The story of proof : logic and the history of mathematics / John Stillwell.

Format:

Book

Author/Creator:

Stillwell, John, 1942- author.

Language:

English

Subjects (All):

Proof theory.

Genre:

History.

Physical Description:

1 online resource (457 pages)

Place of Publication:

Princeton, New Jersey : Princeton University Press, [2022]

Summary:

How the concept of proof has enabled the creation of mathematical knowledgeThe 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.

Contents:

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.

Notes:

Includes bibliographical references and index.

Description based on print version record.

ISBN:

9780691234373

069123437X

OCLC:

1347381552