The n-Category Café

Re: The Mathematical Sublime

The really interesting thing about ‘the sublime’ is how important it is in motivating mathematicians. Surely there’s a wholesome, not–so–sublime pleasure in calculating, sort of like needlepoint or any other careful craft. But I think many mathematicians live for those moments of exaltation that come from suddenly glimpsing a terrifyingly grand vista: sometimes shrouded in mist, sometimes lit by a lightning-bolt of insight.

To me what’s thrilling and scary about the infinite is not the idea of infinitely large numbers or sets, though these can be scary too, even when they’re countable (which somehow makes them more real to me). What’s really thrilling and scary is how our exploration of mathematics seems to endlessly reveal deeper patterns and interconnections, which still keep seeming obvious once we understand them. Your quote of Daniel Davis captures that pretty well:

Behrens and Lawson use stacks, the theory of buildings, homotopy fixed points, the above model category, and other tools to make it possible to use the arithmetic of Shimura varieties to help with understanding the stable homotopy groups of spheres.

I mean: holy moly! Who’d have thought that counting the ways you can wrap one sphere around another would have gotten us into such deep waters? Where’s it all going? Are we going to slowly bog down in the ever-growing complexity of our own thoughts, or will we come to some realization that’s so staggeringly awesome yet obvious that we’ll have to ban the journal it’s published in, because reading it will cause heart attacks?

My series This Week’s Finds is secretly all about the sublime. I don’t talk about it explicitly very much, because this leads to tiresome purple prose. As they say in tips for budding writers: “show, don’t tell”.

So, that’s what I try to do: start with a picture everyone can enjoy, then work up from simple math to something that absolutely flabbergasts me. Then, wrap up with a quote that lays it out quite clearly. There’s an endless supply! From this one attributed to Pythagoras:

There is geometry in the humming of the strings, there is music in the spacing of the spheres.

to this one by Einstein:

A knowledge of the existence of something we cannot penetrate, of the manifestations of the profoundest reason and the most radiant beauty, which are only accessible to our reason in their most elementary forms. It is this knowledge and this emotion that constitute the truly religious attitude; in this sense, and in this alone, I am a deeply religious man.

or this by Shafaverich:

Viewed superficially, mathematics is the result of centuries of effort by thousands of largely unconnected individuals scattered across continents, centuries and millennia. However the internal logic of its development much more closely resembles the work of a single intellect developing its thought in a continuous and systematics way - much as in an orchestra playing a symphony written by some composer the theme moves from one instrument to another, so that as soon as one performer is forced to cut short his part, it is taken up by another player, who continues with due attention to the score.

or this by Bertrand Russell:

The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.

or Grothendieck:

I never once doubted that I would eventually succeed in getting to the bottom of things.

Almost every famous mathematician, logician or physicist has one.

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