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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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‘There is no Ontology Here’: Visual and Structural Geometry in Arithmetic

‘There is no Ontology Here’: Visual and Structural Geometry in Arithmetic

Chapter:
(p.370) 14 ‘There is no Ontology Here’: Visual and Structural Geometry in Arithmetic
Source:
The Philosophy of Mathematical Practice
Author(s):

Colin McLarty

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780199296453.003.0015

Today's number theory solves classical problems by structural tools that violate standard philosophical expectations in ontology. The far-reaching practical demands of this mathematics require on one hand that the tools be fully explicit and more rigorous than many philosophical theories are, and on the other hand that they relate as directly and as concisely as possible to guiding intuitions. The standard tools grew from a century-long trend of unifying algebra, topology, and arithmetic, notably in the Weil conjectures; and they rely on devices that Kronecker produced for his idea of a pure arithmetic. Very large functors serve to organize individually simple kinds of data that can themselves even be depicted in simple pictures. Mathematicians and philosophers have debated issues of individuation and identity raised by these tools.

Keywords:   diophantine equation, Kronecker, Weil conjectures, schemes, ontology, number theory, identity, functor, purity of method

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