Foundations and Fundamental Concepts of Mathematics

1 Mathematics Before Euclid

1(25)

1.1 The Empirical Nature of pre-Hellenic Mathematics

1(4)

1.2 Induction Versus Deduction

5(4)

1.3 Early Greek Mathematics and the Introduction of Deductive Procedures

9(4)

1.4 Material Axiomatics

13(2)

1.5 The Origin of the Axiomatic Method

15(2)

Problems

17(8)

2 Euclid's Elements

26(25)

2.1 The Importance and Formal Nature of Euclid's Elements

26(3)

2.2 Aristotle and Proclus on the Axiomatic Method

29(3)

2.3 Euclid's Definitions, Axioms, and Postulates

32(5)

2.4 Some Logical Shortcomings of Euclid's Elements

37(4)

2.5 The End of the Greek Period and the Transition to Modern Times

41(4)

Problems

45(6)

3 Non-Euclidean Geometry

51(28)

3.1 Euclid's Fifth Postulate

51(3)

3.2 Saccheri and the Reductio ad Absurdum Method

54(4)

3.3 The Work of Lambert and Legendre

58(2)

3.4 The Discovery of Non-Euclidean Geometry

60(5)

3.5 The Consistency and the Significance of Non-Euclidean Geometry

65(5)

Problems

70(9)

4 Hilbert's Grundlagen

79(34)

4.1 The Work of Pasch, Peano, and Pieri

79(3)

4.2 Hilbert's Grundlagen der Geometrie

82(6)

4.3 Poincare's Model and the Consistency of Lobachevskian Geometry

88(4)

4.4 Analytic Geometry

92(6)

4.5 Projective Geometry and the Principle of Duality

98(6)

Problems

104(9)

5 Algebraic Structure

113(34)

5.1 Emergence of Algebraic Structure

113(5)

5.2 The Liberation of Algebra

118(6)

5.3 Groups

124(4)

5.4 The Significance of Groups in Algebra and Geometry

128(4)

5.5 Relations

132(4)

Problems

136(11)

6 Formal Axiomatics

147(26)

6.1 Statement of the Modern Axiomatic Method

147(3)

6.2 A Simple Example of a Branch of Pure Mathematics

150(4)

6.3 Properties of Postulate Sets--Equivalence and Consistency

154(4)

6.4 Properties of Postulate Sets--Independence, Completeness, and Categoricalness

158(4)

6.5 Miscellaneous Comments

162(4)

Problems

166(7)

7 The Real Number System

173(39)

7.1 Significance of the Real Number System for the Foundations of Analysis

173(6)

7.2 The Postulational Approach to the Real Number System

179(4)

7.3 The Natural Numbers and the Principle of Mathematical Induction

183(8)

7.4 The Integers and the Rational Numbers

191(5)

7.5 The Real Numbers and the Complex Numbers

196(6)

Problems

202(10)

8 Sets

212(31)

8.1 Sets and Their Basic Relations and Operations

212(4)

8.2 Boolean Algebra

216(5)

8.3 Sets and the Foundations of Mathematics

221(3)

8.4 Infinite Sets and Transfinite Numbers

224(5)

8.5 Sets and the Fundamental Concepts of Mathematics

229(7)

Problems

236(7)

9 Logic and Philosophy

243(32)

9.1 Symbolic Logic

243(7)

9.2 The Calculus of Propositions

250(7)

9.3 Other Logics

257(5)

9.4 Crises in the Foundations of Mathematics

262(4)

9.5 Philosophies of Mathematics

266(5)

Problems

271(4)

Appendix

275(28)

A.1 The First Twenty-Eight Propositions of Euclid

275(1)

A.2 Euclidean Constructions

276(8)

A.3 Removal of Some Redundancies

284(5)

A.4 Membership Tables

289(1)

A.5 A Constructive Proof of the Existence of Transcendental Numbers

290(2)

A.6 The Eudoxian Resolution of the First Crisis in the Foundations of Mathematics

292(2)

A.7 Nonstandard Analysis

294(2)

A.8 The Axiom of Choice

296(3)

A.9 A Note on Godel's Incompleteness Theorem

299(4)

Bibliography

303(13)

Solution Suggestions for Selected Problems

316(13)

Index

329