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Extremality Assumptions in the Foundations of Mathematics

Published online by Cambridge University Press:  28 February 2022

Jaakko Hintikka
Jaakko Hintikka*
Affiliation:
Florida State University
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Extract

The work by J.H. Conway (1976, ch. 4), Philip Ehrlich (1987, with further references) and others on absolute continua derives much of its philosophical and mathematical interest from its instantiating a more general approach to the foundations of mathematics. This approach focuses on the role of extremality (rnaximality and minimality) assumptions as a part of the basis of mathematical theories. The idea of extremality is potentially much more important than has recently been pointed out in the literature. It is illustrated by the principle of mathematical induction, which can be thought of as an attempt to enforce the requirement of minimality on the models of elementary number theory as well as by Hilbert's Axiom of Completeness (1971), which was calculated to enforce a kind of maximality on the models of axiomatic geometry. (These models were of course intended to be a species of continua.)

Type
Part VII. Mathematics
Information
PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association , Volume 1986 , Issue 2: Volume Two: Symposia and Invited Papers 1986 , 1986 , pp. 247 - 252
Copyright
Copyright © 1987 by the Philosophy of Science Association

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Footnotes

1

This paper should be considered a preliminary report of work very much still in progress. In particular, my observations concerning set theory ought not to be taken at face value before I (or someone else) have had a chance of constructing detailed arguments for them and of checking the details of such arguments. I have taken the liberty of publishing this report in order to honor my promise to PSA and also to stimulate my colleagues to pursue the same line of thought — or perhaps to refute it.

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