Mariano Sigman is in the Cognitive Neuroimaging Research Unit of l'Institut National de la Santé et de la recherche Médicale, Orsay, France, and a fellow of the Human Frontiers Science Program. E-mail:

Mathematical measures of complexity shed light on why some concepts are inherently more difficult to learn than others.

Intuitive categories can be defined by short statements. The universe: circles and triangles, red and yellow, big and small (A). Examples of easy categories: red objects (B); triangles (C). Example of a difficult category: yellow circles and small red circles (D).

This essay is, in a way, about how we avoid becoming Borges's character Funes, who could not understand repeated observations as exemplars of a common rule and thus could not synthesize and categorize. Simply, he could not think. Probably the most disappointing moment of Feldman's paper comes at the very end, where it deals with its (somehow unavoidable) recursive quest. Understanding why some concepts are difficult to learn may itself be difficult to learn. Modern mathematics, together with Kolmogorov complexity and information theory, has taught us another fundamental concept that may be relevant when trying to understand the logic of the mind. In a long series of paradoxes enumerated by Bertrand Russell, Kurt Goedel, and others, we learn that a formal system that looks at itself is bound to fail. At the very end of his paper, Feldman writes, “In a sense, this final conclusion [that psychological complexity is Boolean complexity] may seem negative: human conceptual difficulty reflects intrinsic mathematical complexity after all, rather than some idiosyncratic and uniquely human bias.” Who invented mathematics? The Martians? On the contrary, I believe this result supports a more naturalistic and less Platonic conception of mathematics. Formal ideas in mathematics are not arbitrary constructions of an arbitrary architecture; rather, they reflect the workings of the brain like a massive collective cognitive experiment. Mathematics does not only serve to help us understand biology; mathematics is biology. We are not less original if our thoughts resemble our mental constructions, we are just consistent. It is within this loop, this unavoidable recursion—mathematics understanding the logic of the brain—that we will have an opportunity to test, as some conspire, whether among all the wonders evolution has come out with, the ultimate might be a brain good enough to avoid the risk of understanding itself.