文档介绍：Stewart SHAPIRO
Philosophy of Mathematics
Oxford: Oxford University Press
2000
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Numbers . . . are known only by their laws, the laws
of arithmetic, so that any constructs obeying those
laws—certain sets, for instance—are eligible . . .
explications of number. Sets in turn are known
only by their laws, the laws of set theory . . . arithmetic
is all there is to number . . . there is no saying
absolutely what the numbers are; there is only
arithmetic.
Quine [1969, 44-45]
If in the consideration of a simply infinite system
. . . set in order by a transformation . . . we entirely
neglect the special character of the elements; simply
retaining their distinguishability and taking
into account only the relations to one another in
which they are placed by the order-setting trans
formation . . . , then are these elements called
natural numbers or ordinal numbers or simply
numbers.
Dedekind [1888, §73]
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Contents
Introduction 3
PART I PERSPECTIVE
1 Mathematics and Its Philosophy 21
2 Object and Truth: A Realist Manifesto 36
1 Slogans 36
2 Methodology 38
3 Philosophy 44
4 Interlude on Antirealism 51
5 Quine 52
6 A Role for the External 57
PART II STRUCTURALISM
3 Structure 71
1 Opening 71
2 Ontology: Object 77
3 Ontology: Structure 84
4 Theories of Structure 90
5 Mathematics: Structures, All the Way Down 97
6 Addendum: Function and Structure 106
4 Epistemology and Reference 109
1 Epistemic Preamble 109
2 Small Finite Structure: Abstraction and Pattern Recognition 112
3 Long Strings and Large Natural Numbers 116
4 To the Infinite: The Natural-Number Structure 118
5 Indiscernability, Identity and Object 120
6 Ontological Interlude 126
7 Implicit Definition and Structure 129
8 Existence and Uniqueness: Coherence and Categoricity 132
9 Conclusions: Language, Reference and Deduction 137
5 How We Got Here 143