I feel very strongly that the direction that things like the nLab, Lurie’s /Higher Topos Theory/, and the Stacks project *is the wrong one*. Like Grothendieck’s work it is very hard to consistency check these things and they do tend to doorstopness when published. Errors are subtle and lost in a sea of TeX. Moreover I feel that we’re recapitulating the historical territory that was already encroached on by the story that started with Frege and whose current torchbearer is Chaitin. I get antsy when reading papers that are many pages of category theory without alternate metaphor demonstrations, as if somehow the magic language fairy is going to bop us on the heads with the pixie dust of understanding because now we’re doing things in categories and toposes instead of sets.
Rucker sees gnarl. I agree with him. We don’t know what to make of gnarl yet because many people are waiting for their lucky break at the Forget Functor Lottery: if only I have the right abstractions I can solve everything: hypergeneralization is just as nonproductive as foundationalism, and Chaitin’s comments about randomness aside: there is froth, there is nature.
I get hopeful when I think that future interface rigs to mathematics are going to be like the accountant’s interface to the Hitchhiker’s Guide to the Galaxy’s financial systems (see /Mostly Harmless/).
Furthermore (and this isn’t my torch to bear), I think that the mathematics that is being taught children is the wrong sort: why teach them to be gimcrack calculating machines and later force them to unlearn commandments issued by clueless teachers when we could be using the articulation of the computer to give them intuitive understandings of concepts that have never been available before in history? And think of what kind of mathematicians those children could turn out to be when they grow up? It’s not fair to show kids fractals and then say “it’s time to do this algebra!” for two very important reasons: allowing the child to play with the former would allow them to have a very kinesthetic/visual emotional way about reasoning that would be more useful to developing mathematical competency and interest than dragging them through arbitrary (from the point of the historical contingency of mathematical symbolism) and capricious symbol manipulation rules for which they’re being judged on their ability to apply flawlessly, and it is unfair in the waving a carrot on a stick in front of the kid and then giving them an ant to eat when they have solved the problem to the satisfaction of idiotically backwards grading system. We’re actively reducing the pool size of potential brilliant mathematicians with the current setup.