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The Sublime *June 11, 2008*

*Posted by dcorfield in Uncategorized.*

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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.

[...] I have to use frequently a specific construction, an ultrafilter product of fields. It is pretty sublime in the sense of David Corfield and leads to appearance of very interesting canonical [...]

The sublime is probably in the eye of the beholder, but for whatever it’s worth, here is my list.

1. My first candidate for mathematical sublimity would Euler’s identity. Most of us learn this equation before we’ve learnt anything much about the infinite, and certainly before we hear about different degrees of infinity. Indeed, Euler would have known nothing about transfinite arithmetic (I presume — it is always possible that Euler, in fact, did teach himself such things!)

2. Cantor’s diagonal proof for the non-countability of the reals.

3. That it is possible to develop rigorous, axiomatic geometries without assuming Euclid’s parallel postulate.

4. The existence of space-filling curves — aka, that there are 1:1 mappings between the unit real interval and the unit real square. This result still strikes me as if MC Escher had somehow interposed himself into the mathematics canon.

5. The Central Limit Theorem in statistics.

6. The independence of the Axiom of Choice from the ZF axioms.

7. That the internal logic of a topos is intuitionistic.

8. Monster moonshine.

I guess the infinite seems to play a part in all of these, now that I think about it.

space-filling curves are continuous surjection from unit interval to closed square, but they’re not 1-1.

In certain eighteenth century writers who revived the notion of the sublime, there’s something other than an extreme form of beauty. Several of them seem to have been inspired by their visits to the Alps whose peaks resist all attempts to tame them. Away from the cosy English countryside, these mountains invoke feelings of horror and dread in the travellers, while remaining objects of attraction. Does mathematics ever do something similar, attracting you while discomforting you with its vastness?

I never gave any account of why that quotation from Daniel Davis indicates something sublime for me. Consider also David Ben-Zvi saying

and

So one can recognise pieces of this enormous thing as beautiful, and prospects are for there to be improved such recognition, but there’s also something almost terrifyingly vast here too.

[...] Corfield of the n-category cafe and a dialogue on infinity (and perhaps other blogs I’m unaware of) pointed me to the paper Symplectization, [...]

I came across this idea in GEB and Hofstader’s talking about Myhill.

It reminds me of a Turing Machine churning out digits of Pi.

“We finally come to the prospective. Myhill’s characterization of it

is this: “A prospective character is one by which we cannot either

recognize or create by a series of reasoned but in general

unpredictable acts.” Thus it is neither effective nor constructive.

It eludes production by any finite set of rules. However–and

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

this is important–it can be *approximated* to a higher and

higher degree of accuracy by a series of bigger and better

sets of generative rules.

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

Such rules tell you (or a machine) how to churn out members

of this prospective category. In mathematical logic, works

by Tarski and Goedel establish that *truth* has this open-

ended, prospective character. This means that you can produce

all sorts of examples of truth–unlimitedly many– but no

set of rules is ever sufficient to characterize them *all*.

The prospective character eludes capture in any finite set.

As his prime example outside of mathematical logic of this

quality, Myhill suggest beauty. As he puts it:

“Not only can we not gurantee to recognize it (beauty) when

we encounter it, but also there exists no formula or attitude

such as that in which the romantics believed, which can be

counted upon, even in a hypothetically infinitely protracted

lifetime. to create all the beauty that there is.”

Thus beauty admits of a succession of ever-better approximations,

but is never fully attainable. Beauty and irrationality are

often linked. Is it coincidental that the first example of such

a notion of something approximable but never attainable in a

finite process is called an “irrational” number? Myhill is

bold enough to speculate as follows: The analogue of Goedel’s

theorem for aesthetics would therefore be: There is no school

of art which permits the production of all beauty and excludes

the production of all ugliness.”

S: I’ve also seen “symmetry” prominently mentioned in discussions of

truth and beauty. I think that fractal of the Mandelbrot Set is sublime.