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The Philosophy of Mathematical Practice$
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Paolo Mancosu

Print publication date: 2008

Print ISBN-13: 9780199296453

Published to Oxford Scholarship Online: February 2010

DOI: 10.1093/acprof:oso/9780199296453.001.0001

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Cognition of Structure

Cognition of Structure

(p.43) 2 Cognition of Structure
The Philosophy of Mathematical Practice

Marcus Giaquinto (Contributor Webpage)

Oxford University Press

This chapter investigates the possibility of knowing a structure (an isomorphism type) more directly than as the structure of all models of this or that categorical theory. It starts with grasp of small simple finite structures by means of visual templates and then explains how visual grasp of structures may be possible without templates. The case is then made for the possibility of visual cognition of an infinite structure, an ω-sequence, though this is visual cognition of a more abstract kind, relying on the distinction between a visual category specification and a visual image. It is argued that this kind of cognition is not available for many other structures such as that of the closed unit interval of reals under ‘less-than’, but it is available for some ordinal structures beyond ω. The chapter uses this to explain and support some contentious claims of Kurt Gödel.

Keywords:   structure, grasp of structure, visual template, visual cognition, ω-sequence, visual category specification, visual image, Kurt Gödel

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