Truth in Mathematics

by Dales, H. G.; Oliveri, G.

• ISBN13: 9780198514763
• ISBN10: 019851476X
• Format: Hardcover
• Copyright: 1998-12-10
• Publisher: Clarendon Press

Summary

The nature of truth in mathematics is a problem which has exercised the minds of thinkers from at least the time of the ancient Greeks. The great advances in mathematics and philosophy in the twentieth centuryand in particular the proof of Godel's theorem and the development of the notion ofindependence in mathematicshave led to new viewpoints on this question in our era. This book is the result of the interaction of a number of outstanding mathematicians and philosophersincluding Yurii Manin, Vaughan Jones, and Per Martin-Lofand their discussions of this problem. It provides anoverview of the forefront of current thinking, and is a valuable introduction and reference for researchers in the area.

Table of Contents

1 Truth and the foundations of mathematics. An introduction	1	(40)
H.G. Dales
G. Oliveri
1 The pre-Tarskian debate: Kant	1	(4)
2 The pre-Tarskian debate: Frege	5	(1)
3 The pre-Tarskian debate: the role of Kronecker, Hilbert, and Brouwer	6	(5)
4 Tarski	11	(2)
5 Truth in set theory: preliminaries	13	(2)
6 Truth in set theory: axiomatics and independence	15	(5)
7 The realism/anti-realism debate and the question about mathematical knowledge	20	(2)
Notes	22	(12)
Bibliography	34	(7)
I KNOWABILITY, CONSTRUCTIVITY, AND TRUTH	41	(76)
2 Truth and objectivity from a verificationist point of view	41	(12)
Dag Prawitz
1 Gentzen's idea	42	(2)
2 Basic ideas of verificationism	44	(1)
3 Correctness of an assertion	45	(1)
4 Truth	46	(2)
5 Objectivity	48	(2)
Bibliography	50	(3)
3 Constructive truth in practice	53	(18)
Douglas S. Bridges
1 What is constructive mathematics?	53	(2)
2 Varieties of constructive mathematics	55	(2)
3 Omniscience principles	57	(2)
4 Completeness in constructive analysis	59	(4)
5 Complex analysis	63	(4)
6 The scope of constructive mathematics	67	(1)
Notes	67	(1)
Bibliography	68	(3)
4 On founding the theory of algorithms	71	(34)
Yiannis N. Moschovakis
1 The mergesort algorithm	71	(3)
2 Deconstruction	74	(1)
2.1 The merging algorithm	74	(1)
2.2 The mergesort algorithm	74	(1)
3 How do we define basic notions?	75	(4)
4 Abstract machines and implementations	79	(3)
5 The theory of recursive equations	82	(2)
6 Functionals and recursors	84	(5)
7 Implementations	89	(2)
8 Algorithms	91	(4)
9 Infinitary algorithms	95	(2)
Notes	97	(5)
Bibliography	102	(3)
5 Truth and knowability: on the principles C and K of Michael Dummett	105	(12)
Per Martin-Lof
Acknowledgements	113	(1)
Bibliography	113	(4)
II FORMALISM AND NATURALISM	117	(86)
6 Logical completeness, truth, and proofs	117	(14)
Gabriele Lolli
1 Logical truth	117	(3)
2 Proofs	120	(2)
3 The mathematician as a meta-mathematician	122	(2)
4 The psychology of mathematical thinking	124	(3)
5 Conclusion	127	(1)
Bibliography	128	(3)
7 Mathematics as language	131	(16)
Edward G. Effros
1 Introduction	131	(1)
2 Mathematics is most valued as a language	132	(1)
3 Three challenges to mathematics	132	(2)
4 Language and fluency	134	(2)
5 Computers and mathematics	136	(3)
6 The continuing evolution of mathematics	139	(5)
7 Conclusion	144	(1)
Bibliography	144	(3)
8 Truth, rigour, and common sense	147	(14)
Yu. I. Manin
1 Mathematical truth in history	148	(3)
2 Truth for a working mathematician	151	(4)
3 Materials for three case studies	155	(3)
Bibliography	158	(3)
9 How to be a naturalist about mathematics	161	(20)
Penelope Maddy
1 Naturalism in mathematics	161	(4)
2 The naturalistic methodologist	165	(7)
3 The naturalistic philosopher	172	(4)
4 Conclusion	176	(1)
Notes	177	(2)
Bibliography	179	(2)
10 The mathematician as a formalist	181	(22)
H. G. Dales
1 Introduction	181	(2)
2 Philosophical possibilities	183	(1)
3 Attitudes of mathematicians	184	(2)
4 The style of formalists	186	(5)
5 The choice of axioms; discovering the truth	191	(3)
6 Arguments against realism	194	(2)
7 Summary	196	(1)
Notes	196	(2)
Bibliography	198	(5)
III REALISM IN MATHEMATICS	203	(70)
11 A credo of sorts	203	(12)
V.F.R. Jones
Bibliography	214	(1)
12 Mathematical evidence	215	(18)
Donald A. Martin
1 Introduction	215	(1)
2 Proof and axioms	216	(3)
3 Mathematically proper evidence	219	(2)
4 Determinacy	221	(2)
5 Turing cones	223	(2)
6 Wadge degrees	225	(2)
7 Discussion	227	(3)
Bibliography	230	(3)
13 Mathematical definability	233	(20)
Theodore A. Slaman
1 Introduction	233	(2)
2 The intuitive hierarchy	235	(7)
3 Applications	242	(4)
4 The fine hierarchy	246	(3)
Bibliography	249	(4)
14 True to the pattern	253	(20)
Gianluigi Oliveri
1 Introduction	253	(1)
2 Traditional Kantian teachings concerning concepts and perception	254	(1)
3 Seeing or interpreting?	255	(2)
4 Internal relations	257	(2)
5 The metaphysics of experience	259	(2)
6 Truth	261	(3)
Notes	264	(4)
Bibliography	268	(5)
IV SETS, UNDECIDABILITY, AND THE NATURAL NUMBERS	273	(80)
15 Foundations of set theory	273	(18)
W. W. Tait
1 Iterative conception of set theory	273	(4)
2 First- and second-order reflection	277	(2)
3 Domains and the universe of sets	279	(4)
3.1 Domains	280	(2)
3.2 The universe of all sets	282	(1)
3.3 Higher-order objects	282	(1)
4 Higher-order reflection	283	(6)
5 Appendix: quotient and direct limit types	289	(1)
Notes	289	(1)
Bibliography	290	(1)
16 Which undecidable mathematical sentences have determinate truth values?	291	(20)
Hartry Field
1 Metaphysical preamble	291	(3)
2 The objectivity issue	294	(2)
3 Putnam's `Models and Reality' and the concepts of finiteness and natural number	296	(4)
4 Extreme anti-objectivism	300	(6)
Notes	306	(4)
Bibliography	310	(1)
17 Two conceptions of natural number	311	(18)
Alexander George
Daniel J. Velleman
Notes	318	(8)
Bibliography	326	(3)
18 The tower of Hanoi	329	(24)
W. Hugh Woodin
1 Introduction	329	(1)
2 Preliminaries	330	(2)
3 First-order logic	332	(2)
4 A definition of truth	334	(2)
5 A definition of proof	336	(2)
6 The formula	338	(7)
7 Evidence	345	(2)
8 The standard case, a second sentence	347	(2)
9 Final remarks	349	(1)
10 Appendix	350	(1)
Notes	351	(1)
Bibliography	351	(2)
Complete bibliography	353	(18)
Index	371