Aesthetics and topology

This remains a title looking for an argument, at the moment. Those doing mathematics don’t need the garnish of an extra category to place what they do or what its intended reception is, as would be the case for instance in the realisation of a building project, or the creation of a film, or a work of art. But the practice of mathematics is governed by a severe aesthetic – from setting or defining the problem within a theory, to searching through heuristics for some way to advance it, to achieving and then refining a solution. The nature of this aesthetic is perhaps best revealed in some of the spectacular failures: the failure to secure a logical foundation for analysis; the ultimate failure of euclidean geometry to satisfactorily encompass the world of experience; the failure of the program to formalise mathematics logically. In each case the elusive aesthetic drove mathematics into new territory; an unsatisfactory state demanded resolution.

Rather than closing off however the result was an opening out to new terrains of abstraction, but as well a striking modernisation of mathematics as a tool for advance into new fields of knowledge or practice. The modern world needed the apparatus made available by nonstandard analysis; by noneuclidean geometry (a precursor of quantum physics); of infiinite recursions in which meadow programming flourished. Russell and Whitehead’s program failure lead to spectacular advances in set theory, algebra and number theory. Something similar could be said of the other road blocks listed.

What is this aesthetic then? A striving but never arriving; a fecundity in what is not yet accomplished, viz a viz what is known, what can be demonstrated, what can be mastered. Is it worth spending time on this meta-mathematical whimsy? Can we indeed apply the discipline of meta mathematics to better understand what motivates a proof in the first place; what drives discovery in a practice that eschews speculation, that demands an extreme deductibility, that seems to call out of the air new limits new rules.

Part 2

Let us for a moment step back from the aesthetics of doing mathematics – of creating mathematics, seeing mathematics as a performance – and reorient the question towards a mathematical interpretation of sense experience. There is after all a mathematics of space relations (geometry), a mathematics of sound (Fourier series and derived functional analysis), a mathematics of taste itself? It seems unlikely, or unproductive. Yet this drive to some resolution of experience reaches into mathematics: the music of the spheres; the golden mean; the magic of conic sections  – a theory elucidated by Pascal; the intriguing fixity of regular polygons and how somehow they influence the distribution of the planets (Kepler).

This search for regularity beyond human agency as a reassurance that we make sense in some wider narrative, be it one of numbers or shapes or laws not conditioned on the physicality of things and how they come to occupy the forms they do. In some sense the shape, the number or the law was there first. Behind superficially simple things are profoundly simple things. This is the realm of mathematics as an aesthetic medium, a precursor to experience.

Perhaps it is indeed a universal this search for perfection, be it in form or in time

Further Reading

Tom McCarthy (2014) ‘Ulysses and Its Wake’ London Review of Books pp39-41, 19 June 2014

Philip J. Davis & Reuben Hersh, (1980) The Mathematical Experience, Pelican Books

Jason Socrates Bardi (2006) The Calculus Wars – Newton, Leibniz and the Greatest Mathematical Clash of All Time, Thunder’s Mouth Press

GH Hardy (1940) A Mathematician’s Apology, Cambridge University Press.

R. Thom (1970) Topologie et Linguistique, Essays on Topology, Volume dedie a G. de Rham, Springer

 Hermann Weyl (1952) Symetrie et mathematique moderne, flammarion