Geometric Representations in Mathematical Problem-Solving: Intuition and Creativity

Some mathematicians described geometric representations as a means of making data intuitive and, therefore, their investigation easier. I elucidate one sense in which information is made intuitive: geometric representations evoke an integrated body of knowledge. I show this sense of intuitiveness of geometric representation to play a significant role in mathematical investigations. I find models of reasoning such as analogical reasoning and visual thinking incapable of accounting for this use of geometric representations. I offer an alternative model based on the following cognitive processes: evocation, refinement, and transfer of competence. I link favorable views concerning the use of geometric representations in mathematics to historically important epistemic values like investigational efficiency and allocation of intellectual resources. I consider Hans Hahn's objections against geometric intuition and, more briefly, other objections concerning contrasts within intuition itself. I defend the value of intuition for mathematical investigations vis-à-vis these objections.

Keywords: geometric representation, problem-solving, intuition, creativity, mathematical investigation, analogy, competence transfer, epistemic efficiency, Aleksandr D. Aleksandrov, Jean Dieudonné, Otto Stolz, Giuseppe Veronese, Hans Hahn.