A mathematical language without words

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Recent studies suggest that the brain network associated with mathematical operations is already involved in the sense of number in young children not yet at school. It has a long evolutionary history, being present when macaque monkeys recognize a number of concrete objects. Could it be that the emergence of a mathematical language, in the course of the evolution of the human species, preceded that of spoken language ? This is not impossible, given that mankind has been designing objects and tools with complex symmetries for around two million years (counting : at least 40,000 years ago; geometric symbols : 70,000 to 100,000 years ago ; zigzags : 540,000 years ago ; bifaces with a double plane of symmetry : up to 1.8 million years ago ; spheroids : around 2 million years ago). In this last lecture, we have brought together data suggesting that the human brain is able to manipulate and combine mathematical concepts, even when it has no words to express them. Studies of the Mundurucu Indians of Amazonia, carried out in collaboration with Véronique Izard, Pierre Pica and Elizabeth Spelke, play an essential role here. They show that, in the absence of mathematical education and vocabulary, children and adults alike have complex arithmetical and geometrical intuitions : concepts of number, correspondence between a two-dimensional map and three-dimensional reality, angle, parallelism, curvature, etc.

The lecture ended with the presentation of very recent data on the " language of geometry ". We showed that, even when it comes to remembering a sequence of positions in space, adults and children alike, whatever their level of education, represent this spatial sequence using a mini-language made up of simple primitives (successor, repetition, symmetry) embedded in a recursive fashion. Spatial memory is normally limited to around four positions, but our experiments show that sequences of eight positions are easily retained when they are " compressible ", i.e. regular enough to be encoded in memory using this internal language (which allows us to express, for example, that the positions form " two embedded squares "). Memory capacity is predicted by the complexity of this compressed representation, measured by the length of the minimal description (also known as Kolmogorov complexity). Brain imaging shows that this memory compression occurs in certain prefrontal and dorsal parietal areas, a network that coincides in part with the brain network for mathematics, but shows no overlap with language areas.

In conclusion, it seems that we are able to calculate and do geometry because we inherit, from primate evolution, representations of space and numbers that give us proto-mathematical intuitions. We share these primitives with many animal species, but only our species manages to integrate them into vast systems of symbols, to form precise and coherent formal languages. Symbolic ability enables us to extend our initial repertoire of mathematical concepts. This language of mathematics, which we are just beginning to explore, is not the same as natural language, but involves distinct regions of the brain. However, the role of language in the transmission of mathematical concepts, particularly during the child's early development, remains to be clarified and is certainly not to be overlooked.