Preface 7
Contents 11
Contributors 14
Part I History and Philosophy of Mathematics and Logic 16
1 On Logicist Conceptions of Functions and Classes 17
William Demopoulos 17
1 Introduction 17
2 The Tractarian Background to Ramsey's Extensional Propositional Functions 21
3 Infinity Without Extensional Propositional Functions 22
4 Infinity with Extensional Propositional Functions 25
5 Extensional Propositional Functions and Logicism 29
Bibliography 31
2 Metaphysical Necessity 33
Michael Dummett 33
3 Ibn S谋na and Conflict in Logic 49
Wilfrid Hodges 49
1 To John 49
2 De Interpretatione Chapter 14 50
2.1 The Text Translated Below 50
2.2 Ways of Teaching 51
2.3 Summary of Aristotle's Text 52
3 Ibn S谋na's Semantics 54
3.1 Ideas 54
3.2 Attachments 56
3.3 Meanings of Sentences 59
4 Conflict and Contrariety 60
4.1 Types of Opposition 60
4.2 Aristotle's Question in PH14 61
4.3 The Quickest Contradiction 62
5 The ``Well-Known Commentator'' 63
5.1 Ibn S谋na's Outrageous Analysis 63
5.2 Why? 65
5.3 Who Was the ``Well-Known Commentator''? 65
6 Translation 67
7 Notes 74
Bibliography 81
4 A Minimalist Foundation at Work 82
Giovanni Sambin 82
1 Foundations as a Choice of Abstraction Level 83
1.1 Different Constructive Foundations and Their Incompatibility 83
1.2 The Minimalist Foundation of Constructive Mathematics 85
2 A Minimalist Foundation at Work 90
2.1 Dynamics Between Two Sets 91
2.2 The Definition of Basic Topology 96
3 A Different Topology in Different Foundations 98
4 Benefits of the Minimalist Foundation 102
4.1 Effective Computations and Ideal Structures in One Framework 102
4.2 Creation of New Mathematics 103
4.3 Pluralism as a Source of Richness 106
Bibliography 108
5 The Municipal By-Laws of Thought 110
David DeVidi 110
1 JLB 110
2 The Problem 111
3 The Obvious Approach? 113
4 An Uninteresting Plurality? 115
5 What Makes It Logic? 118
6 The Systems Disagree 120
7 The Systems Are Correct 123
8 Conclusion 124
Bibliography 124
Part II Truth, Consistency and Paradox 126
6 Truth and the Liar 127
Colin Howson 127
1 Introduction 127
2 The Liar Paradox 129
3 Groundedness 137
4 The Strengthened Liar 139
5 Conclusion 144
Bibliography 145
7 Necessary and Sufficient Conditions for Undecidability of the G枚del Sentence and its Truth 147
Daniel Isaacson 147
1 Introduction 147
2 Soundness and Consistency 148
3 Truth of the G枚del Sentence 150
4 G枚del's Notion of 蠅-Consistency 151
5 Kreisel's Notions of n-Consistency 155
5.1 The Notion of 1-Consistency 155
5.2 The Notion of 2-Consistency 158
5.3 Properties of 3-Consistency 160
5.4 Expressing Consistency, 蠅-Consistency, 1-Consistency, and 2-Consistency 160
6 Consistency of S {ConS} 162
7 Comparing G枚del's Incompleteness Theorems with Rosser's Theorem 164
Bibliography 164
8 Paraconsistent Set Theory 165
Graham Priest 165
1 Introduction: For JLB 165
2 Paraconsistent Set Theory: Background 167
3 The Material Strategy 168
4 The Relevant Strategy 170
5 The Model-Theoretic Strategy 175
6 Metatheory of Paraconsistent Logic 177
7 Conclusion: The Shock of the New 179
Bibliography 180
9 Paradox, ZF, and the Axiom of Foundation 182
Adam Rieger 182
1 182
2 183
3 184
4 186
5 188
6 189
7 190
8 191
9 195
10 196
Bibliography 196
10 Absoluteness and the Skolem Paradox 199
Michael Hallett 199
1 Introduction 199
2 ``Impredicative'' Extensions 201
3 The Paradox Internalised 210
4 Absoluteness, Consistency and Independence 217
5 Conclusion 225
Bibliography 225
Part III Logic, Set Theory and Category Theory 229
11 Equalisers of Frames in Constructive Set Theory 230
Peter Aczel 230
1 Introduction 230
2 Proof of Lemma 2 232
3 Proof of Lemma 3 234
Bibliography 237
12 Analogy and Its Surprises: An Eyewitness's Reflections on the Emergence of Real Algebraic Geometry 238
Max Dickmann 238
1 Introduction 238
2 The Late Development of Real Algebraic Geometry 239
3 The Sparse Pieces of a Puzzle 241
3.1 The (Slow) Progress of Real Geometry 241
4 The Puzzle Solved: Invention of the Real Spectrum 244
4.1 Zariski Spectra, Grothendieck Topologies and 茅tale Schemes 244
4.2 ``Real'' Variations on Grothendieck's Theme 247
5 The Inception of Real Algebraic Geometry 249
5.1 Preliminary Remarks 249
5.2 A Simplification 250
5.3 The Geometric Case 250
5.4 Connectedness 252
5.5 Dimension 253
5.6 The Implicit Function Theorem 253
5.7 The Real Spectrum as a Ringed Space 254
5.8 Idempotency of the Real Spectrum and the Substitution Lemma 254
5.9 A Beautiful Application: The Br枚cker--Scheiderer Theorem 255
Bibliography 256
13 Euler's Continuum Functorially Vindicated 257
F. William Lawvere 257
Bibliography 262
14 Natural Numbers and Infinitesimals 263
J.P. Mayberry 263
1 G枚del on Nonstandard Analysis 263
2 Natural Numbers or Finite Sets? 265
3 Euclidean Arithmetic 268
4 Arithmetical Functions and Relations 272
5 Simply Infinite Systems in Euclidean Arithmetic 274
6 Rational Numbers 282
7 Real Numbers 283
8 The Calculus 286
9 A Finitary Version of the Reals 291
10 Conclusions 292
Bibliography 294
15 Logic in Category Theory 295
Alberto Peruzzi 295
1 Motivations 295
2 Perspective Remarks 296
3 The Working Mathematician's Path to Logic 307
4 Layers of Logical Structure 312
5 Further Connections with Type Theoriesand Concluding Remarks 322
Bibliography 329
Part IV Philosophy of Science 333
16 Gunk, Topology and Measure 334
Frank Arntzenius 334
1 Introduction 334
2 The Possibility of Motion and Determinism 335
3 Cutting Things in Half 336
4 Paradoxes of Size 337
5 Quantum Mechanics and Points 338
6 Contact Between Objects 340
7 Now What? 341
8 The Topological Approach to Pointless Spaces 341
9 Objects in a Pointless Topology 344
10 Fields in a Pointless Topology 345
11 The Measure-Theoretic Approach to Pointless Spaces 348
Bibliography 350
17 Pitholes in Space-Time: Structure and Ontology of Physical Geometry 351
Robert DiSalle 351
1 Introduction: The Philosophy of Space-Timeand the Foundations of Mathematics 351
2 Origins of an Ontological Problem 352
3 The Ontological Problem in Its Present State 354
4 The Structure of Space-Time and the Ontologyof the Space-Time Manifold 358
5 Conclusion: Structure from a Pithole 364
Bibliography 365
18 A Silly Answer to a Psillos Question 367
Elaine Landry 367
1 A Psillos Question 368
2 A Category-Theoretic Answer to the Mathematical Structuralist 370
3 Why This Answer Is Silly for a Scientific Structuralist 374
4 An Answer for the Methodological Scientific Structuralist 379
Bibliography 385
19 The Vacuum in Antiquity and in Modern Physics 387
Michael Redhead 387
20 Falsifiability, Empirical Content and the Duhem-Quine Problem 391
Elie Zahar 391
1 Historical Background 391
2 Induction and Direct Proof 392
3 Indirect Method and Crucial Experiments 393
4 Indirect Method and the Duhem-Quine Problem 394
5 Kripke's Challenge 396
6 Quine's Holism and the Analytic-Synthetic Distinction 400
7 Empirical Content and the Duhem-Quine Problem 403
8 Is There a Bayesian Solution of the Duhem-Quine Problem? 406
9 Conclusion 409
Bibliography 410
Part V Decision Theory and Epistemology 411
21 Can Knowledge Be Justified True Belief? 412
Ken Binmore 412
1 Fallibilism or Skepticism? 412
2 Small World Assumptions 413
3 Justification 414
4 What Is an Event? 415
5 All-Encompassing Worlds 416
6 Conclusion 416
Bibliography 417
22 The Stochastic Concept of Economic Equilibrium: A Radical Alternative 418
Mosh茅 Machover 418
1 Introduction 418
1.1 The Equilibrium Price--Profit Equation 419
1.2 The Uniformity Assumption 420
1.3 The Transformation Problem 421
2 Stochastic Equilibrium 422
2.1 Equilibrium in Statistical Mechanics 423
2.2 Economic Equilibrium 424
3 Methodological Comments 425
Bibliography 426
23 Collective Choice as Information Theory: Towards a Theory of Gravitas 427
George Wilmers 427
1 What Can We Learn from a Council of Elders? 427
2 The Notion of Gravitas: A Preliminary Discussion 428
3 Axioms for Gravitas 432
4 Polarity Free Entropy (PFE) 437
5 Conclusions and Open Problems 438
Bibliography 442
24 Scientific Knowledge and Structural Knowledge 443
Peter Clark 443
1 Introduction 443
2 The Modest Defence of Scientific Structural Realism 445
3 Ontic Structural Realism 448
Bibliography 450
Part VI Aesthetics 452
25 Musing on Music 453
Richard Feist 453
1 Introduction: Boethius on Music 453
2 Formalized Music and the Renaissance 455
3 The Turning Point in Aesthetics: Representation 457
4 The Nineteenth Century 458
5 The Twentieth and Twenty-First Centuries 460
6 Representation 463
7 Summary and Conclusion 466
Bibliography 467
26 Inscrutable Harmonies: The Continuous and the Discrete as Reflected in the Playing of Jascha Heifetz and Glenn Gould 469
Joel Bennhall 469
Publications of John L. Bell 474
Name Index 481