Brain-imaging techniques have made it possible to explore the neural foundations of logical and mathematical cognition. These techniques are revealing more than simply where these high-order processes take place in the human cortex. Imaging is beginning to answer some of the oldest questions about what logic and mathematics are, and how they emerge and evolve through visuospatial cognition, language, executive functions and emotion.
Since Aristotle, it has been argued that the essence of the human mind is the 'logos', which encompasses both logic and language. Early in the twentieth century, the psychologist Jean Piaget (Box 1) pointed out that logic is the highest form of biological adaptation1 (see also related speculations about the neurobiology of mathematics2), but it was not until recently, with the advent of brain-imaging techniques, that it became possible to start exploring the neural foundations of logical reasoning and mathematical computation.
Here we examine logical and mathematical cognition together: first, because they are often considered as the two main components of the most abstract form of intelligence3 and second, to compare how they emerge and evolve in children and adults, through basic processes such as visuospatial cognition, language, executive functions and emotion. Rather than a modular view of cognition4,5, which attempts to isolate these processes in the human mind, we focus on interactions between them, as evidenced by brain-imaging techniques. Our aim is not to be exhaustive, but to address core issues and different views, in the hope of providing new research perspectives on the problem.
According to Piaget1 (Box 1), logical cognition, qualified as 'hypotheticodeductive', sets in during adolescence, at about the age of 14 or 15. The most canonical form of logical cognition is the so-called 'if–then' conditional reasoning. In this section, we address three issues6: Do we reason with logic, as argued by Piaget? Why do we make reasoning errors? Can emotions help us reason?
Do we reason with logic? According to Braine and O'Brien7, humans possess a mental or natural logic, which is defined as a set of simple inference rules that are required to understand language and to reason about everyday practical matters: for example, 'if p then q'. The rules of mental logic are universal; they are present in all languages, are fully mastered by adults, and are understood early by children. On the other hand, Johnson-Laird8 has proposed the opposing view that we can reason without logico-linguistic rules. He illustrates his position with syllogisms and conditionals, the terms of which play parts on the stage of a mental model, like actors in a theatre. For example, artists, beekeepers and chemists are the actors in the following syllogism: 'All artists are beekeepers. All beekeepers are chemists. Which conclusion, if any, follows from these two premises?' This reasoning process, so the argument goes, takes place in a visuospatial workspace rather than in a logico-linguistic one. Neither view — mental logic or mental model — has supplanted the other one in cognitive psychology, as both are supported by behavioural data7,8.
What we lacked in the past for directly testing these two opposing theories were brain-imaging techniques. With these new tools, one of the crucial questions is: does correct performance on deductive tasks recruit linguistic or visuospatial brain areas? Reasoning with a mental model, as Johnson-Laird has claimed, should recruit a neural network devoted to mental imagery9.
Going beyond the opposition of both models in cognitive psychology, Goel et al.10,11,12 have obtained compelling imaging data in support of a 'dual-mechanism theory'. According to this theory, different brain networks — logico-linguistic or visuospatial — are activated depending on whether the reasoning problem to be solved has semantic content. On the one hand, they have shown that the left inferior frontal gyrus (Broca's area) is systematically activated by deductive logic, and that other activations that participate in this type of reasoning lie mostly in the left hemisphere, especially in temporal (including Wernicke's area) and frontal regions. These results confirm the strong connection between deductive logic and linguistic brain areas, as predicted by Braine and O'Brien's theory of mental logic7, that is already present in Aristotle's definition of logos (logic and language). They are also in line with neuropsychological data showing that patients with damage to the left hemisphere in regions associated with linguistic ability are impaired at comprehending visuospatial logical relations13. Moreover, the results of Goel et al. are consistent with the fact that, following action and visuospatial cognition in both phylogeny and ontogeny14,15, deductive logic takes a leap forward when verbal formulation becomes possible.
On the other hand, Goel et al. have also shown that when deductive judgement is applied to syllogisms without semantic content (for example, 'All P are B. All B are C. So, all P are C'), a bilateral parieto-frontal network is involved in addition to the previous temporal and frontal activations. This activation pattern is more consistent with a visuospatial representation of the problem in working memory9, as predicted by Johnson-Laird's theory of mental models8.
This example clearly shows that the use of brain-imaging techniques could enable us to go beyond the theoretical opposition between models, which has remained unsolved by purely behavioural studies.
Why do we make reasoning errors? Contrary to Piaget's theory1, which postulated a logical stage of thinking by the age of 14 or 15 (Box 1), new studies on the cognitive psychology of reasoning have shown that adolescents and adults consistently make deduction errors in certain situations, owing to what are called 'reasoning biases'16. So, a crucial task for cognitive and developmental neuroscience is to study not only errors and biases in human reasoning, but also the so-called 'de-biasing mechanisms'16 that redirect our mind to logical thought. In the seventeenth century, the French philosopher René Descartes had already noted this problem when he proposed his method — a set of 'rules for the direction of the mind'17.
The ideal psychological model for this type of study is experimental training. We devised a de-biasing model using a deductive reasoning task with negative conditionals (for example, 'if not p, then q'), where all participants first relied on a perceptual strategy and therefore responded erroneously, obtaining an error rate as high as 90% (Ref. 18) (Fig. 1). The task was to read a rule such as 'If there is not a red square on the left, then there is a yellow circle on the right', and to select two geometrical shapes that would make the rule false. Most people spontaneously put a red square on the left of a yellow circle, thinking they were answering correctly. This logic error is caused by what Evans has called the perceptual-matching bias19. Subjects answer using the shapes that are mentioned in the rule instead of reasoning according to the logical truth table, which would lead them to choose a case where the antecedent of the rule is true (not a red square) and the consequent is false (not a yellow circle); for example, a blue square to the left of a green diamond. The logical response therefore requires resisting the elements perceived in the rule; that is, inhibiting the perceptual-matching bias. This is a good example of abstraction — a controlled and analytical kind of processing that occurs when our brain can break away from perception (which relies more heavily on automatic and heuristic processes). Surprisingly, although logical abstraction is an indisputably crucial feature of human thinking, it has only recently been studied using brain imaging18. Since the pioneering work of Raichle and his colleagues20, imaging research on cognitive training has mainly been devoted to the study of practice-related changes during the setting in of automatic processes (such as verb generation20 and visuomotor sequence learning21) and not to the contrasting study of control resulting from logical abstraction.
To study the perceptual matching bias, we hypothesized that adults, like children during the acquisition of elementary cognitive abilities22,23, have two competing strategies in their mental workspace24 — one logical and one perceptual — but they have trouble inhibiting the perceptual one. This is not really a question of mental logic per se1,7, but one of executive control25,26, in this case, cognitive inhibition. To show this, we used three training conditions: inhibition of the perceptual strategy (including emotional warnings about the error risk), logical explanation and simple task repetition (which corresponds to practice). Only inhibition training proved effective in reducing errors: the success rate increased from 10% to above 90%. This means that an executive inhibitory mechanism is what these adults were lacking, not logic or practice.
We then carried out an imaging study using positron emission tomography (PET) to see what was happening before and after training in the perceptual-matching bias-inhibition condition18. We found a clear shift in activity from the posterior part of the brain before training to the prefrontal part after training (Fig. 1). So, several concurrent reasoning strategies might compete at any time, even during adulthood, in such a way that perceptual responses often override logical ones, and cognitive inhibition turns out to be the key that opens the door to deductive logic. As a whole, these results argue for individual variability in human reasoning, selection by training and dynamic-network cognition27.
In the same vein, Goel and Dolan28 recently conducted a functional magnetic resonance imaging (fMRI) study on bias inhibition and semantic beliefs. Although deductive reasoning is a closed system, our beliefs about the world can influence validity judgements in syllogistic tasks16 (for example, the invalid but believable conclusion 'Some machines are not computers' follows from the premises 'All calculators are machines' and 'All computers are calculators'). In this study, the authors showed that the right lateral prefrontal cortex was specifically activated in the condition where people had to inhibit belief-based responses, in contrast to neutral or facilitatory conditions.
Cognitive inhibition is therefore a key executive function in adult reasoning. Developmental psychology studies also indicate that children often fail to inhibit reasoning biases, especially perceptual ones (or semantic belief biases), and that they are even more receptive to bias-inhibition training than are adults1,23,29. Given the considerable growth of the prefrontal cortex in the course of brain evolution30,31, an important research endeavour in developmental neuroscience is to look into this type of posterior-to-anterior functional reconfiguration18 to find out whether this is one of the processes through which logical cognition emerges and evolves.
This perspective is lacking in some previous accounts32 of functional brain development in humans (Box 2), yet this posterior-to-anterior reconfiguration brought about by inhibitory control might be the neural correlate of human abstraction, the ability to break away from perceptual biases during cognitive development. Recently, Casey and her collaborators explored the neural basis of the development of inhibitory control in children, combining fMRI with a parametric manipulation of a visuomotor go–no-go task33. They showed that successful response inhibition was associated with stronger activation of prefrontal and parietal regions in children than in adults. This new research trend in developmental neuroscience34 should be pursued using logic tasks that tend to trigger perceptual or semantic errors23.
Another challenge for the cognitive neuroscience of logical reasoning is to clarify the topographical layout of deductive logic, language and executive control (especially inhibition) in the left inferior frontal gyrus. As stated above, the left inferior frontal gyrus is involved in deductive logic. But this cortical area is also known to be devoted to language, particularly in semantic-knowledge selection35,36, and to cognitive inhibition in working memory, as suggested by Konishi37,38,39, Jonides40, D'Esposito41 and their collaborators. For all of these functions, authors often report what they call a left ventrolateral prefrontal activation that, in fact — and more exactly — lies in Broca's area (Fig. 2). The implication of the left inferior frontal region during this set of cognitive activities can be understood in different ways. One interpretation would be that several kinds of neural activity support these different cognitive activities. Indeed, one of the limits of brain imaging is that an observed activation signal, although strongly correlated with local field potentials, might correspond to excitatory, inhibitory or both types of synaptic activity42,43.
A second interpretation would be that, in line with Luria44, left inferior frontal involvement illustrates the tight links in the human frontal lobes between language, inhibitory control and action. In our PET experiment18, Broca's area was indeed activated when subjects became able to inhibit their perceptual error after training and to change their mode of action during reasoning.
A third interpretation would be that left frontal activation reflects an underlying neural computation that is required not only for deductive logic, but also for language and cognitive inhibition in working memory. This computation would be modulated by the components of the involved networks, as postulated in 'neural context' theory45. Whatever the actual explanation, the fact that Broca's area is implicated in different cognitive activities should lead us to reconsider the boundaries of what have often been seen as airtight domains. This fact challenges a restricted modular view of brain functioning4,5.
Can emotions help us reason? Except for Damasio's contribution from clinical neuropsychology46,47, this question has not attracted much attention in cognitive neuroscience despite its potentially great implications for educational policies.
Against Descartes' well-known opposition between reason and emotion17, Damasio offers eloquent support for the view that 'good use of reason' depends on emotion and self-feeling. Studying patients with brain injuries, he has shown that emotion probably assists reasoning, especially when it comes to personal and social matters that involve risk and conflict. In this framework, the oldest and most famous case is Phineas Gage48 (Fig. 3a). The study of Gage's lesion and other more recent cases shows that ventromedial prefrontal damage causes a defect in reasoning/decision making, emotion and self-feeling46,47.
In line with Damasio's contribution, we recently hypothesized49 that there might be a close tie between emotion, self-feeling and reasoning error inhibition in the intact human brain. In our previous PET study18, training in our matching bias-inhibition condition included emotional warnings about the error risk, which were not present in the logic-only training condition. We compared the impact of these two training conditions and found that, in the matching bias-inhibition one, activation was observed not only in the left lateral prefrontal cortex, but also in the right ventromedial prefrontal cortex (Fig. 3b). That is, the activation was present exactly in the location of the lesion in Gage (Fig. 3a) and in Damasio's patients46,47,48,50. These data show that, in healthy subjects, this area participates in getting the mind on the 'logical track'51, where it can put the instruments of deduction to good use. Note that the right ventromedial prefrontal cortex was not activated in the logic-only trained subjects who received no emotional warnings and were unable to inhibit their reasoning error. This cortical area could therefore be the emotional component of the brain's error-correction device. To be more precise, it could correspond, together with the anterior cingulate cortex52, to the brain area that detects the conditions under which logical reasoning errors might occur. Other recent imaging studies have stressed the role of the medial part of the prefrontal cortex in the emotional evaluation of error risks in domains related to logical cognition, notably, in the rapid processing of monetary gains and losses during economic reasoning53.
To further our understanding of this issue, cognitive-affective and developmental neuroscience should study exactly how the right ventromedial prefrontal cortex is involved in the emergence of logical cognition.
Do we perform arithmetic operations with language brain areas or with visuospatial brain areas? How might these areas interact in arithmetic computation? Some information on these issues has been obtained through behavioural studies on the mathematical competence of children after two years of age and preverbal infants, and even of monkeys. Bridging the gap between these behavioural studies and imaging data on the arithmetic capabilities of adults is highly fruitful for cognitive neuroscience.
When do mathematics emerge? According to Piaget's theory of developmental stages1 (Box 1), mathematical cognition sets in during childhood, at about the age of 7 or 8. Since then, many authors have reported evidence of precocious mathematical capacities, especially in arithmetic computation23,54. So, Wynn55,56 showed that preverbal human infants can do simple operations such as 1 + 1 = 2 or 2 − 1 = 1, an ability that has also been shown in monkeys by Hauser and his colleagues57,58. Wynn's experiments, which used a violation-of-expectation model, have established that 5-month-old infants can detect the error in violations such as 1 + 1 = 1 and 1 + 1 = 3. Their longer times looking at these 'magical events' are taken as a testimony to the ability of these children to calculate 1 + 1 = 2. According to Wynn, the existence of these numerical abilities so early in infancy indicates that humans might possess an innate capacity to perform simple arithmetic computations, which lays the foundations for the later development of higher mathematical abilities. But authors like Simon59 have refuted this conclusion, arguing for a non-numerical, purely visuospatial account that interprets Wynn's data in terms of object-based attention. Simon contends that with the new tools of brain imaging to add to our experimental methods, we have reached the point where we should be able to discover how mathematical cognition arises from “a brain without numbers”60.
Wynn's findings are robust and consistent56 and show that infants think about numerical events before they speak. At the very least, preverbal infants can be considered to use a 'visuospatial arithmetic'. To address this issue, we conducted cross-linguistic experiments using the same violation-of-expectation model with 2–3-year-old children, just after articulated language emerges in child development61. Measuring children's verbal reactions, we showed that, although there is a general arithmetic ability for small numbers that is shared by preverbal infants55,56 and monkeys57,58, the ontogeny of this initial knowledge in humans follows different performance patterns, depending on what language the children speak (Box 3). This is a cross-linguistic demonstration of the early shift from visuospatial arithmetic (in monkeys and human infants) to symbolic–linguistic arithmetic (in young children). Spelke and Tsivkin62, who conducted a training experiment with Russian–English bilingual college students, also showed that a specific language (the training language) can affect representations of large numbers.
In light of these developmental findings, a key question for imaging in adults is determining whether arithmetic — not only simple additive problems, but also multiplication with multi-digit numbers, for example — recruits language areas, or whether it is still supported mainly by visuospatial systems as in monkeys and infants.
How do adult brains compute arithmetic? According to Dehaene and his colleagues63, exact arithmetic (for example, multiplication tables) is based on a linguistic representation and relies on perisylvian language areas. By contrast, magnitude representations of numbers rely on the parietal cortex, the visuospatial site of a specific biological foundation for 'number sense' in adults, preverbal infants and monkeys. We conducted a PET study aimed at disclosing the functional anatomy of the two basic operations that are involved in mental calculation — arithmetic fact retrieval and actual computation — and the respective roles of the visuospatial and language brain areas64. Subjects had to perform three tasks: read digits, retrieve simple arithmetic facts from memory (for example, what is 2 x 4?), and compute complex operations (for example, what is 32 x 24?). Compared with the reading task, the retrieval task engaged a left parieto-premotor circuit, perhaps the developmental trace of a finger-counting strategy that, by extension, mediates numerical knowledge in adults. Indeed, Siegler65 has stressed that children use various overlapping strategies to solve arithmetic problems: counting with or without fingers, retrieval, decomposition, and so on. In addition to this basic network, the retrieval task involved a naming network that included the left anterior insula and the right cerebellar cortex. But contrary to the predictions of Dehaene and his colleagues63, it did not involve the perisylvian language areas, which were significantly deactivated compared with a rest condition. The arithmetic computation task not only relied on this retrieval network, but also involved a bilateral parietofrontal network (which is probably in charge of holding multi-digit numbers in visuospatial working memory) and the bilateral inferior temporal gyrus (which is probably important for visual mental imagery)9. Overall, these results provide strong evidence for the involvement of visuospatial representations in exact computations that require complex operations. They are consistent with numerous lesion studies in human, showing the crucial role of the parietal lobe in mental calculation and number processing66.
Recent research on the cerebral bases of number in monkeys has confirmed the existence of a 'single-neuron arithmetic' in parietofrontal areas67. Nieder et al.68 reported the discovery of number-encoding neurons in the lateral prefrontal cortex of the macaque brain. They showed monkeys a sequence of two visual displays, each consisting of up to five dots. The monkeys were trained to decide whether the two displays contained the same number of items. Their ability to generalize when presented with various new displays was also tested. The results showed that the macaques attended only to number; it did not matter whether the displays were equivalent in terms of area, shape, linear arrangement or density. Together with an earlier report of number-responsive neurons in the monkey parietal cortex69, this work opens up the possibility of studying the cerebral bases of elementary arithmetic at the single-cell level.
At the highest level of behavioural complexity, we used PET to study the brain of a calculating prodigy, Rüdiger Gamm70,71. Gamm is remarkable in that he has the ability to calculate, for example, the quotient of two primes to 60 decimal places with incredible accuracy. We found that, unlike control subjects, Gamm's calculation processes recruited a system of brain areas that are implicated in episodic memory, which included the right medial frontal and parahippocampal gyri (Fig. 4). This finding indicates that experts might develop a way of exploiting the unlimited storage capacity of long-term memory to retain task-relevant information, such as the sequence of steps and intermediate results that are needed for complex arithmetic operations. By contrast, the rest of us rely on the limited span of working memory. Currently, we do not know whether this process is limited to mathematical cognition or if it is of a general nature. But another important finding from this study that is crucial for the present discussion was obtained by analysing the areas activated both in Gamm's brain and control brains during calculation. Involvement of visuospatial working memory and visual imagery networks was observed in both cases, indicating that a visual short-term representation supports number retention and manipulation during arithmetic calculations.
These imaging data from non-expert and expert adult calculators, along with single-cell data recorded in monkeys, provide new fuel for the debate about the numerical abilities of infants55,56,57,58,59,60. As we have seen, Wynn55,56 argues for the early existence of the capacity to perform simple and exact arithmetic computations, whereas Simon59,60 and many others in the field argue for a visuospatial account of Wynn's data. The violation-of-expectation task used by Wynn primarily requires visual imagery and visuospatial working memory, two cognitive functions that already exist in the infant's brain. But the PET results clearly indicate that these brain functions are recruited by exact arithmetic in adults64, even in a calculating prodigy70. Consequently, visuospatial cognition and mathematical cognition are not mutually exclusive in the human brain. This means that both Wynn and Simon are right, as in monkeys, infants and adults, the recruitment of visuospatial systems to do arithmetic is undeniable. A remaining problem in this debate is to define the breaking point between visuospatial and arithmetical processing. Hauser et al.72 recently proposed that a general kind of recursion might be the marker of high-order mathematical and linguistic abilities in humans, but the neural implementation of this process is still unknown.
We have seen that language acquisition strongly interacts with arithmetic development for both small and large numbers62 (Box 3). So, an important goal for future research is to clarify the verbal/visuospatial interaction by means of which arithmetic evolves and operates in the human brain. The first step is to explore more accurately the fine specialization of the left parietal lobe. In a recent PET study, we pointed out that when adults are making complex mental calculations, they not only recruit visuospatial areas, but also the left supramarginal gyrus — a parietal region close to the language areas that is involved in the phonological component of verbal working memory73. This region would be a good candidate for partially sustaining the interaction between verbalization and visuospatial cognition in arithmetic computation.
Dehaene and his colleagues74 recently conducted an fMRI study in an attempt to clarify the topographical layout of hand-, eye-, calculation- and language-related areas in the human parietal lobe. Subjects had to perform six tasks: grasping, pointing, saccades, attention, calculation and phoneme detection. Examination of task intersections disclosed a systematic anterior-to-posterior organization of activations that were associated with grasping only, grasping and pointing, all visuomotor tasks, attention and saccades, and saccades only. Calculation yielded two distinct activations: one unique to calculation in the bilateral anterior intraparietal sulcus, mesial to the supramarginal gyrus, and the other shared with phoneme detection in the left intraparietal sulcus, mesial to the angular gyrus. According to the authors, these results imply a large cortical expansion of the inferior parietal lobe that correlates with the development of human language and calculation abilities.
Conclusion and future directions
Despite these findings on logical and mathematical cognition, there are many loose ends in the field. It therefore seems appropriate to end with a list of open questions for further research.
Is there a central device for logicomathematical cognition in the human brain? We have seen that deductive logic recruits mainly linguistic brain areas in a left frontotemporal network10,11,12,13, whereas arithmetic computation relies principally on visuospatial brain areas in a bilateral parietofrontal network64,70. In addition to these specific networks, a brain device shared by logic and mathematics remains to be discovered using not only arithmetic tasks but also complex mathematical reasoning tasks that require deductive-logic skills75.
Is the right ventromedial prefrontal cortex involved in mathematical cognition as we know it is for logic46,47,48,49,50,51? In other words, do emotion and self-feeling help get the mind on the 'mathematical track', where the instruments of arithmetic (or other mathematical tools) can be put to good use?
How do language, logic and mathematics interact in the human brain, especially in the left inferior frontal gyrus in the case of deductive logic10,11,12,13, involving reasoning-bias inhibition18, and in the left supramarginal gyrus in the case of arithmetical computation73, which involves visuospatial cognition64,70? Resolving this issue requires finding evidence of a fine-grained anatomical specialization in these brain areas and knowing their neural context45. This question fits well with insights into the relationships between language and cognition in humans and animals, especially for numerical cognition72, which argue that in the course of evolution, the modular, domain-specific4,5 system of recursion (which developed early for specific functions such as spatial navigation) might have become domain-general, operating over a broader range of elements like numbers and words.
Last, how do logical and mathematical cognition emerge and evolve from infancy to adulthood? What are the cognitive stages that involve visuospatial cognition, language, executive functions (especially inhibition) and emotion, which make up this high-order developmental process? This question specifically brings out the importance of a developmental approach in this field (Box 1). The first steps in this imaging approach to logic and mathematics were taken through the study of training18,49 or expertise70 in adults. Although purely behavioural data from developmental psychology will certainly be brought to bear on this issue, brain-imaging studies of pre-school and school children33,34,54 should also provide invaluable information.
We would like to thank S. Moutier, L. Zago and B. Mazoyer for their contribution to our work on logical and mathematical cognition. Support for our work is provided by The Centre National de la Recherche Scientifique, the Commissariat à l'Energie Atomique, Université de Caen, Université Paris-5 (René-Descartes) and the Institut Universitaire de France. We are also grateful to V. Waltz for her help in preparing the manuscript.
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Nature Reviews Neuroscience (2010)