Proof, Logic and Formalization

Detlefsen M. (Ed)0-203-98025-5

Proof, Logic and Formalization addresses the various problems associated with finding a philosophically satisfying account of mathematical proof. It brings together many of the most notable figures currently writing on this issue in an attempt to explain why it is that mathematical proof is given prominence over other forms of mathematical justification. The difficulties that arise in accounts of proof range from the rightful role of logical inference and formalization to questions concerning the place of experience in proof and the possibility of eliminating impredictive reasoning from proof.Students and lecturers of philosophy, philosophy of logic, and philosophy of mathematics will find this to be essential reading. A companion volume entitled Proof and Logic in Mathematics is also available from Routledge.

---

Table of contents :
BOOK COVER……Page 1
HALF-TITLE……Page 3
TITLE……Page 4
COPYRIGHT……Page 5
DEDICATION……Page 6
CONTENTS……Page 7
NOTES ON CONTRIBUTORS……Page 8
PREFACE……Page 9
I……Page 12
II……Page 13
III……Page 14
IV……Page 15
REFERENCES……Page 18
SUMMARY……Page 19
The function of arguments……Page 20
Epistemic biography……Page 21
II. THE STRUCTURE OF PROOFS……Page 22
Abstract proofs……Page 23
Rules of proof……Page 24
Interpreting the abstract structure……Page 26
III. THE STRUCTURE OF EPISTEMIC BIOGRAPHY……Page 27
Reasons and inferences……Page 28
Deductive structure and validity……Page 30
REFERENCES……Page 32
SUMMARY……Page 33
I……Page 34
II……Page 36
III……Page 37
IV……Page 39
REFERENCES……Page 41
I……Page 42
II……Page 43
III……Page 44
V……Page 45
NOTES……Page 47
REFERENCES……Page 49
I. ARITHMETICAL TRUTH……Page 50
II. THE ω-RULE……Page 53
III. CLAIM THAT THE COMPLETENESS OF THE ω-RULE OBLITERATES THE ATTEMPTED DISTINCTION BETWEEN ARITHMETICAL AND NON-ARITHMETICAL TRUTH……Page 55
IV. JUSTIFICATION OF THE ω-RULE FROM ω-CONSISTENCY……Page 57
V. CONSIDERATIONS ON THE FINITELY APPLIED ω-RULE IN RELATION TO ARITHMETICAL TRUTH……Page 60
VI. (ALMOST) FINITISTIC APPLICATION OF THE (ω-RULE IS ARITHMETICAL, BUT IT DOES NOT EXTEND PEANO ARITHMETIC……Page 63
VII. CONCLUDING REMARKS……Page 65
NOTES……Page 66
REFERENCES……Page 68
SUMMARY……Page 71
I……Page 72
II……Page 75
NOTES……Page 78
REFERENCES……Page 80
SUMMARY……Page 82
I. HOW MATHEMATICIANS GET BY (BETTER) WITHOUT FORMAL-LOGICAL RULES……Page 83
II. WHY POSSIBLE PROOFS ARE ACTUAL PROOFS AND THE NECESSITY OF MATHEMATICAL TRUTHS……Page 88
A priori groundedness……Page 89
Relative a priori groundedness and proofs……Page 91
III. PROOFS WITHOUT UNPROVEN ASSUMPTIONS……Page 94
NOTES……Page 97
REFERENCES……Page 98
I. INTRODUCTION……Page 99
II. THE REAL/IDEAL DISTINCTION……Page 100
III…….Page 105
Should ideal mathematics be a conservative extension of real mathematics?……Page 106
IV. CONCLUSIONS……Page 108
NOTES……Page 110
REFERENCES……Page 113
INDEX……Page 115