The Continuum: History, Mathematics, and Philosophy

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2017-12-21 The Continuum: History, Mathematics, and Philosophy

Hayashi, Teppei

Hayashi, T. (2017) The Continuum: History, Mathematics, and Philosophy (Unpublished doctoral thesis). University of Calgary, Calgary, AB. http://hdl.handle.net/1880/106283 doctoral thesis

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The Continuum:

History, Mathematics, and Philosophy

Teppei Hayashi

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE

DEGREE OF DOCTOR OF PHILOSOPHY

GRADUATE PROGRAM IN PHILOSOPHY

CALGARY, ALBERTA

December, 2017

c Teppei Hayashi 2017 Abstract

The main aim of this dissertation is to depict a wide variety of the conceptions of the continuum by tracing the history of the continuum from the ancient Greece to the modern times, and in so doing, to ﬁnd a new way to look at the continuum. In the ﬁrst part, I trace the history of the continuum with a special emphasis on unorthodox views at each period. Basically, the history of the continuum is the history of the rivalry between two views, namely, between the punctiform and the non-punctiform views of the continuum. According to the punctiform view, the continuum is composed of indivisibles; on the other hand, according to the non-punctiform view, the continuum cannot be composed of indivisibles. In the second part, I present Richard Dedekind’s and Georg Can- tor’s standard mathematical theories of the continuum as the modern repre- sentative of the punctiform view of the continuum, and then examine Charles Saunders Peirce’s non-punctiform view of the continuum. In the last chapter, I give some mathematical interpretations to

ii iii

Aristotle’s and Peirce’s theories of the continuum according to both of which the continuum cannot be composed of points. In interpreting Aristotle’s view, I use modern topology and show that Aristotle’s view can be nicely captured by topology. On the other hand, in interpreting Peirce’s view, I appeal to the theory of category and show that in the category-theoretic framework the continuum appears quite diﬀerently from the standard one conceived in the Dedekindian and Cantorian ways. In conclusion, I try to defend a sort of pluralistic view concerning the conceptions of the continuum. Acknowledgements

First of all, I would like to thank my supervisor Richard Zach. Richard has greatly and tirelessly encouraged me throughout my academic journey. I also would like to express my deep gratitude to my committee members: Kristine Bauer, Bernard Linsky, Jack MacIntosh, and Mark Migotti. In my defense, they gave me instructive and useful comments. And I would like to extend my thanks to our graduate program administrator Denise Retzlaﬀ. I am totally sure that I was one of the most troubling graduate students she has ever had. And of course, I would like to thank my wife Asaka. For many years, she has supported this absent-minded husband. Without her, I could not have done this for sure. And I guess I should thank my daughter Kaede for cheering me up by bugging me.

iv To Asaka and Kaede Contents

General Introduction1

I History of the Conceptions and the Theories of the Continuum9

1 Ancient Greece 10 1.1 Aristotle and the Non-Punctiform Continuum...... 12 1.2 Archimedes and the Indivisibles...... 16 1.3 Eudoxus and the Method of Exhaustion...... 24 1.4 Euclid and Incommensurable Magnitudes...... 33 1.5 Summary...... 40 Summary...... 40

2 The Medieval Period 42 2.1 Mathematics and Its Enemies: Scotus, Chatton, and Autrecourt 47

vi CONTENTS vii

2.2 God and Mathematics: Henry of Harclay and Thomas Brad- wardine...... 61 2.3 Propositional Analysis of the Continuum: William of Ockham 71 Summary...... 78

3 Early and Late Modern 80 3.1 Stifel and Stevin...... 80 3.2 Fomalization of the Theory of Real Numbers...... 82 Summary...... 92

II Theories of the Continuum 94

4 Dedekind 95 4.1 Background...... 95 4.2 Continuity and Irrational Numbers ...... 100 Summary...... 108

5 Cantor 109 5.1 Basic Construction: Fundamental Sequences...... 109 5.2 The Numerical Magnitudes and the Straight Line...... 116 5.3 The Second Construction: Limit-Points and Derived Sets... 118 5.4 Perfect and Connected Sets...... 120 Summary...... 122 CONTENTS viii

6 Peirce 124 6.1 Aristotelian Period (Until 1884)...... 126 6.2 Cantorian Period (1884–1895)...... 128 6.3 Peircean Period (After 1895)...... 135 6.4 Summary...... 145 Summary...... 145

III Theories of the Non-Punctiform Continuum 149

7 Some Mathematical Interpretations of the Non-Punctiform Continuum 150 7.1 The Aristotelian Continuum in Topology...... 151 7.2 The Peircean Continuum in Category Theory...... 155 7.2.1 The Construction of the Real Numbers in Category Theory...... 155 7.2.2 Pointless Topology...... 159 7.2.3 Smooth Inﬁnitesimal Analysis...... 161 Concluding Remarks...... 163

Conclusion 165

Bibliography 168 si nemo ex me quaerat, scio; si quaerenti explicare velim, nescio. General Introduction

In today’s mathematics, the continuum is usually identiﬁed with the set of the real numbers; more concretely, it is assumed that there is a one-to-one correspondence between each point on a line and each real number. However prevalent this punctiform conception of the continuum (namely, the conception that the continuum is composed of indivisibles like points) is now, there had been a long and winding haul for the view to attain today’s status. Furthermore, although some think that the problems of the continuum have been settled, at least as far as its mathematical formalization is concerned, once and for all by the works of Dedekind and Cantor, there still remain conceptual problems. The most persistent one would be how and why discontinuous objects (for example, points) become continuous. Thus, it seems to me, the punctiform conception of the continuum left the core problem of the continuum unsolved. My objective in this dissertation is, ﬁrst, to trace such a long and winding history of the conceptions of the continuum with a special emphasis on unorthodox views in each period, and then to investigate non-standard

1 GENERAL INTRODUCTION 2 but mathematically sound ways to capture the concept of the non-punctiform continuum. This dissertation is comprised of three parts. The first part is mostly historical. In the first part, I trace the history of the conceptions of the continuum from the ancient Greece to the late modern period up until 1872. The second part is, compared to the first part, rather theoretical. In this part, first, I present Dedekind’s and Cantor’s theories of the continuum which have become standard today, and then examine Peirce’s rather iconoclastic view of the continuum. The last part is, again, theoretical. In the last chapter, I investigate some non-standard—namely, non-punctiform—but mathematically rigorous ways to formalize the continuum, particularly with Peirce’s ideas in mind. The first part starts with a chapter about ancient Greece. I begin with Aristotle’s conception of the continuum, which had overwhelmingly dominated the way thinkers thought about the continuum at least until the early modern period. Aristotle was perhaps the first person who treated the problem of the continuum explicitly and extensively. Aristotle’s conception of the continuum is first and foremost non-punctiform; that is, he maintained that the continuum cannot be composed of indivisibles like points. After presenting Aristotle’s conception of the continuum, I will examine a few mathematical theories in which indivisibles play an important role; namely, Archimedes’ method of mechanical theorems, Eudoxus’ method of exhaustion, and Euclid’s theory of incommensurable magnitudes. Although these GENERAL INTRODUCTION 3 theories do not deal with the problems of the continuum per se, the existence of such theories shows that the ancient Greeks were not unfamiliar with the non-punctiform conception of the continuum. In Chapter 2, I survey various conceptions of the continuum in the medieval period. I begin by surveying with the argument of Duns Scotus that the continuum cannot be composed of indivisibles. Scotus’ argument is important in the history of the theories of the continuum because his way of arguing is among the first instances which extensively utilizes mathematical arguments for the problems of the continuum. Following Scotus’ argument, I will present Chatton’s and Autrecourt’s views that mathematics is not suitable for capturing the concept of the continuum because, for them, the continuum is first and foremost a physical object and mathematics is a free play of the imagination, even though it takes its materials from the physical world. As a physical object, Chatton and Autrecourt argued, the continuum should be comprised of indivisibles. Strange as it may seem to our modern eyes, the motivation behind the medieval discussions of the continuum was mostly theological; for example, Scotus’ and Chatton’s motivation was to explain the motion of angels and Autrecourt’s motivation is to discuss the eternity of the world. Thus, it is no wonder that some theological elements play an important role in their discussions. As an example of such arguments, I will take up Harclay’s argument in which God plays an important role. Although the influence of theology was prevalent in medieval dis- GENERAL INTRODUCTION 4 cussions of the continuum, not all medievals thinkers discussed the problems of the continuum to settle theological issues; a conspicuous exception to this trend is Thomas Bradwardine. In his Tractatus de continuo, Bradwardine investigated the nature of the continuum in a purely mathematical way and tried to refute all brands of indivisibilisms according to which the continuum is composed of indivisibles. I sketch Bradwardine’s theory of the continuum with particular emphasis on how he tried to refute various indivisibilist conceptions of the continuum. After presenting Bradwardine’s theory of the continuum, I conclude the chapter with Ockham’s analysis of the continuum. Ockham’s way of analysis is, it seems to me, very scholastic (without any derogative implications) and showcases another characteristic way of arguing the continuum in the middle ages. In Chapter 3, I briefly trace the history concerning the conceptions of the continuum from the mid-sixteenth century to right before 1872. In this period, there were two events which had significant meaning to the conceptions, or theorizations, of the continuum. First, around the mid-sixteenth century, irrational numbers started to be recognized as “numbers”. Until then, the term “numbers” had been almost exclusively used for referring the natural numbers; magnitudes which cannot be represented as some ratios of the natural numbers had not been regarded as numbers. However, as is doc- umented by the writings of the German mathematician Michael Stifel and the Flemish mathematician Simon Stevin, which I will briefly present in the GENERAL INTRODUCTION 5 chapter, from around the mid-sixteenth century, irrational numbers started being regarded as numbers. This change in conception had, I think, a huge impact on the later theorizations of the continuum. The second event is the emergence of calculus around the late seventeenth century. However, it seems to have had caused as many problems as it solved; especially, the problem of the concept of limits. I will describe Cauchy’s and Bolzano’s struggles with this problem. The year 1872 is extremely important in the history of the continuum: In 1872, Dedekind and Cantor finally succeeded in formalizing the continuum (more exactly, in their case, the real numbers) in a mathematically satisfiable way. In the first two chapters of the second part (Chapters 4 and 5), I survey Dedekind’s and Cantor’s constructions. Although the main focus here is on their constructions themselves, I also try to make the historical contexts clear in presenting them. After surveying Dedekind’s and Cantor’s constructions of the real numbers, I examine Peirce’s theory of the continuum. As a hard-core Aristotelian in the conception of the continuum, Peirce had throughout his career tried to capture the continuum in the Aristotelian way; namely, Perice had tried to formalize the continuum in a non-punctiform way. However, Peirce was not just a na¨ıve Aristotelian; he was well aware of the works of Dedekind and Cantor, and in light of those works, he tried to develop his own mathematical theory of the continuum which is faithful to the Aristotelian conception. In the last chapter, I examine some non-punctiform conceptions of GENERAL INTRODUCTION 6 the continuum from a modern mathematical point of view. In the first section, I interpret the Aristotelian continuum in terms of modern topology. As will be seen in the section, most of the Aristotelian characteristics of the continuum can be nicely captured by topological terms. Of course, modern topology presupposes the existence of points, and consequently, the continuum captured in it is necessarily punctiform. However, considering how the Aristotelian characteristics of the continuum are captured in topology, we notice that points play no significant role there; in fact, what matters most in capturing the Aristotelian continuum is the concept of open sets. Thus, if we can develop topology solely from the concept of open sets (namely, without presupposing the existence of points), such a development will conform perfectly with the Aristotelian picture. And as will be seen in the second section, there is actually a way of doing topology which has the concept of open sets as one of its primitives and does not presuppose the existence of points. In the second section of the last chapter, I survey some category- theoretic theories which can capture some characteristics which Peirce thought the continuum has to have. First, after briefly presenting the basics of category theory, I outline how the real numbers are constructed in a topos which is a category with some additional structures. The constructions of the real numbers in a topos are done basically as the translations of Dedekind’s and Cantor’s constructions in category-theoretic terms. Surprisingly, those two constructions translated in a topos are not isomorphic (even though they GENERAL INTRODUCTION 7 are in classical mathematics). Moreover, the translation of Dedekind’s construction is not order complete. It seems, as Robert Goldblatt suggests, that we need some more elements in order to make Dedekind reals complete in category theory. Next, I describe a topology which is developed without mentioning points. This topology is, unsurprisingly and somewhat playfully, called pointless topology. Lastly, I will present a category-theoretic version of mathematical analysis which is called smooth infinitesimal analysis. In this category-theoretic analysis, the existence of small lines the square of whose length is 0 is assumed. For writing about such a vast subject as the continuum, I had to leave many topics out. First, I confined myself to the history of the continuum in the Latin West. This means that the Arabic influence on the conception of the continuum has been almost entirely omitted although some aspects of it were touched upon in Chapter 2. Second, I did not discuss Newton’s and Leibniz’s conceptions of the continuum at all. In particular, the lack of the discussion about the view of Leibniz who wrote a lot about le labyrinthe du continu may seem to some a fatal flaw of this dissertation. However, in order to examine and argue about Leibniz’s view of the continuum, I would have to write another dissertation. Third, the only modern figure which I presented as an advocate of the non-punctiform continuum is Peirce. There were, of course, others who sided with the non-punctiform conception of the continuum. Among such figures, the name of Paul du Bois-Reymond should be mentioned. However, du Bois-Reymond’s argument is, it seems to me, GENERAL INTRODUCTION 8 too conceptual; in other words, what is proposed by du Bois-Reymond is less mathematical than what is proposed by Peirce. Thus, I chose Peirce as a medium to lead me into the mathematical discussion of the non-punctiform continuum. Lastly, there are few discussions about the roles of infinity in the conception of the continuum. Again, I omitted the discussion of infinity almost entirely for lack of space. Lastly, even though I mainly took the side of the non-punctiform continuum, it is not my intention to deny the punctiform—namely, standard mathematical—continuum altogether. The aim of this dissertation is, in short, to expand our mathematical and philosophical perspectives about the continuum. Part I

History of the Conceptions and the Theories of the Continuum Chapter 1

Ancient Greece

In ancient Greece, the continuum was thought of almost exclusively in a geometrical way. The main reason for this is that the ancient Greeks did not have the real number system yet. For the ancient Greeks, the term “numbers” always referred to natural numbers which are discrete objects built from a unit number “1”. Therefore, there was no way to think of the continuum in terms of numbers. Nevertheless, they developed sophisticated theories about continuous quantities. In this chapter, I will examine some of such theories. First, I will look at Aristotle’s conception of the continuum. Al- though Aristotle did not treat the continuum as a mathematical object, his conception of the continuum—namely, the conception that the continuum cannot be composed of indivisibles–has had a huge inﬂuence on later gener- ations. Thus, in order to appreciate the later conceptions of the continuum, it is essential to know what Aristotle thought about the continuum. Next, I

10 CHAPTER 1. ANCIENT GREECE 11 will discuss how Archimedes utilized the idea of indivisibles for calculating an area (or a volume) of a plane ﬁgure (or a solid). Although Archimedes seems to have thought that his method is of an experimental nature; this deviation from the canon (namely, the Aristotelian conception of the continuum) is important from the historical point of view. Then, I will move on to Eudoxus’ method of exhaustion. This method, to our eyes, seems quite similar to that of limits in calculus. However, for several reasons, Eudoxus failed to be the originator of the concept of limits. I will examine these reasons, which in turn shed another light on how the ancient Greeks thought of the continuum. Lastly, I will investigate how Euclid treated incommensurable magnitudes. An incommensurable magnitude, when translated in its numerical correspondent, is an indispensable concept for identifying the continuum with real numbers. However, as was said above, the ancient Greeks did not have a number system which can accommodate incommensurable magnitudes. There was an unbridgeable gap for the ancient Greeks between the geometrical and number systems. Still, Euclid’s theory of incommensurable magnitudes is sophisticated enough to enable us to treat irrational numbers properly. Then, it must have been one of the sources of inspiration when later thinkers came to the idea of irrational numbers. Thus, the examination of the Euclidean treatment of incommensurable magnitudes is in order. In writing this chapter, I owe much to Carl B. Boyer’s The History of the Calculus and Its Conceptual Development (Boyer 1949) and J. N. Cross- ley’s The Emergence of Number (Crossley 1987). Although both books are CHAPTER 1. ANCIENT GREECE 12 not entirely dedicated to the continuum problem, they give us nice overviews of the topic.

1.1 Aristotle and the Non-Punctiform Con-

tinuum

In Book V of his Physics, Aristotle (384 bce–322 bce) deﬁnes three types of the “next to each other” relations: in succession, contiguous, and continuous.1 The relation “in succession” means that two objects are next to each other without some other objects of the same kind between them. For example, if two people are standing next to each other and no other person is standing between them, these people are in succession. The relation “contiguous” means that two objects are in succession and their outer extremities touch each other.2 For example, two books next to each other on a bookshelf are contiguous because there is no other book between them and their covers (which are their extremities) touch each other. Lastly, the relation “continuous” means that two objects next to each other share an outer extremity.

1Physics, Book V, Ch. 3 (226b18–227b2); Aristotle 1934, pp. 34–43. In referring Aris- totle’s works, I use the Bekker numbering. For the translations of Aristotle’s works, I mainly consult the Loeb edition unless otherwise noted. 2Things are said to be “touched each other” if their extremities are in the exact same place. It is worth noting that, in Coming-To-Be and Passing-Away, Aristotle says that the term “touching” can be applied to mathematical objects (323a1–2; Aristotle 1955, pp. 224- 225). And in order for something (mathematical or physical) to be in place (a necessary condition for touching), Aristotle continues, it has to have a deﬁnite magnitude (323a3– 6; ibid.). Thus, although their extremities may be indivisible points, points themselves cannot touch each other. CHAPTER 1. ANCIENT GREECE 13

For example, two countries next to each other are continuous because they have only one outer extremity (namely, their border) between them. After deﬁning the above three relations, Aristotle tries to prove his famous thesis: The continuum cannot be composed of indivisibles.3 First, Aristotle assumes, for reductio, that the continuum is composed exclusively of indivisibles. Let us take points as such indivisibles which compose the continuum. Then, two adjacent points must be next to each other; otherwise, there would be a space or some non-point object between them. In either cases, the assumption cannot be retained because, if there were a space between two points, it would mean that something which contains these two points is no longer the continuum; and if there were some non-point object between two points, something which contains these objects cannot be considered as composed exclusively of points. Therefore, if the continuum is composed exclusively of points, the “next to each other” relation which points in this continuum have cannot be that of “in succession”. Then, there are two options left; either two points in the continuum are contiguous or two points are continuous. However, both options are impossible. In both options, two points must have at least one outer extremity. This means that points have parts and therefore are not indivisible. For each option, we ﬁnd a contradiction. Thus, the continuum cannot be composed exclusively of points.4 Moreover, from the indivisibility of points,

3Physics, Book VI, Ch. 1 (231a21–b18); Aristotle 1934, pp. 92–97. 4Actually, this argument cannot be accepted by those who think that there is always at least one point between any two points, namely, the continuum is dense. We will come CHAPTER 1. ANCIENT GREECE 14 it is obvious that points cannot be continuous with anything. This means, in turn, that the continuum cannot be an amalgamation of points and any other things whether they are divisible or not. Therefore, according to Aristotle, the continuum cannot be composed of indivisibles such as points. As a corollary of the claim that the continuum cannot be composed of indivisibles, inﬁnite divisibility of the continuum follows. Aristotle writes:

[I]t is manifest that any continuum is divisible into parts that are divisible without limit—for if the parts were indivisible, we should have one indivisible touching another—since the extremities of things that are continuous meet and become one.5

If we divide the continuum, what we get are two continua which are again divisible. However many times we repeat this process, what we get are always divisible continua.6 In Aristotle’s definition or characterization of the continuum, there are at least two aspects to be noted. First, even though he adamantly and repeatedly denies the thesis that the continuum is composed of indivisibles, Aristotle admits some uses of indivisibles. For instance, back to this point later when I will discuss the views of Cantor and Peirce who thought the density is a necessary condition for anything to be considered continuous. 5Physics, Book VI, Ch. 1 (231b 15–18); Ibid., pp. 96–97. Here, Aristotle bases the continuum’s infinite divisibility on the claim that the continuum cannot be composed of indivisibles. However, Aristotle sometimes defines the continuum by infinite divisibility: “I mean by continuous ‘capable of being divided into parts that can in their turn be divided again, and so on without limit’; and on this definition I say that time is of necessity continuous” (232b24–26; ibid., pp. 106–107). 6Buckley (2012, p. 19) calls this the mirror property. CHAPTER 1. ANCIENT GREECE 15

A point holds together and divides a length because it is the beginning of one part of a length and the end of the other.7

As we have seen in the above, it is an indivisible (a point or a line) at which two things are held together and made continuous.8 And it is also an indivisible which one finds when the continuum is divided into two. In short, an indivisible plays an indispensable role both in making two things continuous and in dividing the continuum into two.9 Second, in the first formulation of the continuum, what Aristotle tells us is not a characteristic of the continuum itself; rather, what Aristotle tells us is the condition under which two things become continuous. And considering the mirror property of the continuum, it seems that two things which become continuous should be continua themselves if such a union of two things can be considered as the continuum at all. Of course, seen from a modern (especially, mathematical) point of view, we would find many theoretical deficiencies in Aristotle’s conception of the continuum. However, as will be seen in Chapter 7, the Aristotelian

7220a10–11; my translation: “καὶ γὰρ ἡ στιγμὴ καὶ συνέχει τὸ μῆκος καὶ ὁρίζει· ἔστι γὰρ τοῦ μὲν ἀρχὴ τοῦ δὲ τελευτή”. I use my own translation here because the Loeb translation has some problems. The Loeb translation reads: “it is a point that both constitutes (by its movement) the continuity of the line it traces and also marks the end of the line that is behind and the beginning of the line in front” (Aristotle 1929, pp. 392–393). First, the translation seems too free. Second, and perhaps more importantly, the verb “συνέχει” is not translated faithfully to its original implication. 8This aspect of an indivisible is emphasized in Categories as well. Aristotle writes in Categories: “A line is, however, continuous. Here we discover that limit of which we have just now been speaking. This limit or term is a point” (5a1–3; Aristotle 1933, pp. 36–37). 9This aspect of indivisibles seems to have a huge inﬂuence on Peirce’s conception of the continuum. See pp. 139 ﬀ. of this dissertation. CHAPTER 1. ANCIENT GREECE 16 continuum is not so absurd even from a modern mathematical view.

1.2 Archimedes and the Indivisibles

Although the majority of thinkers in ancient Greece seems to have thought, perhaps because of the strong inﬂuence of Aristotle, that the continuum cannot be composed of indivisibles, some Greeks asserted that the continuum is in fact composed of indivisibles. We can name Xenocrates10 (c. 396/5 bce – 314/3 bce) and Democritus11 (c. 460 bce – c. 370 bce). However, because most of their works were lost, it is mostly unknown what they actually said. The notable exception to this situation is Archimedes (c. 287 bce – c. 212 bce). In a long-lost text of Archimedes, which was discovered by Johan Ludwig Heiberg in 1906, which is now usually called the Method (Archimedes 1912), Archimedes calculated the area of a segment of a parabola by appealing to the idea that the area of a ﬁgure can be thought of as an aggregation of lines. This means that Archimedes, at least at the time he wrote this text, thought that the continuum (in this case, the area of a segment of a

10A Peripatetic treatise De lineis insecabilibus (“On indivisible lines”), which was for- merly ascribed to Aristotle, is considered to be a response to Xenocrates’ views concerning indivisible lines are partially transmitted to us by the neoplatonist Simplicius (c. 490 ce – c. 560 ce), see Simplicius 2011. 11Some authors do not think that Democritus thought that the continuum is composed of indivisibles. For example, Heath wrote that “Democritus was too good a mathematician to have anything to do with such a theory [as that of the indivisibles continuum]” (Heath 1921, p. 181). However, as Boyer argues (Boyer 1949, p. 22), the idea of the inﬁnite divisibility of the continuum seems inconsistent with Democritus’ atomism. CHAPTER 1. ANCIENT GREECE 17 parabola) can be regarded as composed of indivisibles (in this case, lines). What Archimedes showed in the ﬁrst proposition of Method is:

Any segment of a section of a right angled cone (i.e. a parabola) is four-thirds of the triangle which has the same base and equal height.12

As in Figure 1.1, this can be restated as that the segment ABC of the parabola is four-thirds of the triangle ABC. (The point B is determined as the intersecting point of the arc of the parabola and the line which is perpendicular to the base and passing through the middle point D of AC. Such a point is called the vertex of the triangle.)

B A

D C

Figure 1.1

Next, add some extra lines to the ﬁgure as in Figure 1.2. CF is the tangent to the curve of the parabola at C. Because B is the bisection of DE,13 the extension of CB passes through the middle points N and K of OM and AF . Extend CK to H, making AF the bisector of CH.

12Archimedes 1912, 14; italics in quoted work. 13Archimedes 1897, p. 235. CHAPTER 1. ANCIENT GREECE 18

T H G F

K N E

P A B O D C

Figure 1.2

Now, draw a perpendicular line to the base from any point on AC to CF . Let us pick O in the ﬁgure as one of such arbitrary point and let P be the intersection of OM and the arc of the parabola. Then, the following holds.14

KH OM = . KN OP

In order to appreciate what the above proportional expression means, it is essential to understand two concepts, both of which play an important role in the works of Archimedes: the center of gravity and the law of the lever.

14Archimedes 1912, p. 16. CHAPTER 1. ANCIENT GREECE 19

Oddly enough, despite the fact that Archimedes extensively utilized the concept of the “center of gravity”, he never gave an explicit deﬁnition of it, at least in his extant works.15 However, it can be guessed or reconstructed what Archimedes meant by the term from his extant works. Assis and Magnaghi nicely put it as follows.

The center of gravity of any rigid body is a point such that, if the body be conceived to be suspended from that point, being released from rest and free to rotate in all directions around this point, the body so suspended will remain at rest and preserve its original position, no matter what the initial orientation of the body relative to the ground.16

Archimedes listed the centers of gravity for various figures in the first part of the Method. The followings are those of the figures which concern the investigation here.

• The center of gravity of any straight line is the point of bisection of the straight line.

• The center of gravity of any triangle is the point in which the straight lines drawn from the angular points of the triangle to the middle points of the (opposite) sides cut one another.

15Dijksterhuis 1987, p. 235. 16Assis and Magnaghi 2012, p. 9. CHAPTER 1. ANCIENT GREECE 20

As to the law of the lever, Archimedes gave a clear description in his On the Equilibrium of Planes.

Two magnitudes, whether commensurable [Prop. 6] or incommensurable [Prop. 7], balances at distances reciprocally proportional to the magnitudes.17

For example, think of two planes A and B and let the weights of A and B be WA and WB respectively. Now, suspend them from a bar as in the following ﬁgure.

dA dB

F WA

Figure 1.3

The planes A and B balance, or are in equilibrium, at the fulcrum F if and only if the following holds.

d W A = B . dB WA 17Archimedes 1897, 192; italics in quoted work. CHAPTER 1. ANCIENT GREECE 21

KH OM Now what the expression KN = OP means is clear. That is, if we put a line which has H as its middle point and the same length as OP , OM and GT are in equilibrium at the fulcrum if they are suspended from their centers of gravity from H and N respectively (or put their centers of gravity exactly at H and N respectively as in Figure 1.4) with CH as a horizontal lever and K as the fulcrum. M

T N H G K C

Figure 1.4

Because O could be any point on AC, the area of the segment ABC of the parabola and that of the triangle AF C are in equilibrium as in Figure 1.5.

H K X C F A

C C A

Figure 1.5 CHAPTER 1. ANCIENT GREECE 22

A natural question one might ask would be this: Why is the triangle AF C suspended from the point X? First, remember that all the line segments of the parabola segment ABC are suspended from H. On the other hand, the positions from which the line segments of the triangle AF C are suspended range all over KC. Therefore, we should ﬁnd the center of gravity for the triangle as a whole. As was noted earlier, we know where the center of gravity for a triangle is: It is the point in which the straight lines drawn from the corners of the triangle to the middle points of the (opposite) sides intersect. To see this clearly, take a look at the following ﬁgure.

H K X C

Figure 1.6

In the above ﬁgure, the triangle AF C is at the original position. And if we draw the lines each of which goes from a corner to the middle point of the opposite side of the corner. The meeting point of three lines is, as is clear from the ﬁgure, X. Moreover, the point X divides the line KC to 1 : 2. Seeing that HK = KC, this means that

the parabola segment ABC : the triangle AF C = 1 : 3. CHAPTER 1. ANCIENT GREECE 23

Once this result is obtained, all that is needed to get what Archimedes wanted to show is the fact that the triangle AF C is four times bigger than the triangle ABC.18 Therefore,

the parabola segment ABC : four times the triangle ABC = 1 : 3.

⇒ three times the parabola segment ABC = four times the triangle ABC.

⇒ the parabola segment ABC = four-thirds of the triangle ABC.

This is what had to be shown. Now, the following question must be asked: Did Archimedes really think that the continuum is composed of indivisibles? The answer seems negative, considering what Archimedes told us in Method. He wrote

Now the fact here stated is not actually demonstrated by the argument used; but that argument has given a sort of indication that the conclusion is true.19

Some author argue that Archimedes’ method is purely of experimental or heuristic nature.20 Archimedes might have really thought that

18To see this, think in the following way. First, note that the triangle AF C is divided in half by CK because CK is the bisector of the triangle. Thus, the triangle AKC is half the size of the triangle AF C. In the same fashion, the triangle AKC is divided in half by AB. Consequently, the triangle ABC is half the size of the triangle AKC. Therefore, the triangle ABC is a quarter of the triangle AF C. 19Archimedes 1912, p. 17. 20Dijksterhuis 1987, p. 320; Boyer 1949, p. 51. CHAPTER 1. ANCIENT GREECE 24 this method is merely heuristic. Or he might have actually thought that the method is worth serious consideration but hesitated to say so perhaps because of the intellectual atmosphere at that time. However, one thing is clear: Archimedes was liberal enough, even though he thought that the method does not give a genuine proof of a problem, to apply the idea that the continuum can be thought of as composed of indivisibles to the actual mathematical problems. This alone would be considered to be quite an achievement.

1.3 Eudoxus and the Method of Exhaustion

Right after the cautionary remark just cited in the previous section, Archimedes added the following.

Seeing then that the theorem is not demonstrated, but at the same time suspecting that the conclusion is true, we shall have recourse to the geometrical demonstration which I myself discovered and have already published.21

The “geometrical demonstration” Archimedes mentioned refers to the one in Quadrature of the Parabola (Archimedes 1897, pp. 233–252) and is based on the method of exhaustion. This method has special importance for the investigation here because it is very close to the concept of limit. Needless to say, the concept plays

21Archimedes 1912, pp. 17–18. CHAPTER 1. ANCIENT GREECE 25 an important role in the rigorous deﬁnition of the real numbers although the method was never used for that purpose.

Although Eudoxus (408 bce – 355 bce) is said to have invented the method of exhaustion,22 some Greeks already had a similar idea a little earlier than Eudoxus did. For example, Antiphon (480 bce – 411 bce), who was a contemporary of Socrates (c. 469 bce – 399 bce), had the following idea.

[In order to obtain the area of a circle, Antiphon] inscribed an equilateral triangle in the circle, and on each of the sides set up another triangle, an isosceles triangle with its vertex on the circumference of the circle, and continued this process, thinking that at some time he would make the side of the last triangle, although a straight line, coincide with the circumference.23

This idea can be illustrated as follows.

22Archimedes wrote in the preface to On the Sphere and Cylinder that the theorems in which the method of exhaustion plays an essential role “were in fact unknown to all the many able geometers who lived before Eudoxus and had not been observed by any one” (Archimedes 1897, p. 2). 23Thomas 1939, pp. 312–313. CHAPTER 1. ANCIENT GREECE 26

Figure 1.7

Intuitively, this method, by exhausting the areas between the triangles and the circumference of the circle, seems to give us the area of the circle. However, it must be recalled that the Aristotelian conception of the continuum, namely the thought that the continuum is infinitely divisible, prevailed in ancient Greece. As Simplicius rightly pointed out,24 we can never obtain the area of the circle in this way under the premise of the infinite divisibility of the continuum. What Antiphon’s method gives us is, at best, the approximation of the area of the circle. The genius of Eudoxus is in that he reconciled the method of exhaustion with the conception of the continuum as infinite divisible and gives us the exact value of a given area, not merely its approximation. In order to do so, he first founded the method on a principle which is now called the

24“Now continual division of the space betveen the straight line and the circumference of the circle will never exhaust it nor ever reach the circumference of the circle, if the space is really divisible without limit” (Thomas 1939, p. 315). CHAPTER 1. ANCIENT GREECE 27 postulate, or lemma, of Archimedes.25 In addition to the postulate, another lemma which is derived from the postulate also plays an important role. Uti- lizing these propositions, Eudoxus proved various propositions such as “that every pyramid is one third part of the prism which has the same base with the pyramid and equal height; also, that every cone is one third part of the cylinder having the same base as the cone and equal height”.26 Unfortunately, all the works of Eudoxus were lost, but what he wrote can be reconstructed from the writings of Euclid and Archimedes. Deﬁnition 4 of Book V of Elements reads:

Deﬁnition V.4. Magnitudes are said to have a ratio to one another which are capable, when multiplied, of exceeding one another.27

Using this deﬁnition, the following important proposition is proved.

Proposition X.1. Two unequal magnitudes being set out, if from the greater then be subtracted a magnitude greater than its half, and from, that which is left a magnitude greater than its half, and if this process be repeated continually, there will be left some magnitude which will be less than the lesser magnitude set out.28 25There has been a debate as to whether or not Eudoxus ﬁrst formulated this principle. While Hankel asserted that it must be Hippocrates who ﬁrst formulated the principle (Hankel 1874, p. 122), Heath suggested that it was Eudoxus (Heath 1921, p. 328). Boyer seems to side with Hankel (Boyer 1949, p. 33). 26Archimedes 1897, p. 234. The attribution of the proofs of these propositions to Eu- doxus is stated in the preface to On the Sphere and Cylinder (pp. 1–2). 27Euclid 1908, vol. II, p. 114; bolds in quoted work. 28Euclid 1908, vol. III, p. 14; italics in quoted work. CHAPTER 1. ANCIENT GREECE 28

Why is this proposition so important? It is mainly because the proposition captures the infinite processes of exhausting an area.29 Every- where the method of exhaustion is used in proving propositions,30 so is the proposition. Archimedes’ proof that the area of a parabola is four-thirds of the triangle inscribed within the parabola is no exception to this. Moreover, according to Proposition X.1 (or equivalently, by admit- ting Definition V.4), the existence of infinitesimals is denied. In order to see this, let a and b be unequal magnitudes such that b < a. The process described in the proposition can be expressed as the following sequence.

a a a a a a (= a), a (< 1 ), a (< 2 < 1 ), a (< 1 ), . . . , a (< 1 ),... 1 2 2 3 2 4 4 8 n 2n−1

Proposition X.1 says that we can always find such an that is less than any given b. On the other hand, if there were infinitesimals, a construction like this would be impossible. Therefore, Definition V.4 (and consequently, Proposition X.1) entails the denial of the existence of infinitesimals. As was seen in the previous section, what Archimedes exhibited in Method is not a proof but merely a method of discovery for the fact that the area of a parabola is four-thirds of the inscribed triangle. He proved this fact

29In other words, by assuming the postulate of Archimedes, we can always find a smaller quantity a for every b no matter how small b is. This leads to the denial of the existence of infinitesimals. 30In Elements, the propositions XII. 2, 4–7, 10, and 16–18. Of course, Archimedes also uses it extensively. CHAPTER 1. ANCIENT GREECE 29 in Quadrature of the parabola. This treatise is composed of twenty-four propositions. We have no room here for examining all of those, but let us take a look at the last five propositions just briefly.

Proposition 20. If Qq be the base, and P the vertex, of a parabolic segment, then the triangle P Qq is greater than half the

31 1 segment P Qq. (See Figure 1.8. Note that qV = VQ = 2 qQ.)

P q

V Q

Figure 1.8

What concerns here most is its corollary.

Corollary It is possible to inscribe in the segment a polygon such that the segments left over are together less than any assigned area.32

It is easy to see that the above corollary is derived using Proposition X.1 (hence, the postulate of Archimedes).

31Archimedes 1897, p. 248; italics in quoted work. 32Archimedes 1897, p. 248; italics in quoted work. CHAPTER 1. ANCIENT GREECE 30

Next, in Proposition 21, Archimedes proved that another inscribed triangle, which is made from a side of the original triangle in the same way as before, is one-eighth of the original. (See Figure 1.9. Note that qm =

1 mV = VM = MQ = 4 qQ.)

r P q R m V M Q

Figure 1.9

For example, the sum of the areas of the triangles made at the second stage would be one-fourth of the triangle P Qq (that is, 4P Qq = 4(4PRQ + 4P rq)). Therefore, the sum of the area of the inscribed triangles at the n-th

1 stage is 4n−1 4 P Qq. In Proposition 22, Archimedes proved that no matter how many times we repeat the process of inscribing triangles in a parabola segment, the sum of all the areas of the inscribed triangles never reaches the area of the parabola segment. As Archimedes showed in Proposition 21, the sum of the areas of

1 the triangles inscribed at the n-th stage is 4n−1 4 P Qq (here, 4P Qq refers CHAPTER 1. ANCIENT GREECE 31 to the triangle inscribed at the ﬁrst stage). Therefore, the total sum of the

1 1 1 inscribed triangles to the n-th stage would be (1 + 4 + 16 + ... + 4n−1 ) 4 P Qq. In Proposition 23, Archimedes proved that this total sum amounts to

4 1 1 3 4 P Qq − 3 4n−1 . Now, having obtained the above result, it seems that it can be concluded, by appealing to the concept of limit, that the area of the segment

1 1 of parabola P Qq is four-thirds of the inscribed triangle P Qq because 3 4n−1 in the above expression vanishes at the limit. However, the concept of limit was so foreign to the ancient Greeks that they had to appeal to another way to obtain the result. The area of the parabola segment P Qq is obtained as follows. In Proposition 24, Archimedes ﬁrst assumed, for reductio, that the area of the

4 parabola segment P Qq is not 3 4 P Qq. If the area of the parabola segment 4 4 P Qq is not 3 4 P Qq, it should be either greater or less than 3 4 P Qq. How- ever, contradiction can be derived in either cases. Therefore, it is concluded

4 that the area of the parabola segment P Qq is 3 4 P Qq. Why did Archimedes, or Eudoxus, not appeal to the concept of limit? There seems to be several reason for this: the dependence on sensory intuition in geometrical thinking, the lack of the concept of the real numbers, and the exclusion of infinity from mathematical reasoning. First, the ancient Greeks seem to have thought of geometrical problems almost exclusively in a geometrical way. In other words, the method of exhaustion is based on sensory intuition. Of course, it seems natural to solve CHAPTER 1. ANCIENT GREECE 32 a geometrical problem in the geometrical way. It even sounds tautological. However, especially after the invention of the Cartesian coordinate system, geometrical problems can be solved in algebraic ways, and doing so often makes the solutions easier (and more elegant). For the ancient Greeks, however, geometrical magnitudes and numbers were completely different, and therefore, they never thought of solving geometrical problems in terms of numbers. Seeing that the concept of limit is arithmetical in nature, or more essentially, abstract, it is not attainable by appealing to geometrical intuition. Second, the concept of infinity is required in defining the concept of limit because a limit is defined as that of an infinite sequence. However, the ancient Greeks avoided the use of infinity in mathematics.33 For example, Archimedes did not use the infinite in proving that the area of the parabola segment P Qq is four-thirds of the inscribed triangle P Qq.34 Again, it is impossible to define the concept of limit without that of infinity. Lastly, in order to think of a limit of a number sequence, the number system in which the number sequence is given must be continuous. We need the concept of the arithmetical continuum in addition to that of the geometrical one. However, as I will show in the next section, the word “numbers” exclusively meant the natural numbers for the ancient Greeks. If one

33Stillwell 2010, pp. 53–67. Still, they implicitly utilized the concept of infinity, especially in disguise of the postulate of Archimedes or the derivatives of it (e.g., Proposition X.1 of Euclid). 34Especially note that in Proposition 23 Archimedes calculated the total sum of the areas of a finite number of the inscribed triangles. In fact, he just dealt with five geometrical figures for drawing the conclusion in the original text (Archimedes 1881, pp. 346–348). CHAPTER 1. ANCIENT GREECE 33 only has a discrete number system, there is no way to grasp the concept that a sequence continuously approaches a limit. The above three points are actually three essential aspects of what enabled us to define the real numbers. The ancient Greeks’ inability to define the concept of limits is due to their lack of a concept of the real numbers. In the next section, I will show that, in spite of the fact that the ancient Greeks could not, or did not, define the real numbers explicitly, it is still true that the Greeks had and utilized the concept of real numbers in the guise of a geometrical concept.

1.4 Euclid and Incommensurable Magnitudes

For the ancient Greeks, the word “numbers” exclusively meant natural numbers. Consequently, they did not conceive of rational, irrational, and real numbers. However, it is not entirely accurate to say that the ancient Greeks had no corresponding notion at all. For the ancient Greeks, what is now called rational or irrational numbers fall under the category of “magnitude”. Thus, in order to appreciate what the ancient Greeks thought of what is now called irrational numbers, it is necessary to understand their distinction between the concepts of numbers and magnitudes. Let us start with the Greek conception of numbers. Heath listed various deﬁnitions of numbers by the ancient Greeks as follows.

The first definition of number is attributed to Thales, who defined CHAPTER 1. ANCIENT GREECE 34

it as a collection of units, ‘following the Egyptian view’. The Pythagoreans ‘made number out of one’; some of them called it ‘a progression of multitude beginning from a unit and a regres- sion ending in it’.. . . Eudoxus defined number as a ‘determinate multitude’. Nicomachus has yet another definition, ‘a flow of quantity made up of units’. Aristotle gives a number of definitions equivalent to one or other of those just mentioned, ‘limited multitude’, ‘multitude (or, ‘combination’) of units’, ‘multitude of indivisibles’, ‘several ones’, ‘multitude measurable by one’, ‘multitude measured’, and ‘multitude of measure’ (the measure being the unit).35

Those taken together, the core deﬁnition would be “multitude made out of a unit which is in most cases (what is now called) the number one”.36 Since each number is made out of a unit, a collection of numbers is discrete. On the other hand, a magnitude refers to a continuous quantity which one, two, and three dimensional objects have. For example, a line has a length as its magnitude; a plane has a area; and a solid has a volume. Two magnitudes are said to be commensurable with each other if they can be measured by the same unit. Pythagoras, and the Pythagoreans, ﬁrmly believed that any two magnitudes are commensurable. And this belief, perhaps be-

35Heath 1921, pp. 69–70; all the footnotes and the original Greek words are omitted. 36Note that a unit is not made out of a unit. Therefore, a unit itself was not considered to be a number for the ancient Greeks. The smallest number in the ancient Greek sense is two (“The smallest number, in the strict sense, is two” (Aristotle 1929, 220a27, p. 394 ff)). CHAPTER 1. ANCIENT GREECE 35 cause of the strong influence of Pythagoras, had been broadly shared by the ancient Greeks until “a truly scandalous event in the theory”37 happened: the discovery of incommensurable magnitudes.38 Why was this discovery so shocking? It was not only because their firm belief that everything is made of natural numbers or their ratios39 had been shattered, but also, more fundamentally, because the credibility of proofs which depended on the old Pythagorean theory had been cast into doubt. (How could I trust a system if its fundamental portion is turned out to be untrustworthy?) At any rate, the ancient Greek geometers must have realized that if they stuck to Pythagorean doctrine, they could not advance mathematics any further. The “scandalous” event completely changed the intellectual climate concerning the theory of magnitudes. Following the discovery of incommensurable magnitudes, a new theory which enables the Greek geometers to deal not only with commensurable magnitudes but also with incommensurable ones, was sought. Such a new theory was invented by, again, Eudoxus. Eudoxus’ invention is outlined in Book V of Euclid’s Elements. We already know the statement which is fundamental to the new theory: Def- inition 4 of Book V. Obviously, any incommensurable magnitude, by mul-

37“un véritablescandale logique” (Tannery 1887, p. 98.) 38There are various opinions as to who first discovered incommensurable magnitudes. For example, von Fritz attributes the discovery of incommensurable magnitudes to the Pythagorean Hippasus, not to Pythagoras himself (Von Fritz 1945) while Zeuthen names Theodorus as the discoverer of them (Caveing 1996). 39Note that a ratio itself was not considered to be a number. CHAPTER 1. ANCIENT GREECE 36 tiplying, can exceed a commensurable magnitude which is greater than the incommensurable. Therefore, an incommensurable magnitude can have a ratio to a commensurable magnitude as well as to another incommensurable magnitude. In order for any two multitudes, whether commensurable or incommensurable, to be compared with each other, the following definition is required.

Definition V.5. Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.40

So far, I have not defined “commensurable” and “incommensurable” explicitly. According to the usage in the above, it might seem that “incommensurable” is nothing but a synonym of “irrational”. However, the meaning of “incommensurable” is somewhat different from that of “irrational”. The 40Euclid 1908, vol. II, p. 114. In the modern expression: We call four magnitudes a, b, c, d be in the same ratio and write a : b = c : d if one of the followings holds for any two natural numbers m, n. 1. n · a > m · b if and only if n · c > m · d. 2. n · a = m · b if and only if n · c = m · d. 3. n · a < m · b if and only if n · c < m · d. CHAPTER 1. ANCIENT GREECE 37 following is the definition of “commensurable” and “incommensurable” in Elements.

Deﬁnition X.1. Those magnitudes are said to be commensurable which are measured by the same measure, and those incommensurable which cannot have any common measure.41

According to the above deﬁnition, even irrational numbers can be commen- √ √ √ surable. For example, two irrational numbers 2 and 18 (i.e., 3 2) are commensurable because they are measured by the same measure, that is, √ 2.42 The following deﬁnition states that there are other types of commensurability and incommensurability than those which I have dealt with.

Deﬁnition X.2. Straight lines are commensurable in square when the squares on them are measured by the same area, and incommensurable in square when the squares on them cannot possibly have any area as a common measure.43

√ √ For example, 2 and 3 are not commensurable in length, but commensu- p √ p √ rable in square. On the other hand, 1 + 2 and 1 + 3 are incommensurable both in length and square.

41Euclid 1908, vol. III, p. 10; bolds in quoted work. 42However, in most cases, it seems implicitly assumed that, when a number is said to be incommensurable, it is said to be so with regard to a natural number (or a ratio of natural numbers). 43Euclid 1908, vol. III, p. 10; bolds in quoted work. CHAPTER 1. ANCIENT GREECE 38

We mentioned above the difference between the meanings of incommensurable and irrational. In fact, the Euclidean definitions of “rational” and “irrational” differ from modern ones.

Deﬁnition X.3. With these hypotheses, it is proved that there exist straight lines inﬁnite in multitude which are commensurable and incommensurable respectively, some in length only, and others in square also, with an assigned straight line. Let then the assigned straight line be called rational, and those straight lines which are commensurable with it, whether in length and in square or in square only, rational, but those which are incommensurable with it irrational.44

In short, the Euclidean “rational” means the commensurability in square. The next proposition serves as the test for incommensurable magnitudes.

Proposition X.2. If, when the less of two unequal magnitudes is continually subtracted in turn from the greater, that which is left never measures the one before it, the magnitudes will be incommensurable.45

√ For example, using Proposition X.2, the incommensurability of 2 (with regard to natural numbers or ratios based on them) can be shown as follows.

44Euclid 1908, vol. III, p. 10; bolds in quoted work. 45Euclid 1908, vol. III, p. 17; italics in quoted work. CHAPTER 1. ANCIENT GREECE 39

Let us think of a square ABCD and suppose that the diagonal AC is commensurable with the side AB (see Figure‘1.10 below).

B A

G F

C D

Figure 1.10

Now, put the point F on AC so that AB = AF , draw the perpendicular line to AC at F , and name the intersection of BC and the perpendicular line E. Then, the followings are easily veriﬁed.

BE = EF = FC

CF = AC − AB

CE = CB − CF = AB − (AC − AB)

= 2AB − AC

Because it is supposed that AC and AB are commensurable, so are CF and CE. Note that CF and CE are the side and diagonal of the square CEFG respectively. Thus, in the exact same manner in which we drew CEFG, CHAPTER 1. ANCIENT GREECE 40 we can draw a smaller square based on CEFG. The diagonal and side of this smaller square are also commensurable. We can continue to draw a smaller square whose diagonal and side are commensurable. This means, by Proposition X.2, that AC and AB are incommensurable. The bulk of Book X is devoted to classify incommensurable magnitudes and to give a way of its calculation for each of them. These ways are sophisticated enough to enable us to prove the following equality.

r r 1 1p 1 1p pβ ± γ = β + β2 − γ2 ± β − β2 − γ2 2 2 2 2

However, those methods were still considered to be only applicable to magnitudes. It was not until the late sixteenth century to regard the incommensurable as a number.

1.5 Summary

In this chapter, I have examined how the ancient Greeks understood the continuum and its related concepts. First, in Section 1, I examined Aristotle’s claim that the continuum cannot be composed of indivisibles. His argument depends on the infinite divisibility of the continuum and the distinction of the three types of adjacency. In Sections 2 and 3, the view that the contin- CHAPTER 1. ANCIENT GREECE 41 uum can be thought of as composed of indivisibles was presented. One of such view is that of Archimedes. Archimedes, in calculating the area of a geometric figure, regards such a figure as composed of lines which are two- dimensional indivisibles. And then, he “weighs”46 each line of the figure and calculates a sum of the weights of these lines. This procedure is very similar to that of the integral. Although it is unclear whether Archimedes really thought of the continuum as composed of indivisibles, at least he admitted that regarding the continuum as composed of indivisibles has a considerable heuristic value. The second person who utilized indivisibles is Eudoxus. His method of exhaustion is very similar to that of limits. However, because of the limitation in the way of thinking of the ancient Greeks (the dependence on sensory intuition in geometrical thinking, the lack of the concept of the real numbers, and the exclusion of infinity from mathematical reasoning), Eudoxus did not utilize the concept of limits in solving geometrical problems. Lastly, in Section 4, I examined the concept of incommensurable magnitudes in Euclid’s Elements and showed that Euclid’s theory of incommensurable magnitudes reached a level sophisticated enough to solve an algebraic prob- √ q q 1 1 p 2 2 1 1 p 2 2 lem like β ± γ = 2 β + 2 β − γ ± 2 β − 2 β − γ in a geometrical way. Although the section is not said to be about the continuum per se, still, because of its influence on the later development of the theory of the continuum, Euclid’s theory is worth mentioning here.

46The reason why I put the word “weighs” in quotation marks is that a line cannot be weighed because it has no width, and therefore, has no weight. Chapter 2

The Medieval Period

The philosophical climate in medieval times, even in its later periods, was mainly influenced by the Aristotelian tradition.1 Medieval thinkers usually started their thoughts from Aristotle and developed them. However, some of them started deviating from the Aristotelian tradition in important aspects, and that is exactly the case with the medieval discussions about the continuum. John E. Murdoch summarizes the characteristics, or more precisely, the differences from the Aristotelian tradition, of medieval thought concerning the continuum as follows.2 1Here, by “medieval period”, I refer to the period which ranges from the sixth century (the fall of the Roman Empire) to the fifteenth century (the beginning of the Renaissance), and by “its later periods”, I mean the period ranging from the thirteenth to the fourteenth century. Of course, there can be other periodizations of the middle ages; but it seems to me that this periodization (namely, from the sixth to the fifteenth century) reflects a consensus in the philosophical community. (Some recent handbook and encyclopedias of medieval philosophy (Lagerlund 2010 and Duignan 2011) both have the indication of the period “500–1500” in their titles.) 2Murdoch 1982b, pp. 165–167.

42 CHAPTER 2. THE MEDIEVAL PERIOD 43

According to Murdoch, first, the medieval thinkers directly dealt with the problem of the existence of indivisibles which, according to some, are said to compose the continuum.3 Aristotle did not discuss whether indivisibles actually exist or not; rather, he assumed their existence.4 Second, in trying to settle the question whether the continuum is composed of indivisibles or not, the medievals explicitly appealed to mathematical arguments.5 As was seen in the previous chapter, Aristotle did not use mathematical arguments explicitly in his attempt to show that the continuum is not composed of indivisibles. Third, the medieval tradition considered the question of whether the continuum has a beginning and an end which Aristotle never considered. Fourth, the medievals asked whether all infinities are equal or not. This problem was generally foreign to the ancient thinkers. Fifth, under the assumption that some infinities are not equal, the medievals asked which infinities are logically possible under the absolute power of God.6 Since Aristotle did not ask the question about different infinities, these kinds of problems did not arise for him. Behind all the deviations pointed out by Murdoch, there seems one noticeable trend which brought about a lot of debates in medieval period concerning the continuum: an increasing number of “indivisibilists” claimed

3I will discuss this aspect when I argue about Ockham. 4See Section 1.1, especially pp. 14 ff. 5I will take up Duns Scotus’ and Bradwardine’s arguments as examples of such mathematical arguments. 6These third, fourth, and fifth characteristics will be dealt with when I discuss Harclay’s conception of the continuum. CHAPTER 2. THE MEDIEVAL PERIOD 44 that the continuum is composed of indivisibles. However, medieval indivisibilist views of the continuum were not monolithic. Thomas Bradwardine, who wrote a monograph which was aimed at refuting all brands of indivisibilism once and for all, classified them as follows.7

1. Corporeal indivisibilism (Indivisibles have parts but can never be divided: Nicholas of Autrecourt)8

2. Extensionless indivisibilism

(a) Finitism (The number of indivisibles is ﬁnite: William of Chatton)

(b) Inﬁnitism (The number of indivisibles is inﬁnite)

i. Immediatism (Indivisibles are lined up immediately to one another: Henry of Harclay)

ii. Mediatism (Indivisibles are lined up mediately to one another; namely, between any two indivisibles, there are (inﬁnitely) many indivisibles: Robert Grosseteste)9

In this chapter, I will examine the above varieties of indivisibilism and some counter-arguments against them. In so doing, I will pay special

7The presentation is based on Murdoch 1957, p. 200. A critical text of Bradwardine’s monograph De Continuo is in ibid., pp. 338–471, and the relevant part here is pp. 379–380. 8Bradwardine himself does not name anyone of this type of indivisibilism from the medieval times. 9Although Bradwardine thinks that Grosseteste is a mediatist, I could not find any textual evidence that he has an indivisibilist view of the continuum. Thus, I will not discuss his view. CHAPTER 2. THE MEDIEVAL PERIOD 45 attention to where they parted ways with Aristotle as well as to how the medievals developed and deepened the Aristotelian view of the continuum. I will start with the presentation of Scotus’ arguments for the infinite divisibility of the continuum as an example of mathematical arguments concerning the continuum. Following the presentation of Scotus’ mathematical arguments, I will discuss Walter Chatton’s and Nicholas of Autrecourt’s arguments whose main point is to show that mathematical arguments are in general not suitable for arguments about the continuum. In the second section, I will examine the arguments of Henry of Harclay and Thomas Bradwardine. Harclay’s arguments heavily depend on God (especially his omnipotence) and this represents one of the characteristics of the medieval ways of arguing. On the other hand, in Bradwardine’s arguments, God does not play as important a role as in Harclay’s. Rather, Bradwardine’s arguments are conducted in a rigorous mathematical way and can be considered as one of the pinnacles of mathematical arguments concerning the continuum in the middle ages. Lastly, in the third section, I will examine William Ockham’s arguments for the infinite divisibility of the continuum with a special emphasis on his propositional analysis. In writing this chapter, I greatly owe to John E. Murdoch’s works. Among his writings, Murdoch (1982a) is especially useful for getting an overview of the subject. I also owe much to the following: Murdoch (1981) on Henry of Harclay, Murdoch and Synan (1966) and Celeyrette (2015) on Walter Chatton, Murdoch (1957) on Thomas Bradwardine, and Murdoch CHAPTER 2. THE MEDIEVAL PERIOD 46

(1982b) on William of Ockham. CHAPTER 2. THE MEDIEVAL PERIOD 47

2.1 Mathematics and Its Enemies: Scotus,

Chatton, and Autrecourt

John Duns Scotus

With Aristotle, John Duns Scotus10 maintains that the continuum cannot be composed of indivisibles.11 However, there is a notable diﬀerence between Scotus and Aristotle: To refute the composition of the continuum out of indivisibles, Scotus appeals to mathematics more extensively and explicitly than Aristotle did. He presents two mathematical arguments for that purpose.

10John Duns Scotus was born in a small village of Scotland (hence his name “Scotus”) around 1265/1266. Scotus studied at Oxford from 1288 to 1301. During his school years at Oxford (perhaps around 1298/1299), Scotus commented and lectured on Peter Lombard’s Sentences. After he finished his education, Scotus moved to Paris and lectured on Sen- tences. In 1304, Scotus became a Franciscan master and moved to Cologne. Scotus died there in 1308. The argument with which I deal here is taken from Ordinatio which, based on his lectures on Sentences at Oxford, Scotus prepared for publication around 1300. 11The motivation behind Scotus’ involvement in the problem of the continuum would first seem totally odd to our eyes; he discussed it in order to explain the motion of angels. How are these two topics related? First, note that, according to Aristotle, an indivisible cannot move (Aristotle 1934, pp. 192–203). On the other hand, an angel is a spiritual being (Hebrews 1 : 14), and being spiritual, it is thought to be indivisible. (How can we divide something spiritual?) Thus, based on the Aristotelian theory of motion, it seems to follow that an angel cannot move. To many, this conclusion must have appeared unacceptable; medievals tried to reconcile the indivisibility of angels with the Aristotelian theory of motion. For example, Thomas Aquinas argued that an angel can move either discontinuously or continuously (“an angel can successively quit the divisible place in which he was before, and so his movement will be continuous. And he can all at once quit the whole place, and in the same instant apply himself to the whole of another place, and thus his movement will not be continuous” (Summa Theologica 1.53.1; Thomas Aquinas 1882, p. 425; Thomas Aquinas 1922, pp. 32–33) whereas Scotus thought otherwise. In his attempt to show that an angel moves only continuously, Scotus had to deal with the problem of the continuum. For a general overview of the problems of angels in medieval philosophy, see Hoffmann (2012). For Scotus’ own writing, see his Ordinatio, Lib. II, Dist. 2, Question 9 (Scotus 1968, pp. 219–303). CHAPTER 2. THE MEDIEVAL PERIOD 48

The ﬁrst argument appeals to the A paradox of the part-whole relation.12 In Figure 2.1, there are two circles which share the same center. Now let us suppose that each circle is composed of inﬁnitely many indivisibles. If the C point A on the big circle moves along each indivisible which constitutes the big circle, the Figure 2.1 intersections of the radius AC and the small circle should cover all the indivisibles which constitute the small circle. Therefore, if the circles are composed of indivisibles, the small circle has as many points as the big one. However, the big circle clearly has a longer circumference than the small one. How is this possible? One indivisibilist response is that the radii which are drawn from two points of the big circle can cut the small circle at the same point. In Figure 2.2, the radii AC and BC cut the small circle at the point D. Now let us draw the tangent line at D. Then, since both AC and BC intersect with ED perpendicularly, both ∠ADE and ∠BDE should be the right angle. However, ∠BDE is clearly larger than ∠ADE although ∠ADE is a part of ∠BDE. A part of something cannot be larger than the whole of it. Therefore, Scotus concludes, the continuum cannot be composed of indivisibles. 12Scotus 1968, pp. 230–232; English translation in Grant 1974, pp. 317–318. CHAPTER 2. THE MEDIEVAL PERIOD 49

A B

E D

Figure 2.2

The second argument attempts to show that the assumption that the continuum is composed of indivisibles contradicts the parallel postulate of Euclid.13 First, in a similar A C fashion as the previous argument, one can ar- B gue that the number of indivisibles on the side of a square has as many points as the diago- Figure 2.3 nal of the square by drawing a line parallel to the base of the square and moving it along the side. In order to avoid this, indivisibilists should argue that two points on the side can correspond to one point on the diagonal. Suppose that A and B in Figure 2.3 are such points. Then, according to the parallel postulate, there have to be two parallel lines one of which is drawn from A and the other of which is from B. However, as was supposed above, these two lines meet

13Scotus 1968, pp. 232–233; Grant 1974, pp. 318–319. CHAPTER 2. THE MEDIEVAL PERIOD 50 at C. This is an outright contradiction with the postulate. Thus, Scotus concludes, the continuum cannot be composed of indivisibles.

Walter Chatton

To these mathematical arguments of Scotus, Walter Chatton14 objects that the mathematical arguments cannot apply to the problems of the continuum because the continuum in question is not a mathematical object; mathematical objects, according to Chatton, exist only in our imagination.15 For Chatton, the continuum is a physical object which is composed of a finite number of indivisibles.16 In trying to show that the continuum is composed of (a finite number of) indivisibles, Chatton first argues for the existence of indivisibles.17 Although he proposes several arguments for that purpose, they can be essen-

14Walter Chatton was born in England around 1290 and studied theology at Oxford. During his time at Oxford (perhaps between 1328 and 1330), Chatton gave lectures on Lombard’s Sentences (Walter Chatton 2004) on which my description of Chatton’s arguments are based. Chatton finished his education at Oxford in 1330 and then moved to Avignon where he died around 1343. Although he was critical of Scotus’ arguments in this particular instance, Walter Chatton was a follower of Scotus; and as a realist, he was also a critic of William of Ockham, whose lectures Chatton probably attended at Oxford. His debates with Ockham are perhaps considered to be one of the most constructive and fruitful ones in the history of philosophy, and as such, contributed to the development of Ockham’s thought, especially his famous “Ockham’s razor”. 15The reason why Chatton thinks that mathematical objects reside in the imagination would be that, for Chatton, mathematics is first and foremost geometry which roots in our sensory perception of the world. 16Chatton’s motivation to argue about the problem of the continuum was the same as Scotus’, that is, to explain the motion of angels. This is no coincidence; it was almost customary for philosophers and theologians in the late middle ages to write a commentary to Peter Lombard’s Sentences in which various issues concerning angels are argued. 17Chatton mainly argues against Ockham whose argument will be discussed in Section 2.3. CHAPTER 2. THE MEDIEVAL PERIOD 51 tially summarized in two types: spatial and temporal. The spatial argument goes as follows.18 First, let us think of a perfect sphere and a perfect plane (which God surely can make), and suppose that these two objects touch each other. Then Chatton asks how they touch each other. The answer would be evident: Because they are perfect, they have to touch just in one point. Thus, a point, or indivisible, has to exist. As for the temporal argument, Chatton takes a sinner as an example and claims that there should exist an instant when a sinner has become a sinner (i.e., when they commit a sin).19 Thus, also in this temporal aspect, there exists an indivisible.20 After arguing for the existence of indivisibles, Chatton moves on to a discussion about the number of indivisibles in the continuum. As was stated above, Chatton thinks that there are only a finite number of (potential but actualizable) indivisibles in the continuum. In order to argue for the finiteness of the number of indivisibles in the continuum, Chatton tries to refute the opposite claim. The first argument is a very common medieval

18Murdoch and Synan 1966, p. 249. See also Walter Chatton 2004, p. 127. 19Murdoch and Synan 1966, p. 252. A similar argument is found in Walter Chatton 2004, p. 127. 20Some words on the ontological status of the Chattonian indivisibles is in order. First, for Chatton, indivisibles are not imaginary things (Walter Chatton 2004, p. 123). They are real. However, Chatton also says that indivisibles do not exist in nature (Walter Chatton 2004, p. 114). How should we understand this seemingly contradictory position? Chatton explains this as follows. First, Chatton claims that indivisibles exist potentially in a continuous thing. Because of this potentiality, such a continuous thing retains its totality as one (Walter Chatton 2004, pp. 123, 125). Figuratively speaking, indivisibles are welded to form a continuous whole. Such potential indivisibles, if separated, are actualized and then considered to exist in nature. However, perhaps because Chatton thinks that no one can actually separate indivisibles except God, he says that there is no indivisible per se in nature. CHAPTER 2. THE MEDIEVAL PERIOD 52 way of arguing against the infiniteness of the continuum; that is, if there are infinitely many indivisibles in the continuum, it should follow that there are the same (infinite) number of indivisibles in any continuous things because for those who deny the infiniteness of the continuum there is one and only one type of infinity.21 The second step of the above argument is based on Chatton’s definition of infinity. According to Chatton, a thing is considered as infinite when there is no beginning and no end in the succession of constituents (namely, indivisibles) of the thing.22 Chatton thinks that, in light of this definition, the extension of a thing should be infinitely large (or long) if the number of indivisibles in the thing is infinite. Because the continuum with which Chatton is dealing here is a physical thing and there is no infinitely large (or long) object in nature,23 Chatton concludes that the number of indivisibles

21Walter Chatton 2004, p. 137. 22Walter Chatton 2004, p. 138. 23Here, it should be recalled that in the condemnation of 1277 the following theses were condemned. (The translation is from Parens and Macfarland 2011, pp. 324, 325; the Latin original with detailed commentaries is in Hissette 1977, pp. 60–63, 98–101.)

26. That God has infinite power in duration, not in action, since there is no such infinity except in an infinite body, if there were such a thing. 49. That separated substances are actually infinite [in number]. For infinity is not impossible except for material things.

These theses must have appeared blasphemous to the eyes of the condemner because the theses deny God’s absolute power and state that even God cannot create an infinitely large material thing. To reconcile the absolute power of God and the fact that there is no known infinitely large material thing, it was proposed that God simply decided not to create such an infinitely large material thing because of the very law of nature which God himself had ordained (Grant 2010, pp. 54–55). Thus, the late medieval thinkers might have safely stated that there is no infinitely large material thing without denying the absolute power of God. CHAPTER 2. THE MEDIEVAL PERIOD 53 in any continuous thing should be finite. In the third argument, Chatton claims that dividing a thing infinitely destroys the essence of the thing divided.24 For example,

Let us take the smallest piece of meat which the nature can give and make the situation clear. In the smallest piece of meat, we cannot ﬁnd any natural function which any meat should have. Therefore, in such smallest piece of meat, there is no inﬁnite parts of meat. We arrived at the indivisible of meat.25

Although it might be possible to divide a continuous thing infinitely, there is a limit in such division if we want to retain the essence of the thing.26 After such a limit, the thing divided would lose its essence and cease to be that thing. Chatton also explains the reason why people tend to mistake what is finite for what is infinite. He writes:

It is said that it is possible to divide something infinitely because the creature’s sense and vision are not suitable for indivisibles and 24“It is not possible to divide a continuous thing completely. Before we arrive at its smallest part, the thing would be corrupted” (Walter Chatton 2004, p. 136). It should be noted that, in general, perhaps because of the influence of Aristotle, the medievals did not any problem in infinity by division. 25Murdoch and Synan 1966, p. 240; my translation: “capio istam carnem minimam quam dat naturalis et arguo sic ad intellectum eius : in carne minima non potest haberi operacionem naturalem competentem carni, igitur in illa minima carne non sunt infinite partes carnee, igitur est devenire ad indivisibile carnis”. 26This point may be supported even from the modern scientific view. For example, in order for water to be considered as water, there is a limit in its decomposition; when water is completely decomposed into hydrogen and oxygen, there is no water anymore. CHAPTER 2. THE MEDIEVAL PERIOD 54

consequently people do not try to divide a thing to the extent that there is no smaller part left to be divided. Thus, people say, as in the scripture, “The number of fools is inﬁnite”27 although no one can ever count the number of all the fools. When people do not know what to do and cannot reach a clear conclusion, they simply give up and say: “It is inﬁnite; I don’t want to do that anymore”.28

Lastly, with these conceptions of indivisibles and infinity, Chatton tries to refute the mathematical argument of Scotus.29 First, recall that for Chatton the continuum is a physical object.30 According to Chatton, any mathematical argument for the infinite divisibility of the continuum cannot apply to physical objects because, first, there is no such thing as an infinitely large object in the physical world, and second, even if one can cut something into small pieces infinitely, there has to be a limit size for something to be legitimately considered as the same thing as it was before the cut (recall the meat example). Thus, any argument for the infinite divisibility of the

27Ecclesiastes 1 : 15. 28Murdoch and Synan 1966, pp. 258–259; my translation: “quod dicatur divisio pos- sibibs in infinita, quia tactus vel visus creature non cadit super rem indivisibilem, nec experitur se homo posse dividere quantum in ita minutas porciones quin semper divissam habet partem, admodum quo loquitur scriptura quando dicit: stultorum infinitus est nu- merus, quia nullus homo potest numerare fatuos, similiter, quando homo est perplexus in facto, et non potest devenire ad darum finem, dimittit et dicit: infinitum est; nolo tractare de hoc” (italics in the original). 29Walter Chatton 2004, pp. 140–142. 30In fact, for Scotus as well, the continuum in question does not seem to be a mathematical object. Scotus raises the question of the continuum in the course of discussions about the motion of angels which are not mathematical objects for sure. CHAPTER 2. THE MEDIEVAL PERIOD 55 continuum “is only of mathematical imagination”.31 Furthermore, because any continuous object is composed of a finite number of indivisibles, a larger object always has more indivisibles than a smaller one. Consequently, there is no one-to-one correspondence of indivisibles between a larger and a smaller object. For example, as I presented above, if there are two circles with the same center, the radii drawn from the bigger circle can intersect the smaller one in one point. Similarly, lines drawn from sides of a square can intersect the diagonal in one point. In the latter example, the parallel postulate of Euclid does not hold. Because of this, Scotus has rejected the indivisibilist conception of the continuum. To this rejection, Chatton responds that a mathematical argument or postulation cannot always apply to a physical object. Indeed, for Chatton, the parallel postulate does not hold of physical objects. Thus, Chatton thinks that, even if some mathematical argument or postulation is not consistent with some argument about physical objects, this does not mean that such an argument, which is inconsistent with some mathematical argument or postulation, is wrong. For Chatton, such inconsistency simply implies the inapplicability of mathematics to the physical problems.

31Walter Chatton 2004, p. 138. CHAPTER 2. THE MEDIEVAL PERIOD 56

Nicholas of Autrecourt

Nicholas of Autrecourt is one of the few corporeal indivisibilists in medieval times.32 According to Autrecourt, the continuum is composed of inﬁnitely many indivisibles which line up immediately next to each other and have inﬁnitely small extension. Autrecourt starts his defense for his corporeal indivisibilism by crit- icizing the Aristotelian thesis that the continuum cannot be composed of indivisibles.33 However, most Autrecourt’s counter-arguments against the Aristotelian arguments are not very convincing. For example, let us take a look at one of Autrecourt’s counter-arguments. First, suppose that the continuum is composed of indivisibles. Now, let us think of two objects, one of which moves twice as fast as the other. Then, when the faster object moves three indivisibles, the slower one should be at the half of the second indivisible. However, an object cannot be at the half of an indivisible because of the very essence of indivisibles. Thus, the continuum cannot be composed of indivisibles. Against the above Aristotelian argument, Autrecourt responds that the slower object rests at some indivisibles for some time whereas the faster

32Even though there were few self-acknowledged corporeal indivisibilists in the medieval times, as will become clear in what follows, quite a few thinkers, whether they were indivisibilist or not, often talked about indivisibles as if they have corporeal extension. Such examples will be found in the discussions of Duns Scotus, Walter Chatton, Henry of Harclay, and Thomas Bradwardine. (Yes, they are almost all of the dramatis personae.) 33He develops his corporeal indivisibilism in his Universal Treatise (Nicholas of Autre- court 1971. A Latin text is in O’Donnell 1939), especially in the section entitled “Indivis- ibles” (Nicholas of Autrecourt 1971, pp. 71–87; O’Donnell 1939, pp. 206–217). CHAPTER 2. THE MEDIEVAL PERIOD 57 one does not rest during its movement. However, he does not oﬀer any reason for why the movement should be made in such a way. The argument is ad hoc at best.34 Although Autrecourt’s counter-arguments against the Aristotelian and the mathematical arguments are not very strong, his general reasons that the continuum is composed of indivisibles have some force. First, Autrecourt accuses the Aristotelian and the mathematical arguments against indivisibilism of presupposing, ﬁrst of all, that an indivisible cannot have an extension (and consequently cannot touch another indivisible), and more importantly, that mathematics can capture and describe how our world actually is. According to Autrecourt, the thesis that an indivisible cannot have an extension is just an assumption (which is especially suitable for doing mathematics), not a precise picture of our world. However, there is nothing contradictory in presupposing that an indivisible actually has an extension; rather, Autrecourt thinks, it is more probable35 to presuppose so if one wants to have a more precise picture of our world. Then, why and how does Autrecourt defend his corporeal indivis-

34Autrecourt’s other counter-argument may seem worse. Against the Aristotelian argument that, if the continuum is composed of an odd number of indivisibles, it cannot be divided in half, Autrecourt simply admits the argument and say: we can ignore the difference because indivisibles are so small. 35The frequent occurrence of “probable” in Autrecourt’s treatise is notable. As Leonard A. Kennedy, one of Autrecourt’s translators, points out, this indicates that “[Autrecourt] was dealing only with what was probable, not with what was certain” (Nicholas of Autre- court 1971, p. 2). This might make Autrecourt a kind of fallibilist. CHAPTER 2. THE MEDIEVAL PERIOD 58 ibilism? First of all, Autrecourt points out that indivisibles have to have an extension in order for a collection of them to have an extension. Still, there remains the questions as to how an indivisible can have an extension, and as to how these indivisibles connect with each other and constitute the continuum. To these questions, Autrecourt responds that an indivisible can have an extension and connect to other indivisibles because “[an indivisible] has its own position and its own mode of being”.36 Although it is not clear what Autrecourt means by “its own mode of being”, it is clear that Autrecourt thinks that an indivisible gets an extension and connects to other indivisibles by occupying a place. The reason why Autrecourt thinks that there have to be an infinite number of indivisibles in any continuous thing is, first and foremost, to deflect the criticism that the diagonal of a square, which is composed of indivisibles, would be commensurable with its side which is also composed of indivisibles. Even though, to modern eyes, the assumption that the continuum is composed of infinitely many indivisibles does not seem to dodge the opponent’s argument that the diagonal of a square should be commensurable with its side if a line is composed of indivisibles, Autrecourt himself seems quite confident with his counter-argument that the diagonal a square cannot be commensurable with its side if the continuum is composed of infinitely many indivisibles. Let us grant for now that the continuum is composed of infinitely

36Nicholas of Autrecourt 1971, p. 73; O’Donnell 1939, p. 208 CHAPTER 2. THE MEDIEVAL PERIOD 59 many indivisibles each of which has its extension and connects with other indivisibles. Then, it seems to follow that the continuum is infinitely large. Concerning this, although Autrecourt does not respond in a straightforward manner, we can reconstruct why he thinks that infinite many indivisibles do not constitute an infinitely large continuum. Autrecourt draws the following two conclusions concerning the composition of the continuum: “First, the continuum is not composed of parts which can always be further divided . . . . Secondly, a continuum demonstra- ble to sense or imagination is not composed of a finite number of points . . . ”.37 And from these conclusions, Autrecourt claims, the following two corollaries follow: “For every magnitude which is given or which is pointed out to sense or imagination, there is a smaller one (at least, nothing seems to pre- vent this), and yet, along with this, it will be true to say that there is in a thing some magnitude such that a smaller cannot be found”.38 At first sight, the two corollaries above seem contradictory to each other. However, if one interpret them in the light of Autrecourt’s distinction among the senses, the intellect, and the imagination, the corollaries do not necessarily constitute contradictory propositions. First, according to Autrecourt, we know from the senses how things39 appear to us. However, the senses do not tell us how things actually are, and

37Nicholas of Autrecourt 1971, p. 82; O’Donnell 1939, p. 213 38Ibid. 39By “things” Autrecourt seems to mean, ﬁrst and foremost, corporeal ones; and for Autrecourt, our world is composed of such corporeal things. CHAPTER 2. THE MEDIEVAL PERIOD 60 more importantly, how things have to be according to their nature; we need something other than the senses in order to know such a nature. We accomplish this—to get to the nature of things—by the intellect which, according to Autrecourt, tells us the nature of things by abstracting it from the appearances of things. On the other hand, in the imagination, what is imagined does not have to be bounded by the nature of things; the imagination is allowed to use the sense data freely to create what actually is not (for example, a chimera).40 It can be however said that the imagination has the same function as the intellect in the sense that both receive the data from the senses and process them. Now, in the realm of the imagination, for every magnitude we can think of a smaller one. However, the mode of the existence of such imaginable beings does not necessarily conform with how things in the world actually are; in fact, Autrecourt thinks that the imagination is quite misleading in ﬁnding out the nature of things. In order to know the nature of things, Autrecourt claims, we have to appeal to the intellect; and by the intellect, we know that there is the smallest magnitude that things can have. From Autrecourt’s distinction among the senses, the intellect, and the imagination, we also understand why Autrecourt thinks why mathematics is not an appropriate means for knowing the nature of things and of the

40One may have to note that Autrecourt does not deny the existence of such imaginable beings; he clearly states that imaginary beings truly exist although he hastens to add that the existence of imaginable beings is diﬀerent from that of things (Nicholas of Autrecourt 1971, p. 124; O’Donnell 1939, p. 241). CHAPTER 2. THE MEDIEVAL PERIOD 61 world. For Autrecourt, mathematics is not the activity of the intellect; it is the activity of the imagination. As such, mathematics can create its own world independently of how our world actually is. The inﬁnitely divisible continuum is one such products of the mathematical imagination; it exists in the mathematical world but it does not in our world.

2.2 God and Mathematics: Henry of Harclay

and Thomas Bradwardine

English theologian and philosopher Henry of Harclay41 is one of the propo- nents of the claim that the continuum is composed of infinitely many indivisibles.42 He presents two arguments for the claim. God plays a significant role in both.43 The first argument proceeds as follows. First, let us think of a finite line. God can perceive all the points in it even if there are infinitely many

41Henry of Harclay was born in England around 1270. He studied theology and lectured on Lombard’s Sentences at the University of Paris around 1300. At the time Harclay was in Paris, Duns Scotus was lecturing on Sentences there as well. Harclay must have been attending Scotus’ lectures; he was hugely influenced by Scotus. After going back to England, Harclay became a master of theology around 1310, and then, the chancellor of Oxford University in 1312 until he died in 1317. In his late years at Oxford, Harclay had gradually become critical of Scotus and sympathetic of Ockham perhaps because he attended Ockham’s lectures on Sentences at Oxford. The arguments which I am concerning here are written in his late Oxford era. 42Harclay’s motivation for arguing about the continuum was, again, theological. How- ever, he did not deal with the problems of the continuum in discussing the motion of angels; he did so in discussing the eternity of the world. 43Henry of Harclay 2008, pp. 1052–1055. CHAPTER 2. THE MEDIEVAL PERIOD 62 points in the line.44 Thus, God can surely perceive the first point of the line. Let us choose an arbitrary point on the line. Then, Harclay asks whether or not we can always find another line between the first point of the original line and the point we chose. If the answer to this question is negative, there should be a point immediate to the first point of the line. On the other hand, if the answer is affirmative, we should be able to find a line between the first point and the most immediate point to the first one. Since there is at least one point in any line, there should be at least one point which God could not perceive. However, this conclusion contradicts the assumption that God can perceive all the points in a line. Therefore, in either cases, it follows that the continuum is composed of indivisibles. Harclay examines some possible objections to his arguments. One of such objections is that lenghthless indivisibles cannot constitute a line which has surely some length.45 To this objection Harclay replies that it is possible for indivisibles to constitute a line. He argues that the opponents mistakenly think that the only way to put two points together is to put them in a superposed position, that is, to put them in the same location.46 If the opponents were right, there would be no way to make a line out of points.

44In this respect, Harclay appeals to the writing of Robert Grosseteste. Grosseteste writes, “every infinite number is finite, more so than two is finite to me” (Henry of Harclay 2008, pp. 1030–1031). This passage originally appears in Grosseteste’s commentary on Aristotle’s Physics (Robert Grosseteste 1963, p. 92). This conception (“every number is finite to God”) can be traced back to Augustine. 45Henry of Harclay 2008, pp. 1054–1067. 46Harclay has Aristotle in mind as such an opponent. In Aristotle’s own words: “[S]ince the indivisible has no parts, if two indivisibles touched each other at all it must be in their entirety” (Aristotle 1934, pp. 92–93). CHAPTER 2. THE MEDIEVAL PERIOD 63

However, according to Harclay, superposing a point on another is not the only way to put two points together. Harclay claims that we can put two points immediately next to each other and make a line out of points. Another objection Harclay considers is exactly the same as one of Scotus’ arguments which was described in the previous section.47 If a line is composed of infinitely many points, the length of a side of a square and that of a diagonal of the square should be the same because all infinities are equal. To this objection Harclay maintains that some infinity can be larger than some others although he does not say much about how this is possible. Yet another, perhaps the most effective objections to Harclay’s argument comes from the English mathematician Thomas Bradwardine.48 In his Tractatus de continuo, probably the most sophisticated and thorough treatise on the problems of the continuum in the middle ages, Bradwardine not only tried to refute all the variations of indivisibilism, but also to clarify the nature of indivisibles and the continuum. Tractatus de continuo is written in a highly axiomatic manner rem- iniscent of Euclid’s Elements; that is, it is comprised of the series of definitions, suppositions and conclusions. Let us start our examination of Tractatus with his definition of the continuum. 47Henry of Harclay 2008, pp. 1066–1069 48Thomas Bradwardine was born in England around 1300 and studied theology at Mer- ton College. Back then at Merton, Bradwardine was known as one of the “Calculators” who claimed the importance of the mathematical reasonings (as in the Euclid’s Elements) in theological and philosophical issues. Soon after he was elected as archbishop of Can- terbury in 1349, Bradwardine died of the black death. Tractatus de continuo which I will argue about is thought to be written around 1325. CHAPTER 2. THE MEDIEVAL PERIOD 64

Deﬁnition 1 The continuum is a quantity whose parts are connected to one another.49

The important point of the above definition is that Bradwardine does not exclude the possibility that the continuum is composed of indivisibles.50 Bradwardine’s strategy for refuting various indivisibilist positions is, first, to leave room for indivisibilists to argue for their positions, and then, to refute each indivisibilist position. After Bradwardine defined the continuum as above, he defines indivisibles as follows.

Deﬁnition 7 An indivisible is that which cannot be divided.51

Now, in order for indivisibilists to maintain that the continuum is composed of indivisibles, they have to show that indivisibles can connect with each other. Bradwardine deﬁnes two ways of connecting geometrical objects: superposition and imposition. These concepts are deﬁned as follows.

Deﬁnition 15 A line superposed to another line is a line ad- hering directly to another line without anything between them. Deﬁnition 16 A line imposed to another line is a line continued in another line.52

49Murdoch 1957, p. 339; my translation: “1 – Continuum est quantum cuius partes ad invicem copulantur”. 50Recall that Aristotle defines the continuum by the mirror property and the infinite divisibility. If the continuum were defined in this way, there would be no way for the continuum to be composed of indivisibles. 51Murdoch 1957, p. 339; my translation: “7 – Indivisibile est quod numquam dividi potest”. 52Murdoch 1957; my translation: “15 – Lineam linee superponi partialiter vel total- CHAPTER 2. THE MEDIEVAL PERIOD 65

Let us clarify these definitions by ex- A B amples. In Figure 2.4, there are two lines AB and CD, and when these lines are superposed, C D the result looks like just one longer line as depicted under the arrow of Figure 2.4. However, ⇓ there are actually still two lines after superpo- A B sition. If we had an advanced magnifying glass C D and took a look at superposed lines closely, we Figure 2.4 would see something like Figure 2.5. They are B two discontinuous lines although they look like C one continuous line for mere mortals. Figure 2.5 On the other hand, if the lines AB and CD are imposed with each other, the result is actually one continuous line. In the case of imposition, two lines are somehow fused together and became a continuous one. Based on these definitions, Bradwardine raises his first objection to the thesis that the continuum can be composed of indivisibles. At first sight, it seems possible that lines each of which is composed of indivisibles are imposed with each other and become one continuous line. However, according to Bradwardine, the only relation which is allowed between indivisibles is iter est ipsam secundum longitudinem totius vel partis simpliciter sine media adherere alteri. 16 – Lineam linee secundum partem vel secundum totum imponi est ipsam secundum longitudinem ipsius totius vel partis in aliam continuari”. In the original definitions, Bradwardine defines total and partial versions for each concept. However, these distinctions are not important here so I omit them. CHAPTER 2. THE MEDIEVAL PERIOD 66 superposition. This is a consequence of the following conclusion.

Conclusion 3 More than one indivisible in the continuum cannot situate in the same indivisible place.53

The reason why Bradwardine claims this conclusion is that, if more than one indivisible can be in the same indivisible place, it is possible for some continuum to have no quantity because, with the assumption that the continuum can be composed of indivisibles, all the indivisibles can be in the same indivisible place. This contradicts the above-mentioned definition of the continuum as a quantity. Bradwardine’s argument above seems flawed in several aspects. First, Bradwardine assumes that all indivisibles are extensionless and consequently they do not have a quantity. However, he admits the possibility that indivisibles have a quantity as in the Democritean indivisibles.54 Thus, there is no reason that Bradwardine has to assume the extensionlessness of indivisibles in this particular instance. Secondly, although it is surely possible that, with the assumption that indivisibles here are extensionless, even when infinitely many indivisibles are together in the same indivisible place, it does not follow at all that all the collections of indivisibles must be in the same indivisible place. Presumably, in very many cases, each indivisible would be in its own indivisible place. All that has to be done in the case Bradwardine proposes

53Murdoch 1957, p. 350; my translation: “Nullius continui multa indivisibilia in eodem situ indivisibili situari”. 54See Footnote2 above. CHAPTER 2. THE MEDIEVAL PERIOD 67 above is just dismiss such a case as the continuum. Lastly, Bradwardine does not say anything about why continuous geometrical objects like lines can be in the same place and imposed to one another. Here, it seems that he simply assumes that and consequently falls into the fallacy of begging the question. The next objection Bradwardine raises seems a bit strange to our modern eyes. In this objection, Bradwardine assumes that the relation which indivisibles bear to one another is the immediate relation; namely, he tries to refute immediate indivisibilism. Now, Figure 2.6 let us think of an indivisible. It has two immediate indivisibles in a horizontal direction. It also has two more immediate indivisibles as neighbors in a vertical direction each of which has two immediate indivisibles in a horizontal direction. Thus, the situation can be depicted in Figure 2.6. This is asserted in the following conclusion.

Conclusion 38 [If an atom, i.e. indivisible or point, in the continuum is immediate to another atom,] there are no more than 8 points immediate to each other in a two-dimensional plane.55

If this is so, there are only 8 lines which can be drawn from an arbitrary point. At ﬁrst sight, it seems that we can draw more than 8 lines

55Murdoch 1957, p. 386; my translation: “Si sic, puncto in medio superﬁciei plane situato 8 puncta immediata, et non plura”. CHAPTER 2. THE MEDIEVAL PERIOD 68

Figure 2.7

from any point as in Figure 2.7. However, if we admit Conclusion 35, we cannot draw such an extra line. The reason of this is as follows. First, let us assume that we can draw a line from A to B as in Figure 2.8. Then, there B should be an immediate point to each of A and B. However, if we put an immediate point to each of A and B, we ﬁnd out that these im- A mediate points overlap (Figure 2.9). This is Figure 2.8 impossible because indivisibles cannot overlap. B Therefore, Bradwardine concludes that, if the continuum is composed of immediate points, there are only 8 lines which can be drawn from A any arbitrary point. Consequently, he denies Figure 2.9 the possibility that the continuum can be composed of immediate points because we should be able to draw a lot more than 8 lines from a point. CHAPTER 2. THE MEDIEVAL PERIOD 69

In the above argument, Bradwardine seems to implicitly assume that indivisibles have an extension. If the main target here were the Dem- ocritean indivisibilism, such an assumption would be admissible. However, what Bradwardine intends to accomplish is to refute immediate indivisibilism in which indivisibles are supposed to be extensionless. If indivisibles are extensionless, there is no way for them to overlap and then we should be able to put immediate points to each of A and B. The most successful argument against immediate indivisibilism would be the one which is presented in the way of proving the following corollary.

Corollary 20 Every straight line can be divided into many straight lines.56

In what follows, let us see how Bradwardine proves this corollary and refutes immediate indivisibilism. Let us think of the following construction.

1. Draw a line CD vertically. This is the line which is to be divided inﬁnitely.

2. Draw a circle which passes through D.

3. Draw a horizontal line which passes through C.

56Murdoch 1957, p. 371; my translation: “Omnem lineam rectam in multa rectas posse dividi”. Although the original text just says “many” as to the number of straight lines into which a straight line is divided, it seems reasonable to think of the number as inﬁnite because Conclusion 66, which is directly proved from the corollary, states that every straight line has inﬁnitely many lines as its parts. CHAPTER 2. THE MEDIEVAL PERIOD 70

4. Name the intersections of the circle and the horizontal line A and B.

5. Draw a circle which passes through A and B and is bigger than the circle drawn in the step 2.

6. Name the intersection of the circle drawn above and the vertical line E.

The result of this construction is depicted in Figure 2.10. The important point D to note here is that EC becomes smaller as E A B the circle drawn in the step 5 becomes bigger. C Then, it follows that EC becomes smaller and smaller infinitely if the circle can become bigger and bigger infinitely. And this is guaranteed by Figure 2.10 Postulate 3 of Euclid’s Elements which allows us to draw a circle with any center and radius.57 To the above argument one can object as follows. Let us suppose that the line CD is the minimal line.58 Then, there would not be a circle 57Euclid 1908, p. 154. Here, one might want to ask why Bradwardine proceeded in this rather cumbersome way in spite of the fact that there is a proposition in Euclid’s Elements which seems to enable us to argue in a more straightforward way. According to the tenth proposition of Book I of Elements (Euclid 1908, p. 267), we can bisect a given straight line. If we repeat this bisecting procedure infinitely, it seems to follow that a straight line can be divided infinitely. However, even if Euclid states that we can bisect a give straight line, he does not tell us how small a straight line can be. This is a question worth asking because there would be a limit to the process of bisection if there is the minimal line whose initial and end points are immediate to one another (Euclid leaves room for the possibility of the existence of such a minimal line). On the other hand, the question concerning how big a straight line can be seems less critical. 58See Footnote 57. CHAPTER 2. THE MEDIEVAL PERIOD 71 which can cut CD into two line segments. Therefore, it follows that CD cannot be divided infinitely. Bradwardine responds to this objection by saying that it is the existence of such a minimal line, not Euclid’s postulate, to be abandoned.

2.3 Propositional Analysis of the Continuum:

William of Ockham

As was seen in the previous chapter, Aristotle claims that the continuum cannot be composed of indivisibles. However, Aristotle does not say much about the existence of indivisibles themselves. Do indivisibles exist? There are some passages in Physics where Aristotle seems to commit himself to the existence of indivisibles. Aristotle writes:

[I]f, as they [i.e., the Pythagoreans] say, there were such things as sejunct [i.e., separate] points and monads [i.e., natural numbers], then the point and the monad could not be identical; for two points could touch each other, but two monads can only be next- in-succession to each other. And between any two points there can be found intermediate points, for between every two points there is a line, and in every line there are points. . . .59

In the above, Aristotle seems to admit the existence of indivisibles.

59Aristotle 1934, Book V, 227a29–32. CHAPTER 2. THE MEDIEVAL PERIOD 72

William of Ockham,60 who considered himself as a faithful follower of Aris- totle and assiduously denied the existence of indivisibles,61 maintains that most of the appearances of the term “point” in Aristotle’s writings are in conditional sentences, and since Aristotle uses some terms in such conditional contexts, he does not necessarily commit himself to the existence of the reference of the term. However, as Ockham himself admits, there are many cases where the term “point” appears not conditionally. Even in such cases, still, Ockham claims, there is no need to suppose the existence of points.62 According to Ockham, in order for something to be legitimately considered as existing, it has to have a three-dimensional body.63 In this conception, there is no room for the zero-dimensional indivisible, namely a point, to exist.64 However, Ockham does not deny the use of the term “point” in natural philosophy. For Ockham, the term “point” is just a short-hand expression for “a line of such and such a length”.65

60William of Ockham was born in Ockham, a small village in the southwest of London, around 1285. Ockham started his study of theology at Oxford around 1310 and lectured on Peter Lombard’s Sentences from 1317 to 1319. In 1324, Ockham was summoned to Avignon for clearing up the doubt that he had a heretic thought, and until 1328 when Pope John XXII was about to issue a condemnation against him, he had to stay there. Ockham died in Munich in 1347. The writings which I’ll deal with here are all thought to be written in his Avignon era. 61The main reason why Ockham denied the existence of indivisibles is, once again, theological. To explain how the bread and wine turn into the body of Christ, Ockham thought that he had to ﬁrst ﬁgure out what is considered as really existing. In such an attempt, Ockham examined whether a point, a line, a surface, and a body really exist. As will be shown below, Ockham’s conclusion is that only a three-dimensional body can be said to exist. 62For the full discussion, see William of Ockham 1985a, pp. 452–462. 63Ockham calls such a real existing body res permanens. 64For the same reason, Ockham denies the real existence of lines and surfaces. 65William of Ockham 1986, p. 22. Since a line is not a res permanens, it has to be CHAPTER 2. THE MEDIEVAL PERIOD 73

Denying the existence of indivisibles, Ockham naturally affirms the infinite divisibility of the continuum. His attitude toward this issue is mostly Aristotelian. However, his conception of parts in such an infinitely divisible continuum significantly deviates from the Aristotelian conception. At least it seems so.66 As I have shown in the previous chapter, Aristotle thinks that the infinite division of the continuum can be done only potentially. He also seems to think that the parts of the continuum exist only potentially. Whereas Ock- ham agrees with Aristotle as to the potential divisibility of the continuum, he has a different view concerning the potential existence of the parts of the continuum. One of the reasons why it is usually thought that Aristotle denies the actuality of the existence of the parts in the continuum would be sought in the following passage of his Physics.

Now things are said to exist as potentialities or as actualities; and there is no limit to the addition (or subtraction) of terms in a convergent series; and though we have seen that a magnitude cannot actually be increased beyond limit by multiplication, it may be divided into something smaller yet than any magnitude you choose to mention—for there is no diﬃculty in refuting the thought of as a short-hand for “a surface of such and such a width”. The same is applied to the term “surface”. Thus, the full expression of the term “point” would be pretty complex. Luckily, Ockham does not require such pedantry. 66Ockham himself thinks, as I will describe in what follows, that he rightly understand the real intention of Aristotle. CHAPTER 2. THE MEDIEVAL PERIOD 74

doctrine that there are such things as atomic lines. It results that the unlimited potentialities exists.67

Ockham maintains that the only conclusion we can draw from the above passage is that some aspects of the parts in the continuum are indeterminate. As to the existence of such parts, he claims that it is actual.68 For Ockham, in order for something to be considered as having an actual existence, it is required that it has a three-dimensional body. Thus, as long as it has a three-dimensional body, something continuous is considered as existing actually. Any part of something that actually exists is, Ockham claims, also actual.69 Ockham argues that the misreading of Aristotle results from confusion of the potentiality of relations with that of existence.70 For example, Aristotle writes that “if a mass is continuous and homogeneous, its parts are only potentially in places-proper”.71 In the passage just cited, it is the ﬁrst kind of potentialities that is asserted. We simply do not know where some part of a continuum situates in it until we specify about which part we are talking. After such designation, the position of a part is actualized.72 Ockham applies this method of analysis to the problem of the inﬁ-

67Aristotle 1929, Book III, 206a14–19. Interestingly, there is no corresponding Greek expression to “a convergent series”. 68A detailed discussion concerning this issue is in Murdoch 1982b, pp. 186–190. 69William of Ockham 1985b, p. 562. 70William of Ockham 1985b, p. 564. 71Aristotle 1929, Book IV, 212b3–7. 72“[I]f they were so divided as to be in mutual contact (as if in a heap), they would have actualized places-proper” (Aristotle 1929, Book IV, 212b3–7). CHAPTER 2. THE MEDIEVAL PERIOD 75 nite divisibility of the continuum. With Aristotle, Ockham thinks that the continuum can be divided infinitely. However, special attention should be paid to the mode of this statement. In other words, the infinite divisibility of the actual continuum must not be confused with the actual infinity of such divisions. The actual continuum can be divided infinitely but it cannot be actually divided into an infinite number of its parts. What is denied here is the actual existence of an infinite number of parts in the continuum, not the actual existence of parts themselves. When we are given the actual continuum, Ockham says, we are given its parts actually as well.73 The next task Ockham sets himself is the propositional analysis of the infinite divisibility of the continuum.74 Ockham starts his analysis by distinguishing two kinds of de possibili propositions, that is, propositions in which the auxiliary verb “can” appears.75 The first of these two kinds is a proposition which can be rewritten without the auxiliary verb “can” and still can be true after this rewriting. For example, let us think of the proposition “Bronze can be a statue”.76 Bronze, as a whole, can be a statue. Then, it is possible to say “Bronze is a statue” without the auxiliary verb “can”. In

73An intuitive argument Ockham gives here is as follows. First, suppose that we are given the actual continuum. Then, ask this: Does the half of the continuum exist? Surely it does. How about the half of that half? One would agree that the half of the half of the continuum exists. Ockham pushes this thought-experiment to the limit and concludes that the parts of the continuum actually exist. See Murdoch 1982b, p. 185. 74For a summary of this propositional analysis, see Murdoch 1982b, pp. 190–199. 75William of Ockham 1985a, pp. 541–545. 76This “bronze” example was first brought up by Aristotle in his Physics (Aristotle 1934, pp. 246–247). Since then, it has become almost customary to take this example up when talking about potentiality in Aristotle’s Physics. CHAPTER 2. THE MEDIEVAL PERIOD 76 short, this kind of de possibili propositions can be actualized. On the other hand, the other type of de possibili propositions cannot be actualized. This kind has the property that rewriting a de possibili proposition which belongs to the second type produces more than two propositions at least one of which still has the auxiliary verb “can”. For example, let us think of the very sentence which represents the infinite divisibility of the continuum: “A line can be infinitely divided”. This sentence cannot be rendered into the first type because the rewritten sentence would be “A line is infinitely divided” but no line has ever been, and will ever be, divided infinitely. The right rewriting is: A line is divided, and after this division, there is still another part which can be divided. This signals us that the initial sentence is a genuine de possibili one. The next propositional analysis Ockham offers attempts to clarify not only the meaning of the infinite divisibility of the continuum, but also the relation of parts in the continuum to the whole.77 Ockham begins his analysis by pointing out that there are some modifiers which, according to their position in a sentence, change the meaning of a sentence in which they appear. In Latin, the word “infinitum” is exactly that sort of a modifier. When the word “infinitum” proceeds a term it modifies, it gives potentiality to a sentence in which it appears. Thus, for example, the sentence “in infinitum continuum est divisibile” means that the continuum is divided so numerously but it can be divided more. On the other hand, if the word “in-

77William of Ockham 1985a, pp. 554–561. CHAPTER 2. THE MEDIEVAL PERIOD 77

finitum” follows a term it modifies, it gives actuality to a sentence in which it appears. Thus, the sentence “continuum est divisibile in infinitum” means that the continuum is actually divided into infinitely many parts. There is another kind of words which, according to their positions in a sentence, change the meaning of the sentence. The paradigmatic example of this would be a pair of the words “every” (omnis) and “some” (aliqua). Let us take a look at the following pair of sentences.

1. Omni magnitudine est aliqua magnitudo minor.

2. Aliqua magnitudo est minor omni magnitudine.

In the ﬁrst sentence of the pair, the word “omni” precedes “aliqua”. In this case, the sentence means that, for any multitude, there is still smaller magnitude. The second sentence has the word “aliqua” preceding “omni”. In this case, the sentence means that there is a magnitude which is smaller than every magnitude.78 Provided that the continuum can be divided inﬁnitely, the former is true whereas the latter is false.79

78As is easily seen, this difference exactly corresponds to the difference between the ∀∃-sentences and ∃∀-sentences. 79William of Alnwick, in his counter-argument against Harclay, utilizes this difference between the omnis-aliqua sentences and the aliqua-omnis sentences. See Grant 1974, p. 322. CHAPTER 2. THE MEDIEVAL PERIOD 78

Summary

In this chapter, I have presented how the medievals understand the continuum. In the first section, I discussed Scotus’ mathematical arguments for the infinite divisibility of the continuum. His arguments first assume the existence of indivisibles and show that absurdities are derived from the assumption. The arguments were so influential that almost every medieval writer who argued about the continuum touched on them. After the discussion of Scotus’ argument, I examined Chatton’s counter-arguments against Scotus’ mathematical arguments. First, Chatton’s conceptions of indivisibles and infinity were presented, and then, the counter-argument based on these conceptions were discussed. Chatton’s main point is that mathematical arguments are not suitable for arguing the continuum. Although the arguments do not seem very convincing to our modern eyes, they were thought of as very powerful arguments against the infinite divisibility of the continuum by the medievals. Following Chatton, I then presented Autrecourt’s argument against Scotus’ type of mathematical arguments. Autrecourt’s argument, which appeals to the distinction of the intellect and the imagination, seems more persuasive than Chatton’s. In the second section, I described Harclay’s arguments for indivisibilism and Bradwardine’s arguments against it. In Har- clay’s argument, God plays an important role, and perhaps because of that, Harclay’s arguments do not seem so convincing to our eyes. On the other hand, Bradwardine’s arguments seems to hold a special place in the medieval discussions of the continuum because of their rigorous mathematical struc- CHAPTER 2. THE MEDIEVAL PERIOD 79 ture. Finally, in the third section, I discussed Ockham’s arguments against indivisibilism with a special emphasis on propositional analysis which is a very typical way of arguing for Ockham. Chapter 3

Early and Late Modern

3.1 Stifel and Stevin

The climate concerning the irrational numbers began to change in the sixteenth century. In his Arithmetica Integra (1544), the german mathematician Michael Stifel (1487–1567) wrote as follows.

It is rightly disputed of irrational numbers whether they are true numbers or false. For because in proving with geometric ﬁgures, where rational numbers desert us, irrationals take their place, and they prove precisely what rational numbers were not able to prove, at least from the demonstrations which we know of: we are moved and compelled to admit they are correct, as is clear from their eﬀects, which we feel to be real, certain and constant.1

1Cited in Crossley 1987, p. 137.

80 CHAPTER 3. EARLY AND LATE MODERN 81

However, Stifel still retained the sharp distinction between magnitudes and numbers, and argued that magnitudes cannot be captured by numbers. For Stifel, the irrational numbers can be represented only as magnitudes, never as numbers. Therefore, although he admitted the utility of the irrational numbers, he concluded that “just as an inﬁnite number is not a number, so too an irrational number is not a real number”.2 It was the Flemish mathematician Simon Stevin (1548–1620) who ﬁnally asserted that the irrational and any other numbers fall under the same category; namely, they are all numbers. Stevin wrote:

It is a very common thing amongst authors of arithmetics to √ treat numbers like 8 and similar ones, which they call absurd, Irrational, irregular, inexplicable, surds &c. and which we deny to be the case for any number which turns up: But by what reason will the adversary prove it unreasonable?3

Stevin’s stance to admit the irrational and all other kinds of magnitudes as genuine numbers is well expressed in the manifesto-like theses in the very last part of his Arithm´etique (1585).

2Crossley 1987, p. 137. Stifel even denied that the circumference of a circle can be expressed by a number at all, whether it is rational or irrational. The point here is his distinction between physical and mathematical circles. Whereas physical circles can be measured by some physical means, there is no way to measure mathematical ones physically. In this sense, for Stifel, the irrational numbers were not real, but some kind of “mathematical ﬁction”. 3Cited in Crossley 1987, p. 96. CHAPTER 3. EARLY AND LATE MODERN 82

Thesis I That unity is a number.

Thesis II That any given numbers can be square, cubes, fourth powers, &c.

Thesis III That any given root is a number.

Thesis IV That there are no absurd, irrational, irregular, inexplicable or surd numbers.4

However, Stevin did not deﬁne the irrational numbers per se. Rather, they were supposed to be already given. The systematic ways of deﬁning the real numbers did not appear until the nineteenth century.

3.2 Fomalization of the Theory of Real Num-

bers

By the beginning of the nineteenth century, mainly due to the French mathematician Augustin-Louis Cauchy (1789–1857) and the Bohemian mathematician Bernhard Bolzano (1781–1848), the machinery to treat the real numbers

4Cited in Crossley 1987, p. 141. CHAPTER 3. EARLY AND LATE MODERN 83 systematically was almost ready. First, in his Cours d’analyse (1821), Cauchy presented the idea of Cauchy sequences of rationals by which we can express all the real numbers. Still, Cauchy himself seems to have just presupposed the existence of the real numbers and consequently did not try to define the real numbers in terms of his sequences. There seems to be some evidences of that. First of all, in Preliminaries of his Cours d’analyse, Cauchy gave definitions to many terms. Among them, he defined the concept of limit as follows.

When the values successively attributed to a particular variable indefinitely approach a fixed value in such a way as to end up by differing from it by as little as we wish, this fixed value is called the limit of all the other values.5

Following the above deﬁnition, Cauchy wrote

Thus, for example, an irrational number is the limit of the various fractions that give better and better approximations to it.6

It seems as if Cauchy is giving a deﬁnition of the irrational numbers by the concept of limit. However, we should pay attention to the phrase “for example”. Here, what Cauchy is doing is just to give an example of how the concept of limit works, not to deﬁne the irrational numbers by the concept.

5Cauchy 2009, p. 6. 6Cauchy 2009, p. 6. CHAPTER 3. EARLY AND LATE MODERN 84

And more fundamentally, Cauchy’s definition of limit tells us that the limit to which values of a sequence converge must be previously known or defined (there is no way to take difference between each value of the sequence and an unknown value). Another piece of evidence concerns the famous Cauchy criterion for convergence. In order to see this, we need some definitions: Cauchy sequences and convergence.

Deﬁnition We call a sequence sn = s1, s2, s3,... a Cauchy sequence if

∀ε > 0 ∃N ≥ 1 ∀n ≥ N ∀k ≥ 1 |sn − sn+k| < ε

holds. (N, n, and k are natural numbers.)

Informally, a Cauchy sequence is a sequence such that we can make the diﬀerence between sn and sn+k as small as we want for suﬃciently large N.

Deﬁnition We say that a sequence sn converges if

∃s ∀ε > 0 ∃N ≥ 1 ∀n ≥ N |sn − s| < ε

holds. (Again, N and n are natural numbers.) Otherwise, we say

that sn diverges. CHAPTER 3. EARLY AND LATE MODERN 85

Informally, convergence is a situation where we can make the value of sn indefinitely close to a fixed value s for sufficiently large N. With these definitions, we can state the following theorem.

Theorem A sequence sn converges if and only if it is a Cauchy sequence.

In Cours d’analyse, Cauchy did not give a proof for the above theorem. He just stated it as if it were an axiom.7 Maybe Cauchy thought it an easy theorem and left its proof to the reader as an exercise. However, as a matter of fact, it was impossible for him to prove it because in order to prove the theorem one has to appeal to one of the fundamental properties of the real numbers of which Cauchy does not seem to have had a clear and solid understanding. We cannot know for sure how much Cauchy realized his lack of understanding about the fundamental properties of the real numbers. As we have just seen above, he might have implicitly presupposed those properties of the real numbers because he thought that they are trivial. Or he might have realized that he needed to know something more about the real numbers but was unable to clarify what. Either way, he did not try to define the real numbers explicitly even though he got almost everything which is needed for defining the real numbers. Unlike Cauchy, who does not seem to have had an interest in defining the real numbers, Bolzano explicitly tried to do so at least in two places: 7Cauchy 2009, p. 87. CHAPTER 3. EARLY AND LATE MODERN 86 one in his 1817 book Rein anlytischer Beweis (Purely Analytic Proof),8 the other in his posthumous work Reine Zahlenlehre (Pure Theory of Numbers).9 In both works, Bolzano tried to prove that what we now call the greatest lower bound property (or more commonly, its dual, the least upper bound property) holds for every sequence of real numbers. And in Rein analytischer Beweis, he used the Cauchy criterion (or more precisely, the Bolzano-Cauchy theorem) in the proof.10 However, there seems a vicious circle in his proof because in order to prove the Cauchy criterion, one needs to use the greatest lower bound property (or its equivalents) which is to be proved here using the criterion. The greatest lower bound property expresses one of the properties —perhaps the most important one of them—which characterizes the real numbers. The property tells us that the real numbers are complete; in other words, it expresses the continuity of the real numbers. Then, what exactly does the greatest lower bound property state? Let us look at it in Bolzano’s own words.

Theorem. If a property M does not apply to all values of a variable quantity x but does apply to all values smaller than a certain u, then there is always a quantity U which is the greatest of those of which it can be asserted that all smaller x possess the

8Bolzano 1905. 9Bolzano 1976. 10Note that Bolzano’s use of the criterion preceded Cauchy’s use of it. Because of this fact, some authors argue that Cauchy might have plagiarized Bolzano. For example, see Grattan-Guinness 1970. CHAPTER 3. EARLY AND LATE MODERN 87

property M.11

Bolzano appealed to the Bolzano-Cauchy theorem in proving the above theorem and tried to prove the Bolzano-Cauchy theorem itself in § 7 of Rein analytischer Beweis. If the proof were legitimate, Bolzano would escape the criticism of being circular in proving the least upper bound theorem. Bolzano’s attempted proof of the Bolzano-Cauchy theorem roughly

12 goes as follows. Let xn be a Cauchy sequence. Let us suppose that there exists a limit for the sequence xn and call this limit X. Since xn is a Cauchy sequence, we can always ﬁnd such N that |xn − xm| < ε for any ε > 0 and for all n, m ≥ N. On the other hand, for any d, |X − xn+m| < d if n + m

11Bolzano 1905, p. 25; Russ 2004, p. 269. Although what the statement means should be clear, it might not be clear at ﬁrst sight why the greatest lower bound property expresses the completeness (or the continuity) of the real numbers. To make this point clear, let us think of a case in which the property fails to hold: a case for Q. Theorem. Q is not complete. Proof. Instead of showing some sequences or subsets of Q do not have the greatest lower bound property, we will show its dual, namely, that some sequences or subsets of Q do not have the least upper bound property. Now, let the property M for x in Q be “the square of x is less than 2”. Then, the set X the elements of which meet this property would be {x ∈ Q : x2 < 2}. Clearly, X is bounded above (by 2). In fact, X is also bounded below (by −2). Then, the property M holds for all x smaller than a certain u and greater than −u. Suppose that there is the least upper bound of X. Let us call that least upper bound r. As is easily seen, r has to be greater than zero and its square has to be either greater than 2 or less than 2 2r+2 2 2. Now suppose that r < 2. Let t = r+2 . Clearly, t > 0. Moreover, t < 2 because 2 −2r2+4 2 − t = (r+2)2 > 0. Therefore, t has the property M and is in the set X. Now it can be easily veriﬁed that t − r > 0 which means that t is greater than r. However, if r is less than t, it loses its status as an upper bound for X. Thus, a contradiction has been drawn from the assumption that there is the least upper bound which is less than 2. As we can draw a contradiction for the case of 2 < r2 in a very similar manner, it is concluded that there is no least upper bound for X in the domain Q and hence Q is not complete. 12Bolzano used F x for expressing a sequence. In the following, I’ll modernize Bolzano’s notations without notice. CHAPTER 3. EARLY AND LATE MODERN 88

is suﬃciently large. Note that |X − xn| = |(X − xn+m) − (xn − xn+m)| ≤

|X − xn+m| + |xn − xn+m| < d + ε. Since ε can be as small as we desire if m is taken suﬃciently large, we obtain |X − xn| ≤ d. Also, we can make d as small as we want if n is taken suﬃciently large. This means that X is a limit of the sequence xn. After this, Bolzano went on to show the uniqueness of such an X. However, we do not need to bother to follow it because where Bolzano erred in his attempted proof is already obvious at this point: He presupposed what

13 to be proved, namely, the existence of the limit for the sequence xn. Rusnock, in his Bolzano’s Philosophy and the Emergence of Modern Mathematics (2000), tries to amend Bolzano’s attempt by getting rid of the presupposition that there exists a limit X for a Cauchy sequence xn. Rusnock’s amendment is as follows.

Let d > 0 be given and choose N so that |Xn − xn+r| < d for

all n ≥ N and r > 0. Then all the terms of the sequence {xn}

with n ≥ N are between xN − d and xN + d; Then, too, clearly, if

there is a limit, it will be between xN − d and xN + d (or possibly equal to one of these values). Since the possible range of values

13Actually, Bolzano proposed an argument for why the limit for a Cauchy sequence should exist. He argues: “[T]he hypothesis that there exists a quantity X which the terms of this series approach as closely as we please when it is continued ever further certainly contains nothing impossible.. . . [A]lso the assumption of a constant quantity with this property of proximity to the terms of our series contains nothing impossible because on this assumption it is possible to determine this quantity as accurately as we please” (Bolzano 1905, pp. 21–22; Russ 2004, p. 267). Still, there is a huge gap between the possibility and the actuality. CHAPTER 3. EARLY AND LATE MODERN 89

may be made as small as desired by ﬁrst taking d small enough, a real limit exists.14

At ﬁrst sight, this amendment seems to work. However, there still remains a problem even with the amendment. As Rusnock himself points out, in order for a limit of the sequence {xn} to exist at all, the intersection

of the closed intervals [xNi − di, xNi + di] should be non-empty. However, this is just another way of stating the completeness of the real numbers.15 Therefore, the above amendment is still begging the question. As is probably seen from the above discussion, the fundamental problem in Bolzano’s attempt to prove the Bolzano-Cauchy theorem lies in the lack of the theory or the definition of the real numbers. In his posthu- mously published work Reine Zahlenlehre (1976), Bolzano finally addressed the issue head-on by inventing the idea of measurable numbers. First, suppose that the natural numbers, the integers, the rational numbers, and the operations on them (addition, subtraction, multiplication, and division) are given.16 Then, an infinite number concept and an expression of such a concept are defined.

Deﬁnition. Let me call every number concept in which an inﬁnite number of operations is required—these may be additions, sub-

14Rusnock 2000, p. 82. 15The proposition that the intersection of nested closed intervals should be non-empty ∞ \ (in symbols, [xn − dn, xn + dn] 6= ∅) is usually called the Cantor axiom. n=1 16The definitions for these concepts are done in the first five chapter of Reine Zahlenlehre. CHAPTER 3. EARLY AND LATE MODERN 90

tractions, multiplications or divisions, or all of them together—an inﬁnite quantity concept, and an expression by which such a concept is represented, an inﬁnite quantity expression.17

For example, the infinite sum of 1, 2, 3, . . . is an infinite number concept and expressed as 1 + 2 + 3 + ··· in inf . Among the infinite number concepts, there are some infinite number concepts S which meet the following condition: For all non-zero natural

p numbers q there exists an integer p such that two equations S = q + P and p+1 S = q − P1 hold where, in these equations, P is a non-negative infinite number concept (which could be zero) and P1 is a positive infinite number concept. If an infinite number concept meets this condition, it is called

p measurable. In these two equations, q is called the measuring fraction and p+1 q the next greater fraction. Let us take a look at some examples. Consider an inﬁnite num-

b ber expression S = a + 1+1+1+···in inf. (here, a and b are positive inte- 18 p b gers). If we put q = a and P = 1+1+1+···in inf. , the expression turns p b into S = q + P . On the other hand, we can rewrite a + 1+1+1+···in inf. p+1 1 b 1 b as q − ( q − 1+1+1+···in inf. ). Further, we can rewrite q − 1+1+1+···in inf. as (1+1+1+···in inf.)−qb q(1+1+1+···in inf) which is an inﬁnite number expression and strictly posi- (1+1+1+···in inf.)−qb p+1 tive. By putting P1 = q(1+1+1+···in inf) , s can be expressed as q − P1. b Therefore, S = a + 1+1+1+···in inf. is measurable.

17Bolzano 1976, p. 100; Russ 2004, p. 357; italics in the original. 18This example is taken from Bolzano 1976, pp. 102–103 (Russ 2004, pp. 358 ﬀ). CHAPTER 3. EARLY AND LATE MODERN 91

Another example: an inﬁnite number expression S = 1 + 2 + 3 + ··· in inf. Although it is easy to ﬁnd a way to rewrite this in the form of p p q + P (just put q = 1 and P = 2 + 3 + 4 + ··· in inf.), it is impossible to p+1 rewrite this in the form of q − P1. Therefore, S = 1 + 2 + 3 + ··· in inf. is not measurable. With the concept of measurable numbers, Bolzano seems to have succeeded in proving the Bolzano-Cauchy theorem this time without falling into vicious circle. However, the very concept of measurable numbers has

3 its own problems. For example, consider inﬁnite number concept 1 − 2 + 3 3 3 3·(−1)n 4 − 8 + 16 − · · · + 2n + ··· in inf. which clearly converges to zero. This concept is not measurable. However, it can be expressed as the sum of two

1 1 1 measurable inﬁnite expressions, that is, 1 − 1 + 4 − 4 + 16 − · · · in inf. and 1 1 1 1 0 − 2 + 2 − 8 + 8 − · · · in inf. This means that the measurable numbers are not closed under very simple and fundamental operations such as addition.19 That is surely not a nice property to have. Besides Cauchy and Bolzano, the name of Joseph Liouville (1809– 1882) should be mentioned concerning the discussions of the real numbers around the mid-eighteenth century; the existence of the transcendental numbers was ﬁnally proved by Liouville in 1844. Up until this time, the numbers which mathematicians had been dealing with were basically those that can be obtained by addition, subtraction, multiplication, division, and taking n- th roots for some positive integer, although it had long been guessed that π

19I owe this point to Rusnock 2000, p. 182. CHAPTER 3. EARLY AND LATE MODERN 92 and e should be transcendental numbers.

Summary

Irrational magnitudes had long not been considered as numbers. However, from around the mid-sixteenth century, irrational magnitudes had started being considered as numbers by mathematicians such as Stifel and Stevin. This was a crucial move in regard to the theorization (more particularly, mathematization) of the real number system. In the nineteenth century, mainly because of the need for making calculus rigorous, mathematicians began to try to give an explicit definition of the real numbers. For example, Cauchy formalized the concept of limits, and based on that, provided the concept of Cauchy sequences which plays an extremely important in one of the Cantorian constructions of the real numbers. However, it is unclear if Cauchy sought a rigorous definition of the real numbers; as we have seen in the above, Cauchy seem to have just presupposed the existence of the real numbers without defining them. On the other hand, Bolzano was well aware of the necessity of defining the real numbers explicitly and tried to find a way to define them by inventing the concept of measurable numbers without much success. From around 1850 on, more and more mathematicians had started to try to define the real numbers explicitly. This current culminated in 1872 when Cantor and Dedekind independently published papers which finally CHAPTER 3. EARLY AND LATE MODERN 93 gave satisfactory definition to the real numbers. These two figures, or two approaches to the theory of the real numbers, will be discussed in the next chapters. Part II

Theories of the Continuum Chapter 4

Dedekind

4.1 Background

As Dedekind writes in the preface of Continuity and Irrational Numbers (1872),1 his direct motivation to write this small pamphlet came from his teaching experience at the Polytechnikum in Z¨urich:

As a professor in the Polytechnic School in Zürich I found myself for the first time obliged to lecture upon the elements of the dif- ferential calculus and felt more keenly than ever before the lack of a really scientific foundation for arithmetic. In discussing the notion of the approach of a variable magnitude to a fixed limiting value, and especially in proving the theorem that every magnitude which grows continually, but not beyond all limits, must

1Hereafter, I abbreviate this work as Continuity.

95 CHAPTER 4. DEDEKIND 96

certainly approach a limiting value, I had recourse to geometric evidences. [. . . ] But that this form of introduction into the dif- ferential calculus can make no claim to being scientiﬁc, no one will deny.2

Although Cauchy had given an arithmetic foundation to the notion of limits at the end of the eighteenth century, he still had to resort to a geometrical intuition in proving some theorems. As Dedekind himself admits, there is nothing wrong with appealing to geometry as an educational means in teaching calculus3 as long as there are firm foundations for what is stated in geometrical terms. To give such firm foundations to the concept of continuity on which calculus is to be founded is exactly what Dedekind aims to do in Continuity. In addition to the above motivation, there seems to be a deeper reason which made Dedekind set out on this foundational enterprise: Dedekind’s intellectual environment around the time he was attempting to give an arithmetic foundation to the concept of continuity. Dedekind entered the University of Göttingenin 1850, and two years later he earned a doctoral degree under the supervision of Gauss.4 After his graduation, however, not being qualified to remain at the university as a

2Dedekind 1901, p. 1; Dedekind 1932, p. 315. 3“Even now such resort to geometric intuition in a first presentation of the differen- tial calculus, I regard as exceedingly useful, from the didactic standpoint, and indeed indispensable” (Dedekind 1901, p. 1; Dedekind 1932, p. 315). 4In regard to the biography of Dedekind, I referred to the Dictionary of Scientific Biography (Gillispie 1981, pp. 1–5). CHAPTER 4. DEDEKIND 97 researcher, Dedekind spent two years to gain knowledge required for the qualification. In 1854, he was qualified and went back to the university. A year later, Peter Gustav Lejeune Dirichlet (1805–1859) came to the university as a successor of Gauss. Dedekind attended Dirichlet’s lectures assiduously and was hugely influenced by him. The following description by Gotthold Eisenstein (1823–1852) tells us Dirichlet’s distinct traits well.

In contrast to the older school, the fundamental principle of the new school, founded by Gauss, Jacobi and Dirichlet, is this. Where the old school sought to attain its ends through long and involved calculation and deductions (as is the case even in Gauss’s Disquisitiones), the new avoids it by the use of a brilliant expe- dient; it comprehends a whole area in a single main idea, and in one stroke presents the ﬁnal result with utmost elegance.5

In dealing with mathematical problems, Dirichlet did not merely resort to brute-force calculation.6 In many cases, he devised a new tool to tackle the problem at hand.7 Moreover, as a corollary to this aspect, when Dirichlet analyzed a problem, he drew some basic concepts from it which in turn he utilized for solving the problem.8 In this way, Dirichlet brought a

5Wussing 1984, p. 270 6However, this does not mean at all that Dirichlet underrated the power of calculation. In his obituary of Carl Gustav Jacob Jacobi (1804–1851), Dirichlet says: “There is a certain field in which calculation is suitable” (“so es giebt doch gewisse Gebiete, in denen die Rechnung ihr Recht behält”(Dirichlet 1897, p. 245); my translation). 7For example, Dirichlet proposed the pigeonhole principle for proving a theorem about Diophantine approximation, now called Dirichlet’s approximation theorem. 8For example, Dirichlet analyzed periodic functions, drew some conditions which should CHAPTER 4. DEDEKIND 98 new level of rigor to mathematics.9 There was also another person who exerted a deep influence on Dedekind: Bernhard Riemann (1826–1866). Riemann, in a sense, pushed ahead with one aspect of Dirichlet’s research style. In his dissertation written in 1851, Riemann wrote:

A theory of those functions on the basis of the foundations here established would determine the configuration of the function (i.e., its value for each value of the argument) independently of any definition by means of operations. Therefore one would add, to the general notion of a function of a complex variable, only those characteristics that are necessary for determining the function, and only then would one go over to the different expressions that the function can be given.10

Following Dirichlet, Riemann also stressed the importance of general concepts. However, in Riemann, there is an aspect which Dirichlet does not seem to have emphasized: the importance of inner properties rather than outer expressions. This aspect greatly influenced Dedekind. Dedekind wrote in a letter to Rudolf Lipschitz:11 be met by a periodic function to be expressed by the sum of Fourier series, and proved a theorem about Fourier series. 9Jacobi says, in a letter to Alexander von Humboldt, that “only Dirichlet [. . . ] knows what a complete rigorous mathematical proof is” (cited in Ferreirós 1999, p. 10). 10Cited in Ferreirós 1999, p. 30. 11Lipschitz was also a student of Dirichlet. CHAPTER 4. DEDEKIND 99

My aim in the theory of numbers is to found the research neither on contingent forms of representation or of expressions but on simple basic concepts, and in so doing, [. . . ] to achieve in this ﬁeld something similar to what Riemann achieved in the ﬁeld of the theory of functions.12

As JoséFerreiróssuccinctly puts, Dedekind cultivated “an abstract, conceptual vision of mathematics” in his time at Göttingen.13 As was seen above, Dedekind’s motivation for constructing his theory of the real numbers is to give the arithmetic foundation for analysis. In the following sections, I will examine Dedekind’s theory of the real numbers in three parts. First, I will summarize the theory of the real numbers described in Continuity and Irrational Numbers. Then, I will show that Dedekind’s real number system meets the requirements of the standard formalization of the real number system (namely, the real numbers Dedekind constructs form the complete totally ordered Archimedean field).

12Dedekind 1932, p. 468; my translation: “Mein Streben in der Zahlentheorie geht dahin, die Forschung nicht auf zufälligeDarstellungsformen oder Ausdrücke sondern auf einfache Grundbegriffe zu stützenund hierdurch [. . . ] auf diesem Gebiete etwas Ahnliches¨ zu erreichen, wie Riemann auf dem Gebiete der Functionentheorie [. . . ]”. 13Ferreirós 1999, p. 26. CHAPTER 4. DEDEKIND 100

4.2 Continuity and Irrational Numbers

In Continuity and Irrational Numbers, Dedekind presupposes the existence of the rational numbers.14 He calls the set15 of the rational numbers the rational number system R16 and characterizes it as a totally ordered infinite complete field; that is, first, any two rational numbers are comparable (totally ordered), second, for any rational number there is always a greater (or less) rational number (infinite), and third, the result of the four arithmetic operations (addition, subtraction, multiplication, and division) for any two rational numbers (except the case of the division by 0) is a rational number again (complete field).17

14However, Dedekind gives us a sketch of how to construct the rational numbers. He writes: “I regard the whole of arithmetic as a necessary, or at least natural, consequence of the simplest arithmetic act, that of counting, and counting itself as nothing else than the successive creation of the infinite series of positive integers in which each individual is defined by the one immediately preceding. [. . . ] Addition is the combination of any arbitrary repetitions of the above-mentioned simplest act into a single act ; from it in a similar way arises multiplication. While the performance of these two operations is always possible, that of the inverse operations, subtraction and division, proves to be limited. [. . . ] thus negative and fractional numbers have been created by the human mind” (Dedekind 1901, p. 4; Dedekind 1932, pp. 317–318). How Dedekind constructs the natural number system will be surveyed in some detail when I will argue the philosophical aspects of Dedekind’s theory in Section 3 of this chapter. 15Dedekind does not use the term “set” (Menge) in Continuity. Instead, he uses the term “system” (System) for expressing the set-concept. Since there is no significant difference in their meanings, I occasionally use “set” when there is no danger of confusion. 16It might seem a little confusing to use R for denoting the rational number system. However, since Dedekind uses < for denoting the real number system, there should be no danger of confusion. 17In Dedekind’s own words: “This system, which I shall denote by R, possesses first of all a completeness and self-containedness which I have designated in another place [Dirichlet’s Lectures on the Theory of Numbers] as characteristic of a body of numbers [Zahlkörper] and which consists in this that the four fundamental operations are always performable with any two individuals in R, i.e., the result is always an individual of R, the single case of division by the number zero being excepted. / For our immediate purpose, CHAPTER 4. DEDEKIND 101

After characterizing the set of the rational numbers as above, Dedekind states three laws for R. The ﬁrst law says that the rational number system R follows the law of transitivity (i.e., for any three rational numbers a, b, c, if a < b and b < c, then a < c). The second law tells us that R is dense (i.e., between any two rational numbers, there are inﬁnitely many rational numbers). The third law merits a full citation.

If a is any deﬁnite number, then all numbers of the system R

fall into two classes, A1 and A2, each of which contains inﬁnitely

many individuals; the ﬁrst class A1 comprises all numbers a1 that

are < a, the second class A2, comprises all numbers a2 that are > a; the number a itself may be assigned at pleasure to the ﬁrst or second class, being respectively the greatest number of the ﬁrst class or the least of the second. In every case the separation of the

system R into the two classes A1, A2 is such that every number

of the ﬁrst class A1 is less than every number of the second class

18 A2.

This seemingly obvious law is, as I will show shortly, the very core of Dedekind cuts. however, another property of the system R is still more important ; it may be expressed by saying that the system R forms a well-arranged domain of one dimension extending to inﬁnity on two opposite sides.” (Dedekind 1901, pp. 4–5; Dedekind 1932, p. 318). In the above translation, “well-arranged” is a translation of wohlgeordent which is now usually translated as “well-ordered”. However, the usage of wohlgeordnet is diﬀerent from today’s usage. Judging from what Dedekind writes right after this passage, it would better be translated as “totally-ordered”. 18Dedekind 1901, p. 6; Dedekind 1932, p. 319. CHAPTER 4. DEDEKIND 102

In the second section of Continuity, Dedekind points out the par- allelism between a straight line and the rational number system R. In fact, if we interpret “p < q” as “a point p is to the left of a point q” (or “q is to the right of p”), any two points on a straight line are comparable (totally ordered), and for any point p there is always another point q such that p < q (or q < p) (infinite). Moreover, the three laws for the system R hold true of a straight line: If q is to the left of r and p is to the left of q, p is to the left of r; there are infinitely many points between any two points p, q; and if a straight line is cut in two, all the points in the left line segment are to the left of all the points in the right one. However, there is a crucial difference between a straight line and the rational number system R: “The straight line L is infinitely richer in point-individuals than the domain R of rational numbers in number-individuals.”.19 In other words, the rational number system R is full of gaps (lückenhaft). These remarks seem, as Dedekind himself admits, quite evident and superfluous.20 Furthermore, judging from how Dedekind has brought forth his argument so far, it might appear that Dedekind is trying to base his theory of the real numbers on a geometrical straight line. Rather, Dedekind thinks that the theory of magnitudes (i.e., geometry) should be founded on the that of numbers (i.e., arithmetic), not vice versa.21 In order to succeed

19Dedekind 1901, p. 9; Dedekind 1932, p. 321. 20“The previous considerations are so familiar and well known to all that many will regard their repetition quite superﬂuous” (Dedekind 1901, p. 9; Dedekind 1932, p. 321. 21“In my conception, the concept of relation between two homogeneous magnitudes can, to the contrary, be developed in a clear way only after the irrational numbers are CHAPTER 4. DEDEKIND 103 in such an enterprise, one must make clear what the essence of continuity is. Dedekind thinks that the essence of continuity is in the following principle.

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.22

After stating the above principle, Dedekind makes some important remarks. First, he says that “[t]he assumption of this property of the line is nothing else than an axiom”,23 and thanks to this axiom, he goes on, “we find continuity in the line”.24 It should be noted that Dedekind does not say that the property or principle itself is an axiom. If Dedekind considered the principle as an axiom, it would follow that he also thinks of continuity as inherent in a line. However, it is we that give continuity to a line. For Dedekind, continuity is not something extracted from a geometrical figure. The above interpretation is supported by the following remark which Dedekind makes immediately after the comment on the assumption of the introduced” (Dedekind 1901, p. 10; Dedekind 1932, p. 321. Here, I use my own translation because it seems to me that the translator of Dedekind 1901 made a critical mistake by translating “Größen” as “numbers” and muddled the contrast Dedekind wanted to emphasize. The original is as follows: “Nach meiner Auffassung kann umgekehrt der Begriff des Verhältnisseszwischen zwei gleichartigen Größenerst dann klar entwickelt werden, wenn die irrationalen Zahlen schon eingeführtsind”). Dedekind even mentions the possibility that space is not continuous. I will come back to this issue below. 22Dedekind 1901, p. 11; Dedekind 1932, p. 322. 23Dedekind 1901, p. 12; Dedekind 1932, p. 323. 24Dedekind 1901, p. 12; Dedekind 1932, p. 323. CHAPTER 4. DEDEKIND 104 principle of continuity as an axiom: “If space has at all a real existence it is not necessary for it to be continuous; many of its properties would remain the same even if were it discontinuous”.25 And he goes on to say that we can make a discontinuous space continuous by creating new points which fill the gaps. This is exactly the core idea of Dedekind cuts. The essence of Dedekind cuts has already appeared in the third law for the rational number system R.26 In short, this third law states that any number can divide the set of all rational numbers into two classes in such a way that all the rational numbers in one class are less than all the numbers in another. In the fourth section of Continuity which is entitled “Creation of Irrational Numbers”, Dedekind calls such a division a cut (Schnitt) and

27 denotes it by (A1,A2). Then, he remarks that, if a division (A1,A2) is produced by a rational number a, one of the following cases holds.

1. A1 has a greatest number while A2 does not have the least number. (In

this case, the greatest number in A1 is a.)

2. A2 has a least number while A1 does not have the greatest number. (In

this case, the least number in A2 is a.)

25Dedekind 1901, p. 12; Dedekind 1932, p. 323. Here, “many of its properties” undoubt- edly refer to those of Euclid’s Elements. In fact, in a letter to Rudolf Lipschitz, Dedekind writes that “Euclid’s whole system remains even without continuity” (Dedekind 1932, p. 479; my translation: “sein ganzes System bleibt bestehen auch ohne die Stetigkeit”). He takes this point up again in his 1888 Was sind und was sollen die Zahlen? and argues that algebraic numbers suﬃce for all the constructions found in Euclid’s Elements (Dedekind 1932, pp. 339–340). 26See p. 101. 27 Dedekind 1932, p. 323. Here, it is stipulated that all the rational numbers in A1 are less than all rational numbers in A2. CHAPTER 4. DEDEKIND 105

Between these two cases, there is no essential difference. The only (inessen- tial) difference is whether the rational number which produces the cut belongs to one class or another. However, there are other cases which are essentially different from the above two. Such cases are paradigmatically stated as follows.

3. There is neither a greatest number in A1 nor a least number in A2.

In order to see that there is a cut of which the above case 3 holds true, let us consider the cut (P,Q) where Q is deﬁned as {q : 0 < q and 2 < q2}. In this cut, there is no least number in Q. In other words, we can always ﬁnd a rational number s in Q such that s < q for any q in Q. To

2q+2 see this, let us think of s = q+2 for q chosen arbitrarily from Q. Clearly 2 2 2(q2−2) s > 0. Also, s > 2 since s − 2 = (q+2)2 > 0. Therefore, s is in Q. Now q2−2 q − s = q+2 > 0. Since the choice of q is arbitrary, this means that q > s for all q in Q. In a similar fashion, it can be shown that there is no greatest number in P either.28

If there is neither the greatest number in A1 nor the least number in A2, this cut (A1,A2) cannot be produced by a rational number. And if the cut (A1,A2) cannot be produced by a rational number, there has to be a gap between the classes A1 and A2. In order to make the union of the classes

A1 ∪ A2 continuous, the gap has to be ﬁlled. But how? Dedekind answers: By creating a new number. He writes:

28 2p+2 For s = p+2 for any p in P , it can be shown that p < s. CHAPTER 4. DEDEKIND 106

Whenever, then, we have to do with a cut (A1,A2) produced by no rational number, we create a new, 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.29

After giving the above crucial statement, Dedekind defines order relations between any two numbers purely based on cuts, and shows that any two numbers defined with cuts are comparable; that is, the set of the numbers obtained from cuts is totally ordered.30 Dedekind calls this set <. From the totally-orderedness of the real number system < and the very fact that this set is created (or defined) by cuts, it can be shown that < has the same properties as the rational number system R; that is, < is transitive, dense, and has the property that all the numbers of the first class of a cut produced by a certain (real) number are less that all the numbers of the second class. And naturally, < has the essence of continuity as the straight line has. However, the status of the essence of continuity in < is quite different from the case for the straight line. For the straight line, the essence of continuity is an axiom. For <, it is a theorem31 whose statement

29Dedekind 1901, p. 15; Dedekind 1932, p. 325. 30Dedekind 1901, pp. 15–19; Dedekind 1932, pp. 325–328. 31Concerning the essence of continuity’s being a theorem in <, Lipschitz writes, in his letter to Dedekind dated 6 July 1876, “What you mention about the completeness of the domains, which is derived from your principle, essentially coincides with the basic property of a line, without which no one can think of a line” (Dugac 1976, pp. 219–220; my translation: “Was Sie von der Vollständigkeit des Gebietes erwähnen,die aus Ihren Principien abgeleitet wird, so fälltdieselbe in der Sache mit der Grundeigenschaft einer CHAPTER 4. DEDEKIND 107 goes as follows.

If the system < of all real numbers breaks up into two classes

A1, A2 such that every number α1 of the class A1 is less than

every number α2 of the class A2 then there exists one and only one number α by which this separation is produced.32

With the above theorem, it is established that Dedekind’s <’ is complete.33

Linie zusammen, ohne die kein Mensch sich eine Linie vorstellen kann”). For Lipschitz, the continuity of a line is so evident that there is no need for a cumbersome proof that the domain of the real numbers is continuous. Lipschitz thinks that the domain of the real numbers is based on, or derived from, a straight line which is, for Lipschitz, necessarily continuous, and consequently that the domain of the real numbers is clearly continuous. Here, Lipschitz misunderstands the presupposition and aim of Continuity. First of all, as already explained (p. 104), Dedekind does not presuppose the continuity of a line. Rather, for Dedekind, the continuity of a line, if a line is supposed to be continuous at all, has to be based on his principle of continuity. After making clear what the principle of continuity is, the domain of the real numbers (i.e., the real number system <) defined with cuts can be shown to be continuous. Then, only at this point, with the assumption that there is an isomorphism between the real numbers and a line (the Dedekind-Cantor axiom), a line is shown to be continuous. For Dedekind, the continuity of a line should be deduced from that of <. 32Dedekind 1901; Dedekind 1932, p. 329. Its proof is straightforward from <’s being totally ordered and its three laws. 33It should be noted that, for Dedekind, the completeness of < means that < is the maximal extension of the totally ordered field of the rational number system R. In other words, once we add an extra element (for example, an infinitesimal) to <, it loses its properties inherited from R. This understanding is supported from the following passage in Dedekind’s letter to Lipschitz dated 10 June 1876: “[O]ne can find nowhere in Euclid’s writings or any others’ writings written after Euclid the achievement of such completeness, the concept of the continuous, i.e., of the most complete thinkable domain of magnitudes, whose essence is in the following property: “If all magnitudes in a continuously graded domain of magnitudes into two classes in such a way that every magnitudes in the first class are smaller than every magnitudes in the second class, then there exists either the greatest magnitude in the first class or the smallest in the second” ” (Dedekind 1932, pp. 473–474; my translation: “[N]irgends findet sich bei Euklid oder einem späteren Schriftsteller der A b s c h l u ß solcher Vervollständigung, der Begriff des stetigen d. h. denkbar vollständigsten Größen-Gebietes,dessen Wesen in der Eigenschaft besteht: zer- ” fallen alle Größeneines stetig abgestuften Größen-Gebietes in zwei Classen von der Art, CHAPTER 4. DEDEKIND 108

The next task Dedekind sets out to accomplish is the definition of the four basic arithmetic operations purely in terms of cuts. For example, addition is defined as follows.34 First, suppose that we are given two real numbers α, β. These real numbers produce cuts (A1,A2) and (B1,B2). What has to be done here is to define the sum γ of α and β in terms of the cuts (A1,A2) and (B1,B2).

Now, let us deﬁne the cut (C1,C2) as follows. If a rational number c is less than the sum of some numbers a1 in A1 and b1 in B1, we put it into C1; if c is greater than the sum of any choice of a1 and b1, we put it into C2. Clearly, these two classes C1 and C2 form a cut.

Summary

In 1872, Dedekind published the monograph in which he explicitly gave the definition of the real numbers. For defining the real numbers, he appealed to the method of cuts. However, the most important point of Dedekind’s construction of the real numbers is, I think, not in the method itself; rather, it is in his conception of continuity. And then, based on this conception, he “created” the real numbers, and gave the real number system the status of the complete ordered field.

daß jede Größeder ersten Classe kleiner ist als jede Größeder zweiten Classe, so exi- stirt entweder in der ersten Classe eine größte,oder in der zweiten Classe eine kleinste Größe“ ”). 34Dedekind 1901, pp. 21–22; Dedekind 1932, pp. 329–330. Chapter 5

Cantor

Cantor’s constructions of the real numbers are important in two ways here. One importance is that, needless to say, one of Cantor’s constructions of the reals has now become the standard in mathematics. The other importance is, more importantly for my purpose, that Charles Saunders Peirce, who will be the subject of Chapter 6, took Cantor’s construction as a pivotal reference point. Unlike the previous chapter on Dedekind, this chapter will almost exclusively focus on the theoretical aspects of Cantor’s works.

5.1 Basic Construction: Fundamental Sequences

Cantor’s motivation for his construction of the real numbers was, at ﬁrst, in his attempt to solve the uniqueness problem of the representation of a

109 CHAPTER 5. CANTOR 110 function in a trigonometric series. The history of the uniqueness problem goes back to the early nineteenth century when Fourier discovered that every function can be expressed as a sum of trigonometric series. Like Dedekind, Cantor also presupposes the existence of the rational numbers in constructing the real numbers.1 First Cantor takes the domain [Gebiet] of the rational numbers (including 0) which he calls A, and he tells us that we can think of an inﬁnite sequence of rational numbers a1, a2, . . . , an,... which is given by the following law:

The diﬀerence an+m −an becomes indeﬁnitely small as n increases with an arbitrary positive whole number m, or in other words, for an arbitrarily chosen (positive rational) ε there exists a whole

number n1 such that |an+m − an| < ε, if n = n1 and if m is an arbitrary positive whole number.2

A sequence with the above property is said to “have a deﬁnite limit b”.3 However, as Cantor cautions,4 we should take this limit not as a number,

1“The rational numbers form the foundation for establishing the larger concept of a numerical magnitude” (Cantor 1872, p. 92; my translation: “Die rationalen Zahlen bilden die Grundlage fürdie Feststellung des weiteren Begriffes einer Zahlengröße”. All the page numbers of the German originals are of Cantor’s Gesammelte Abhandlungen (Cantor 1932)). 2 Cantor 1872, p. 93; my translation: “[D]ie Differenz an+m − an mit wachsendem n unendlich klein wird, was auch die positive ganze Zahl m sei, oder mit anderen Worten, daß bei beliebig angenommenem (positiven, rationalen) ε eine ganze Zahl n1 vorhanden ist, so daß |an+m − an| < ε, wenn n = n1 und wenn m eine beliebige positive ganze Zahl ist”. Cantor calls sequences with this property fundamental sequences [Fundamentalreihe] in his 1883 Grundlagen (Cantor 1932, p. 186; Ewald 1996, p. 899). He does not use the term “Cauchy sequences” which is commonly used today for this kind of sequences. 3Ibid.; italics in the original and my translation: “[Die Reihe] hat bestimmt Grenze b”. 4Ibid. CHAPTER 5. CANTOR 111 but as a mere sign [Zeichen] assigned to a sequence; if we thought of such a limit as a number at this point, we would make a logical mistake.5 Only after we establish the order relations and the arithmetic operations on such signs, Cantor says,6 the status as a number of a limit b is justified. The order relations between signs are defined as follows. Let b

0 0 be a sign for a sequence {an} and b be for another sequence {an}. (Both sequences of course are supposed to have the above property.) The relation

0 between these sequences {an} and {an} falls under one and only one case of the following.

0 7 1. The diﬀerence an − an becomes inﬁnitely small as n increases; if this is the case, we say b = b0.

0 2. The diﬀerence an −an remains greater than a positive (rational) ε from a certain n; if this is the case, we say b > b0.

0 3. The diﬀerence an − an remains less than a negative (rational) −ε from a certain n; if this is the case, we say b < b0.

Note that the above still holds true if one of the limits of the sequences is

5Cantor says, in his Grundlagen, that “the number b to be defined is not set at the beginning as equal to the sum Σaν of the infinite series (aν ); this would be a logical error, because the definition of the sum Σaν is only reached by equating it with the finished number b which is necessarily defined earlier” (Cantor 1883a, p. 185; italics in the original and translation from Ewald 1996, p. 898). Although Cantor speaks of Weierstrass’ construction of the real numbers in this quotation, the same holds true of Cantor’s construction of the real numbers here. 6Cantor 1872, p. 93. 7 0 That is, for any rational number ε, there exists some n such that |an − an| < ε. CHAPTER 5. CANTOR 112

rational (for example, a sequence of the form {a1 = r, . . . , an = r, . . .} with a rational number r). With the above definition of the order relations, each limit b can be thought of as having some numerical magnitude [Zahlengröße].8 Cantor calls the aggregate [Gesamtheit] of such numerical magnitudes B. The arithmetic operations (that is, addition, subtraction, multiplication, and division) are defined for numerical magnitudes of A and B. Let b, b0, b00 be numerical magnitudes of B which are the limits of the sequences

0 00 0 00 0 {an}, {an}, {an} respectively. The arithmetic operations b ± b = b , bb = b00, b/b0 = b00 are deﬁned as follows.

0 00 0 00 9 b ± b = b : lim(an ± an − an) = 0,

0 00 0 00 bb = b : lim(anan − an) = 0,

b 00 an 00 0 = b : lim( 0 − an) = 0. b an As in the order relations, we can perform the arithmetic operations to elements from A and B. After expanding the arithmetic operation to the domain B, Cantor takes a sequence each term of which is from B and calls its limit c. The

8This usage of the term “Zahlengröße”may be from Weierstrass. See, for example, a lecture note of Weierstrass compiled by Adolf Hurwitz (in Dugac 1973, pp. 96–118). There, Weierstrass seems to reserve the use of the term “numbers” [Zahlen] almost exclusively for the natural numbers and uses “numerical magnitudes” for designating the rational and real numbers. As for Cantor, while avoiding the term “numbers” for designating the real numbers, he uses the expression “rational numbers” [die rationalen Zahlen]. See footnote 1 above. 9As to this use of the sign “lim”, Cantors says: “I need not explain the meaning of the lim sign in detail since its meaning should be clear from the above description” (Cantor 1872, p. 94; my translation: “ich auf die Bedeutung des lim-Zeichens nach dem Vorhergehenden nicht nähereinzugehen brauche”). CHAPTER 5. CANTOR 113 aggregate of such limits c is called C. For such limits c, we can define the order relations and the arithmetic operations between them; thus, C has the order relations and the arithmetic operations exactly as in A and B. The above process of creating a new domain from old domains can be repeated indefinitely; we have the new domain D from A, B, C and E from A, B, C, D, and so on. Cantor calls a stage at which a domain is created a kind [Art]; for example, when we got the domain L (namely the twelfth domain), we say that it belongs to the λ-th kind.10 The domain C constructed from B of course contains all the limits of A and B. Furthermore, the limits in C are exactly the same as those in B; namely, B is complete in the sense that we cannot expand B as we expanded A (there are limits in B which are not in A). Thus, Cantor’s construction gives us the complete totally-ordered domain in which we can perform the arithmetic operation, just as Dedekind’s does.11 While we may be able to identify the domains B and C in terms of the limits contained in them, Cantor cautions, there is a conceptual difference between B and C. The paragraph in which Cantor makes that caution is worth citing in full.

Although the domains B and C coincide with each other in some sense, it is essential in the theory expressed here (in which the

10For some reasons, in the 1872 paper Cantor does not use a numerical value for designating which stage a domain is in. In his 1883 Grundlagen Cantor starts using the ordinal numbers for that purpose. He also replaces the terminology “kind” with “order” [Ordnung] in his Grundlagen. See Cantor 1883a, p. 187; Ewald 1996, pp. 900–901. 11In this sense, Cantor’s and Dedekind’s constructions can be considered the same. CHAPTER 5. CANTOR 114

numerical magnitude, which has initially and generally no objectivity in itself, appears only as constituent part of propositions which have objectivity, for example, of the proposition that the sequence in question has the numerical magnitude as limit) to note the conceptual diﬀerence between the domains B and C. In fact, the identiﬁcation between two numerical magnitudes b, b0 of B does not entail their identity; it only expresses a certain relation between the sequences to which the numerical magnitudes correspond.12

Unfortunately, it is not clear in this 1872 paper why the conceptual diﬀerence between B and C matters. Of course, as Cantor explains in the above, there is a conceptual diﬀerence even between any two limits b, b0 of B which have the same numerical magnitude but not the same generating sequence. Furthermore, even if b of B and c of C have the same numerical magnitude, c may refer to a limit which is comprised of limits from B.13

12Cantor 1872, p. 95; my translation: “Obgleich hierdurch die Gebiete B und C sich gewissermaßen gegenseitig decken, ist es bei der hier dargelegten Theorie (in welcher die Zahlengröße,zunächst an sich im allgemeinen gegenstandslos, nur als Bestandteil von Sätzen erscheint, welchen Gegenständlichkeit zukommt, des Satzes z. B., daß die entsprechende Reihe die Zahlengrößezur Grenze hat) wesentlich, an dem begrifflichen Unterschiede der beiden Gebiete B und C festzuhalten, indem ja schon die Gleichsetzung zweier Zahlengrößen b, b0 aus B ihre Identitätnicht einschließt, sondern nur eine bestimmte Relation ausdrückt, welche zwischen den Reihen stattfindet, auf welche sie sich beziehen”. 13For this matter, what Cantor writes in his 1883 Grundlagen may be elucidating. There, he writes: “I therefore use the expression: the numerical quantity b is given by a fundamental sequence of the nth (respectively αth) order. If one agrees to this, one thereby acquires an extraordinarily flexible and at the same time intelligible idiom for describing in the simplest and most significant way the richness of the protean and often complicated webs of analysis. One also gains clarity and lucidity, which, in my opinion, is CHAPTER 5. CANTOR 115

Even though it is not clear why we should care about the conceptual difference between two domains of different kinds with the very same aggregate of limits, it is important to note that there is a difference between those domains of different kinds, and even between two limits from the same domain but with different generating sequences. This is especially so for the latter kind of differences because we somehow need to suppress such a difference in order to have a number system in which there is one and only one number for each limit; we need to think of the set of the real numbers as the equivalence class of sequences.14 Lastly, Cantor says that a numerical magnitude does not have objectivity [Gegenständigkeit] in itself at first. This way of saying (“initially” not to be undervalued. I herewith oppose the reservation which Dedekind, in the preface to his article ‘Continuity and irrational numbers’, expressed concerning these distinctions. It was not my intention to introduce new numbers by these fundamental sequences of the second, third order, etc., which could not already be represented by the fundamental sequences of the first order. Rather, I had in view only a conceptually different form of presentation; this appears clearly in various places in my work” (Cantor 1883a, p. 188; italics in the original and English translation from Ewald 1996, p. 901). In this passage, on the one hand, Cantor seems to stress the importance of the conceptual difference between the domains of different orders (like B and C); however, on the other hand, he also seems to say that the difference is only a matter of presentation (thus, it is not so important). It seems to me that the importance of this conceptual difference is not in the theory of the real numbers itself; rather, the difference may be important as a driving force for Cantor to build his theory of transfinite numbers. The following passage from the 1872 paper seems to indicate that Cantor had some embryonic form of his theory of transfinite numbers by 1872: “the number-concept, if it is to be developed as in here, would bring the germ of of an extension [of the number-concept] necessary and absolutely never-ending in itself” (Cantor 1872, p. 95; my translation: “der Zahlenbegriff, soweit er hier entwickelt ist, den Keim zu einer in sich notwendigen und absolut unendlichen Erweiterung in sich trägt”). 14However, this conception of the set of the real numbers, namely thinking of the set as the equivalence class, is foreign to Cantor; thus, attributing this way of the construction (taking the equivalent class of sequences) to Cantor is historically wrong. Maybe the first person who thought of the set of the real numbers as the equivalent class is Bolzano. See Bentley, Herrlich, and Huˇsek 1998, p. 580. CHAPTER 5. CANTOR 116

[zun¨achst]) implies that a numerical magnitude can eventually have objectivity in some relation. When and how a numerical magnitude gets objectivity will be made clear in the second section of the 1872 paper.

5.2 The Numerical Magnitudes and the Straight

Line

In the second section of the 1872 paper, Cantor starts taking about the relation between the numerical magnitudes and the straight line. First, in order to think about the relation between the numerical magnitudes and the straight line, one needs to ﬁx one point as a basic reference point and a unit [Maßeinheit] for measuring the position of a point; as a basic reference point, we take 0 on the line, and as a unit, we take the set of rational numbers (namely, we measure the magnitude of a line segment by a rational number). If a point is on the left side of 0, we add the negative sign − in front of a numerical magnitude which designate the point’s distance from 0; if a point is on the right-hand side of 0, we add the positive sign + (which is usually omitted) in front of a numerical magnitude. If the distance of a point can be represented by a rational number, no complications arise; for example, if a point is on the left-hand side of 0 and its distance from 0 is 1/2, we simply say that the point is at −1/2 of the line. However, how can we designate a point whose distance cannot be represented by any rational numbers? CHAPTER 5. CANTOR 117

First, note that the position of a point whose distance from 0 is a rational number can be approximated by a series of rational numbers. Let us suppose that the following series approximates a point P .

a1, a2, . . . , an,...

As n increases, so does the precision of the representation. It is clear that the above series is nothing but a sequence investigated in the previous section; we can represent the position of a point whose distance is not of any rational numbers by such a sequence, and consequently, by a limit b which denotes the sequence. Now the mapping from a point to a numerical magnitude is established. How about the mapping from a numerical magnitude to a point? Is there any rigorous way to establish such a mapping? Cantor writes as follows regarding this matter.

In order to make complete the relation explained in this § of the domains of the numerical magnitudes deﬁned in § 1 with the geometry of the straight line, it would suﬃce to add an axiom which can be stated simply as follows: conversely as well, to each numerical magnitude belongs a certain point of the line, whose coordinate is equal to that numerical magnitude in the same sense as explained in this §. I call this proposition an axiom because it is in its nature to CHAPTER 5. CANTOR 118

be not provable generally.15

And then Cantor adds the following remark.

By this [axiom] a certain objectivity is acquired for the numerical magnitudes; however, they are completely independent of this [geometrical objectivity].16

At this point, the numerical magnitudes, which are initially said to have no objectivity in itself, gain some objectivity.

5.3 The Second Construction: Limit-Points

and Derived Sets

After stating the above axiom, Cantor presents an important concept: a derived point-set [abgeleitete Punktmenge]. First, he deﬁnes the concept of

15Cantor 1872, p. 97; italics in the original and my translation: “Um aber den in diesem § dargelegten Zusammenhang der Gebiete der in § 1 definierten Zahlengrößenmit der Geometrie der geraden Linie vollständigzu machen, ist nur noch ein Axiom hinzuzufügen, welches einfach darin besteht, daß auch umgekehrt zu jeder Zahlengrößeein bestimmter Punkt der Geraden gehört,dessen Koordinate gleich ist jener Zahlengröße,und zwar in dem Sinne gleich, wie solches in diesem § erklärtwird./Ich nenne diesen Satz ein Axiom, weil es in seiner Natur liegt, nicht allgemein beweisbar zu sein”. This axiom is now called the Cantor-Dedekind axiom perhaps because of Dedekind’s following statement: “the axiom given in Section II. of that paper, aside from the form of presentation, agrees with what I designate in Section III. as the essence of continuity” (Dedekind 1872, p. 317; English translation from Dedekind 1901, p. 3). However, as Dauben notes, it is doubtful that Dedekind’s essence of continuity really states the one-to-one correspondence between the points in the line and the real numbers. See Dauben 1979, p. 320, note 34. 16Cantor 1932, p. 97; my translation: “Durch ihn wird denn auch nachträglich für die Zahlengrößeneine gewisse Gegenständlichkeit gewonnen, von welcher sie jedoch ganz unabhängigsind”. CHAPTER 5. CANTOR 119 a limit-point of a point-set P [Grenzpunkt einer Punktmenge P ]. A limit- point of a point-set P is a point in whose neighborhood (which is defined as any interval in which the point in consideration is contained) there are infinitely many points of P . By the Bolzano-Weierstrass theorem,17 it is easy to see that there is always at least one limit-point in a point-set with infinitely many points. It should also be noted that a limit-point of P may not belong to P (for example, 0 is a limit-point of the open interval (0, 1) but does not belong to the interval itself). The first derived-set of a point-set P is defined as the set of the limit-points of P and denoted as P 0. Now, the set of all the rational points of the open interval (0, 1) clearly contain infinitely many points and therefore there is at least one limit-point; and in fact, the first derived set of the limit-points of the point- set of all the rational points in (0, 1) turns out to be the point-set of all the points in the interval since the neighborhood of any point (which does not have to be a rational point) contains infinitely many points because of the density of the rational points. As in the process of creating a domain from limits, this process of creating a new derived set from a point-set can be repeated. For example, the above process of creating a new derived set from the point-set of (0, 1) can be repeated indefinitely and we have P 00, P 000 and so on. However, while the process of creating a new domain from limits can be always repeated indefinitely, the process of creating a derived set sometimes stops. For ex-

17The theorem shows that any bounded point-sequence has a convergent subsequence. CHAPTER 5. CANTOR 120 ample, let us think of the following point-set: {1, 1/2, 1/3,..., 1/n, . . .}. The derived set of this point-set contains only one point: 0. And because there is just one point in this derived set, there cannot be any limit-point of it and consequently there cannot be any derived set of it either. In short, if a point-set contains only a ﬁnite number of points, there is no derived set of it.

5.4 Perfect and Connected Sets

Cantor presents yet another characterization of the real numbers/continuumin his 1883 paper;18 this time, Cantor characterizes the continuum (more precisely, a point-continuum) as a perfect and connected set of points. For defining the concept of a perfect set, let us recall the definitions of a limit-point and of a derived set. A limit-point of a point-set P is a point whose neighborhood contains infinitely many points of P ; and a derived set of P , denoted as P 0, is the set of all limit-points of P . A point-set P is called perfect if it coincides with its derived set (namely, if P = P 0).19 After defining the concept of a perfect set, Cantor remarks that a

18This is the fifth of a series of papers entitled “Uber¨ unendliche lineare Punktman- nigfaltigkeiten” (“On infinite linear point-manifolds”) and was published separately as Grundlagen einer allgemeinen Mannigfaltigkeitslehre (Foundations of a General Theory of Manifolds) usually abbreviated as Grundlagen. 19Cantor 1883a, p. 193. In Cantor 1883b (p. 247), this equality between P and P 0 is expressed by two-ways inclusion relations between them; namely, a point-set P is perfect if and only if every point of P is a limit-point of P and every limit-point of P is a point of P . In Beiträge (Cantor 1895, p. 309; Cantor 1915, p. 132), Cantor calls the former condition dense-in-itself [insichdicht] and the latter closed [abgeschlossen]. CHAPTER 5. CANTOR 121 perfect point-set is not always “everywhere-dense” [überalldicht] and therefore being perfect is not sufficient for a set to be continuous.20 The concept of “everywhere-dense” is defined in Cantor 1879.

If P is partially or entirely in the interval (α . . . β), there can be the remarkable case such that every interval (γ . . . δ) contained in (α . . . β), no matter how small it is, contains points of P .21

In order for a perfect set to be everywhere-dense, it has to meet another condition: connectedness. Cantor deﬁnes a connected set as follows.

We call T a connected point-set if, for any two of its points t and t0 and for any arbitrarily small number ε there always exists a

ﬁnite number [Anzahl] of points t1, t2, . . . , tν of T , such that the

0 22 distances tt1, t1t2, t2t3,..., tνt are all less than ε.

And then, Cantor enthusiastically declares:

20A classic example of a perfect set which is not everywhere-dense would be the Cantor set. The Cantor set is constructed as follows. Let A0 be the closed interval [0, 1] of R. A1 is deﬁned as A0 without its “middle” part; namely, A1 is the union of the closed intervals 1 2 [0, 3 ] and [ 3 , 1]. In a similar fashion, A2 is deﬁned as the union of A1 without its middle 1 2 1 2 7 8 parts; namely, A2 is [0, 9 ] ∪ [ 9 , 3 ] ∪ [ 3 , 9 ] ∪ [ 9 , 1]. The general form of An is

∞ [ 1 + 3k 2 + 3k A = A − ( , ). n n−1 3n 3n k=0

∞ T And the Cantor set is deﬁned as An. The Cantor set is perfect but nowhere dense. n=1 21Cantor 1879, p. 140; italics in the original and my translation: “Liegt P teilweise oder ganz im Intervalle (α . . . β), so kann der bemerkenswerte Fall eintreten, daß jedes noch so kleine in (α . . . β) enthaltene Intervall (γ . . . δ) Punkte von P enth¨alt”. 22Cantor 1883a, p. 194; Ewald 1996, p. 906; italics in the original. CHAPTER 5. CANTOR 122

Now, all the geometric point-continua known to us fall under this concept of connected point-set, as it is easy to see; I be- lieve that in these two predicates ‘perfect’ and ‘connected’ I have discovered the necessary and suﬃcient properties of a point-

continuum. I therefore deﬁne a point-continuum inside Gn as 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.23

Summary

In 1872 when Dedekind published his monograph in which he gave the definition of the real numbers, Cantor also published a paper in which he presented another way of defining the real numbers. Cantor’s construction of the real numbers is drastically different from that of Dedekind; whereas Dedekind appealed to the method of cuts in defining the real numbers, Cantor appealed to the concept of Cauchy sequences. In his 1872 paper, Cantor also proposed an important axiom: there is a one-to-one correspondence between the real numbers and points on a line. Based on this axiom, Cantor presented another construction in his 1872 paper which depends on the concepts of limit points and derived sets. This construction is considered to be one of the sources 23Cantor 1883a, p. 194; Ewald 1996, p. 906; italics in the original. CHAPTER 5. CANTOR 123 of modern topology. In 1883, Cantor offered yet another construction of the real numbers which is based on the concepts of perfect and connected sets. Although this construction is not very popular in today’s mathematics, it had a huge impact on Peirce who will be the main figure in the next chapter. Chapter 6

Peirce

As a philosopher, Peirce placed particular stress on the importance of continuity in his late philosophical system.1 And as a mathematician, Peirce tried to develop his own mathematical theory of continuity.2 In this chapter, I will describe and examine Peirce’s mathematical theory (or, more precisely,

1In Peirce’s late philosophical system, two subsystems play an important role. One is called phenomenology whose business is to investigate what is present to our experience. Peirce calls what is present to our experience phenomena and classiﬁes such phenomena into three categories. Of these categories, perhaps the most important one is thirdness which roughly represents the cognitive aspects—representation and thought—of our experience. And Peirce says, concerning thirdness, “Continuity represents Thirdness almost to perfection” (Peirce 1958-1960, Volume I, p. 170, paragraph 337). The other important subsystem of Peirce’s late philosophical system is called evolutionary cosmology whose main task is to clarify the structure of the universe. Peirce thinks that the universe is operating according to three principles: tychism (the doctrine of chance), agapism (the doctrine of love), and synechism (the doctrine of continuity). And of synechism, Peirce says, “[Synechism is] that tendency of philosophical thought which insists upon the idea of continuity as of prime importance in philosophy and, in particular, upon the necessity of hypotheses involving true continuity” (CP 6.169). 2The interrelationship between Peirce’s philosophical and mathematical conceptions of continuity is a very interesting topic but to investigate it thoroughly and is simply beyond the scope of this chapter.

124 CHAPTER 6. PEIRCE 125 theories) of continuity. As is often the case with Peirce, his mathematical theory of continuity went thorough several significant changes. I will divide the development of Peirce’s mathematical theory of continuity into three stages: Aristotelian (until 1884), Cantorian (1884–1895), and Peircean proper (after 1895).3 In Section 1, the early stage of Peirce’s development in the theory of continuity will be discussed. At this early stage, Peirce did not develop a theory of continuity per se. He just touched upon the problems of the continuum in discussing other topics. First, as a theorist of continuity, Peirce started as an Aristotelian according to the position of which a continuum has parts which are themselves continua. Then, he changed his position a bit although he still remained Aristotelian after the change. Namely, Peirce claimed that continuity can be characterized by infinite divisibility. Clearly, being infinitely divisible is not enough for something to be considered as continuous because the set of the rational numbers has this property as well. In this sense, Peirce did not capture continuity well at this stage. Section 2 will deal with an im-

3There are, of course, other periodizations. For example, Potter and P. B. Shields (1977), which is perhaps the most influential paper on this topic (JérômeHavenel says that “[t]his article is still considered by most Peirce scholars as the reference on this topic” in Havenel 2008), divide the development of Peirce’s mathematical theory of continuity into four stages: Pre-Cantorian (until 1884), Cantorian (1884–1894), Kantistic (1895– 1908), and Post-Cantorian (1908–1911). On the other hand, in Havenel (2008) (Matthew E. Moore describes this paper as a “finer-grained revision” of Potter and P. B. Shields 1977 in Moore 2015), Havenel divides it into five stages: Anti-Nominalistic (1868–1884), Cantorian (1884–1892), Infinitesimal (1892–1897), Supermultitudinous (1897–1907), and Topological (1908–1913). I divide the development of Peirce’s theory of the continuum into three stages described above because it seems to me that the development of Peirce’s theory of the continuum can be well portrayed as the approach to and the parting from the Cantorian conception of the continuum. CHAPTER 6. PEIRCE 126 portant period for Peirce in the development of his own theory of continuity. In this period, a decisive event occurred: Peirce read Cantor. This experience without a doubt made a huge impact on Peirce and from then on Peirce tried to develop his own theory of continuity with Cantor’s works as a reference point. In Section 3, the period in which Peirce parted ways with Cantor in some crucial aspects will be discussed. In this period, Peirce abandoned the view that a continuum is composed of points. Because of this significant change in his thought, Peirce’s conception of the continuum started deviating hugely from the ordinary—namely, Dedekindian/Cantorian—conception in mathematics.

6.1 Aristotelian Period (Until 1884)

In a sense, Peirce had been Aristotelian throughout his career as to his conception of continuity. For example, in 1868, Peirce writes that “a continuum is precisely that, every part of which has parts, in the same sense”.4 Here, Peirce clearly characterizes a continuum by the mirror property which Aris- totle thinks of as the main characteristic of a continuum.5 As a consequence, Peirce thinks that a continuum cannot be composed of indivisibles.6

4“Grounds of Validity of the Laws of Logic: Further Consequences of Four Incapacities” (1868) in Peirce 1958-1960, Volume V, p. 205, paragraph 335. 5See Section 1 of Chapter 1 of this dissertation. 6As to this point, the following remark on a logical atom would be illuminating: “A logical atom, like a point in space, would involve for its precise determination an endless process. . . . The absolute indivisible [like a logical atom or a point] can not only not be realized in sense or thought, but cannot exist, properly speaking” (“Description of a Notation for the Logic of Relatives, Resulting from an Ampliﬁcation of the Conceptions CHAPTER 6. PEIRCE 127

In 1878, Peirce changes his view of continuity slightly; namely, he starts claiming that continuity can be characterized by infinite divisibility. He writes that “[continuity is] the passage from one form to another by insensible degrees”.7 Although it is hard to see why this characterization expresses infinite divisibility, it can be interpreted so according to a note written in 1893. The 1893 note says that what Peirce means by the expression is an idea of “limitless intermediation; i.e., of a series between every two members of which there is another member of it”.8 This characterization is nothing but that of infinite divisibility. Infinite divisibility does neither characterize a continuum sufficiently nor imply that a continuum cannot be composed of indivisibles because, even though the set of rational numbers is infinitely divisible, it is clearly discontinuous and made of indivisibles. Still, Peirce keeps characterizing continuity in this way at least until 1881. In “On the Logic of Numbers” published in 1881, Peirce defines continuity as follows.

[A] continuous system is one in which every quantity greater than another is also greater than some intermediate quantity greater than that other.9

In sum, at this early stage of his development, Peirce does not seem of Boole’s Calculus of Logic” (1870) in Peirce 1958-1960, Volume III, p. 58, paragraph 93; my italics and insertion). 7Peirce 1958-1960, Volume II, p. 391, paragraph 646; my insertion. 8Peirce 1958-1960, Volume II, p. 391, footnote 1. 9Peirce 1958-1960, Volume III, p. 159, paragraph 256. CHAPTER 6. PEIRCE 128 to have a solid grasp of how he should deﬁne continuity even though he realizes the need for a thorough study on the topic.10

6.2 Cantorian Period (1884–1895)

In 1884, a crucial event for Peirce’s development of the theory of continuity occurred: the encounter with Cantor.11 However, it seems to have taken a little while for Peirce to absorb Cantor’s ideas completely. It is probably in 1889 that for the ﬁrst time Peirce explicitly mentions Cantor in discussing continuity. In an article written for the Century Dictionary, Peirce writes:

[Continuous means] in mathematics and philosophy a connection of points (or other elements) as intimate as that of the instants or points of an interval of time: thus, the continuity of space consists in this, that a point can move from any one position to any other so that at each instant it shall have a definite and distinct position in space. This statement is not, however, a proper 10Even though Peirce’s definition of continuity is insufficient there, the 1881 paper cited above includes two very important ideas for developing the theory of continuity: the idea of cardinality and the criteria of distinguishing infinite from finite. In particular, the Peircean criteria of distinguishing infinite from finite is essentially the same as Dedekind’s one which was proposed in Was sind und was sollen die Zahlen? published in 1888 (Dedekind 1932, pp. 335–391; Dedekind 1901, pp. 29–115). Moreover, both of Peirce’s and Dedekind’s works aimed at the same objective (the axiomatization of the natural number system) and proven to be equivalent to each other. See P. Shields (2012) as to their equivalence. 11In a letter to Philip E. B. Jourdain, Peirce testifies that “I say I cannot well have made the least acquaintance with Cantor’s work until the winter of 1883–4 or later” (Peirce 1976, Volume III, p. 883). Here, Cantor’s works which Peirce mentions are those published in Acta Mathematica in 1883–1884. Considering that the 1883 issue of Acta was published in December of 1883, 1884 would be the most probable year when Peirce read Cantor’s works for the first time. CHAPTER 6. PEIRCE 129

definition of continuity, but only an exemplification drawn from time. The old definitions — the fact that adjacent parts have their limits in common (Aristotle), infinite divisibility (Kant), the fact that between any two points there is a third (which is true of the system of rational numbers) — are inadequate. The less unsatisfactory definition is that of G. Cantor, that continuity is the perfect concatenation of a system of points — words which must be understood in special senses.12

Before examining how well Peirce understand Cantor’s definition of continuity, a few remarks on the following points are called for. First, in the above article, Peirce dismisses his own former characterization of continuity — continuity as infinitely divisible — as inadequate and he relates this characterization to Kant.13 Second, Peirce brings up a new characterization that “adjacent parts [in a continuum] have their limits in common”14 although he swiftly dismisses it as inadequate. At any rate, it is clear that in 1889 Peirce thinks that his former characterization of continuity is insufficient. Now let us look at how Peirce explains Cantor’s definition of continuity. Peirce starts his explanation with the concept of concatenation. He writes:

Cantor calls a system of points concatenated when any two of 12The Century Dictionary article “Continuity” (1889) in Peirce 1958-1960, Volume VI, p. 114, paragraph 164; italics and insertion in the original. 13Later I will explain what this property — inﬁnite divisibility – has to do with Kant. 14In Chapter 1, I expressed this characteristic as sharing an outer extremity between two parts. CHAPTER 6. PEIRCE 130

them being given, and also any finite distance, however small, it is always possible to find a finite number of other points of the system through which by successive steps, each less than the given distance, it would be possible to proceed from one of the given points to the other.15

As was explained in Chapter 5, a set S is called connected if, for any two points t, t0 of S and for any non-zero positive number ε, there are a ﬁnite number of points t1, t2, . . . , tn of S such that all the distances

|tt1|, |t1t2|,..., |tnt| are less than ε. Following this deﬁnition of concatenated, Peirce deﬁnes the concept of perfect as follows.

[Cantor] terms a system of points perfect when, whatever point belonging to the system be given, it is not possible to find a finite distance so small that there are not an infinite number of points of the system within that distance of the given point.16

In the above definition, Peirce defines a perfect set as a set whose points are all limit-points. Although Cantor himself defines a perfect set in terms of derived sets,17 considering that a derived set is in turn defined in terms of limit-points, Peirce’s definition seems to be correct at first sight. However, as a definition of a perfect set, the above one is insufficient. As

15Peirce 1958-1960, Volume VI, p. 114, paragraph 164; italics in the original. 16Peirce 1958-1960, Volume VI, p. 114, paragraph 164; italics in the original. 17For derived sets, see 118. CHAPTER 6. PEIRCE 131

Cantor clearly states in another paper published in the same volume of Acta Mathematica as the French translation of Grundlagen, in order for a set P to be perfect, it is required not only that every point of the set is a limit-point, but also that every limit-point of the set is in P .18 Although Peirce’s understanding of Cantor has some defects as was just seen above, it is certain that Peirce tries hard to absorb Cantor’s ideas and incorporate them into his own definition of continuity. On the other hand, Peirce is not completely satisfied with Cantor’s definition as is obvious from Peirce’s phrasing “less unsatisfactory”. The reason why Peirce hesi- tates to fully accept Cantor’s definition becomes clear in “The Law of Mind” (1892). In §5 of “The Law of Mind”, Peirce expresses his criticism of Can- tor’s definition of continuity. After presenting Cantor’s definition in a similar fashion as in the Century Dictionary,19 Peirce points out three defects of the definition as follows.

[Cantor’s definition of continuity] has some serious defects. In the first place, it turns upon metrical considerations; while the distinction between a continuous and a discontinuous series is manifestly non-metrical. In the next place, a perfect series is defined as one containing “every point” of a certain description.

18“[I]f a set P is perfect, not only each point of P has to be a limit-point of P , but also each limit-point of P has to be included in P ” (Cantor 1883b, p. 410; my translation). In Beitr¨age (Cantor 1895, p. 510; English translation is Cantor 1915, p. 132), Cantor calls the former condition insichdicht (“dense-in-itself”), the latter abgeschlossen (“closed”). 19This time Peirce deﬁnes a perfect set as being closed. CHAPTER 6. PEIRCE 132

But no positive idea is conveyed of what all the points are. [. . . ] Finally, Cantor’s deﬁnition does not convey a distinct notion of what the components of the conception of continuity are. It inge- niously wraps up its properties in two separate parcels, but does not display them to our intelligence.20

First, Peirce criticizes Cantor for appealing to metrical notions such as “connected” in defining continuity. For Peirce, continuity should be defined exclusively in qualitative terms. Second, the reference to “all the points” in Cantor’s definition is accused of being a definition by negation. Peirce argues that, if such a reference were to be accepted, it would be possible to define continuity just as “one which contains every point of the line between its extremities”.21 Lastly, Peirce blames Cantor for his definition’s redundancy.22 Putting aside the question of whether Peirce’s criticism is valid or not, it points to what Peirce wants his definition of continuity to be like. Then, what kind of definition does Peirce propose in “The Law of Mind”? In the paper, he resurrects the concepts once rejected as inadequate in his definition in the Century Dictionary — namely, Aristotle’s and Kant’s definitions, which Peirce now calls Aristotelicity and Kanticity respectively in “The Law of Mind” — and claims that, together, they provide a satisfactory

20Peirce 1958-1960, Volume VI, pp. 96–97, paragraph 121; my insertion and omission. 21Peirce 1958-1960, Volume VI, p. 97, paragraph 121. 22In fact, the property of being connected implies that of being dense-in-itself. In this sense, Cantor’s definition can be said to state the latter property twice. However, as was seen above, being dense-in-itself alone is not enough for warranting a set to be perfect. CHAPTER 6. PEIRCE 133 definition of continuity.23 Pace Peirce, it seems that all three points of his criticism of Cantor’s definition apply equally to Peirce’s new definition. First, Aristotelicity still depends on the metrical notion of distance. Second, both Aristotelicity and Kanticity utilize the “all the points” type of expression.24 Lastly, Peirce himself states that Aristotelicity implies Kanticity. He writes:

Our ideas will find expression more conveniently if, instead of points upon a line, we speak of real numbers. Every real number is, in one sense, the limit of a series, for it can be indefinitely approximated to. [. . . ] [T]he series referred to in the definition of Aristotelicity must be understood as including all series. [. . . ] Consequently, it is implied that between any two points an innu- merable series of points can be taken.25

Another important remark in “The Law of Mind” is that the concept of continuity presupposes inﬁnitesimals. Peirce’s argument goes as follows.26 As was hinted in the above quotation, each point of a line can be

23Peirce 1958-1960, Volume VI, pp. 97–98, paragraph 122. In that paragraph, Peirce says: “[. . . ] Aristotelicity of the series, together with Kant’s property, or its Kanticity, completes the definition of a continuous series” (p.98). Peirce refers to the following passage of Kritik der reinen Vernunft as his reason to call infinite divisibility Kanticity: “The property of quantities according to which no part of quantities is the smallest among them (i.e., no part is simple) is called continuity” (Kant 1956, p. 223 (A169/B212); my translation). Peirce later changes his interpretation of what Kant says about continuity. I will come back to this issue later in this chapter. 24In the case of Kanticity, all the points are referred in the guise of “any two points”. 25Peirce 1958-1960, Volume VI, p. 98, paragraph 124. 26Peirce 1958-1960, Volume VI, p. 98, paragraph 125. CHAPTER 6. PEIRCE 134 represented by some real number. And it is clear that there are points which can be represented only by irrational numbers. Irrational numbers are, when expressed in decimals, those which have an infinite series of decimals after the decimal point. Now Peirce points out that the term “infinitesimal” just means “infinitieth” in Latin. This suggests, Peirce continues, that there is an infinitesimal quantity in the sense that there is an infinitesimal difference at the infinitieth place of decimals between two infinitesimal numbers. Thus, Peirce concludes that “[t]here is nothing contradictory about the idea of such quantities”.27 Following the above argument, Peirce expresses his preference for the method of infinitesimals over that of limits by saying that “”[a]s a mathematician, I prefer the method of infinitesimals to that of limits, as far easier and less infested with snares”.28 However, Peirce does not develop any theory of continuity based on infinitesimals. Overall, Peirce’s conception of continuity in this period is mostly Cantorian; that is, Peirce understands continuity in terms of points. Despite his dissatisfaction with some aspects of Cantor’s definition and his preference for the method of infinitesimals, Peirce cannot find a way to resolve his dissatisfaction and incorporate infinitesimals into his own theory of continuity at this stage of his development.

27Peirce 1958-1960, Volume VI, p. 98, paragraph 125. 28Peirce 1958-1960, Volume VI, pp. 98–99, paragraph 125. CHAPTER 6. PEIRCE 135

6.3 Peircean Period (After 1895)

In a manuscript written around 1895, Peirce proposes the following idea with which he departs from the Cantorian conception of continuity: “The very word continuity implies that the instants of time or the points of a line are everywhere welded together”.29 Peirce develops this idea in his 1897 paper “Multitude and Number”. “Multitude and Number” opens with a series of definitions. First, the relation “be u’d by” is defined for making clear a part-whole relation.30 For example, “B is u’d by A” means that A is a unit of a collective whole B. Next, the term collection is defined as referring to anything u’d by something with a definite quality.31 For example, if it is assumed that a continuum is composed of points, it is said that a continuum is a collection of points. Then, the term multitude is defined as follows.

I shall use the word multitude to denote that character of a collection by virtue of which it is greater than some collections and less than others, provided the collection is discrete, that is, provided the constituent units of the collection are or may be distinct.32

After deﬁning multitudes as above, Peirce remarks that when the units which comprise a collection lose their individual identity, the term

29“On Quantity with Special Reference to Collectional and Mathematical Inﬁnity” (c. 1895) in Peirce 1976, p. 61; italics in the original. 30Peirce 1958-1960, Volume IV, p. 145, paragraph 170. 31Peirce 1958-1960, Volume IV, p. 145, paragraph 171. 32Peirce 1958-1960, Volume IV, p. 148, paragraph 175; italics in the original. CHAPTER 6. PEIRCE 136

“multitude” cannot be applied to the collection anymore.33 In such cases, another term, multiplicity is applied. Based on the concepts explained above, Peirce defines (or constructs) enumerable, denumerable, and postnumeral (or abnumeral) collections. These correspond to, in modern terms, finite, countably infinite, and uncountably infinite collections respectively. And then, later in the paper, Peirce poses a crucial question: Is there any multitude greater than all other multitudes?34 Peirce admits that, within the realm of multitudes, there is no greatest one because, as Peirce clearly states in the paper,35 2µ > µ for any multitude µ. However, he asserts that, when the units which comprises a collection lose their identity as individuals, the size of such a collection exceeds that of any multitude.36 Namely, there is a multiplicity greater than any other multitude. The paradigmatic example of a collection whose units lose their individual identity is a continuum. Peirce writes:

[In a continuum] the individual identities of the units are completely merged, so that not a single one of them can be identiﬁed, even approximately [. . . ].37

And according to Peirce’s qualiﬁcation, the size of a continuum exceeds any

33Peirce 1958-1960, Volume IV, p. 148, paragraph 175. 34Peirce 1958-1960, Volume IV, p. 183, paragraph 218. 35Peirce 1958-1960, Volume IV, p. 172, paragraph 204. 36Peirce seems to be using the term “aggregate” instead of “collection” in such cases although he does not explicitly define the term as such. 37Peirce 1958-1960, Volume IV, p. 184, paragraph 219; my insertion and omission. CHAPTER 6. PEIRCE 137 multitude. It is understandable that Peirce wants the size of a continuum to be greater than any multitude because, if there is no end in the series of sizes of collections, there are (infinitely many) sizes which exceed that of a continuum. Peirce cannot accept such a situation where the sizes of discrete collections are larger than that of a continuum. However, it is completely unclear why the size of a collection whose units lose their individual identity should exceed any multitude. Moreover, in the first place, it is also unclear how and when the units in a collection lose their individual identity. Peirce does not say much about these points in the paper.38 As to the first difficulty, it might be argued as follows. First, recall that Peirce thinks that there is a gap between any two points no matter how close they are to each other in a line which is made of points. Let us call such a line whose members are points discrete. For Peirce, the real line is a discrete line. Now, in order to make a discrete line (particularly, the real line) continuous, gaps in it should be filled. Peirce seems to think that a continuous line (or collection) is more multitudinous than any discrete line (or collection) because of such fillings.39

38In addition to the above conceptual difficulties, there seems another — this time, prac- tical — difficulty: Peirce seems to suggest that the ordinary arithmetic operations cannot be applied to continuous collections (Peirce 1958-1960, Volume IV, p. 187, paragraph 221). Because of this, some argue that the Peircean continuum has no place in mathematics. For example, Dauben writes: “[Peirce] was not concerned to develop the arithmetic properties of his ideas [. . . ]. He was interested in illuminating a deep philosophical problem of long standing, namely that of the continuum” (Dauben 1977, p. 131). The same view is also expressed by Buckley (2012, pp. 121–123). I will come back to this issue in the next chapter. 39Obviously, since λ + µ = max(λ, µ) for any infinite cardinals λ, µ, and since Peirce CHAPTER 6. PEIRCE 138

As to the second difficulty, Peirce proposes a drastic solution in the 1898 Cambridge Conferences lectures (Peirce 1992): a continuum is not a collection of points at all, at least in the usual sense of the word. And this solution sheds some light on the first difficulty. In the eighth of the 1898 Cambridge Conferences lectures which is entitled “The Logic of Continuity”,40 Peirce asks the question whether there is a size which is larger than any multitude. As was seen above, Peirce thinks that the size of a continuum is larger than that of any discrete collection and calls such a size multiplicity instead of multitude. This time, Peirce add another characterization to such a size which is larger than any multitude: It has to be that of a potential aggregate. Peirce says:

That which is possible is in so far general, and as general, it ceases to be individual. Hence, remembering that the word “potential” means indeterminate yet capable of determination in any special case, there may be a potential aggregate of all the possibilities that are consistent with certain general conditions; and this may be such that given any collection of distinct individuals what- soever, out of that potential aggregate there may be actualized a more multitudinous collection than the given collection. Thus the potential aggregate is with the strictest exactitude greater in wants the cardinality κ of any continuous collection to be more than those which can be obtainable by the repeated power set operations from 2ℵ0 , the cardinality of ﬁllings has to be a strong limit cardinal other than ℵ0. 40Peirce 1992, pp. 242–268. CHAPTER 6. PEIRCE 139

multitude than any possible multitude of individuals.41

As a potential aggregate, a continuum does not have any points as its constituents.42 Thus, it does not make sense to ask how points in a continuum lose their individual identity and constitute one uniﬁed object. First, there is a continuum, and then, potential points are actualized in some special cases (for example, when they are designated or a continuum is cut; in short, when a continuum is made discontinuous in some point(s)).43 Points obtain their individual identity by actualization. However, even when actualized, those points seem quite diﬀerent from ordinal ones. Let us think of the straight line depicted below.44

As was stated above, this line is a potential collection. As a potential collection, it has potential points which are “indeterminate yet capable of determination”. Those potential points can be actualized by making a continuum

41Peirce 1992, p. 247; italics in the original. It must be noted that in the above quotation Peirce uses the term “multitude” for a size which exceeds any multitude of discrete collections. One of the possible hypotheses would be that Peirce prepared a manuscript for this lecture before he wrote the 1897 “Multitude and Number”. However, if so, it would be strange that he did not mention the potentiality of collections in the 1897 paper. Another hypothesis would be that Peirce might have thought that there are (infinitely many) grades of the sizes of continuous collections. Although this is plausible, there is no textual evidence that Peirce actually thinks so. 42“[A potential aggregate] does not contain any individuals at all” (Peirce 1992, p. 247). 43Still, in another place of the lecture, Peirce talks as if there are points in the first place (“[Distinct individuals] become welded into one another”, Peirce 1992, p. 160). However, this way of talking should be taken as figurative. When Peirce asks where a continuum has come from, he obviously denies the answer that it has comes from points (Peirce 1992, p. 258). 44The actual example Peirce gives deals with a circle, not with a line but this does not make any difference as to essential points. CHAPTER 6. PEIRCE 140 discontinuous at places where points are to be actualized. For example, designating a point, say P , on the line makes it discontinuous as in the following figure.

A P

Note that there are also other discontinuities in the line: the start and end points. Let us assign the labels A and B to these extreme points respectively.

A P B

If the line is split into two at the point P , there appear the new extreme points. Let us call these new points C and D.

A C D B

Now ask this: Where do these new extreme points C and D come from? A straight-forward answer would be “from P ”. Actually Peirce thinks so.45 However, how is it possible for the point P , which is supposed to have no parts in it, to become two points? A straight-forward answer would be that a point has parts. And Peirce seems to think so.46 In the above example, the point P is split into two points C and

45“[A point] might burst into any discrete multitude of points whatever, and they would all have been one point before the explosion” (Peirce 1992, p. 160). 46“Points might ﬂy oﬀ, in multitude and order like all the real irrational quantities from 0 to 1” (Peirce 1992, p. 160). CHAPTER 6. PEIRCE 141

D. Peirce also argues about the reverse case where two points C and D are merged into the point P . In that reverse case, according to Peirce, the order relation between C and D is retained in the point P even after their merger: “[Points] might all have had that order of succession in the line and yet all have been at one point”.47 As Peirce himself admits, this seems implausible.48 However, once the assumption that a point has no parts is once abandoned, the explosion and merger of points becomes not so implausible. And Peirce seems to talk about such a point with some parts. Obviously, what Peirce has in mind would be an infinitesimal quantity although he is not explicit about this point in the lectures. Further questions arise concerning the above conception of continuity. First, even though Peirce talks about points (whether infinitesimal or not), they are potential in the first place. In a continuum, they lose their individual identity. Then, if so, does it make sense to use the term “collection” for designating something which does not have actual individuals as its constituents?49 Second, Peirce assumes that we can always find some kind of ordering in a collection (or aggregate) whose cardinality is far more than that of the power set of the real numbers. Is this a legitimate assumption? As to the first question, Peirce admits that the term “collection” is

47Peirce 1992, p. 160 48“Men will say this is self-contradictory” (Peirce 1992, p. 160). 49Peirce seems to try not to use the term “collection” for something which has individual parts only in their potential form. He uses in most cases the term “aggregate” instead. CHAPTER 6. PEIRCE 142 not suitable for those which do not have actual individuals in them. In 1900, he writes in a letter to the editor of Science as follows.

Since [. . . ] any multitude of points whatever are determinable on the line (not, of course, by us, but of their own nature), and since there is no maximum multitude, it follows that the points cannot be regarded as constituent parts of the line, existing on it by virtue of the line’s existence. For if they were so, they would form a collection; and there would be a multitude greater than that of the points determinable on a line. We must, therefore, conceive that there are only so many points on the line as have been marked, or otherwise determined, upon it. Those do form a collection; but ever a greater collection remains determinable upon the line. All the determinable points cannot form a collection, since, by the postulate, if they did, the multitude of that collection would not be less than another multitude.50

First, note that Peirce changes his conception of the size of a continuum here. In “Multitude and Number”, the size of a continuum is said to be greater than any multitude and called “multiplicity”. In the above quotation, the distinction of “multitude” and “multiplicity” seems to be abandoned and the size of a continuum is argued in terms of multitude. And the inference that there would be the maximum multitude (and then a contradiction would

50Peirce 1958-1960, Volume III, p. 363, paragraph 568; italics in the original and my omission. CHAPTER 6. PEIRCE 143 be derived) if all the potential points could be actualized is the very reason of the claim that a continuum cannot be a collection. The term “collection” is used exclusively for something made of (actualized) individuals. As to the second question, Peirce seems to give up investigating the order relation among those whose cardinality is more than that of the power set of the real number. In an undated manuscript which is thought to have been written around 1900,51 Peirce says:

When the scale of numbers, rational and irrational, is applied to a line, the numbers are insuﬃcient for exactitude; and it [is] in- trinsecally [sic] doubtful precisely where each number is placed.52

Certainly, we cannot be sure about the order relation of inﬁnitesimals if we cannot be sure about that of ordinal numbers. Another important change in Peirce’s conception of continuity is observed around 1900. In the letter to the editor of Science quoted above, Peirce writes:

Although Kant confuses continuity with inﬁnite divisibility, yet it is noticeable that he always deﬁnes a continuum as that of which every part (not every echter Theil [sic]) has itself parts.53

In “The Law of Mind”, Peirce defines Kanticity as infinite divisibility. This time, he attributes another definition of continuity to Kant. In a 51Havenel 2008, p. 108. 52“On Continuous Series and the Infinitesimal” (n.d.) in Peirce 1976, Volume III, p. 127; insertion in the original. 53Peirce 1958-1960, Volume III, p. 364, paragraph 589; italics in the original. CHAPTER 6. PEIRCE 144 marginal note written in his personal copy of the Century Dictionary, Peirce tries to make the way Kant defines continuity more precise.

[Peirce’s definition in Century Dictionary] involves a misunder- standing of Kant’s definition which he himself likewise fell into. Namely he defines a continuum as that all of whose parts have parts of the same kind.54

The passages of Kritik der reinen Vernunft which made Peirce correct his interpretation of what Kant says about continuity is worth citing.

Space and time are quanta continua [continuous quantities], since no part of them can be given without being enclosed between limits (such as points or instants) so that this part is again a space or a time. Space comprises only of spaces, time of times.55

What is said in the above is nothing but one of the Aristotelian definitions of continuity which Peirce states as the definition of continuity in 1868. It is a bit surprising to see that Peirce missed this part of Kant’s definition considering the above appears right after the passage which Peirce once took to express infinite divisibility. After re-realizing that this Aristotelian/Kantian aspect is indispensable in defining continuity, Peirce seems to have had kept it until his very

54Peirce 1958-1960, Volume VI, p. 115, paragraph 168; italics in the original, my insertion. This note was written in 1903. 55Kant 1956, pp. 223–224 (A169/B212); italics in the original, my insertion and translation. CHAPTER 6. PEIRCE 145 last stage of his attempt to deﬁne continuity.56 At this stage, he comes back where he started. As I said in the opening sentence of Section 1, it could be said that Peirce has been Aristotelian throughout his career in his attempt to deﬁne continuity.

6.4 Summary

In this chapter, I have been presenting and examining the development of Peirce’s theory of continuity. In Section 1, I described the early stage of the development which I called the pre-Cantorian Period up until 1884. Peirce started his attempt to define continuity with the very Aristotelian idea that a continuum has parts which are themselves continua. At some point during this early stage, Peirce changed his conception of continuity from one form of the Aristotelian definition to another, namely, to the one that continuity is characterized by its infinite divisibility. Although it is clear that continuity cannot be characterized by infinite divisibility alone, Peirce does not seem to have had a problem with characterizing continuity so. In sum, in this

56In 1906, Peirce writes as follows: “Whatever is continuous has material parts. I begin by defining these thus: The material parts of a thing or other object, W, that is composed of such parts, are whatever things are, firstly, each and every one of them, other than W ; secondly, are all of some one internal nature [. . . ]; thirdly, form together a collection of objects in which no one occurs twice over and, fourthly, are such that the Being of each of them together with the modes of connexion between all subcollections of them, constitute the being of W ” (“Continuity Redefined” in Peirce 1958-1960, Volume VI, pp. 118–119, paragraph 174; italics in the original, my omission). He repeats essentially the same definition in the 1908 “The Amazing Mazes: The First Curiosity”: “my notion of the essential character of a perfect continuum is the absolute generality with which two rules hold good, first, that every part has parts” (Peirce 1958-1960, Volume IV, p. 541, paragraph 642). CHAPTER 6. PEIRCE 146 early stage, Peirce’s conception of continuity had been mostly a na¨ıve and uncritical form of the Aristotelian conception. The na¨ıvetéin Peirce’s conception during his pre-Cantorian period drastically changed by the encounter with Cantor in 1884. In Section 2, I dealt with the second developmental stage of Peirce’s attempt to define continuity which ranged from 1884 to 1895. I called this period Cantorian because of the huge influence of Cantor upon Peirce’s thought of continuity. The first manifestation of that influence was in an article written for Century Dictionary which was published in 1889. In there, Peirce mostly followed the Cantorian way of presenting continuity. Namely, he presented continuity as a collection of limit-points. Although he was under the huge influence of Cantor, Peirce had not been satisfied with the Cantorian definition of continuity even at the early stage of his Cantorian period. In the 1892 paper “The Law of Mind”, Peirce started expressing his own ideas about continuity. First, in “The Law of Mind”, Peirce characterized continuity by what he called Aristotelicity and Kanticity. Aristotelicity means the property that parts next to each other share their limit, Kanticity simply infinite divisibility. Also in “The Law of Mind”, Peirce expressed his preference of the method of infinitesimals over that of limits although he did not develop any theory or characterization of continuity with infinitesimals in the paper. From around 1895, Peirce started deviating from the Cantorian conception of continuity in some important aspects. In Section 3, I examined the period after 1895 which I called the post-Cantorian. The first sign of the devi- CHAPTER 6. PEIRCE 147 ations from the Cantorian conception of continuity was seen in a manuscript written around 1895. There Peirce wrote that “[t]he very word continuity implies that the instants of time or the points in a line are everywhere welded together”. The meaning of this enigmatic statement was somewhat clarified in the 1897 paper “Multitude and Number”. In that paper, after developing the theory of size (in modern parlance, the theory of cardinals), Peirce claimed that the size of a continuous collection is larger than the size of any discrete one. Thus, according to Peirce, the size of a continuum is larger than 2µ for any cardinality µ. Peirce called such a size multiplicity instead of multitude which is used exclusively for designating the size of discrete collections. He further asserted that in a continuous collection its constituent elements lose their individual identity and are completely merged. Although Peirce did not explicitly explain why such losing and merging happens, it can be thought that it happens because of the extreme abundance of elements in a continuous collection. In the 1898 Cambridge Conferences lectures, Peirce finally abandoned one of the core these of the Cantorian definition of continuity, that is, the thesis that a continuum is composed of points. First, in one of the lectures, Peirce proposed the concept of potential aggregates. In such aggregates, individuals can only exist potentially. In other words, individuals in potential aggregates do not exist unless they are actualized. Understood in this way, the question as to how points can be welded together becomes meaningless. In another lecture, Peirce proposed a very surprising characteristic of points: A point can be split into two points and two points CHAPTER 6. PEIRCE 148 can be merged into one. Although Peirce did not make it clear, this characteristic can be thought of as that of infinitesimals. Towards the end of his career, Peirce seems to have returned to his starting point: the Aristotelian conception that the parts of a continuum are themselves continua. How- ever, unfortunately, Peirce did not have time to incorporate the Aristotelian conception into his drastic new ideas. Regrettably, I could not explore the interplay between Peirce’s philosophical and mathematical theories/conceptions of continuity but it is simply beyond the scope of this chapter. I might have been able to point out some similarities between Peirce’s philosophy and mathematics of continuity with relative ease.57 However, to point out the similarity is one thing, to determine the influence-relation is quite another. In the next chapter, I will re-interpret (or try to make sense of) Peirce’s theories of continuity with the help of modern mathematics.

57For example, there is a similarity between how the universe evolves in Peirce’s cosmology and how a continuum becomes definite. In the Peircean cosmology, the universe starts in a completely indeterminate state. And then, it evolves towards the complete determination. However, such complete determination can never be achieved because it means the disappearance of the universe. On the other hand, a continuum is an indeterminate (in other words, vague) existence in the sense that there are infinitely many potential points waiting to be actualized. A continuum can be made determinate by the actualization of its potential points (for example, by designating a point). However, just as in the case of the determination process of the universe, this process of actualization can never be completed because the complete actualization of potential points means nothing but the disappearance of a continuum. As just described, one can relatively easily find similarities between Peirce’s philosophy and mathematics of continuity. However, in order to determine its influence-relation, one needs rigorous textual examinations. Part III

Theories of the Non-Punctiform Continuum Chapter 7

Some Mathematical Interpretations of the Non-Punctiform Continuum

In this last chapter, I will sketch several mathematical theories which ar- guably capture the notion of a non-punctiform continuum. First, following Michael J. White’s paper “On Continuity: Aristotle versus Topology?” (1988), I will present the possibility of capturing the Aristotelian continuum in topological terms. Second, in the next section, I will present category theory as the most suitable mathematical theory in which the Peircean continuum can be formalized. Although these expositions are made for particular conceptions of the continuum (one is the Aristotelian, the other the Peircean), my aim

150 CHAPTER 7. NON-PUNCTIFORM CONTINUUM 151 here is not restricted to the presentation of the theories for those particular conceptions; rather, in this chapter, I would like to provide another way of looking at the continuum and think about the nature of the continuum with a fresh perspective.

7.1 The Aristotelian Continuum in Topology

As we have seen in Chapter 1, for Aristotle the continuum is an entity whose parts are connected at some common boundary; and consequently, Aristo- tle argues, the continuum cannot be composed of indivisibles like points. This Aristotelian view seems, on the surface, incompatible with the (main- stream) modern mathematical conception of the continuum according to which the continuum is composed of indivisibles. However, as Michael J. White argues,1 the basic Aristotelian conception of the continuum is not so far from the modern mathematical conception of it. We can find some similarities between the Aristotelian and the modern mathematical—especially topological—conceptions of the continuum. In this section, after defining some core concepts of topology which will be needed for capturing the concept of the continuum in topological terms, I will present White’s topological interpretation of the Aristotelian continuum. Let us start directly with the definition of a topological space.A

1White 1988. CHAPTER 7. NON-PUNCTIFORM CONTINUUM 152 topology is deﬁned as a collection T of subsets of a set X with the following properties.

1. The empty set ∅ and X are members of T .

2. The arbitrary union of members of T belongs to T .

3. The ﬁnite intersection of members of T belongs to T .

Members of T are called open sets and the pair (X, T ) are called a topological space.2 If there is no danger of confusion, a topological space (X, T ) is simply denoted as X. The union of all open sets in a subset A of a topological space X is called the interior of A. The dual concept of open sets is that of closed sets. A subset A of a topological space is called closed if the diﬀerence of X and A, X − A, is open. The intersection of all closed sets in a subset A of a topological space X is called the closure of A.3 One of the important concepts (perhaps the most important one)

2Intuitively, an open set can be thought of as an open interval like (a, b) in a line (or an open area in a plane). If we interpret an open set as such, it would be clear why we need the adjective “finite” in the above condition 3. Let us think of the intersection of the sets of open intervals (−1/n, 1/n) for n ∈ N; by taking the intersection, we have {0} as the intersection of the open intervals. As I will show later, {0} is not an open interval (actually, it is not an interval at all). This shows that the infinite intersection of open sets is not necessarily open. 3The concept of closed sets can be defined in terms of limit points. A point p of a topological space X is called a limit point of a subset A of X if a point p0 which is different from p is contained in every open sets containing p. A subset A of a topological space X is closed if and only if A contains all of its limit points. Note that limit points in modern topology is different from those in Cantor’s theory although they share the core idea. CHAPTER 7. NON-PUNCTIFORM CONTINUUM 153 in expressing continuity in topological terms is connectedness.4 A topological space X is said to be connected if there is no pair of open sets U and V in X such that U ∪ V = X and U ∩ V = ∅; namely, connectedness requires a topological space not to be separated disjointly.5 By the concept of connectedness, the two aspects which Aristotle regards as the necessary conditions for something to be continuous can be explained. First, if a topological space is connected, we can cut it in half in the way that the “parts” share a common boundary; namely, we can divide a topological space X into two subspaces U and V such that their union is X itself and their intersection is a common boundary (if we are thinking of a line for a topological space, this common boundary would be a point).6 Second, by connectedness together with the concept of a Hausdorff space, we can show that in a Hausdorff space, two points cannot be connected. A Hausdorff space is defined as a topological space X for any two distinct points x1, x2 of which there are disjoint open sets U, V such that x1 ∈ U

7 and x2 ∈ V . In such a space, any singleton {x} is closed. For a set to be connected to any other, it has to be open; thus, any singleton cannot be connected to any other set.8

4Another important concept in capturing continuity in topological terms is compact- ness. A subset A of a topological space X is called compact if A is in some (not necessarily finite) union of open sets, one can find a finite union of open sets of which A is a subset. 5It would be worth noting that, as we have seen in Chapter 5, Cantor also emphasizes the importance of connectedness in the definition of the real numbers. 6White 1988, pp. 3–4. 7This can be shown by the fact that X \{x} is open. 8Ibid, p. 7. CHAPTER 7. NON-PUNCTIFORM CONTINUUM 154

Furthermore, the Aristotelian conception of “parts” can be captured by the concept of regular closed sets. A regular closed set A is equal to the closure of the interior of A. Thus, a singleton cannot be a part of the continuum.9 Thusly, the Aristotelian continuum seems to be perfectly modeled in topology. However, White notices, there is a crucial difference between the Aristotelian continuum and the topologically-modeled continuum; that is, perhaps needless to say, the topological continuum can be thought of as the collection of indivisibles whereas the (pure) Aristotelian one is definitely not. However, I think, the roles played by indivisibles (especially, points) in modeling the continuum do not seem so substantial; the crucial topological concepts for modeling the continuum are those of open sets and connectedness the definitions of which do not even require the concept of points. It seems that we can model the Aristotelian continuum in topological terms even without mentioning points at all. And actually, there is such a topological language which will be the theme of Section 7.2.2.

9Ibid., p. 11. CHAPTER 7. NON-PUNCTIFORM CONTINUUM 155

7.2 The Peircean Continuum in Category The-

ory

As we have seen in the previous chapter, Peirce’s late conception of the continuum can be summarized as follows.

1. The continuum is not composed of points. (Non-punctiform)

2. The size of the continuum exceeds any size of the sets comprised of points. (Supermultitudinousness)

3. We cannot completely determine the continuum. (Indeterminateness)

Is there any mathematical theory which is faithful to the above characteristics? Category theory may provide us the most promising candidates for such a theory. In what follows, I will brieﬂy describe those theories.

7.2.1 The Construction of the Real Numbers in Cate-

gory Theory

7.2.1.1 Basic deﬁnitions

In set theory, the objects of study are completely determined by what elements a set contains. As such, set theory can be said to capture the mathematical structure very statically.10 On the other hand, in category theory, it

10In set theory, even a function, which is supposedly a dynamic concept, is captured by a set of ordered pairs. CHAPTER 7. NON-PUNCTIFORM CONTINUUM 156 is not enough to specify what objects we are dealing with; besides this static aspect, we need to specify its dynamic aspect; that is, we also need to specify what relation(s) objects have to each other. Thus, ﬁrst of all, we need the following deﬁnitions.

A category C is comprised of

1. a collection of objects, and

2. a collection of arrows between objects.

There are two operations which assign to each arrow f an object from which f comes and an object to which f goes; the ﬁrst operation is denoted as dom and called the domain operation and the second is denoted as cod and called the codomain operation. Thus, dom f denotes an object from which f comes and cod f denotes an object to which f goes. If dom f = a and cod f = b, we write f : a → b. For two arrows f, g, if cod f = dom g, we can deﬁne their composition and denote it as g ◦ f. And for any object a, there exists an arrow whose domain and codomain are a; we call such an arrow an identity arrow and denote it as 1a. Moreover, we assume that the following laws hold.

Associativity For three arrows f, g, h, if cod f = dom g and cod g = dom h, (h ◦ g) ◦ f = h ◦ (g ◦ f).

Identity Given an arrow f : a → b and two identity arrows 1a and 1b,

f ◦ 1a = 1b ◦ f. CHAPTER 7. NON-PUNCTIFORM CONTINUUM 157

With these deﬁnitions, we can deﬁne various operations (products, co-products, pullbacks, pushout, and exponentials).

7.2.1.2 Topos

In constructing the real numbers in the category-theoretical framework (and in fact, in other two theories which may capture some aspect of Peirce’s conception of the continuum), a special kind of categories plays a central role: a topos. A topos E is deﬁned as a category with the following characteristics.11

1. E is ﬁnitely complete.

2. E is ﬁnitely co-complete.

3. E has an exponential for any two objects in E .

4. E has a subobject classiﬁer.

For now, let us not bother about what (finitely) (co-)complete mean. Basically, what the conditions 1–3 tells us is we can express the basic set- theoretical operations (unions, intersections, set-differences etc) in E .12 A subobject classifier is a machinery which assigns a truth value to each result of an operation. It should be noted that in a topos the set of truth values is not necessarily comprised of true and false.

11The deﬁnition of a topos here is taken from Goldblatt 2006, p. 84. There are various ways to deﬁne a topos but they are all equivalent. 12In arithmetical terms, these conditions tell us that we can express basic operations (addition, subtraction, multiplication, division, and exponentiation) and have the units (denoted as 0 and 1) as to addition and multiplication. CHAPTER 7. NON-PUNCTIFORM CONTINUUM 158

We can think of the internal logic of a topos. As it turns out, the most natural candidate for such an internal logic of a topos is not classical.13 This fact will play an important role in what follows.

7.2.1.3 Construction of the Real Numbers in a Topos

The two well-known constructions of the real numbers, that is the Cantor and the Dedekind constructions, are formalized in set theory.14 Thus, we should be able to formalize those two constructions in a topos.15 As is well known, the Cantor and Dedekind constructions give us the same objects in the sense that the set of the real numbers constructed in the Cantor way (hereafter, I denote those real numbers as Rc) is isomorphic to that of those constructed in the Dedekind way (hereafter, I denote those real numbers as Rd). However, Rc and Rd constructed in a topos are generally not isomorphic. Moreover, Rd in a topos may not be complete in the sense that there may be some non-empty subsets of the reals which do not have the least upper bound. In other words, from the topos-theoretical point of view, a graph which is considered to be continuous in the classical setting may actually be “jumpy”. In order to make it complete, we somehow need to ﬁll these gaps. This seems to capture some aspect of supermultitudinousness

13More formally, the internal logic of a topos is expressed as a Heyting algebra.A Heyting algebras is a lattice (Ω, v) with the implication relation a ⇒ b for any a, b in Ω. I will explain what a lattice is in the next section. 14Of course, in order to express these constructions, we need the quantiﬁers ∀ and ∃. Luckily, we can express the quantiﬁer with the help of another important category- theoretic concept adjoints. 15In writing this section, I mainly referred to Goldblatt (2006, pp. 413–437). CHAPTER 7. NON-PUNCTIFORM CONTINUUM 159 of Peirce’s continuum.

7.2.2 Pointless Topology

In the previous section, we have seen that the constructions of the real numbers in a topos may capture the supermultitudinous aspect of Peirce’s continuum. However, in constructing the real numbers in a topos, we ﬁrst need the set of the natural numbers.16 This way of constructing the real numbers is hardly what Peirce would admit because the non-punctiform aspect of the continuum is perhaps the most important one in Peirce’s requirements for the continuum. Pointless topology gives us a way to think of the continuum in the non-punctiform way.17 Before stating what pointless topology is, we need to recall what a topology is. A topology is deﬁned as a collection T of subsets of a set X with the following properties.

1. The empty set ∅ and X are members of T .

2. The arbitrary union of T belongs to T .

3. The ﬁnite intersection of T belongs to T .

Members of T are called open sets and the pair (X, T ) are called a topological space.

16In the category-theoretical term, the natural number objects 17In writing this section, I referred to Johnstone (1983). CHAPTER 7. NON-PUNCTIFORM CONTINUUM 160

Now note that there exist inclusion relations among open sets such that

1. For any open set A, A ⊆ A (reﬂexivity);

2. For any open sets A, B, C, if A ⊆ B and B ⊆ C, then A ⊆ C (transitivity); and,

3. For any open sets A, B, if A ⊆ B and B ⊆ A, then A = B (antisym- metry).

If a set meets the above conditions, it is called a partial order; thus, the set of open sets of X is a partial order. If we want to stress the fact that a set is a partial order, we use the symbol v instead of ⊆. Also note that for any two open sets there exist their least upper bound and greatest lower bound. We call a partial order any two members of which have their least upper bound and greatest lower bound a lattice. Then, the set of open sets forms a lattice. In a lattice, the least upper bound of a and b is denoted as a ∨ b and called the join of a and b; similarly, the greatest lower bound of a and b is denoted as a ∧ b and called the meet of a and b. In a topology, if we think of the set of open set as a lattice, meets distribute over joins; namely, for any open set a and a join of (possibly infinitely many) open sets W B, a ∧ W B = W{a ∧ b : b ∈ B} CHAPTER 7. NON-PUNCTIFORM CONTINUUM 161 holds. In the way described above, a topological space in which we talk about the continuum is formalized without mentioning points at all. How- ever, we may want to assume the existence of points in a topological space especially when we want to talk about a topological space which is Hausdorff because, in order to characterize some topological space as Hausdorff, we need the concept of points. In pointless topology, we can extract a point from each open set residing a set X. However, there may be infinitely many open sets in a set; thus, in order to extract a point from each of those infinitely many open sets, we clearly need the axiom of choice. Unfortunately, our working space which is seen from the lattice- theoretical point of view is not equipped with the axiom of choice. The reason of this is that the lattice-theoretical version of a topological space is essentially the same as a Heyting algebra18, and as such, it lacks the law of the excluded middle which is implied by the axiom of choice (Diaconescu’s theorem).

7.2.3 Smooth Inﬁnitesimal Analysis

There is another candidate for a theory which does not presuppose the existence of points in talking about the continuum: Smooth inﬁnitesimal analy-

18A Heyting algebra is basically deﬁned as a lattice equipped with the implication relation ⇒ for any two elements a, b of the lattice; and the implication relation in a lattice is equivalent to the condition that meets distribute over joins. CHAPTER 7. NON-PUNCTIFORM CONTINUUM 162 sis.19 The basic idea of smooth inﬁnitesimal analysis (hereafter, I abbreviate it as SIA) is very simple. Instead of assuming the existence of points in developing the theory, SIA assumes the existence of lines whose length is so small that its square is equal to 0. Such lines are called microstraight lines. From this, we have the following principle.

Principle of Microaﬃneness Let ∆ be the set of microstraight lines. Then, for any function g : ∆ → R, there is a unique b ∈ R such that, for all ε ∈ ∆, g(ε) = g(0) + bε.20

From the microstraightness, we can also deduce that all functions on R are continuous. One of the signiﬁcant consequences of the principle of microaﬃne- ness is that the law of excluded middle does not always hold. To show this, we need the following principle.

Principle of Microcancellation If εa = εb for all ε ∈ ∆, then a = b.

To see that the law of excluded middle sometimes fails in smooth inﬁnitesimal analysis, ﬁrst, note that if x 6= 0, then x2 6= 0; thus, taking its

19In writing this section, I referred to Bell (2008). 20This follows from the microstraightness of ε (for all ε ∈ ∆, ε2 = 0) and Taylor’s 2 n theorem (every function f can be represented as a0 + a1x + a2x + ... + anx ). CHAPTER 7. NON-PUNCTIFORM CONTINUUM 163 contrapositive, if x2 = 0, then x = 0. Now recall that ε2 = 0 for all ε ∈ ∆. Therefore, it has to be the case not ε 6= 0. This means that, if the law of excluded middle were the case, ε = 0. Furthermore, if ε = 0, ε · 1 = ε · 0. However, if ε · 1 = ε · 0, by the principle of microcancellation, we have 1 = 0 which is of course a sheer contradiction. Thus, for ε ∈ ∆, the law of excluded middle does not hold. What is the implication of this? First of all, as we have seen above, we cannot decide whether a microstraight line is 0 or not; and the set of the real numbers in SIA contains microquantities which are the lengths of microstraight lines. In short, R in SIA is indeterminate which is one of the characteristics of Peirce’s continuum.

Concluding Remarks

In the above, I briefly presented how some of the non-punctiform views of the continuum can be captured within modern mathematics. The Aristotelian continuum can be nicely captured in topological terms although the Aristotelian and the topological conceptions of the continuum differ with one another in one obvious and crucial aspect of the continuum; the Aristotelian continuum is non-punctiform whereas the topological one is punctiform. However, considering that in capturing the Aristotelian continuum, the existence of points does not seem to play a significant role, those views are not so far away as we expected. CHAPTER 7. NON-PUNCTIFORM CONTINUUM 164

The above presentation of the category-theoretic formulations of the continuum is, I admit, sketchy at best. However, as Robert Goldblatt points out,21 the deep structure of the continuum in the category-theoretic framework is still unknown. We know little about, for example, the implication of the result that the Dedekind and the Cantor reals are not isomorphic in the category-theoretic framework. Perhaps, the reason why they are not isomorphic might be their diﬀerence in what sense these are considered to be complete; the Dedekind reals are order-complete whereas the Cantor reals are metric-complete. If so, however, what does such a diﬀerence in the meaning of completeness mean? Again, there are so many unknown aspects in the category-theoretically conceived continuum. In a sense, we are still wandering in the labyrinth of the continuum.

21Goldblatt 2006, p. xii. Conclusion

In this dissertation, I have been trying to depict a wide variety of the conceptions of the continuum, and in so doing, also trying to expand the conceptions and to ﬁnd a new way to look at the continuum. I started my discussion with, perhaps a bit too conventionally, Aris- totle. One might think that Aristotle’s non-punctiform conception of the continuum has completely become obsolete with the arrival of Cantor’s and Dedekind’s theories of the continuum. Still, in the Aristotelian conception of the continuum, there is an irresistible appeal to our intuitive sense: in order for two things to connect with each other, their joint has to be one; and extensionless indivisibles cannot constitute a continuum with some magnitude. Of course, intuition can be deceiving; this is exactly why Dedekind and Cantor dismissed the intuitive geometrical conception of the continuum. However, it seems also true that without intuition of the continuum we cannot even start to construct a theory of it although it would be still true that we sometimes err because of intuition. The best-case scenario would be, then, to construct a robust theory of the continuum which is also faithful to

165 CHAPTER 7. NON-PUNCTIFORM CONTINUUM 166 the intuitive conception of it. To keep intuitive naturalness and theoretical robustness at the same time—this is exactly what Peirce wanted to achieve in constructing his own theory of the continuum. Thus, it would have been a natural choice for Peirce to attempt to synthesize the Aristotelian and the Cantorian views of the continuum in his theory-making. Unfortunately, Peirce’s attempt cannot be said to be very successful. Still, Peirce left us an insightful and unique view of the continuum; namely, the continuum as a multitudinous vague object. As I sketched in the last chapter, it is not impossible at all to theorize these Aristotelian and Peircean continua in a robust mathematical way. First, we saw that Aristotle’s view is mot so far from the modern view when it is interpreted with the language of topology. In fact, most aspect of the Aristotelian conception of the continuum can be translatable into the language of topology. Of course, there is an acute difference between the Aristotelian and the topological views of the continuum; namely, whereas the Aristotelian continuum is thoroughly non-punctiform, the topologically conceived continuum is thoroughly punctiform. However, as as is sketched in the chapter, there is a way to do topology without presupposing the existence of points: pointless topology. It is surprising to see that many theories which can capture the continuum without presupposing the existence of points are based on category theory. This might be because the internal logic of category theory is CHAPTER 7. NON-PUNCTIFORM CONTINUUM 167 not classical. Whatever the reason, the theories based on category theory can capture the continuum in an intuitively natural and theoretically robust way. However, as I said in the last chapter, the investigation of the continuum in the category-theoretic frameworks has just begun. There are a lot of unknown aspects of the category-theoretically conceived continuum. Turning our eyes to the medieval conceptions of the continuum, we are struck by the close relationship between their conceptions of the continuum and the reality; however strange they seem to our modern eyes, their problems concerning the continuum were so real and might well have been life-and-death matters. This seems more so considering the Parisian condemnation of 1277. Perhaps there will not be such a tension between the reality and the conceptions of the continuum. However, we might do better to look at the continuum in relation to the reality. My main aim in this dissertation is, as I made clear in General Introduction, not to deny the standard mathematical view of the continuum (àla Dedekind et Cantor); rather, my aim is to find a new way of looking at the continuum which can be coexistent with the standard view. Of course I am not claiming that I am close to finding such a new way; actually, far from it. I just hope that my dissertation would do a little good in opening a new chapter of the continuum. Bibliography

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