Introducing 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 of Philosophy at George Washington University, Washington DC

Acknowledgements	p. vii
Preface	p. ix
Infinity	p. 1
Introduction	p. 1
Zeno's paradoxes	p. 2
Potential versus actual infinity	p. 7
The ordinal notion of infinity	p. 12
The cardinal notion of infinity	p. 13
Summary	p. 22
Mathematical Platonism and realism	p. 23
Introduction	p. 23
Historical origins	p. 23
Realism in general	p. 26
Kurt Godel	p. 35
Penelope Maddy	p. 37
General problems with set-theoretic realism	p. 41
Conclusion	p. 46
Summary	p. 47
Logicism	p. 49
Introduction	p. 49
Frege's logicism: technical accomplishments	p. 52
Frege's logicism: philosophical accomplishments	p. 58
Problems with Frege's logicism	p. 63
Whitehead and Russell's logicism	p. 66
Philosophically, what is wrong with Whitehead and Russell's type theory?	p. 71
Other attempts at logicism	p. 78
Conclusion	p. 78
Summary	p. 79
Structuralism	p. 81
Introduction	p. 81
The motivation for structuralism: Benacerraf's puzzle	p. 83
The philosophy of structuralism: Hellman	p. 85
The philosophy of structuralism: Resnik and Shapiro	p. 90
Critique	p. 96
Summary	p. 100
Constructivism	p. 101
Introduction	p. 101
Intuitionist logic	p. 106
Prima facie motivations for constructivism	p. 113
Deeper motivations for constructivism	p. 114
The semantics of intuitionist logic: Dummett	p. 121
Problems with constructivism	p. 123
Summary	p. 124
A pot-pourri of philosophies of mathematics	p. 127
Introduction	p. 127
Empiricism and naturalism	p. 130
Fictionalism	p. 134
Psychologism	p. 137
Husserl	p. 141
Formalism	p. 147
Hilbert	p. 153
Meinongian Philosophy of Mathematics	p. 157
Lakatos	p. 163
Proof: ex falso quod libet	p. 167
Glossary	p. 169
Notes	p. 177
Guide to further reading	p. 191
Bibliography	p. 195
Index	p. 201
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Introducing Philosophy of Mathematics

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