and Cauchy completeness. (b). Abraham Fraenkel, in his famous Einleitung, defined ‘continuum’ thus: The set of all points on a ‘continuous’ interval or of a ‘continuous’ straight line is designated simply as ‘continuum’. [Fraenkel 1946 60–61]

This time, a continuum is a geometrical and set-theoretic item, a set of points from or on a line. (c). In Stephen Willard’s standard textbook General Topology, one finds, A continuum is a compact, connected Hausdorff space. [Willard 1970 203]

Here too, the continuum fails to exist. Continua comprise a whole category of topological spaces, with neither the entire collection of Dedekind’s reals nor such an interval as open (0, 1) falling into it. (d). Now, we go lowbrow. In response to the query ‘continuum meaning’, Google generated the set of real numbers

as its second option. Of course, there is no one such set. (I return to the first Google definition anon.) (e). The promise Hermann Weyl made on the title page of The Continuum [1918] is not kept once the reader ventures within the book’s covers. Weyl tried to distinguish—within a single paragraph on his page 67—between ‘an intuitively given continuum’, such as that of ‘the most fundamental continuum, phenomenal (as opposed to objective) time’, and ‘the concept of number’. So, the latter, Weyl’s arithmetic continuum, is no set, class, or category of spaces, but a concept.

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part 1 285 (f). Outside of analysis, the recognized extent of continuous phenomena grows even wider. It is interesting to note that all possible sensations of color, of tone, and of temperature constitute as many groups of qualitative continua. By ‘continuum’ is here meant a series of presentations changing gradually in quality, i.e., so that any two differ less the more they approximate in the series. [Ward 1886 56]

(g). By the way, this is in the same ballpark as the sense of ‘continuum’ Google churns out as a first option: a continuous sequence in which adjacent elements are not perceptibly different from each other, although the extremes are quite distinct.

I offer three remarks concerning the manifold variety of continua and attempted definitions of the word for that variety. (1) Premiere among the many phenomenal continua is a multidimensional continuum (or even continua) of bodily motion gauged internally, a proprioceptive space [Heuer 2003 329]. Proprioceptive and kinaesthetic sensors in our muscles and joints map that space. Close your eyes, raise your arm, swing it wide, as if to toast the bride and groom. This motion of your arm and hand passes before your inner eye as perfectly smooth, without pause, jerk, bump, or stop-and-restart. Your arm does not disappear, for even a single nanosecond, to reappear, farther along its trajectory, one wink-of-an-eye later. Its forward path does not dissolve into separate pulses, but flows gently—like Afton. Continuous proprioceptive space stands in a homeomorphic but not isometric relation to intuited physical space [Longo & Haggard 2010] [Penfield & Boldrey 1937]. Incidentally, I find the proprioceptive body map of Penfield and Boldrey eerily reminiscent of ancient seafarers’ maps of the eastern Mediterranean. (2) In addition, there are continua of a strictly intelligible variety wherein relative locations are picked out thanks in part to the intelligible or noetic matter detected by Aristotle [Aristotle 1984 1635, 1637] and, later, by Proclus [Proclus 192 54.14–56.22]. It is against the backdrop of continua of this sort that, for example, one triangle in a plane gets distinguished, before the mind’s eye, from a different triangle congruent to the first in the same plane. Without continua of noetic matter, it would make no sense to speak of two particular copies of, say, the standard natural numbers, one within a designated model of set theory, another wholly outside the first and in a second, isomorphic model of the same theory. The models and the copies maintain their separation, individuality, and ‘outsidedness’ within a continuous, intelligible space.

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286 what is a number? continua, magnitudes, quantities (3) It concerns me greatly that phenomenal and intelligible continua seem to have vanished from recent studies in the foundations of mathematics.

Part 2 Right from the opening sections of his Stetigkeit und Irrationale Zahlen [Continuity and Irrational Numbers], Dedekind insisted upon his version of the Cantor– Dedekind Axiom, roughly, that gaps discernible via analytical or geometric means in intuitional objects such as geometric lines are to be filled by taking appropriate nongeometrical limits and inserting corresponding points, generated by the limits, into the gaps. In that way, he claimed, a resulting geometric linear continuum, so extended, becomes order-isomorphic to the Dedekind real numbers. By Dedekind’s lights, we thereby make the intuited geometrical line continuous by imposing, in thought, the axiom, so ensuring a perfect correspondence between geometric points on a line and Dedekind’s cuts. We simply ‘plug the gaps’. Perhaps Hermite’s 1873 proof that Napier’s number e is transcendental [Hermite 1912] (given earlier results of [Wantzel 1837]) would exemplify for us Dedekind’s idea of an analytically discernible gap in the line. It follows from these theorems that e among the real numbers corresponds, given the usual unit on the line, to a gap in the Euclidean linear continuum; one could not construct a point with displacement e from the origin employing strictly Euclidean tools. I emphasize that, according to Dedekind, the Cantor–Dedekind Axiom and its consequence, the Upper Bound Theorem (to be introduced in a moment), were intended to rest for their truth ultimately upon an imposition on the character of objects of intuition, continuous geometrical magnitudes, an intuition that all humans supposedly share, at least subconsciously. (Recall that, for Kant, axioms always report upon matters exhibited in intuition [Kant 1974 117].) Here again is Dedekind: I am glad if every one finds the above principle [the axiom] so evident and so concordant with his own ideas of a line; for I am in no position to adduce any proof of its correctness, and no one is in that position. The assumption of this property of the line is nothing else than an axiom through which we first award its continuity to the line, through which we think continuity into the line. [Dedekind 1872 18]

From the Cantor–Dedekind Axiom, interpreted as Dedekind wished, one deduces easily an Upper Bound Theorem for the Euclidean linear continuum: Every nonempty set of points in a geometric linear continuum all lying to the left of a single point has a single point bounding it most closely on the right.

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part 2 287 Dedekind’s contemporary, mathematician and philosopher Paul du BoisReymond (1831–89), demonstrated that the axiom and attendant theorem are no truths, but express, in his words, eine nachträgliche Anpassung [Du Bois-Reymond 1877 150], an adjustment ex post facto, resultant upon any attempt to shoehorn intuitions of multivalent geometric continuity into a single arithmetic mould. The axiom is not merely mathematically untrue; it is an incorrect representation of our thoughtful dealings with continua, magnitudes, and quantities. According to du Bois-Reymond, no direct insight into continuous magnitude alone can yield up conclusive reasons for the axiom, since there are genuine continua that are, as far as the geometric eye and geometric mind can see, indistinguishable from the continuous line and yet, for them, the theorem and, hence, the axiom fail demonstrably. In effect, Dedekind had already admitted that continuity in the geometrical line may well be an adjustment ex post facto. However, he did not admit that it is both a falsification of and a mathematically unfair limitation upon the variety and exuberance of continuous phenomena. One such continuum, an extension of the Euclidean line, arises from an analysis of the rates of growth of functions that du Bois-Reymond set out as early as his [1870–1]. He treated mathematical functions over the real numbers as first-class entities, on the same ontological plane as real or fractional numbers themselves. By comparing their rates of growth, he defined orders of infinity, conceived as infinitesimal, finite, and infinitary magnitudes sorted into distinct infinite sizes. He came thereby to recognize that there is a higher infinity of those sizes. In his On asymptotic values, infinitary approximations, and infinitary solutions of equations [du Bois-Reymond 1875, 363–4], he wrote, I have distinguished the various infinities of functions according to their magnitudes, so that they form a domain of magnitudes (the infinitary). This is achieved by laying down that the infinity of f(x) is considered greater than that of g(x) or equal to g(x) according to whether the [limit, as x becomes arbitrarily large, of the] quotient f(x)/g(x) is infinite or finite, respectively . . . Between the two domains [the infinitary and that of ordinary real numbers] there exist a number of analogies. Instead of the numbers as fixed markers in the number domain one finds, in the infinitary domain, an unbounded number of simple functions: exponentials, powers, and logarithmic functions that afford fixed points of comparison . . . These functions serve as numbers.

On the original conception, du Bois-Reymond’s rates of growth were not linearly ordered under the relation just described. In his monograph [1954] on du BoisReymond’s ideas, G.H. Hardy reinstated linearity by restricting consideration to piecewise algebraic combinations of logarithmic and exponential functions he termed ‘L-functions’. (See his Theorem 13, page 17.) Following Hardy, I do not here reproduce du Bois-Reymond’s own treatment, but offer a reconstruction.

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288 what is a number? continua, magnitudes, quantities Consider rates of growth of Hardy’s L-functions, all strictly positive, monotonically increasing, and real-valued. For L-functions f and g, one says that f ≺ g, read ‘f is of (strictly) lesser infinity than g’ just in case limx→∞ [g(x)/f(x)] exists and is equal to ∞. I can put it another way: f ≺ g if and only if, for every natural number n, there is a right-cofinal segment s of the positive real numbers such that, for any x from s, g(x) > n × f(x). Equivalently, one can think of functions f as nonstandard values that generate linear 𝜔-chains by repeated function addition, as follows. f < f + f < 3 × f < 4 × f... Then, think of g as having strictly greater infinity than f when g lies above, at least eventually, every member of the 𝜔-chain determined by f. We know therefore that, as orders of infinity, 1 ≺ x ≺ x2 ≺ x3 ≺ x4 . . . Du Bois-Reymond not only took these functional orders to be legitimate finite, infinite, and infinitesimal quantities, analytically discovered, but also, and importantly, held that we have at our disposal geometrical insight into their properties as comprising a linear continuum. That insight can be conveyed kinematically: envision the rate of growth of one function, in relation to others, by comparing the graphs or curves associated with them. Make the comparison vivid by cutting the graphs with a line parallel to the y-axis. Now, to visualize relative rates of growth, watch the motion of the intersection point between that vertical line and the function graphs when the former is moved continuously and rapidly rightwards. One sees, for example, the curve y = x2 increasing much more rapidly—the intersection point racing up the vertical line—than the line given by y = x, just ambling along. So envisioned, the differing rates of growth, du Bois-Reymond’s infinites, form an intuitional and geometrical continuum extending that of the real numbers. This infinitary continuum is certainly dense in itself: between any two such quantities, one can intercalate an infinite number of others. Moreover, given any infinite order, one can approximate it using rates of growth that are distinct and, yet, as close as one would like intuitively to the original. So, as du Bois-Reymond noted, the number and proximity of orders is at least the number and proximity of real numbers, which can be embedded into the structure of orders in any number of ways. These plain facts du Bois-Reymond understood as

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part 2 289 verdicts of a properly geometrical insight [Du Bois-Reymond 1875 151–2] yielding us the information that the apparent geometrical properties of the infinitary rates of growth include those the ancients identified as earmarks of continuity: unity, density, infinite divisibility, seemingly gapless, strongly homogeneous, having no first or last member. However, and importantly, the general analytical profile, the limit structure on infinitary rates of growth, diverges markedly from that of the Dedekindian real numbers. There are no geometrically discernible gaps to be filled, yet the Upper Bound Theorem fails—and is seen to fail analytically. Theorem (du Bois-Reymond) Given L-functions fi , for each natural number i, and g such that f0 ≺ f1 ≺ f2 ≺ . . . g, there is always a suitable L-function h such that its infinity lies strictly between those of all the fi ’s and that of g:  f0 ≺ f1 ≺ f2 ≺ . . . ≺ h ≺ g. Du Bois-Reymond constructed several distinct and interesting proofs of this result, one of which was, recognizably, in the form of a diagonal argument, perhaps the first argument of that kind to feature in the foundations of mathematics. From du Bois-Reymond’s Theorem, it follows that the most penetrating intuition into the geometric features of the line cannot support the Cantor–Dedekind Axiom: a geometric construction with the same relevant intuitional properties as the Euclidean line does not satisfy the Upper Bound Theorem and, hence, the Cantor– Dedekind Axiom. Moreover, Dedekind failed to give an accurate account of the way in which limits arise. He wrote, And if we knew for certain that space was discontinuous [in the sense of Dedekind] there would indeed be nothing to hinder us, in case we chose, from filling up its gaps, in thought, and thus making it continuous; however, this filling up would consist in a creation of new point-individuals and would have to be carried out in accordance with the above principle [i.e., the Cantor–Dedekind Axiom]. [Dedekind 1872 19]

In general, processes of repeated ‘plugging’ into nonarithmetical continua cannot close the ‘gaps’ such limits as e create. Although limits certainly exist among the infinite orders, no finite or countable number of ‘plugs’ will close the ‘gap’ in du Bois-Reymond’s continuum between the orders fi and their non-minimal upper bound g.

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290 what is a number? continua, magnitudes, quantities

Part 3 Du Bois-Reymond’s magnum opus was Die allgemeine Functionentheorie [General Function Theory] [1882]. It opens with what may seem digressive reflections on continuous, measurable quantities as they feature in a panoply of scientific fields. Perhaps Kant’s writings inspired du Bois-Reymond to set the fact of continuous magnitude at the centre of his thinking, for Kant gave to Euclidean lines a special prominence in his treatment of extensive magnitudes at A163 in the First Critique [Kant 1929 198]. Another of du Bois-Reymond’s guiding spirits, Leonhard Euler, proclaimed mathematics the very Grössenlehre, the very doctrine of magnitude: Mathematics in general is nothing else but a science of magnitudes that discovers those means by which magnitudes are supposed to be measured. [Euler 1911 9]

For du Bois-Reymond, analysis was very much and more specifically eine lineare Grössenlehre, a doctrine of linear magnitude. A single, straight, connected, onedimensional extract from a one-dimensional geometrical continuum, a line or line segment, was for him the mathematical Urphänomen, the protofact of higher mathematics. (This notion of linear magnitude is reminiscent of Fraenkel’s definition of ‘continuum’, set out above.) All else, number included, is adventitious. Consequently, it was for charting the realm foundational to analysis that du Bois-Reymond commenced, in Chapter 1 of his General Function Theory— entitled On Mathematical Magnitudes—a survey of linear continuous magnitudes in science. The survey ranged from ancient land mensuration through the physics and physiology of his day including a detailed analysis of the Weber–Fechner Law [du Bois-Reymond 1882 32–34]. Here is du Bois-Reymond on linear magnitude: The linear mathematical magnitudes—which I name the basic form of magnitude—are the true roots of analysis, through which she continually draws new nourishment from her natural ground. We will also grasp the natures of mathematical magnitudes otherwise constituted if we compare them with the linear magnitudes, and assay those properties through which they deviate from the linear. [du Bois-Reymond 1882 15] If one speaks of mathematical magnitudes, one certainly thinks first of geometrical magnitudes only, particularly of the bounded line or length, to which one attempts to reduce the other magnitudes. For these are, without question, the simplest, most invariable, and most widely manifested of such representations. [du Bois-Reymond 1882 20]

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part 3 291 He intended his survey of continuous scientific magnitude first to guarantee the generality of his treatment of continua, magnitude, and quantity, and second to underscore a main theme of the work: that advanced mathematics finds its foundational bedrock in neither logic nor a more primitive mathematics but in a study of magnitude, in particular, in the practices of mensuration that convert magnitudes in continua into quantities in arithmetic. Du Bois-Reymond’s linear magnitudes are themselves either geometrical or readily geometrizable quantities. They embrace those, such as time and weight, easily represented, plotted, and compared as line lengths. All are continuous in nature. Du Bois-Reymond took continuous linear magnitudes also to incorporate those measurable quantities that vary by degree and to possess a governing homogeneity. Moreover, every aliquot part of a linear magnitude is another linear magnitude, as are regular concatenations of linear magnitudes. We apply the expression ‘different according to degree’, where we take the changes (increases or decreases) of the magnitude to be magnitudes of the same kind, where therefore the differences within a kind of magnitude appear to us as magnitudes of the same kind. [du Bois-Reymond 1882 25–26]

On the basis of these reflections, du Bois-Reymond isolated six fundamental properties of linear magnitudes. 1. Linear magnitudes are open to comparison, one with another, and can thereby be judged equal or unequal. 2. No one linear magnitude has special priority. Hence, there is no upper bound on the largeness and no lower bound on the smallness of linear magnitudes. 3. Two or more linear magnitudes can be joined together to produce a third linear magnitude greater than both. Similarly, any linear magnitude can be subdivided into arbitrarily many others, each smaller than the original. 4. When one linear magnitude is greater than another, there is a linear magnitude that is the difference between the two. 5. One can always assemble, in sufficient number, equal or unequal linear magnitudes, the least of which is not supposed to fall below an arbitrarily selected small magnitude, to obtain one not less than any specified magnitude of the same type. 6. The division of a magnitude can be continued until all the parts become less than any magnitude of the same type selected to be arbitrarily small. No matter how far the division may be thought to be pushed, the parts are always magnitudes of the same type. [du Bois-Reymond 1882 44–47] The fifth axiom appears to imply a version of the Archimedean Principle for linear magnitudes. However, these axioms for linear magnitude do not include

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292 what is a number? continua, magnitudes, quantities any that approach or imply Dedekind completeness. Du Bois-Reymond did not insist that linear magnitudes universally be closed under the limits of classes or sequences of magnitudes of that same kind. As stated, he knew already that his orders of infinity constituted a geometric continuum of magnitudes violating the Cantor–Dedekind Axiom. Du Bois-Reymond maintained that, even were all intuitional reasons to become magically available, they would not extend to supporting the postulation of arbitrary limits. Nothing in geometry’s character or results, as well as nothing in a fair survey of scientific measurement, implies the truth of the Cantor–Dedekind Axiom. The (lengthy) first and second chapters of General Function Theory—especially the famous dialogue between Empirist and Idealist [du Bois-Reymond 1882 58–156]—are devoted to arguing precisely that conclusion.

Part 4 Otto Hölder (1859–1937) was one of du Bois-Reymond’s PhD students. Hölder wrote his dissertation while his research supervisor was putting together the Functionentheorie. After submitting that dissertation, Hölder held faculty positions at Tübingen, Königsberg, and Leipzig. Over his long career, Hölder published significantly on Fourier series, finite groups, factor groups, projective geometry, and the methodology of mathematics. Today, mathematicians refer to the Hölder inequality, Hölder condition, Hölder mean, Hölder summation, as well as the Jordan–Hölder Theorem. In 1901, he published a result often—but I think unfairly—interpreted so as to undermine du Bois-Reymond’s vision of finite, infinite, and infinitesimal linear magnitudes. I believe that Hölder’s Theorem, properly construed, offers marvellous confirmation of his teacher’s outlook. Here is a contemporary statement of the Hölder Theorem of 1901. Theorem: (Hölder) Whenever 𝔄 = ⟨A, ∘, 0,

do not exist for intuitionists or are not well-defined (or substitute other mindless waffle). In truth, such functions exist and are perfectly well-defined, as the above diagram attests. In truth, Brouwer’s Continuity Theorem entails that fcase and its ilk are not total functions and that ∀r 𝜀 ℝ (r ≤ 0 ∨ r > 0) is a false statement. By the same token, equality over ℝ fails to satisfy TND; in other words, ¬∀r, s 𝜀 ℝ (r = s ∨ r ≠ s). With sharply focussed use of a choice axiom and some constructively OK principles, mathematicians have shown how to extend Brouwer’s Theorem, obtaining— for example—the result [Bridges & Richman 1984 109] that Theorem Every total function from an inhabited, complete, separable metric space into a metric space is continuous.  A form of Brouwer’s Continuity Theorem reappears in the semantics of computer programs [McCarty 1984b 349] as Theorem Every total function from a presented Scott information system into another is Scott continuous.  Information systems afford a presentation of basic semantical entities alternative to Scott domains. For definitions of information system and Scott continuous, plus explanations of their roles in denotational semantics, vide [Scott 1982] or [Winskel 1993]. A striking (and strikingly useful) consequence of the Continuity Theorem, first proved by Brouwer [1928], is the Unzerlegbarkeit or Indecomposability Theorem. Theorem (Unzerlegbarkeit) Let A be a subset of ℝ. If ∀r 𝜀 ℝ (r 𝜀 A ∨ r 𝜀 ̸ A),

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foundations of intuitionistic analysis 307 then A = ℝ ∨ A = ∅. Therefore, ℝ cannot be decomposed nontrivially: every set of real numbers that is decidable over ℝ is either all of ℝ or empty. Thanks to the above theorem, every nontrivial subset of ℝ is undecidable: ℕ, ℚ, 𝕀, even {0}. There is marked similarity between Unzerlegbarkeit and the connectedness of ℝ in conventional mathematics: that ℝ cannot be divided into nontrivial, pairwise disjoint, open subsets. Over topological models of intuitionistic set theory, the similarity becomes other than accidental. Dirk van Dalen [1997] obtained remarkable extensions of Unzerlegbarkeit, among them, Theorem (Van Dalen) 𝕀 is itself indecomposable.



In his proof, van Dalen called upon several strong intuitionistic assumptions, including one inspired by Brouwer’s theory of the creative subject. Vide [Van Dalen 1997].

3.3 Analogues of Continuity and Unzerlegbarkeit in Recursive Mathematics and Realizability Readers familiar with recursive function theory will note at once the strong resemblance between Unzerlegbarkeit and Rice’s Theorem for partial recursive functions [Rice 1953]. Let Rec be the set of all unary partial recursive functions and S be a subset of Rec. DS , the decision problem for S, is the set of natural numbers {e ∶ fe 𝜀 S}. Theorem (Rice’s Theorem) If DS is decidable, then either S = Rec or S = ∅.



Hence, every decidable, extensional collection of Turing machine indices is either ℕ or ∅. This time, the term ‘decidable’ means ‘its characteristic function is Turing computable’. S ⊆ ℕ ‘is extensional’ means that, if n 𝜀 S and fn is the same partial function as fm , then m 𝜀 S. A natural generalization of Rice’s Theorem is the Rice–Shapiro Theorem [Shapiro 1956]: Theorem (Rice–Shapiro Theorem) Let S be subset of Rec whose decision problem is recursively enumerable. For every f 𝜀 Rec, f 𝜀 S just in case there is a finite function 𝜙 ⊆ f such that 𝜙 𝜀 S. 

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308 continuity in intuitionism Over the universe for intuitionistic set theory inspired by Kleene’s realizability interpretation for elementary arithmetic [Kleene 1945], otherwise vague statements about resemblances between intuitionistic facts and facts about computability become perfectly precise. First, both BPN and Unzerlegbarkeit hold in the realizability universe [McCarty 1984a]. Second, a proof that BPN is realizable rests crucially upon Rice–Shapiro, while a proof that Unzerlegbarkeit is realizable relies upon Rice’s Theorem. Over realizability, BPN is intuitionistic analogue to the Kreisel–Lacombe Shoenfield Theorem or KLS [Kreisel, Lacombe & Shoenfield 1959] [Čeitin 1959]. An index i for a Turing machine is total whenever fi is total on ℕ. A partial recursive function g is extensional on total indices i and j just in case, if the functions fi and fj are equal, then g(i) = g(j). KLS states that Theorem (Kreisel–Lacombe-Shoenfield Theorem KLS) Every partial recursive function defined on every total index and extensional on total indices is continuous.  Equally reminiscent of the Continuity Theorem—but in a logical, rather than a computational, respect—is the Uniformity Principle of set theory or UP, first enunciated in [Troelstra 1973]: Principle (Uniformity UP) Let R be a binary relation holding between members of the powerset P(ℕ) of ℕ and members of ℕ that is total on the powerset, that is, ∀S 𝜀 P(ℕ) ∃n 𝜀 ℕ R(S, n). It follows that there is at least one natural number that R relates to every subset: ∃n 𝜀 ℕ ∀S 𝜀 P(ℕ) R(S, n). It follows immediately from the above that the entire universe 𝕍 of sets, as well as every nontrivial powerset, are unzerlegbar. Uniformity holds as well in the Kleene realizability universe [McCarty 1984a]. (We will return to a consideration of UP and logical objects in section 6.2.)

3.4 Uniform Continuity and the Fan Theorem To show that closed intervals of ℝ such as [0, 1] ⊆ ℝ are compact, intuitionists employ (what they call) the Fan Theorem, following [Brouwer 1923]. Let ℂ be the subset of Baire space containing all and only 0, 1-valued sequences of natural numbers. We will call a decidable set of initial segments of members

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foundations of intuitionistic analysis 309 of ℂ a subtree of C. Such a subtree T is bounded in case, for each 𝜙 𝜀 T, there is an n 𝜀 ℕ such that n is greater than the length of 𝜙. T is uniformly bounded when there is an n 𝜀 ℕ that bounds every segment in T. The Fan Theorem gets its name from standard usage in intuitionism where finitarily branching trees get called ‘fans’. Theorem (The Fan Theorem FT) Every bounded subtree is uniformly bounded. Plainly, FT is, in conventional mathematics, a contrapositive to König’s Lemma: that every subtree containing an infinite number of initial segments also has an infinite branch. (The reader will recall that inferences by contraposition removing negatives, such as from ¬p → ¬q, deduce q → p are not generally valid in intuitionistic mathematics.) In the presence of BPN, FT suffices to show that Theorem (Brouwer’s Uniform Continuity Theorem) Every total function from [0, 1] into ℝ is uniformly continuous. Sometimes, in the literature, this result is called ‘Brouwer’s Continuity Theorem’. Intuitionists can also prove that [0, 1] is a compact subset of ℝ, specifically, that every cover of [0, 1] by open subsets has a finite open subcover. More generally, one can show [Bridges & Richman 1987 114] that Theorem Every metric space that is a total functional image of

is compact.

The reader should note that Arend Heyting, in his justly famous book [1956], does not distinguish FT from BPN [Heyting 1956 42–43] and a few recent expositions follow Heyting in this.

3.5 Recursive Mathematics, Realizability, and the Fan Theorem The obvious recursive analogue to FT false. As Kleene showed in his [1952], Theorem (Kleene) There is a primitive recursive subtree that is not uniformly bounded but contains no infinite recursive branch. It follows that FT is not true under the recursive realizability interpretation for set theory. In accord with Kleene’s result, [0, 1] is compact neither in recursive analysis nor in the realizability universe. Vide [Beeson 1985 68].

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310 continuity in intuitionism

4. Brouwer on Intuition and Choice Sequences—also du Bois-Reymond 4.1 The Intuitive Continuum, Inner Time, and Primordial Intuition At the beginning of his career [Brouwer 1907] and to an extent later [Brouwer 1981 4–5], Brouwer countenanced the existence of a whangdoodle he called (translating his Dutch and German) ‘the intuitive continuum’. He meant that intuitive continuum to afford the ontic, conceptual, and epistemic fons et origo of both ℕ and ℝ as subjects of mathematical enquiry. Had he meant by the misbegotten phrase ‘the intuitive continuum’ what Dedekind [1872] had in mind when writing of the traditional geometrical line, I might start to understand him on this topic. However, it seems that Brouwer did not mean to refer at all to formal geometry, and I do not understand him. As should be plain from my previous remarks on this subject, Brouwer’s desired ontology is unavailable: the intuitive continuum does not exist, just as the number or arithmetical continuum does not exist [McCarty 2020]. Brouwer imagined this fictive one and only intuitive continuum to serve as the source of our ken of natural and real numbers by being a premier object of a primordial intuition into mathematics. Here’s roughly how this was supposed to work: Brouwer thought his mythic continuum to be the pure structure behind yet another phantasm, a neoKantian revenant he called ‘internal time’, open and available to view by some infinitely discerning inner eye. Brouwer came to believe, via a genetic fallacy of jaw-dropping fatuity, that the intuitive continuum arises and appears through the action of a wholly chimerical two-stage process of mental abstraction. At the first stage of abstraction, the substratum of internal time needs to be—in his words—‘divested of all quality’ [Brouwer 1907]. As van Atten and van Dalen (seek to) explain [2002], The reason for divesting this ‘substratum’ [scare quotes in the original] of all quality is simple. In order to do mathematics, we need to focus on the form or structure of the phenomenon. Thus, acts of abstraction or idealization are already at work in the primordial intuition of mathematics. [Van Atten, van Dalen, & Tieszen 2002 3]

We hardly need a Bishop Berkeley to remind us that, were we to abstract or divest all qualities—even all secondary qualities—from any appearance of internal time, there would be nothing left! There would remain nothing whatsoever to appear, nothing for the inner eye to scan. It would be like removing all the notes from a musical performance and expecting something wonderful and rarified, its ‘pure structure’, to hover before an inner ear. Moreover, and more pertinently here, until we do some mathematics—and do quite a bit of it!—we have absolutely no idea what is to count as form, to be

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brouwer on intuition and choice sequences 311 retained throughout the abstraction process, and what to count as content, to be discarded along the way. Appearances and phenomena do not come to us with readily discernible labels—‘content’ or ‘form’, ‘retain’ or ‘discard’—already stitched onto their variegated and tangled properties. Are we, for example, to abstract away from a bit of boredom we experience over a particular interval of internal time? What is there to tell us, before we know mathematics, that bored-feeling is not a mathematical property? Any presumptive intuition that requires a whole bunch of mathematics be done ahead of time and before intuition can get off the ground hardly counts as primordial. Brouwer’s second stage of abstraction was meant to hand us natural numbers and their concepts. It appeared to him that even the total mathematical naïf, someone ignorant of all mathematics, can abstract from that aforementioned pure structure all the various basic number concepts, the concept ‘three’, for instance. In other words, Brouwer held that a mere introspection of the passage of time (whatever that would amount to) suffices to convey the concept ‘three’ to someone who lacks it utterly. [I]ntuitionistic mathematics is a languageless activity of the mind having its origin in the perception of a move of time, i.e.„ of the falling apart of a life moment into two distinct things, one of which gives way to the other, but is retained by memory. This ‘two-ity,’ divested of all quality, is the empty form of the common substratum of all two-ities. In this common substratum, this empty form of all two-ities, lies the basis of the discrete aspect of the primordial intuition of mathematics. It successively generates each natural number and arbitrary finite sequences. It is natural to suppose that it lies at the basis of the finite combinatorial objects that could be generated from the natural numbers. [Van Atten, van Dalen, & Tieszen 2002 3]

This is more utter nonsense. Once again, an abstraction of all qualities cannot leave behind something apparent, a spectral form. Larded atop that error is the questionbegging assumption that the primordial intuition somehow generates each natural number and, thereby, all finite combinatorial objects. From the outset, what is in question foundationally at this turn is precisely the nature of finite generation. Once I learn what finite generation is, I know what the natural numbers are. And once I know what the natural numbers are, I know what finite generation is. Hence, after all is said and done, the double abstraction is vapid hoo-ha, wholly unnecessary. To have and to know the natural numbers is to have and to know the nature of finite generation. No processes of abstraction are required. Lastly, my final point on this topic: Gottlob Frege [1884] had already, long before Brouwer, held up such inanity as the abstraction of numbers and number concepts from simple multitudes—whether timeless or evolving in time—to justified ridicule.

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312 continuity in intuitionism One of Frege’s objections is also mine: without a prior grasp of a number concept, the abstractor has no idea what to abstract! In sum, there is neither phenomenological basis nor primordial intuition behind counting. For there is nothing behind counting. There are only the crude and clumsy processes of humans counting and learning to count, by reciting rhythmically, in singsong fashion, ‘one’, ‘two’, ‘three’, and so on.

4.2 Brouwer’s Choice Sequences Central to Brouwer’s mature (after 1917) treatment of arithmetical continua is, famously, a notion of Wahlfolgen or choice sequences, the picture of a sequence generated step-by-step over time, that generation governed by either a rule or rules set down along the way, temporary or permanent, a series of restrictions open to possible updates or abolition, throws of a die [Troelstra 1977 12] [Troelstra & van Dalen 1988 645], arbitrary choices of outputs wholly ungoverned by rule or restriction, or arithmetic and functional combinations of the foregoing. As mentioned above, detailed investigations of each of these conceptions have not resulted in strict proofs of BPN generally acceptable to intuitionists everywhere [Troelstra 1983] [Van Atten & van Dalen 2002]. When a rule is set down permanently to govern a sequence, it is said to be lawlike. The penultimate manner of sequence just listed, called ‘lawless sequence’ in the literature, features large—as described above—in Brouwer’s efforts to prove BPN by suggesting (falsely) that assignments of natural numbers to choice sequences can be made only extensionally and only as if the sequences were lawless, i.e., as if there were no rules or restrictions governing them and, hence, that everything known of them, at the moment of the assignment, is their respective finite initial segments. If the sequences in question were not lawless, but governed by rules, one could make assignments dependent upon the existence and characters of those rules. For instance, if the sequences to be assigned natural numbers were recursive, then the number to be assigned could be a code or Turing machine number for the sequence. And that assignment, since the same sequence will have in general an infinite numbers of such codes, need not be extensional. In short, one could say that, for purposes of arguing for BPN, Brouwer treated all choice sequences as lawless. (By the way, to point this out is not to assert that the sole benefit to intuitionists of lawless sequences is the thin disguise of an otherwise obvious fallacy. Vide [Kreisel 1958].) The entire mathematics bound up with Brouwer’s doctrine of choice sequences is clouded, as often, by his unfortunate choice of terms. His nomenclature is

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brouwer on intuition and choice sequences 313 always rebarbative: spreads, species, arrows, fleeing properties—a Derridean gaggle of idiosyncratic technicalities. Worse, some later intuitionists readily adopted these and yet other bizarre titles ad lib., giving the impression to all the world that their doctrines need to be couched in a word-magic almost Marxian in its opacity. Such nutty naming is otiose; clear and straightforward expositions in familiar terms— such as that of [Bridges & Richman 1987 103–119]—so and vividly demonstrate.

4.3 Du Bois-Reymond on Choice Sequences Neither Emil Borel—as suggested in [Troelstra & van Dalen 1988 639–642]—nor Brouwer introduced into mathematics the notion of a Cauchy sequence whose terms are generated over time by a series of unfettered choices or by, say, imagined throws of a cosmic die, a die with infinitely many faces no less! This idea, including the image of shooting cosmic craps, goes back at least as far as Die allgemeine Functionentheorie [du Bois-Reymond, P. 1882 90ff.]. Because of its historical import and since du Bois-Reymond seems to have anticipated so many insights later associated with the names of Borel and Brouwer [McCarty 2004], not to mention Bernays and Kleene, I here quote the relevant passages from du Bois-Reymond in full. These paragraphs represent an episode in du Bois-Reymond’s imaginary foundational dialogue between an Idealist and an Empirist [du Bois-Reymond, P. 1882 58–176]. To a first and possibly misleading approximation, the Idealist may be compared with today’s full-bore set-theoretic realist, and the Empirist with an Hilbertian finitist, under the proviso that du Bois-Reymond’s characters are plainly practising mathematicians rather than wool-gathering philosophers. The ‘I’ below represents du Bois-Reymond’s Idealist speaking. I am ready for these consequences of my basic assumptions; they do not frighten me at all. The Empirist concludes with perfect correctness that I need to allow number sequences that, although they can never reveal to us laws for their generation, yet continue into the infinite. This is certainly unimaginable—but no more unimaginable in themselves than the infinite and the eternal. Numbers whose digits cannot be written out are as much unavoidable consequences of general human conceptions as are the infinity of space and the eternity of time. Consider restrictions on the number of possible decimal expansions via laws that infinite operations per se demand. Such restrictions are obviously not contained a priori in the concept of all infinite sequences. For these sequences are given by whatever digits from the list 0, 1, 2, . . . 9 are written one after another. These restrictions are merely attached to that concept retrospectively, via speculation. If one starts from the idea of quantity, then all lengths other than the unit length are of the same type in the imagination. Let one think of the unit interval plus another

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314 continuity in intuitionism length already laid out. If we measure to determine the relation between the second length and the unit, we will ascertain one decimal place after another, and be able to continue this enterprize indefinitely, obviously under the assumption of sufficient instrumentation. One can do this as long as one cares to, and thereby develop an unbounded decimal expansion empirically. However, nothing in this generation process for a decimal expansion requires a law that rules its sequence of digits. Through the strict relation of lengths to one another, an endless decimal expansion is fully determined. This happens without a need for a rule to exist such that, for example, if the first n digits are given, then every digit after the nth can be calculated. Also, one could think about generating an endless and lawless sequence in the following way. Every term in the sequence is determined simply by throwing a die. [This is precisely the image Troelstra and van Dalen [1988 645] provide to illustrate their idea of lawless sequence.] Were we to assume that the die-throwing takes place from eternity or through all eternity, the idea of a lawless number would thereby be produced. However, observation of nature delivers even better examples. It is undisputed that many constants of nature are determined by the state of the universe from eternity. Certainly, this includes the temperature of space, its optical constants, and, above all, its electrical potential. [Du Bois-Reymond seems to be referring here to the electric potential V of a point in space.] When it comes to potential and similar quantities, it is true to assert that they are determined by the totality of the masses in space, under the obviously rather questionable assumption that the powers exerted by a point get exercised everywhere simultaneously. By contrast, the other constants of space are produced via the influence of everything that is and was in space. For instance, the temperature of space is the result of all conditions in spatial infinity and from temporal eternity. Should we think of matter as finite—coming to an end in every spatial direction—and its state given at any time, the numerical representation of these natural constants must lead to a law. However, should we think of matter as infinite, a constant such as the temperature of space is dependent upon influences that are not required to cease at any decimal place. Were its sequence of digits to continue according to a law of formation, then this law would reveal the history and the picture of both temporal eternity and spatial infinity. Therefore, such physical deliberations suggest the existence of lawless irrational numbers. [du Bois-Reymond, P. 1882 90–92]

In the final sentences of this passage, one sees immediately an anticipation not only of the later (fallacious) weak counterexamples of Brouwer, but also the (nonfallacious) noncomputability arguments by reduction. Du Bois-Reymond’s Idealist suggests that numbers such as that giving the temperature of space exist but cannot be governed by any known law. Were we to have such a law at our

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brouwer on intuition and choice sequences 315 disposal, we could approximate the background temperature to any decimal digit whatsoever, and hence make detailed predictions on the state of the universe at any moment past or future—a epistemic ability the Idealist believed we do not and cannot ever possess. This kind of ‘we will never know’ argumentation is very much apiece with Ignorabimus—a studied agnosticism about fundamental physical conditions and quantities—extolled by Paul’s brother, physiologist Emil du Bois-Reymond [du Bois-Reymond, E. 1886].

4.4 The Law of the Excluded Third Some intuitionists have insisted that the invalidity of the law of the excluded third could be demonstrated from the existence and basic concept of choice sequences. Here are intuitionists van Atten and van Dalen, and their collaborator Tieszen on the subject. This unfinished character of choice sequences has repercussions for logic. It means that a sequence cannot, at any stage, have (or lack) a certain property if that could not be demonstrated from the information available at that stage. [B]ivalence, and hence the Principle of the Excluded Middle, does not hold generally for statements about choice sequences. For example, consider a lawless sequence 𝛼 of which so far the initial segment ⟨1, 2, 3⟩ has been generated, and the statement P = ‘The number 4 occurs in 𝛼.’ Then, we cannot say that P ∨ ¬P holds . . . Acceptance of choice sequences as mathematical objects forces a revision of logic. [Van Atten, van Dalen, & Tieszen 2002 11]

The reasoning here is as hilarious as that surrounding ‘twoities’ and ‘the intuitive continuum’. If 𝛼 is a lawless (or any) sequence, and P means ‘the number 4 occurs in 𝛼’, then it is plain that either 4 appears in 𝛼’s generation or it does not. Hence, P ∨ ¬P holds for the given P and sequence 𝛼. If, as the authors suggest earlier in the passage, P really means One can demonstrate ab initio for a particular lawless 𝛼 that 4 occurs in it

then P ∨ ¬P still holds good, since one can either demonstrate ab initio that 4 occurs in 𝛼 or one cannot so demonstrate. If, in a desperate effort to explain why choice sequences somehow oblige us to cast standard logic aside, one takes the futile option of changing the meaning of the connective signs ¬ from ‘not’ and ∨ from ‘or’ to some other connectives phrases, then one is no longer constructing a counterexample to the law of the excluded third—a sentential property of disjunction and negation. One is criticizing some other scheme, one which (who knows?) may be of no foundational interest at all! Besides, it is absurd to think

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316 continuity in intuitionism that the inclusion of a certain brand of object into one’s ontology will, all by itself, demand a change in the way one reasons about those, and other, objects.

5. Weyl’s Intuitionism and Arithmetic Continuity 5.1 Weyl’s Treachery To be honest, I have never taken the contributions of Hermann Weyl to the foundations of intuitionistic mathematics at all seriously. For one thing, he was a turncoat, the Benedict Arnold of mathematical intuitionism. First, around 1919, consequent to holiday conversations with Brouwer in the Engadin of Switzerland [Van Dalen 1995 147], Weyl embraced (a ragged selection from) Brouwer’s ideas with his usual grandiose fanfare. However—by 1924 or so—he had slunk away from them ignominiously. It appears that, for Weyl, being an intuitionist turned out to be just a teeny bit too awkward. Reflecting back in 1927 upon his betrayal, Weyl asserted, Mathematics with Brouwer gains its highest intuitive clarity . . . It cannot be denied, however, that in advancing to higher and more general [intuitionistic] theories, the inapplicability of the simple laws of classical logic eventually results in an almost unbearable awkwardness. And the mathematician watches with pain the greater part of his towering edifice which he believed to be built of concrete blocks dissolve into mist before his eyes. [Weyl 1963 54]

(Needless to remark, we now know that all three of Weyl’s principal assertions above are wholly false and demonstrably so.) It is no exoneration of Weyl (cf. [Van Dalen 1995 163]) to point to an occasion in 1927 when Weyl sought to ‘defend intuitionism’ against Hilbert’s unfair critical assaults. In that so-called defence, Weyl asserted, That from this [intuitionistic] point of view only a part, only a wretched part, of classical mathematics is tenable is a bitter but inevitable fact. [Weyl 1928 483]

Errett Bishop [Bishop 1967] and others have proved that this is no fact at all! Worse, Weyl there trumpeted his freshly adopted reactionary standpoint: And, as I am very glad to confirm, there is nothing that separates me from Hilbert in the epistemological appraisal of the new situation thus created [with the introduction of Hilbert’s Beweistheorie]. [Weyl 1928 483]

As if he had not inflicted enough damage upon Brouwer’s intuitionism, Weyl set out to clarify the murk of the Brouwerian philosophy by folding into it liberal

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weyl’s intuitionism and arithmetic continuity 317 doses of the intellectually poisonous mire of Husserlian phenomenology. In that, Weyl only compounded his treachery and, like his latterday followers [Van Atten, D. van Dalen, & Tieszen 2002], ramped up the crazy. Without question, there is a hell for mathematicians—and Hermann Weyl now lies deep in its Ninth Circle.

5.2 Weyl on Choice Sequences The current article is aimed primarily at examining works of historical intuitionists; so, I limit further consideration of Weyl’s ideas to his brief flirtation with that mathematics and its expression in the influential [Weyl 1921]. Weyl’s distinctive version of intuitionism, developed by him over the years 1918 through 1924 or 1925 (at least up to the composition of [Weyl 1925]) differs markedly from that of Brouwer. I shall focus here upon the salient differences, i.e., on the errors that Weyl committed and the confusions he engendered. Also, I note that there is in [Weyl 1921], unlike [Weyl 1918], no extended treatment of non-arithmetical continua. In [1921], Weyl repudiated his earlier predicativism, announced in The Continuum [Weyl 1918]. He endorsed notions of both lawlike and lawless sequences— but not, it seems, the whole panoply of choice sequences imagined by Brouwer. Weyl appeared to allow only the lawlike and lawless, those classes of sequences at opposite ends of the Brouwerian spectrum. (At times, Weyl at least toyed with notions of non-lawless sequences constructed arithmetically from lawless ones. But he seems never to have embraced the idea fully.) At one point, Weyl suggested that his sequences are all either wholly lawlike or wholly lawless: An infinite determined sequence (and, I mean, determined in infinitum) can be defined only by a law. If however, a sequence is created step-by-step through free acts of choice, then it must be seen as a developing one. A developing choice sequence can meaningfully be said to have properties for which the decision ‘Yes’ or ‘No’—Does the sequence possess the property or not?—can already be obtained only when the sequence has reached a certain stage such that the further development . . . cannot reverse this decision. [Weyl 1921 94]

Here, Weyl conflated, with help from the potentially confusing expression ‘can meaningfully be said’, the meaning of a statement with its truth. Weyl conceived of lawlike, and not lawless, sequences as determining individual real numbers. For him, as for the later Russian constructivists A.A. Markov, Jr. and followers [Kushner 2006], all individual mathematical items must be encoded as or with natural numbers. By contrast, Weyl intended lawless sequences to convey the concept of the entire (arithmetic) continuum:

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318 continuity in intuitionism [The] continuum is represented by the choice sequence unrestricted in the freedom of its development by any law. [Weyl 1921 94]

Although Weyl distinguished particular classes of functions and functionals in types over the natural numbers, he dismissed any hope for a unitary theory of functions and functionals (as well as sets), Brouwerian or otherwise: Sets of functions and sets of sets, however, shall be wholly banished from our minds. There is therefore no place in our analysis for a general set theory, as little as there is room for general statements about functions. [Weyl 1921 109]

In this regard, Weyl himself admitted a major difference between his outlook and that of Brouwer: As far as I understand, I no longer completely concur with Brouwer in the radical conclusions drawn here. After all, he begins immediately with a general theory of functions . . . He looks at properties of functions, properties of properties of functions, etc., and applies the identity principle to them. (I am unable to find a sense for many of his statements.) [Weyl 1921 109]

In [1921], Weyl stated (what we have called) Brouwer’s Continuity Theorem governing total functions from ℝ into ℝ, not as theorem or principle, but as (part of) the definition of functions of the relevant type, claiming Such a function . . . is determined by a law according to which each dual [rational] interval once it has become sufficiently small [as an approximation to the function’s input] generates an interval [i.e., a given approximation to the function’s output] . . . As soon as the argument is given with a certain degree of accuracy— and in application it is never given in any other way—the value of the function is also known with a corresponding degree of accuracy. [Weyl 1921 114]

In the paragraph to follow, he concluded, One can see that the concept of a continuous function over a bounded interval cannot be defined without simultaneously including uniform continuity and boundedness in the definition. Above all, however, there cannot be any functions in a continuum other than continuous functions. [Weyl 1921 114]

Thus, there is a sense in which Weyl anticipated the publication of both Brouwer’s Continuity Theorem and his Uniform Continuity Theorem. He also asserted, correctly but without a shred of proof, that the Weylian continuum has to be unzerlegbar:

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weyl’s intuitionism and arithmetic continuity 319 The point is that a genuine continuum is something connected in itself, and it cannot be divided into separate fragments; this conflicts with its nature. [Weyl 1921 111]

5.3 The Law of the Excluded Third, Again In tandem with his intuitionism and his approach to the sequences undergirding arithmetic continua, Weyl put forward his own, idiosyncractic interpretations of the logical signs. These were distinct from those Heyting would later popularize [Heyting 1956 97–99] but suggestive of Hilbert’s ‘finitistic’ rendering of the meanings of the familiar quantifiers; vide [Hilbert 1926]. According to Weyl, the existential quantifier in ‘there are real numbers such that’ ranges only over lawlike sequences: ∃𝛼 𝜀 𝔹 P(𝛼) means that there is a lawlike sequence 𝛽 𝜀 𝔹 that has been successfully constructed and of which one can prove that P(𝛽). By contrast, the universal quantifier in ‘for all real numbers’ ranges over lawless sequences only. Given these interpretations, only the careless might now suggest that the assertion ¬∀𝛼P(𝛼) → ∃𝛼¬P(𝛼) is no longer true and hence affords a counterexample to the logical scheme ¬∀xA(x) → ∃x¬A(x). Curiouser and curiouser: Weyl also thought any conclusion derived from such a universal quantification (with meaning as reconstrued by him) by instantiation to be a statement about lawlike sequences exclusively. He maintained as well that the negations of unbounded existential and of universal statements over the real numbers are both meaningless. He did allow that, as an alternative to such a Procrustean outlook on negation, the statement ‘it is false that there is a sequence with property P’ can be rendered as ‘every sequence fails to have the property P’, which is a claim about all lawless sequences and is true—by Weyl’s lights—only if it is demonstrated on the basis of the very notion of lawless sequence. Weyl took it to follow from these strange and conflicting re-interpretations of quantifiers and negation that the law of the excluded third is invalid. [I]t would be absurd to think of a complete disjunction in this context. In this way, one will understand Brouwer’s claim that there are no grounds for believing in the logical principle of the excluded middle. [Weyl 1921 96]

The reader will realize at once that Weyl has in no way here described any sort of counterexample to the law of the excluded middle or any sort of good reason

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320 continuity in intuitionism for thinking it invalid. That there are ‘instances’ of TND that are meaningless because the substituends were meaningless to start with, e.g., (with apologies to Lewis Carroll) Either the slithy tove did gyre or the slithy tove did not gyre

shows nothing about the validity or invalidity of the law. One can take any standard expression of a valid logical scheme and substitute uniformly into its sentential variables nonsensical strings of words and get further nonsense out! Surprise! On the other hand, to assert that the negation of ∃𝛼P(𝛼) is ‘one can demonstrate from the nature of all lawless sequences 𝛽 that ¬ P(𝛽)’ is already to ascribe to the sign ¬ a meaning that it does not possess. Once the sign ¬ takes on an imagined meaning, one different from the standard meaning, the scheme P ∨ ¬P no longer expresses the traditional law TND. Counterexamples to an expression featuring nonstandard negation cannot count as counterexamples to TND.

6. In Sum 6.1 Continua On the one hand, we find—consequent largely to Herculean efforts by nineteenthcentury foundationalists—a clutch of arithmetic continua, including ℝ and 𝔻, inter alia. On the other hand, we see all around us a booming, buzzing phantasmagoria of continua, both mathematical and nonmathematical, ranging from proprioceptic space through the lines of Euclidean geometers. Is there any robust mathematical relation between pairs of them? Cantor postulated CDA, the Cantor– Dedekind Axiom, the existence of an order-isomorphism between his arithmetic continuum and ‘the geometric line’, and claimed that its truth, although apparent to a trained analytical eye, is unprovable [Waisman 1936 147]. Paul du BoisReymond’s diagonalization showed CDA and the reasons Cantor and Dedekind gave for it to be stuff and nonsense [du Bois-Reymond, P. 1875]. Du BoisReymond’s PhD student Otto Hölder underscored and furthered his teacher’s result by showing, in his [1901], that a straightforward mathematical argument for CDA from the axioms of measurable geometrical magnitudes is questionbegging in that it requires the assumption of Dedekindian order completeness for geometric magnitudes to show that the putative isomorphism between the line and 𝔻 is surjective. Apart from some remarkably unintelligible lip service—on the parts of both Brouwer and Weyl—there is little in the writings of sainted, ole-time intuitionists

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in sum 321 that represents a full and theoretically informed attempt, comparable to that apparent in the tradition running from Paul du Bois-Reymond through Hermann von Helmholtz to Otto Hölder, to understand and grasp the formal structures of non-numerical continua, arguably the original and originary continuous phenomena. In its place, one finds only empty and witless blather about ‘twoities’, and no serious effort to take on some main points, e.g., the truth—even the meaning—of CDA. This is not just the intuitionists of the day taking a scientific step backwards. It is more than sacrificium intellectus on the part of mathematicians; it is an intentional plunge from the ship of sanity into the dark waters of nonsense. I therefore encourage intuitionists everywhere now to do better; cease the hagiolatrous repetitions of the confusions of their forebears and take a serious crack at analysing Hölder’s Measurement Theorem, attempting to probe and even to prove it on the grounds of strictly intuitionistic principles.

6.2 Continuity Theorems First Brouwer, and then Troelstra and van Dalen after him, made strenuous efforts to found a characteristic intuitionistic analysis on some intuitionistically proprietary notion of sequence so that at least BPN would be deducible. As mentioned earlier, the best of those efforts have not been successful, and intuitionists admit as much [Van Atten & van Dalen 2002 §4]. Now, I cannot offer conclusive reasons in favour of what I am about to suggest. Perhaps I am not even formulating the crux of it incisively or effectively. With caution thrown to the winds, though, here goes. In an infinity of intuitionistic universes for set theory, one finds, displayed prominently in the mathematics, logical objects roped off from arithmetic objects. (I do not mean to suggest that this is an exhaustive dichotomy.) Among the former, there are objects that are defined and articulated in markedly class- or purely set-theoretic terms. The premier examples are the set-theoretic universe 𝕍 and all powersets P(A) of inhabited sets A. One of the earmarks of logical objects is the lack of apartness relations over them. An apartness relation ♯ is a binary relation that is symmetric, antireflective, and comparative, wherein the last means that, for all a, b, and c elements of the field of ♯ , a ♯ b → (a ♯ c ∨ b ♯ c). As you can see, apartness is an intuitionistic improvement over the non-identity relation ≠. Arithmetic objects, among them ℕ and ℚ, sport appropriate apartness relations as well as the decidability of equality, ∀a, b 𝜀 A(a = b ∨ a ≠ b).

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322 continuity in intuitionism These conditions on arithmetic objects manifest themselves in the fact that there is no significant continuity constraint on functions between arithmetic objects. Since all of them can support discrete topologies, every function from A to B, with A and B arithmetic, is in this sense continuous—and this fact is trivial. However and importantly, governing relations between logical and arithmetic objects is Troelstra’s Uniformity Principle UP (see section 3.3) from which it follows, in addition to Unzerlegbarkeit for nontrivial powersets, that every total function from a powerset P(A), for A inhabited, into a set with apartness is constant.

Obviously, this theorem bespeaks a kind of relatively nontrivial continuity, if we think of the powersets as bearing something (roughly!) akin to a trivial topology and ℕ its discrete topology. Second, in the intuitionistic universes I have in mind, crucial to goings-on there are often witnesses w 𝜀 W in virtue of which items a are members of a set A, represented w ⊩ a 𝜀 A. In the case of such a logical object as the powerset of {0}, P({0}), when S is a subset of {0}, anything capable of being a witness at all, any member of W, is a witness for S is a subset of {0}. Take it from me: this fact is absolutely key to the proof that the Uniformity Principle holds in this universe. Third, 𝔹 is a set that stands smack in between the two classes of objects. Since its members are (infinitary) sequences over ℕ, 𝔹 partakes of higher-order settheoretic or logical structure in its very definition. Since 𝔹 sports an apartness relation, namely, for f, g 𝜀 𝔹, f ♯ g if and only if ∃n 𝜀 ℕ f(n) ≠ g(n), it is a near relative to arithmetic objects. So, given the strong continuity governing logical objects like P({0}), we would expect continuity to appear over 𝔹. Fourth and finally, central to the proof that BPN obtains in these universes is the fact that the collection of witnesses w such that, for f 𝜀 𝔹, F a functional from 𝔹 into ℕ, and n 𝜀 ℕ, w ⊩ F(f) = n

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bibliography 323 is essentially a recursively enumerable subset of Rec (or analogue thereof). Therefore, something akin to the Rice–Shapiro Theorem applies and the collection of witnesses {w ∶ w ⊩ F(f) = n} is governed by the initial segment condition. It is this condition that leads immediately to BPN. From these remarks, there are two morals to be drawn, I think. First, nontrivial continuity theorems may be rooted in that part of the universe containing logical objects, rather than the part featuring arithmetic ones exclusively. Moreover, the proofs of those theorems arise not so much from analysing either natures of sequences per se or concepts of continuity, but from mathematical features of the truth- or witness-conditions of such statements as S 𝜀 P({0}) or f 𝜀 𝔹 ∧ n 𝜀 ℕ ∧ F 𝜀 (𝔹 ⇒ ℕ) ∧ F(f) = n.

6.3 Logic From Brouwer onward, intuitionists and fellow travellers (e.g., [Dummett 1977]) have struggled to construct some kind or other of convincing argument to the conclusion that the classical law of the excluded third TND is invalid. As we have seen, many but not all of those arguments try to give counterexamples to TND via reflections on choice sequences or via changes in the meanings of logical connectives and quantifiers. Neither ploy has much of a track record. They both beg the question against TND flagrantly or commit plain equivocations. Both are fallacies that these selfsame philosophers or intuitionistic mathematicians would not let their first-year students commit. Honest-to-God demonstrations that TND is invalid do indeed follow—correctly and immediately—from principles like BPN and UP. Therefore, formulating plausible objections to TND may reduce to finding plausibility arguments for, say, BPN. Now, loop back to the preceding subsection.

Bibliography Bibliographic references are in brackets by author name and year, and if required, page or section numbers. Aristotle [1984] The Complete Works of Aristotle. The Revised Oxford Translation. Volume 2. J. Barnes (ed.) Bollingen Series LXXI/2. Princeton, NJ: Princeton University Press. vii+1251–2488. Beeson, M. [1985] Foundations of Constructive Mathematics. Berlin: SpringerVerlag. xxiii+466.

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324 continuity in intuitionism Bell, J. [2008] A Primer of Infinitesimal Analysis. Second edition. Cambridge, UK: Cambridge University Press. xi+124. Bishop, E. [1967] Foundations of Constructive Analysis. New York, NY: McGraw-Hill Book Company. x+370. Bridges, D. & F. Richman [1987] Varieties of Constructive Mathematics. London Mathematical Society Lecture Notes Series. Volume 97. Cambridge, UK: Cambridge University Press. x+149. Brouwer, L. [1907] Over de Grondslagen der Wiskunde. [On the foundations of mathematics.] Ph.D. thesis. Amsterdam: University of Amsterdam. (English quotations from the translation in [Brouwer 1975].) Brouwer, L. [1913] Intuitionisme en formalisme. [Intuitionism and formalism.] Wiskundig Tijdschrift. Volume 9. pp. 180–211. Brouwer, L. [1918] Begründung der Mengenlehre unabhängig vom vom logischen Satz vom ausgeschlossenen Dritten. Erster Teil. Allgemeine Mengenlehre. [Foundation of set theory independent of the logical law of the excluded third. First part. General set theory.] Konink. Nederl. Akad. Wetensch. Verhandelingen. Volume 12. Number 5. pp. 1–43. (Reprinted in [Brouwer 1975].) Brouwer, L. [1923] Begründung der Functionenlehre unabhängig vom logischen Satz vom ausgescholossenen Dritten. [Foundation of function theory independent of the logical law of the excluded third.] Nederl. Akad. Wetensch. Verhandelingen. Tweede. Afd. Nat. Volume 13. Number 2. Brouwer, L. [1928] Intuitionistische Betrachtungen über den Formalismus. [Intuitionistic considerations concerning formalism.] Sitzungsberichte der Preuszischen Akademie der Wissenschaften zu Berlin. pp. 48–52. (English translation of its first section appears in [van Heijenoort 1967 490–492].) Brouwer, L. [1975] A. Heyting (ed.) Collected Works I. Philosophy and Foundations of Mathematics. Amsterdam: North-Holland Publishing Company. xv+628. Brouwer, L. [1981] Brouwer’s Cambridge Lectures on Intuitionism. D. van Dalen (ed.). Cambridge, UK: Cambridge University Press. xii+109. Čeitin, G. [1959] Algorithmic operations in constructive complete separable metric spaces. (in Russian) Doklady Akad. Nauk. Volume 128. pp. 49–52. Dedekind, R. [1872] Stetigkeit und irrationale Zahlen. [Continuity and Irrational Numbers]. Braunschweig: Friedrich Vieweg und Sohn. 31 pp. Diaconesu, R. [1975] Axiom of choice and complementation. Proceedings of the American Mathematical Society. Volume 51. pp. 176–178. du Bois-Reymond, E. [1886] Reden. Erste Folge. Leipzig: Verlag von Veit und Comp. viii+550. du Bois-Reymond, P. [1875] Ueber asymptotische Werthe, infinitäre Approximationen und infinitäre Auflösung von Gleichungen. [On asymptotic values, infinitary approximations, and infinitary solutions to equations.] Mathematische Annalen. Volume 8. pp. 363–414.

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bibliography 325 du Bois-Reymond, P. [1882] Die allgemeine Functionentheorie. Erster Theil. [General Function Theory. First Part.] Tübingen: Verlag der H. Laupp’schen Buchhandlung. xiv+292. (Translations from this source by the author.) Dummett, M. [1977] Elements of Intuitionism. Oxford Logic Guides Series. Oxford: The Clarendon Press. xii+467. Fourman, M. & M. Hyland [1979] Sheaf models for analysis. M. Fourman et al. (eds.) Applications of Sheaves. Proceedings of the Research Symposium on Applications of Sheaf Theory to Logic, Algebra, and Analysis. Durham, July 9–21, 1977. Springer Lecture Notes in Mathematics. Volume 753. Berlin: Springer-Verlag. pp. 280–301. Frege, G. [1884] Die Grundlagen der Arithmetik: Eine logisch mathematische Untersuchung über den Begriff der Zahl. [The Foundations of Arithmetic: A Logically Mathematical Investigation of the Concept of Number.] Breslau: Verlag von Wilhelm Koebner. xi+119. Goodman, N. & J. Myhill [1978] Choice implies excluded middle. Zeitschrift für mathematische Logik und Grundlagen der Mathematik. Volume 24. p. 461. Heuer, H. [2003] Motor control. Weiner, I. et al. (ed.) Handbook of Psychology. Volume 4. Experimental Psychology. Hoboken, NJ: John Wiley and Sons, Inc. pp. 317–356. Heyting, A. [1956] Intuitionism: An Introduction. Studies in Logic and the Foundations of Mathematics. Amsterdam: North-Holland Publishing Company. viii+132. Hilbert, D. [1926] Über das Unendliche. [On the infinite.] Mathematische Annalen. Volume 95. pp. 161–190. Hölder, O. [1901] Die Axiome der Quantität und die Lehre vom Mass. [The axioms of quantity and the doctrine of measurement.] Berichte über die Verhandlungen der königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig. Mathematisch-Physikalische Classe. Band 53. pp. 1–64. Kleene, S. [1945] On the interpretation of intuitionistic number theory. The Journal of Symbolic Logic. Volume 10. Number 4. December 1945. pp. 109–124. Kleene, S. [1952] Recursive functions and intuitionistic mathematics. L. Graves et al. (eds.) Proceedings of the International Congress of Mathematicians. August 1950. Cambridge, Mass. Providence, RI: American Mathematical Society. pp. 679–685. Koch, A. [1981] Synthetic Differential Geometry. London Mathematical Society Lecture Notes Series. Volume 51. Cambridge, UK: Cambridge University Press. 311 pp. Kreisel, G. [1958] A remark on free choice sequences and topological completeness proofs. The Journal of Symbolic Logic. Volume 23. Number 4. pp. 369–388. Kreisel, G., D. Lacombe, & J. Shoenfield [1959] Partial recursive functions and effective operations. A. Heyting (ed.) Constructivity in Mathematics. Proceedings of the Colloquium at Amsterdam, 1957. Amsterdam: North-Holland Publishing Company. pp. 195–207.

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326 continuity in intuitionism Kushner, B. [2006] The constructive mathematics of A.A. Markov. American Mathematical Monthly. Volume 113. Number 6. pp. 559–566. Longo, M.R. & P. Haggard [2010] An implicit body representation underlying human position sense. Proceedings of the National Academy of Sciences of the USA. Volume 107. pp. 11727–11732. Martin-Löf, P. [1975] An intuitionistic theory of types: predicative part. H. Rose et al. (eds.) Logic Colloquium ’73. Amsterdam: North-Holland. pp. 73–118. McCarty, C. [1984a] Realizability and Recursive Mathematics. D.Phil. dissertation. University of Oxford. iii+281. McCarty, C. [1984b] Information systems, continuity, and realizability. E. Clark & D. Kozen (eds.) Logic of Programs. Lecture Notes in Computer Science. Volume 164. Berlin: Springer-Verlag. pp. 341–359. McCarty, C. [2004] David Hilbert and Paul du Bois-Reymond: limits and ideals. G. Link (ed.) One Hundred Years of Russell’s Paradox. Berlin: De Gruyter. pp. 517–532. McCarty, C. [2020] What is a number? Continua, magnitudes, quantities. This volume. Penfield, W. & E. Boldrey [1937] Somatic motor and sensory representation in the cerebral cortex of man as studied by electrical stimulation. Brain. Volume 60. Number 4. pp. 389–443. Proclus [1992] A Commentary on the First Book of Euclid’s Elements. Reprint Edition. G.R. Morrow (tr.) Princeton, NJ: Princeton University Press. lxix+355. Rice, H. [1953] Classes of recursively enumerable sets and their decision problems. Transactions of the American Mathematical Society. Volume 74. pp. 358–366. Scott, D. [1982] Domains for denotational semantics. M. Nielsen et al. Automata, Languages, and Programming. Proceedings of ICALP ’82. Aarhus, Denmark, July 12–16, 1982. Springer Lecture Notes in Computer Science. Volume 140. Berlin: Springer-Verlag. pp. 577–610. Shapiro, N. [1956] Degrees of computability. Transactions of the American Mathematical Society. Volume 82. pp. 281–299. Troelstra, A. [1973] Notes on intuitionistic second-order arithmetic. A. Mathias et al. (eds.) Cambridge Summer School in Mathematical Logic. Lecture Notes in Mathematics. Volume 337. Berlin: Springer-Verlag. pp. 171–205. Troelstra, A. [1977] Choice Sequences: A Chapter of Intuitionistic Mathematics. Oxford: Clarendon Press. ix+170. Troelstra, A. [1983] Analyzing choice sequences. Journal of Philosophical Logic. Volume 10. Number 2. May 1983. pp. 197–260. Troelstra, A. & D. van Dalen [1988] Constructivism in Mathematics. Volumes I and Volume II. Amsterdam: North-Holland Publishing Company. Van Atten, M. & D. van Dalen [2002] Arguments for the continuity principle. Bulletin of Symbolic Logic. Volume 8. Number 3. pp. 329–347. Van Atten, M, D. van Dalen, & R. Tieszen [2002] Brouwer and Weyl: the phenomenology and mathematics of the intuitive continuum. Philosophia Mathematica. Volume 10. Number 2. pp. 203–226.

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bibliography 327 Van Dalen, D. [1995] Hermann Weyl’s intuitionistic mathematics. The Bulletin of Symbolic Logic. Volume 1. Issue 2. June 1995. pp. 145–169. Van Dalen, D. [1997] How connected is the intuitionistic continuum? The Journal of Symbolic Logic. Volume 62. Number 4. pp. 1147–1150. Van Dalen, D. [2009] The return of the flowing continuum. Intellectica. Volume 51. Issue 1. pp.135–144. Van Heijenoort, J. (ed.) [1967] From Frege to Gödel: A Sourcebook in Mathematical Logic, 1879–1931. Cambridge, MA: Harvard University Press. x+660. Waismann, F. [1936] Einführung in das mathematische Denken: Die Begriffsbildung der modernen Mathematik. [Introduction to Mathematical Thinking: The Concept-Formation of Modern Mathematics.] Vienna: Gerold & Co. vii+188. Weyl, H. [1918] Das Kontinuum: Kritische Untersuchungen über die Grundlagen der Analysis. [The Continuum: Critical Investigations in the Foundations of Analysis]. Leipzig: Verlag von Veit & Comp. vi+83. (Translated and reprinted as [Weyl 1994].) Weyl, H. [1921] Über die neue Grundlagenkrise in der Mathematik. [Concerning the new foundations crisis in mathematics.] Mathematische Zeitschrift. Volume 10. pp. 37– 79. (Quotations and pagination from the translation On the new foundational crisis of mathematics. B. Müller (tr.) From Brouwer to Hilbert. Oxford: Oxford University Press. 1998. pp. 86–118.) Weyl, H. [1927] Philosophie der Mathematik und Naturwissenschaft. [Philosophy of mathematics and natural science.] A. Bäumler et al. (eds.) Handbuch der Philosophie. Abteilung II: Natur/Geist/Gott. [Handbook of Philosophy. Section II: Nature/Spirit/God.] Munich: R. Oldenbourg. pp. 1–162. Weyl, H. [1928] Diskussionsbermerkungen zu dem zweiten Hilbertschen Vortrag über die Grundlagen der Mathematik. [Comments on Hilbert’s second lecture on the foundations of mathematics.] Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität. Volume 6. pp. 86–88. (Quotations and pagination from the translation Comments on Hilbert’s second lecture on the foundations of mathematics. J. van Heijenoort (ed.) From Frege to Gödel: A Sourcebook in Mathematical Logic, 1879–1931. Cambridge, MA: Harvard University Press. 1967. pp. 482–484.) Weyl, H. [1963] Philosophy of Mathematics and Natural Science. O. Helmer et al. (trs.) New York, NY: Atheneum. x+312. (A translation and expansion of Weyl [1927].) Weyl, H. [1985] Axiomatic versus constructive procedures in mathematics. The Mathematical Intelligencer. Volume 7. Number 4. December 1985. pp. 3–17, 38. Weyl, H. [1994] The Continuum: A Critical Examination of the Foundation of Analysis. S. Pollard & T. Bole (trs.) New York, NY: Dover Publications, Inc. xxvi+130. Winskel, G. [1993] The Formal Semantics of Programming Languages: An Introduction. Cambridge, MA: MIT Press. xviii+361.

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13 The Peircean Continuum Francisco Vargas and Matthew E. Moore

Charles Sanders Peirce’s views on continuity, the concept he lionized as ‘the master-key which … unlocks the arcana of philosophy’ (Peirce 1893, p. 155), are of vital importance for students of his philosophy, but have received much less attention from historians and philosophers of continuity. This is partly because Peirce’s mathematics of continuity was still very much a work in progress when he died over a century ago. In the first and principal section of this essay, the first author summarizes the defining features of the mathematical theory of continuity that Peirce made the most progress on, and constructs a model for that theory in Zermelo–Fraenkel Set Theory with Choice (ZFC); this model provides a fuller mathematical vindication of Peirce’s conception than any reconstruction offered to date. The second section is a historical appendix, in which the second author briefly summarizes Peirce’s own attempts to put his conception into a rigorous form.1

1. A Model for Peirce’s Continuum The extraordinary success of analytic geometry and the calculus during the seventeenth and eighteenth centuries posed the foundational problem of the interrelationship between the geometric and the algebraic domains in mathematics, particularly with regard to the notion of limit.2 This led, as is well known, to a process of increasing arithmetization of analysis, and to an identification of the points on the line with numbers. This process culminates in the construction of the real numbers by Dedekind and Cantor. The rigour and usefulness of these constructions have made them predominant in today’s mainstream mathematics, but have also encouraged the uncritical ‘obvious’ assumption that we can identify the points in a line with the real numbers.3 In other words, what is often referred as the

1 We have not attempted to iron out all differences of exegetical and evaluative opinion. The reader should interpret the pronoun ‘we’, in the body of the paper, accordingly. 2 See Levey’s, Jesseph’s, Sutherland’s, Kanamori’s, and Schlimm/Haffner’s contributions to this volume. 3 We refer here to a predominant paradigm. Of course, this identification has not been exempt from perplexities and scrutiny, even by Dedekind and Cantor themselves (see Cavaillès 1965). Francisco Vargas and Matthew E. Moore, The Peircean Continuum In: The History of Continua: Philosophical and Mathematical Perspectives. Edited by: Stewart Shapiro and Geoffrey Hellman, Oxford University Press (2021). © Stewart Shapiro and Geoffrey Hellman. DOI: 10.1093/OSO/9780198809647.003.0014

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a model for peirce’s continuum 329 Cantor–Dedekind axiom establishes that there is a one-to-one order-preserving correspondence between the real numbers and the points in the geometrical line. The predominance of this standard view of ‘the continuum’ may well have posed a major obstacle to the understanding of Peirce’s ideas on the subject. But this does not entirely account for that lack of understanding; we must also note, in addition, that Peirce’s views on continuity evolved throughout his life and may seem perplexing, or even plainly contradictory, when not examined in perspective.⁴ The development of the distinctively Peircean conception of continuity reaches maturity roughly at the turn of the century, even if no definitive formulation (or even one satisfactory to Peirce himself) was ever reached.⁵ This can be largely attributed to the germinal stage of development of set theory at that time, in addition to the imperfect knowledge of it that Peirce himself managed to acquire. Cantorian transfinite cardinals are in fact at the very basis of the notion of ‘supermultitudinousness’ (see below). Nevertheless, we see Peirce still puzzled, in the last years of his life, about the possibility of linearly ordering arbitrary nondenumerable sets.⁶ As is now well known, of course, this possibility is established by Zermelo’s Theorem, an equivalent of the Axiom of Choice. As noted at the beginning of this essay, this section, and the construction to be offered in it, focus on the distinctive features of the concept of what we have been calling the ‘Peircean conception’ of continuity (developed by Peirce in what Havenel calls the ‘Supermultitudinous Period’).⁷ The major challenge, if we wish to vindicate Peirce’s ideas, is in fact the construction of an actual mathematical model which could show the total or partial realization of those ideas, in the same way that the real numbers are a model for the axioms of dense, complete linear orders. Is this feasible? In order to address this question we must look beyond the real number construction; the natural thing to do is to examine the alternative views and models for the continuum developed in the course of the twentieth century (see Zalamea (2012a) for a wide account of these developments). With regard to Peirce’s views, ⁴ An accurate chronological account, based upon close examination of the manuscripts, provided in Havenel’s (Havenel (2008)) subdivision of Peirce’s views into five major periods: Anti-nominalistic Period (1868–84), Cantorian Period (1884–92), Infinitesimal Period (1892–7), Supermultitudinous Period (1897–1907), and Topological Period (1908–13). We will refer to these periods in what follows, without extensive explanation. The reader should bear in mind that these periods should not be taken in a too literal or exclusive way; anti-nominalism is not found only in the ‘Anti-nominalistic’ period, and the same is true for infinitesimals and the period with that name. ⁵ Over the same period continuity became more closely intertwined with other aspects of Peirce’s philosophy: anti-nominalism, fallibilism, evolutionary cosmology, and the community-oriented idea of transcendence of our identification as individuals (which are posited as fictions, much as we consider ‘points’ as indivisibles whose existence is prior to that of the continuum). All this converges in his characterization of his whole philosophy as ‘Synechism’. ⁶ See selection 28 in Peirce (2010), and the final section below. ⁷ Even so, this treatment should cast light on most of the formulations and elaborations of the other periods.

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330 the peircean continuum we can actually see some aspects of these models as partial realizations of his conception, as we will now briefly argue. There are some significant analogies between Peirce’s continuum and intuitionistic conceptions, notably in their potential character and in the idea that the continuum’s existence is prior to that of the ‘points’ eventually determinable on it. Nevertheless, intuitionism is diametrically opposed to the actual infinities of the Cantorian universe, and even more so to Peirce’s idiosyncratic notion of ‘supermultitudinousness’.⁸ A more popular source nowadays of alternative models for the continuum is to be found in Abraham Robinson’s nonstandard analysis. Robinsonian models contain infinitesimals, which is another feature Peirce strongly endorsed. Nevertheless, because Robinsonian hyperreals are constructed by the addition of infinitesimals to the standard real numbers, they fail from the outset to provide an account of a continuum not composed of individuals or points (that is, they lack the ‘Inextensibility’ to be discussed in the next section).⁹ Furthermore, Peirce’s infinitesimals are better understood as ‘infinitesimal micro-segments’ than as ‘infinitesimal numbers’.1⁰ Similar remarks apply to the use of Conway’s surreal numbers in the reconstruction of the Peircean conception, as in the model ⟨NoP , y ≥ z) ⟹ x > z ¬(x > y) ⇔ y ≥ x ¬¬(x ≥ y) ⇔ ¬¬(y > x) (x ≥ y ≥ z) ⟹ x ≥ z (x ≥ y ∧ y ≥ x) ⟹ x = y ¬(x > y ∧ x = y) x ≥ 0 ⇔ ∀𝜀 < 0 (x > 𝜀) x + y > 0 ⟹ (x > 0 ∨ y > 0) x > 0 ⟹ −x < 0 (x > y ∧ z > 0) ⟹ yz > xz x#0 ⟹ x2 > 0 1 > 0 x2 > 0 0 < x < 1 ⟹ x > x2 x2 > 0 ⟹ x#0 n ∈ ℕ+ ⟹ n−1 > 0 if x > 0 and y ≥ 0, then there exists n ∈ ℤ such that nx > y x > 0 ⟹ x−1 > 0 xy > 0 ⟹ (x ≠ 0 ∨ y ≠ 0) if a < b, then there exists r ∈ ℚ such that a < r < b The constructive real line R as introduced above is a model of CA.

7. Smooth Infinitesimal Analysis Finally, we describe a remarkable new approach to the continuum, smooth infinitesimal analysis, which is the core of the synthetic differential geometry made possible by intuitionistic logic and realized through category theory.23 Smooth infinitesimal analysis and constructive analysis have entirely different origins, but they are linked in that they both require the use of intuitionistic logic. We describe a simple route into smooth infinitesimal analysis, through which infinitesimals—and intuitionistic logic—emerge naturally. In the usual development of the calculus, for any differentiable function f on the real line ℝ, y = f(x), it follows from Taylor’s theorem that the increment 𝛿y = f(x + 𝛿x) − f(x) in y

22 Bridges (1999), pp. 103–105. 23 For a detailed technical account of the development of synthetic differential geometry, see Kock (1981) and Moerdijk and Reyes (1991). Informal accounts are given in Bell (1998) and Bell (2019).

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smooth infinitesimal analysis 491 attendant on an increment 𝛿x in x is determined by an equation of the form 𝛿y = f ′ (x)𝛿x + A(𝛿x)2 ,

(1)

where f ′ (x) is the derivative of f(x) and A is a quantity whose value depends on both x and 𝛿x. Now if it were possible to take 𝛿x so small (but not demonstrably identical with 0) that (𝛿x)2 = 0 then (1) would assume the simple form f(x + 𝛿x) − f(x) = 𝛿y = f ′ (x)𝛿x.

(2)

We shall call a quantity having the property that its square is zero a microquantity2⁴ (also known as a nilsquare infinitesimal). In smooth infinitesimal analysis (SIA) sufficient microquantities are ‘present’ to ensure that equation (2) holds nontrivially for arbitrary functions f ∶ ℝ → ℝ. (Of course (2) holds trivially in standard mathematical analysis because there 0 is the sole microquantity.) The meaning of the term ‘nontrivial’ here may be explicated in the following way. If we replace 𝛿x by the letter 𝜀 standing for an arbitrary microquantity, a convention that we adopt in the sequel, (2) assumes the form f(x + 𝜀) − f(x) = 𝜀f ′ (x).

(3)

Ideally, we want the validity of this equation to be independent of 𝜀, that is, given x, for it to hold for all microquantities 𝜀. In that case the derivative f ′ (x) may be defined as the unique quantity D such that the equation f(x + 𝜀) − f(x) = 𝜀D holds for all microquantities 𝜀. Setting x = 0 in this equation, we get in particular f(𝜀) = f(0) + 𝜀D,

(4)

for all 𝜀. It is equation (4) that is taken as axiomatic in SIA. Let us write Δ for the set of microquantities, that is, Δ = {x ∶ x ∈ ℝ & x2 = 0}. Then it is postulated that, for any f ∶ Δ → ℝ, there is a unique D ∈ ℝ such that equation (4) holds for all 𝜀. This says that the graph of f is a straight line passing 2⁴ The use of the term ‘microquantity’ would make it natural to call our framework smooth microanalysis.

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492 intuitionistic accounts of the continuum today y = f(x) image under f of ∆ + a

∆

∆+a

Figure 17.3 Microintervals.

through (0, f(0)) with slope D. Thus any function on Δ is what mathematicians term affine, and so this postulate is naturally termed the principle of microaffineness.2⁵ It means that Δ cannot be bent or broken: it is subject only to translations and rotations—and yet is not (as it would have to be in ordinary analysis) identical with a point. Δ may be thought of as a tiny straight line possessing position and direction, but lacking true extension. If we think of a function y = f(x) as defining a curve, then, for any a, the image under f of the ‘microinterval’ Δ + a obtained by translating Δ to a is straight and coincides with the tangent to the curve at x = a (see Figure 17.3). In this sense each curve is ‘microstraight.’ From the principle of microaffineness we deduce the important Principle of microcancellation. If 𝜀a = 𝜀b for all 𝜀, then a = b. For the premise asserts that the graph of the function g ∶ Δ → ℝ defined by g(𝜀) = a𝜀 has both slope a and slope b: the uniqueness condition in the principle of microaffineness then gives a = b. The principle of microcancellation supplies the exact sense in which sufficient microquantities are ‘present’ in SIA. From the principle of microcancellation it follows that Δ is nondegenerate, i.e., not identical with {0}. For if Δ = {0}, we would have 𝜀.0 = 𝜀.1 = 0 for all 𝜀, and infinitesimal cancellation would give 0 = 1. From the principle of microaffineness it also follows that all functions on ℝ are continuous in the sense that each sends neighbouring points to neighbouring points. Here two points x, y on ℝ are said to be neighbours if x − y is in Δ, that is, if x and y differ by a microquantity. To see this, given f ∶ ℝ → ℝ and neighbouring points x, y, note that y = x + 𝜀 with 𝜀 in Δ, so that f(y) − f(x) = f(x + 𝜀) − f(x) = 𝜀f ′ (x). But clearly any multiple of a microquantity is also a microquantity, so 𝜀f ′ (x) is a microquantity, and the result follows. 2⁵ It is usually known as the Kock–Lawvere axiom, after its inventors, F.W. Lawvere and Anders Kock.

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smooth infinitesimal analysis 493 In fact, since equation (3) holds for any f, it also holds for its derivative f ′ ; it follows that functions in SIA are differentiable arbitrarily many times, thereby justifying the use of the term ‘smooth’. Let us derive in SIA a basic law of the differential calculus, the product rule: (fg)′ = f ′ g + fg′ . To do this we compute (fg)(x + 𝜀) = (fg)(x) + (fg)′ (x) = f(x)g(x) + (fg)′ (x), (fg)(x + 𝜀) = f(x + 𝜖)g(x + 𝜀) = [f(x) + f ′ (x)] . [g(x) + g′(x)] = f(x)g(x) + 𝜀(f ′ g + fg′ ) + 𝜀2 f ′ g′ = f(x)g(x) + 𝜀(f ′ g + fg′ ), since 𝜀2 = 0. Therefore 𝜀(fg)′ = 𝜀(f ′ g + fg′ ), and the result follows by microcancellation. We observe that the postulates of SIA are incompatible with the law of excluded middle of classical logic. This incompatibility can be demonstrated in two ways, one informal and the other rigorous. First the informal argument. Consider the function f defined for real numbers x by f(x) = 1 if x = 0 and f(x) = 0 whenever x ≠ 0. If the law of excluded middle held, each real number would then be either equal or unequal to 0, so that the function f would be defined on the whole of ℝ. But, considered as a function with domain ℝ, f is clearly discontinuous. Since, as we know, in SIA every function on ℝ is continuous, f cannot have domain ℝ there.2⁶ So the law of excluded middle fails in SIA. To put it succinctly, universal continuity in SIA implies the failure of the law of excluded middle.2⁷ Here is the rigorous argument. We show that the failure of the law of excluded middle can be derived from the principle of microcancellation. To begin with, if x ≠ 0, then x2 ≠ 0, so that, if x2 = 0, then necessarily not x ≠ 0. This means that for all microquantities 𝜀, not 𝜀 ≠ 0.

(*)

2⁶ The domain of f is in fact (ℝ − {0}) ∪ {0}, which, because of the failure of the law of excluded middle in SIA, is provably unequal to ℝ. 2⁷ Thus the reasons for abandoning the law of excluded middle in SIA and CA are entirely different. In SIA the law is actually refutable for the ‘objective’ reason that in a world in which all change is actually continuous, the validity of that law would produce an instance of discontinuous change. In CA, by contrast, the law of excluded middle fails to be affirmable essentially on the epistemic ‘subjective’ grounds that the truth or falsity of a proposition may not, in principle, be knowable.

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494 intuitionistic accounts of the continuum today Now suppose that the law of excluded middle were to hold. Then we would have, for any 𝜀, either 𝜀 = 0 or 𝜀 ≠ 0. But (*) allows us to eliminate the second alternative, and we infer that, for all 𝜀, 𝜀 = 0. This may be written for all 𝜀, 𝜀.1 = 𝜀.0, from which we derive by microcancellation the falsehood 1 = 0. So again the law of excluded middle must fail. The ‘internal’ logic of SIA is accordingly not full classical logic. It is, in fact, intuitionistic logic. In our brief sketch we did not notice this ‘change of logic’ because, like much of elementary mathematics, the topics we discussed are naturally treated by constructive means such as direct computation. Note that it follows from the principle of microaffineness that ℝ can be identified as the subset V of Δ∆ consisting of all maps vanishing at 0. In this sense ℝ is ‘generated’ by Δ. Explicitly, Δ∆ is a monoid under composition which may be regarded as acting on Δ by application: for f ∈ Δ∆ , f ⋅ 𝜀 = f(𝜀). Then V is a submonoid naturally identified as the set of ratios of infinitesimals. The identification of ℝ and V made possible by the principle of microaffineness thus leads to the characterization of ℝ itself as the set of ratios of infinitesimals. This was essentially the view of Euler, who regarded infinitesimals as formal zeros and 0 real numbers as representing the possible values of . For this reason Lawvere2⁸ 0 has suggested that ℝ in SIA should be called the space of Euler reals. Once one has ℝ, Euclidean spaces of all dimensions may be obtained as powers of ℝ, and arbitrary manifolds may be obtained by patching together subspaces of these.

8. Algebraic and Order Structure of ℝ in SIA What are the algebraic and order structures on ℝ in SIA? As far as the former is concerned, there is little difference from the classical situation: in SIA ℝ is equipped with the usual addition and multiplication operations under which it is a field. In particular, ℝ satisfies the condition that each x ≠ 0 has a multiplicative inverse. Notice, however, that since in SIA no microquantity (apart from 0 itself) is provably ≠ 0, microquantities are not required to have multiplicative inverses (a requirement which would lead to inconsistency). From a strictly algebraic standpoint, ℝ in SIA differs from its classical counterpart only in being required to satisfy the principle of microcancellation.

2⁸ Lawvere (2011).

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comparing sia and ca 495 J I ∆ (

(

(

)

)

)

0

Figure 17.4 Microneighbourhoods.

The situation is different, however, as regards the order structure of ℝ in SIA. Because of the failure of the law of excluded middle, the order relation < on ℝ in SIA cannot satisfy the trichotomy law x < y ∨ y < x ∨ x = y, and accordingly < must be a partial, rather than a total ordering. Since microquantities do not have multiplicative inverses, and ℝ is a field, any microquantity 𝜀 must satisfy ¬𝜀 < 0 ∧ ¬𝜀 > 0. Accordingly, if we define the relation ≤ (‘not less than’) by x ≤ y iff ¬(y < x), then, for any microquantity 𝜀 we have 𝜀 ≤ 0 ∧ 𝜀 ≥ 0. Using these ideas we can identify three distinct microneighbourhoods of 0 on ℝ in SIA, each of which is included in its successor (Figure 17.4). First, we have the set Δ of microquantities itself, next, the set I = {x ∈ ℝ ∶ ¬x ≠ 0} of elements indistinguishable from 0; finally, the set J = {x ∈ ℝ ∶ x ≤ 0 ∧ x ≥ 0} of elements neither less nor greater than 0. These three may be thought of as the microneighbourhoods of 0 defined algebraically, logically, and order-theoretically, respectively. Observe that none of these is degenerate.

9. Comparing SIA and CA SIA may be furnished with the following axiomatic description: Axioms for the continuum, or smooth real line R. These are the usual axioms for a field expressed in terms of two operations + and ⋅ and two distinguished elements 0, 1. In particular every nonzero element of R is invertible.

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496 intuitionistic accounts of the continuum today Axioms for the strict order relation < on R. These are: 1. 2. 3. 4. 5.

a < b and b < c implies a < c. ¬(a < a) a < b implies a + c < b + c for any c. a < b and 0 < c implies a.c < b.c. if a < b, then, for any x, either a < x or x < b.

The subset Δ = {x ∶ x2 = 0} of R is subject to the Microaffineness Principle. For any map g ∶ Δ → R there exist unique a, b ∈ R such that, for all 𝜀, we have g(𝜀) = a + b.𝜀. From these three axioms it follows that the continuum in SIA differs in certain key respects from its counterpart in CA. To begin with, a basic property of the strict ordering relation < in CA, namely, ¬(x < y ∨ y < x) → x = y

(*)

is incompatible with the axioms of SIA. For (*) implies ∀x ¬(x < 0 ∨ 0 < x) → x = 0.

(**)

Thus in CA the set Δ of microquantities would be degenerate (i.e., identical with {0}), while, as we have seen, the nondegeneracy of Δ in SIA is one of its characteristic features. Call a binary relation S on R stable if it satisfies ∀x ∀y (¬¬xRy → xRy). As we have observed, in CA, the equality relation is stable. But in SIA it is not stable, for, if it were, I, and so also Δ, would be degenerate, which we have observed is not the case in SIA.

10. Cohesiveness of the Continuum in SIA We have remarked that the intuitionistic continuum is cohesive, that is, cannot be split into two nonempty disjoint parts. This is also true of the continuum and closed intervals in (most versions of) SIA. This is a consequence of the Constancy

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cohesiveness of the continuum in sia 497 Principle, a postulate concerning stationary points of functions adopted in (most versions) of SIA. In SIA a stationary point of a function f ∶ R → R is defined to be a point in whose vicinity the value of f remains unchanged under ‘microvariations’, that is, an a ∈ R such that f(a + 𝜀) = f(a) for all 𝜀. This means that f(a) + 𝜀f ′ (a) = f(a), so that 𝜀f ′ (a) = 0 for all 𝜀, whence it follows from microcancellation that f ′ (a) = 0. This is a version of Fermat’s rule. The Constancy Principle may be stated: if every point in an interval J is a stationary point of f ∶ J → R (that is, if f ′ is identically 0), then f is constant. Put succinctly, ‘universal local constancy implies global constancy’. It follows from this that two functions with identical derivatives differ by at most a constant. The cohesiveness of R, and also of arbitrary closed intervals, in SIA is a straightforward consequence of the Constancy Principle. For suppose R = U ∪ V with U ∩ V = ∅. Define f ∶ R → {0, 1} by f(x) = 1 if x ∈ U, f(x) = 0 if x ∈ V. We claim that f is constant. For we have (f(x) = 0 or f(x) = 1) & (f(x + 𝜀) = 0 or f(x + 𝜀) = 1). This gives four possibilities: (i) (ii) (iii) (iv)

f(x) = 0 & f(x + 𝜀) = 0 f(x) = 0 & f(x + 𝜀) = 1 f(x) = 1 & f(x + 𝜀) = 0 f(x) = 1 & f(x + 𝜀) = 1

Possibilities (ii) and (iii) may be ruled out because f is continuous. This leaves (i) and (iv), in either of which f(x) = f(x + 𝜀). So f is locally, and hence globally, constant, that is, constantly 1 or 0. In the first case V = ∅, and in the second U = ∅. The argument for an arbitrary closed interval is similar. From the cohesiveness of closed intervals it follows without difficulty that all intervals in R are cohesive. In some versions of SIA the ordering of R is subject to the axiom of distinguishability: x ≠ y ⟹ x < y ∨ y < x.

(*)

Aside from certain microsubsets to be discussed below, in these versions of SIA cohesive subsets of R correspond to connected subsets of R in classical analysis, that is, to intervals. In particular, in versions of SIA subject to (*) any puncturing of R is decomposable, i.e., not cohesive, for it follows immediately from (*) that R − a = {x ∶ x > a} ∪ {x ∶ x < a}.

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498 intuitionistic accounts of the continuum today Similarly, the set R − ℚ of irrational numbers is decomposable as R − ℚ = {{x ∶ x > 0} − ℚ} ∪ {{x ∶ x < 0} − ℚ}. This is in sharp contrast to the intuitionistic continuum. For, as we have observed above, not only is any puncturing of the intuitionistic continuum cohesive, but this is even the case for the set of irrational numbers. This would seem to indicate that in some sense the continuum in SIA is considerably less ‘syrupy’2⁹ than its intuitionistic counterpart. It can be shown that the various microneighbourhoods of 0 are cohesive. For example, the cohesiveness of Δ can be established as follows. Suppose f ∶ Δ → {0, 1}. Then by the Microaffineness Principle there are unique a, b ∈ R such that f(𝜀) = a + b.𝜀 for all 𝜀. Now a = f(0) = 0 or 1; if a = 0, then b.𝜀 = f(𝜀) = 0 or 1, and clearly b.𝜀 ≠ 1. So in this case f(𝜀) = 0 for all 𝜀. If on the other hand a = 1, then 1 + b.𝜀 = f(𝜀) = 0 or 1; but 1 + b.𝜀 = 0 would imply b.𝜀 = −1 which is again impossible. So in this case f(𝜀) = 1 for all 𝜀. Therefore f is constant and Δ cohesive. In SIA nilpotent infinitesimals are defined to be the members of the sets Δk = {x ∈ R ∶ xk+1 = 0}, for k = 1, 2, . . . , each of which may be considered a microneighbourhood of 0. These are subject to the Micropolynomiality Principle. For any k ≥ 1 and any g ∶ Δk → R, there exist unique a, b1 , . . . , bk ∈ R such that for all 𝛿 ∈ Δk we have g(𝛿) = a + b1 𝛿 + b2 𝛿 2 + . . . + bk 𝛿 k . Micropolynomiality implies that no Δk coincides with {0}. An argument similar to that establishing the cohesiveness of Δ does the same for each Δk . Thus let f ∶ Δk → {0, 1}; Micropolynomiality implies the existence of a, b1 , . . . , bk ∈ R such that f(𝛿) = a + 𝜁(𝛿), where 𝜁(𝛿) = b1 𝛿 + b2 𝛿 2 + . . . + bk 𝛿 k . Notice that 𝜁(𝛿) ∈ Δk , that is, 𝜁(𝛿) is nilpotent. Now a = f(0) = 0 or 1; if a = 0 then 𝜁(𝛿) = f(𝛿) = 0 or 1, but since 𝜁(𝛿) is nilpotent it cannot = 1. Accordingly in this case f(𝛿) = 0 for all 𝛿 ∈ Δk . If on the other hand a = 1, then 1 + 𝜁(𝛿) = f(𝛿) = 0 or 1, but 1 + 𝜁(𝛿) = 0 would imply 𝜁(𝛿) = −1 which is again impossible. Accordingly f is constant and Δk cohesive.

2⁹ It should be emphasized that this phenomenon is a consequence of (*): it cannot necessarily be affirmed in versions of SIA not including this axiom.

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natural numbers and invertible infinitesimals in sia 499 The union D of all the Δk is the set of nilpotent infinitesimals, another microneighbourhood of 0. The cohesiveness of D follows immediately by applying the lemma above. The next microneighbourhood of 0 is the closed interval [0, 0], which, as a closed interval, is cohesive. It is easily shown that [0, 0] includes D, so that it does not coincide with {0}. It is also easily shown, using axioms 2 and 6, that [0, 0] coincides with the set I = {x ∈ R ∶ ¬¬x = 0}. So I is cohesive. (In fact the cohesiveness of I can be proved independently of axioms 1–6 through the general observation that, if A is cohesive, then so is the set A∗ = {x ∶ ¬¬x ∈ A}.) Finally, we observe that the sequence of microneighbourhoods of 0 generates a strictly ascending sequence of decomposable subsets containing R − {0}, namely: R − {0} ⊂ (R − {0}) ∪ {0} ⊂ (R − {0}) ∪ Δ1 ⊂ . . . ⊂ (R − {0}) ∪ D ⊂ (R − {0}) ∪ [0, 0].

⊂

(R − {0}) ∪ Δ2

11. Natural Numbers and Invertible Infinitesimals in SIA In SIA the system of natural numbers can possess some subtle and intriguing features which make it possible to introduce another type of infinitesimal—the socalled invertible infinitesimals—resembling those of nonstandard analysis, whose presence engenders yet another infinitesimal microneighbourhood of 0 properly containing all those introduced above. In SIA the set ℕ of natural numbers can be defined to be the smallest subset of R which contains 0 and is closed under the operation of adding 1. In some models of SIA, R satisfies the Archimedean principle that every real number is majorized by a natural number. However, models of SIA have been constructed in which R is not Archimedean in this sense. In these models it is more natural to consider, in place of ℕ, the set ℕ∗ of smooth natural numbers defined by ℕ∗ = {x ∈ R ∶ 0 ≤ x ∧ sin𝜋x = 0}. ℕ∗ is the set of points of intersection of the smooth curve y = sin𝜋x with the positive x-axis (Figure 17.5). In these models R can be shown to possess the ∗ is replaced by . In Archimedean property provided that in the definition ∗ ∗ these models, then, ℕ is a proper subset of ℕ : the members of ℕ − ℕ may be considered nonstandard integers. Multiplicative inverses of nonstandard integers are infinitesimals, but, being themselves invertible, they are of a different type from

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500 intuitionistic accounts of the continuum today y = sin πx

0

1

2

Figure 17.5 Defining smooth natural numbers.

the ones we have considered so far. It is quite easy to show that they, as well as the infinitesimals in J (and so also those in Δ and I), are all contained in the set—a further microneighbourhood of 0— K = {x ∈ R ∶ ∀n ∈ ℕ

−1 1 y.⁵

2. The Emergence of Non-Archimedean Systems of Magnitudes Even before Cantor (1872) and Dedekind (1872) had published the modern theories of real numbers that would be employed to all but banish infinitesimals from late nineteenth- and pre-Robinsonian, twentieth-century analysis, Johannes Thomae (1870) and Paul du Bois-Reymond (1870–1) were beginning the process that would in the years bracketing the turn of the century not only establish consistent and relatively sophisticated theories of infinitesimals in mainstream mathematics but make them the focal point of great interest and a mathematically profound and philosophically significant research programme. One theory grew out of the pioneering investigations of non-Archimedean geometry of Giuseppi Veronese (1889, 1891, 1894), Tullio Levi-Civita (1892/3, 1898), and David Hilbert (1899), and led to the celebrated algebraico-set-theoretic work of

⁴ For NBG and its relation to ZFC, see (Mendelson 2010). ⁵ The concepts of ordered class, divisible ordered abelian group, ordered commutative ring with identity, ordered field, and positive cone of an ordered abelian group play prominent roles in portions of the text. For the reader’s convenience, the definitions of these and some related concepts are collected below, where “ordered” is understood to mean “totally ordered”. An ordered class is a structure ⟨A, ≤⟩, where A is a class (a set or proper class) and ≤ is a binary relation on A that satisfies the conditions: ∀xy(x ≤ y ∨ y ≤ x); ∀xyz((x ≤ y ∧ y ≤ z) → x ≤ z); ∀xy((x ≤ y ∧ y ≤ x) → x = y). If x ≤ y and x ≠ y, we write x < y or y > x. An ordered abelian group (written additively) is a structure ⟨A, ≤, +, 0⟩, where ⟨A, ≤⟩ is an ordered class and + is a commutative, associative binary operation on A satisfying the conditions: ∀x(x+0 = x); ∀x∃y(x+y = 0); ∀xyz(x ≤ y → (x+z ≤ y+z)). An ordered abelian group A is divisible, if for each x ∈ A and each positive integer n, there is an a ∈ A such that na = x. An ordered abelian group ⟨A, ≤, ⋅, 1⟩ (written multiplicatively) is defined similarly using ⋅ and 1 in place of + and 0, and divisibility is defined similarly by the condition: for each x ∈ A and each positive integer n, there is an a ∈ A such that an = x. An ordered commutative ring with identity (or with unity) is a structure ⟨A, ≤, +, ⋅, 0, 1⟩, where ⟨A, ≤, +, 0⟩ is an ordered abelian group, 0 ≠ 1, and ⋅ is a commutative, associative binary operation on A that satisfies the conditions: ∀x(x⋅1 = x); ∀xyz[(x⋅(y+z) = (x⋅y)+(x⋅z)]; ∀xy((0 ≤ x∧0 ≤ y) → 0 ≤ xy). An ordered field is an ordered commutative ring with identity that satisfies the further condition: ∀x[x ≠ 0 → ∃y(x ⋅ y = 1)]. The ordered additive group of every ordered field is divisible. The positive cone of an ordered additive abelian group A is {x ∈ A ∶ x > 0}. A nonzero element x of an ordered commutative ring A with identity is said to be a proper zerodivisor if for some nonzero y ∈ A, xy = 0. An ordered commutative ring with identity is said to be an ordered integral domain if it has no proper zero-divisor. In the case of an ordered integral domain the condition ∀xy((0 ≤ x ∧ 0 ≤ y) → 0 ≤ xy) can be replaced by ∀xy((0 < x ∧ 0 < y) → 0 < xy). An ordered field is an ordered integral domain.

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the archimedean axiom 505 Hans Hahn (1907). And another emerged from a parallel development of du BoisReymond’s (1870–1, 1875, 1877, 1882) groundbreaking work on the rates of growth of real functions and led in the same period to the famous works of G. H. Hardy (1910, 1912) and some less well known though important work of Felix Hausdorff (1907, 1909). Each of these research programmes, which collectively gave rise to modern-day non-Archimedean mathematics, led to nonstandard theories of continua that have come to play important roles in the mathematics of our day, albeit not always in the guise of theories of continua. Before turning to these research programmes and the nonstandard theories of continua they gave rise to, we will first provide some historical background on the Archimedean axiom.

3. The Archimedean Axiom In his historically important paper Zur Geometrie der Alten, insbesondere über ein Axiom des Archimedes (On the Geometry of the Greeks, in Particular, on the Axiom of Archimedes) Otto Stolz observed that: It has often been noted that Euclid implicitly used the principle: a magnitude can be so often multiplied that it exceeds any other of the same kind . . . Archimedes employed this principle as an explicit axiom in some of his works . . . For brevity, we will therefore henceforth refer to this principle as the Axiom of Archimedes. To investigate whether or not this is a necessary proposition, requires us first to have agreement on a characterization of the concept of ‘magnitude’. (1883, p. 504)

Such agreement was required for, as Stolz emphasized, the term “magnitude” occurs in Euclid’s Elements, but he nowhere explains the concept. In response to his query, Stolz provided an axiomatization for the type of ordered additive systems of line segments occurring in the Elements—an ordered, additive system of equivalence classes (of congruent line segments) constituting what we today call the positive cone of a divisible (additively written) ordered abelian group, and therewith established the following result: whereas every such system of magnitudes that is continuous in the sense of Dedekind is Archimedean, there are such systems of magnitudes that are non-Archimedean. To establish the existence of a latter such system he made use of an algebraic development of a fragment of du Bois-Reymond’s system of orders of infinity that emerged from the latter’s aforementioned work on the rates of growth of real functions (see section 6).⁶

⁶ See (Ehrlich 2012, pp. 24–27) for Stolz’s construction.

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506 contemporary infinitesimalist theories of continua It was with these and related discoveries that Stolz (1883, 1884, 1885), Rodolfo Bettazzi (1890), Veronese (1889), and Otto Hölder (1901) laid the groundwork for the modern theory of magnitudes, the branch of late nineteenth- and early twentieth-century mathematical philosophy that would, in the decades that followed, evolve into the theories of Archimedean and non-Archimedean ordered algebraic systems, theories in which talk of systems of magnitudes would be replaced by talk of ordered abelian groups, ordered fields, the positive cones of such structures, and so on. The non-Archimedean systems of magnitudes studied in the just-mentioned works are additive structures that sometimes have modest multiplicative structures as well. Unlike the system of real numbers, however, none of them is an ordered field. Just as ordered fields of real numbers arose in conjunction with the study of Euclidean geometry, it was from the study of non-Archimedean geometry that non-Archimedean ordered fields emerged. It was also from nonArchimedean geometry that the first well-developed non-Archimedean theory of continua emerged.

4. Veronese’s Theory of Continua Following Wallis’s and Newton’s incorporation of directed segments into Cartesian geometry, it became loosely understood that given a unit segment AB of a line L of a classical Euclidean space, the collection of directed segments of L emanating from A, including the degenerate segment AA itself, constitutes an Archimedean ordered field with AA and AB the additive and multiplicative identities of the field and addition and multiplication of segments suitably defined. This idea was made precise by Veronese (1891, 1894) and Hilbert (1899) in their pioneering works on the foundations of geometry. Also emerging from these works, and inspired in part by the aforementioned work of Stolz, was the idea that it is possible to construct an axiomatization for the central theorems of Euclidean geometry that is independent of the Archimedean axiom and for which the aforementioned system of line segments in a model of the geometry continues to be an ordered field; however, in those models of the geometry in which the Archimedean axiom fails, the ordered fields in question are non-Archimedean ordered fields. Following Veronese,⁷ two segments s and s′ are said to be finite relative to one another if there are positive integers m and n such that s′ ≺ ms and s ≺ ns′ , where a ≺ b indicates that a is congruent to a proper subsegment of b, i.e., a subsegment ⁷ For the remainder of this section, except when referring to an ordered field of segments, we employ the terms “segment” and “subsegment” without the modifiers “directed” or “degenerate” to refer to a nondegenerate line segment without a direction.

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veronese’s theory of continua 507 of b that is not identical to b. In accordance with this terminology, a system S of segments may be said to be Archimedean if every pair of such segments is finite relative to one another. If S is non-Archimedean, there are segments s and s′ that are not finite relative to one another, in which case s is said to be infinitesimal relative to s′ and s′ is said to be infinite relative to s, if s ≺ s′ . Thus, s is infinitesimal relative to s′ and s′ is infinite relative to s just in case ns ≺ s′ for all positive integers n. If S is the positive cone of a non-Archimedean ordered field, then for each segment s in S there are segments s′ and s″ in S such that s is infinite relative to s′ and infinitesimal relative to s″ . Unlike the analytic constructions of Hilbert, Veronese’s construction of a nonArchimedean ordered field of line segments is clumsy and quite complicated, though to some extent the complexity is a by-product of what he is attempting to achieve. After all, Veronese is not merely attempting to construct a nonArchimedean ordered field of segments that is appropriate to the geometry in question, but moreover an ordered field of such segments that models his novel theory of non-Archimedean continua, and one that is synthetically constructed to boot. Veronese had a distinctive, well-developed philosophy of geometry that underlay his philosophy of the continuum. For the sake of space, we simply note that in addition to holding that our conception of geometric continua should be grounded in our conception of magnitude rather than in number or in points, Veronese held that just as it is compatible with geometrical intuition that the parallel postulate fails, it is compatible with our geometrical intuition that continua need not be Archimedean. It was for these reasons that, unlike Cantor and Dedekind, Veronese sought a segment-based, synthetic theory of geometric continua that is independent of the Archimedean condition.⁸ Despite the lack of elegance in its presentation and elements of obscurity in its formulation, the theory of rectilinear continua developed in Veronese’s Fondamenti di Geometria (1891) is a profound and relatively sophisticated scheme, several of whose central concepts and ideas permeate the twentieth-century theory of ordered algebraic systems and through it nonstandard analysis, the theory of the rates of growth of functions, and the theory of surreal numbers. For our purpose here, however, we limit our attention to its two continuity conditions, each of which, unlike the Dedekind continuity condition, is satisfiable by Archimedean as well as non-Archimedean ordered abelian groups and ordered fields. In fact, as Veronese (1889, 1891) and Levi-Civita (1898) collectively demonstrate, each of ⁸ While Veronese has a segment-based, rather than a point-set based, theory of continua, the notion of a point is a primitive of his geometric system. For further details on Veronese’s philosophy of geometry, see (Veronese 1909/1994) and (Cantù 1999).

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508 contemporary infinitesimalist theories of continua the conditions is equivalent to the Dedekind continuity condition if and only if the Archimedean axiom is assumed. Veronese formulates his two continuity conditions as follows. Relative Continuity Condition. Every segment XX′ whose ends vary in opposite directions and becomes indefinitely small contains an element outside the domains of variability of its ends (Veronese 1891, p. 128). Absolute Continuity Condition. Every segment XX′ whose ends vary in opposite directions and becomes indefinitely small in the absolute sense contains an element outside the domains of variability of its ends (Veronese 1891, p. 150). Veronese unpacks his continuity conditions in terms of a variable segment XX′ that is the difference AX′ −AX of a pair of subsegments of a segment AB, where AX is a proper subsegment of AX′ , henceforth written AX ≺∗ AX′ . While keeping A fixed and preserving the condition that AX ≺∗ AX′ , X is envisioned to increase in a strict monotonic fashion (without a greatest member) as X′ decreases in a strict monotonic fashion (without a least member), ⃗ ⃖′ ..........B A..........X.......... X subject to the following conditions that flesh out Veronese’s conceptions of becoming “indefinitely small” and “indefinitely small in the absolute sense”, respectively. Indefinitely Small. For each segment s that is finite relative to an arbitrarily given unit segment which could be taken to be AB, X and X′ take on values Xs and X′s , respectively, where Xs X′s ≺ s. Indefinitely Small in the Absolute Sense. For each segment s, X and X′ take on values Xs and X′s , respectively, where Xs X′s ≺ s. Given the satisfaction of these respective conditions, Veronese’s continuity conditions assert that there is a segment AY such that AXs ≺∗ AY ≺∗ AX′s for all such values Xs of X and all such values X′s of X′ . Veronese refers to his first continuity condition as a relative continuity condition since it is concerned with families of segments that grow arbitrarily small subject to the proviso that they remain finite relative to a given unit segment. The absolute continuity condition, by contrast, is concerned with families of segments that grow arbitrarily small subject to the limits of the geometric space itself.

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veronese’s theory of continua 509 Veronese’s relative continuity condition ensures that if one limits oneself to the segments that are finite relative to an arbitrarily selected segment s and if one collects together into equivalence classes all such segments that differ from one another by amounts that are infinitesimal relative to s, the resulting system of equivalence classes with order defined in the expected manner is isomorphic to the standard continuum. Moreover, if (as in Veronese’s geometry) the system of directed segments on a line emanating from a point is an ordered field, then if one takes the equivalence class containing s as the unit and defines addition and multiplication of the equivalence classes in the familiar geometrical fashion, the resulting system is isomorphic to the positive cone ℝ+ of the ordered field of real numbers. That is, Veronese’s non-Archimedean continuum is indistinguishable from the classical continuum when infinite and infinitesimal differences are ignored.⁹ Unlike the segment AY in the relative continuity condition, which is unique if and only if the Archimedean axiom holds, the segment AY in the absolute continuity hypothesis is invariably unique, as Veronese was well aware. For this reason, Veronese’s absolute continuity condition may also be stated in the following algebraic form that more clearly highlights its relation to the continuity condition of Dedekind. Absolute Continuity: Algebraic Formulation. Let G be an ordered abelian group (or the positive cone thereof). If (A, B) is a Dedekind cut of G and if for each positive 𝜖 ∈ G there are elements a of A and b of B for which b − a < 𝜖, then either A has a greatest member or B has a least member, but not both. It is a simple matter to show that in the Archimedean case, and only in the Archimedean case, Veronese’s metrical condition on cuts is invariably satisfied. Thus, for Veronese, unlike for Dedekind, continuous systems of magnitudes need not be completely devoid of Dedekind gaps, though they must be devoid of those Dedekind gaps that satisfy the metrical condition satisfied in the classical case. Veronese maintained that insofar as the intuitive conception of a continuum does not require the Archimedean axiom, it is his absolute continuity condition, rather than Dedekind’s continuity condition, that is intuitively more justifiable. The algebraic and geometric formulations of Veronese’s absolute continuity condition were widely discussed during the first decade of the twentieth century by authors such as Hölder, A. Schoenflies, L. E. J. Brouwer, K. Vahlen, G. Vitali, F. Enriques, and Hahn1⁰ before sinking into relative obscurity as the Cantor– Dedekind conception of the continuum solidified its status as the standard ⁹ In addition to a non-Archimedean Euclidean space, which is what we are considering above, Veronese considers a non-Archimedean elliptic space, where the system of directed segments emanating from a point has a somewhat different structure. However, in the elliptic case, the import of his relative continuity condition is much the same. 1⁰ For references, see (Ehrlich 2006, pp. 66–70).

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510 contemporary infinitesimalist theories of continua conception. However, it was resurrected (without reference to Veronese) by a number of authors including R. Baer (1929, 1970), L. W. Cohen and C. Goffman (1949), K. Hauschild (1966), and Dana Scott (1969a), who (along with others) carefully studied it as a completeness condition. Among the results that emerged from these investigations is that every nontrivial densely ordered abelian group (ordered field) admits an extension, unique up to isomorphsim, to a least ordered abelian group (ordered field) that satisfies Veronese’s absolute continuity condition. Motivated by the work of Cohen and Goffman and especially Scott, and unaware of the Veronesean roots of the condition, Zakon (1969, p. 226) asked if nonstandard models of analysis are complete in the absolute sense of Veronese. In response it was found that some are (e.g., Keisler 1974; Keisler and Schmerl 1991; Jin and Keisler 1993) and others are not (e.g., Kamo 1981, 1981a; Keisler and Schmerl 1991; Ozawa 1995), something we will return to in section 8. Those that are are usually said to be Scott Complete.11 On the other hand, since (as can be readily shown) an ordered field is continuous in the relative sense of Veronese if and only if it contains an isomorphic copy of ℝ, every nonstandard model of analysis satisfies Veronese’s relative continuity condition. The same is also true of the ordered fields due to Hahn discussed in the following section, the first group of which drew inspiration from the Cantor–Dedekind continuum as well as the non-Archimedean continua of Veronese.

5. Hahn’s Non-Archimedean Generalizations of the Archimedean Arithmetic Continuum Though Veronese’s construction of his non-Archimedean geometric continuum is synthetic, he represents the line segments that emerge from his construction using a loosely defined, complicated system of numbers consisting of finite and transfinite series of the form y

y

y

∞11 r1 + ∞12 r2 + ∞13 r3 + ⋯ y

y

y

where r1 , r2 , r3 , … are real numbers, and ∞11 , ∞12 , ∞13 , … is a sequence of units, each of which is infinitesimal relative to the preceding units, ∞1 being the number (of “infinite order 1” (1891, p. 101)) introduced by Veronese to represent the infinitely large line segment whose existence is postulated by his “hypothesis on the existence of bounded infinitely large segments” (1891, p. 84). Veronese’s number system was provided an analytic foundation by Levi-Civita (1892–3, 1898), 11 Veronese’s absolute continuity condition has emerged as a standard concept in the theory of ordered algebraic systems, albeit usually under a variety of other names. For further references, see (Ehrlich 1997, p. 224).

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hahn’s non-archimedean continua 511 who therewith provided the first analytic constructions of non-Archimedean ordered fields. Building on the work of Levi-Civita, Hahn (1907) constructed nonArchimedean ordered fields (and ordered abelian groups more generally) having properties that generalize the familiar continuity properties of Dedekind and Hilbert (Ehrlich 1995, 1997, 1997a), and he demonstrated (vis-à-vis his celebrated Hahn Embedding Theorem for ordered abelian groups12) that his number systems provide a panorama of the finite, infinite, and infinitesimal numbers that can enter into a non-Archimedean theory of continua based on the concept of an ordered field (Ehrlich 1995, 1997, 1997a). This idea was later brought into sharper focus when it was demonstrated that every ordered field can be embedded in a perspicuous fashion in a suitable Hahn field,13 the general construction of the latter of which is given as follows. Hahn Field 1 (Hahn 1907) Let ℝ be the ordered field of real numbers and G be a nontrivial ordered abelian group. The collection, ℝ((tG )) of all series ∑ tg𝛼 ⋅ r𝛼 , 𝛼 a and n|a| > b; if a and b are not Archimedean equivalent, then a is said to be infinitesimal (in absolute value) relative to b and b is said to be infinite (in absolute value) relative to a, if |a| < |b|. In accordance with these conventions, 0 is infinitesimal (in absolute value) relative to every other member of G. Moreover, if G is the additive group of an ordered field or of an ordered ring with a unit or identity more generally, the elements are simply said to be infinite (in absolute value) and infinitesimal (in absolute value), respectively, if they are infinite (in absolute value) relative to and infinitesimal (in absolute value) relative to the identity. In virtue of the lexicographical ordering, every nonzero member of a Hahn field ℝ((tG )) is Archimedean equivalent to exactly one element of the form tg (g ∈ G), the latter of which may be regarded as a canonical “unit” element of the Archimedean class. Two such elements x and y are Archimedean equivalent if and only if their zeroth exponents (g0 in the statement of Hahn Field 1) are equal, and x is infinitesimal (in absolute value) relative to y if and only if the zeroth exponent

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the pantachies of du bois-reymond and hausdorff 513 of x is less than the zeroth exponent of y. Thus, like the numbers representing Veronese’s non-Archimedean continuum, Hahn’s numbers are formal sums of terms, each being infinitesimal (in absolute value) relative to the preceding terms, where each term is a nonzero real multiple of the canonical unit of its Archimedean class. Furthermore, every nonzero x ∈ ℝ((tG )) is the sum of three components: the purely infinite part of x, whose terms have positive exponents; the real part of x, whose sole term has exponent 0; and the infinitesimal part of x, whose terms have negative exponents. The appellation “real part” is motivated by the fact that {rt0 ∶ r ∈ ℝ} is a canonical copy of the ordered field of reals in ℝ((tG )).

6. The Pantachies of du Bois-Reymond and Hausdorff Although interest in the rates of growth of real functions is already found in Euler’s De infinities infinitis gradibus tam infinite magnorum quam infinite parvorum (On the infinite degrees of infinity of the infinitely large and infinitely small) (1778), their systematic study was first undertaken by Paul du Bois-Reymond (cf. 1870– 1, 1875, 1877, 1882), under the rubric Infinitärcalcül (infinitary calculus).1⁴ And while du Bois-Reymond never attempted to employ this work to develop a nonArchimedean theory of continua, he and others including Poincaré (1893/1952, pp. 28–9) believed it provided an intimation of the possibility of such a theory. Moreover, as we shall see in section 8, there are intriguing historical and conceptual relations between his theory and one of the most important such theories of our day. Though du Bois-Reymond’s contribution to the Infinitärcalcül is concerned solely with functions and is analytic by nature, it is intimately related to his ideas on quantity, in general, and the geometric linear continuum, in particular. Unlike Veronese, who admitted the possibility of an Archimedean geometric linear continuum, for du Bois-Reymond the geometric linear continuum is necessarily non-Archimedean. Indeed, on the basis of a misguided argument that has been aptly characterized as “breathtaking” (Fisher 1981 p. 114), du Bois-Reymond maintained that the infinite divisibility of the line implies that ‘the unit segment decomposes into infinitely many subsegments of which none is finite’ (1882, p. 72). Moreover, as in Veronese’s non-Archimedean continuum, for every segment of du Bois-Reymond’s continuum, there are segments that are infinitesimal relative to it. However, unlike Veronese’s non-Archimedean continuum, du BoisReymond’s continuum of segments is not the positive cone of an ordered field (or even of an ordered abelian group) since, according to du Bois-Reymond, ‘two finite segments are equal when there is no finite difference between them. [That 1⁴ For a complete list of du Bois-Reymond’s writings on his Infinitärcalcül and a survey of the contents thereof, see (Fisher 1981). Also see McCarty’s chapter in this volume.

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514 contemporary infinitesimalist theories of continua is,] a finite quantity does not change if an infinitely small quantity is added to it or taken away from it’ (1882, pp. 73–4), even when the infinitely small quantity is nondegenerate. It was the comparison of quantities (of the same type) having different orders of magnitude that du Bois-Reymond took to be the primary object of his infinitary calculus (1882, pp. 66–75), but as was mentioned above, he only developed the theory for functions. In particular, du Bois-Reymond erects his Infinitärcalcül primarily on families of continuous, monotonically increasing functions from ℝ+ = {x ∈ ℝ ∶ x > 0} to ℝ+ (sometimes with differentiability and integrability properties) such that for each function f of a given family, lim f(x) = +∞, and for x→∞

each pair of functions f and g of the family, 0 ≤ lim f(x)/g(x) ≤ +∞. He assigns to x→∞

each such function f a so-called infinity, and defines an ordering on the infinities of such functions by stipulating that for each pair of such functions f and g: f(x) has an infinity greater than that of g(x), if lim f(x)/g(x) = ∞; x→∞

f(x) has an infinity equal to that of g(x), if lim f(x)/g(x) = a ∈ ℝ+ ; x→∞

f(x) has an infinity less than that of g(x), if lim f(x)/g(x) = 0. x→∞

In accordance with this scheme, the infinities of the following functions x

… , ln(ln x), ln x, … , x1/n , … , x1/3 , x1/2 , x, x2 , x3 … , xn , … , ex , ee , … increase as we move from left to right. Moreover, as the comparative graphs of several of these functions illustrate (see Figure 18.1), given any two functions f and g having different infinities from a family of the just-said kind, f(x) has a greater infinity than g(x) if and only if f(x) > g(x) for all x > some x0 . Furthermore, since, for example, x2 has a greater infinity than x and lim (x2 + x)/x2 = 1,

x→∞

x2 +x has the same infinity as x2 , which illustrates for functions du Bois-Reymond’s corresponding idea for segments, that if a segment s is infinitesimal relative to a segment s′ , then the segment resulting from adding s to s′ is equal in length to s′ .1⁵ While du Bois-Reymond usually restricted his investigations to families of functions of the above said kind, on occasion he mistakenly assumed that each pair of continuous, monotonically increasing functions f and g from ℝ+ to ℝ+ 1⁵ Stolz developed another number system (Stolz 1884) that models this absorptive aspect of du Bois-Reymond’s conception. For a discussion of this little-known system of “moments”, see (Ehrlich 2006, §3).

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the pantachies of du bois-reymond and hausdorff 515

y = x2 y=x y = x3

y = ex 1

y = x2 1

y = x3

4

3.6

3.2

2.8

2.4

2

1.6

1.2

0.8

0.4

y = lnx

Figure 18.1 Comparative graphs of select real functions.

for which lim f(x) = +∞ and lim g(x) = +∞ could be compared in the manner x→∞

x→∞

described above. This led him to postulate the existence of an all-inclusive ordering of the infinities of such real functions—an infinitary pantachie, as he called it (1882, p. 220).1⁶ Such a pantachie, according to du Bois-Reymond, would provide a conception of a numerical linear continuum “denser” than that of Cantor and Dedekind. Georg Cantor, however, who was an ardent opponent of infinitesimals and nonCantoriean infinities (cf. Ehrlich 2006), would have no part of this. Indeed, having demonstrated as Stolz (1879, p. 232) and Pincherle (1884, p. 742) had before him that there could be no such all-inclusive ordering of the infinities of such functions, Cantor proclaimed: ‘the “infinitary pantachie,” of du Bois-Reymond, belongs in the 1⁶ Du Bois-Reymond explains that his adjective “pantachie” derives from the Greek words for “everywhere”.

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516 contemporary infinitesimalist theories of continua wastebasket as nothing but paper numbers!’ (1895, p. 107). Felix Hausdorff, on the other hand, suggested, ‘[t]here is no reason to reject the entire theory because of the possibility of incomparable functions as G. Cantor has done’ (1907, p. 107), and in its place undertook the study of maximally inclusive sets of pairwise comparable real functions, each of which, retaining du Bois-Reymond’s term, he calls an infinitary pantachie or a pantachie for short. This led him to his well-known investigation of 𝜂𝛼 -orderings (1907), and to the following less well-known theorem. Pantachie 1 (Hausdorff 1907, 1909) Infinitary pantachies exist. If P is an infinitary pantachie, then P is an 𝜂1 -ordering of power 2ℵ0 ; in fact, P is (up to isomorphism) the unique 𝜂1 -ordering of power ℵ1 , assuming (the Continuum Hypothesis) CH. An ordered set L is said to be an 𝜂𝛼 -ordering, if for all subsets A and B of L of power < ℵ𝛼 , where A < B (i.e., every member of A precedes every member of B), there is a y ∈ L such that A < {y} < B. The ordered field ℝ of real numbers is an 𝜂0 ordering, though not an 𝜂1 -ordering, and as such Hausdorff ’s pantachies are in a precise sense more dense than ℝ, thereby lending precision to du Bois-Reymond’s intimation. As was noted above, du Bois-Reymond further believed that pantachies exhibit numerical aspects, but he never undertook an arithmetization of them. Motivated by different considerations, however, Hausdorff did. Moreover, like his theory of pantachies, more generally, he did so in the following broader setting. In addition to modifying du Bois-Reymond’s conception of an infinitary pantachie, Hausdorff redirected du Bois-Reymond’s investigation by investigating numerical sequences rather than continuous functions, though he showed that the desired results about the latter can be obtained as corollaries from results about the former. He also deleted the monotonicity assumption and replaced the infinitary rank ordering with the final rank ordering (which was illustrated above). That is, Hausdorff redirected du Bois-Reymond’s investigation to the study of subsets of the set 𝔅 of all numerical sequences A = (a1 , a2 , a3 , … , an , … ) in which the an are real numbers, and he defines the “final ordering” on 𝔅 (and subsets thereof) by the conditions A < B if eventually an < bn , A = B if eventually an = bn , A > B if eventually an > bn , and A ∥ B (i.e., A is incomparable with B) in all other cases, where “eventually” means for all values of n with the exception of a finite number, thus for all n ≥ some n0 [1909 in (Plotkin 2005, p. 276)].1⁷ Hausdorff, who bases his theory on representative elements of the equivalences classes of eventually

1⁷ Strictly speaking, in his (1907), unlike his (1909), Hausdorff only considers numerical sequences in which the an are positive real numbers. However, the proof of the above theorem carries over to the more general case. Moreover, in his (1907), unlike his later works, he uses the term “finally” instead of “eventually” in the definitions of , = and ∥.

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elementary continua 517 equal numerical sequences rather than on the equivalence classes themselves, calls a subset 𝔅′ of 𝔅 totally ordered by the final order a pantachie if it is not properly contained in another subset of 𝔅 totally ordered by the final order. In his aforementioned investigation of 1907, Hausdorff first raised the question of the existence of a pantachie that is algebraically a field, but he only made partial headway in providing an answer. However, in 1909 he returned to the problem and provided a stunning positive answer. Indeed, beginning with the ordered set of numerical sequences of the form (r, r, r, … , r, … ) where r is a rational number, and utilizing what appears to be the very first algebraic application of his maximal principle, Hausdorff proves the following little-known, remarkable result. Pantachie 2 (Hausdorff 1909) There is a pantachie of numerical sequences that is an ordered field, whose field operations are given by A + B = (a1 + b1 , a2 + b2 , … , an + bn , … ), A − B = (a1 − b1 , a2 − b2 , … , an − bn , … ), AB = (a1 b1 , a2 b2 , … , an bn , … ), A/B = (a1 /b1 , a2 /b2 , … , an /bn , … ), where A + B, A − B, AB and A/B are defined up to final equality. Any ordered field that is a pantachie is, in fact, real-closed.1⁸ Writing before Artin and Schreier (1926), Hausdorff does not employ the term “real-closed”, though as Ehrlich (2012, p. 29) observed, Hausdorff proves enough to show that an ordered field that is a pantachie is real-closed. However, to fully appreciate the significance of Hausdorff ’s result as well as the extent of its intimate relation to certain contemporary non-Archimedean continua, we require some background about real-closed fields.1⁹

7. Elementary Continua In their groundbreaking work on “real algebra” Emil Artin and Otto Schreier (1926) sought to characterize the algebraic content of a kind of ordered field of 1⁸ This result had been all but forgotten until it was resurrected in (Plotkin 2005, pp. 271–301) and (Ehrlich 2012, p. 29). 1⁹ By a classical result of R. Baer (1927) and W. Krull (1932), the collection of Archimedean classes of an ordered field A constitutes an ordered abelian group induced by the order and multiplication in A (see Ehrlich 1995, p. 186). Though we will not further pursue the matter here, we note that the system of orders of infinity associated with the members of a Hausdorff pantachie ℍp is isomorphic to the positive cone of the ordered abelian group of Archimedean classes of ℍp , the latter being (up to isomorphism) the unique divisible ordered abelian group that is an 𝜂1 -ordering of power ℵ1 , assuming CH (e.g., (Ehrlich 1988)).

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518 contemporary infinitesimalist theories of continua which the arithmetic continuum is paradigmatic. This led to their theory of realclosed fields, which is a jewel of twentieth-century mathematics. Following Artin and Schreier, an ordered field K may be said to be real-closed if it admits no extension to a more inclusive ordered field that results from supplementing K with solutions to polynomial equations with coefficients in K. Intuitively speaking, real-closed ordered fields are precisely those ordered fields having no “holes” that can be filled by algebraic means alone. Among a number of important properties of the arithmetic continuum they showed are satisfied by real-closed ordered fields more generally is the intermediate value theorem for polynomials of a single variable. In fact, as soon became clear, for an ordered field K, the satisfaction of the intermediate value theorem for polynomials of a single variable over K is equivalent to K being real-closed (cf. Tarski 1939; Warner 1965, pp. 492–493). Accordingly, the idea of a real-closed ordered field is equivalent for the case of polynomials of a single variable to satisfying what has traditionally been regarded as one of the quintessential properties of a continuous function—the formalization of the intuitive idea that a continuous curve connecting points on the opposite sides of a straightline intersects the given line. A similar equivalence was later established for the extreme value theorem for polynomials in a single variable (Gamboa 1987), another prototypical classical property of continuous functions. Shedding still more light on the relation between real-closed ordered fields and ℝ, Tarski (1948, 1959) demonstrated that real-closed ordered fields are precisely the ordered fields that are first-order indistinguishable from ℝ, or, to put this another way, they are precisely the ordered fields that satisfy the elementary (first-order) content of the Dedekind continuity axiom. For this reason they are sometimes called elementary continua. When Tarski’s result is combined with a classical result of Artin and Schreier (1926) on real-closed ordered fields, one obtains the critical fact that every ordered field is contained in a (to within isomorphism) smallest elementary continuum, the latter having the same cardinality as the given field. While ℝ is the best known elementary continuum, as is evident from the above it is not the only one. Some elementary continua, like ℝ, are Archimedean, though most are non-Archimedean, and among the latter many are extensions of ℝ. In the following section, we will discuss a class of real-closed extensions of ℝ that are among the foremost non-Archimedean continua of our day, and in the subsequent section we will draw attention to some of the interesting connections between them, Hausdorff ’s pantachies, the surreals, and some of the elementary continua that emerge from the constructions of Hahn. Unlike Veronese’s non-Archimedean continuum, which was motivated by generalizing the geometrical properties of the classical geometric continuum, those to which we now turn are motivated by the idea of providing a non-Archimedean treatment of the analytic properties of ℝ.

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nonstandard (robinsonian) continua 519

8. Nonstandard (Robinsonian) Continua In the early 1960s Abraham Robinson (1961, 1966) made the momentous discovery that among the real-closed extensions of the reals there are number systems that can provide the basis for a consistent and entirely satisfactory nonstandard approach to analysis based on infinitesimals, a possibility that had been called into question by many since the latter decades of the nineteenth century.2⁰ By analogy with Thoralf Skolem’s (1934) nonstandard model of arithmetic, a number system from which Robinson drew inspiration, Robinson called his number systems nonstandard models of analysis. These number systems, which are now more often called hyperreal number systems (Keisler 1976, 1994), may be characterized as follows: let ⟨ℝ, S ∶ S ∈ 𝔉⟩ be a relational structure where 𝔉 is the set of all finitary relations defined on ℝ (including all functions). Furthermore, let ∗ ℝ be a proper extension of ℝ and for each n-ary relation S ∈ 𝔉 let ∗S be an n-ary relation on ∗ ℝ that is an extension of S. The structure ⟨∗ ℝ, ℝ, ∗S ∶ S ∈ 𝔉⟩ is said to be a hyperreal number system if it satisfies the transfer principle: every n-tuple of real numbers satisfies the same first-order formulas in ⟨ℝ, S ∶ S ∈ 𝔉⟩ as it satisfies in ⟨∗ ℝ, ℝ, ∗S ∶ S ∈ 𝔉⟩. The existence of hyperreal number systems is a consequence of the compactness theorem of first-order logic and there are a number of algebraic techniques that can be employed to construct such systems. The earliest and still one of the most commonly employed such techniques is the ultrapower construction (e.g., Keisler 1976, pp. 48–57; Goldblatt 1998, ch. 3), a construction that was introduced in full generality by Łoś (1955), identified as a source of hyperreal number systems by Robinson (1961, p. 3), and popularized as such by Luxemburg (1962) and Stroyan and Luxemburg (1976).21 Not all hyperreal number systems can be obtained this way, however. By the results of H. J. Keisler (1963, 1976, pp. 58–59), on the other hand, every hyperreal number system is isomorphic to a limit ultrapower. Using the transfer principle, one can develop satisfactory nonstandard conceptions and treatments of all of the concepts and theorems of the calculus (e.g., Keisler 1976; Goldblatt 1998; Loeb 2000). For example, it follows from the transfer principle that:

2⁰ There were some earlier successes in this direction, including the modest contributions of LeviCivita (1892–3), Neder (1941, 1943), and Gesztelyi (1958). The most successful of these is the theory of Schmieden and Laugwitz (1958; see also Laugwitz 1983, 1992, 2001). Unlike Robinson, Schmieden and Laugwitz make use of a partially ordered number system containing zero-divisors, with the result that many of the classical results need to be reformulated. As Laugwitz aptly acknowledged, Robinson’s system ‘was much more powerful when it came to applications in contemporary research’ (2001, p. 128). 21 Hewitt (1948) had already employed the ultrapower construction to obtain a non-Archimedean real-closed extension of the reals, but unlike Łoś he did not establish the transfer principle that accrues from the construction, the latter being crucial to hyperreal number systems in the sense used above.

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520 contemporary infinitesimalist theories of continua A real-valued function f is continuous at a ∈ ℝ (in the standard sense) if and only if ∗f(x) is infinitely close to ∗f(a) whenever x is infinitely close to a, for all x ∈ ∗ ℝ; and on the basis of this one may prove the familiar classical results concerning the continuity of real-valued functions including the intermediate and extreme value theorems (e.g., Goldblatt 1998, pp. 79–80). Of course modern analysis goes well beyond the traditional province of the calculus, dealing with arbitrary sets of reals, sets of sets of reals, sets of functions from sets of reals to sets of reals, and the like. For example, it is commonplace for analysts to prove theorems about the set of all continuous functions on the reals or about the set of all open subsets of the reals, sets to which the justsaid transfer principle does not apply. To obtain nonstandard treatments of these aspects of analysis a more general transfer principle is required. For this purpose Robinson (1966) originally employed a type-theoretical version of higher order logic, but it proved to be unpopular. Since then, following Robinson and Zakon (1969), it has become most common to employ a transfer principle associated with a structure ⟨V(∗ ℝ), V(ℝ), ∗⟩ that generalizes the corresponding hyperreal number system ⟨∗ ℝ, ℝ, ∗S ∶ S ∈ 𝔉⟩. In this setup the transfer principle emerges from an elementary embedding ∗ that relates the members of the superstructure V(ℝ) over ℝ with those of the superstructure V(∗ ℝ) over ∗ ℝ, where for any set of individuals X, the superstructure V(X) over X is defined by V(X) = ⋃n