Table Of ContentREASON’S NEAREST KIN
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Reason’s Nearest Kin
PHILOSOPHIES OF ARITHMETIC FROM
KANT TO CARNAP
MICHAEL POTTER
3
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Preface
If one is to choose a fifty-year period in the history of mathematical
philosophy for concentrated study, one hardly needs to apologize for
choosingtheonethatstartedwiththeGrundlagen in1884andfinished
with Logische Syntax der Sprache in 1934. It was a period of exhil-
arating progress in the subject. Frege, Dedekind, Russell, Wittgen-
stein, Hilbert, and Carnap produced accounts of arithmetic that were
brilliantly innovative both technically and philosophically. All are de-
scribedhere. ButalloftheseauthorsstoodinKant’sshadow,andallof
ustodaystandinGo¨del’s. Sotheworkofthesetwoauthorsisdiscussed
as well.
The writing was greatly assisted by two periods of leave, at the De-
partment of Logic and Metaphysics at the University of St. Andrews,
and at the Department of Philosophy at Harvard University. I am
grateful to Fitzwilliam College for funding the first of these periods,
and to the British Academy for funding the second under its Research
Leave Scheme. The concluding chapter is based with the Editor’s per-
missiononmyarticleintheAristotelian Society Supplementary Volume
for1999. InthecourseofwritingthebookIconsultedunpublishedma-
terial in the Modern Research Archives of King’s College Cambridge,
the Russell Archive at McMaster University, and the Moore Archives
at Cambridge University Library. I am grateful for the assistance I
received from librarians at all these institutions. The extract from
Ramsey’s diary quoted here is copyright of the Provost and Fellows
of King’s College, Cambridge; Russell’s letters are quoted by courtesy
of the William Ready Division of Archives and Research Collections,
McMaster University Library, Hamilton, Ontario; I have been unable
to trace the owner of copyright in Whitehead’s letters to Russell.
For detailed and perceptive comments on earlier drafts I am very
greatly indebted to Alex Oliver, Timothy Smiley, Naomi Goulder, Ian
Proops, CharlesParsons,WarrenGoldfarb, StevenGross, MichaelDet-
lefsen, BobHanna, andPeterSullivan. Iamsorrythatthefinalversion
does not do justice to all the points they raised.
M.D.P.
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Contents
Introduction 1
0.1 Arithmetic 1
0.2 The a priori 4
0.3 Empiricism 6
0.4 Psychologism 9
0.5 Pure formalism 10
0.6 Trivial formalism 12
0.7 Reflexive formalism 15
0.8 Arithmetic and reason 17
1 Kant 20
1.1 Intuitions and concepts 21
1.2 Geometrical propositions 24
1.3 Arithmetical propositions 25
1.4 The Transcendental Deduction 27
1.5 Analytic and synthetic 30
1.6 The principle of analytic judgements 31
1.7 Geometry is not analytic 35
1.8 Arithmetic is not analytic 37
1.9 The principle of synthetic judgements 39
1.10 Geometry as synthetic 42
1.11 Arithmetic as synthetic 50
1.12 Arithmetic and sensibility 52
2 Grundlagen 55
2.1 Axiomatization 56
2.2 Arithmetic independent of sensibility 60
2.3 The Begriffsschrift 62
2.4 Frege’s conception of analyticity 65
2.5 Numerically definite quantifiers 69
2.6 The numerical equivalence 72
viii Contents
2.7 Frege’s explicit definition 75
2.8 The context principle again 78
2.9 The analyticity of the numerical equivalence 79
3 Dedekind 81
3.1 Dedekind’s recursion theorem 81
3.2 Frege and Dedekind 83
3.3 Axiomatic structuralism 85
3.4 Existence 87
3.5 Uniqueness 91
3.6 Implicationism 95
3.7 Systems 97
3.8 Dedekind on existence 99
3.9 Dedekind on uniqueness 102
4 Frege’s account of classes 105
4.1 The Julius Caesar problem yet again 105
4.2 The context principle in Grundgesetze 109
4.3 Russell’s paradox 112
4.4 Numbers as concepts 115
4.5 The status of the numerical equivalence 117
5 Russell’s account of classes 119
5.1 Propositions 119
5.2 The old theory of denoting 122
5.3 The new theory of denoting 125
5.4 The substitutional theory 128
5.5 Russell’s propositional paradox 131
5.6 Frege’s hierarchy of senses 134
5.7 Mathematical logic as based on the theory of types 136
5.8 Elementary propositions 139
∗
5.9 The hierarchy of propositional functions in 12 140
5.10 The hierarchy of propositional functions in the Introduction 141
5.11 Typical ambiguity 144
5.12 Cumulative types 146
5.13 The hierarchy of classes 147
5.14 Numbers 150
5.15 The axiom of reducibility 152
5.16 Propositional functions and reducibility 154
5.17 The regressive method 157
5.18 The Introduction to Mathematical Philosophy 160
Contents ix
6 The Tractatus 164
6.1 Sign and symbol 164
6.2 The hierarchy of types 166
6.3 The doctrine of inexpressibility 168
6.4 Operations and functions 171
6.5 Sense 174
6.6 The rejection of class-theoretic foundations for mathematics 176
6.7 Number as the exponent of an operation 177
6.8 The adjectival strategy 179
6.9 Equations 181
6.10 Numerical identities 184
6.11 Generalization 185
6.12 The axiom of infinity 187
6.13 A transcendental argument 189
6.14 Another transcendental argument 192
7 The second edition of Principia 195
7.1 Logical atomism and empiricism 195
7.2 The hierarchy of propositional functions 197
7.3 Mathematical induction 199
7.4 The definition of identity 201
8 Ramsey 206
8.1 Propositions 206
8.2 Predicating functions 208
8.3 Extending Wittgenstein’s account of identity 213
8.4 Propositional functions in extension 216
8.5 Wittgenstein’s objections 218
8.6 The axiom of infinity 221
9 Hilbert’s programme 223
9.1 Formal consistency 223
9.2 Real arithmetic 228
9.3 Schematic arithmetic 232
9.4 Ideal arithmetic 237
9.5 Metamathematics 239
9.6 Hilbert’s programme 241
