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Why a Little Bit Goes a Long Way: Logical Foundations of Scientifically Applicable Mathematics

Solomon Feferman
PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association
Vol. 1992, Volume Two: Symposia and Invited Papers (1992), pp. 442-455
Stable URL: http://www.jstor.org/stable/192856
Page Count: 14
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Why a Little Bit Goes a Long Way: Logical Foundations of Scientifically Applicable Mathematics
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Abstract

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.

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