On Understanding Advanced Mathematics

What’s it like to have an understanding of very advanced mathematics? A very detailed answer in a Quora post:

  • You can answer many seemingly difficult questions quickly. But you are not very impressed by what can look like magic, because you know the trick. The trick is that your brain can quickly decide if question is answerable by one of a small number of powerful general purpose “machines” (e.g. continuity arguments, combinatorial arguments, correspondence between geometric and algebraic objects, linear algebra, compactness arguments that reduce the infinite to the finite, dynamical systems, etc.). The number of fundamental ideas and techniques that people use to solve problems is pretty small — see http://www.tricki.org/tricki/map for a partial list, maintained by Tim Gowers.
  • You are often confident that something is true long before you have an airtight proof for it (this happens especially often in geometry). The main reason is that you have a large catalogue of connections between concepts, and you can quickly intuit that if X were to be false, that would create tensions with other things you know to be true, so you are inclined to believe X is probably true to maintain the harmony of the conceptual space. It’s not so much that you can “imagine” the situation perfectly, but you can quickly imagine many other things that are logically connected to it.
  • Your intuitive thinking about a problem is productive and usefully structured, wasting little time on being puzzled. For example, when answering a question about a high-dimensional space (e.g., whether a certain kind of rotation of a five-dimensional object has a “fixed point” which does not move during the rotation), you do not spend much time straining to visualize those things that do not have obvious analogues in two and three dimensions. (Violating this principle is a huge source of frustration for beginning maths students who don’t know that they shouldn’t be straining.) Instead…
  • When trying to understand a new thing, you automatically focus on very simple examples that are easy to think about, and then you leverage intuition about simple examples into much more impressive insights. For example, you might imagine two- and three- dimensional rotations that are analogous to the one you really care about, and think about whether they clearly do or don’t have the desired property. Then you think about what was important to those examples and try to distill those ideas into symbols. Often, you see that the key idea in those symbolic manipulations doesn’t depend on anything about two or three dimensions, and you know how to answer your hard question.
    As you get more mathematically advanced, the examples you consider easy are actually complex insights built up from many easier examples; the “simple case” you think about now took you two years to become comfortable with. But at any given stage, you do not strain to obtain a magical illumination about something intractable; you work to reduce it to the things that feel friendly.
  • You go up in abstraction, “higher and higher”. The main object of study yesterday becomes just an example or a tiny part of what you are considering today. For example, in calculus classes you think about functions or curves. In functional analysis or algebraic geometry, you think of spaces whose points are functions or curves — that is, you “zoom out” so that every function is just a point in a space, surrounded by many other “nearby” functions. Using this kind of “zooming out” technique, you can say very complex things in very short sentences — things that, if unpacked and said at the “zoomed in” level, would take up pages. Abstracting and compressing in this way allows you to consider very complicated issues while using your limited memory and processing power.
  • Understanding something abstract or proving that something is true becomes a task a lot like building something. You think: “First I will lay this foundation, then I will build this framework using these familiar pieces, but leave the walls to fill in later, then I will test the beams…” All these steps have mathematical analogues, and structuring things in a modular way allows you to spend several days thinking about something without feeling lost or frustrated. Andrew Wiles, who proved Fermat’s Last Theorem, used an “exploring” metaphor: “Perhaps I can best describe my experience of doing mathematics in terms of a journey through a dark unexplored mansion. You enter the first room of the mansion and it’s completely dark. You stumble around bumping into the furniture, but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it’s all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they’re momentary, sometimes over a period of a day or two, they are the culmination of—and couldn’t exist without—the many months of stumbling around in the dark that proceed them.”
  • You are humble about your knowledge because you are aware of how weak maths is, and you are comfortable with the fact that you can say nothing intelligent about most problems. There are only very few mathematical questions to which we have reasonably insightful answers. There are even fewer questions, obviously, to which any given mathematician can give a good answer. After two or three years of a standard university curriculum, a good maths undergraduate can effortlessly write down hundreds of mathematical questions to which the very best mathematicians could not venture even a tentative answer. This makes it more comfortable to be stumped by most problems; a sense that you know roughly what questions are tractable and which are currently far beyond our abilities is humbling, but also frees you from being intimidated, because you do know you are familiar with the most powerful apparatus we have for dealing with these kinds of problems.

(Hat tip: Chris Dixon)

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