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• Michele Friend　　Introducing Philosophy of Mathematics, McGill-Queen's University Press/Acumen Publishing, 2007

A concise introduction to the central debates in the philosophy of mathematics.

Michele Friend provides an introduction to the standard theories of mathematics - platonism and realism, logicism, formalism, constructivism, and structuralism - as well as to some of the less standard theories, such as psychologism, fictionalism, and Meinongian philosophy of mathematics.

The author explains what characterises each theory, the differences between them, and some of the arguments in favour of and against the different positions. Introducing Philosophy of Mathematics also explores questions that occupy present-day philosophers and mathematicians, such as the relationship between good reasoning and mathematics, the problem of infinity, and whether we are more certain of mathematics than we are of everyday sense experience or science. Friend strikes a nice balance between conceptual accessibility and clear representation of the issues to enable readers to challenge existing positions.

Michele Friend is assistant professor, philosophy, George Washington University.

Table of Contents

1.Infinity 1

1. Introduction 1
2. Zeno's paradoxes 2
3. Potential versus actual infinity 7
4. The ordinal notion of infinity 12
5. The cardinal notion of infinity 13
6. Summary 22

2. Mathematical Platonism and realism 23

1. Introduction 23
2. Historical origins 23
3. Realism in general 26
4. Kurt Godel 35
5. Penelope Maddy 37
6. General problems with set-theoretic realism 41
7. Conclusion 46
8. Summary 47

3. Logicism 49

1. Introduction 49
2. Frege's logicism: technical accomplishments 52
3. Frege's logicism: philosophical accomplishments 58
4. Problems with Frege's logicism 63
5. Whitehead and Russell's logicism 66
6. Philosophically, what is wrong with Whitehead and Russell's type theory? 71
7. Other attempts at logicism 78
8. Conclusion 78
9. Summary 79

4. Structuralism 81

1. Introduction 81
2. The motivation for structuralism: Benacerraf's puzzle 83
3. The philosophy of structuralism: Hellman 85
4. The philosophy of structuralism: Resnik and Shapiro 90
5. Critique 96
6. Summary 100

5. Constructivism 101

1. Introduction 101
2. Intuitionist logic 106
3. Primafacie motivations for constructivism 113
4. Deeper motivations for constructivism 114
5. The semantics of intuitionist logic: Dummett 121
6. Problems with constructivism 123
7. Summary 124

6. A pot-pourri of philosophies of mathematics 127

1. Introduction 127
2. Empiricism and naturalism 130
3. Fictionalism 134
4. Psychologism 137
5. Husserl 141
6. Formalism 147
7. Hilbert 153
8. Meinongian Philosophy of Mathematics 157