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][1] to gain some insight into how people think when they are doing mathematics.


  [1]: http://gowers.wordpress.com/2010/04/22/a-little-experiment/