The origins of human abilities for mathematics are debated: Some theories suggest that they are founded upon evolutionarily ancient brain circuits for number and space and others that they are grounded in language competence. To evaluate what brain systems underlie higher mathematics, we scanned professional mathematicians and mathematically naive subjects of equal academic standing as they evaluated the truth of advanced mathematical and nonmathematical statements. In professional mathematicians only, mathematical statements, whether in algebra, analysis, topology or geometry, activated a reproducible set of bilateral frontal, Intraparietal, and ventrolateral temporal regions. Crucially, these activations spared areas related to language and to general-knowledge semantics. Rather, mathematical judgments were related to an amplification of brain activity at sites that are activated by numbers and formulas in nonmathematicians, with a corresponding reduction in nearby face responses. The evidence suggests that high-level mathematical expertise and basic number sense share common roots in a nonlinguistic brain circuit.

During language processing, humans form complex embedded representations from sequential inputs. Here, we ask whether a “geometrical language” with recursive embedding also underlies the human ability to encode sequences of spatial locations. We introduce a novel paradigm in which subjects are exposed to a sequence of spatial locations on an octagon, and are asked to predict future locations. The sequences vary in complexity according to a well-defined language comprising elementary primitives and recursive rules. A detailed analysis of error patterns indicates that primitives of symmetry and rotation are spontaneously detected and used by adults, preschoolers, and adult members of an indigene group in the Amazon, the Munduruku, who have a restricted numerical and geometrical lexicon and limited access to schooling. Furthermore, subjects readily combine these geometrical primitives into hierarchically organized expressions. By evaluating a large set of such combinations, we obtained a first view of the language needed to account for the representation of visuospatial sequences in humans, and conclude that they encode visuospatial sequences by minimizing the complexity of the structured expressions that capture them.

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Is mathematical language similar to natural language? Are language areas used by mathematicians when they do mathematics? And does the brain comprise a generic semantic system that stores mathematical knowledge alongside knowledge of history, geography or famous people? Here, we refute those views by reviewing three functional MRI studies of the representation and manipulation of high-level mathematical knowledge in professional mathematicians. The results reveal that brain activity during professional mathematical reflection spares perisylvian language-related brain regions as well as temporal lobe areas classically involved in general semantic knowledge. Instead, mathematical reflection recycles bilateral intraparietal and ventral temporal regions involved in elementary number sense. Even simple fact retrieval, such as remembering that 'the sine function is periodical' or that 'London buses are red', activates dissociated areas for math versus non-math knowledge. Together with other fMRI and recent intracranial studies, our results indicated a major separation between two brain networks for mathematical and non-mathematical semantics, which goes a long way to explain a variety of facts in neuroimaging, neuropsychology and developmental disorders.This article is part of a discussion meeting issue 'The origins of numerical abilities'.

Memory for spatial sequences does not depend solely on the number of locations to be stored, but also on the presence of spatial regularities. Here, we show that the human brain quickly stores spatial sequences by detecting geometrical regularities at multiple time scales and encoding them in a format akin to a programming language. We measured gaze-anticipation behavior while spatial sequences of variable regularity were repeated. Participants' behavior suggested that they quickly discovered the most compact description of each sequence in a language comprising nested rules, and used these rules to compress the sequence in memory and predict the next items. Activity in dorsal inferior prefrontal cortex correlated with the amount of compression, while right dorsolateral prefrontal cortex encoded the presence of embedded structures. Sequence learning was accompanied by a progressive differentiation of multi-voxel activity patterns in these regions. We propose that humans are endowed with a simple "language of geometry" which recruits a dorsal prefrontal circuit for geometrical rules, distinct from but close to areas involved in natural language processing.

Among primates, humans are special in their ability to create and manipulate highly elaborate structures of language, mathematics, and music. Here we show that this sensitivity to abstract structure is already present in a much simpler domain: the visual perception of regular geometric shapes such as squares, rectangles, and parallelograms. We asked human subjects to detect an intruder shape among six quadrilaterals. Although the intruder was always defined by an identical amount of displacement of a single vertex, the results revealed a geometric regularity effect: detection was considerably easier when either the base shape or the intruder was a regular figure comprising right angles, parallelism, or symmetry rather than a more irregular shape. This effect was replicated in several tasks and in all human populations tested, including uneducated Himba adults and French kindergartners. Baboons, however, showed no such geometric regularity effect, even after extensive training. Baboon behavior was captured by convolutional neural networks (CNNs), but neither CNNs nor a variational autoencoder captured the human geometric regularity effect. However, a symbolic model, based on exact properties of Euclidean geometry, closely fitted human behavior. Our results indicate that the human propensity for symbolic abstraction permeates even elementary shape perception. They suggest a putative signature of human singularity and provide a challenge for nonsymbolic models of human shape perception.

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Advanced mathematical reasoning, regardless of domain or difficulty, activates a reproducible set of bilateral brain areas including intraparietal, inferior temporal and dorsal prefrontal cortex. The respective roles of genetics, experience and education in the development of this math-responsive network, however, remain unresolved. Here, we investigate the role of visual experience by studying the exceptional case of three professional mathematicians who were blind from birth (n = 1) or became blind during childhood (n = 2). Subjects were scanned with fMRI while they judged the truth value of spoken mathematical and nonmathematical statements. Blind mathematicians activated the classical network of math-related areas during mathematical reflection, similar to that found in a group of sighted professional mathematicians. Thus, brain networks for advanced mathematical reasoning can develop in the absence of visual experience. Additional activations were found in occipital cortex, even in individuals who became blind during childhood, suggesting that either mental imagery or a more radical repurposing of visual cortex may occur in blind mathematicians.