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# Beyond Core Knowledge: Natural Geometry

The origins and nature of human intuitions about the properties of points, lines, and figures on the Euclidean plane have been a subject of philosophical and scientific inquiry for many centuries. The majority of these thinkers have proposed that a set of Euclidean concepts is either innate or is the result of general learning processes. A distinct perspective may be supported by recent findings in cognitive and developmental psychology, cognitive anthropology, animal cognition, and cognitive neuroscience.

Elizabeth Spelke is a cognitive scientist and developmental psychologist known for her work on the development of core knowledge in infants and young children. One of her key areas of research is in the field of geometry and spatial reasoning.

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).

Elizabeth Spelke

Spelke’s research has shown that infants as young as six months old have a basic understanding of geometric principles, such as the concept of continuity and the ability to perceive geometric shapes. This suggests that the ability to understand and reason about geometric information is innate and not something that is learned through experience.

In addition, her studies on the development of spatial reasoning in older children and adults found that children’s understanding of geometric principles becomes more sophisticated as they grow older and that this understanding is closely tied to their ability to think and reason about the physical world.

Figure 1: A geometrical theory can be fully characterized by the set of transformations that leave the properties of figures invariant (Klein, 1893). In Euclidean plane geometry, two forms are identical if they can be made to coincide through a rigid displacement (rotation and/or translation) on the plane. Forms that cannot be made to coincide differ in one or more of the properties of (A) distance (the lengths of parts and distances between them), (B) angle (the orientations at which parts meet), and (C) sense (the left–right directions of parts with respect to one another). © Elizabeth Spelke

Figure 4: Schematic depiction of the core systems of geometry (A) and number (B). © Elizabeth Spelke

One of Spelke’s key contributions to the field of cognitive development is her theory of core knowledge. This theory proposes that the human mind is equipped with a set of innate, domain-specific cognitive systems that are dedicated to processing information in specific areas, such as mathematics, physics, and social cognition.

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According to Spelke, the core knowledge system for geometry and spatial reasoning is responsible for the ability to understand and reason about the properties and relationships of geometric shapes and objects in the physical world. This system allows us to navigate and manipulate objects in the environment, as well as to make predictions about how objects will behave in different situations.

There are at least two core systems of geometry that are present and functional early in human development, that predate the evolution of humans as a species, and that continue to be universally present in human adults, according to research on young human children, nonhuman animals, and human adults in different cultures.

Evidence for the development of these capacities comes from studies on older children. At the age of 4, children seem to simply associate the forms of items and the arrangement of surfaces in terms of their usual distance relationships. But as they grow, youngsters begin to associate these images based on angle, and by maturity, direction.

Beyond Core Knowledge: Natural Geometry, Elizabeth Spelke, Sang Ah Lee, Véronique Izard
Published: July, 2010
DOI: 10.1111/j.1551-6709.2010.01110.x

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