Jump to ContentJump to Main Navigation
Science without Numbers$
Users without a subscription are not able to see the full content.

Hartry Field

Print publication date: 2016

Print ISBN-13: 9780198777915

Published to Oxford Scholarship Online: December 2016

DOI: 10.1093/acprof:oso/9780198777915.001.0001

Show Summary Details
Page of

PRINTED FROM OXFORD SCHOLARSHIP ONLINE (www.oxfordscholarship.com). (c) Copyright Oxford University Press, 2017. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter of a monograph in OSO for personal use (for details see http://www.oxfordscholarship.com/page/privacy-policy).date: 16 December 2017

Second Illustration of Why Mathematical Entities are Useful: Geometry and Distance

Second Illustration of Why Mathematical Entities are Useful: Geometry and Distance

Chapter:
(p.26) 3 Second Illustration of Why Mathematical Entities are Useful: Geometry and Distance
Source:
Science without Numbers
Author(s):

Hartry Field

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198777915.003.0003

Shows that the use of real numbers in geometry can be accounted for by the conservativeness of mathematics, without assuming the truth of the theory of real numbers. Focuses on Hilbert’s axiomatization of Euclidean geometry, which, since it doesn’t involve real numbers, shows that real numbers are theoretically dispensable in geometry. Discusses Hilbert’s representation and uniqueness theorems. Shows how the representation theorem explains the utility of real numbers in geometric reasoning (without requiring that the theory of real numbers be true), while the uniqueness theorem establishes that the axiomatization without numbers has certain quite desirable properties.

Keywords:   application of mathematics, conservativeness, geometry, Hilbert, mathematical truth, real numbers, representation and uniqueness theorems

Oxford Scholarship Online requires a subscription or purchase to access the full text of books within the service. Public users can however freely search the site and view the abstracts and keywords for each book and chapter.

Please, subscribe or login to access full text content.

If you think you should have access to this title, please contact your librarian.

To troubleshoot, please check our FAQs , and if you can't find the answer there, please contact us .