For many centuries, philosophers and scientists have pondered the origins and nature of human intuitions about the properties of points, lines, and figures on the Euclidean plane, with most hypothesizing that a system of Euclidean concepts either is innate or is assembled by general learning processes. Recent research from cognitive and developmental psychology, cognitive anthropology, animal cognition, and cognitive neuroscience suggests a different view. Knowledge of geometry may be founded on at least two distinct, evolutionarily ancient, core cognitive systems for representing the shapes of large-scale, navigable surface layouts and of small-scale, movable forms and objects. Each of these systems applies to some but not all perceptible arrays and captures some but not all of the three fundamental Euclidean relationships of distance (or length), angle, and direction (or sense). Like natural number (Carey, 2009), Euclidean geometry may be constructed through the productive combination of representations from these core systems, through the use of uniquely human symbolic systems.

journal homepage: www.elsevier.com/locate/cogpsych

journal homepage: www.elsevier.com/locate/cogpsych

How does the human brain support abstract concepts such as seven or square? Studies of nonhuman 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. 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”

and two anonymous reviewers for their helpful comments on the manuscript. Supported by NIH

journal homepage: www.elsevier.com/locate/COGNIT

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright Author's personal copy