The Search for Certainty

Marcus Giaquinto traces the story of the search for firm foundations for mathematics. The nineteenth century saw a movement to make higher mathematics rigorous; this seemed to be on the brink of success when it was thrown into confusion by the discovery of the class paradoxes. That initiated a period of intense research into the foundations of mathematics, and with it the birth of mathematical logic and a new, sharper debate in the philosophy of mathematics. The Search for Certainty focuses mainly on two major twentieth-century programmes: Russell's logicism and Hilbert's finitism. Giaquinto examines their philosophical underpinnings and their outcomes, asking how successful they were, and how successful they could be, in placing mathematics on a sound footing. He sets these questions in the context of a clear, non-technical exposition and assessment of the most important discoveries in mathematical logic, above all Godel's underivability theorems. More than six decades after those discoveries Giaquinto asks what our present perspective should be on the question of certainty in mathematics. Taking recent developments into account, he gives reasons for a surprisingly positive response.

ISBN 0198752458   Pages 298 ISBN13 9780198752455 (What's this?)   Volumes 1 Publisher Oxford University Press   Weight (grammes) 472 Imprint Oxford University Press   Published in Oxford Format Paperback   Height (mm) 234 Publication date 27 May 2004   Width (mm) 156 Library of Congress QA8.4   Spine width (mm) 17 DEWEY 510.1   Academic level Professional / Scholarly DEWEY edition DC22

I		Setting		1
1		Clarifying mathematical analysis		3
2		Numbers and classes		15
II		The class paradoxes and early responses		35
1		The class paradoxes		37
2		Cantor's approach to the class paradoxes		42
3		Frege's logicism and his response to Russell's paradox		49
4		Type theory as a response to the class paradoxes		58
III		The language paradoxes and Principia Mathematica		67
1		The definability paradoxes and the vicious circle principle		69
2		Principia Mathematica		85
3		Paradoxes of truth		98
4		Ramsey's attempt to rescue logicism		104
IV		Axiomatic set theory and Hilbert's programme		117
1		Zermelo's axiomatic set theory		119
2		Blitz on paradise		130
3		Hilbert's finitism		142
4		Hilbert's programme		158
V		Godel's underivability theorems		165
1		Incompleteness, and undefinability of truth		167
2		Underivability of 'consistency'		182
VI		Aftermath		199
1		Paradise restored?		201
2		Solving the class paradoxes		214
3		The search for certainty : a reckoning		222
4		Outlook		230

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