[Math] How to read an article and make it actually useful

Question

I've been wondering for a while: how should mathematicians read an article in order to "take most" from it?

For example, when I did my Master's thesis I based it on an article (I'm into analysis) and of course I analyzed every and each part of it, extending some results in there and filling some gaps that were "left to the reader" I guess, or were thought to be sufficiently trivial by the authors.

Now I'm doing my phd and I have to choose a specific topic (more or less I have an idea but I haven't exactly set up my mind yet). The fact is I've had to look up some articles to get some ideas and possible topics of research, and by now and I quickly understood that reading and trying to understand the whole thing is impossible. I mean they often have like 60+ articles/books in the bibliography, so I mainly read the introduction and the results, skipping proofs entirely (or almost entirely). Basically what I try to do is to get an idea of the general path taken by the authors, skipping all the technicalities are ok and noting on the side the techniques/results mentioned that I don't know about. Then I quickly look up on the internet what the idea of these techniques is, and that's it. Obviously, a couple of days pass and most of it is gone, except maybe the very very general idea/result it obtained (but only if it isn't too technical).

I mean it seems a bit shallow, but I can't come up with with better ways to read them, they are so stuffed I really can't keep up. So how would a professional mathematician read an article about a topic he's interested in without going crazy, trying to learn most from it? Maybe in n years from now if one continues this path one can expect to be so well-versed in a very specific topic so that research articles about it become way easier to read?

Answer 1

I like this anecdote involving Hassler Whitney.

I worked with Hassler Whitney for two years at the Institute for Advanced Study, and the mark he left on me goes deep. I guess I'm attracted to unconventional, original sorts, and Hass was surely that. His undergraduate days were at Yale, so one might expect that here was just about the most incredible math major ever. But his major was music, not math. Well, then, he must have taken all sorts of math courses, and.... Actually, he took almost no math courses. Mathematically, he was largely self-taught.

Anyway, one day, in his office, I happened to mention Bézout's theorem, which basically says that two curves of degree $n$ and $m$ intersect in $nm$ points. He says he never heard of it (Bézout's theorem is in fact highly under-appreciated), and seems galvanized by it. He jumps up and heads toward the blackboard, saying "Let's see if I can disprove that!" Disprove it?! "Wait a minute!" I say, "That theorem is nearly two centuries old! You can't disprove anything... really..." As he begins working on some counterexamples at the blackboard, I see that my well-meant words are simply static.

His first tries were easy to demolish, but he was a fast learner, and ideas soon surfaced about the complex line at infinity, and how to count multiple points of intersection. After a while, it got harder for me to justify the theorem, and when he asked, "What about two concentric circles?" I had no answer. He argued his way through, and eventually found all four points. Finally he was satisfied, and the piece of chalk was given a rest. He backed away from the blackboard and said. "Well, well—that is quite a theorem, isn't it?"

I think I mostly kept my cool during all this, but after I left his office, I realized I was pretty shaken. I remember thinking to myself, "Golly, Kendig, you just saw how one of the giants does it!" He'd taken the theorem to the mat, wrestled it, and the theorem won. I'd known about that result for at least two years, and I realized that in 15 or 20 minutes, he'd gained a deeper appreciation of it than I'd ever had. In retrospect, it represented a turning point for me: I began to think examples, examples. Whitney worked by finding an example that contained the essential crux of a problem, and then worked relentlessly on it until he cracked it. He left it to others to generalize. It is to Hass that I affectionately dedicate this book.