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. |
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abstract objects actual infinity analytic anti-realist argument arithmetic axiom of choice basic law cardinal numbers Chapter classical logic concepts constructive constructivism constructivist contradiction definition developed discuss disjunctive empiricist empty set epistemological example excluded middle exist false fiction finite first-order logic formalist Frege geometry Gödel Hellman Hilbert Husserl idea independent infinite cardinal infinite number infinite set intuitionist logic justify Lakatos language law of excluded logicist Maddy mathematical objects mathematical structures mathematical theory mathematical truths mathematicians meaning model theory natural numbers notion numbers principle objects of mathematics one-to-one correspondence ontology ordinals Oxford paradox philosophy of mathematics physical objects Platonism platonist possible worlds potential infinity problem proof properties proposition prove quantifier rational numbers real numbers realist reasoning rules of inference Russell's second-order logic semantic sentence set-theoretic sort structuralist subset symbols theorems triangle true truth-value type theory understanding University Press well-formed formula Whitehead and Russell Zermelo-Fraenkel set theory