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Alternatives to the Calculus: Nonstandard Analysis and Smooth Infinitesimal Analysis

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Alternatives to the Calculus: Nonstandard Analysis and Smooth Infinitesimal Analysis Alternatives to the Calculus: Nonstandard Analysis and Smooth Infinitesimal Analysis A thesis presented to the faculty of the College of Arts and Sciences of Ohio University In partial fulfillment of the requirement for the degree Master of Arts Jesse P. Houchens May 2013 © 2013 Jesse P. Houchens. All Rights Reserved. 2 This thesis titled Alternatives to the Calculus: Nonstandard Analysis and Smooth Infinitesimal Analysis by JESSE P. HOUCHENS has been approved for the Department of Philosophy and the College of Arts and Sciences by Philip W. Ehrlich Professor of Philosophy Robert Frank Dean, College of Arts and Sciences 3 ABSTRACT HOUCHENS, JESSE P., M.A., May 2013, Philosophy Alternatives to the Calculus: Nonstandard Analysis and Smooth Infinitesimal Analysis Director of Thesis: Philip W. Ehrlich We attempt to clarify and evaluate what shall be called Mac Lane’s thesis—the thesis that nonstandard analysis (NSA) and smooth infinitesimal analysis (SIA) are alternatives to the standard approach to the calculus. In doing so, we outline the historical approaches to the calculus, the standard approach to the calculus, and two nonstandard approaches, namely NSA and SIA; we also attempt to clarify and evaluate a set of comparisons of NSA and SIA, namely Bell’s 5 mathematico-philosophical contentions and Bell’s historical contention. 4 For my parents who continually remind me “While it may not always be easy, it will always be worth it.” 5 ACKNOWLEDGMENTS First, I would like to thank the members of my thesis committee. Todd Eisworth provided insightful comments from the perspective of a working mathematician. In addition, his set theory course allowed me to clarify many of the ideas underlying this thesis. Stewart Shapiro travelled down to Athens for the defense as well as allowed me to participate in his in his seminar on continuity and infinitesimals at The Ohio State University. His comments were very helpful in clarifying many of my arguments. In addition, it was in his seminar that I was first exposed to NSA and SIA and that many of the questions raised in section II.A were first voiced. Second, I would like to express my sincere gratitude to Philip Ehrlich for advising this project. Each week, as I returned from Dr. Shapiro’s seminar, he would ask me what was discussed and my thoughts on the matter. When coming up with the topic of this thesis, he gave me invaluble advice for making the project manageable. In addition, he directed me to excellent sources on the history of the calculus as well as on each of the mathematical theories. Finally, his patience as I attempted to understand each of the theories as well as his willingness to always correct my mistakes and provide a surplus of comments on previous drafts was absolutely beneficial. Overall, one could say that Phil’s guidance contributed significantly in improving the quality of this paper. Finally, I would like to thank my family, former professors, and fellow graduate students as they listened to my struggles and constantly voiced their confidence in my abilities. 6 TABLE OF CONTENTS Abstract..............................................................................................................................3 Dedication..........................................................................................................................4 Acknowledgements............................................................................................................5 List of Figures....................................................................................................................7 Introduction........................................................................................................................8 1. Historical and Theoretical Background........................................................................11 1.1. Historical Approaches to the Calculus.........................................................11 1.2. The Standard Approach to the Calculus.......................................................21 1.3. Nonstandard Analysis..................................................................................30 1.4. Smooth Infinitesimal Analysis.....................................................................47 2. Clarifications and Evaluations......................................................................................57 2.1. Bell’s 5 Mathematico-Philosophical Contentions.........................................57 2.2. Bell’s Historical Contention..........................................................................70 2.3. Mac Lane’s Thesis.........................................................................................73 Conclusion.........................................................................................................................79 Works Cited.......................................................................................................................80 7 LIST OF FIGURES Figure 1. Newton’s differential calculus………………….……………….……….13 Figure 2. Newton’s integral calculus...…………………………………………......14 Figure 3. Leibniz’ calculus………..………………………………………………..15 Figure 4. Toricelli’s trumpet………………………………………………………..21 Figure 5. Infinitesimal difference between dy /x and y /x …...…………...…..44 Figure 6. Motivation for nondegeneracy………….……………………….………..58 Figure 7. Local straightness SIA………………..…………………………………..62 Figure 8. Local straightness NSA 1………………………..………………………..63 Figure 9. Local straighntess NSA 2……………………………..…………………..64 8 INTRODUCTION After a historically-sensitive discussion of integration and differentiation in Mathematics, Form and Function, Saunders Mac Lane concludes with the following: This discussion of alternative views of the calculus may serve to support our thesis that Mathematics is not just formalism and is not just empirically convenient ideas, but consists in formalizable intuitive or empirical ideas. Calculus fits this thesis as to the nature of Mathematics because it starts from problems (such as the calculation of areas or of rates of change) and from these problems develops suggestive ideas which ultimately can be fully formalized… Another thesis asserts that the same intuitive idea can be variously formalized. For the calculus, we know that this is the case. Indeed, the earlier and originally wholly speculative use of infinitesimals can be rigorously formalized in at least two different ways: By using Abraham Robinson’s non-standard model of the reals, as presented for example in Keisler’s text [1976] on calculus, or by using Lawvere’s proposals to use ‘elementary topoi’ in which the real line R is presented not as a field but as a ring in which there is a suitable infinitesimal neighborhood of 0 (see Kock [1981]). (Mac Lane 155) The passage above contains two theses: A) The historical development of the calculus consisted of different attempts to formalize intuitive ideas suggested by basic problems; B) There are at least two different ways to rigorously formalize the “originally wholly speculative use of infinitesimals”, specifically the ways employed in nonstandard analysis (NSA) and smooth infinitesimal analysis (SIA)1. These two theses, we believe, motivate a third, namely that NSA and SIA are two alternatives to the calculus. Let this third thesis be called Mac Lane’s thesis. John L. Bell has made a remark similar to Mac Lane’s thesis restricted to SIA: “As we show in this book, within [SIA] the basic calculus...can be developed along traditional ‘infinitesimal’ lines—with full rigor—using straightforward calculations with infinitesimals in place of the limit concept.” ([2008] 4) 1 While Mac Lane references Kock’s book on synthetic differential geometry (SDG), the fragment of SDG that addresses the calculus is SIA. 9 The purpose of this essay is to explicate and evaluate Mac Lane’s thesis. This will involve first discussing some of the differing historical approaches to the calculus as well as the problems that motivated its development. After doing so, the standard ε—δ approach to the calculus as well as the two nonstandard approaches, NSA and SIA, will be outlined to facilitate comparison among the theories. According to John L. Bell, some of the differences between NSA and SIA are the following: 1. In models of [SIA], only smooth maps between objects are present. In models of nonstandard analysis, all set-theoretically definable maps (including discontinuous ones) appear. 2. The logic of [SIA] is intuitionistic, making possible the nondegeneracy of the microneighbourhoods ∆ and Mi, i = 1, 2, 3. The logic of nonstandard analysis is classical, causing all these microneighborhoods to collapse to zero. 3. In [SIA], the Principle of Microaffineness entails that all curves are ‘locally straight’. Nothing resembling this is possible in [NSA]. 4. The property of nilpotency of the microquantities of [SIA] enables the differential calculus to be reduced to simple algebra. In [NSA] the use of infinitesimals is a disguised form of the classical limit method. 5. In any model of [NSA] R* has exactly the same set theoretically expressible properties as R does: in the sense of that model, therefore, R* is in particular an Archimedean ordered field. This means that the ‘infinitesimals’ and ‘infinite numbers’ of nonstandard analysis are so not in the sense of the model in which they ‘live’, but only relative to the ‘standard’ model with which the construction began. That is, speaking figuratively, a ‘denizen’ of a model of nonstandard analysis would be unable to detect
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