How Do You Actually Do Your Mathematics?

In many ways, I am atypical in the way that I approach a problem, but it works for me. Specifically, I try to understand an example in as much detail as I possibly can. If the example, is too complicated, then I make a simpler example. As much of the intricate detail that I can bring to bear on the example is brought.

For example, instead of trying to understand Lie groups and Lie algebras in general, start with the circle and the line that is tangent at the point (1,0). What is the exponential map? Oh, OK. Now how about $SU(2)$ and $su(2)$? Can you understand that the Lie group is the $3$-dimensional sphere? Can you understand the coordinates? Can you understand the equators? How do $i,j$ and $k$ really work? What is the difference between the multiplication rule $i\times i =0$ and $i^2=-1$?

I spend time pondering. And often my notebooks will contain tangential problems or specific computations. I will keep doing the computation until I get it right! If necessary, I will write a program to complete the computation. When I understand the example completely, it is usually easy to abstract.

Then I follow up, usually writing in a notebook or several notebooks before I begin writing on the computer. I have an advantage in that I have long-distance collaborators, so it becomes necessary to explain the idea to the collaborator(s). That is the first writing stage: write for someone who knows your short-hand and your metaphors. the second stage is to write for someone who does not. Then I write with a set of colleagues in mind, but I assume the colleagues do not remember anything from the previous work. I also try to explicate the notation writing for example "the function $f$, the knot $k$, or the tubular neighborhood $N$.

A complex analytical colleague only uses $z$ for a complex number, $x$ for a real variable, and $n$ for an integer. These variable choices are culturally determined, and so one keeps with the culture of the discipline unless there is good reason to deviate. As a final example of this, the variable $A$ in the bracket polynomial is known to everyone in the field. The variables $q$, $t$, $X$ etc. are less known and involve different normalizations. So it is the burden of the author to relate these to the more well known choices.

Though I left the comment about Gauss saying that "Notions are more important then Notations", I find that when I do mathematics (in my case more of computational and applied mathematics) having good notations/symbols helps me out. Good notations/symbols in some sense makes you "lazy" and if your notations/symbols is good enough, all you might need to do is some "blind" algebra without actually thinking deep. It is like having good variable names when you code up a computer program. You want the variable names to be suggestive and at the same time not too long or messy. I sometimes get too finicky with notations and symbols. If I find that I should have used some other notation or symbol at the beginning, then I actually take pains and the efforts to go back to the beginning and change them throughout. Having good notations/symbols I believe lets your mathematical thought process flow smoothly, especially when you are explaining to others.

Another habit which I believe is helpful is to always write out your thoughts and ideas. I always face the problem that if I don't write them out, all the thoughts and ideas in my head gets disorganized. If you want to learn a language, you need to speak in the language to learn it. Similarly, if you want to learn Mathematics, you need to think and write mathematics to learn it.

Another thing which I usually do (which sometimes lead others to frown upon me) is to write out the details and pay attention to details, even though it might seem trivial. I understand that this is an infinite process i.e. writing out all the details. Somewhere we need to draw a line, between paying attention to details and keeping track of the overall problem.

Also once every quarter, I take a look at all I have done over the last quarter and "try" to get them in order and organize them. (though I have often failed on this front of organizing them in some nice order).

Another thing I find, especially when I study pure mathematics on my own, is that I am totally lost to find out why certain things are even done or discussed about in the first place. I still remember the day when I did my first rigorous real analysis course when they defined, a closed set as "A set which contains all its limit points". I was like "what?". I was asking myself "What has an innocent looking sequence and its limit got to do with defining a closed set". Where and how did the limits come into picture?

I think this is where a good teacher can help by providing the right motivation for why things are done in a specific way. In fact this is one of the main reasons why I visit this website daily. You have a lot of wonderful Professors and some great students here actively participating and answering questions raised by students like me. Reading some of the questions and answers are really enlightening even though you might have already looked at the problem. Each answer gives you a different perspective of the problem which essentially helps to understand the problem and its intricacies better and helps to understand why things are done in a certain way.

Another important thing to do when learning or doing mathematics is to look for counter examples. I really believe counterexamples play a crucial role in understanding a problem better. Sometimes counter examples help to understand a problem/proof better than the actual proof itself.

How do good mathematicians format their work on paper so as not to get lost in the is, js, and ks and keep track of what they're investigating?

To be honest, part of this is just getting used to juggling several things in your head at once. Fortunately, as with many other skills, this is trainable: try, for example, doing Sudoku puzzles with a pen.

There are, of course, other ways to keep your work organized. The first thing to do when solving a problem is to write down all of the data you're given and write down the goal of the problem. The second thing is to unravel all of the definitions, working through all the quantifiers. This will get you surprisingly far, at least in real analysis. And then the third thing is to actually think about it. If you can't hold all the quantifiers in your head at once, practice reading the statements aloud slowly until you can (and see the first paragraph).

When good mathematicians get intuitionistic ideas, what (explicit) steps do they take to formalize them, especially when it is likely that first idea is murky or wrong?

Write down an argument that follows the lines of the intuition, at least for a simpler case or version of the problem. If it doesn't work on the simple version, it probably won't work on the hard version. If it does, you can try to figure out whether it extends to the original question and, if not, what the barrier is.

This is easier said than done. The bottom line is you need to practice the skill of turning your intuitions into proofs. This comes in a few steps: first you need to learn how to prove things, then you need to learn how to train your intuition to help you prove things more easily. Like anything else, this takes hard work and practice and there isn't a magical shortcut.

Terence Tao has written some very clear stuff on this and related subjects.

How do you [professional and aspiring mathematicians] organize your math "notebook", and what perhaps idiosyncratic methods do you employ to be original and clever within it?

I'm not completely sure what this means.

There seems to me a bit of a jump between the dryly algorithmic way one is taught to do math in high school and the more abstruse methods at the undergraduate level. I'd be interested if anyone here could help me bridge that gap, if only for my personal fulfillment.

Many people I know bridged the gap through competitions such as the AMC. Training for competitions is not everybody's style, but it is a chance to be exposed to interesting topics not covered in the high school curriculum and also a chance to hone problem-solving and proof-writing skills (at the Olympiad level). A book I benefited from enormously while doing this is Engel's Problem-Solving Strategies, which is geared fairly specifically to Olympiad preparation but is a great source of problems and elementary techniques for solving them. For more general advice, Polya's How to Solve It comes highly recommended (although I have not read it myself).