Table of Contents
Preface ix
Duality, Epistemic Efficiency & Consistency Michael Detlefsen 1
1 Introduction 1
2 Abstract Duality or Dualization? 5
3 The Contentual Addition Model of Dualization 7
4 Proofs & Proof Developments 8
5 The Contentual Addition Model & The Traditional Contentualist View of Proof 10
6 Contentual Addition in an Abstract Setting 12
7 Non-Trivial Axiom Systems 17
8 Conclusion 19
Frege on Quantities and Real Numbers in Consideration of the Theories of Cantor, Russell and Others Matthias Schirn 25
1 Introduction 26
2 The concept of quantity in Frege's writings between 1874 and 1884 35
3 Cantor's theory of irrational numbers and Frege's critique 49
4 Russell on quantities and real numbers in Principles of Mathematics and Principia Malhematica 56
5 Quantities and real numbers in Grundgesetze 59
6 Frege's plan carried out: von Kutschcra's account 89
Frege on Formality and the 1906 Independence-Test Patricia A. Blanchette 97
1 Introduction 97
2 The Proposal 98
3 The Import of the 1910 Notes 104
4 The Anti-Metatheory Explanation 107
5 The Similarity with Hilbert 110
6 Conclusion 115
Formal Discourse in Russell: From Metaphysics to Philosophical Logic Godehard Link 119
1 Introduction 119
2 Setting the Stage: Russell's Early Ontology 124
3 On the Nature of Functions 139
4 The Substitutional Theory 152
5 Principia Mathematica 155
6 Ramification: Gödel's Gestalt Switch 163
7 Lessons for Ontology 166
8 Conclusion 175
On Live and Dead Signs in Mathematics Felix Mühlhöker 183
1 A Mess Concerning the Reference. Interpretation and Application 184
2 How can Intended Models be Singled Out? 195
3 Strings of Strokes in Hilbertian Finitism 201
Generalization and the Impossible Paul Ztche 209
1 "Contradictions are emotions": The example of the complex numbers 209
2 Russell on symbolism: Making the simple complicated 214
3 Ways into logic: Generalization and abstraction 215
4 " Pure logic and meta-scientific induction 219
5 Scientistic liberalism and interesting generalizations 221
Assumptions of Infinity Karl-Georg Niebergall 229
1 Introduction 229
2 "I makes an assumption of infinity": "I assumes merely the finite" 234
3 Expressing infinity: a preliminary suggestion 237
4 Axioms for and definitions of "finite" 242
5 Elaboration of (DIiii) 250
6 The potentially infinite 256
7 Conclusion 261
8 Appendix 266
The Interpretation of Classes in Axiomatic Set Theory Daniel Roth Gregor Schneider 275
1 Introduction 275
2 Set Theories 275
3 Interpreting Classes 296
4 Concluding Remarks 308
Purity in Arithmetic: some Formal and Informal Issues Andrew Arana 315
1 Introduction 315
2 Topical purity 316
3 The infinitude of primes 318
4 Incompleteness and the possibility of purity 331
5 Closing thoughts 333
Domain Extensions and Higher-Order Syntactical Interpretations Marek Polanski 337
1 Introductory remarks 337
2 Domain extensions: some paradigmatic examples 338
3 L-operations and L-constructions 340
4 Higher-order syntactical interpretations and their constructions 344
5 Concluding remarks 349
Finite Methods in Mathematical Practice Laura Crosilla Peter Schuster 351
1 Introduction 351
2 Hilbert's programme now and then 352
3 Finite methods for constructive algebra 365
4 Geometric formulas and dynamical proofs 369
5 Realising Hubert's programme in commutative algebra 372
6 Appendix 398
List of Contributors 411
Name Index 413
