Natural Number and Natural Geometry

concepts in general, and the concepts of natural number and geometry in particular, present a longstanding puzzle to the brain and cognitive sciences. Nevertheless, a rich array of studies in cognitive development, animal cognition, cognitive psychology, comparative cultural psychology, and cognitive neuroscience is providing clues that may lead to its solution. Four cognitive systems found in infants, animals, and human adults across widely varying cultures give rise to representations with true numerical or geometrical content. The systems develop on the basis of little or no experience in numerically or geometrically structured environments, and therefore are innate. Abstract concepts of natural number and Euclidean geometry build on these systems of core knowledge. Nevertheless, each of these four systems is limited in its domain of application (none is fully abstract) and in the information that it makes available (none has the power of the system of integers or of Euclidean geometry). The limits on the two core number systems, and on the two core geometry systems, are complementary: richer systems of number and geometry could be constructed if representations from the different core systems could be productively combined. Research suggests that children begin to make these combinations in the preschool years. Thus, the most intuitive, abstract geometrical and numerical concepts that we possess as adults may not be given to us as infants; they may develop as children come to combine their core representations productively. The processes that give rise to fundamental human conceptual integrations are only beginning to be explored. Some of the research reviewed in this chapter suggests that natural language plays a pivotal role in the development of abstract numerical and geometric concepts, and does so by serving as the primary medium for combining information productively across distinct systems of core knowledge. These suggestions raise many questions, however, concerning the aspects of language that play this role, and the ways in which language interfaces with non-linguistic conceptual representations. Research is also needed to probe whether other symbolic devices can substitute for language and serve to combine core representations productively. Finally, research into other abstract concepts, in domains such as morality, politics or economics, is needed to explore whether core systems and productive combinatorial abilities produce a broad range of abstract concepts or apply more narrowly to the concepts of mathematics. By addressing such questions, research in cognitive science promises to elucidate the mechanisms by which humans go beyond the core knowledge systems that we share with other animals and construct truly abstract knowledge systems that are unique in the living world. 18. NATURAL NUMBER AND NATURAL GEOMETRy 314