# Studying theorems in mathematics

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?

• 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. Mar 7 at 9:41
• 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. Mar 7 at 9:51
• +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. Mar 7 at 10:06