Most scholars think of David Hilbert's program as the most demanding and ideologically motivated attempt to provide a foundation for mathematics, and because they see technical obstacles in the way of realizing the program's goals, they regard it as a failure. Against this view, Curtis Franks argues that Hilbert's deepest and most central insight was that mathematical techniques and practices do not need grounding in any philosophical principles. He weaves together an original historical account, philosophical analysis, and his own development of the meta-mathematics of weak systems of arithmetic to show that the true philosophical significance of Hilbert's program is that it makes the autonomy of mathematics evident. The result is a vision of the early history of modern logic that highlights the rich interaction between its conceptual problems and technical development.
Author(s): Curtis Franks
Edition: 1
Publisher: Cambridge University Press
Year: 2009
Language: English
Pages: 228
Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 11
Acknowledgments......Page 14
1.1 Recovering Hilbert's thought......Page 17
1.2 Freedom from nature......Page 25
1.3 Freedom from philosophy......Page 30
1.4 The wrong conclusion......Page 41
2.1 Introduction......Page 45
2.2 Anti-foundationalism......Page 48
2.3 Mathematical autonomy......Page 57
2.4 Formalism and finitism......Page 63
2.5 Conclusion......Page 75
3.1 Introduction......Page 77
3.2 Gödel's work as a contribution to Hilbert's program......Page 80
3.3 The Grundlagen der Geometrie......Page 84
3.4 Hilbert's proto-proof theory......Page 89
3.5 Herbrand's reception of Hilbert......Page 99
3.6 Investigating metatheory with arithmetic......Page 105
3.7 Progress towards purity......Page 114
4.1 Introduction......Page 121
4.1.1 First pass......Page 122
4.1.2 Second pass......Page 124
4.1.3 Third pass......Page 126
4.2 Gödel's second theorem and Hilbert's program......Page 128
4.3 Feferman's approach......Page 135
4.4 The inside/outside distinction......Page 139
4.5 A theory-dependent interpretation of consistency......Page 143
4.6 Conclusion......Page 152
5.1 Historical background......Page 155
5.2 Pudlák's argument......Page 158
5.3 Two routes to intensionality......Page 161
5.4 Ambiguity in interpretation......Page 168
5.5 Concluding remark......Page 183
6.1 The naturalistic turn......Page 185
6.2 Wittgenstein's critique of the second-order......Page 190
6.3 First-order meta-mathematics......Page 200
6.4 Evidence of autonomy......Page 207
References......Page 216
Index......Page 225
