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 | |
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
able abstract acceptable activity actual allows answer argument arithmetic axioms basic law called cardinality Chapter choice classical logic computers concepts concerning consider consistent constructive constructivist contradiction contrast count definition depend derive developed discuss example exist experience expression fact false fiction finite follows formal system formalist formula Frege further geometry give idea important independent inference infinite infinite number infinite set infinity interested intuition intuitionist intuitionist logic knowledge language look mathematicians meaning natural numbers notion numbers principle objects ontology Oxford paradox particular philosophical philosophy of mathematics physical Platonism position possible Press principle problem proof propositional prove quantifier question rational realist reasoning reducing refer relation rules Russell sense sentence set theory simply sort structuralist structures symbol talk tell theorems things thought tion true truths understanding University