Realism
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What is it?
The view that the truths of mathematics are thus independent of human activities, i.e., they are objective.
Where is it found?
BLANCHETTE, PATRICIA A. (1998). Realism in the philosophy of mathematics. In E. Craig (Ed.), Routledge Encyclopedia of Philosophy. London: Routledge. Retrieved February 27, 2008, from http://www.rep.routledge.com.virtual.anu.edu.au/article/Y066
MOORE, A.W. (1998, 2004). Antirealism in the philosophy of mathematics. In E. Craig (Ed.), Routledge Encyclopedia of Philosophy. London: Routledge. Retrieved February 27, 2008, from http://www.rep.routledge.com.virtual.anu.edu.au/article/Y065
Horsten, Leon, “Philosophy of Mathematics”, The Stanford Encyclopedia of Philosophy (Winter 2007 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/win2007/entries/philosophy-mathematics/>
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Realism and Platonism
Platonism Platonism is a form of math. realism in which the existence of independent math. objects is posited. Thus reasoning involving maths is exactly the same as reasoning involving concrete objects and both are held to exist independently of our thoughts regarding them.
Problems for the Platonist - Epistemological: If math objects are abstract, how do we come to have knowledge of them? - Practical: how do these abstract mathematical objects (or even truths in the general realist case) relate to the practices that we have for determining mathematical truths (i.e. proof procedures)?
Problem Responses - Epist.: - Godel: as empirical data are to concrete object knowledge, math. intuition is to math object knowledge — prob. re: giving a clear account of math. intuition - Frege: knowledge of math objects is just knowledge of propositions involving math objects. - Hence just need to explain justification for these propositions. — prob re: is this position actually Platonist (i.e. should knowledge of suitable propositions count as knowledge of the objects they contain)?
Arguments for Platonism - Colyvan Indispensability
Non-Platonist Realism Talking about objects in math is just a convenient way of speaking. Maths is really about general features of some less controversial/abstract kind of thing (e.g. collections, measurements, structures, etc.). - e.g. structuralism - math. statements about collections of entities with similar structural properties or these abstract structures themselves. - Problems: — structuralism still postulates the existence of abstract objects, just a different kind. — {[green There is a point in Routledge concerning the need to phrase things in terms of possibility (in order for things to be true in a finite world?) that I don’t really get. ]}
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What do I think?
- While on the face of it mathematical realism seems to be the position that is most compatible with a sort of “common sense” view of mathematics, upon further examination I’m not so sure that this is case. When you actually think about them, actually existing abstract mathematical entities (indeed, any sort of abstract entity) are very strange things indeed. Thus, I don’t think the realist can really claim any support from some compatibility with the “common view”.
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