Philosophy of Mathematics
One of the most striking features of mathematics is the fact that we are much more certain about the mathematical knowledge we have than about what mathematical knowledge is knowledge of. Are numbers, sets, functions and groups physical entities of some kind? Are they objectively existing objects in some non-physical, mathematical realm? Are they ideas that are present only in the mind? Or do mathematical truths not involve referents of any kind?
It is these kinds of questions that have encouraged philosophers and mathematicians alike to focus their attention on issues in the philosophy of mathematics. Over the centuries a number of reasonably well-defined positions about the nature of mathematics have been developed and it is these positions (both historical and current) that are surveyed in the current volume.
Traditional theories (Platonism, Aristotelianism, Kantianism), as well as dominant modern theories (logicism, formalism, constructivism, fictionalism, etc.), are all analyzed and evaluated. Leading-edge research in related fields (set theory, computability theory, probability theory, paraconsistency) is also discussed.
The result is a handbook that not only provides a comprehensive overview of recent developments but that also serves as an indispensable resource for anyone wanting to learn about current developments in the philosophy of mathematics.
-Comprehensive coverage of all main theories in the philosophy of mathematics
-Clearly written expositions of fundamental ideas and concepts
-Definitive discussions by leading researchers in the field
-Summaries of leading-edge research in related fields (set theory, computability theory, probability theory, paraconsistency) are also included
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Chapter 3 Aristotelian Realism
Chapter 4 Empiricism in the Philosophy of Mathematics
Chapter 5 A Kantian Perspective on the Philosophy of Mathematics
Chapter 6 Logicism
Chapter 7 Formalism
Chapter 8 Constructivism in Mathematics
Chapter 9 Fictionalism
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abstract objects analysis applied argue arithmetic Axiom of Choice axiomatic axiomatic set theory axioms Benacerraf Brouwer calculable Cambridge Cantor cardinal claim Colyvan computability concept confirmational holism consistency consistency proof construction continuum Dedekind defined definition degrees of belief ematics empirical example existence fictionalism fictionalists finite finitist first-order logic formal foundations of mathematics Frege function geometry Gödel Hilbert indispensability argument infinite interpretation intuition intuitionism intuitionistic intuitionistic logic Kant knowledge L. E. J. Brouwer language large cardinals Maddy math mathematical entities mathematical objects mathematical realm mathematical theories mathematicians natural numbers nominalistic notion objective Bayesianism Oxford paradoxes philosophy of mathematics physical platonism platonists possible predicate principle priori probability problem proof properties propositions quantifiers question Quine real numbers reason recursive relation Reprinted result Russell semantics sense sentences sequence set theory structure subsets theorem thesis transfinite true truth Turing machine Turing's University Press Zermelo