Synthese

, Volume 190, Issue 1, pp 3–19 | Cite as

Mathematical symbols as epistemic actions

Article

Abstract

Recent experimental evidence from developmental psychology and cognitive neuroscience indicates that humans are equipped with unlearned elementary mathematical skills. However, formal mathematics has properties that cannot be reduced to these elementary cognitive capacities. The question then arises how human beings cognitively deal with more advanced mathematical ideas. This paper draws on the extended mind thesis to suggest that mathematical symbols enable us to delegate some mathematical operations to the external environment. In this view, mathematical symbols are not only used to express mathematical concepts—they are constitutive of the mathematical concepts themselves. Mathematical symbols are epistemic actions, because they enable us to represent concepts that are literally unthinkable with our bare brains. Using case-studies from the history of mathematics and from educational psychology, we argue for an intimate relationship between mathematical symbols and mathematical cognition.

Keywords

Mathematical symbols Extended mind Symbolic cognition History of mathematics 

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References

  1. Adams F., Aizawa K. (2001) The bounds of cognition. Philosophical Psychology 14: 43–64CrossRefGoogle Scholar
  2. Barabashev A. G. (1997) In support of significant modernization of original mathematical texts (in defense of presentism). Philosophia Mathematica 5: 21–41CrossRefGoogle Scholar
  3. Benacerraf P. (1973) Mathematical truth. Journal of Philosophy 70: 661–680CrossRefGoogle Scholar
  4. Biro D., Matsuzawa T. (2001) Use of numerical symbols by the chimpanzee (Pan troglodytes): Cardinals, ordinals, and the introduction of zero. Animal Cognition 4: 193–199CrossRefGoogle Scholar
  5. Boyer P. (2001) Religion explained. The evolutionary origins of religious thought. Basic Books, New YorkGoogle Scholar
  6. Buzaglo M. (2002) The logic of concept expansion. Cambridge University Press, CambridgeGoogle Scholar
  7. Cantlon J. F., Brannon E. M., Carter E. J., Pelphrey K. A. (2006) Functional imaging of numerical processing in adults and 4-y-old children. PLoS Biology 4: e125CrossRefGoogle Scholar
  8. Chemla K. (2003) Generality above abstraction: The general expressed in terms of the paradigmatic in mathematics in ancient China. Science in Context 16: 413–458CrossRefGoogle Scholar
  9. Chrisomalis S. (2004) A cognitive typology for numerical notation. Cambridge Archaeological Journal 14: 37–52CrossRefGoogle Scholar
  10. Clark A. (2006) Material symbols. Philosophical Psychology 19: 291–307CrossRefGoogle Scholar
  11. Clark A., Chalmers D. (1998) The extended mind. Analysis 58: 7–19CrossRefGoogle Scholar
  12. Dacke M., Srinivasan M. V. (2008) Evidence for counting in insects. Animal Cognition 11: 683–689CrossRefGoogle Scholar
  13. De Cruz H. (2006) Why are some numerical concepts more successful than others? An evolutionary perspective on the history of number concepts. Evolution and Human Behavior 27: 306–323CrossRefGoogle Scholar
  14. De Cruz H. (2008) An extended mind perspective on natural number representation. Philosophical Psychology 21: 475–490CrossRefGoogle Scholar
  15. DeLoache J. S. (2004) Becoming symbol-minded. Trends in Cognitive Sciences 8: 66–70CrossRefGoogle Scholar
  16. De Morgan A. (1830) On the study and difficulties of mathematics. Paul Kegan, LondonGoogle Scholar
  17. Eger E., Sterzer P., Russ M. O., Giraud A.-L., Kleinschmidt A. (2003) A supramodal number representation in human intraparietal cortex. Neuron 37: 1–20CrossRefGoogle Scholar
  18. Ekert A. (2008) Complex and unpredictable Cardano. International Journal of Theoretical Physics 47: 2101–2119CrossRefGoogle Scholar
  19. Ernest P. (1998) Social constructivism as a philosophy of mathematics. State University of New York Press, AlbanyGoogle Scholar
  20. Feigenson L., Dehaene S., Spelke E. S. (2004) Core systems of number. Trends in Cognitive Sciences 8: 307–314CrossRefGoogle Scholar
  21. Féron J., Gentaz E., Streri A. (2006) Evidence of amodal representation of small numbers across visuo-tactile modalities in 5-month-old infants. Cognitive Development 21: 81–92CrossRefGoogle Scholar
  22. Fischer M. H. (2003) Cognitive representation of negative numbers. Psychological Science 14: 278–282CrossRefGoogle Scholar
  23. Goodman N. D. (1981) The experiential foundations of mathematical knowledge. History and Philosophy of Logic 2: 55–65CrossRefGoogle Scholar
  24. Harper E. (1987) Ghosts of Diophantus. Educational Studies in Mathematics 18: 75–90CrossRefGoogle Scholar
  25. Jordan K. E., Brannon E. M. (2006) The multisensory representation of number in infancy. Proceedings of the National Academy of Sciences of the United States of America 103: 3486–3489CrossRefGoogle Scholar
  26. Joseph G. G. (2000) The crest of the peacock: Non-European roots of mathematics (2nd ed.). Princeton University Press, PrincetonGoogle Scholar
  27. Kirsh D. (1996) Adapting the environment instead of oneself. Adaptive Behavior 4: 415–452CrossRefGoogle Scholar
  28. Kirsh D., Maglio P. (1994) On distinguishing epistemic from pragmatic action. Cognitive Science 18: 513–549CrossRefGoogle Scholar
  29. Krieger M. H. (1991) Theorems as meaningful cultural artifacts: Making the world additive. Synthese 144: 135–154CrossRefGoogle Scholar
  30. Margolis E., Laurence S. (2007) The ontology of concepts—abstract objects or mental representations?. Noûs 41: 561–593CrossRefGoogle Scholar
  31. Meck W. H., Church R. M. (1983) A mode control model of counting and timing processes. Journal of Experimental Psychology: Animal Behavior Processes 9: 320–334CrossRefGoogle Scholar
  32. Millikan R. G. (1998) A common structure for concepts of individuals, stuffs, and real kinds: More mama, more milk, and more mouse. Behavioral and Brain Sciences 21: 55–65Google Scholar
  33. Muntersbjorn M. M. (2003) Representational innovation and mathematical ontology. Synthese 134: 159–180CrossRefGoogle Scholar
  34. Naets J. (2010) How to define a number? A general epistemological account of Simon Stevin’s art of defining. Topoi 29: 77–86CrossRefGoogle Scholar
  35. Netz R. (1999) Linguistic formulae as cognitive tools. Pragmatics and Cognition 7: 147–176CrossRefGoogle Scholar
  36. Nieder A., Miller E. K. (2003) Coding of cognitive magnitude: Compressed scaling of numerical information in the primate prefrontal cortex. Neuron 37: 149–157CrossRefGoogle Scholar
  37. Oaks J. A. (2007) Medieval Arabic algebra as an artificial language. Journal of Indian Philosophy 35: 543–575CrossRefGoogle Scholar
  38. Petersson K. M., Silva C., Castro-Caldas A., Ingvar M., Reis A. (2007) Literacy: A cultural influence on functional left-right differences in the inferior parietal cortex. European Journal of Neuroscience 26: 791–799CrossRefGoogle Scholar
  39. Popper K. (1994) In search of a better world. Lectures and essays from thirty years. Routledge, LondonGoogle Scholar
  40. Preissler M., Bloom P. (2007) Two-year-olds appreciate the dual nature of pictures. Psychological Science 18: 1–2CrossRefGoogle Scholar
  41. Qin Y., Carter C. S., Silk E. M., Stenger V. A., Fissell K., Goode A., Anderson J. R. (2004) The change of the brain activation patterns as children learn algebra equation solving. Proceedings of the National Academy of Sciences of the United States of America 101: 5686–5691CrossRefGoogle Scholar
  42. Rips L., Bloomfield A., Asmuth J. (2008) From numerical concepts to concepts of number. Behavioral and Brain Sciences 31: 623–642CrossRefGoogle Scholar
  43. Schlimm D., Neth H. (2008) Modeling ancient and modern arithmetic practices: Addition and multiplication with Arabic and Roman numerals. In: Sloutsky V., Love B., McRae K. (eds) Proceedings of the 30th annual meeting of the Cognitive Science Society. Cognitive Science Society, AustinGoogle Scholar
  44. Schwartz D. L., Martin T., Pfaffman J. (2005) How mathematics propels the development of physical knowledge. Journal of Cognition and Development 6: 65–88CrossRefGoogle Scholar
  45. Sfard A. (1991) On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics 22: 1–36CrossRefGoogle Scholar
  46. Siegler R. S., Booth J. L. (2004) Development of numerical estimation in young children. Child Development 75: 428–444CrossRefGoogle Scholar
  47. Spelke E. S., Tsivkin A. (2001) Language and number: A bilingual training study. Cognition 78: 45–88CrossRefGoogle Scholar
  48. Sperber D. (1996) Explaining culture. A naturalistic approach. Blackwell, OxfordGoogle Scholar
  49. Staal F. (2006) Artificial languages across sciences and civilizations. Journal of Indian Philosophy 34: 89–141CrossRefGoogle Scholar
  50. Stedall J. A. (2001) Of our own nation: John Wallis’s account of mathematical learning in medieval England. Historia Mathematica 2: 73–122CrossRefGoogle Scholar
  51. Tan L. H., Feng C. M., Fox P. T., Gao J. H. (2001) An fMRI study with written Chinese. NeuroReport 12: 83–88CrossRefGoogle Scholar
  52. Tang Y., Zhang W., Chen K., Feng S., Ji Y., Shen J., Reiman E., Liu Y. (2006) Arithmetic processing in the brain shaped by cultures. Proceedings of the National Academy of Sciences of the United States of America 103: 10775–10780CrossRefGoogle Scholar
  53. Temple E., Posner M. I. (1998) Brain mechanisms of quantity are similar in 5-year-old children and adults. Proceedings of the National Academy of Sciences of the United States of America 95: 7836–7841CrossRefGoogle Scholar
  54. Thurston W. (2006) On proof and progress in mathematics. In: Hersh R. (Ed.) 18 unconventional essays on the nature of mathematics. Springer, New York, pp 37–55CrossRefGoogle Scholar
  55. Tratman E. K. (1976) A late Upper Palaeolithic calculator (?), Gough’s cave, Cheddar, Somerset. Proceedings of the University of Bristol Spelæological Society 14: 123–129Google Scholar
  56. Uttal D. H., Scudder K. V., DeLoache J. S. (1997) Manipulatives as symbols: A new perspective on the use of concrete objects to teach mathematics. Journal of Applied Developmental Psychology 18: 37–54CrossRefGoogle Scholar
  57. Vlassis J. (2004) Making sense of the minus sign or becoming flexible in ’negativity’. Learning and Instruction 14: 469–484CrossRefGoogle Scholar
  58. Vlassis J. (2008) The role of mathematical symbols in the development of number conceptualization: The case of the minus sign. Philosophical Psychology 21: 555–570CrossRefGoogle Scholar
  59. Wellman H. M., Miller K. F. (1986) Thinking about nothing: Development of concepts of zero. British Journal of Developmental Psychology 4: 31–42CrossRefGoogle Scholar
  60. Whitehead A. N. (1911) An introduction to mathematics. Williams & Northgate, LondonGoogle Scholar
  61. Xu F., Spelke E. S. (2000) Large number discrimination in 6-month-old infants. Cognition 74: B1–B11CrossRefGoogle Scholar
  62. Zhang J., Norman D. A. (1995) A representational analysis of numeration systems. Cognition 57: 271–295CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Centre for Logic and Analytical PhilosophyUniversity of LeuvenLeuvenBelgium
  2. 2.Department of Philosophy and EthicsGhent UniversityGentBelgium

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