Why a Little Bit Goes a Long Way: Logical Foundations of Scientifically Applicable Mathematics
Why a Little Bit Goes a Long Way: Logical Foundations of Scientifically Applicable Mathematics
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为什么一点点就能大有帮助：科学适用数学的逻辑基础
DOI：
10.1086/psaprocbienmeetp.1992.2.192856
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发表时间：
1992
期刊：
影响因子：
--
通讯作者：
S. Feferman
中科院分区：
文献类型：
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作者：
S. Feferman
科学是否证明了数学的任何部分，如果是，是哪一部分？这些问题与奎因和普特南等人提出的所谓不可或缺的论点有关；而且，他们都在这个基础上接受了集合论的重要部分。然而，集合论建立在柏拉图式实在论的基础上，这种实在论作为数学的基础受到了各种各样的批评，并且与科学实在论不一致。最近的逻辑结果表明，有可能直接形式化几乎所有，如果不是全部，科学上适用的数学，在一个形式系统，证明是由皮亚诺算术（通过证明-理论简化）。有人认为，这实质上削弱了不可或缺的论点。
Does science justify any part of mathematics and, if so, what part? These questions are related to the so-called indispensability arguments propounded, among others, by Quine and Putnam; moreover, both were led to accept significant portions of set theory on that basis. However, set theory rests on a strong form of Platonic realism which has been variously criticized as a foundation of mathematics and is at odds with scientific realism. Recent logical results show that it is possible to directly formalize almost all, if not all, scientifically applicable mathematics in a formal system that is justified simply by Peano Arithmetic (via a proof-theoretical reduction). It is argued that this substantially vitiates the indispensability arguments.