Chapter

Mathematical Objects as Positions in Patterns

Michael D. Resnik

in Mathematics as a Science of Patterns

Published in print December 1999 | ISBN: 9780198250142
Published online November 2003 | e-ISBN: 9780191598296 | DOI: http://dx.doi.org/10.1093/0198250142.003.0010
 Mathematical Objects as Positions in Patterns

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It is usual to regard mathematical objects as entities that can be identified, characterized, and known in isolation. In this chapter, I propose a contrasting view according to which mathematical entities are structureless points or positions in structures (patterns) that are not distinguishable or identifiable outside the structure. By analysing the various relations that can hold between patterns, like congruence, equivalence, mutual occurrence, I also account for the incompleteness of mathematical objects, for mathematics turns out to be a conglomeration of theories, each dealing with its own structure and each forgoing identities leading outside its structure. My suggestion that mathematical objects are positions in structures or patterns is not intended as an ontological reduction, but rather as a way of viewing mathematical objects and theories that put the phenomena of multiple reductions and ontological and referential relativity in a clearer light.

Keywords: congruence; equivalence; identity; incompleteness; mathematical object; occurrence; patterns; positions; reduction; relativity; structures

Chapter.  9031 words.  Illustrated.

Subjects: Philosophy of Mathematics and Logic

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