[ajkjk]

I happen to think that novel facts and theorems and proofs are way overrated. If you find a new fact it just goes into the giant pile of facts that are sitting around uselessly. The useful progress in math is comes from "refactoring" efforts to make things simpler and more intuitive.

I don't mean that this is necessarily the case, but that it is where we are now: we have found ourself in a situation where we have way too many facts and not enough simple perspectives that make them useful and accessible.

Just my opinion, though.

[kfse]

I agree that a lot of progress in math comes from refactorings and novel concepts that generalize neatly. My point is that those breakthroughs don't happen through refactoring in a vacuum, they happen because the refactoring is undertaken with a specific motivation. It often ends up having more general applications than the original motivation, but doing it without a specific motivation doesn't usually yield progress.

It's the same as with software refactoring. If you refactor without a sense of what you want to get out of the refactor, how do you know whether you're refactoring the right things?

(Also just my opinion)

[ajkjk]

I don't think that argument is actually true? You can refactor for purely aesthetic reasons and it may well turn out to help. Maybe you don't even know, going in, what the goal is; just that the way it works now isn't good and maybe if you start looking you'll find a better approach. Happens all the time, I think. Of course you can also refactor for aesthetic reasons and not help, but that's a skill issue.

My clue-finding and pattern-matching and such is all based on philosophical aesthetics: something feels amiss when these patterns exist without being obvious; therefore they should be extracted and examined from various sides to see if a connection is found.