Natural Number and Natural Geometry

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 uni-versal across humans: systems of core knowledge. Two of these systems—for tracking small num-bers 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. For millenia, philosophers and scientists have pondered the existence, nature and origins of abstract numerical and geometrical concepts, because these concepts have striking features. First, the integers, and the figures of the Euclidean plane, are so intuitive to human adults that the systems underlying them are called “natural number ” and, by some, “natural geometry” [1]. Second, these two systems are extremely useful: it is hard to find any important human cultural achievement—from money to measurement, to the arts, sciences, and mathemat-ics—that does not depend on them. Third, these conceptual systems are simple: five postu-lates, together with some general axioms of logic, suffice to define all the objects of Euclidean geometry, and an even smaller set of postulates defines the positive integers. It is perhaps not surprising, therefore, that explorations of the origins of abstract concepts, from Plato [2] and Kant [3] to Piaget [4] and Carey [5], often use number or geometry as case studies