This is sort of a general question but how does a mathematician do his/her work? Specifically, does he try to memorize every theorem he comes across or just the important ones? (If the latter, how does he say which are more important?) If he uses a reference book of some sort daily to do his work, then what exactly does he learn by going to school to learn all of this for, if he can just as easily look it up (I've gotten the idea that he learns some kind of "mathematical maturity" but what exactly is this)? Well, I've got a lot more questions but my biggest one is whether a mathematician tries to memorize all the (important) theorems he comes across and how he decides which are important.
I'm not a mathematician [ yet? ], but after spending time here in PF, I kind of got the impression that a student gains an interest in a certain field of mathematics during the undergraduate years. Then he/she pursues and studies that [ in a more specific fashion ] in graduate school, and by the time he/she obtains a Ph.D., he/she would be familiar with the major and other unresolved questions in the field. And after getting the Ph.D., it's not like the process of learning is over already.
I'm also not a mathematician (yet?). But I think I can answer your main question -- which seems to be the question of mathematicians memorizing theorems... Theorems are memorized throughout the process of learning, studying and doing mathematics. New theorems are forged in the same manner, through the process of doing mathematics. I can say for myself that as I go along studying and doing mathematics many theorems are sort of taken for granted. I mean they become almost intuitive and we forget just how important and beautiful the proofs of said theorems are. A few good examples (for me) are the fundamental theorem of calculus and L'hospital's rule. These things were huge topics in calc I and now have become almost second nature to me; so that when I see a definite integral or a limit with an indeterminate form I just use these theorems right away, without much though. But, that certainly does not mean that the proofs for these theorems are any less important or less beautiful than others. This may be an aspect of the "mathematical maturity" that you spoke of.
I'm by no means qualified to answer this question, but I can tell you the impressions I've gathered from talking to my professors. The person I consider my math mentor, basically hates to memorize things. His view of mathematics is that there a few core concepts/ideas in mathematics, that everything else is built on in some fashion or another. The pattern of thought, the creative development that he seems most interested in, and this does not involve memorization. Of course occassionally it would seem you would need to refer to other published works to keep up with the twists of your field, but in my personal opinion a person who spends more time reading other people's papers and looking up theorems is not doing mathematics. Which leads me into my view of mathematical maturity. Mathematical maturity is the ability to create mathematics. It is the ability to of course do proofs, but proofs in their raw form are not structured, a -> b -> c. They are the mathematician noticing something in the cosmos of mathematics, this pattern, something that recurs. Then they ask why this happens. A good proof will answer this why, a mature mathematician will be able to answer this why. Sadly, all too often in text books, we get the refined proof product. This is like walking over a bridge, but the original mathematician had to build the bridge. Text books never explain why. They usually use some bag of tricks to get to some desired conclusion. And in my opinion, that sort of mathematics is not mathematics at all. Mathematical maturity is being creative. Its knowing the why. That is just my opinion. EDIT: Also, I don't know anyone who given a list of definitions can still understand what the hell something means. Theorems can be complicated pieces of information, and you have to know how to forumlate the ideas. How to make sense of them. The poincare conjecture, what the hell does this mean: (I, of course, have no topology experience, but I am trying to make my point.) Consider a compact 3-dimensional manifold V without boundary. Is it possible that the fundamental group of V could be trivial, even though V is not homeomorphic to the 3-dimensional sphere? Understanding what the theorem is getting at, is a much more sophisticated process than knowing what the theorem says.
This is quite a well-said post. I believe, whole-heartedly, that mathematics is a creative, almost artistic, process -- more so than a scientific one. Many disagree with me, but for me this proves true. I don't do math to prove anything, more or less, but to solve a problem using the most creative method that I can. And to me, there is true beauty in that process and the results that are produced.
I completely agree with someperson05. Of course, you won't get very far in mathematics if you view it solely as an individual endeavor. Part of being a mathematician is learning about and understanding the structures other mathematicians have developed before you. I heard a story once which reflects a philosophy I once had towards life. Basically, once there was a man who lived in some cave in the mountains without formal education in mathematics. One day he stumbled upon a basic algebra textbook and took to reading it. As he learned more and more, he began to see what math was all about, and began to be able to predict what the next ideas in the book were going to be. Even after he finished the book, he continued to construct idea after idea on top of what he had already figured out. When people found him later, he had deduced in this manner years and years of mathematical knowledge all on his own. While one can certainly say that such a man has intelligence, I don't believe anymore that this is the way to approach things. An important part of mathematics is humility: to recognize the achievements of others. If you believe yourself to be so intelligent, then you should take as your primary motive to understand, as quickly and wholly as possible, the fields to which you are to make your brilliant contributions. An opposition to "memorization" can very easily turn into an opposition to learning things which aren't immediately intuitive to you. This is a grave mistake, and so I make it a rule to NEVER let myself not understand a mathematical idea that I stumble upon (unless there is a lack of time and/or prerequisite knowledge I lack).