# Spaced proof review is not about memorizing proofs

Spaced proof review involves proving a result at spaced intervals with the help of Anki. And Anki is about memorizing things. So at first glance, it seems like the goal of spaced proof review is to memorize proofs. But this is not how I like to think about it. It's true that you want to be able to prove the same results over a long period of time, but the way you attain this ability isn't to memorize the proofs. Instead, you're trying to acquire a generic ability to prove things.

Why not just solve *new* problems, if you want the generic ability to prove things? Isn't that much better? Here are some reasons for preferring spaced proof review:

- There are only so many important theorems in a given subfield of math. In real analysis, you have things like the intermediate value theorem, Bolzano-Weierstrass theorem, etc. There are only so many of these "classic" theorems. Any new problems you will have access to will necessarily be not as deep as these fundamental results.
- You want to make sure you can still prove things you were able to prove several months or years ago. It's satisfying to know that you aren't forgetting anything important.
- If you set things up right, there's less friction to do these problems. You don't need to go searching for solutions to check your work (because it's on the back side of the card). (Of course, the downside is that you need to put in some initial effort to make the cards.)

And of course, even if you do spaced proof review, you will be adding new proofs all the time, so you're still solving new problems. The difference is thus in whether to invest the time into reviewing the problems you've already seen. I do think that in the future, it would be nice to have something sort of like Anki which would show problems you haven't seen before (and where the problems are selected to be interesting to you/based on your tastes).

Even extremely competent math grad students seem to forget a lot of problems:^{[1]}

I have, on more occasions than I’m willing to admit, run into the following situation. I solve some exercise in a textbook. Sometime later, I am reading about some other result, and I need some intermediate result, which looks like it could be true but I don’t how to prove it immediately. So I look it up, and then find out it was the exercise I did (and then have to re-do the exercise again because I didn’t write up the solution).

I think you can argue that if you don’t even recognize the statement later, you didn’t learn anything from it. So I think the following is a good summarizing test:

how likely is the student to actually remember it later?

This strikes me as extremely sad. Evan Chen's solution is to pick better exercises, but you could also work from the opposite end: come up with a better system to prevent you from forgetting even the duller exercises.

using anki as a problem bank (with my proofs deck): the spacing is more to make sure seeing the same problem over and over again does not trigger your annoyance threshold, than to memorize optimally. actually you're not even trying to memorize the proofs, but to "absorb the rhythm" of the underlying concepts. When you're proving things, there's a bunch of non-verbal instincts/"mental moves" that go on (which are pretty hard to consciously observe/reason about). The point isn't to do like supervised learning on the text of the proof and be able to reproduce it perfectly. instead, you're training yourself to practice these mental moves, in a spaced out way (spacing it out makes it non-annoying, and verifies that the ability is an instinct rather than a reflex/rote-memorized).

See also Tim Gowers on memorizing proofs, e.g. ^{[2]}, ^{[3]}