The Proof is in the Pudding A Look at the Changing Nature of Mathematical Proof Steven G. Krantz July 25, 2007 To Jerry Lyons, mentor and friend. Table of Contents Preface ix 0 What is a Proof and Why? 3 0.1 What is a Mathematician? . 4 0.2 The Concept of Proof . 7 0.3 The Foundations of Logic . 14 0.3.1 The Law of the Excluded Middle . 16 0.3.2 Modus Ponendo Ponens and Friends . 17 0.4 What Does a Proof Consist Of? . 21 0.5 The Purpose of Proof . 22 0.6 The Logical Basis for Mathematics . 27 0.7 The Experimental Nature of Mathematics . 29 0.8 The Role of Conjectures . 30 0.8.1 Applied Mathematics . 32 0.9 Mathematical Uncertainty . 36 0.10 The Publication of Mathematics . 40 0.11 Closing Thoughts . 42 1 The Ancients 45 1.1 Eudoxus and the Concept of Theorem . 46 1.2 Euclid the Geometer . 47 1.2.1 Euclid the Number Theorist . 51 1.3 Pythagoras . 53 2 The Middle Ages and Calculation 59 2.1 The Arabs and Algebra . 60 2.2 The Development of Algebra . 60 iii iv 2.2.1 Al-Khwarizmi and the Basics of Algebra . 60 2.2.2 The Life of Al-Khwarizmi . 62 2.2.3 The Ideas of Al-Khwarizmi . 66 2.2.4 Concluding Thoughts about the Arabs . 70 2.3 Investigations of Zero . 71 2.4 The Idea of Infinity . 73 3 The Dawn of the Modern Age 75 3.1 Euler and the Profundity of Intuition . 76 3.2 Dirichlet and Heuristics . 77 3.3 The Pigeonhole Principle . 81 3.4 The Golden Age of the Nineteenth Century . 82 4 Hilbert and the Twentieth Century 85 4.1DavidHilbert........................... 86 4.2 Birkhoff, Wiener, and American Mathematics . 87 4.3 L. E. J. Brouwer and Proof by Contradiction . 96 4.4 The Generalized Ham-Sandwich Theorem . 107 4.4.1 Classical Ham Sandwiches . 107 4.4.2 Generalized Ham Sandwiches . 109 4.5 Much Ado About Proofs by Contradiction . 111 4.6 Errett Bishop and Constructive Analysis . 116 4.7 Nicolas Bourbaki . 117 4.8 Perplexities and Paradoxes . 129 4.8.1 Bertrand’s Paradox . 130 4.8.2 The Banach-Tarski Paradox . 134 4.8.3 The Monty Hall Problem . 136 5 The Four-Color Theorem 141 5.1 Humble Beginnings . 142 6 Computer-Generated Proofs 153 6.1 A Brief History of Computing . 154 6.2 The Difference Between Mathematics and Computer Science . 162 6.3 How the Computer Generates a Proof . 163 6.4 How the Computer Generates a Proof . 166 v 7 The Computer as a Mathematical Aid 171 7.1 Geometer’s Sketchpad . 172 7.2 Mathematica, Maple, and MatLab . 172 7.3 Numerical Analysis . 175 7.4 Computer Imaging and Proofs . 176 7.5 Mathematical Communication . 178 8 The Sociology of Mathematical Proof 185 8.1 The Classification of the Finite, Simple groups . 186 8.2 de Branges and the Bieberbach Conjecture . 193 8.3 Wu-Yi Hsiang and Kepler Sphere-Packing . 195 8.4 Thurston’s Geometrization Program . 201 8.5 Grisha Perelman and the Poincar´eConjecture . 209 9 A Legacy of Elusive Proofs 223 9.1 The Riemann Hypothesis . 224 9.2 The Goldbach Conjecture . 229 9.3 The Twin-Prime Conjecture . 233 9.4 Stephen Wolfram and A New Kind of Science .........234 9.5 Benoit Mandelbrot and Fractals . 239 9.6 The P/NP Problem.......................241 9.6.1 The Complexity of a Problem . 242 9.6.2 Comparing Polynomial and Exponential Complexity . 243 9.6.3 Polynomial Complexity . 244 9.6.4 Assertions that Can Be Verified in Polynomial Time . 245 9.6.5 Nondeterministic Turing Machines . 246 9.6.6 Foundations of NP-Completeness . 246 9.6.7 Polynomial Equivalence . 247 9.6.8 Definition of NP-Completeness . 247 9.6.9 Intractable Problems and NP-Complete Problems . 247 9.6.10 Examples of NP-Complete Problems . 247 9.7 Andrew Wiles and Fermat’s Last Theorem . 249 9.8 The Elusive Infinitesimal . 257 9.9 A Miscellany of Misunderstood Proofs . 259 9.9.1 Frustration and Misunderstanding . 261 vi 10 “The Death of Proof?” 267 10.1 Horgan’s Thesis . 268 10.2 Will “Proof” Remain the Benchmark? . 271 11 Methods of Mathematical Proof 273 11.1 Direct Proof . 274 11.2 Proof by Contradiction . 279 11.3 Proof by Induction . 282 12 Closing Thoughts 287 12.1 Why Proofs are Important . 288 12.2 Why Proof Must Evolve . 290 12.3 What Will Be Considered a Proof in 100 Years? . 292 References 295 viii Preface The title of this book is not entirely frivolous. There are many who will claim that the correct aphorism is “The proof of the pudding is in the eat- ing.” That it makes no sense to say, “The proof is in the pudding.” Yet people say it all the time, and the intended meaning is always clear. So it is with mathematical proof. A proof in mathematics is a psychological device for convincing some person, or some audience, that a certain mathematical assertion is true. The structure, and the language used, in formulating that proof will be a product of the person creating it; but it also must be tailored to the audience that will be receiving it and evaluating it. Thus there is no “unique” or “right” or “best” proof of any given result. A proof is part of a situational ethic. Situations change, mathematical values and standards develop and evolve, and thus the very way that we do mathematics will alter and grow. This is a book about the changing and growing nature of mathemati- cal proof. In the earliest days of mathematics, “truths” were established heuristically and/or empirically. There was a heavy emphasis on calculation. There was almost no theory, and there was little in the way of mathematical notation as we know it today. Those who wanted to consider mathemat- ical questions were thereby hindered: they had difficulty expressing their thoughts. They had particular trouble formulating general statements about mathematical ideas. Thus it was virtually impossible that they could state theorems and prove them. Although there are some indications of proofs even on ancient Babylo- nian tablets from 1000 B.C.E., it seems that it is in ancient Greece that we find the identifiable provenance of the concept of proof. The earliest math- ematical tablets contained numbers and elementary calculations. Because ix x of the paucity of texts that have survived, we do not know how it came about that someone decided that some of these mathematical procedures required logical justification. And we really do not know how the formal con- cept of proof evolved. The Republic of Plato contains a clear articulation of the proof concept. The Physics of Aristotle not only discusses proofs, but treats minute distinctions of proof methodology (see our Chapter 11). Many other of the ancient Greeks, including Eudoxus, Theaetetus, Thales, Euclid, and Pythagoras, either used proofs or referred to proofs. Protagoras was a sophist, whose work was recognized by Plato. His Antilogies were tightly knit logical arguments that could be thought of as the germs of proofs. But it must be acknowledged that Euclid was the first to systematically use precise definitions, axioms, and strict rules of logic. And to systemat- ically prove every statement (i.e., every theorem). Euclid’s formalism, and his methodology, has become the model—even to the present day—for es- tablishing mathematical facts. What is interesting is that a mathematical statement of fact is a free- standing entity with intrinsic merit and value. But a proof is a device of communication. The creator or discoverer of this new mathematical result wants others to believe it and accept it. In the physical sciences—chemistry, biology, or physics for example—the method for achieving this end is the re- producible experiment.1 For the mathematician, the reproducible experiment is a proof that others can read and understand and validate. Thus a “proof” can, in principle, take many different forms. To be ef- fective, it will have to depend on the language, training, and values of the “receiver” of the proof. A calculus student has little experience with rigor and formalism; thus a “proof” for a calculus student will take one form. A professional mathematician will have a different set of values and experiences, and certainly different training; so a proof for the mathematician will take a different form. In today’s world there is considerable discussion—among mathematicians—about what constitutes a proof. And for physicists, who are our intellectual cousins, matters are even more confused. There are those workers in physics (such as Arthur Jaffe of Harvard, Charles Fefferman of Princeton, Ed Witten of the Institute for Advanced Study, Frank Wilczek of MIT, and Roger Penrose of Oxford) who believe that physical concepts 1More precisely, it is the reproducible experiment with control. For the careful scientist compares the results of his/her experiment with some standard or norm. That is the means of evaluating the result. xi should be derived from first principles, just like theorems. There are other physicists—probably in the majority—who reject such a theoretical approach and instead insist that physics is an empirical mode of discourse. These two camps are in a protracted and never-ending battle over the turf of their sub- ject. Roger Penrose’s new book The Road to Reality: A Complete Guide to the Laws of the Universe, and the vehement reviews of it that have appeared, is but one symptom of the ongoing battle.