i will unfortunately give you a half-assed response. Because the truth will take up a lot of time & space & is very inflammatory. So here's what I honestly believe - there are only isolated tricks. The ones who sit in judgement of us, like savants, and claim some sort of coalesced mastery of these tricks into some larger theoretical construct, are faking. They are good at faking, so we listen to them. They have better rhetoric, better optics. But beneath it all, it is a grab-bag of tricks. I worked with several professors. When trying to prove something they will start with their favorite trick. One guy was like - okay let me see if i can cauchy schwarz this. I'm like, why ? Why cauchy schwarz always? There are million inequality bounds. Why pester cs all the time ? He actually looked at me curiously then said I don't know I just have some favorite tricks it seems to work! Atleast he was honest. So that's the deal. If you probe deeply there are a few isolated tricks. They seem to work well. In reality we don't know why they work. But they do. Its like why do you not use spoon for spaghetti ? Why fork ? Ok you can cut lines into a spoon and call it a spork, but then you can't drink soup with the spork. So you reach for the spoon if you want to sip soup, use your fork for spaghetti. Don't try to design some universal theory of cookware. Just buy 1 spoon & 1 fork. That's basically all of mathematical statistics. Few tricks. They work. Why ? I mean who gives a fuck. Maybe bickel & doksum do but I personally don't.

One of my teachers taught me this: "If you apply a trick once, it's a trick, if you apply it twice, it becomes a method."

Rota said something similar in this essay about mathematicians and their bags of tricks: https://www.ams.org/notices/199701/comm-rota.pdf

But I think they also have their intuitions built from experience in the sense of Tao's post-rigorous stage: https://terrytao.wordpress.com/career-advice/theres-more-to-...

:).

Someone once wrote this. If someone gives you Excalibur, you don't fucking ask where it came from, who made it, etc. Just go use it and have fun.

What about creating/finding tricks? I wonder if there a larger theoretical sense of things involved there? What about in a weaker sense, like: "I think I'll start my trial-and-error with this"?

(a mostly-hypothetical question)

Mathematics is just a collection of cheap tricks and bad jokes - Lipman Bers