Spaced proof review

Spaced proof review is the general idea of trying to memorize/deeply understand a theorem using spaced repetition. This is currently one of my main interests with spaced repetition and with learning in general.

I also want to use this term to mean that particular method of proof review I've settled on? which involves adding the theorem statement on front, then proof solution on back (plus a whole bunch of subtleties about deck options, how to grade yourself, how to split things by deck, etc).

Rationale

Four steps of insights leading to my current workflow of theorem cards:

1. it's a good idea to store big cards in the first place, rather than just smaller cards — this lets you actually test your knowledge
2. separating big cards into their own deck is a good idea, because it prevents breaking your workflow/rhythm while reviewing small cards
3. changing the deck settings to make review of big cards not as onerous is important for keeping the habit.
4. when you get stuck on big cards, make life easy on yourself by isolating the difficulty and making smaller cards that help you navigate those difficult points (e.g. difficult steps/insights in proofs).

Analysis

The main problem with "state this theorem" and "prove this theorem" cards is that if your theorems are actually non-trivial, then it will take some time to write them down and so forth — so if you mix them in with your typical anki cards, then it will break the flow of review. So my current idea is to have a separate deck that's really slowly moving (only 1 card per day (edit: a better idea is to do a fake "review" as soon as you add a card, and to add cards on the day you first do an exercise)) and with a long time interval (i'm thinking something like 1 month, 1.5 months, 2.5 months, etc. (edit: this turned out to be horrible)) that just contains some important theorems and their statements. Previously my idea was to do this separately from anki — maybe in an orgmode file or setting up an email reminder — because i had made this theorems deck in anki many months ago but felt ugh-y about actually using it. Then i realized that the reason i felt this way was that the 5 cards per day or whatever, plus the shorter intervals meant i would be proving many things many times, which i didn't want. The idea is to make things fun! and one way to make them fun is to make the intervals longer, reduce your work load, and add only fun theorems.

i start to notice a bunch of subtleties about a proof once i put it in the deck. There are a couple of reasons for this: (1) i'm forced to actually do the proof now, instead of just reading, so by doing the proof and comparing with the one in the book, i notice the differences, in particular the places where my proof is lacking. (2) because i forget some of the details of the proof, i start coming up with some alternative ways to do some things. This is like how andrew wiles says that having a bad memory is important for doing math, so that you forget how you got stuck so that you can do things a little differently next time. To give an example of this that i just experienced: i was proving the "length of linearly independent list <= length of spanning list" in axler. i didn't notice until now how subtle to use of the linear dependence lemma was. in particular, you need both parts (a) and (b) of the lemma, and it totally matters that the j is the same in both (a) and (b). until now, it wasn't clear to me why the j had to be the same, and also (b) seemed like an immediate corollary of (a) so i didn't see why it mattered that axler stated both. But now i see how all the parts of that lemma are used in this proof, which motivates things.

I think a key difficulty of mathematics is "combinatorial explosion" or the problem of alternatives. e.g. if a proof uses ≤ in one place instead of < or some other inequality (e.g. reversed direction) then you, as the reader of the proof, must be prepared to explain why. With each piece of the proof, you have to be able to say why the author did one thing instead of the other. The end result is that you're not just memorizing the short string representing the proof — rather, you're internalizing this much larger space of "why the author did this thing instead of this other thing which doesn't work", which can guide you when you write the proof yourself. Why is this relevant to Anki? Well, i feel like one really good way to actually experience all these alternatives in proof-writing is to attempt the proof yourself at spaced intervals. As you start to forget pieces of the proof, your mind goes like "wait, so here, should i do this or this other thing?" and so you check both, and then you get this feeling of like "oh! so that's why the book does it like that".

the spacing part matters for two reasons. first, you want to be able to prove these things at any point in the future, so the only solution to that is to just keep reviewing it over the course of your life. second, it's important that you forget some things in the proof. you might never forget the main idea, but there might be a small trick somewhere, and you can appreciate it way more if you forget it once and then reinvent it yourself later.

when does a problem deck not work? e.g. i'm worried that this style only works for things like real analysis where there are relatively simple theorems you can prove. what about something like belief propagation, where the "meat" of the content is in a long derivation of the algorithm/formula? and also i'm still not sure about things like godel's first incompleteness theorem: you can put in the diagonalization lemma, but what about all the tedious arithmetization work, which is different in every single textbook? do i just pick one and really internalize it? or should i just forget about internalizing it?

one of the mistakes i made earlier is to add theorems from books that i hadn't fully processed. this meant that when it came up in the deck, i had more work to do then. but now i think it's better to do all the work when first making the card. then you know that everything in the deck is something you've deeply processed. there is a kind of relief, rather than a slight panic/dread of "oh boy, is this the day when i'll get that horrible card?" i also now think that putting the full solution on the card, rather than just a reference to a textbook or web page, is better. it keeps everything self-contained, so when you're reviewing there is less work to do. in general, the idea to make reviewing fun.

See also

• Spaced proof review as a way to invent novel proofs
• Spaced proof review as a way to understand key insights in a proof
• Linked list proof card -- this is sort of the opposite of the "spaced proof review" method