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I am an (almost independent [1] learner) mathematics student. I believe in the only way to get used to the ideas is to derive them by yourself, but when I try to derive some good and deep result, my feeling is extremely like Grothendieck here:

(...) much more "gifted" than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle—while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things that I had to learn (so I was assured), things I felt incapable of understanding the essentials or following through to the end.

While rediscovering the main ideas by myself, mainly I lack clarity and depth and of thinking. Most of the time [2] feel that my thoughts are too shallow to do actual mathematics [3], much like the ox quotation. The main question is not how to have clarity and depth in thinking (although I would be happy if someone says something about those), but to how to get rid of those feelings ?

Note:

[1] By independant, I mean I do not directly interact with any other human for learning. My "school" is excessively stupid (it suppresses original thinking) and propagates "plug-n-chug" method. The other day, I asked why the Real number allows limits (or, say, partial derivatives commute)- and I ended up having an reply "It is because it is, don't ask such stupid questions" .

[2] This happens when I try to back-fill the tendrils also (as suggested by Ravi Vakil), but in a bit less degree.

[3] Suppose I want to learn (or derive the idea) of "A". I, after much struggling, do that (Occasionally I end up with a generalization or two (after given some hints)). Then, when I see the vastness and depth of generalization of "A" (by great mathematicians, usually Euler), which is ton times deeper and broad than me, I feel I am even stupid than the ox.

  • I'd be hard pressed to find anyone who isn't impressed with the "vastness and depth" of Euler's discoveries. I had a physics professor once who regularly joked that Euler must have been an alien because no human could do all the stuff he did. – wgrenard Apr 10 '16 at 05:50
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    Whilst I sympathise, this isn't really a question about mathematics. – Karl Apr 10 '16 at 06:07

2 Answers2

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If you can rediscover or even just prove main results by yourself without referring to the existing proofs, then it is more than good enough. Most students don't even care, even when their school does not suppress personal exploration. Furthermore, you can't expect to be better than the very top mathematicians (by definition almost everyone is not as prolific as them or 'clever' as them anyway). Finally, don't forget that a lot of well-known mathematicians actually read tons of books and writings by other mathematicians, which teaches them a broad foundation that they then build further upon.

And what's wrong with not knowing a lot of things? No one should feel stupid for not knowing anything. Stupidity is only when one wilfully ignores truth, not when one is ignorant.

user21820
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    @ArkaKarmakar: I have no idea what on earth you are talking about. I don't have any assumption about natural aristocracy. If you don't want answers, don't ask! – user21820 Apr 10 '16 at 08:48
  • @ArkaKarmakar: What most people label as "top mathematicians" are invariably those that are extremely 'clever' (they can solve problems that nobody had been able to solve for a long time) or are prolific (they have the capacity or speed to produce a vast amount of mathematical work). So by definition these are very rare. It's not realistic to expect to be like them, which is what you seem to feel you ought to be. – user21820 Apr 10 '16 at 14:04
  • I don't excepet to be like them - don't make unnecessary assumptions. –  Apr 11 '16 at 02:40
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    @ArkaKarmakar: Stop deleting your comments! And I didn't make assumptions. I said you seem to feel you ought to be, not that you did. You said that when you look at what great mathematicians like Euler came up with, you feel even more stupid than an ox. This is insulting to most people, who are not even as good as you, since if you are more stupid than an ox, than how about the rest of us?!?! It is your question that is stupid, not you! – user21820 Apr 11 '16 at 05:08
  • I am really sorry to do stupid thing (writing blather), but really I can't understand i.Why you are assuming that the mathematicians are rare ? (Genetically mutated ? I probably dismiss that - and assuming the mutation is true was what I mentioned earlier as assuming the (gibberish) natural aristocracy ) and ii."if you are more stupid than an ox, than how about the rest of us?!?!" . –  Apr 11 '16 at 06:38
  • @ArkaKarmakar: Read what I wrote properly. (i) I said top mathematicians are rare. What part of my comment 21 hours ago did you not understand? (ii) If you don't expect to be a top mathematician, then you've no reason to feel stupid like an ox, because if you actually had a good reason for that, then likewise all of those people who aren't even as good as you ought to feel stupid too. If you don't think so, then you should not feel stupid. But if you think we all should feel stupid like oxen, then your opinion is frankly a stupid one. – user21820 Apr 11 '16 at 11:44
  • I don't understand the part why you mention they are rare (don't just answer it dully by saying "by definition") –  Apr 12 '16 at 06:26
  • @ArkaKarmakar: That's your problem, not mine. You asked about great mathematicians. Go and look up the definition of the English word "great" in a dictionary. If you don't want to ask about "great/top mathematicians", then your question is totally meaningless. Moreover, the vast majority of mathematicians do not feel stupid for not being able to solve the hardest problems or find the most general theorems, and so neither should you, unless you're saying we ought to as well. Bottom line is: If you don't call us stupid, don't call yourself stupid! – user21820 Apr 12 '16 at 11:25
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You have to stand on the shoulders of giants. You cant go through life trying to discover everything yourself. Learn from what others have done in the past ... theres a reason you can learn 100 years of maths in one calculus class in high school. If you like you can spend 100+ years trying to figure it out yourself but I think thats stupid. Hopefully you agree.

terrace
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  • I can't find out what you are trying to mean by this answer. Add this as a comment, not as an answer. –  Apr 10 '16 at 06:27