10 - Higher-dimensional algebra
Published online by Cambridge University Press: 22 September 2009
Summary
Mathematical diagrams may well have been the first diagrams. The diagram is not a representation of something else; it is the thing itself. It is not like a representation of a building, it is like a building, acted upon and constructed.
(Netz 1999: 60)Angular momentum and the topology of knots and links are a fantasy and fugue on the theme of pattern in a formal plane. The plane sings its song of distinction, unfolding into complex topological and quantum mechanical structures.
(Kauffman 1991: 621)INTRODUCTION
Mathematicians are in the business of interpretation. Their lives are spent on open-ended quests for improved reformulations and reconceptualisations, where the familiar and taken-for-granted may at any moment be cast in a surprising light. The real numbers are seen at one time as the natural completion of the set of rational numbers, and later as just one such completion – the completion ‘at infinity’ – alongside infinitely many p-adic completions for prime p. The Euler characteristic is first seen as a regularity holding between the number of vertices, edges and faces of a polyhedron, later understood to be a topological invariant of a triangulable space of any dimension, and is now also seen by geometric probability theorists as a valuation on the algebra of sets generated by polytopes in Rn.
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- Towards a Philosophy of Real Mathematics , pp. 237 - 270Publisher: Cambridge University PressPrint publication year: 2003