Frege's Theorem and Mathematical Cognition

Frege’s theorem states that Peano Arithmetic can be interpreted in Frege Arithmetic, a second-order system with one extra-logical axiom, Hume’s Principle. The epistemological implications of this result have been controversial. Starting from Heck’s epistemological analysis of the conceptual relations between cardinality, counting, and equinumerosity and expanding on his discussion of empirical results on numerical cognition, I assess the relative importance of equinumerosity and the successor relation in empirical results on numerical cognition in developmental psychology and linguistic anthropology. I conclude that both concepts independently play a role in the development of number concepts. However, the cognitive explanations of the acquisition of numerical concepts arguably do not lead to a full understanding of numerical concepts and are in various ways problematic from a methodological point of view. I conclude that for the analysis of numerical concepts in mathematical practice, a modest provisional apsychologism is commendable.