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.)