The Story of Proof : Logic and the History of Mathematics

**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. 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 "The proof is one of the most significant and defining features of mathematical thought. Many of the most significant advances in mathematics have been advances in the concept of proof, and in many ways, the history of mathematics can be viewed as a history of the proof. Yet proof itself is not considered an interesting topic by many mathematicians. In the United States, proof is not deemed an essential part of mathematics education until the upper-undergraduate level. In The Story of Proof, John Stillwell argues that the proof is an important object of study in itself. He provides insight into the history, theory, and practice of proof in order to shed light on the way that mathematics works at its most fundamental levels. The book begins with the proof of one of mathematics' most foundational theorems, the Pythagorean Theorem. The book then proceeds to look at the role of proof in the various fields of mathematics, including geometry and number theory, algebra, algebraic geometry, and calculus. By examining the history of the proof, Stillwell sheds new light on age-old questions in mathematics, including how logic, computation, and abstraction are connected with one another and with the rest of mathematics"-- Provided by publisher