University of Calgary PRISM: University of Calgary's Digital Repository Graduate Studies The Vault: Electronic Theses and Dissertations 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Mathematics, and Philosophy Hayashi, Teppei Hayashi, T. (2017) The Continuum: History, Mathematics, and Philosophy (Unpublished doctoral thesis).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 University of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission. Downloaded from PRISM: https://prism.ucalgary.ca UNIVERSITY OF CALGARY The Continuum: History, Mathematics, and Philosophy by 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 concep- tion 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 dis- continuous 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 con- tinuum 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 mathemat- ically rigorous ways to formalize the continuum, particularly with Peirce's ideas in mind. The first part starts with a chapter about ancient Greece. I be- gin 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 exhaus- tion, 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