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Terrence Tao commented of internalizing [here: https://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/]

"It is true that some mathematicians can be vastly more efficient than others at learning material, but I feel this is more due to experience and an efficient means of study than to any innate genius ability, though of course innate talent is still a contributing factor. For instance, among the graduate students I have advised, the first paper they read in a subject often takes a month or so to read (and they have a question on almost every page on the paper); but after a few years, they can get the gist of a new paper in the subject within a day, skimming past all the “standard” (or at least “plausible”) portions of the argument and focusing on the key new ideas. The key, I think, is to find one or more efficient ways to internalise the subject – either by using formalism, or geometric intuition, or physical intuition, or some other analogy or heuristic. Each mathematician has their own different way of doing this. Ramanujan, for instance, apparently performed a tremendous number of numerical computations, and derived much of his intuition from the patterns he observed from those computations. The intuition wasn’t always correct (for instance, he famously gave an incorrect formula for the n^th prime), but he did discover a number of amazing results this way, some of which took a long time to prove rigorously."

I could not make any sense of what's written in the bold letters (not the literal meaning, of the methods). How one does that ?

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closed as unclear what you're asking by Jack, BruceET, Frits Veerman, Gabriel Romon, Lee Mosher Nov 18 '16 at 20:17

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    $\begingroup$ The title suggests you literally don't know what the quote means, the body of the question suggests you want advice achieving that state. They're two very different questions, which are you asking? $\endgroup$ – pjs36 Mar 15 '16 at 11:48
  • $\begingroup$ @pjs36 Both, (The first one because I'm stupid; the second because I don't want to be) but focusing on the latter. $\endgroup$ – user185498 Mar 15 '16 at 15:12
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Practise, practise and more practise. The more you study and the more questions you answer the more you'll get a feel for the subject you are looking at. What Terry is pointing out is that the way in which you get that intuition varies for different people, some people will end up 'seeing' their subject in a geometric way, e.g. viewing homology as the search for holes in mathematical objects and spaces, others will work towards physical intuitions e.g. when proving very abstract solutions to nonlinear PDE often ideas from physics like conservation of 'mass' and 'energy' type equations are used in an intuitive way.

The essence of all of these ways of gaining a deeper understanding of a subject is that you take the formal definitions, which might be very abstract indeed and by considering them in a geometric, physical or other intuitive light you gain a real understanding of what the definitions $mean$ rather than just what they say.

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  • $\begingroup$ How you gain deeper understanding in something ? Rote practice will not probably suffice. $\endgroup$ – user185498 Mar 15 '16 at 11:41
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    $\begingroup$ Sorry if I was unclear, I meant that you need to practise developing the intuition for a given area and part of this comes from solving a lot of problems and doing lots of reading into a subject to really try and develop an idea of what is happening. I agree that rote learning won't do it but as part of developing an intuition doing problems and proofs etc. is key. $\endgroup$ – EHH Mar 15 '16 at 12:03