And now from far out in right field, where the math nerds live..
I recently had the unhappy experience of having to review my understanding, or utter lack of it, of category theory. Yikes - but, with all due respect to Eilenberg et al, I'd now like to at least try to write down an appropriately childish thought about making sense of it all.
It seems to me that Category Theory, abstruse nonsense as it may be, offers a way to unify all of mathematics through descriptive generalization: making it more an application of the principles of math to epistemology than actual math (and maybe a case of moving into the intellectual vacuum left when philosophy abandoned analysis for politics?). Langlands, in any case, makes more sense within math by taking the opposite approach: showing the unity of math by metaphorically zooming in rather than out - effectively instancing and eliminating the need for functors and their associated idiomorphs at the same time.
And if that isn't a sufficiently arrogant statement - here's Murphy's Axiom:
1 - all of mathematics can be thought of as the search for ways to find the value of expressions subject to some set of conditions; and,
1 - all of physics can be thought of the search for ways to define real world conditions and expressions so that the conditions allow the evaluation of the expressions.
Profound huh?
Sure, but once upon a time a kid came running back to the cave to tell his dad about 15 fat ducks down at the pond. Luckily, for us, dad was out hunting and mom didn't want him to use up all the firewood just to record a number. So he just used three piles of five sticks - thereby not only saving seven sticks and creating the prime numbers, but implicitly inventing category theory by mapping one arithmetical method to another in order to explain the idea. Compare that to the efforts of a man who generalized an algebra of expressions and conditions largely because he thought learning to spell his last name wasn't punishment enough for grad students, and you see the problem: Navier-Stokes may very well fit into the same quiver with Euler's beta function, but there's no useful there, there - nothing to help you count ducks or differentiate 10 variables from 11.
And that's the difference: category theory may be fun and internally consistent, but so was string theory - and the cave kid's invention does something category theory doesn't: it helps me in what I was trying to do when this bit of antler rubbing got started: find some conditions under which a collection of expressions might sprout something as helpful as an equals sign and so become at least partially tractable.
Well, Yes, I am hiding from visiting relatives; so, Coffee?
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