I have read a similar question posted on MSE, but honestly, I did not get the answer to what I was thinking. Suppose I am new to conic sections. Now, my book provides me an insight to what conic sections are, and their analytical focus-directrix definition, but doesn't really give me a proof of this focus-directrix property. I searched the internet to find a proof, and saw the ideas of Dandelin's spheres. I didn't read the whole proof, I still haven't(I might after getting the answer), so that I can try to explore conics my own way, and maybe create my own proof of the property. But, provided that the property is already proved by someone, should I really do it, or reading the already existing proof is good for me? Because, if I try to do this for every mathematical theory, it would take me years to do so.
So, what is good for a mathematician or a physicist? Re-invent the wheel(because you never know, maybe you discover something new by doing so) or read the works of others and use it as a black box?

  • $\begingroup$ Stand on the shoulders of giants $\endgroup$ – ÍgjøgnumMeg Oct 31 '16 at 19:32
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    $\begingroup$ I think it is misleading to phrase the dichotomy as being between building the whole of mathematics on your own and using black boxes. "Black box" suggests something you are using mechanically, with no understanding of how it works. Why do that? You can learn and understand standard techniques and standard theories. Doing everything from scratch, on your own, is hopeless. Dandelin spheres, for example, weren't discovered until 1820 or so...a mere two thousand years after Archimedes introduced conic sections. $\endgroup$ – lulu Oct 31 '16 at 19:32
  • $\begingroup$ Do you mean "theorem" rather than "theory"? $\endgroup$ – littleO Oct 31 '16 at 19:34
  • $\begingroup$ @littleO, by a "theory", I mean a set of "theorems" related to that theory $\endgroup$ – codetalker Oct 31 '16 at 19:35
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    $\begingroup$ I'm interested in this question too. It seems to me that there is a difficult "calibration" question here -- certainly we should spend some time trying to figure out some proofs before we read them, but the crucial and difficult question is "how much time?" How does one answer a calibration question like this? $\endgroup$ – littleO Oct 31 '16 at 19:38

I used to worry about this as well. I eventually got my Ph.D. So, maybe I'm not wholly unqualified to answer.

My advice would be to study the things that you like to study (or need to study for class). As you work through books, articles, and problems, try to think of examples that you think help explain the subject and maybe conjecture a few results and try to prove them to yourself. If you get stuck or bored, move on. You can always revisit these things later and take another crack at them. Or, you might later discover that your particular example or conjecture is actually much more difficult than you first expected. And, it is often a good idea to hear another person's perspective on a subject, especially someone that has spent a lot of time thinking about it.

The important thing is to not push yourself to the point of boredom or mathematical exhaustion. Have fun playing with the math. Try to guess which things would be true in the subject you are studying. Try to test your grasp on the subject with examples that you come up with. But, keep your mind engaged.

How much of each subject you want/need to develop on your own versus how much you will learn from someone else will become evident to you with enough experience.

That last sentence is a good thing to recognize. Getting good at math requires a LOT of time and experience. Just make sure you don't kill your interest by pressing yourself unnaturally.

As to avoid (more) rambling, I will draw my answer to an end. Ask any questions, and I will edit my responses in.


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