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This is a soft question concerning the mental processes at work when doing maths. I hope this question is not too vague, and I believe I haven't found a similar one previously posted on MathSE.

In some of the answers I received on previous posts, I've been confronted to very abstract concepts which I find difficult to grab, because I don't see any physical meaning to them. In the same way, Grothendieck's programmes in his biography, which I read once out of curiosity, leave me completely puzzled.

My question is : do mathematicians have a concrete/visual mental picture of the (sometimes very abstract) concepts they manipulate, which can help them finding new leads/paths/theorems ?

(For example, in this interview (in French) of Benoit Mandelbrot, he says that during his math education, he realized he could turn any math problem (even algebra or arithmetics) into a geometrical problem, and that was his way of solving it.)

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    Some do, some don't. I don't think you'll get an accurate answer that applies to all mathematicians.2011-09-07
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    Depends very much on the mathematician. I don’t visualize in any literal sense, but I do have mental models of some kind for some things.2011-09-07
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    Personally I don't think mathematics would be anywhere near as far or large as it is today without mathematicians generally having a minimal mental "picture" of what it is they're doing. Rearranging symbols like mere puzzle pieces (what I think some people view math as close to being) is outpaced by creative conceptual insight. Of course, if you're asking if all of mathematics can be understood pictorially in 2 or 3-dimensions as we normally "see" things in our visual consciousness, I say no: this is much too narrow to capture the form of deeper or more sophisticated math.2011-09-07
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    @AlexPof I had a real analysis lecturer saying that visualisation has its limitation in mathematics, e.g. maybe you can draw open balls and visualise the meaning of an open set but then what happens in higher dimensions?2011-09-07

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