Pure geometry’s cognitive foundation is mostly unknown. Even the ‘simpler’ question of what kind of geometric object representation we have. Mario Bacelar Valente proposes a model of geometric object representation at a neurological level for the case of Euclid’s pure geometry in his work. He considers historical characteristics of practical and pure geometry together to arrive at the model. This allows for a consistent representation of geometric objects based on past data.

To develop the model consistent with the previous geometrical practices, we have considered a historically informed account of practical geometry. The objective was to provide a basic characterization of practical geometry. Taking into account these basic ‘characteristics’ we build a model of the neural concept representation of geometric figure in practical geometry using in a very simple way the hub-and-spoke theory of neural concept representation.

M.B. Valente

The hub-and-spoke idea underpins the models (Ralph, Jefferies, Patterson, and Rogers 2017). The neural representation of concepts, according to this hypothesis, is made up of spokes, which are modality-specific brain areas that encode modal properties of concepts. The spokes, for example, encode visual, verbal (speaking), and physical representations. There are also integrative areas – the hub – that integrate the different components encoded in the spokes in an amodal style to produce coherent thoughts. Also, the hub enables a modality-free codification of further aspects of concepts.

By addressing the practical geometry of ancient Greece and taking into account elements of the practical geometry of ancient cultures, a more general characterization of geometry is built. Valente then builds a neural representations of geometric objects for Hippocrates and Aristotle’s models for pure geometry.

He reconstructs some aspects of the Euclidean practice taking into account how these are different from the corresponding aspects in the Hippocratic practice. Taking these differences into account together with the models of neural concept representation in practical geometry and Hippocrates’ pure geometry, a simple model of abstract geometric objects is then proposed.

By taking into account the encoding of the verbal spoke, the representations in the visual, praxis, and symbolic number-magnitude spokes are expressed in a highly abstract fashion. When we look at a figure, we have a certain indifference and responsiveness to its qualities that are related to the linguistic definition, thus we re-conceive the figure as an abstract geometric entity rather than a perfect figure (as mentioned, the figure becomes for us a representation of the geometric object). At this approach, the verbal spoke influences how the visual, physical, and symbolic number-magnitude spokes are interpreted and recombined in the hub, resulting in a ‘higher order’ representation of geometric abstract objects.

In our view, the hub-and-spoke models of the neural representation of geometric figure/object make more intelligible what a geometric object might be since we can relate its neural representation to that of a geometric figure in Hippocrates’ pure geometry and in practical geometry.

*Pure geometry and geometric cognition*, Mario Bacelar Valente

Published: March 2021