The Nature of Mathematics: A Critical Survey

Title Page......Page 1
Contents......Page 3
Preface......Page 10
Introduction......Page 12
Logistic......Page 18
Formalism......Page 19
Intuitionism......Page 20
Mutual Relations of the Three Schools......Page 22
History of Logistic Views of Mathematics......Page 25
The Tasks of a Philosophy of Mathematics......Page 29
Supplementary Note on Logical Analysis......Page 34
The Formal Character of Pure Mathematics......Page 47
The Propositional Calculus......Page 51
The Calculus of Propositional Functions......Page 58
Variable and Function in Mathematics......Page 60
Various Usages of Variable Symbols......Page 61
Definitions of Mathematical Functions......Page 63
Propositional Functions......Page 70
Quantifiers, Truth-Values, etc......Page 72
The Algebra of Propositional Functions......Page 76
Propositional Functions of Functions and the General Functional Calculus......Page 77
Extensional Functions of Functions......Page 78
Derivation of Mathematical Functions from Propositional Descriptions......Page 79
Plural Descriptive Phrases......Page 81
Definitions of Descriptions and Classes......Page 84
Complete and Incomplete Symbols......Page 86
Definition of Incomplete Symbol......Page 87
Importance of Incomplete Symbols......Page 88
Nature of Principia Classes......Page 89
Consistency of Definition of Classes as Incomplete Symbols......Page 90
The Real Number......Page 95
Intuitions of Continuity in a Sensory Field......Page 97
Continuity in Geometrical Space......Page 99
Dedekind's Definition of Real Number......Page 103
The Logistico-Mathematical Paradoxes......Page 107
Solution of the Paradoxes......Page 110
Note on Types and Orders......Page 111
Connection Between the 'Extended Infinite' and the Paradoxes......Page 114
Confusion of Types in the Theorem of the Upper Bound......Page 117
The Axiom of Reducibility and the Logistic Definition of Real Number......Page 119
The Axiom of Reducibility......Page 121
Arguments for the Axiom of Reducibility......Page 122
Axiom of Reducibility Equivalent to the Assertion of the Existence of c Propositional Functions......Page 125
Other Criticisms of the Axiom of Reducibility......Page 127
F. P. Ramsey......Page 129
Note on the Thesis of Extensionality......Page 132
H. Weyl......Page 134
L. Wittgenstein......Page 139
L. Chwistek......Page 145
Conclusions......Page 149
Section II: Formalism......Page 156
The Development of Geometry......Page 161
The Formalist View of Mathematics......Page 170
The Formalist Programme in Detail......Page 171
Note on Godel's Theorem......Page 176
Section III: Intuitionism......Page 178
The Mathematical Predecessors of the Intuitionists......Page 183
Digression on the Theory of Sets of Points......Page 187
The Mathematical Controversy......Page 193
Intuitionism......Page 195
Kant and Brouwer......Page 196
The Sociological Basis of Mathematics......Page 200
The Denial of the Law of the Excluded Middle......Page 204
The Intuitionist Continuum......Page 205
Supplementary Note on the Intuitionist Calculus of Propositions......Page 210
Intuitionist Theory of Cardinal Numbers......Page 216
Bibliography......Page 220
Index......Page 224