Study Strategy: broad/depth; abstract/intuitive

Question

Research Interest: Chaos, Dynamical System, PDE.

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Study style:

• Shallowly Broad: Quickly absorb as many branch as possible [master more 'toolbox']

OR

• Narrowed Depth: Carefully focus on narrow interest as much deeply as possible.

It's said one should master many tools on hand to solve complicated problems, but will it decrease the depth/mastery level ?

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• Purely Abstract: Go for abstractions theoretically as highly as possible[math developed from itself]

OR

• Applied Intuitive: Go back and forth to the nature, to try to get new idea[math developed from nature] ?

We all saw in recent decades/centuries the rigorous/abstraction has developed a lot, it's reasonable. But some giant mathematicians also gave the warning for that

e.g. Felix Klein[He means maths, for its beauty, placed in billboard is admired by 'connoisseur', but it's originally sharp weapons to fight against the heavy enemy, but people gradually forget this original use.]; Arnold[On teaching mathematics]; Kolmogorov[He more think the maths as a whole organic entity, should not be mastered seperately]

Terry Tao Blog - There’s more to mathematics than rigour and proofs

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In all, it seems to be a dilemma, should we say the better way is to do mathematics both broad and deep, both purely abstract and applied intuitive ??

Answer 1

If by "swallow dates whole" you mean "studying some subject or doing something without really seeking to understand it", as I found at http://www.cherriyuen.com/Idioms.php?idiom=123&keyword=& , then in general this is not good advice for mathematics. In order to apply a mathematical technique effectively, especially in a new area, a good understanding of it is often necessary: in particular you need to know what conditions are needed for it to work. You often need to modify the technique somewhat to apply it to a new situation, and so you really need to know how it works in order to see whether a modification will still work.

Answer 2

Every area of research is competitive because there so many people doing research nowadays compared to, say 18th century. Many people can be working on the same problem or conjecture and so it is important to go to conferences otherwise you would not know what other people are doing in your area of interest. Many undergrads usually think that if some area of math or physics is well-known then mathematicians or physicists must have discovered everything and there is nothing left. This is not always true. Consider quantum mechanics. Some time ago physicists thought there was nothing else to discover -- classical mechanics was perfect. But things like the photoelectric effect did not make sense in terms of classical mechanics. Physicists did not even want to believe in atoms until Einstein convinced them with his paper on Brownian motion (not many people know about this). Now let us take the area you are interested in -- chaos and dynamical systems. When do you think people started to do research in this area? What about fractals? Many people, for a long time, did not realize that almost all object in nature are not those perfect shapes mathematicians study, they are something else entirely. Sometimes people just look through the old journals and find something important, for example, Lyapunov's work was not appreciated until we started to study dynamical systems and chaos. Let us consider Evariste Galois. His paper on solvability of equations by radicals was rejected because mathematicians did not understand what groups were. Regarding the second part of your question, I think you need to understand the fundamental areas of math such as analysis, algebra, topology, etc. but once you specialize, one of them becomes more important. The reason why you need to know about other areas is because you could be proving something and techniques from analysis, for instance, might not be enough for you to make any progress.