Journal of Logic, Language and Information

, Volume 22, Issue 4, pp 421–448 | Cite as

Prolegomena to a Cognitive Investigation of Euclidean Diagrammatic Reasoning

  • Yacin HamamiEmail author
  • John Mumma


Euclidean diagrammatic reasoning refers to the diagrammatic inferential practice that originated in the geometrical proofs of Euclid’s Elements. A seminal philosophical analysis of this practice by Manders (‘The Euclidean diagram’, 2008) has revealed that a systematic method of reasoning underlies the use of diagrams in Euclid’s proofs, leading in turn to a logical analysis aiming to capture this method formally via proof systems. The central premise of this paper is that our understanding of Euclidean diagrammatic reasoning can be fruitfully advanced by confronting these logical and philosophical analyses with the field of cognitive science. Surprisingly, central aspects of the philosophical and logical analyses resonate in very natural ways with research topics in mathematical cognition, spatial cognition and the psychology of reasoning. The paper develops these connections, concentrating on four issues: (1) the cognitive origins of Euclidean diagrammatic reasoning, (2) the cognitive representations of spatial relations in Euclidean diagrams, (3) the nature of the cognitive processes and cognitive representations involved in Euclidean diagrammatic reasoning seen as a form of visuospatial relational reasoning and (4) the complexity of Euclidean diagrammatic reasoning for the human cognitive system. For each of these issues, our analysis generates concrete experiment proposals, opening thereby the way for further empirical investigations. The paper is thus a prolegomenon to a research program on Euclidean diagrammatic reasoning at the crossroads of logic, philosophy and cognitive science.


Diagrammatic reasoning Euclidean geometry Mathematical cognition Visuospatial relational reasoning Categorical and coordinate spatial relations Relational complexity 



Parts of this paper have been presented at the Logic and Cognition 2012 Conference in Poznań (Poland, May 2012), the International Conference on Thinking in London (UK, July 2012) and the Logic and Cognition Workshop at ESSLLI in Opole (Poland, August 2012). We would like to thank the audiences of these events for helpful feedback and comments, in particular Markus Knauff and Michiel van Lambalgen for stimulating discussions. We are thankful to Jean Paul van Bendegem for providing comments on the whole manuscript. Finally, we are very grateful to Jakub Szymanik, Rineke Verbrugge and three anonymous referees for providing extensive comments that led to substantial improvements of the paper. Yacin Hamami acknowledges support from a doctoral fellowship of the Research Foundation Flanders (FWO).


  1. Avigad, J., Dean, E., & Mumma, J. (2009). A formal system for Euclid’s Elements. Review of Symbolic Logic, 2(4), 700–768.CrossRefGoogle Scholar
  2. Baciu, M., Koenig, O., Vernier, M., Bedoin, N., Rubin, C., & Segebarth, C. (1999). Categorical and coordinate spatial relations: fMRI evidence for hemispheric specialization. NeuroReport, 10(6), 1373–1378.CrossRefGoogle Scholar
  3. Barsalou, L. (1999). Perceptual symbol systems. Behavioral and Brain Sciences, 22(04), 577–660.Google Scholar
  4. Barsalou, L. (2008). Grounded cognition. Annual Review of Psychology, 59, 617–645.CrossRefGoogle Scholar
  5. Christoff, K., Prabhakaran, V., Dorfman, J., Zhao, Z., Kroger, J., Holyoak, K., et al. (2001). Rostrolateral prefrontal cortex involvement in relational integration during reasoning. NeuroImage, 14(5), 1136–1149.CrossRefGoogle Scholar
  6. Clark, A., & Chalmers, D. (1998). The extended mind. Analysis, 58, 7–19.CrossRefGoogle Scholar
  7. De Cruz, H. (2009). An enhanced argument for innate elementary geometric knowledge and its philosophical implications. In B. van Kerkhove (Ed.), New perspectives on mathematical practices, essays in philosophy and history of mathematics (pp. 185–206). New Jersey: World Scientific.CrossRefGoogle Scholar
  8. Dehaene, S., Izard, V., Pica, P., & Spelke, E. (2006). Core knowledge of geometry in an Amazonian indigene group. Science, 311(5759), 381–384.CrossRefGoogle Scholar
  9. Euclid (anc.) The thirteen books of Euclid’s elements.Google Scholar
  10. Fangmeier, T., Knauff, M., Ruff, C., & Sloutsky, V. (2006). fMRI evidence for a three-stage model of deductive reasoning. Journal of Cognitive Neuroscience, 18(3), 320–334.CrossRefGoogle Scholar
  11. Giaquinto, M. (2007). Visual thinking in mathematics: An epistemological study. Oxford: Oxford University Press.CrossRefGoogle Scholar
  12. Glasgow, J., & Papadias, D. (1992). Computational imagery. Cognitive Science, 16(3), 355–394.CrossRefGoogle Scholar
  13. Goel, V., Gold, B., Kapur, S., & Houle, S. (1998). Neuroanatomical correlates of human reasoning. Journal of Cognitive Neuroscience, 10(3), 293–302.CrossRefGoogle Scholar
  14. Goodwin, G., & Johnson-Laird, P. (2005). Reasoning about relations. Psychological Review, 112(2), 468–493.CrossRefGoogle Scholar
  15. Halford, G., Wilson, W., & Phillips, S. (1998). Processing capacity defined by relational complexity: Implications for comparative, developmental, and cognitive psychology. Behavioral and Brain Sciences, 21(06), 803–831.Google Scholar
  16. Halford, G., Baker, R., McCredden, J., & Bain, J. (2005). How many variables can humans process? Psychological Science, 16(1), 70–76.CrossRefGoogle Scholar
  17. Halford, G., Wilson, W., & Phillips, S. (2010). Relational knowledge: The foundation of higher cognition. Trends in Cognitive Sciences, 14(11), 497–505.CrossRefGoogle Scholar
  18. Hegarty, M., & Stull, A. (2012). Visuospatial thinking. In K. Holyoak & R. Morrison (Eds.), The Oxford handbook of thinking and reasoning (pp. 606–630). Oxford: Oxford University Press.Google Scholar
  19. Hutchins, E. (1999). Cognitive artifacts. In R. Wilson & F. Keil (Eds.), The MIT encyclopedia of the cognitive sciences (pp. 126–128). Cambridge, MA: MIT Press.Google Scholar
  20. Izard, V., Pica, P., Dehaene, S., Hinchey, D., & Spelke, E. (2011a). Geometry as a universal mental construction. In E. Brannon & S. Dehaene (Eds.), Space, time and number in the brain: Searching for the foundations of mathematical thought, no. XXIV in attention and performance (pp. 319–332). Oxford: Oxford University Press.CrossRefGoogle Scholar
  21. Izard, V., Pica, P., Spelke, E., & Dehaene, S. (2011b). Flexible intuitions of Euclidean geometry in an Amazonian indigene group. Proceedings of the National Academy of Sciences, 108(24), 9782–9787.CrossRefGoogle Scholar
  22. Jager, G., Postma, A., et al. (2003). On the hemispheric specialization for categorical and coordinate spatial relations: A review of the current evidence. Neuropsychologia, 41(4), 504–515.CrossRefGoogle Scholar
  23. Johnson-Laird, P., Byrne, R., & Tabossi, P. (1989). Reasoning by model: The case of multiple quantification. Psychological Review, 96(4), 658–673.CrossRefGoogle Scholar
  24. Knauff, M. (2009). A neuro-cognitive theory of deductive relational reasoning with mental models and visual images. Spatial Cognition & Computation, 9(2), 109–137.CrossRefGoogle Scholar
  25. Knauff, M. (2013). Space to reason: A spatial theory of human thought. Cambridge, MA: MIT Press.CrossRefGoogle Scholar
  26. Koedinger, K., & Anderson, J. (1990). Abstract planning and perceptual chunks: Elements of expertise in geometry. Cognitive Science, 14(4), 511–550.CrossRefGoogle Scholar
  27. Kosslyn, S. (1987). Seeing and imagining in the cerebral hemispheres: A computational approach. Psychological Review, 94(2), 148–175.CrossRefGoogle Scholar
  28. Kosslyn, S. (1994). Image and brain: The resolution of the imagery debate. Cambridge, MA: MIT press.Google Scholar
  29. Kosslyn, S., Koenig, O., Barrett, A., Cave, C., Tang, J., & Gabrieli, J. (1989). Evidence for two types of spatial representations: Hemispheric specialization for categorical and coordinate relations. Journal of Experimental Psychology: Human Perception and Performance, 15(4), 723–735.Google Scholar
  30. Kosslyn, S., Chabris, C., Marsolek, C., & Koenig, O. (1992). Categorical versus coordinate spatial relations: Computational analyses and computer simulations. Journal of Experimental Psychology: Human Perception and Performance, 18(2), 562–577.Google Scholar
  31. Kosslyn, S., Thompson, W., Gitelman, D., & Alpert, N. (1998). Neural systems that encode categorical versus coordinate spatial relations: PET investigations. Psychobiology, 26(4), 333–347.Google Scholar
  32. Laeng, B. (1994). Lateralization of categorical and coordinate spatial functions: A study of unilateral stroke patients. Journal of Cognitive Neuroscience, 6(3), 189–203.CrossRefGoogle Scholar
  33. Manders, K. (2008). The Euclidean diagram. In P. Mancosu (Ed.), Philosophy of mathematical practice. Oxford: Oxford University Press.Google Scholar
  34. Margolis, E., & Laurence, S. (2007). Creations of the mind: Theories of artifacts and their representation. Oxford: Oxford University Press.Google Scholar
  35. Marr, D. (1982). Vision: A computational investigation into the human representation and processing of visual information. San Francisco, CA: Freeman.Google Scholar
  36. Miller, N. (2007). Euclid and his twentieth century rivals: Diagrams in the logic of Euclidean geometry. Stanford: CSLI Publications.Google Scholar
  37. Miller, N. (2012). On the inconsistency of Mumma’s Eu. Notre Dame Journal of Formal Logic, 53(1), 27–52.CrossRefGoogle Scholar
  38. Mumma, J. (2006). Intuition formalized: ancient and modern methods of proof in elementary geometry. Ph.D. Thesis, Carnegie Mellon University.Google Scholar
  39. Mumma, J. (2010). Proofs, pictures, and Euclid. Synthese, 175(2), 255–287.CrossRefGoogle Scholar
  40. Mumma, J. (2013). Refining and elaborating Eu: a response to Miller’s ‘on the inconsistency of Eu’, (submitted).Google Scholar
  41. Norman, D. (1991). Cognitive artifacts. In J. Carroll (Ed.), Designing interaction: Psychology at the human–computer interface (pp. 17–38). Cambridge: Cambridge University Press.Google Scholar
  42. Palermo, L., Bureca, I., Matano, A., & Guariglia, C. (2008). Hemispheric contribution to categorical and coordinate representational processes: A study on brain-damaged patients. Neuropsychologia, 46(11), 2802–2807.CrossRefGoogle Scholar
  43. Randell, D., Cohn, A. & Cui, Z. (1992). A spatial logic based on regions and connection. In Nebel, B., Swartout, W., & Rich, C. (eds) Proceedings of the 3rd conference on principles of knowledge representation and reasoning, vol. 92, (pp 165–176). Cambridge, MA: Morgan Kaufmann.Google Scholar
  44. Renz, J., & Nebel, B. (2007). Qualitative spatial reasoning using constraint calculi. In M. Aiello, I. Pratt-Hartmann, & J. van Benthem (Eds.), Handbook of spatial logics (pp. 161–215). Berlin: Springer.CrossRefGoogle Scholar
  45. Rips, L. (1994). The psychology of proof: Deductive reasoning in human thinking. Cambridge, MA: The MIT Press.Google Scholar
  46. Spelke, E. (2011). Natural number and natural geometry. In E. Brannon & S. Dehaene (Eds.), Space, time and number in the brain: Searching for the foundations of mathematical thought, no XXIV in attention and performance (pp. 287–317). Oxford: Oxford University Press.CrossRefGoogle Scholar
  47. Spelke, E., & Kinzler, K. (2007). Core knowledge. Developmental Science, 10(1), 89–96.CrossRefGoogle Scholar
  48. Stenning, K. (2002). Seeing reason: Image and language in learning to think. Oxford: Oxford University Press.CrossRefGoogle Scholar
  49. Tarski, A. (1959). What is elementary geometry? In P. Suppes, A. Tarski, & L. Henkin (Eds.), The axiomatic method: With special reference to geometry and physics (1st ed., pp. 16–29). Amsterdam: North-Holland.CrossRefGoogle Scholar
  50. Trojano, L., Conson, M., Maffei, R., & Grossi, D. (2006). Categorical and coordinate spatial processing in the imagery domain investigated by rTMS. Neuropsychologia, 44(9), 1569–1574.CrossRefGoogle Scholar
  51. Tversky, B. (2005). Visuospatial reasoning. In K. Holyoak & R. Morrison (Eds.), The Cambridge handbook of thinking and reasoning (pp. 209–240). Cambridge: Cambridge University Press.Google Scholar
  52. van der Ham, C. (2010). Thinking left and right: Neurocognitive studies on spatial relation processing. Ph.D. Thesis, Utrecht University.Google Scholar
  53. Waltz, J., Knowlton, B., Holyoak, K., Boone, K., Mishkin, F., de Menezes, Santos M., et al. (1999). A system for relational reasoning in human prefrontal cortex. Psychological Science, 10(2), 119–125.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Centre for Logic and Philosophy of ScienceVrije Universiteit BrusselBrusselsBelgium
  2. 2.Department of PhilosophyCalifornia State University, San BernardinoSan BernardinoUSA

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