I don’t trust category theory. I’ve read Jacob Lurie’s /Higher Topos Theory/ and it leaves a bad taste in my mouth. It seems to me that this is the modern fundamentalism, somewhat akin to logic in the Fregean era. I don’t really trust reasoning that you can’t support numerically or visually. I find the direction that Flajolet’s /Analytic Combinatorics/ works in to be much more appealing, and I would like to see category theory brought in that direction. Similarly I can’t read pages of ncatlab without feeling that it isn’t the right general direction for mathematics to take.
What would be nice, and doesn’t exist, if there were an “analytic category theory” much the same way as there is analytic group theory.
There is thread on mathoverflow Your favorite surprising connections in mathematics, and I think this begins to broach the conversation about big data and the discovery of new mathematics in a way that needs to become more and more meaningful as we continue to throw scads and scads of computing power at mathematics.
What if we could discover things like Monstrous Moonshine on a regular basis? What if we had the tools that allowed us to ferret out unusual functors? What i we could apply something like Don Swanson’s Arrowsmith to the arxiv? We need better tools for directed discovery of unusual connections because they tend to stimulate progress better than usual connections.