Does one have to be a genius to do maths? (2007)

https://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/

by vinchuco • 11 years ago

52 7 11 years ago

terrytao.wordpress.com

Lejendary• 11 years ago

I believe, not per se a genius, but very discipline to practice a lot.

agumonkey• 11 years ago

And a taste for this generic beauty, twisted compaction, underneath mathematics.

gruez• 11 years ago

needs a (2007) tag.

lovelearning• 11 years ago

I think year tags are only for technology related articles, since they tend to become outdated quickly.

clintonc• 11 years ago

I disagree; I clicked this to read it and was dismayed to learn that I had read this YEARS ago. Most posts are brand new content, so all old content should be marked, I think.

eli_gottlieb• 11 years ago

Math sort of has to be learned in the opposite way from how it's written. It's best written "short and sweet", compressing as much meaning as one can handle precisely into a single theorem for maximum generality. Human minds, though, learn mental models of how things work from relatively concrete examples, ones which require only a little new information to be absorbed in order to gain the new concept.

A lot of things you thought were separate and overly-elaborate start seeming simple and small once you actually work your way up enough levels, to have seen enough examples, that the generalizations make sense to you. But that requires a lot of work!

Billesper• 11 years ago

Agreed.

I remember being a freshman and having my first real exposure to proofs and pure math, a calculus class. We used the text "Calculus" by Spivak. The first actual calculus topic was one of, if not the most difficult concepts in the course: the epsilon-delta definition of a limit. That was a struggle. I had no idea about the significance of logical qualifiers and precise mathematical language like "for all" and "there exists," or what it meant to "choose" or "fix" a value, much less the ability to parse all of the those things combined in a complex statement. Even familiar and elementary concepts like absolute value and inequalities were sort of difficult in that context. We very briefly covered basic proof techniques and formal logic, but overall it was intractable at first. But one by one, you learn individual concepts and notation, such that you are eventually able to understand their fully significance without really having to think. Looking back, and after taking other courses and studying proofs/logic/sets/functions more in depth, epsilon-delta makes perfect sense.

I've learned to tackle theorems and proofs (or any kind of problem, really - in math or otherwise) by starting with working out a few of the simplest possible meaningful example(s), or just any valid example if you can't quickly determine that. If you are learning a new formal concept or idea, I find that it's often very helpful to do a few applied problems first if you have any trouble. If you can identify individual concepts used in a theorem, try to master them first in isolation before tackling the overall problem, if need be. The key idea is to break things down to their smallest units of understanding.

I find this comment by Terry Tao fascinating:

"Ramanujan, for instance, apparently performed a tremendous number of numerical computations, and derived much of his intuition from the patterns he observed from those computations."

I think I used to assume that most of the best mathematicians could read a new theorem or idea, and internalize all of the significance and intuition by simply using logic and working only with the abstractions. People have different ways of thinking, but now I'm not sure if anyone, even a genius, can do that on a truly difficult problem. One needs to avoid blindly "plugging in numbers" or over-generalizing from examples, but working out concrete things and then asking questions, considering cases, inferring, and making hypotheses based on examples seems to generally be the most efficient way to tackle a new problem or idea.

Does one have to be a genius to do maths? (2007)

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