The Sublime June 11, 2008Posted by dcorfield in Uncategorized.
There’s an interesting post – Whatever Happened to Sublimity? – at the blog Siris. It includes a quotation from Edmund Burke
But let it be considered that hardly any thing can strike the mind with its greatness which does not make some sort of approach toward infinity; which nothing can do while we are able to perceive its bounds; but to see an object distinctly, and to perceive its bounds, are one and the same thing. A clear idea is, therefore, another name for a little idea. (A Philosophical Inquiry into the Origin of Our Ideas of the Sublime and the Beautiful, Part II, Section V.)
A natural question to ask, then, is where do we encounter the sublime in mathematics? And an obvious answer, you might think, would be the mathematical infinite.
Joseph Dauben has an interesting section in his book – Georg Cantor: his mathematics and philosophy of the infinite, Harvard University Press, 1979 – on how Cantor, receiving such discouragement from his mathematical colleagues, found an audience in certain thinkers within the Catholic church. Where earlier in the nineteenth century any attempt to describe a completed infinity was viewed as a sacrilegious attempt to circumscribe God, some theologians were open to Cantor’s new hierarchy of infinities, with its unreachable Absolute Infinite leaving room for the divine.
Personally, set theory has rarely invoked in me a sense of the sublime. On the other hand, the following comment by Daniel Davis does:
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.
There’s plenty about the infinite in that statement, it’s true, but there’s so much more to it than that.
Brandon Watson, the blogger at Siris, writes
…what often strikes me when I look around at the philosophical scene today is how foreign this has all become. There are a few exceptions, but sublimity has vanished as a serious concern.
With regard to the philosophy of mathematics in particular I couldn’t agree more.