

Does one have to be a genius to do maths? (2007) - vinchuco
https://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/

======
eli_gottlieb
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
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.

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

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

------
gruez
needs a (2007) tag.

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become outdated quickly.

~~~
clintonc
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.

