The Story of Proof

How the concept of proof has enabled the creation of mathematical knowledge

The Story of Proof investigates the evolution of the concept of proof―one of the most significant and defining features of mathematical thought―through critical episodes in its history. From the Pythagorean theorem to modern times, and across all major mathematical disciplines, John Stillwell demonstrat...

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How the concept of proof has enabled the creation of mathematical knowledge

The Story of Proof investigates the evolution of the concept of proof―one of the most significant and defining features of mathematical thought―through critical episodes in its history. From the Pythagorean theorem to modern times, and across all major mathematical disciplines, John Stillwell demonstrates that proof is a mathematically vital concept, inspiring innovation and playing a critical role in generating knowledge.

Stillwell begins with Euclid and his influence on the development of geometry and its methods of proof, followed by algebra, which began as a self-contained discipline but later came to rival geometry in its mathematical impact. In particular, the infinite processes of calculus were at first viewed as “infinitesimal algebra,” and calculus became an arena for algebraic, computational proofs rather than axiomatic proofs in the style of Euclid. Stillwell proceeds to the areas of number theory, non-Euclidean geometry, topology, and logic, and peers into the deep chasm between natural number arithmetic and the real numbers. In its depths, Cantor, Gödel, Turing, and others found that the concept of proof is ultimately part of arithmetic. This startling fact imposes fundamental limits on what theorems can be proved and what problems can be solved.

Shedding light on the workings of mathematics at its most fundamental levels, The Story of Proof offers a compelling new perspective on the field’s power and progress.

Table of Contents
Preface xi
1 Before Euclid 1
1.1 The Pythagorean Theorem 2
1.2 Pythagorean Triples 4
1.3 Irrationality 6
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Table of Contents
Preface xi
1 Before Euclid 1
1.1 The Pythagorean Theorem 2
1.2 Pythagorean Triples 4
1.3 Irrationality 6
1.4 From Irrationals to Infinity 7
1.5 Fear of Infinity 10
1.6 Eudoxus 12
1.7 Remarks 15
2 Euclid 16
2.1 Definition, Theorem, and Proof 17
2.2 The Isosceles Triangle Theorem and SAS 20
2.3 Variants of the Parallel Axiom 22
2.4 The Pythagorean Theorem 25
2.5 Glimpses of Algebra 26
2.6 Number Theory and Induction 29
2.7 Geometric Series 32
2.8 Remarks 36
3 After Euclid 39
3.1 Incidence 40
3.2 Order 41
3.3 Congruence 44
3.4 Completeness 45
3.5 The Euclidean. Plane 47
3.6 The Triangle Inequality 50
3.7 Projective Geometry 51
3.8 The Pappus and Desargues Theorems 55
3.9 Remarks 59
4 Algebra 61
4.1 Quadratic Equations 62
4.2 Cubic Equations 64
4.3 Algebra as "Universal Arithmetick" 68
4.4 Polynomials and Symmetric Functions 69
4.5 Modern Algebra: Groups 73
4.6 Modern Algebra: Fields and Rings 77
4.7 Linear Algebra 81
4.8 Modern Algebra: Vector Spaces 82
4.9 Remarks 85
5 Algebraic Geometry 92
5.1 Conic Sections 93
5.2 Fermat and Descartes 95
5.3 Algebraic Curves 97
5.4 Cubic Curves 100
5.5 Bézout's Theorem 103
5.6 Linear Algebra and Geometry 105
5.7 Remarks 108
6 Calculus 110
6.1 From Leonardo to Harriot 111
6.2 Infinite Sums 113
6.3 Newton's Binomial Series 117
6.4 Euler's Solution of the Base! Problem 119
6.5 Rates of Change 122
6.6 Area and Volume 126
6.7 Infinitesimal Algebra and Geometry 130
6.8 The Calculus of Series 136
6.9 Algebraic Functions and Their Integrals 138
6.10 Remarks 142
7 Number Theory 145
7.1 Elementary Number Theory 146
7.2 Pythagorean Triples 150
7.3 Fermat's Last Theorem 154
7.4 Geometry and Calculus in Number Theory 158
7.5 Gaussian Integers 164
7.6 Algebraic Number Theory 171
7.7 Algebraic Number Fields 174
7.8 Rings and Ideals 178
7.9 Divisibility and Prime Ideals 183
7.10 Remarks 186
8 The Fundamental Theorem of Algebra 191
8.1 The Theorem before Its Proof 192
8.2 Early "Proofs" of FTA and Their Gaps 194
8.3 Continuity and the Real Numbers 195
8.4 Dedekind's Definition of Real Numbers 197
8.5 The Algebraist's Fundamental Theorem 199
8.6 Remarks 201
9 Non-Euclidean Geometry 202
9.1 The Parallel Axiom 203
9.2 Spherical Geometry 204
9.3 A Planar Model of Spherical Geometry 207
9.4 Differential Geometry 210
9.5 Geometry of Constant Curvature 215
9.6 Beltrami's Models of Hyperbolic Geometry 219
9.7 Geometry of Complex Numbers 223
9.8 Remarks 226
10 Topology 228
10.1 Graphs 229
10.2 The Filler Polyhedron Formula 234
10.3 Euler Characteristic and Genus 239
10.4 Algebraic Curves as Surfaces 241
10.5 Topology of Surfaces 244
10.6 Curve Singularities and Knots 250
10.7 Reidemeister Moves 253
10.8 Simple Knot Invariants 256
10.9 Remarks 261
11 Arithmetization 263
11.1 The Completeness of R 264
11.2 The Line, the Plane, and Space 265
11.3 Continuous Functions 266
11.4 Defining "Function" and "Integral" 268
11.5 Continuity and Differentiability 273
11.6 Uniformity 276
11.7 Compactness 279
11.8 Encoding Continuous Functions 284
11.6 Remarks 286
12 Set Theory 291
12.1 A Very Brief History of Infinity 292
12.2 Equinumerous Sets 294
12.3 Sets Equinumerous with R 300
12.4 Ordinal Numbers 305
12.5 Realizing Ordinals by Sets 305
12.6 Ordering Sets by Rank 308
12.7 Inaccessibility 310
12.8 Paradoxes of the Infinite 311
12.9 Remarks 312
13 Axioms for Numbers, Geometry, and Sets 316
13.1 Peano Arithmetic 317
13.2 Geometry Axioms 320
13.3 Axioms for Real Numbers 322
13.4 Axioms for Set Theory 324
13.5 Remarks 327
14 The Axiom of Choice 329
14.1 AC and Infinity 330
14.2 AC and Graph Theory 331
14.3 AC and Analysis 332
14.4 AC and Measure Theory 334
14.5 AC and Set Theory 337
14.6 AC and Algebra 339
14.7 Weaker Axioms of Choice 342
14.8 Remarks 344
15 Logic and Computation 347
15.1 Prepositional Logic 348
15.2 Axioms for Prepositional Logic 351
15.3 Predicate Logic 355
15.4 Godels Completeness Theorem 357
15.5 Reducing Logic to Computation 361
15.6 Computably Enumerable Sets 363
15.7 Turing Machines 365
15.8 The Word Problem for Semigroups 371
15.9 Remarks 376
16 Incompleteness 381
16.1 From Unsolvability to Unprovability 382
16.2 The Arithmetization of Syntax 383
16.3 Gentzen's Consistency Proof for PA 386
16.4 Hidden Occurrences of ε0 in Arithmetic 390
16.5 Constructivity 393
16.6 Arithmetic Comprehension 396
16.7 The Weak K?nig Lemma 399
16.8 The Big Five 400
16.9 Remarks 403
Bibliography 405
Index 419
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