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Neural foundations of logical and mathematical cognition


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.

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Figure 1: Imaging error-inhibition training in a deductive logic task.
Figure 2: Prefrontal activation during deductive-logic and cognitive-inhibition tasks.
Figure 3: The right ventromedial prefrontal cortex and its relation to emotion and reason.
Figure 4: Imaging cortical activity in a calculating prodigy.


  1. 1

    Fischer, K. & Kaplan, U. in The Encyclopedia of Cognitive Science Vol. 3 (ed. Nadel, L.) 679–682 (Nature Publishing Group, Macmillan, London, 2003).

    Google Scholar 

  2. 2

    Changeux, J.-P. & Connes, A. Conversations on Mind, Matter, and Mathematics (Princeton Univ. Press, Princeton, 1998).

    Google Scholar 

  3. 3

    Gardner, H. Frames of Mind: The Theory of Multiple Intelligences (Basic Books, New York, 1993).

    Google Scholar 

  4. 4

    Fodor, J. The Modularity of Mind (The MIT Press, Cambridge, 1983).

    Google Scholar 

  5. 5

    Fodor, J. The Mind Doesn't Work That Way (The MIT Press, Cambridge, 2000).

    Book  Google Scholar 

  6. 6

    Kosslyn, S. M. & Rosenberg, R. Psychology: the Brain, the Person, the World (Allyn and Bacon, Boston, 2001).

    Google Scholar 

  7. 7

    Braine, M. D. S. & O'Brien, D. P. (eds) Mental Logic (Erlbaum, Hove, 1998).

    Book  Google Scholar 

  8. 8

    Johnson-Laird, P. N. Mental models and deduction. Trends Cogn. Sci. 5, 434–442 (2001).

    Article  Google Scholar 

  9. 9

    Mellet, E., Petit, L., Denis, M. & Tzourio-Mazoyer, N. Reopening the mental imagery debate: lessons from functional anatomy. NeuroImage 8, 129–139 (1998).

    CAS  Article  Google Scholar 

  10. 10

    Goel, V., Gold, B., Kapur, S. & Houle, S. The seats of reason? An imaging study of deductive and inductive reasoning. NeuroReport 8, 1305–1310 (1997).

    CAS  Article  Google Scholar 

  11. 11

    Goel, V., Gold, B., Kapur, S. & Houle, S. Neuroanatomical correlates of human reasoning. J. Cogn. Neurosci. 10, 293–302 (1998).

    CAS  Article  Google Scholar 

  12. 12

    Goel V., Buchel, C., Frith, C. & Dolan, R. Dissociation of mechanisms underlying syllogistic reasoning. NeuroImage 12, 504–514 (2000).

    CAS  Article  Google Scholar 

  13. 13

    Wharton, C. M. & Grafman, J. Deductive reasoning and the brain. Trends Cogn. Sci. 2, 54–59 (1998).

    CAS  Article  Google Scholar 

  14. 14

    Langer, J. The descent of cognitive development. Dev. Sci. 3, 361–388 (2000).

    Article  Google Scholar 

  15. 15

    Baillargeon, R. & Wang, S. Event categorization in infancy. Trends Cogn. Sci. 6, 85–92 (2002).

    Article  Google Scholar 

  16. 16

    Evans, J. St. B. T. Bias in Human Reasoning: Causes and Consequences (Erlbaum, London, 1989).

    Google Scholar 

  17. 17

    Gaukroger, S. in The Encyclopedia of Cognitive Science Vol. 1 (ed. Nadel, L.) 947–950 (Nature Publishing Group, Macmillan, London, 2003).

    Google Scholar 

  18. 18

    Houdé, O. et al. Shifting from the perceptual brain to the logical brain: the neural impact of cognitive inhibition training. J. Cogn. Neurosci. 12, 721–728 (2000).

    Article  Google Scholar 

  19. 19

    Evans, J. St B. T. Matching bias in conditional reasoning. Thinking Reasoning 4, 45–82 (1998).

    Article  Google Scholar 

  20. 20

    Raichle, M. E. et al. Practice-related changes in human brain functional anatomy during non-motor learning. Cereb. Cortex 4, 8–26 (1994).

    CAS  Article  Google Scholar 

  21. 21

    Sakai, K. et al. Transition of brain activation from frontal to parietal areas in visuomotor sequence learning. J. Neurosci. 18, 1827–1840 (1998).

    CAS  Article  Google Scholar 

  22. 22

    Diamond, A., Kirkham, N. & Amso, D. Conditions under which young children can hold two rules in mind and inhibit a prepotent response. Dev. Psychol. 38, 352–362 (2002).

    Article  Google Scholar 

  23. 23

    Houdé, O. Inhibition and cognitive development: object, number, categorization, and reasoning. Cogn. Dev. 15, 63–73 (2000).

    Article  Google Scholar 

  24. 24

    Dehaene S., Kerszberg M. & Changeux, J.-P. A neuronal model of a global workspace in effortful cognitive tasks. Proc. Natl Acad. Sci. USA 95, 14529–14534 (1998).

    CAS  Article  Google Scholar 

  25. 25

    Smith, E. E. & Jonides, J. Storage and executive processes in the frontal lobes. Science 283, 1657–1661 (1999).

    CAS  Article  Google Scholar 

  26. 26

    Miller, E. K. The prefrontal cortex and cognitive control. Nature Rev. Neurosci. 1, 59–65 (2000).

    CAS  Article  Google Scholar 

  27. 27

    Fuster, J. M. Linkage at the top. Neuron 21, 1223–1229 (1998).

    CAS  Article  Google Scholar 

  28. 28

    Goel, V. & Dolan, R. J. Explaining modulation of reasoning by belief. Cognition 87, 11–22 (2003).

    Article  Google Scholar 

  29. 29

    Moutier, S. Deductive reasoning and matching-bias inhibition training in school children. Curr. Psychol. Cogn. 19, 429–452 (2000).

    Google Scholar 

  30. 30

    Fuster, J. The Prefrontal Cortex: Anatomy, Physiology, and Neuropsychology of the Frontal Lobe (Lippincott, New York, 1997).

    Google Scholar 

  31. 31

    Bjorklund, D. F. In search of a metatheory for cognitive development (or, Piaget is dead and I don't feel so good). Child Dev. 68, 144–148 (1997).

    CAS  Article  Google Scholar 

  32. 32

    Johnson, M. H. Functional brain development in humans. Nature Rev. Neurosci. 2, 475–483 (2001).

    CAS  Article  Google Scholar 

  33. 33

    Durston, S. et al. A neural basis for the development of inhibitory control. Dev. Sci. 5, 9–16 (2002).

    Article  Google Scholar 

  34. 34

    Casey, B. J., Davidson, M. & Rosen, B. Functional magnetic resonance imaging: basic principles of and application to developmental science. Dev. Sci. 5, 301–309 (2002).

    Article  Google Scholar 

  35. 35

    Thompson-Schill, S. L., D'Esposito, M., Aguirre, G. K. & Farah, M. J. Role of left inferior prefrontal cortex in retrieval of semantic knowledge: a reevaluation. Proc. Natl Acad. Sci. USA 94, 14792–14797 (1997).

    CAS  Article  Google Scholar 

  36. 36

    Thompson-Schill, S. L., D'Esposito, M. & Kan, I. P. Effects of repetition and competition on activity in left prefrontal cortex during word generation. Neuron 23, 513–522 (1999).

    CAS  Article  Google Scholar 

  37. 37

    Konishi, S. et al. Transient activation of inferior prefrontal cortex during cognitive set shifting. Nature Neurosci. 1, 80–84 (1998).

    CAS  Article  Google Scholar 

  38. 38

    Konishi, S. et al. Common inhibitory mechanism in human inferior prefrontal cortex revealed by event-related functional MRI. Brain 122, 981–991 (1999).

    Article  Google Scholar 

  39. 39

    Nakahara, K., Hayashi, T., Konishi, S. & Miyashita, Y. Functional MRI of macaque monkeys performing a cognitive set-shifting task. Science 295, 1532–1536 (2002).

    CAS  Article  Google Scholar 

  40. 40

    Jonides, J., Smith, E. E., Marshuetz, C. & Koeppe, R. A. Inhibition in verbal working memory revealed by brain activation. Proc. Natl Acad. Sci. USA 95, 8410–8413 (1998).

    CAS  Article  Google Scholar 

  41. 41

    D'Esposito, M., Postle, B. R. & Rypma, B. Prefrontal cortical contributions to working memory: evidence from event-related fMRI studies. Exp. Brain Res. 133, 2–11 (2000).

    Article  Google Scholar 

  42. 42

    Logothetis, N. K., Pauls, J., Augath, M. A., Trinath, T. & Oeltermann, A. Neurophysiological investigation of the basis of the fMRI signal. Nature 412, 150–157 (2001).

    CAS  Article  Google Scholar 

  43. 43

    Caesar, K., Gold, L. & Lauritzen, M. Context sensitivity of activity-dependent increases in cerebral blood flow. Proc. Natl Acad. Sci. USA 100, 4239–4244 (2003).

    CAS  Article  Google Scholar 

  44. 44

    Tupper, D. E. in The Encyclopedia of Cognitive Science Vol. 2 (ed. Nadel, L.) 965–969 (Nature Publishing Group, Macmillan, London, 2003).

    Google Scholar 

  45. 45

    McIntosh, A. R., Rajah, M. N. & Lobaugh, N. J. Interactions of prefrontal cortex in relation to awareness in sensory learning. Science 284, 1531–1533 (1999).

    CAS  Article  Google Scholar 

  46. 46

    Damasio, A. R. Descartes' Error: Emotion, Reason, and the Human Brain (Grosset, Putnam, New York, 1994).

    Google Scholar 

  47. 47

    Damasio, A. R. The Feeling of What Happens: Body and Emotion in the Making of Consciousness (Harcourt Brace, New York, 1999).

    Google Scholar 

  48. 48

    Damasio, H, Grabowski, T., Frank, R., Galaburda, A. & Damasio, A. R. The return of Phineas Gage: clues about the brain from the skull of a famous patient. Science 264, 1102–1105 (1994).

    CAS  Article  Google Scholar 

  49. 49

    Houdé, O. et al. Access to deductive logic depends on a right ventromedial prefrontal area devoted to emotion and feeling: evidence from a training paradigm. NeuroImage 14, 1486–1492 (2001).

    Article  Google Scholar 

  50. 50

    Tranel, D., Bechara, A. & Denburg, N. L. Asymmetric functional roles of right and left ventromedial prefrontal cortices in social conduct, decision-making, and emotional processing. Cortex 38, 589–612 (2002).

    Article  Google Scholar 

  51. 51

    Houdé, O. Consciousness and unconsciousness of logical reasoning errors in the human brain. Behav. Brain Sci. (in the press).

  52. 52

    Bush, G., Luu, P. & Posner, M. I. Cognitive and emotional influences in anterior cingulate cortex. Trends Cogn. Sci. 4, 215–222 (2000).

    CAS  Article  Google Scholar 

  53. 53

    Gehring, W. & Willoughby, A. The medial frontal cortex and the rapid processing of monetary gains and losses. Science 295, 2279–2282 (2002).

    CAS  Article  Google Scholar 

  54. 54

    Carey, S. in Developmental Cognitive Neuroscience (eds Nelson, C. A. & Luciana, M.) 415–431 (The MIT Press, Cambridge, 2001).

    Google Scholar 

  55. 55

    Wynn, K. Addition and subtraction by human infants. Nature 358, 749–750 (1992).

    CAS  Article  Google Scholar 

  56. 56

    Wynn, K. Findings of addition and subtraction in infants are robust and consistent. Child Dev. 71, 1535–1536 (2000).

    CAS  Article  Google Scholar 

  57. 57

    Hauser, M. D., MacNeilage, P. & Ware, M. Numerical representations in primates. Proc. Natl Acad. Sci. USA 93, 1514–1517 (1996).

    CAS  Article  Google Scholar 

  58. 58

    Hauser, M. D. Wild Minds: What Animals Really Think (Henry Holt, New York, 2000).

    Google Scholar 

  59. 59

    Simon, T. J. Reconceptualizing the origins of number knowledge: a 'non-numerical' account. Cogn. Dev. 12, 349–372 (1997).

    Article  Google Scholar 

  60. 60

    Simon, T. J. The foundations of numerical thinking in a brain without numbers. Trends Cogn. Sci. 3, 363–365 (1998).

    Article  Google Scholar 

  61. 61

    Houdé, O. Numerical development: from the infant to the child. Wynn's (1992) paradigm in 2- and 3-year-olds. Cogn. Dev. 12, 373–392 (1997).

    Article  Google Scholar 

  62. 62

    Spelke, E. S. & Tsivkin, S. Language and number: a bilingual training study. Cognition 78, 45–88 (2001).

    CAS  Article  Google Scholar 

  63. 63

    Dehaene, S., Spelke, E., Pinel, P., Stanescu, R. & Tsivkin, S. Sources of mathematical thinking: behavioral and brain-imaging evidence. Science 284, 970–974 (1999).

    CAS  Article  Google Scholar 

  64. 64

    Zago, L. et al. Neural correlates of simple and complex mental calculation. NeuroImage 13, 314–327 (2001).

    CAS  Article  Google Scholar 

  65. 65

    Siegler, R. S. Emerging Minds: The Process of Change in Children's Thinking (Oxford Univ. Press, New York, 1996).

    Google Scholar 

  66. 66

    Dehaene, S., Dehaene, G. & Cohen, L. Abstract representations of numbers in the animal and human brain. Trends Neurosci. 21, 355–361 (1998).

    CAS  Article  Google Scholar 

  67. 67

    Dehaene, S. Single-neuron arithmetic. Science 297, 1652–1653 (2002).

    CAS  Article  Google Scholar 

  68. 68

    Nieder, A., Freedman, D. J. & Miller, E. K. Representation of the quantity of visual items in the primate prefrontal cortex. Science 297, 1708–1711 (2002).

    CAS  Article  Google Scholar 

  69. 69

    Sawamura, H., Shima, K. & Tanji, J. Numerical representation for action in the parietal cortex of the monkey. Nature 415, 918–922 (2002).

    CAS  Article  Google Scholar 

  70. 70

    Pesenti, M. et al. Mental calculation in a prodigy is sustained by right prefrontal and medial temporal areas. Nature Neurosci. 4, 103–108 (2001).

    CAS  Article  Google Scholar 

  71. 71

    Butterworth, B. What makes a prodigy? Nature Neurosci. 4, 11–12 (2001).

    CAS  Article  Google Scholar 

  72. 72

    Hauser, M. D., Chomsky, N. & Fitch, W. T. The faculty of language: what is it, who has it, and how did it evolve? Science 298, 1569–1579 (2002).

    CAS  Article  Google Scholar 

  73. 73

    Zago, L. & Tzourio-Mazoyer, N. Distinguishing visuospatial working memory and complex mental calculation areas within the parietal lobes. Neurosci. Lett. 331, 45–49 (2002).

    CAS  Article  Google Scholar 

  74. 74

    Simon, O., Mangin, J.-F., Cohen, L., Le Bihan, D. & Dehaene, S. Topographical layout of hand, eye, calculation, and language-related areas in the human parietal lobe. Neuron 33, 475–487 (2002).

    CAS  Article  Google Scholar 

  75. 75

    Lakoff, G. & Nunez, R. Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being (Basic Books, New York, 2000).

    Google Scholar 

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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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Correspondence to Olivier Houdé.

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Houdé, O., Tzourio-Mazoyer, N. Neural foundations of logical and mathematical cognition. Nat Rev Neurosci 4, 507–514 (2003).

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