Summary: How does the human brain support abstract concepts such as seven or square? Studies of non-human animals, of human infants, and of children and adults in diverse cultures suggest these concepts arise from a set of cognitive systems that are phylogenetically ancient, innate, and universal across humans: systems of core knowledge. Two of these systems --- for tracking small numbers of objects and for assessing, comparing and combining the approximate cardinal values of sets --- capture the primary information in the system of positive integers. Two other systems --- for representing the shapes of small-scale forms and the distances and directions of surfaces in the large-scale navigable layout --- capture the primary information in the system of Euclidean plane geometry. As children learn language and other symbol systems, they begin to combine their core numerical and geometrical representations productively, in uniquely human ways. These combinations may give rise to the first truly abstract concepts at the foundations of mathematics.
Abstract: Dehaene (this volume) articulates a naturalistic approach to the cognitive foundations of mathematics. Further, he argues that the ‘number line’ (analog magnitude) system of representation is the evolutionary and ontogenetic foundation of numerical concepts. Here I endorse Dehaene’s naturalistic stance and also his characterization of analog magnitude number representations. Although analog magnitude representations are part of the evolutionary foundations of numerical concepts, I argue that they are unlikely to be part of the ontogenetic foundations of the capacity to represent natural number. Rather, the developmental source of explicit integer list representations of number are more likely to be systems such as the object-file representations that articulate mid-level object based attention, systems that build parallel representations of small sets of individuals.