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.
Author(s): John Stillwell
Publisher: Princeton University Press
Year: 2022
Language: English
Pages: 456
City: Princeton
Cover
Contents
Preface
1. Before Euclid
1.1 The Pythagorean Theorem
1.2 Pythagorean Triples
1.3 Irrationality
1.4 From Irrationals to Infinity
1.5 Fear of Infinity
1.6 Eudoxus
1.7 Remarks
2. Euclid
2.1 Definition, Theorem, and Proof
2.2 The Isosceles Triangle Theorem and SAS
2.3 Variants of the Parallel Axiom
2.4 The Pythagorean Theorem
2.5 Glimpses of Algebra
2.6 Number Theory and Induction
2.7 Geometric Series
2.8 Remarks
3. After Euclid
3.1 Incidence
3.2 Order
3.3 Congruence
3.4 Completeness
3.5 The Euclidean Plane
3.6 The Triangle Inequality
3.7 Projective Geometry
3.8 The Pappus and Desargues Theorems
3.9 Remarks
4. Algebra
4.1 Quadratic Equations
4.2 Cubic Equations
4.3 Algebra as “Universal Arithmetick”
4.4 Polynomials and Symmetric Functions
4.5 Modern Algebra: Groups
4.6 Modern Algebra: Fields and Rings
4.7 Linear Algebra
4.8 Modern Algebra: Vector Spaces
4.9 Remarks
5. Algebraic Geometry
5.1 Conic Sections
5.2 Fermat and Descartes
5.3 Algebraic Curves
5.4 Cubic Curves
5.5 Bézout’s Theorem
5.6 Linear Algebra and Geometry
5.7 Remarks
6. Calculus
6.1 From Leonardo to Harriot
6.2 Infinite Sums
6.3 Newton’s Binomial Series
6.4 Euler’s Solution of the Basel Problem
6.5 Rates of Change
6.6 Area and Volume
6.7 Infinitesimal Algebra and Geometry
6.8 The Calculus of Series
6.9 Algebraic Functions and Their Integrals
6.10 Remarks
7. Number Theory
7.1 Elementary Number Theory
7.2 Pythagorean Triples
7.3 Fermat’s Last Theorem
7.4 Geometry and Calculus in Number Theory
7.5 Gaussian Integers
7.6 Algebraic Number Theory
7.7 Algebraic Number Fields
7.8 Rings and Ideals
7.9 Divisibility and Prime Ideals
7.10 Remarks
8. The Fundamental Theorem of Algebra
8.1 The Theorem before Its Proof
8.2 Early “Proofs” of FTA and Their Gaps
8.3 Continuity and the Real Numbers
8.4 Dedekind’s Definition of Real Numbers
8.5 The Algebraist’s Fundamental Theorem
8.6 Remarks
9. Non-Euclidean Geometry
9.1 The Parallel Axiom
9.2 Spherical Geometry
9.3 A Planar Model of Spherical Geometry
9.4 Differential Geometry
9.5 Geometry of Constant Curvature
9.6 Beltrami’s Models of Hyperbolic Geometry
9.7 Geometry of Complex Numbers
9.8 Remarks
10. Topology
10.1 Graphs
10.2 The Euler Polyhedron Formula
10.3 Euler Characteristic and Genus
10.4 Algebraic Curves as Surfaces
10.5 Topology of Surfaces
10.6 Curve Singularities and Knots
10.7 Reidemeister Moves
10.8 Simple Knot Invariants
10.9 Remarks
11. Arithmetization
11.1 The Completeness of R
11.2 The Line, the Plane, and Space
11.3 Continuous Functions
11.4 Defining “Function” and “Integral”
11.5 Continuity and Differentiability
11.6 Uniformity
11.7 Compactness
11.8 Encoding Continuous Functions
11.9 Remarks
12. Set Theory
12.1 A Very Brief History of Infinity
12.2 Equinumerous Sets
12.3 Sets Equinumerous with R
12.4 Ordinal Numbers
12.5 Realizing Ordinals by Sets
12.6 Ordering Sets by Rank
12.7 Inaccessibility
12.8 Paradoxes of the Infinite
12.9 Remarks
13. Axioms for Numbers, Geometry, and Sets
13.1 Peano Arithmetic
13.2 Geometry Axioms
13.3 Axioms for Real Numbers
13.4 Axioms for Set Theory
13.5 Remarks
14. The Axiom of Choice
14.1 AC and Infinity
14.2 AC and Graph Theory
14.3 AC and Analysis
14.4 AC and Measure Theory
14.5 AC and Set Theory
14.6 AC and Algebra
14.7 Weaker Axioms of Choice
14.8 Remarks
15. Logic and Computation
15.1 Propositional Logic
15.2 Axioms for Propositional Logic
15.3 Predicate Logic
15.4 Gödel’s Completeness Theorem
15.5 Reducing Logic to Computation
15.6 Computably Enumerable Sets
15.7 Turing Machines
15.8 TheWord Problem for Semigroups
15.9 Remarks
16. Incompleteness
16.1 From Unsolvability to Unprovability
16.2 The Arithmetization of Syntax
16.3 Gentzen’s Consistency Proof for PA
16.4 Hidden Occurrences of ε0 in Arithmetic
16.5 Constructivity
16.6 Arithmetic Comprehension
16.7 TheWeak Kőnig Lemma
16.8 The Big Five
16.9 Remarks
Bibliography
Index
