[baggers]

This question is as someone who is mathematically curious but not yet adept.

In programming we slowly gain a big grab-bag of patterns and approaches to certain problems and build an intuition of what to apply where. It doesn't nearly cover your full experience but helps break problems down.

Do you find there is an analog to this with theorems? If so what's the essentials from your 'grab bag' and, beyond just reading more, what practices help build your feeling of where to use them?

[CurtMonash]

I haven't been a mathematician for decades, and in particular I don't recall the details in the examples I'm about to give. :) But I'd say yes up to a point. In particular, one matches problems to generalized patterns, which in math often has the feel of transforming a problem into another form.

1. If you're going to take a test for school, you can recognize patterns pretty easily. For example, Harvard has a 3-day Qualifying Exam as a PhD requirement, given twice a year. Once, after looking at the exam pages for the first two days, I said "Hmm, there hasn't been a Schwarz Reflection Principle question yet."

The next evening, I got a nice hug from Lisa Mantini. :)

2. If you're doing actual research math, it's of course much harder. For example, the theorem in my thesis was scooped by a few months by Neyman and Mertens. What's more, they did it as a special case of a fixed-point theorem, while I just proved it by brute force.

And now some simpler anecdotes from my undergraduate days.

3. In my E&M course, we were supposed to work out fields for a cylindrical wire with an off-center cylindrical hole. This was a LOT easier in curvilinear coordinates than Euclidean. When I showed that to the professor in his office, he literally stood up and applauded.

4. In an abstract algebra course, there was a typo in the book for one of the exercises as to whether a certain polynomial was reducible. I solved the hard form by transforming the problem into one about a polynomial with matrix variables.

I took that story with me when I went to graduate school. Ron Livne's eyes lit up, and he transformed it again into something about -- well, into something about transformations on the polynomial's complex roots.

[ColinWright]

Yes, there is, but I've never thought of it in those terms, and I can't enumerate them easily. There are techniques to apply, approaches to try, and connections to make that can help. But I don't know any way of building intuition other than by actual doing.

Tim Gowers writes well about how to build mathematical knowledge, techniques, and a library of tools.

[baggers]

Thanks, that's good to know.