Most math classes are 1. Learn the big concepts/definitions 2. Present theorems/corollaries 3. Prove them.
I presume it is essential to memorize the definitions and big ideas.
However, real mathematicians, do you use the theorems so much that you just memorized them all and they come to you as obvious, or do you understand the concepts so well that the theorems are naturally deduced when thinking about them and you don't think of them as "theorem 1, theorem 2, etc.", or do you actually have to refer to text when doing math? For myself, I know the main definitions - there is nothing to understand there, but when solving problems, I often search through the textbook looking for theorems that will be used to solve my problem. I feel that this is not a good method and the ultimate goal is to understand the concepts so well that the theorems are "obvious". I was wondering if this is indeed the case for mathematicians.
Lastly, how important is it to know the proofs of all the theorems? Is it important to be able to prove them without reference, or is it okay to simply take a theorem for granted once you have seen the proof once?
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9$\begingroup$ You are confusing learning with memorization. Studying math analogizes to learning to become fluent in a foreign language (e.g. French). When conversing in French, you don't want to have to translate what you hear into English, think of the reply in English, and then translate it back into French. Instead, you want to think in French, just like young French children do. Same thing in math -- you want to develop math fluency on a specific topic so that you think in that topic. $\endgroup$– user2661923Mar 7, 2021 at 9:41
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5$\begingroup$ I disagree that there's nothing to understand about definitions. I'd even say that understanding definitions is core to understanding anything at all. You can know definitions by heart without even an inkling of understanding about the intuitive idea those definitions are supposed to formalize. How can you even hope to understand the deeper connections between objects if you don't even understand what the objects are? If you forgot the formal definition of a concept, but you could still write down an at least equivalent definition just by thinking about it, then you understand the definition. $\endgroup$– VercassivelaunosMar 7, 2021 at 9:51
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2$\begingroup$ +1 : for your query, from (-2) back to (-1). Surprising that others would downvote a deep and very relevant question, as far as the study of math is concerned. Personally, when upvoting/downvoting your query, the fact that you don't have the perspective of a professional mathematician is irrelevant. $\endgroup$– user2661923Mar 7, 2021 at 10:06
1 Answer
I think it is really close to know the theorems and their proofs in the following sense.
When doing mathematics (whether it is an exercise sheet or research) you often want to prove things by reducing it to things you know. So it is helpful if not necessary to know the theorems of the subject you are working in and have an idea how and why they work. By this I don’t mean remembering proofs in full detail, but rather a sketch like extend a basis of the subspace to the whole space. Some theorems are crucial for a theory to work out (eg. existence of bases, extending bases and being able to define linear functions on a basis) and not knowing them is equivalent to not knowing the subject at all. But even if you cannot apply a (maybe less crucial) theorem per se, its proof may use methods/tricks applicable to your problem.
In fact it is useful to have this sort of knowledge of every bit of mathematics you learn. Great mathematics often comes by connecting seemingly different areas of mathematics. I mean what on earth has something rigid algebraic like group theory to do with analytic things like measure theory and Fourier-transformations? Some mathematicians knew enough about both topics to notice similarities of the objects/concepts/proofs and made a connection. These sort of connections allow new ideas to flow from one subject to the other, which is a really great thing.
There is a last kind of mathematical knowledge that is very common in research. It can become impossible to know each and every paper in and around your research topic, as new ones appear on a daily basis. But you can have an overview over the progress in the area in that by reading abstracts and attending conferences you are up to date in what progress and what connections were made. If you happen to need some statement you might recall that you read/heard something along the lines, locate and study it in more detail.
TLDR: As a student know the theorems and have a sketch in mind of how to prove them. You will need the ideas and methods later, sometimes even from different subjects combined. In research it is often more important to know that something is true than how it was proved.
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$\begingroup$ Would it not then help to have every math test be open book. Where you get a list of the all the theorems, corollaries, etc. At the end of the day, math is not about memorization. $\endgroup$ May 7, 2023 at 8:19
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$\begingroup$ But you need to memorize things. Overexagerating, your suggestion sounds like knowing the reference is the only thing needed. But just knowing of the stacks-project or SGA doesn't make you a master of algebraic geometry. You need to know how things work and since math is quite a big construction this involves memorization. Of cause you need to understand and memorize. $\endgroup$ May 7, 2023 at 12:52
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$\begingroup$ Personally, I think giving enough time to think and admitting a homemade cheat-sheet is the way to go. For example imo it is not important to be able to write out every formula from the top of your head. You should vaguely know how they look like and what they are used for. Writing your own cheat-sheet ensures that you do. But not being able to look everything up ensures that you have a broad understanding of the subject. $\endgroup$ May 7, 2023 at 12:57
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$\begingroup$ When I say open book I mean only allowing to bring certain things with you. For example key theorems, corollaries, definitions. Sometimes, I found in undergraduate courses I would understand the material but found it would be harder to solve a 'proof problem' without a list of such things, especially for final exams or tests that covered a lot of content. You also have to remember that some people are not great at memorization/don't have a great memory in general but are excellent problem-solvers and at the end of the day math is about problem solving. $\endgroup$ May 11, 2023 at 22:23
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$\begingroup$ Then I misunderstood. I was used to the term "open book" meaning that you can literally use anything printed (lecture notes, books etc) and I don't think this is a good thing... $\endgroup$ May 12, 2023 at 9:01