Acknowledgments ................................................ xi
Introduction ................................................. xiii
I Proof and How it is Changing ................................ 1
1 Proof: Its Nature and Significance, Michael Detlefsen .... 3
2 Implications of Experimental Mathematics for the
Philosophy of Mathematics, Jonathan Borwein ............. 33
3 On the Roles of Proof in Mathematics, Joseph
Auslander ............................................... 61
II SociaC Constructivist Views of Mathematics ................. 79
4 When Is a Problem Solved?, Philip J. Davis .............. 81
5 Mathematical Practice as a Scientific Problem, Reuben
Hersh ................................................... 95
6 Mathematical Domains: Social Constructs?, Julian
Cole ................................................... 109
III The Nature of Mathematical Objects and Mathematical
Knowledge ................................................. 129
7 The Existence of Mathematical Objects, Charles
Chihara ................................................ 131
8 Mathematical Objects, Stewart Shapiro .................. 157
9 Mathematical Platonism, Mark Balaguer .................. 179
10 The Nature of Mathematical Objects, Øystein Linnebo .... 205
11 When is One Thing Equal to Some Other Thing?, Barry
Mazur .................................................. 221
IV The Nature of Mathematics and its Applications ............ 243
12 Extreme Science: Mathematics as the Science of
Relations as Such, R.S.D. Thomas ....................... 245
13 What is Mathematics? A Pedagogical Answer to a
Philosophical Question, Guershon Harel ................. 265
14 What Will Count as Mathematics in 2100?, Keith
Devlin ................................................. 291
15 Mathematics Applied: The Case of Addition, Mark
Steiner ................................................ 313
16 Probability - A Philosophical Overview, Alan Hájek ..... 323
Glossary of Common Philosophical Terms ........................ 341
About the Editors ............................................. 345
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