How Do You Actually Do Your Mathematics?

Better yet, what I’m asking is how do you actually write your mathematics?

I think I need to give brief background: Through most of my childhood, I’d considered myself pretty good at math, up through the high school level. I easily followed mathematical concepts introduced in my classes and even did a few competitions; I definitely wouldn’t say I was a star of the caliber one meets when one ventures out into the bigger ponds, but I thought I was decent and convinced myself I would major in math when I entered college.

That changed after a couple of years when I hit my first Real Analysis class that used Rudin’s book; that was the first class, I think, I took that really required more than “expand-a-definition” type proofs and my struggle to find intuition and understanding there impacted my mathematical self-confidence. I eventually switched majors, with a bit of regret.

One thing that got me, I think, was the veritable explosion of superscripts and subscripts that one encounters for the first time in Real Analysis. I’d often find myself struggling to set up the machinery of what I was trying to prove, lost in the notation. How do good mathematicians format their work on paper so as not to get lost in the $i$s, $j$s, and $k$s and keep track of what they’re investigating? I remember dealing with subsequences of sequences to show that limits did or did not exist got especially hairy in this way…writing things like $s_{n_{k_{\epsilon}}}$ and remembering what my goal at each “level” was difficult. I’d be interested in knowing if aspiring mathematicians and/or professional mathematicians scribble marginalia or have a system to overcome such problems.

Another thing that got me were what I personally called “consider…” statements. Many times, on this site, the most talented commenters will say “Consider $f(n)$” or “Consider transformation $T:$ $U \rightarrow V$” that in the first case gives a summation that wonderfully telescopes/has an obvious bound, or in the second case transforms the problem into a trivial application of the rank-nullity theorem, or something like that. Mathematics is a subject replete with geniuses , I understand that, but how do mere mortals investigate such functions and “massage” them into doing what they want? 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? (Aside: I’ve been given “use numerical examples” as advice before, but sometimes I think to myself, “I’ve been dealing with $\mathbb{Z}$ since I was 6 years old, and not so much with Dedekind’s definition of the real numbers…”)

There’s lots more I could ask, but I want to keep this question tractable, so I guess I might summarize by asking: 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 know there will be no strict formulas anyone can give; mathematicians are scientists of the abstract; I understand that the subject is acclaimed partly because it’s so intellectually and individually demanding. But I think even acclaimed scientists draw on Springer’s Protocols and Nature Methods…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.

(Apologies in advance if the question is ill-posed or too subjective in its current form to meet the requirements of the FAQ; I’d certainly appreciate any suggestions for its modification if need be.)

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.