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Mathematics without NumbersTowards a Modal-Structural Interpretation$
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Geoffrey Hellman

Print publication date: 1993

Print ISBN-13: 9780198240341

Published to Oxford Scholarship Online: November 2003

DOI: 10.1093/0198240341.001.0001

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The Natural Numbers and Analysis

The Natural Numbers and Analysis

Chapter:
(p.11) 1 The Natural Numbers and Analysis
Source:
Mathematics without Numbers
Author(s):

Geoffrey Hellman (Contributor Webpage)

Publisher:
Oxford University Press
DOI:10.1093/0198240341.003.0002

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

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