Chapter

The Natural Numbers and Analysis

Geoffrey Hellman

in Mathematics without Numbers

Published in print November 1993 | ISBN: 9780198240341
Published online November 2003 | e-ISBN: 9780191597664 | DOI: http://dx.doi.org/10.1093/0198240341.003.0002

Series: Clarendon Paperbacks

 The Natural Numbers and Analysis

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Motivation and details of a systematic translation of number‐theoretic sentences into a suitable modal‐logical language are provided, eliminating reference to numbers as objects and expressing the structuralist viewpoint. Modal existence postulates, although minimal, distinguish the present approach from ‘“if‐then”ism’ or ‘deductivism’. The translation scheme is then justified as accurate and adequate. Extension of the methods to more powerful systems, e.g. real analysis, is sketched. Finally, it is shown how the methods so far can be adapted to the concerns of nominalism, avoiding functions and relations as objects.

Keywords: deductivism; modal existence; natural numbers; nominalism; real analysis; structuralism

Chapter.  17484 words. 

Subjects: Philosophy of Mathematics and Logic

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