Foundations and Fundamental Concepts of Mathematics

Samenvatting

1 Mathematics Before Euclid1(25)1.1 The Empirical Nature of pre-Hellenic Mathematics1(4)1.2 Induction Versus Deduction5(4)1.3 Early Greek Mathematics and the Introduction of Deductive Procedures9(4)1.4 Material Axiomatics13(2)1.5 The Origin of the Axiomatic Method15(2)Problems17(8)2 Euclid\'s Elements26(25)2.1 The Importance and Formal Nature of Euclid\'s Elements26(3)2.2 Aristotle and Proclus on the Axiomatic Method29(3)2.3 Euclid\'s Definitions, Axioms, and Postulates32(5)2.4 Some Logical Shortcomings of Euclid\'s Elements37(4)2.5 The End of the Greek Period and the Transition to Modern Times41(4)Problems45(6)3 Non-Euclidean Geometry51(28)3.1 ... Euclid\'s Fifth Postulate51(3)3.2 Saccheri and the Reductio ad Absurdum Method54(4)3.3 The Work of Lambert and Legendre58(2)3.4 The Discovery of Non-Euclidean Geometry60(5)3.5 The Consistency and the Significance of Non-Euclidean Geometry65(5)Problems70(9)4 Hilbert\'s Grundlagen79(34)4.1 The Work of Pasch, Peano, and Pieri79(3)4.2 Hilbert\'s Grundlagen der Geometrie82(6)4.3 Poincare\'s Model and the Consistency of Lobachevskian Geometry88(4)4.4 Analytic Geometry92(6)4.5 Projective Geometry and the Principle of Duality98(6)Problems104(9)5 Algebraic Structure113(34)5.1 Emergence of Algebraic Structure113(5)5.2 The Liberation of Algebra118(6)5.3 Groups124(4)5.4 The Significance of Groups in Algebra and Geometry128(4)5.5 Relations132(4)Problems136(11)6 Formal Axiomatics147(26)6.1 Statement of the Modern Axiomatic Method147(3)6.2 A Simple Example of a Branch of Pure Mathematics150(4)6.3 Properties of Postulate Sets--Equivalence and Consistency154(4)6.4 Properties of Postulate Sets--Independence, Completeness, and Categoricalness158(4)6.5 Miscellaneous Comments162(4)Problems166(7)7 The Real Number System173(39)7.1 Significance of the Real Number System for the Foundations of Analysis173(6)7.2 The Postulational Approach to the Real Number System179(4)7.3 The Natural Numbers and the Principle of Mathematical Induction183(8)7.4 The Integers and the Rational Numbers191(5)7.5 The Real Numbers and the Complex Numbers196(6)Problems202(10)8 Sets212(31)8.1 Sets and Their Basic Relations and Operations212(4)8.2 Boolean Algebra216(5)8.3 Sets and the Foundations of Mathematics221(3)8.4 Infinite Sets and Transfinite Numbers224(5)8.5 Sets and the Fundamental Concepts of Mathematics229(7)Problems236(7)9 Logic and Philosophy243(32)9.1 Symbolic Logic243(7)9.2 The Calculus of Propositions250(7)9.3 Other Logics257(5)9.4 Crises in the Foundations of Mathematics262(4)9.5 Philosophies of Mathematics266(5)Problems271(4)Appendix275(28)A.1 The First Twenty-Eight Propositions of Euclid275(1)A.2 Euclidean Constructions276(8)A.3 Removal of Some Redundancies284(5)A.4 Membership Tables289(1)A.5 A Constructive Proof of the Existence of Transcendental Numbers290(2)A.6 The Eudoxian Resolution of the First Crisis in the Foundations of Mathematics292(2)A.7 Nonstandard Analysis294(2)A.8 The Axiom of Choice296(3)A.9 A Note on Godel\'s Incompleteness Theorem299(4)Bibliography303(13)Solution Suggestions for Selected Problems316(13)Index329