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Someone who is really good at math (and did algebraic geometry in high school, 3x Putnam Fellow, 2x IMO Gold and currently doing world-class research in algebraic geometry) wrote (while he was still in high school !) this in regarding to learning math somewhere

Virtually all of undergraduate math can be learned by people with no actual ability to understand anything, hence why there are quite a few graduate students who have no real understanding of what they learned as an undergrad.

EDIT: To show what I mean by this:

For example, the proof of most theorems in later real analysis/basic functional analysis can be looked at in two ways:

  1. This proof is a bunch of lines such that each one proves the one after it!
  2. This proof is a set of ideas, relying on this hypothesis here and this hypothesis here, that cannot be generalized without overcoming _________, that has importance because of it's relevance in ________, is motivated because __________, and shows the power and limitations of ____________.

A lot of undergraduate education is focused on "rigor", so you end up with students who can see 1 but not 2.

I am reading basic group theory (from Herstein) and Analysis (from Abbott), and it's clear to me that I'm doing only 1. What to do so that I can learn to do 2 ?


Also, here's another thing he wrote (regarding contest math, viz IMO preparation)

You're thinking too much about PREPARATION.

You learn these ideas by just exploring.... if you see a probabilistic method with 1/2 for the first time, you should be asking 50 thousand questions, and one of them should be "what if p is a value that isn't 1/2?"

So many people spend 5 hours a day preparing and then lose to people who prepare 0 hours a day, because those people who prepare 0 hours a day, even though they don't do any formal practice, they just explore and find stuff.

How do you learn to explore and find stuff ?

PS: Please don't reveal his name here if you recognize him from his description.

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    $\begingroup$ He (I bet he's not a she) sounds like a bit of a smartarse.... Anyway you only get to stage 2 by passing through stage 1, so don't worry.... $\endgroup$ – Lord Shark the Unknown Jul 9 '18 at 6:40
  • $\begingroup$ @LordSharktheUnknown No well as far as I know him he is definitely not n smartarse. $\endgroup$ – cdt Jul 9 '18 at 6:44
  • $\begingroup$ You might also be interested in reading about Terence Tao's three stages of mathematical understanding. ("There's more to math than just rigor and proofs.") $\endgroup$ – littleO Jul 14 '18 at 10:54
  • $\begingroup$ @littleO That's actually a very good link, so thanks for pointing that ! But unfortunately I'm not a grad student (but a high school student) so I heavily lack the mathematical experince/maturity for doing so. For example, sentences like Among other things, this can impact one’s ability to read mathematical papers, Without one or the other, you will spend a lot of time blundering around in the dark (which can be instructive, but is highly inefficient) looks to much abstract to me because I've no such experince. ... $\endgroup$ – cdt Jul 16 '18 at 16:06
  • $\begingroup$ @littleO So do you have know any other links like this which actually teaches to find the big picture at a relatively basic level (eg abstract algebra/beginning real/complex analysis etc) ? $\endgroup$ – cdt Jul 16 '18 at 16:11
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Except for passing requirements for graduation and prerequisites for courses, there is no point in studying mathematics unless you are interested in it. If you are interested in mathematics, then explore interesting topics on your own and read books on its history. I suggest reading The Princeton Companion to Mathematics to get a sense of the big picture. In any area of mathematics look for simple examples that motivate the development of the subject. Ask yourself natural questions about what you read and then try to answer them. For example, if you read a theorem, ask why all of the hypotheses are required. Look for examples and potential counterexamples to theorems.

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  • $\begingroup$ I would like to expand a bit the idea of asking natural questions, in addition to analyzing whether all hypothesis are necessary. In combinatorics particularly, there appear some threshold numbers, above which some phenomena start or stop to happen. Ask whether you can improve that threshold. For graph coloring, It can be shown that one can color a map using 5 colors such that no neighbor countries have the same color. Can the number 5 be improved? Yes, to 4 $\endgroup$ – Newton fan 01 Aug 2 '18 at 9:26
  • $\begingroup$ A second extension: in an analogy to polishing a brute diamond into a processed diamond, theorems can also be polished, mainly by reducing their size and making them more intuitive. Ask yourself if a proof can be shortened or be made more intuitive. As a fact, most of the theorems found in books are rarely in the same form as they were first discovered; instead, the proof was smoothed in time by many other mathematicians. Credit for both ideas (this and the above comment) belongs to Dan Schwarz - imo-official.org/participant_r.aspx?id=16318 $\endgroup$ – Newton fan 01 Aug 2 '18 at 9:45

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