18 Unconventional Essays on the Nature of Mathematics

Cover Page......Page 1
Title Page......Page 5
ISBN: 0387257179......Page 6
Contents......Page 7
Introduction......Page 9
About the Authors......Page 19
1 A Socratic Dialogue on Mathematics......Page 25
2 “Introduction” to Filosofia e matematica......Page 41
1. What is it that mathematicians accomplish?......Page 61
2. How do people understand mathematics?......Page 63
3. How is mathematical understanding communicated?......Page 66
4. What is a proof?......Page 69
5. What motivates people to do mathematics?......Page 72
6. Some personal experiences......Page 74
4 The Informal Logic of Mathematical Proof......Page 80
1. Toulmin’s pattern of argument......Page 81
2 Applying Toulmin to mathematics......Page 82
3 Walton’s new dialectic......Page 85
4 Applying Walton to mathematics......Page 88
5 Proof dialogues......Page 92
Introduction......Page 95
The Main Tenets of Evolutionary Epistemology......Page 97
Some Perennial Questions in the Philosophy of Mathematics......Page 99
MATHEMATICS AND REALITY......Page 102
The Trilemma of a Finitary Logic and Infinitary Mathematics......Page 106
Invention Versus Discovery......Page 108
Recapitulation and Concluding Remarks......Page 110
NOTES......Page 112
6 Toward a Semiotics of Mathematics......Page 121
Introduction......Page 123
A semiotic model of mathematics......Page 126
Formalism, intuitionism, platonism......Page 135
What Is mathematics ‘about’?......Page 147
1. Introduction......Page 152
2. Computers, Inductive Testing and Deductive Proof......Page 154
3. Proofs Using Computers......Page 156
4. A Prediction and its (Near) Confirmation......Page 157
5. Mathematical Proof, Intentions and the Material World......Page 158
6. Formal Proof and Rigorous Argument......Page 159
7. Proof and Disciplinary Authority......Page 162
8. Logics, Bugs and Certainty......Page 163
9. Conclusion......Page 166
10. Appendix: An Outline Chronology......Page 167
Hardy and Littlewood: A Study in Collaboration......Page 171
Towards Foundational Pluralism.........Page 172
The Mathematical Community......Page 174
From Scribal Culture to Typographic Culture......Page 177
From Typographic Culture to Electronic Culture......Page 178
Implications and a Proposal......Page 180
Sequel.........Page 182
Conclusion......Page 183
1. A challenge to embodiment: The nature of Mathematics......Page 184
a) Limits of infinite series......Page 186
c) Continuity......Page 187
3. Looking at pure Mathematics......Page 189
4. Embodied Cognition......Page 193
5. Fictive Motion......Page 195
5. Dead Metaphors?......Page 196
6. Gesture as Cognition......Page 197
7. Conclusion......Page 202
Introduction......Page 206
1. What is 2+2?......Page 208
2. The empty set......Page 210
3. Subsets of the natural numbers......Page 211
4. Some terminology......Page 213
5. Ordered pairs......Page 215
6. Truth and provability......Page 217
7. The axiom of choice......Page 220
8. Concluding remarks......Page 222
Brief additions in response to the discussion after the talk......Page 223
I......Page 225
II......Page 226
III......Page 232
IV......Page 233
V......Page 236
VI......Page 239
The Double Life of Mathematics......Page 244
The Double Life of Philosophy......Page 245
The Loss of Autonomy......Page 246
Mathematics and Philosophy: Success and Failure......Page 247
Misunderstanding the Axiomatic Method......Page 248
“Define Your Terms!”......Page 249
The Appeal to Psychology......Page 250
The Reductionist Concept of Mind......Page 251
The Illusion of Definitiveness......Page 252
NOTES......Page 254
13 The Pernicious Influence of Mathematics on Science......Page 255
1. Dialogue of the deaf......Page 260
2. Mathematical reasoning about mathematics......Page 262
3. Philosophical reasoning about mathematics......Page 266
4. Conclusion......Page 271
15 Concepts and the Mangle of Practice Constructing Quaternions......Page 274
1. Disciplinary Agency......Page 275
2. From Complex Numbers to Triplets......Page 281
3. Constructing Quaternions......Page 287
4. Concepts and the Mangle......Page 297
5. Science and The Mangle......Page 301
6. Mathematics, Metaphysics and the Social......Page 304
16 Mathematics as Objective Knowledge and as Human Practice......Page 313
1. Fallibilism......Page 314
2. Objectivity......Page 315
3. Interaction......Page 317
4. Popperian Dialectic......Page 319
5. Socially Conditioned Change......Page 321
6. Alternative Practices......Page 323
7. Concluding Discussion......Page 325
17 The Locus of Mathematical Reality: An Anthropological Footnote......Page 328
18 Inner Vision, Outer Truth......Page 344