Significance Our work addresses the long-standing issue of the relationship between mathematics and language. By scanning professional mathematicians, we show that high-level mathematical reasoning rests on a set of brain areas that do not overlap with the classical left-hemisphere regions involved in language processing or verbal semantics. Instead, all domains of mathematics we tested (algebra, analysis, geometry, and topology) recruit a bilateral network, of prefrontal, parietal, and inferior temporal regions, which is also activated when mathematicians or nonmathematicians recognize and manipulate numbers mentally. Our results suggest that high-level mathematical thinking makes minimal use of language areas and instead recruits circuits initially involved in space and number. This result may explain why knowledge of number and space, during early childhood, predicts mathematical achievement.

Abstract 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.

The human brain is unique in the animal kingdom in its ability to gain access to abstract mathematical truths. How this singular cognitive ability evolved in the primate lineage is currently unknown. According to one hypothesis, mathematics, like other cultural abilities that appeared suddenly with modern humans in the upper Paleolithic, is an offshoot of the human language faculty—for Noam Chomsky, for instance, “the origin of the mathematical capacity [lies in] an abstraction from linguistic operations” (1). Many mathematicians and physicists, however, disagree and insist that mathematical reflection is primarily nonlinguistic—Albert Einstein, for instance, stated: “Words and language, whether written or spoken, do not seem to play any part in my thought processes.” (2).

An alternative to the language hypothesis has emerged from recent cognitive neuroscience research, according to which mathematics arose from an abstraction over evolutionarily ancient and nonlinguistic intuitions of space, time, and number (3, 4). Indeed, even infants and uneducated adults with a drastically impoverished language for mathematics may possess abstract protomathematical intuitions of number, space, and time (5, 6). Such “core knowledge” is predictive of later mathematical skills (7⇓–9) and may therefore serve as a foundation for the construction of abstract mathematical concepts (10). Advanced mathematics would arise from core representations of number and space through the drawing of a series of systematic links, analogies, and inductive generalizations (11⇓⇓–14).

The linguistic and core-knowledge hypotheses are not necessarily mutually exclusive. Linguistic symbols may play a role, possibly transiently, in the scaffolding process by which core systems are orchestrated and integrated (10, 15). Furthermore, mathematics encompasses multiple domains, and it seems possible that only some of them may depend on language. For instance, geometry and topology arguably call primarily upon visuospatial skills whereas algebra, with its nested structures akin to natural language syntax, might putatively build upon language skills.

Contemporary cognitive neuroscience has only begun to investigate the origins of mathematical concepts, primarily through studies of basic arithmetic. Two sets of brain areas have been associated with number processing. Bilateral intraparietal and prefrontal areas are systematically activated during number perception and calculation (16), a circuit already present in infants and even in untrained monkeys (17). Additionally, a bilateral inferior temporal region is activated by the sight of number symbols, such as Arabic numerals, but not by visually similar letters (18). Those regions lie outside of classical language areas, and several functional MRI (fMRI) studies have confirmed a double dissociation between the areas involved in number sense and language (19, 20). Only a small part of our arithmetic knowledge, namely the rote memory for arithmetic facts, encoded in linguistic form (16, 21). The bulk of number comprehension and even algebraic manipulations can remain preserved in patients with global aphasia or semantic dementia (22⇓–24). Contrary to intuition, brain-imaging studies of the processing of nested arithmetic expressions show little or no overlap with language areas (25⇓–27). Thus, conceptual understanding of arithmetic, at least in adults, seems independent of language.

Many mathematicians, however, argue that number concepts are too simple to be representative of advanced mathematics. To address this criticism, here we study the cerebral representation of high-level mathematical concepts in professional mathematicians. We collected fMRIs in 15 professional mathematicians and 15 nonmathematician controls of equal academic standing while participants performed fast semantic judgments on mathematical and nonmathematical statements (Fig. 1A). On each trial, a short spoken sentence was followed by a 4-s reflection period during which the participants decided whether the statement was true, false, or meaningless. Meaningful and meaningless statements were matched on duration and lexical content, but meaningless statements could be quickly dismissed, whereas meaningful statements required in-depth thinking, thus presumably activating brain areas involved in conceptual knowledge. Statements were generated with the help of professional mathematicians and probed four domains of higher mathematics: analysis, algebra, topology, and geometry. A fifth category of nonmath sentences, matched in length and complexity, probed general knowledge of nature and history. Two additional fMRI runs evaluated sentence processing and calculation (28) and the visual recognition of faces, bodies, tools, houses, numbers, letters, and written mathematical expressions.

Fig. 1. Main paradigm and behavioral results. (A) On each trial, subjects listened to a spoken statement and, 4 s later, classified it as true, false, or meaningless. (B) Performance in this task (% correct). (C and D) Mean d′ values for discrimination of meaningful versus meaningless statements (C) and, within meaningful statements, of true versus false statements (D). *P < 0.05 (Student t tests). Error bars represent one SEM.

Discussion Using high-resolution whole-brain fMRI, we observed the activation of a restricted and consistent network of brain areas whenever mathematicians engaged in high-level mathematical reflection. This network comprised bilateral intraparietal, inferior temporal, and dorsal prefrontal sites. It was activated by all domains of mathematics tested (analysis, algebra, topology, and geometry) and even, transiently, by meaningless mathematical statements. It remained silent, however, to nonmathematical statements of matched complexity. Instead, such problems activated distinct bilateral anterior temporal and angular regions. Our main goal was to explore the relationships between high-level mathematics, language, and core number networks. In mathematicians, we found essentially no overlap of the math-responsive network with the areas activated by sentence comprehension and general semantic knowledge. We observed, however, a strong overlap and within-subject similarity of the math-responsive network with parietal and inferior temporal areas activated during arithmetic calculation and number recognition (SI Appendix, Table S7). In particular, bilateral ventral inferior temporal areas corresponding to the visual number form area (18, 37) were activated by high-level mathematics as well as by the mere sight of numbers and mathematical formulas. The latter activations were enhanced in mathematicians. Correspondingly, a reduced activation to faces was seen in the right fusiform gyrus. Those results are analogous to previous findings on literacy, showing that the acquisition of expertise in reading shifts the responses of left ventral visual cortex toward letters and away from faces (38⇓–40). Our findings shed light on the roots of mathematical abilities. Some authors have argued that mathematics rests on a recent and specifically human ability for language and syntax (1) whereas others have hypothesized that it is a cultural construction grounded upon evolutionary ancient representations of space, time, and number (3, 4, 12). In our task, language areas were activated only transiently during the presentation of auditory statements, whether mathematical or nonmathematical. Rather, the activations that we observed during mathematical reflection occurred in areas previously associated with number coding in humans and other animals. Bilateral intraparietal and dorsal prefrontal regions are active during a variety of number-processing and calculation tasks (16) and contain neurons tuned to numerical quantities (17). Bilateral inferior temporal regions have been termed “visual number form areas” (VNFAs) because they activate to written Arabic numerals much more than to letter strings or other pictures (18, 37). The VNFAs were previously difficult to detect with fMRI because they lie close to a zone of fMRI signal loss (18). However, using a fast high-resolution fMRI sequence that mitigates these difficulties, we found that the VNFAs are easily detectable and are activated bilaterally not only by Arabic numerals, but also by algebraic formulas, arithmetic problems, and, in mathematicians only, during high-level mathematical reasoning. Although we investigated, within our subjects, only the relationship between the cortical territories for high-level mathematics, formulas, and number processing, previous work strongly suggests that the representation of geometrical relationships and visuo-spatial analogies also calls upon a similar bilateral dorsal prefrontal and intraparietal network (41, 42). Indeed, representations of cardinal number, ordinal knowledge, and spatial extent overlap in parietal cortex (43, 44). Given those prior findings, our results should not be taken to imply that number is the sole or even the main foundation of higher mathematical abilities; more likely, a complex integration of numerical, ordinal, logical, and spatial concepts is involved (12). Although one might have thought that the relationship between language and math would depend strongly on the domain of mathematics under consideration, we found no support for this hypothesis. Except for a small additional activation in posterior inferotemporal and posterior parietal cortex for geometry statements, all problems in algebra, analysis, topology, and geometry induced correlated and overlapping activations that systematically spared language areas. Using elementary algebraic and arithmetic stimuli, previous fMRI and neuropsychological research in nonmathematicians also revealed a dissociation between mathematical and syntactic knowledge (19, 22, 26, 45). Together, those results are inconsistent with the hypothesis that language syntax plays a specific role in the algebraic abilities of expert adults. Importantly, however, they do not exclude a transient role for these areas in the acquisition of mathematical concepts in children (10). Imaging studies of the learning process would be needed to resolve this point. Our results should not be taken to imply that the IPS, IT, and PFC areas that activated during mathematical reflection are specific to mathematics. In fact, they coincide with regions previously associated with a “multiple-demand” system (29) active in many effortful problem-solving tasks (30) and dissociable from language-related areas (46). Some have suggested that these regions form a “general problem solving” or “general purpose network” active in all effortful cognitive tasks (47). Several arguments, however, question the idea that this network is fully domain-general. First, we found no activation of this network during equally difficult reasoning with nonmathematical semantic knowledge. In fact, the easiest mathematical problems caused more activation than the most difficult nonmathematical problems (Fig. 5), and even meaningless mathematical problems caused more activation than meaningful general-knowledge problems (Fig. 4). Second, other studies have found a dissociation between tightly matched conditions of linguistic versus logical or arithmetical problem solving (19, 48). Overall the existing literature suggests that the network we identified engages in a variety of flexible, abstract, and novel reasoning processes that lie at the core of mathematical thinking, while contributing little to other forms of reasoning or problem solving based on stored linguistic or semantic knowledge. Our conclusions rest primarily on within-subject comparisons within the group of professional mathematicians (e.g., between math and nonmath reasoning, meaningful and meaningless math, etc.). As an additional control, we also presented the same stimuli to a gender- and age-matched group of nonmathematically trained but equally talented researchers and professors in humanities and related disciplines. Although mathematicians and controls may still differ on dimensions such as IQ, musical talent, hobbies, etc., such putative differences are irrelevant to our main conclusion of a dissociation between general-knowledge and mathematical reasoning within the mathematicians. They also seem unlikely to account for the enhanced ventral visual responses to numbers and math formulas, which most plausibly reflect the much higher frequency with which mathematicians process such symbols. Previous explorations of the brain mechanisms underlying professional-level mathematics are scarce. One fMRI study scanned 15 professional mathematicians, focusing entirely on their subjective sense of beauty for math expressions (49). The results revealed a medial orbito-frontal correlate for this subjective feeling but could not determine which brain areas are responsible for the mathematical computations that precede it. The network we observed seems to be a plausible candidate that should be tested in further work. The regions we observed also fit with the sites showing increased gray matter in mathematicians relative to control subjects of equal academic standing (50). During elementary problem-solving tasks, fronto-parietal activations at locations similar to ours were enhanced in mathematically gifted subjects (51). Interindividual variations in this network predict corresponding variations in fluid intelligence (29, 52), which is a major correlate of mathematical skills independently of other language skills. The connectivity between those regions, mediated by the superior longitudinal fasciculus, also increases in the course of normal numerical and mathematical education and in mathematically gifted students relative to others (53⇓–55). The fact that these brain areas are jointly involved in higher mathematics and basic arithmetic may explain the bidirectional developmental relationships that have been reported between prelinguistic number skills and later mathematical skills, whereby intuitive number sense predicts subsequent mathematical scores at school (7⇓–9, 56) and, conversely, mathematical education enhances the precision of the nonverbal approximate number system (57). Educational research also provides ample correlational and interventional evidence suggesting that early visuo-spatial and numerical skills can predict later performance in mathematics. The present results provide a putative brain mechanism through which such links may arise.

Methods Participants. We scanned a total of 30 French adult participants. Fifteen were professional mathematicians (11 male, 4 female, age range 24–39 y, mean = 28.1 y), and 15 were humanities specialists (10 male, 5 female, age range 24–50 y, mean = 30.1 y). Their ages did not significantly differ (t = 0.8397, P = 0.41). Professional mathematicians were full-time researchers and/or professors of mathematics. All had a PhD in mathematics and/or had passed the French national examination called “aggregation,” which is the last qualification examination for professorship. The 15 control subjects had the same education level but had specialized in humanities and had never received any mathematical courses since high school. Their disciplines were as follows: literature (n = 3), history (n = 3), philosophy (n = 1), linguistics (n = 2), antiquity (n = 1), graphic arts and theater (n = 3), communication (n = 1), and heritage conservation (n = 1). All subjects gave written informed consent and were paid for their participation. The experiment was approved by the regional ethical committee for biomedical research (Comité de Protection des Personnes, Hôpital de Bicêtre). Visual Runs. Seven categories of images were presented: faces, houses, tools, bodies, words, numbers, and mathematical formulas, plus a control condition consisting of circular checkerboards whose retinotopic extent exceeded that of all other stimuli (see SI Appendix for details). Auditory Runs. Subjects were presented with 72 mathematical statements (18 in each of the fields of analysis, algebra, topology, and geometry) and 18 nonmathematical statements. Within each category, 6 statements were true, 6 were false, and 6 were meaningless. All meaningless statements (in math or nonmath) were grammatically correct but consisted in meaningless associations of words extracted from unrelated meaningful statements. All meaningful statements bore upon nontrivial facts that were judged unlikely to be stored in rote long-term memory and therefore required logical reflection. Reference to numbers or to other mathematical concepts (e.g., geometrical shapes) was purposely excluded. A complete list of statements, translated from the original French, is presented in SI Appendix. All statements were recorded by a female native French speaker who was familiar with mathematical concepts. Statements from the different categories were matched in syntactic construction, length (mean number of words: math = 12.4, nonmath = 12.6, t = 0.24, P = 0.81), and duration (mean duration in seconds: math = 4.70, nonmath = 4.22, t = 1.93, P = 0.056). The experiment was divided into six runs of 15 statements each, which included one exemplar of each subcategory of statements [5 categories (analysis, algebra, geometry, topology, or general knowledge) × 3 levels (true, false, or meaningless)]. On screen, the only display was a fixation cross on a black background. Each trial started with a beep and a color change of the fixation cross (which turned to red), announcing the onset of the statement. After auditory presentation, a fixed-duration reflection period (4 s) allowed subjects to decide whether the statement was true, false, or meaningless. The end of the reflection period was signaled with a beep and the fixation cross turning to green. Only then, for 2 s, could subjects give their evaluation of the sentence (true, false, or meaningless) by pressing one of three corresponding buttons (held in the right hand). Each trial ended with a 7-s resting period (Fig. 1A). Localizer Scan. This 5-min fMRI scan is described in detail elsewhere (20). For present purposes, only two contrasts were used: language processing (sentence reading plus sentence listening relative to rest) and mental calculation (mental processing of simple subtraction problems, such as 7 − 2, presented visually or auditorily, and contrasted to the processing of nonnumerical visual or auditory sentences of equivalent duration and complexity). Post-MRI Questionnaire. Immediately after fMRI, all of the statements that had been presented during fMRI were reexamined in the same order. For each of them, participants were asked to rate the following: their comprehension of the problem itself within the noisy environment of the fMRI machine, their confidence in their answer, whether the response was a well-known fact or not (variable hereafter termed “immediacy”), the difficulty of the statement, its “imageability,” and the kind of reasoning that they had used on an axis going from pure intuition to the use of a formal proof. fMRI Data Acquisition and Analysis. We used a 3-Tesla whole body system (Siemens Trio) with a 32-channel head-coil and high-resolution multiband imaging sequences developed by the Center for Magnetic Resonance Research (CMRR) (58) [multiband factor = 4, Grappa factor = 2, 80 interleaved axial slices, 1.5-mm thickness and 1.5-mm isotropic in-plane resolution, matrix = 128 × 128, repetition time (RT) = 1,500 ms, echo time (ET) = 32 ms]. Using SPM8 software, functional images were first realigned, normalized to the standard Montreal Neurological Institute (MNI) brain space, and spatially smoothed with an isotropic Gaussian filter of 2 mm FWHM. A two-level analysis was then implemented in SPM8 software. For each participant, fMRI images were high-pass filtered at 128 s. Then, time series from visual runs were modeled by regressors obtained by convolution of the eight categories of pictures plus the button presses with the canonical SPM8 hemodynamic response function (HRF) and its time derivative. Data from the auditory runs were modeled by two regressors for each sentence, one capturing the activation to the sentence itself (kernel = sentence duration) and the other capturing the activation during the reflection period (4-s rectangular kernel). We then defined subject-specific contrasts over specific sentences, either comparing the activation evoked by any two subsets of sentences (during sentence presentation or during the postsentence reflection period) or evaluating the impact of a continuous variable, such as subjective difficulty, on a subset of sentences. Regressors of noninterest included the six movement parameters for each run. Within each auditory run, two additional regressors of noninterest were added to model activation to the auditory beeps and to the button presses. For the second-level group analysis, individual contrast images for each of the experimental conditions relative to rest were smoothed with an isotropic Gaussian filter of 5 mm FWHM and, separately for visual and auditory runs, entered into a second-level whole-brain ANOVA with stimulus category as within-subject factor. All brain-activation results are reported with a clusterwise threshold of P < 0.05 corrected for multiple comparisons across the whole brain, using an uncorrected voxelwise threshold of P < 0.001.

Acknowledgments We thank Saab Abou-Jaoudé and Vincent Pecastaing for help in writing the mathematical statements; Max Fathi and Vincent Pecastaing for translating those statements into English; Alexis Amadon for high-resolution fMRI scanning; Kamil Ugurbil, Essa Yacoub, Steen Moeller, Eddie Auerbach, and Gordon Junqian Xu (Center for Magnetic Resonance Research, University of Minnesota) for sharing pulse sequence and reconstruction algorithms; Véronique Izard, Manuela Piazza, Elizabeth Spelke, and Bertrand Thirion for useful suggestions; and Ghislaine Dehaene-Lambertz, Lucie Hertz-Pannier, and the NeuroSpin teams for technical support. This research was funded by INSERM, CEA, Collège de France, Université Paris-Sud, the Bettencourt-Schueller Foundation, and a PhD award from Région Ile-de-France (to M.A.).