I have a basically limitless pile of papers to read, a large fraction of which (I’ve not counted) contain a heavy math element. To take one example basically at random: Stein’s paper on “The Inadmissibility Of The Usual Estimator For The Mean Of A Multivariate Normal Distribution”. I don’t really get that paper. In particular, I don’t have the intuition about the topic that I do about, say, Unix or even economics: I can’t come to the topic and consume the whole thing in big chunks, which I can understand as a unit; instead I’m groveling through it line by line, and maybe understanding one tree at a time while the forest remains unassaulted. My reading of mathematics is a lot like my reading of French, while my reading of other topics is fluid like my reading of English.
Backing up from academic papers: it’s been a long-term goal of mine to learn measure theory, which underlies probability theory. I have ever so many books that cover aspects of measure theory: this one, and also this one, not to mention this one, and of course we’d be remiss if we didn’t mention this one. They’re all good, but I find them all hard. To pick another random mathematics book off the shelf: Körner’s book on Fourier analysis is fantastic at the high level at which I’ve been able to understand it, but digging into the details has always felt to me like an impossible slog.
Is the answer no more complicated than
- go to a library with just those books (no phone, no laptop), a pad of paper, and a pen
- bang your head against the problem sets for hours and hours
- GOTO 1
? It’s always felt to me like, no matter how much I bang my head against them, I’m going to be unable to prove theorems. I wonder if that’s true, or if that’s just the wrong side of my brain talking. I wonder if it’s the side of my brain that Ira Glass mentioned when we saw him in Boston a while back; a canonical form of the quote seems to be the one here:
All of us who do creative work, we get into it because we have good taste. But there is this gap. For the first couple years you make stuff, it’s just not that good. It’s trying to be good, it has potential, but it’s not. But your taste, the thing that got you into the game, is still killer. And your taste is why your work disappoints you. A lot of people never get past this phase, they quit. Most people I know who do interesting, creative work went through years of this. We know our work doesn’t have this special thing that we want it to have. We all go through this. And if you are just starting out or you are still in this phase, you gotta know its normal and the most important thing you can do is do a lot of work. Put yourself on a deadline so that every week you will finish one story. It is only by going through a volume of work that you will close that gap, and your work will be as good as your ambitions. And I took longer to figure out how to do this than anyone I’ve ever met. It’s gonna take awhile. It’s normal to take awhile. You’ve just gotta fight your way through.
An unspoken part of this is that not everyone is able to do creative work at a high level. (And mathematics is certainly included within “creative work”). Something similar would have to be said about, say, basketball: I watched a lot of the Bulls during the Michael Jordan era, and I knew from quite early on that my playing basketball at that level was just not an option. Or maybe it was — maybe I just needed to put in the hours, hire the coaches, etc. But it would always be harder for me to reach that level than it would be for Jordan. If I put in 9 billion hours of work, maybe I could get where he got with the mythical 10,000 hours. Probably not, though. There I stood on this side of the chasm, and there he stood on the other, and nothing I did was going to get me to his side.
Obviously I’m closer to being a professional mathematician than I am to being a professional basketball player. I feel like I’ve got more inherent writing talent than I do inherent mathematical talent. This isn’t to say that I’m John McPhee, but writing comes to me more fluidly than mathematics does.
Anyway, so yeah: do I just go to the library with some books and let the rest happen naturally? An obvious alternative is to pay someone to sit by my side while I read books and do problem sets; here in Boston, anyway, we call this “college”. So do I go to this mythical “college” to learn some more math? Or do I adopt some good self-study methods and do it on my own? Or is there some better way to learn math on one’s own?
On the topic of measurement, is Lockhart’s “Measurement” worth looking at (I haven’t read it)?
As for studying math, I like the idea of finding a starving grad student with a knack for teaching and paying them to give little nudges here and there when you’re stuck. That’s *way* better value for money than taking a college class, I’ll wager, and far more productive than just sweating it out on your own.
One thing I can recommend is looking up the “cited by” in google Scholar and see how a paper was used *forward* instead of backward. This breaks the Kuhn-Lakatos rules about looking at papers by their history but that’s good – they’re overrated. The trouble with the linked paper is that it isn’t quite effective, it doesn’t give an example function h(x^2).
Reflective papers like this (and wikipedia) are helpful:
Click to access 2012b%20A%20Geometrical%20Explanation%20of%20Stein%20Shrinkage.pdf
I think the trouble is that this example is in three (or more) dimensions, and it’s hard to get an intuition about three (or more) dimensional shapes. Maybe the best thing to do is to code up some random oviods and compare the naive average ovoid to the James ovoid given by equation 8 in the above paper. Once you’ve convinced yourself the James ovoid is better by sight, re-write James’s arguments in vector notation and with some help from the paper I linked I think you’ll be able to read James’s more difficult paper.