Indispensability argument and anti-realism in philosophy of mathematics

Frontiers of Philosophy in China 2 (4):614-628 (2007)
Abstract
The indispensability argument for abstract mathematical entities has been an important issue in the philosophy of mathematics. The argument relies on several assumptions. Some objections have been made against these assumptions, but there are several serious defects in these objections. Ameliorating these defects leads to a new anti-realistic philosophy of mathematics, mainly: first, in mathematical applications, what really exist and can be used as tools are not abstract mathematical entities, but our inner representations that we create in imagining abstract mathematical entities; second, the thoughts that we create in imagining infinite mathematical entities are bounded by external conditions.
Keywords philosophy of mathematics  indispensability argument  anti-realism  nominalism  realism  Quine  数学哲学  不可或缺性论证  反实在论  唯名论  实在论  奎因
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