The Body of/in Proof: An Embodied Analysis of Mathematical Reasoning

  • Laurie D. EdwardsEmail author
Part of the Mathematics in Mind book series (MATHMIN)


Mathematics is, in part, a search for structure and certainty, and the primary practice of mathematicians toward that end consists of creating and communicating mathematical proofs. The purpose of this chapter is to examine mathematical proof and logical reasoning from the perspective of embodied cognition (Clark 1998; Gibbs 2005; Johnson 1987, 2012; Lakoff and Johnson 1980; Shapiro 2010, 2014; Varela et al. 1991). Although there are multiple theoretical and methodological approaches to embodiment, Varela (1999: 11–12) characterized its essential aspects as follows:


Embodiment Mathematical proof Gesture Infant cognition Logical deduction 


  1. Abrahamson, D., and Lindgren, R. (2014). Embodiment and embodied design. The Cambridge handbook of the learning sciences, 2, 358–376.CrossRefGoogle Scholar
  2. Balacheff, N. (1991) Benefits and limits of social interaction: The case of teaching mathematical proof. In: A. Bishop, S. Mellin-Olsen and J. Van Dormolen J. (eds.) Mathematical knowledge: Its growth through teaching, pp. 175–192. Dordrecht, The Netherlands: Kluwer.Google Scholar
  3. Bazzini, L. (1991). From grounding metaphors to technological devices: A call for legitimacy in school mathematics. Educational Studies in Mathematics, 47: 259–271.CrossRefGoogle Scholar
  4. Best, J. (2005). Recognition of proofs in conditional reasoning. Thinking and Reasoning 11: 326–348.CrossRefGoogle Scholar
  5. Calbris, G. (2008). From left to right…Coverbal gestures and their symbolic use of space. In: A. Cienki and C. Müller (eds.) Metaphor and gesture, pp. 27–54. Amsterdam, The Netherlands: John Benjamins Publishing.CrossRefGoogle Scholar
  6. Chazan, D. (1993). High school geometry students’ justification for their views of empirical evidence and mathematical proof. Educational Studies in Mathematics 24(4): 359–387.CrossRefGoogle Scholar
  7. Clark, A. (1998). Being there: Putting brain, body, and world together again. Cambridge, MA: MIT Press.Google Scholar
  8. Dancygier, B., and Sweetser, E. (2005). Mental spaces in grammar: Conditional constructions. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
  9. Edwards, L. D. (2009). Gestures and conceptual integration in mathematical talk. Educational Studies in Mathematics 70(2): 127–141.CrossRefGoogle Scholar
  10. Edwards, L. D. (2010). Doctoral students, embodied discourse and proof. In: M. M. F. Pinto and T. F. Kawasaki (eds.), Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education, Vol. 2, pp. 329–336. Belo Horizonte, Brazil: PME.Google Scholar
  11. Edwards, L. D. (2011). Embodied cognitive science and mathematics. In: B. Ubuz (ed.), Proceedings of the 35th Conference of the International Group for the Psychology of Mathematics Education, Vol. 2, pp. 297–304. Ankara, Turkey: PME.Google Scholar
  12. Edwards, L. D., Moore-Russo, D., and Ferrara, F. (eds.) (2014). Emerging perspectives on gesture and embodiment in mathematics. Charlotte, NC: Information Age Publishing.Google Scholar
  13. Evans, V., and Green, M. (2006). Cognitive linguistics: An introduction. Mahwah, NJ: Lawrence Erlbaum.Google Scholar
  14. Evans, J. S. B., Newstead, S. E., and Byrne, R. M. (1993). Human reasoning: The psychology of deduction. Hove, UK: Lawrence Erlbaum.Google Scholar
  15. Fauconnier, G. (1994). Mental spaces: Aspects of meaning construction in natural language. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  16. Fauconnier, G. and Lakoff, G. (2009). On metaphor and blending. Cognitive Semiotics 5, (1–2): 393–399.Google Scholar
  17. Fauconnier, G. and Turner, M. (2002). The way we think: Conceptual blending and the mind’s hidden complexities. New York, NY: Basic Books.Google Scholar
  18. Gibbs Jr., R. W. (2005). Embodiment and cognitive science. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
  19. Gopnik, A., and Schulz, L. (eds.) (2007). Causal learning: Psychology, philosophy, and computation. Oxford, UK: Oxford University Press.Google Scholar
  20. Hall, R., and Nemirovsky, R. (2012). Introduction to the special issue: Modalities of body engagement in mathematical activity and learning. Journal of the Learning Sciences 21(2): 207–215.CrossRefGoogle Scholar
  21. Hanna, G. (1990). Some pedagogical aspects of proof. Interchange 21(1): 6–13.CrossRefGoogle Scholar
  22. Hanna, G. (1995). Challenges to the importance of proof. For the Learning of mathematics 15(3): 42–49.Google Scholar
  23. Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational Studies in Mathematics 44(1–2): 5–23.CrossRefGoogle Scholar
  24. Harel, G., and Sowder, L. (2007). Toward comprehensive perspectives on the learning and teaching of proof. In: F. Lester (ed.), Second handbook of research on mathematics teaching and learning, pp. 805–842. Charlotte, NC: Information Age Publishing.Google Scholar
  25. Johnson, M. (1987). The body in the mind: The bodily basis of reason and imagination. Chicago, IL: University of Chicago Press.CrossRefGoogle Scholar
  26. Johnson, M. (2012). The meaning of the body: Aesthetics of human understanding. Chicago, IL: University of Chicago Press.Google Scholar
  27. Johnson-Laird, P. N. (1999). Deductive reasoning. Annual Review of Psychology 50: 100–135.CrossRefGoogle Scholar
  28. Kahneman, D., Slovic, P., and Tversky, A. (1982). Judgment under uncertainty: Heuristics and biases. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
  29. Kiefer, M., and Trumpp, N. M. (2012). Embodiment theory and education: The foundations of cognition in perception and action. Trends in Neuroscience and Education 1(1): 15–20.CrossRefGoogle Scholar
  30. Laborde, C. (2000). Dynamic geometry environments as a source of rich learning contexts for the complex activity of proving. Educational Studies in Mathematics 44(1–2): 151–161.CrossRefGoogle Scholar
  31. Lakoff, G., and Johnson, M. (1980). Metaphors we live by. Chicago, IL: University of Chicago Press.Google Scholar
  32. Lakoff, G., and Núñez, R. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York, NY: Basic Books.zbMATHGoogle Scholar
  33. Leslie, A. M., and Keeble, S. (1987). Do six-month-old infants perceive causality? Cognition 25(3): 265–288.CrossRefGoogle Scholar
  34. Lin, F., Hsieh, F., Hanna, G., and de Villiers, M. (Eds.) (2009). Proceedings of ICMI STUDY 19: Proof and proving in mathematics education. Taipei, Taiwan: The Department of Mathematics, National Taiwan Normal University.Google Scholar
  35. Mandler, J. M. (2004). The foundations of mind: Origins of conceptual thought. Oxford, UK: Oxford University Press.Google Scholar
  36. Manin, Y. (1998). Truth, rigour, and common sense. In: H.G. Dales and G. Oliveri (eds.). Truth in mathematics (pp. 147–159). Oxford, UK: Oxford University Press.Google Scholar
  37. Manin, I. I., Manin, J. I., and Manin, Y. I. (2007). Mathematics as metaphor: Selected essays of Yuri I. Manin (Vol. 20). Providence, RI: American Mathematical Society.zbMATHGoogle Scholar
  38. Marghetis, T., Edwards, L. D., and Núñez, R. (2014). More than mere handwaving. In: L. D. Edwards, D. Moore-Russo, and F. Ferrara (eds.), Emerging perspectives on gesture and embodiment in mathematics, pp. 227–246. Charlotte, NC: Information Age Publishing.Google Scholar
  39. McNeill, D. (1992). Hand and mind: What gestures reveal about thought. Chicago, IL: Chicago University Press.Google Scholar
  40. McNeill, D. (2005). Gesture and thought. Chicago, IL: Chicago University Press.CrossRefGoogle Scholar
  41. Nardi, E., and Knuth, E. (2017). Changing classroom culture, curricula, and instruction for proof and proving: How amenable to scaling up, practicable for curricular integration, and capable of producing long-lasting effects are current interventions? Educational Studies in Mathematics 96(2): 267–274.CrossRefGoogle Scholar
  42. New York Times (July 16, 2017). Maryam Mirzakhani, Only Woman to Win a Fields Medal, Dies at 40. Retrieved December 18, 2018 from:
  43. Núñez, R., Edwards, L. D., and Matos, J. (1999). Embodied cognition as grounding for situatedness and context in mathematics education. Educational Studies in Mathematics 39(1–3): 45–65.CrossRefGoogle Scholar
  44. Oakley, T. (2007). Image schema. In: D. Geeraerts and H. Cuyckens (eds.), Handbook of cognitive linguistics, pp. 214–235. Oxford, UK: Oxford University Press.Google Scholar
  45. Piaget, J. and Inhelder, B. (1964). The early growth of logic in the child: Classification and seriation. London, UK: Routledge.Google Scholar
  46. Radford, L., Edwards, L., and Arzarello, F. (2009). Introduction: Beyond words (Special Issue: Gestures and Multimodality in the Construction of Mathematical Meaning). Educational Studies in Mathematics 70(2): 91–95.CrossRefGoogle Scholar
  47. Rasmussen, C., Stephan, M., and Allen, K. (2004). Classroom mathematical practices and gesturing. The Journal of Mathematical Behavior 23(3): 301–323.CrossRefGoogle Scholar
  48. Rotman, B. (1993). Taking God out of mathematics and putting the body back in: An essay in corporeal semiotics. Palo Alto, CA: Stanford University Press.zbMATHGoogle Scholar
  49. Roy, S., Inglis, M., and Alcock, L. (2017). Multimedia resources designed to support learning from written proofs: An eye-movement study. Educational Studies in Mathematics 96(2): 249–266.CrossRefGoogle Scholar
  50. Shapiro, L. (2010). Embodied cognition. New York, NY: Routledge.CrossRefGoogle Scholar
  51. Shapiro, L. (ed.). (2014). The Routledge handbook of embodied cognition. New York, NY: Routledge.Google Scholar
  52. Sperber, D., Premack, A., and Premack, J. (eds.) (1996). Causal cognition: A multidisciplinary debate. Oxford, UK: Oxford University Press.Google Scholar
  53. Sweetser, E. (1996). Mental spaces and the grammar of conditional constructions. In: G. Fauconnier and E. Sweetser, (eds.) Spaces, worlds, and grammar, pp. 318–333. Chicago, IL: University of Chicago Press.Google Scholar
  54. Sweetser, E. (1998). Regular metaphoricity in gesture: Bodily-based models of speech interaction. In: Actes du 16e Congrès International des Linguistes. Amsterdam, The Netherlands: Elsevier (CD-ROM).Google Scholar
  55. Sweetser, E., and Smith, I. (2015, July). Conditionals, mental spaces and gesture. Paper presented at the 13th International Cognitive Linguistic Conference, Newcastle-on-Tyne, England.Google Scholar
  56. Tall, D. (2008). The transition to formal thinking in mathematics. Mathematics Education Research Journal 20(2): 5–24.CrossRefGoogle Scholar
  57. Talmy, L. (1988). Force dynamics in language and cognition. Cognitive Science 12: 49–100.CrossRefGoogle Scholar
  58. Thurston, W. P. (1994) On proof and progress in mathematics. Bulletin of the American Mathematical Society 30(2):161–177MathSciNetCrossRefGoogle Scholar
  59. Varela, F. J. (1999). Ethical know-how: Action, wisdom, and cognition. Palo Alto, CA: Stanford University Press.Google Scholar
  60. Varela, F. J., Thompson, E., and Rosch, E. (1991). The embodied mind: Cognitive science and human experience. Cambridge, MA: MIT Press.CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.St. Mary’s College of CaliforniaMoragaUSA

Personalised recommendations