As you pursue this topic, I think you will discover that the distinction you make between visualization and reason is specious. Consider, on the one hand, the way in which Courant and Robbins in What is Mathematics? use sets of dots in rectangular boxes to explain the laws of the arithmetic of integers, and on the other hand, the interesting remark made by the logician Dale Jacquette: "Like many another logician, therefore, I can report that I was drawn to study logic in part by the 'beauty', or, as I prefer to say, the absorbing visual interest, of logical syntax" (Masses of Formal Philosophy 59).

You may be interested in four "experiments" Timothy Gowers conducted on his WordPress blog to gain some insight into how people think when they are doing mathematics.