Biological Theory

, Volume 4, Issue 1, pp 68–83 | Cite as

Numbers and Arithmetic: Neither Hardwired Nor Out There

  • Rafael NúñezEmail author


What is the nature of number systems and arithmetic that we use in science for quantification, analysis, and modeling? I argue that number concepts and arithmetic are neither hardwired in the brain, nor do they exist out there in the universe. Innate subitizing and early cognitive preconditions for number— which we share with many other species—cannot provide the foundations for the precision, richness, and range of number concepts and simple arithmetic, let alone that of more complex mathematical concepts. Numbers and arithmetic, and mathematics in general, have unique features—precision, objectivity, rigor, generalizability, stability, symbolizability, and applicability to the real world—that must be accounted for. They are sophisticated concepts that developed culturally only in recent human history. I suggest that numbers and arithmetic are realized through precise combinations of non-mathematical everyday cognitive mechanisms that make human imagination and abstraction possible. One such mechanism, conceptual metaphor, is a neurally instantiated inference-preserving cross-domain mapping that allows the conceptualization of abstract entities in terms of grounded bodily experience. I analyze how the inferential organization of the properties and “laws” of arithmetic emerge metaphorically from everyday meaningful actions. Numbers and arithmetic are thus—outside of natural selection—the product of the biologically constrained interaction of individuals with the appropriate cultural and historical phenotypic variation supported by language, writing systems, and education.


abstraction arithmetic conceptual metaphor conceptual systems embodiment imagination inferential organization mathematics numerical cognition number concepts 


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  1. Antell SE, Keating DP (1983) Perception of numerical invariance in neonates. Child Development 54: 695–701.CrossRefGoogle Scholar
  2. Bächtold D, Baumöller M, Brugger P (1998) Stimulus-response compatibility in representational space. Neuropsychologia 36: 731–735.CrossRefGoogle Scholar
  3. Bates E, Benigni L, Bretherton I, Camaioni L, Volterra V (1979) The Emergence of Symbols: Cognition and Communication in Infancy. New York: Academic Press.Google Scholar
  4. Bates E, Dick F (2002) Language, gesture, and the developing brain. Developmental Psychobiology 40: 293–310.CrossRefGoogle Scholar
  5. Bates E, Snyder LS (1987) The cognitive hypothesis in language development. In: Infant Performance and Experience: New Findings with the Ordinal Scales (Ina E, Uzgiris C, McVicker Hunt EJ, eds), 168–204. Urbana, IL: University of Illinois Press.Google Scholar
  6. Bates E, Thal D, Whitesell K, Fenson L, Oakes L (1989) Integrating language and gesture in infancy. Developmental Psychology 25: 1004–1019.CrossRefGoogle Scholar
  7. Bideaud J (1996) La construction du nombre chez le jeune enfant: Une bonne raison d’affûter le rasoir d’Occam. Bulletin de Psychologie 50: 19–28.Google Scholar
  8. Bijeljac-Babic R, Bertoncini J, Mehler J (1991) How do four-day-old infants categorize multisyllabic utterances? Developmental Psychology 29: 711–721.CrossRefGoogle Scholar
  9. Boroditski L (2000) Metaphoric structuring: Understanding time through spatial metaphors. Cognition 75: 1–28.CrossRefGoogle Scholar
  10. Brannon E (2002) The development of ordinal numerical knowledge in infancy. Cognition 83: 223–240.CrossRefGoogle Scholar
  11. Butterworth B (1999) What Counts: How Every Brain is Hardwired for Math. New York: Free Press.Google Scholar
  12. Carey S (2004) Bootstrapping and the origin of concepts. Daedalus 133: 59–68.CrossRefGoogle Scholar
  13. Caselli MC (1990) Communicative gestures and first words. In: From Gesture to Language in Hearing and Deaf Children (Volterra V, Erting C, eds), 56–68. New York: Springer.CrossRefGoogle Scholar
  14. Changeux J-P, Connes A (1998) Conversations on Mind, Matter, and Mathematics. Princeton: Princeton University Press.Google Scholar
  15. Connes A, Lichnerowicz A, Schötzenberger MP (2000) Triangle de pensées. Paris: Odile Jacob.Google Scholar
  16. Cooperrider K, Núñez R (2009). Across time, across the body: Transversal temporal gestures. Gesture 9: 181–206.CrossRefGoogle Scholar
  17. Davis H, Pérusse R (1988) Numerical competence in animals: Definitional issues, current evidence, and new research agenda. Behavioral and Brain Sciences 11: 561–615.CrossRefGoogle Scholar
  18. Dehaene S (1997) The Number Sense: How the Mind Creates Mathematics. New York: Oxford University Press.Google Scholar
  19. Dehaene S (2002) Single-neuron arithmetic. Science 297: 1652–1653.CrossRefGoogle Scholar
  20. Dehaene S (2003) The neural basis of the Weber-Fechner law: A logarithmic mental number line. Trends in Cognitive Sciences 7: 145–147.CrossRefGoogle Scholar
  21. Dehaene S, Cohen L (1994) Dissociable mechanisms of subitizing and counting: Neuropsychological evidence from simultanagnosic patients. Journal of Experimental Psychology: Human Perception and Performance 20: 958–975.Google Scholar
  22. Dehaene S, Izard V, Spelke E, Pica P (2008a) Log or linear? Distinctintuitions of the number scale in Western and Amazonian indigene cultures. Science 320: 1217–1220.CrossRefGoogle Scholar
  23. Dehaene S, Izard V, Spelke E, Pica P (2008b) Supplementary material to “Log or linear? Distinct intuitions of the number scale in Western and Amazonian indigene cultures.” Science online
  24. Dehaene S, Piazza M, Pinel P, Cohen L (2003) Three parietal circuits for number processing. Cognitive Neuropsychology 20: 487–506.CrossRefGoogle Scholar
  25. Fauconnier G, Turner M (1998) Conceptual integration networks. Cognitive Science 22: 133–187.CrossRefGoogle Scholar
  26. Fauconnier G, Turner M (2002) The Way We Think: Conceptual Blending and the Mind’s Hidden Complexities. New York: Basic Books.Google Scholar
  27. Fischer MH (2006) The future for SNARC could be stark. Cortex 42: 1066–1068.CrossRefGoogle Scholar
  28. Fowler DH, Robson ER (1998) Square root approximations in Old Babylonian mathematics: YBC 7289 in context. Historia Mathematica 25: 366–378.CrossRefGoogle Scholar
  29. Gallistel CR, Gelman R, Cordes S (2006) The cultural and evolutionary history of the real numbers. In: Evolution and Culture (Levinson SC, Jaisson P, eds), 247–274. Cambridge, MA: MIT Press.Google Scholar
  30. Gelman R, Gallistel CR (1978) The Child’s Understanding of Number. Cambridge, MA: Harvard University Press.Google Scholar
  31. Gentner D (2001) Spatial metaphors in temporal reasoning. In: Spatial Schemas and Abstract Thought (Gattis M, ed), 203–222. Cambridge, MA: MIT Press.Google Scholar
  32. Gibbs R (ed) (2008) The Cambridge Handbook of Metaphor and Thought. Cambridge, UK: Cambridge University Press.Google Scholar
  33. Kaufmann EL, Lord MW, Reese TW, Volkmann J (1949) The discrimination of visual number. American Journal of Psychology 62: 498–525.CrossRefGoogle Scholar
  34. Lakoff G (1993) The contemporary theory of metaphor. In: Metaphor and Thought (Ortony A, ed), 2nd ed, 201–251. New York: Cambridge University Press.Google Scholar
  35. Lakoff G, Johnson M (1980) Metaphors We Live By. Chicago: University of Chicago Press.Google Scholar
  36. Lakoff G, Johnson M (1999) Philosophy in the Flesh. New York: Basic Books.Google Scholar
  37. Lakoff G, Núúez R (1997) The metaphorical structure of mathematics: Sketching out cognitive foundations for a mind-based mathematics. In: Mathematical Reasoning: Analogies, Metaphors, and Images (English L, ed), 267–280. Mahwah, NJ: Erlbaum.Google Scholar
  38. Lakoff G, Núñez R (2000) Where Mathematics Comes From: How the Embodied Mind Brings Mathematics Into Being. New York: Basic Books.Google Scholar
  39. Mandler G, Shebo BJ (1982) Subitizing: An analysis of its component processes. Journal of Experimental Psychology: General, 111: 1–22.CrossRefGoogle Scholar
  40. Menninger K (1969) Number Words and Number Symbols. Cambridge: MIT Press.Google Scholar
  41. Nunes T, Schliemann AL, Carraher D (1993) Street Mathemathics and School Mathematics. Cambridge, UK: Cambridge University Press.Google Scholar
  42. Núñez R (1995) What brain for God’s-eye? Biological naturalism, ontological objectivism, and Searle. Journal of Consciousness Studies 2: 149–166.Google Scholar
  43. Núñez R (1997) Eating soup with chopsticks: Dogmas, difficulties, and alternatives in the study of conscious experience. Journal of Consciousness Studies 4: 143–166.Google Scholar
  44. Núñez R (1999) Could the future taste purple? Reclaiming mind, body, and cognition. In: Reclaiming Cognition: The Primacy of Action, Intention, and Emotion (Núñez R, Freeman WJ, eds), 41–60. Thorverton, UK: Imprint Academic.Google Scholar
  45. Núñez R (2005) Creating mathematical infinities: The beauty of transfinite cardinals. Journal of Pragmatics 37: 1717–1741.CrossRefGoogle Scholar
  46. Núñez R (2006) Do real numbers really move? Language, thought, and gesture: The embodied cognitive foundations of mathematics. In: 18 Unconventional Essays on the Nature of Mathematics (Hersh R, ed), 160–181. New York: Springer.CrossRefGoogle Scholar
  47. Núñez R (2008a) Proto-numerosities and concepts of number: Biologically plausible and culturally mediated top-down mathematical schemas. Behavioral and Brain Sciences 31: 665–666.CrossRefGoogle Scholar
  48. Núñez R (2008b) Reading between the number lines. Science 231: 1293.CrossRefGoogle Scholar
  49. Núñez R (2008c) Mathematics, the ultimate challenge to embodiment: Truth and the grounding of axiomatic systems. In: Handbook of Cognitive Science: An Embodied Approach (Calvo P, Gomila T, eds), 333–353. Amsterdam: Elsevier.CrossRefGoogle Scholar
  50. Núñez R (in press) Enacting infinity: Bringing transfinite cardinals into being. In: Enaction: Towards a New Paradigm in Cognitive Science (Stewart J, Gapenne O, Di Paolo E, eds). Cambridge, MA: MIT Press.Google Scholar
  51. Núñez R (submitted) No innate number line in the human brain.Google Scholar
  52. Núñez R, Lakoff G (1998) What did Weierstrass really define? The cognitive structure of natural and ε-δ continuity. Mathematical Cognition 4(2): 85–101.CrossRefGoogle Scholar
  53. Núñez R, Lakoff G (2005) The cognitive foundations of mathematics: The role of conceptual metaphor. In: Handbook of Mathematical Cognition (Campbell J, ed), 109–124. New York: Psychology Press.Google Scholar
  54. Núñez R, Motz B, Teuscher U (2006) Time after time: The psychological reality of the ego- and time-reference-point distinction in metaphorical construals of time. Metaphor and Symbol 21: 133–146.CrossRefGoogle Scholar
  55. Núñez R, Sweetser E (2006) With the future behind them: Convergent evidence from Aymara language and gesture in the cross-linguistic comparison of spatial construals of time. Cognitive Science 30: 401–450.CrossRefGoogle Scholar
  56. Rips LJ, Bloomfield A, Asmuth J (2008) From numerical concepts to concepts of number. Behavioral and Brain Sciences 31: 623–687.CrossRefGoogle Scholar
  57. Ristic J, Wright A, Kingstone A (2006) The number line effect reflects top-down control. Psychonomic Bulletin and Review 13: 862–868.CrossRefGoogle Scholar
  58. Robson E (2008) Mathematics in Ancient Irak. Princeton, NJ: Princeton University Press.Google Scholar
  59. Rotman B (1987) Sygnifying Nothing: The Semiotics of Zero. New York: St. Martin’s Press.Google Scholar
  60. Santens S, Gevers W (2008) The SNARC effect does not imply a mental number line. Cognition 108: 263–270.CrossRefGoogle Scholar
  61. Shepard, R (2001) Perceptual-cognitive universals as reflections of the world. Behavioral and Brain Sciences 24: 581–601.Google Scholar
  62. Shore C, Bates E, Bretherton I, Beeghly M, O’Connell B (1990) Vocal and gestural symbols: Similarities and differences from 13 to 28 months. In: From Gesture to Language in Hearing and Deaf Children (Volterra V, Erting C, eds), 79–91. New York: Springer.CrossRefGoogle Scholar
  63. Strauss MS, Curtis LE (1981) Infant perception of numerosity. Child Development 52: 1146–1152.CrossRefGoogle Scholar
  64. Sweetser E (1990) From Etymology to Pragmatics: Metaphorical and Cultural Aspects of Semantic Structure. New York: Cambridge University Press.CrossRefGoogle Scholar
  65. Talmy L (1988) Force dynamics in language and cognition. Cognitive Science 12: 49–100.CrossRefGoogle Scholar
  66. Talmy L (2003) Toward a Cognitive Semantics. Vol. 1: Concept Structuring Systems. Cambridge, MA: MIT Press.Google Scholar
  67. van Loosbroek E, Smitsman AW (1990) Visual perception of numerosity in infancy. Developmental Psychology 26: 916–922.CrossRefGoogle Scholar
  68. Wilden A (1972) System and Structure: Essays in Communication and Exchange. New York: Harper & Row.Google Scholar
  69. Wynn K (1992) Addition and subtraction by human infants. Nature 358: 749–750.CrossRefGoogle Scholar
  70. Xu F, Spelke E (2000) Large number discrimination in 6-month-old infants. Cognition 74: B1–B11.CrossRefGoogle Scholar

Copyright information

© Konrad Lorenz Institute for Evolution and Cognition Research 2009

Authors and Affiliations

  1. 1.Department of Cognitive ScienceUniversity of California, San DiegoLa JollaUSA

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