Visualization vs memorization of mathematical knowledge As far as math research is concerned, what kind of understanding level of mathematical knowledge is required in order to truly master a topic and leverage on it? Can top-level researchers fully visualize everything or do they also rely on memory/mechanical/symbolic application of knowledge ?
I would distinguish between the 4 following stages (in decreasing order of mastery):
1- You can prove it and you can fully visualize it (down to the most elementary steps)
2- You can prove it and you can only partially visualize it (down to some “high-level” steps, but you can’t visualize all the way down to the most elementary steps)
3- You can prove it and only mechanically use it because but you fail to visualize it (at best you have some intuition about it)
4- You don’t know how to prove it (at this stage this is not acquired knowledge anymore)
My hesitation is about level 2 – because I find that I sometimes fall in that level for topics that are a little “too deep” (for me, maybe). For instance, I can’t fully visualize a theorem such as Schwarz’ theorem on partial derivatives, which is definitely a simple one (although I can prove it and fully understand each step and visualize each elementary step). Meaning that such knowledge has a purely symbolical representation in my mind (rather than an intuitive one), which taxes my memory...
 A: I wouldn't insist so much on visualisation. In some cases, mathematical machinery has been developed precisely to handle cases which are difficult to visualise. Think of linear algebra, I presume nobody is able to really visualise (rather then just going by analogy) $n$-dimensional vector spaces if $n>3$, but operating with them is nevertheless often no more difficult than handling the 3d case.
What you call visualisation I prefer to think of as intuition. This comes  with practice, working through many examples, seeing the main theorems into play several times...
I am not saying that the ability to come up with a picture/geometric understanding should not be valued - to the contrary it is very valuable and satisfying. However, I would not say that true understanding is necessarily geometric/visual understanding. In fact how one thinks of a  mathematical idea and "understand" it is probably quite personal and could be more geometric or algebraic or something else depending on one's inclination.
