How to learn a great number of theorems by heart?

Imagine you have ten definitions and you want to learn them by heart. It is easy - definitions are somehow unique. But, imagine 40 (60,100,1000) theorems that all look somehow similar and are all important. How would you learn them by heart? What the word "learn" mean here for you?

Of course, you can learn them by numbers, and everytime I will say "Theorem 147" you will say the right one. But somehow it is not right. We do not call people with their birth date, do we?

If so, how would you categorize the (limited number of) theorems? What would be your approach to learn tens of theorems by heart and still clearly distinguish between them? What makes a theorem unique? Could you draw a graph or a tree of theorems?

• Repetition. Do exercises that force you to recall and apply results as well. Nothing will imprint a result in your head like sitting down and trying to recall the needed hypotheses. – Rellek Jun 9 '17 at 22:41
• There's no magic trick. Don't bother learning theorems word-for-word. Try to remember the tricks and techniques more than the theorems. The best way to remember the hypotheses for a theorem is to think of counterexamples when the hypotheses are removed - it helps to keep a bank of objects with strange properties, like, the Cantor set. – Jair Taylor Jun 9 '17 at 22:45
• Also, another guiding principle is to try to imprint an image in your head that represents a technical result. Two excellent examples of this are with Urysohn's Lemma and the Rising Sun Lemma. Both of these have rather unilluminating statements that are easily represented by a simple mental image, allowing you to recall the needed hypotheses by deducing what the image represents. – Rellek Jun 9 '17 at 22:47
• I feel like the point of collecting textbooks and references is precisely so you don't need to know all of the theorems by heart. Maybe some mathematicians never need to look things up at all when they're working, but they're probably in a small minority. – Mark S. Jun 9 '17 at 23:56
• @Mirko, the point is - I am given 10 theorems / sentences a day. On the one hand, I mention that I forget the theorems that I have gone through some days ago. On the other hand, during my homework I mention that I miss something, and start to search in the lecture notes until I find a theorem that could be a nice start for some proof. On the "third" hand, I do not manage to get all the proofs. So, I thought that if I cannot get them now, I could learn them by heart and understand later - when I will have time. – Kirill Jun 10 '17 at 13:35

Two things I think are important here:

1. Mathematics is/has a language. Learn the language by heart and then you will be able to formulate by heart a lot of sentences (i.e. propositions and theorems) without having to memorize almost anything. A lot of theorems that you're supposed to learn by heart are just one idea, once you're able to formulate this idea in the correct mathematical language you're almost done.

2. Try to teach the theorems you want to memorize. And try to teach them in the most simple way you can. This will certainly clarify to you what you know and what you missed of the proofs. After one or two complete attempts you should be able to demonstrate the theorem by heart.

• Hello, @Daco, thank you for your answer! Could you please explain the 1. point more or give an example? I suppose what you mean, but am not sure about the "language" you noticed. Do you mean logic, proof theory, or something else? Could you also give some example for "one idea" for different theorems? Thank you. – Kirill Jun 10 '17 at 13:20
• The mathematical language in a broad sense or mathematical formalism is just a quite exact way of expressing concepts. It involves quantifiers, basic tools, logic, way of constructing senteces, etc... as a normal language is made of grammar, syntax etc... – Dac0 Jun 10 '17 at 15:25
• If you study calculus 1 you will notice that most of the difficulties are only in using correctly mathematical formalism: once you put the correct formalism you'll usually have a straight forward proof. This can give you an hint of what I'm talking about. Once you will be graduated and seen a lot of different proofs along with some effectively big and hard-to-proof theorems, then you will easily understand what are the true ideas behind theorems and what are only plain developing of definitions using mathematical language. – Dac0 Jun 10 '17 at 15:25

"Memorization" is not the best way to try and learn theorems. And furthermore, the number of theorems you know is not actually that important. The knowledge of the proofs of theorems is infinitely more useful.

If you try to prove some of the theorems that you need to know, (or read a thorough proof, following all of the steps) that is much more effective than trying to memorize it. The beauty of mathematical theorems is that, even if you forget one, you can "reinvent" it by arriving at it again through a logical argument. Thus if you forget the exact wording of a theorem, you can most likely deduce it again from what you already know.

So my answer to you is that you should not try to learn them by heart. Try to understand them. You will likely even develop an intuition of your own about the proofs of theorems, and the proofs will come more and more easily to you the longer you continue to do this.

Happy mathing!

• But that is exactly what I mean: if you can deduce a theorem, so there is some sort of connection (a logical argument as you have said). But then you can stretch it in time and draw some graph where one follows from another. Or, you can use logic and build implications. Still, if you see that the mean value theorem follows from the Rolle's theorem - it gives understanding. The another point - theorems are often named by their number in the book or some lecture note. Could you imagine a better way of sorting the theorems? Imagine you were Google. How would you decide what to put first i.t.list? – Kirill Jun 9 '17 at 22:49
• If it helps you to visualize the "hierarchy" of theorems, then drawing a tree would be a good idea. What I'm trying to emphasize here is that memorization by brute force is not the way to go. – Frpzzd Jun 9 '17 at 22:51
• Thank you! I will think about it. – Kirill Jun 9 '17 at 22:53

There is really no point in memorizing $1000$ theorems. For one thing, different expositions of the same subject will organize the theorems somewhat differently. A particular theorem in textbook A might correspond to parts of several different theorems in textbook B, or might just be an exercise in textbook C.

• Hello @Robert Israel and thank you for the answer! If so, what does learning means? 10 theorems are given to me every day, and I am expected to know and use them in my homework. In my natural tempo I would spend one day for 2 or 3 theorems to think about them and get them, find some exercises etc. But I have to do much more than 3. So, if the "by heart" learning is not productive, what could be productive? Thank you. – Kirill Jun 10 '17 at 13:27
• Mathematics is about ideas. You should concentrate first on the basic framework of ideas and how they fit together. – Robert Israel Jun 11 '17 at 19:20
• As I see now, mathematics is about formalism. I have been studying it for almost one year and haven't seen any exercise where my creativity or ideas were asked. It is all about learning tons of definitions and theorems that you have not "invented" and about thinking how we can glue all this together. So I see more tricks than ideas and cannot say that this kind of studying stimulates the building of ideas. – Kirill Jun 15 '17 at 6:29