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.
摘要
关于抽象数学对象的不可或缺性论证, 是近几十年来数学哲学中的争论焦点。 该论证依赖于一些假设。 目前已有人对这些假设提出质疑, 但这些质疑还存在重要的缺陷, 对这些缺陷的修补, 可以引出一种新的反实在论数学哲学, 其要点包括: 1. 在数学应用中, 真正存在的、 被用作工具的, 不是抽象数学对象, 而是我们想象抽象数学对象时在大脑中创造的内在表征; 2. 我们对数学对象的想象是受客观性条件制约的。
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Translated from Zhexue Yanjiu 哲学研究 (Philosophical Researches), 2006, (8): 74–83
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Ye, F. Indispensability argument and anti-realism in philosophy of mathematics. Front. Philos. China 2, 614–628 (2007). https://doi.org/10.1007/s11466-007-0040-0
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DOI: https://doi.org/10.1007/s11466-007-0040-0