December 29, 2011 AT 8:31 am

What is it Like to Have an Understanding of Very Advanced Mathematics?

(above: Paul Dirac, who was totally rad and probably could’ve answered this question)

My friend Bill pointed out a terrific question recently asked on Quora: “what is it like to have an understanding of very advanced mathematics?” — something I’ve definitely wondered myself. I have a pretty solid understanding of calculus and linear algebra, but the really advanced stuff just seems kind of like magic to me, and the people who understand it are sort of like superheroes.

There are about a dozen answers given to the Quora post, but the first one — an anonymous response — is just superb:

  • 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 few powerful general purpose “machines” (e.g., continuity arguments, the correspondences between geometric and algebraic objects, linear algebra, ways to reduce the infinite to the finite through various forms of”compactness) combined with specific facts you have learned about your area. The number of fundamental ideas and techniques that people use to solve problems is, perhaps surprisingly, pretty small — see 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.
  • You are comfortable with feeling like you have no deep understanding of the problem you are studying. Indeed, when you do have a deep understanding, you have solved the problem and it is time to do something else. This makes the total time you spend in life reveling in your mastery of something quite brief. One of the main skills of research scientists of any type is knowing how to work comfortably and productively in a state of confusion. More on this in the next few bullets.

You can read the whole thing here. It’s quite thorough and informative.

Thank you, anonymous mathematician(s), whoever you are!

[h/t Daniel Lemire via Bill Ward]

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  1. I like the 2nd answer, in particular it reflects exactly how one feels with a mastery of anything IMO. I feel this way with computers and particularly Linux-based IT systems, and I often feel that way about cars (although with a healthy helping of #3…)

  2. Paul Dirac would probably have mumbled the answer then rudely ignore your reply while wandering off.

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