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, erika.com Follow this and additional works at: https://scholarcommons.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.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 scholarcommons@usf.
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. Co-Major Professor: Douglas Jesseph, Ph. Roger Ariew, Ph. William Goodwin, Ph.
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.1 Chapter One: The Importance of History. History as Understood by Historians.
History Used in Mathematical Research. How Philosophers Can Use History. Example: The Axiom of Choice. 30 Chapter Two: Mathematical Explanation Revisited.
Against Explanatory Proofs. Explanation and Understanding. 57 Chapter Three: A Historical Approach to Mathematical Explanation. Proofs in Training and Research.
Genealogy of a Theorem. Example: Lewy's Theorem. Kneser's Proof of Lewy's Theorem. Recapulation and Analysis.
From Demonstration to Explanation. What Can We Gain from Looking at Successive Proofs?. 84 Chapter Four: Case Study—The Four Color Theorem. A Very Brief Sketch of the Proof.
Philosophical Issues with the Proof. Mathematicians‘ Concerns with the Proof. Two Later Proofs of the Theorem. 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 of mathematicians. Contrasted to the traditional philosophies of mathematics of the early to mid-twentieth century, which concentrated on mathematical foundations, ontology, and truth of mathematical propositions, many philosophers now tackle questions concerning what mathematicians do and the historical development of mathematics.
This approach requires philosophers to depend on the history of mathematics, because it reflects on why and how mathematics has developed in the way it has. Studying mathematical practices also benefits the philosopher, because the successes of mathematics demonstrate the overall correctness of its practices, thereby suggesting broader epistemological lessons to be learned. There are still some ties to the traditional philosophical views. For example, the Quine-Putnam Indispensability Argument formulated over forty years ago is still influential today.
According to the argument, we are justified in believing in the existence of the mathematical objects found in the parts of mathematics that are used in our best scientific theories. The problem with this view is that it implies that the sciences determine what exists in mathematics. The objects in the applied parts of mathematics exist, while the others have to wait until the sciences have a use for them. 1 1 Mathematics is commonly divided into two parts: pure and applied.
On the surface, the differences between the two may seem clear—applied mathematics is used outside of mathematics, and pure mathematics is not. However, there are branches of mathematics 1 However, there is much more to mathematics than its applicability, and most mathematicians are not concerned with how their work is used outside of their discipline. Thus, the indispensability argument ignores mathematical practices. In spite of this problem, philosophers who have written on mathematical practices such as Alan Baker, Mark Colyvan, and Christopher Pincock have their own versions of the indispensability argument to support their views on mathematical realism.
The history of mathematics provides us with a rich source on how mathematicians come to know their subjects and why mathematics has developed in the way it has. When we take historical events into account, we can examine the changes that have occurred within mathematical practices to form a better understanding of what mathematicians do. Philosophers José Ferreirós, Phillip Kitcher, and Penelope Maddy explain that there are different key factors that can change over time, contributing to the development of mathematics. These factors include what language is used, which questions are important, and which methods of reasoning are most salient.
Although these philosophers ascribe different degrees of importance to each component, and other factors are also considered, it is generally accepted by them that all three are highly dependent on the history of mathematics. There are two types of history to consider when thinking about changes in these important factors. The first type is the history that is written by historians. Ideally, sequences of events are presented as accurately as possible, leaving little trace of the present in their narratives.
Details external to the technical details of mathematical that could belong to both. For instance, harmonic analysis is based on methods from analysis (which tends to be thought of as pure mathematics) but has many applications in the sciences and engineering. The famous number theorist, G. Hardy, praised his field as the purest branch of mathematics, but it has many applications in cryptology.
2 development are vital to understanding how a piece of mathematics came about. Such details include the backgrounds of mathematicians, the locations where developments took place, as well as major concurrent events. The second type of history is the one typically used in mathematical research. It focuses more on the progress of a piece of mathematics.
The technical developments are key here, while there is no use for any external details. This type of history tends to be Whiggish; only major developments are highlighted. Developments are presented as if they are compatible with our present mathematics—no changes in language or methods are addressed. Although history used for research purposes is not an accurate presentation of past events, it focuses on the relevant technical details the mathematician uses in her research.
The philosopher benefits from knowing both types of history. The history used in mathematical research reflects the ―final products‖ of mathematical developments. These are papers found in journals, reference books, and other sources that represent polished versions of mathematics. From these sources, philosophers can gain an understanding of what mathematicians consider important in their research.
History as told by historians provides the details of what actually occurred; it gives us a narrative of how mathematicians developed their discipline through the twists and turns of the past that are usually ignored in the history used in mathematical research. Since it is a very difficult (if not impossible) task to write on mathematics and its history in general, I will focus on mathematical proof. Specifically, I will investigate what makes a proof explanatory to mathematicians. A proof is thought to be explanatory if it answers why its corresponding theorem is true, as opposed to only providing justification that it is true.
There has been recent philosophical discussion on explanatory proofs. However, it has mostly concentrated on the contents and forms of 3 such proofs without much regard to history, although examples from the distant past are chosen above contemporary ones. Unfortunately, assuming that explanation depends only on the form or content of a proof leaves out the proof‘s audience. What may be explanatory to one audience may be confusing for another.
This can easily be imagined: a research mathematician may find a given proof to be explanatory though it remains very difficult to follow for a first-year undergraduate student. Although philosophical theories on explanation are intended to reflect the practices of research mathematicians, philosophers have chosen case studies that are very simple to follow. This has the consequence of making it seem as if an explanatory proof is accessible to everyone, which is hardly the case.