The proof is in the pudding. A look at the changing nature of mathematical proof

334 Pages·2007·2.591 MB·English

by  Krantz S.

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
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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
4.1 David Hilbert . . . . . . . . . . . . . . . . . . . . . . . . . . . 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
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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´e Conjecture . . . . . . . . . 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
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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
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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
heuristicallyand/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
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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 mathematicianwillhaveadifferentsetofvaluesandexperiences,
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 intellectualcousins, matters are evenmore 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,itisthe reproducible experiment withcontrol. Forthecarefulscientist
compares the results of his/her experiment with some standard or norm. That is the
means of evaluatingthe result.
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should be derived from first principles, just like theorems. There are other
physicists—probablyinthemajority—whorejectsucha theoreticalapproach
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 vehementreviewsof it that haveappeared,
is but one symptom of the ongoing battle.
The idea of “proof” certainly appears in many aspects of life other than
mathematics. In the courtroom, a lawyer (either for the prosecution or the
defense)mustestablishhis/hercasebymeans ofan acceptedversionof proof.
For a criminal case this is “beyond a reasonable doubt” while for a civil case
itis“thepreponderanceof evidenceshows”. Neitheroftheseismathematical
proof, noranythinglikeit. Fortherealworldhasno formaldefinitionsandno
axioms; there is no sense of establishing facts by strict logical exegesis. The
lawyer certainly uses logic—such as “the defendant is blind so he could not
have drivento Topanga Canyon on the night of March 23” or “the defendant
has no education and therefore could not have built the atomic bomb that
was used to ...”—but his/her principal tools are facts. The lawyer proves
thecase beyond a reasonable doubt byamassing a preponderance of evidence
in favor of that case.
At the same time, in ordinary, family-style parlance there is a notion
of proof that is different from mathematical proof. A husband might say,
“I believe that my wife is pregnant” while the wife may know that she is
pregnant. Her pregnancy is not a permanent and immutable fact (like the
Pythagorean theorem), but instead is a “temporary fact” that will be false
after several months. Thus, in this circumstance, the concept of truth has a
different meaning from the one that we use in mathematics, and the means
of verification of a truth are also rather different. What we are really seeing
here is the difference between knowledge and belief—something that never
plays a formal role in mathematics.
It is also common for people to offer “proof of their love” for another
individual. Clearlysuch a “proof” willnot consist of a tightlylinkedchain of
logical reasoning. Rather, it will involve emotions and events and promises
and plans. There may be discussions of children, and care for aging parents,
and relations with siblings. This is an entirely different kind of proof from
the kind treated in the present book. It is in the spirit of this book in the
sense that it is a “device for convincing someone that something is true.”
But it is not a mathematical proof.