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To start really getting somewhere with attacking hard problems in the wild, I would guess we need to have a cursory understanding of a wide variety of math topics, and how they link together.

How much do we really need to know to get somewhere, and in what order do we need to learn it? Where can we learn how the different topics in Maths link together?

For example, graph theory, number theory, calculus, trig, complex, ... We learn each topic but rarely discuss how they all weave together.

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Graph theory seems like an outsider to me, maybe someone can correct me? – quanta Apr 9 '11 at 13:25
When you learn each topic, you usually notice and see how it connects with others... There is no eed to separately discuss this! – Mariano Suárez-Alvarez Apr 9 '11 at 15:14
It is hard to see what kind of answer you are looking for? There are standard answers: for example, read any list of requirements for a math degree and for a math PhD. Other than that, I am sorry to say that this is not a real question... – Mariano Suárez-Alvarez Apr 9 '11 at 15:16
"We learn each topic but rarely discuss how they all weave together." If I were looking for a one sentence indictment of the math curriculum in America (say from the K-16 level, although it does not magically get better as soon as you hit graduate school), I think that I would have just found it. – Pete L. Clark Apr 9 '11 at 21:02
@PeteL.Clark: I think there are many sources, not just curricular (which I strongly agree is poor). Recently my daughter (6th grade at the time) was inundated with math. homework and I offered to help; the homework was obviously (to me) intended to illustrate the $\gcd$/$\operatorname{lcm}$ relationship, but because of workload, her focus was on grinding through rather than extracting insight. She is currently dealing with lines & graphing which are presented with the same numbingly boring high-workload style devoid of 'ah-ha's. – copper.hat Feb 9 '14 at 5:17

Of course, all mathematics is the same, it just takes a while to notice the similarity between some parts.

Two suggestions:

  • learn more math particular subareas and compare them with each other/seek out applications of one to the other yourself

  • learn category theory which is an overarching extrapolation of abstract properties of different areas of mathematics and is among other things geared towards showing similarities between diverse areas of mathematics.

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+1 for category theory. – Eivind Dahl Apr 9 '11 at 15:06

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