There is often a dichotomy posed between verbal thinking and visual thinking. I'm definitely very very visual. I hardly think in terms of words at all. However, within the context of math, there are two different mental mechanisms that come into play when I'm solving a math problem. There is the visual understanding where I am at my strongest. Then there is the logical aspect of math - the formal application of operations on mathematical objects. I am very very strongly inclined to visual understanding of math. In fact, I don't even consider myself as understanding something until I can visualize what the operators are "doing" to the objects. I can apply operators to objects according to the rules without coming up with visualizations of the concept, but then I have no real idea of what I'm doing or what the result means. I can keep going ad nauseum, barring mistakes, but finding my way back to seeing what it means, what I'm doing, is difficult after two or three operations. The "visualizations" don't even have to be real 3 dimensional pictures (for example, I can be operating on 4+ dimensional objects, but I can tell I'm using that part of my brain, and not the part where I logicaly operate on symbol-strings) The logical approach seems to be favored by modern formal mathematics. It isn't considered rigourous until there is some logical definition to the concept. It seems similar to me to what you can do with computers functionally operating on strings, and reducible to some Turing problem. I think there was work done on searching spaces of "theorems" derivable from "axioms" using computers. ----- How do you approach mathematics? Is your visual reasoning primary to your understanding of the problems, or are you a primarily logical thinker? Do you understand the problem when you can visualize an answer? Or when you have an algorithm that operates on the mathematical objects? Were any of you extraordinarily precocious in mathematics? Did you acheive this via visual or symbolic reasoning? How do you think various historical mathematicians and physicists reasoned? Do you suspect that Einstein was a visual thinker, or a symbolic thinker? How about Reimann? Euler? Gauss? Newton?, ect. Also, if you are strong in one style of thinking, have you ever managed to build the other up to par, and how did you go about it?