The Origins of Psychological Axioms of Arithmetic and Geometry

1. The Problem The papers by Giaquinto and Wynn concern different aspects of mathematical cognition; Giaquinto examines a cognitive process through which people can acquire novel geometrical knowledge, while Wynn focuses on fundamental arithmetical knowledge present in human infants. This commentary considers an issue that arises with both of these research projects-the evolutionary origin of unlearned mathematical knowledge. The existence of unlearned mathematical knowledge is explicit in Wynn's discussion, but is present as well as part of the visualization process outlined in Giaquinto's paper. In particular, inferences that lead to geometric insights of the kind described by Giaquinto require some 'visual' knowledge that cannot itself be obtained by a visualization process, but which must be taken as given. In the example he presents, this is the knowledge that the diagonal of a square cuts the square into identical, mirror-image triangles. Without accepting this fact, a person would not know, and presumably would be unable to visualize, that the outer comers fold in to exactly cover the inner square without any overlaps or gaps, and would thus not appreciate the proof that the inner square is half the area of the square that it is contained within. But there