Introducing Philosophy of MathematicsWhat is mathematics about? Does the subject-matter of mathematics exist independently of the mind or are they mental constructions? How do we know mathematics? Is mathematical knowledge logical knowledge? And how is mathematics applied to the material world? In this introduction to the philosophy of mathematics, Michele Friend examines these and other ontological and epistemological problems raised by the content and practice of mathematics. Aimed at a readership with limited proficiency in mathematics but with some experience of formal logic it seeks to strike a balance between conceptual accessibility and correct representation of the issues. Friend examines the standard theories of mathematics - Platonism, realism, logicism, formalism, constructivism and structuralism - as well as some less standard theories such as psychologism, fictionalism and Meinongian philosophy of mathematics. In each case Friend explains what characterises the position and where the divisions between them lie, including some of the arguments in favour and against each. This book also explores particular questions that occupy present-day philosophers and mathematicians such as the problem of infinity, mathematical intuition and the relationship, if any, between the philosophy of mathematics and the practice of mathematics. Taking in the canonical ideas of Aristotle, Kant, Frege and Whitehead and Russell as well as the challenging and innovative work of recent philosophers like Benacerraf, Hellman, Maddy and Shapiro, Friend provides a balanced and accessible introduction suitable for upper-level undergraduate courses and the non-specialist. |
Contents
1 Infinity | 1 |
2 Mathematical Platonism and realism | 23 |
3 Logicism | 49 |
4 Structuralism | 81 |
5 Constructivism | 101 |
6 A potpourri of philosophies of mathematics | 127 |
Proof ex falso quod libet | 167 |
Glossary | 169 |
Notes | 177 |
Guide to further reading | 191 |
| 195 | |
| 201 | |
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Common terms and phrases
abstract objects actual infinity analytic anti-realist argument arithmetic axiom of choice basic law cardinal numbers Chapter classical logic computers concepts consistent constructive constructivism constructivist contradiction definition developed discuss disjunctive ematics empty set epistemology example excluded middle exist false fiction finite first-order logic formal system formalist Frege give Gödel Hellman Hilbert Husserl idea independent infinite cardinal infinite number infinite set intuitionist logic Kant Lakatos language law of excluded logicist Maddy math mathematical objects mathematical theory mathematical truths mathematicians meaning model theory natural numbers notion numbers principle one-to-one correspondence ontology ordinals Oxford paradox philosophy of mathematics physical objects Platonism platonist possible worlds potential infinity problem proof prove quantifier rational numbers real numbers realist reasoning rules of inference second-order logic semantic sentence sort structuralist subset symbol theorems tion true truth-value type theory understanding University Press well-formed formula Whitehead and Russell Zermelo–Fraenkel set theory