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A Historical Approach to Understanding Explanatory Proofs Based on Mathematical Practices

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A Historical Approach to Understanding Explanatory Proofs Based on Mathematical Practices University of South Florida Scholar Commons Graduate Theses and Dissertations Graduate School February 2019 A Historical Approach to Understanding Explanatory Proofs Based on Mathematical Practices Erika Oshiro University of South Florida, [email protected] Follow this and additional works at: https://scholarcommons.usf.edu/etd Part of the Philosophy Commons Scholar Commons Citation Oshiro, Erika, "A Historical Approach to Understanding Explanatory Proofs Based on Mathematical Practices" (2019). Graduate Theses and Dissertations. https://scholarcommons.usf.edu/etd/7882 This Dissertation is brought to you for free and open access by the Graduate School at Scholar Commons. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Scholar Commons. For more information, please contact [email protected]. A Historical Approach to Understanding Explanatory Proofs Based on Mathematical Practices by: Erika Oshiro A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Philosophy College of Arts and Sciences University of South Florida Co-Major Professor: Alexander Levine, Ph.D. Co-Major Professor: Douglas Jesseph, Ph.D. Roger Ariew, Ph.D. William Goodwin, Ph.D. Date of Approval: December 5, 2018 Keywords: Philosophy of Mathematics, Explanation, History, Four Color Theorem Copyright © 2018, Erika Oshiro DEDICATION To Himeko. ACKNOWLEDGMENTS I would like to express my gratitude to Dr. Alexander Levine, who was always very encouraging and supportive throughout my time at USF. I would also like to thank Dr. Douglas Jesseph, Dr. Roger Ariew, and Dr. William Goodwin for their guidance and insights. I am grateful to Dr. Dmitry Khavinson for sharing his knowledge and motivation during my time as a math student. He always gave me great advice both math and non-math related. Thanks goes to Matt Fleeman, my math buddy, who provided very valuable feedback on part of this dissertation. With his help, I was able to make my arguments stronger and the mathematical parts clearer. Thank you to my friends who have proof-read my chapters, given me suggestions, and helped make the most of my downtime: Alex Shillito, Dahlia Guzman, Dan Collette, Dwight Lewis, Greg McCreery, Julie Krupa, Mark Castricone, Megan Flocken, Mike Sinclair, Patrick Miller, Simon Dutton, William Parkhurst, the late night custodians, and Fancy Pants. And last, but not least, thank you to Aaron Spink, who always encourages me to do my best in everything. TABLE OF CONTENTS Abstract............................................................................................................................... iii Introduction ......................................................................................................................... 1 Chapter One: The Importance of History ........................................................................... 9 I. History as Understood by Historians ................................................................. 10 II. History Used in Mathematical Research .......................................................... 16 III. How Philosophers Can Use History ................................................................ 19 IV. Example: The Axiom of Choice ........................................................................ 22 V. Conclusion.......................................................................................................... 30 Chapter Two: Mathematical Explanation Revisited ........................................................ 34 I. Explanatory Proofs ............................................................................................. 36 II. Against Explanatory Proofs ............................................................................... 44 III. Explanation and Understanding ...................................................................... 51 IV. Conclusion ........................................................................................................ 57 Chapter Three: A Historical Approach to Mathematical Explanation ............................ 59 I. Proofs in Training and Research ........................................................................ 61 II. Genealogy of a Theorem .................................................................................... 63 III. Example: Lewy's Theorem ............................................................................... 66 III.a. Kneser's Proof of Lewy's Theorem ..................................................... 68 III.b. Lewy's Proof ....................................................................................... 70 III.c. Bers' Proof .......................................................................................... 72 III.d. Duren's Proof ..................................................................................... 73 IV. Recapulation and Analysis ............................................................................... 74 V. From Demonstration to Explanation ................................................................ 79 VI. What Can We Gain from Looking at Successive Proofs?................................. 82 VII. Conclusion ....................................................................................................... 84 Chapter Four: Case Study—The Four Color Theorem ...................................................... 86 I. A Very Brief Sketch of the Proof ......................................................................... 87 II. Philosophical Issues with the Proof .................................................................. 89 III. Mathematicians‘ Concerns with the Proof .................................................... 103 IV. Two Later Proofs of the Theorem .................................................................. 107 V. Conclusion......................................................................................................... 110 Conclusion ........................................................................................................................ 113 i References ........................................................................................................................ 116 ii ABSTRACT My dissertation focuses on mathematical explanation found in proofs looked at from a historical point of view, while stressing the importance of mathematical practices. Current philosophical theories on explanatory proofs emphasize the structure and content of proofs without any regard to external factors that influence a proof‘s explanatory power. As a result, the major philosophical views have been shown to be inadequate in capturing general aspects of explanation. I argue that, in addition to form and content, a proof‘s explanatory power depends on its targeted audience. History is useful here, because from it, we are able to follow the transition from a first-generation proof, which is usually non-explanatory, into its explanatory version. By tracking the similarities and differences between these proofs, we are able to gain a better understanding of what makes a proof explanatory according to mathematicians who have the relevant background to evaluate it as so. My first chapter discusses why history is important for understanding mathematical practices. I describe two kinds of history: one that presents a narrative of events, which influenced developments in mathematics both directly and indirectly, and another, typically used in mathematical research, which concentrates only on technical developments. I contend that both versions of the past benefit the philosopher. History used in research gives us an idea of what mathematicians desire or find to be important, iii while history written by historians shows us what effects these have on mathematical practices. The next two chapters are about explanatory proofs. My second chapter examines the main theories of mathematical explanation. I argue that these theories are short-sighted as they only consider what appears in a proof without considering the proof‘s purported audience or background knowledge necessary to understand the proof. In the third chapter, I propose an alternative way of analyzing explanatory proofs. Here, I suggest looking at a theorem‘s history, which includes its successive proofs, as well as the mathematicians who wrote them. From this, we can better understand how and why mathematicians prove theorems in multiple ways, which depends on the purposes of these theorems. The last chapter is a case study on the computer proof of the Four Color Theorem by Appel and Haken. Here, I compare and contrast what philosophers and mathematicians have had to say about the proof. I argue that the main philosophical worry regarding the theorem—its unsurveyability—did not make a strong impact on the mathematical community and would have hindered mathematical development in computer-assisted proofs. By studying the history of the theorem, we learn that Appel and Haken relied on the strategy of Kempe‘s flawed proof from the 1800s (which, obviously, did not involve a computer). Two later proofs, also aided by computer, were developed using similar methods. None of these proofs are explanatory, but not because of their massive lengths. Rather, the methods used in these proofs are a series of calculations that exhaust all possible configurations of maps. iv INTRODUCTION Much of contemporary philosophy of mathematics has been focused on the practices
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