Cover

Title Page

Copyright Page

General Preface

Contents

Preface

Contributors

Les Liaisons Dangereuses

Acknowledgements

Realism and Anti-Realism in Mathematics

1 A Survey of Positions

1.1 Mathematical Realism

1.1.1 Anti-platonistic realism (physicalism and psychologism)

1.1.2 Mathematical platonism

1.2 Mathematical Anti-Realism

2 Critique of the Various Views

2.1 Critique of Platonism

2.1.1 The Epistemological Argument Against Platonism

2.1.2 The Non-Uniqueness Objection to Platonism

2.1.3 Responses to Some Recent Objections to FBP-NUP

2.2 Critique of Anti-Platonism

2.2.1 Introduction: The Fregean Argument Against Anti-Platonism

2.2.2 Critique of Non-Fictionalistic Versions of Anti-Realistic Anti-Platonism

2.2.3 Critique of Realistic Anti-Platonism (i.e., Physicalism and Psychologism)

2.2.4 Indispensability

2.3 Critique of Platonism Revisited: Ockham's Razor

3 Conclusions: The Unsolvability of the Problem and a Kinder, Gentler Positivism

3.1 The Strong Epistemic Conclusion

3.2 The Metaphysical Conclusion

3.3 My Official View

Bibliography

Aristotelian Realism

1 Introduction

2 The Aristotelian Realist Point of View

3 Mathematics as the Science of Quantity and Structure

4 Necessary Truths About Reality

5 The Formal Sciences

6 Comparison with Platonism and Nominalism

7 Epistemology

8 Experimental Mathematics and Evidence for Conjectures

9 Conclusion

Bibliography

Empiricism in the Philosophy of Mathematics

1 Introduction

2 Aristotle

3 John Stuart Mill

3.1 Mill on geometry

Appendix: non-Euclidean geometry

3.2 Mill on arithmetic

4 Mill's Modern Supporters

4.1 Kitcher against apriorism

4.2 Kitcher on arithmetic

4.3 Maddy on arithmetic

5 Quine, Putnam and Field

5.1 The indispensability of mathematics

5.2 How much mathematics is indispensable?

5.3 Digression: Nominalism

5.4 Doing Without Numbers

6 Logic and Analysis

6.1 Analysis

6.2 Logic

6.3 Coda

Bibliography

A Kantian Perspective on the Philosophy of Mathematics

1 Mathematics, Science of Forms

2 Individual Objects — Why Mathematics Cannot Be Reduced to Logic

3 Formal Rules — Why Mathematics Cannot Be Reduced to Manipulation of Marks on Paper

4 Rules and Forms of Representation — Hilbertian Formalism

5 Axiomatization and Structures — Changing the Object of Mathematics

6 Does Set Theory Provide a Pure Theory of Manifolds?

7 Ordinal, Cardinal and Two Kinds of Infinite

8 Intuition and the Theory of Pure Manifolds

9 Manifolds as Aggregates

9.1 Aggregates as Manifolds

10 Maxima, Minima — Totalities and Quantifiers

11 What Is a Kantian Approach?

Appendix

A Non-Euclidean Geometry and Einstein's Relativity Theories

Bibliography

Logicism

1 What Is Logicism?

2 What Is Mathematics?

3 What Is the Logic of Logicism?

4 Frege the First Logicist

5 Frege's Logic of Quantifiers

6 Defining Real Numbers

7 Frege's Higher-Order Logic

8 Axiomatic Set Theory vs. Logicism

9 Principia Mathematica and its Aftermath

10 Logicism vs. Metamathematics

11 The Transformation of Logicism

12 Correcting Frege's Theory of Quantification

13 Reduction to the First-Order Level

14 Logicism Vindicated?

Bibliography

Formalism

1 Preliminaries

1.1 Problem of Definition

1.2 Hilbert

1.3 Working Mathematicians

2 The Old Formalism and its Refutation

2.1 Contentless Manipulation

2.2 Frege's Critique

3 The New Axiomatics

3.1 Hilbert's Grundlagen

3.2 Implicit Definition and Contextual Meaning

3.3 Dispute with Frege

3.4 The Axioms of Real Numbers

4 The Crisis of Content

4.1 Logicism's Waterloo and other Paradoxes

4.2 Self-Restriction

5 The Classical Period

5.1 Preparations

5.2 Hilbert's Maximal Conservatism

5.3 Finitism

5.4 Syntacticism and Meaning

6 Gödel's Bombshell

7 The Legacy of Formalism

7.1 Proof Theory

7.2 Consistency Proofs

7.3 Bourbakism

8 Conclusion

Bibliography

Constructivism in Mathematics

1 Introduction: Varieties of Constructivism

1.1 Crucial Statements

1.2 Appropriateness

1.3 Constructions

1.4 Constructive Logics and Proof Conditions

1.5 Formalizations of Constructive Logic and Mathematics

2 Constructivism in the 19th Century: du Bois-Reymond and Kronecker

2.1 Paul du Bois-Reymond

2.2 Leopold Kronecker

3 Intuitionism and L. E. J. Brouwer

4 Heyting and Formal Intuitionistic Logic

5 Markovian or Russian Constructivism

6 Bishop's New Constructivism

7 Predicativism

8 Finitism

Bibliography

Fictionalism

1 Kinds of Fictionalism

1.1 Truth

1.2 Interpretation

1.3 Elimination

2 Motivations for Fictionalism in the Philosophy of Mathematics

2.1 Benacerraf's Dilemma

2.2 Yablo's Comparative Advantage Argument

3 A Brief History of Fictionalism

3.1 William of Ockham: Reductive Fictionalism

3.2 Jeremy Bentham: Instrumentalist Fictionalism

3.3 C. S. Peirce: Representational Fictionalism

3.4 Hans Vaihinger: Free-range Fictionalism

4 Science Without Numbers

5 Balaguer's Fictionalism

6 Yablo's Figuralism

7 Semantic Strategies

7.1 Truth in a Fiction

7.2 Constructive Free-range Fictionalism

7.3 Definitions

7.4 Settled Models

7.5 Minervan Constructions

7.6 Marsupial Constructions

7.7 Open Models

7.8 Modal Translations

Bibliography

Set Theory from Cantor to Cohen

1 Cantor

1.1 Real Numbers and Countability

1.2 Continuum Hypothesis and Transfinite Numbers

1.3 Diagonalization and Cardinal Numbers

2 Mathematization

2.1 Axiom of Choice and Axiomatization

2.2 Logic and Paradox

2.3 Measure, Category, and Borel Hierarchy

2.4 Hausdorff and Functions

2.5 Analytic and Projective Sets

2.6 Equivalences and Consequences

3 Consolidation

3.1 Ordinals and Replacement

3.2 Well-Foundedness and the Cumulative Hierarchy

3.3 First-Order Logic and Extensionalization

3.4 Relative Consistency

3.5 Combinatorics

3.6 Model-Theoretic Methods

4 Independence

4.1 Forcing

4.2 Envoi

Acknowledgements

Bibliography

Alternative Set Theories

Introduction

Part I: Topological Solutions to the Fregean Problem

1 The Naïve Notion of Set

2 The Abstraction Process

3 Sets and Membership

4 First-Order Versions

5 Russell's Paradox

6 Solution Routes

7 Frege Structure

8 The Limitation of Size Doctrine

9 Adding Structure

10 Topology and Indiscernibility

11 Indiscernibility as a Lightning Discharger (?)

Part II: Partial, Paradoxical and Double Sets

12 Introduction

13 Partial Sets

14 Positive Sets

15 Paradoxical Sets

16 Double Sets

Part III: Proximity Spaces of Exact Sets

17 Introduction

18 Towards Modal Set Theory

19 Proximity Structures

20 Proximal Frege Structures

21 The Ortholattice of Exact Sets

22 Models of PFS

23 On the Discernibility of the Disjoint

24 Plenitude

25 Conclusion

Bibliography

Philosophies of Probability

1 Introduction

Part I: Frameworks for Probability

2 Variables

3 Events

4 Sentences

Part II: Interpretations of Probability

5 Interpretations and Distinctions

6 Frequency

7 Propensity

8 Chance

9 Bayesianism

10 Chance as Ultimate Belief

11 Applying Probability

Part III: Objective Bayesianism

12 Subjective and Objective Bayesianism

13 Objective Bayesianism Outlined

14 Challenges

15 Motivation

16 Language Dependence

17 Computation

18 Qualitative Knowledge

19 Infinite Domains

20 Fully Objective Probability

21 Probability Logic

Part IV: Implications for the Philosophy of Mathematics

22 The Role of Interpretation

23 The Epistemic View of Mathematics

24 Conclusion

Acknowledgements

Bibliography

On Computability

1 Introduction

1.1 Foundational contexts

1.2 Overview

1.3 Connections

2 Decidability and Calculability

2.1 Decidability

2.2 Finitist mathematics

2.3 (Primitive) Recursion

2.4 Formalizability and calculability

3 Recursiveness and Church's Thesis

3.1 Relative consistency

3.2 Uniform calculations

3.3 Elementary steps

3.4 Absoluteness

3.5 Reckonable functions

4 Computations and Combinatory Processes

4.1 Machines and workers

4.2 Mechanical computors

4.3 Turing's central thesis

4.4 Stronger theses

4.5 Machine computability

5 Axioms for Computability

5.1 Discrete dynamical systems

5.2 Gandy machines

5.3 Global assembly

5.4 Models

5.5 Tieferlegung

6 Outlook on Machines and Mind

6.1 Mechanical computability

6.2 Beyond calculation

6.3 Beyond discipline

6.4 (Supra-) Mechanical devices

Acknowledgments

Bibliography

Inconsistent Mathematics: Some Philosophical Implications

1 Introduction: The Paradoxes

2 The Role of Logic

3 Pure Mathematics

4 Geometry

5 Applied Mathematics

6 Logicism and Foundationalism Revisited

7 Revisionism and Duality

8 The Role of Text

9 Conclusions

Bibliography

Mathematics and the World

1 The Indispensability Argument

1.1 Realism and Anti-realism in Mathematics

1.2 Indispensability Arguments

2 What Is it to Be Indispensable?

3 Naturalism and Holism

3.1 Introducing Naturalism

3.2 Quinean Naturalism

3.3 Holism

3.4 The First Premise Revisited

4 The Hard Road to Nominalism: Field's Project

4.1 Science without Numbers

5 The Easy Road to Nominalism: Rejecting Holism

5.1 Maddy

5.2 Sober

6 The Unreasonable Effectiveness of Mathematics

6.1 What is the Puzzle?

6.2 Is the Puzzle Due to a Particular Philosophy of Mathematics?

7 Applied Mathematics: The Philosophical Lessons and Future Directions

Bibliography

Index