After I took my first analysis course and learned how to be truly "pedantic," I have always been having a dilemma of balancing myself between being "detailed" and "intuitive". I would like to ask how other Math SE people manage this problem. (If this is a duplicate, I would appreciate the link to the similar question, but I also want to ask some following questions if anyone answers this, so it is not completely duplicate.)
Of course, being detailed and intuitive should not be a duality, and I think anyone needs both aspects when it comes to learning mathematics. So, more specific question is: How do you fill up your details?
This question may be understood easily if one considers a very fast paced course or a book that skips many details.
For me, I have tried
- To write (TeX) up entire topic in my own notations and details (e.g. http://gycheong.wordpress.com/)
- Make details that I could not directly see as easy exercises and just prove them (on papers neatly and collect them).
- Read a chapter very briefly and accept whatever the author says without too much thinking and reread it very carefully.
So far, Method 1 (which I have been using for more than a year) hasn't been working very well. It took too much time and, to me, it started to seem like understanding and writing were not exactly the same. I think Method 1 is good when someone wants to review a course that he/she understood (but not always a good way for the first learning). I have switched to Method 2 for about a month and just started to use Method 3 for some subjects that interest me but are sides. Method 2 has been working very well and it also gives a solid background for dealing with more difficult problems that do not trivially follow from theories (again, this is just my thought as the word "trivial" is different for everyone).
I think I am safe to say I am an "average" undergraduate senior in math and I am hoping to continue mathematics as graduate level this September. I really hope to see various opinions from a variety of people in different stages of mathematics.
Best Answer
Let me first refer you to Terrence Tao's wonderful post on this subject.
As a person who is just starting to exit the hyper-pedantic stage in a few sub-fields I both feel your pain and feel that I might have a little bit of insight from the other side. Let me start off by answering your question: I fill in the details slowly and repeatedly. I am particularly fond of von Neumann's famous line that in mathematics we 'do not understand things [we] get used to them'. Learning anything in mathematics is a process, not a result--it never ends. I do not feel there is much of a point in trying to do everything at once. I am not sure that there is a best approach to this, but I can offer the one I typically use.
To start with, I always try to find the intuition. Forget rigor until you have a good moral reason why you think the statement is true or false. If you get stuck, ask someone who works in that field how they think about it (I pretty much always do this even if I don't get stuck). Find a special case you think might tell the whole story or draw a picture you think is indicative of the general result, then see if you can make the intuition rigorous yourself. This may take a while, so don't hesitate to read and try to understand the proof in your book if you need to move on, but keep working on the problem. I really want to stress this: it is much more useful to find out where your intuition fails than to learn another theorem.
Now suppose that you have found your own argument and proven it. Look at the author's proof and ask why they chose the argument they chose. If your intuition is good or the result is trivial, then your argument should be essentially the same as the book's. If not, compare the two proofs. Is the book's solution cleaner than yours? Is it faster? Is it more elementary? Double check to make sure you didn't miss any details. It's worth going line by line and asking yourself why the author is doing all the thing they are doing. If you didn't come up with a rigorous proof of the intuition, this is even more important. Look at the proof and see if you can work backwards to the intuition that led the author to write the proof the way they did.
While I think that it is worthwhile to fill in the details of the author's argument, that is much less important than learning how to think like a professional mathematician. This is where the repetition comes in: come up with a different heuristic and see if you can make that rigorous too. Constantly compare what you are doing with what the book does and ask yourself why the author takes the approach they take. Keep reading then periodically go back and see if you understand better why the material is structured the way it is. In class, don't be afraid to ask (yourself or the professor) why you are using one argument style over another.
Let me also comment that there will come a point when you cannot fill in all of the details in your textbooks. Some of the time, it will be because the textbook is just plain wrong. This is especially common in higher level books, though there were some notorious examples in the last version of Royden (which were only removed a couple of years ago). If you get stuck, don't be afraid to work with others or ask questions of people who understand the material well. Mathematics is collaborative. Very few people succeed in isolation.
As a final note, I spend an enormous amount of my study time thinking about mathematical aesthetics. Style is real and it is important. Typically beautiful arguments are the ones that are clean and generalize--in short, they are the arguments worth learning. Don't be afraid to use secondary sources to supplement your learning, but make sure you find authors who have a good sense of mathematical style. Try to use works from authors who are known to write well. This will also help you develop a better big-picture view and learn new proof techniques (both of which help your intuition and rigor).