Chapter

Patterns and Mathematical Knowledge

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/10.1093/0198250142.003.0011
 Patterns and Mathematical Knowledge

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I present a hypothetical account of how the ancients might have come to introduce mathematical objects in order to describe patterns, and I explain how working with patterns can generate information about the mathematical realm. The ancients might have started using what I call templates, i.e. concrete devices, like blueprints or drawings, to represent how things are shaped or structured, and this could have evolved into representing the abstract patterns that concrete things might fit. In this way, they might have come to believe that written constructions and computations could provide information about the mathematical realm, for by their very nature, patterns should be structurally analogous to their templates and in positing that they are, one simply projects onto structures and features already attributed to templates. By reflecting on systems of dots representing cardinalities, the ancients could generate a body of results that then evolved into a systematic theory of numbers, but the approach fails when there is no direct connection between the computations and the patterns they are supposed to concern, e.g. those concerning trigonometric functions or transfinite ordinal numbers. In these cases, we forge a connection between proofs and patterns by positing that the premises of the proofs state uncontroversial features of the patterns, i.e. the premises constitute implicit definitions of the patterns. I explain how this position does not entail that mathematics is analytic or a priori.

Keywords: a priori; analytic; computation; implicit definition; mathematics; patterns; proof; structures; templates

Chapter.  7162 words.  Illustrated.

Subjects: Philosophy of Mathematics and Logic

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