Models of Geometric Perception
Abstract
This latest lecture takes a critical look at the diversity of models that have been proposed for the mental representation of geometric shapes. How can we model the mental representations of geometry specific to the human species ? My research and that of my collaborators has led me to propose the existence of a " language of geometry ". According to this hypothesis, only the human species has the syntactic or compositional capacity to organize sequences of operations, either by repeating them, concatenating them or embedding them recursively. Repetition, concatenation and embedding are the only three syntactic operations of this internal language, which resembles a programming language, and whose combinations reproduce all the simple geometric shapes drawn by children and adults of all cultures. Several experiments have shown that the perception and memory of shapes are determined by a simple measure : the " minimal descriptionlength ", i.e. the size of the shortest program that reproduces the shape.
Advances in artificial intelligence suggest that deep neural networks of sufficient size can acquire impressive mathematical knowledge. Do they offer an alternative to the hypothesis of a language of thought ? The laboratory's work shows that this is not the case: until now, these networks have struggled to acquire geometric intuitions comparable to those of a young child. They often seem to lack symmetry, parallelism and the logic that governs these discrete, symbolic geometric properties.
The lecture also looks at a third interesting model : the hypothesis of extracting the median axes of a shape, for which there is a great deal of concordant data. However, this is a complementary model, not an alternative, to the hypothesis of a language of geometry : even if our visual system extracts the median axis, this is a skill that seems to be shared with other non-human primates, and is not sufficient to explain the effect of geometric regularity specific to the human species.
