Cognitive Processing

, Volume 14, Issue 3, pp 217–229 | Cite as

Metaphorical motion in mathematical reasoning: further evidence for pre-motor implementation of structure mapping in abstract domains

  • Chris Fields


The theory of computation and category theory both employ arrow-based notations that suggest that the basic metaphor “state changes are like motions” plays a fundamental role in all mathematical reasoning involving formal manipulations. If this is correct, structure-mapping inferences implemented by the pre-motor action planning system can be expected to be involved in solving any mathematics problems not solvable by table lookups and number line manipulations alone. Available functional imaging studies of multi-digit arithmetic, algebra, geometry and calculus problem solving are consistent with this expectation.


Analogy Category theory Computation Embodied cognition Event files Parietal cortex 



Thanks to Action Editor Martin Fischer and two anonymous referees for helpful comments on an earlier version of this paper.

Conflict of interest

The author declares that he has no financial or other conflicts of interest with regard to the research reported here.


  1. Adámek J, Herrlich H, Strecker GE (2004). Abstract and concrete categories (Web edition). Available from Accessed Nov. 26, 2012
  2. Anderson JR (2005) Human symbol manipulation within an integrated cognitive architecture. Cogn Sci 29:313–341PubMedCrossRefGoogle Scholar
  3. Anderson JR, Betts S, Ferris JL, Fincham JM (2011) Cognitive and metacognitive activity in mathematical problem solving: prefrontal and parietal patterns. Cogn Affect Behav Neurosci 11:52–67PubMedCrossRefGoogle Scholar
  4. Aziz-Zadeh L, Damasio A (2008) Embodied semantics for actions: findings from functional brain imaging. J Physiol Paris 102:35–39PubMedCrossRefGoogle Scholar
  5. Baez JC, Stay M (2011) Physics, topology, logic and computation: a Rosetta stone. New structures for physics: lecture notes in physics 813. Springer, Berlin, pp. 95–172Google Scholar
  6. Bahlmann J, Schubotz RI, Friederici AD (2008) Hierarchical artificial grammar processing engages Broca’s area. NeuroImage 42:525–534PubMedCrossRefGoogle Scholar
  7. Blair KP, Schwartz DL (2012) A value of concrete learning materials in adolescence. In: Reyna VF, Chapman S, Dougherty M, Confrey J (eds) The adolescent brain: learning, reasoning, and decision making. APA, Washington, pp 95–122CrossRefGoogle Scholar
  8. Bubic A, von Cramon DY, Schubotz RI (2010) Prediction, cognition and the brain. Front Psychol Human Neurosci 4:25. doi: 10.3389/fnhum.2010.00025 Google Scholar
  9. Bunge SA, Wendelken C, Badre D, Wagner AD (2005) Analogical reasoning and prefrontal cortex: evidence for separate retrieval and integration mechanisms. Cereb Cortex 5:239–249Google Scholar
  10. Butterworth B (2010) Foundational numerical capacities and the origins of dyscalculia. Trends Cogn Sci 14:534–541PubMedCrossRefGoogle Scholar
  11. Cantlon J, Brannon E, Carter E, Pelphrey K (2006) Functional imaging of numerical processing in adults and 4-yr-old children. PLoS Biol 4:0844–0854CrossRefGoogle Scholar
  12. Catmur C, Walsh V, Heyes C (2007) Sensorimotor learning configures the human mirror system. Curr Biol 17:1527–1531PubMedCrossRefGoogle Scholar
  13. Cattell RB (1971) Abilities: their structure, growth and action. Houghton Mifflin, BostonGoogle Scholar
  14. Chatterjee A (2010) Disembodying cognition. Lang Cogn 2:79–116PubMedGoogle Scholar
  15. Cho S, Moody TD, Fernandino F, Mumford JA, Poldrack RA, Cannon TD, Knowlton BJ, Holyoak KJ (2010) Common and dissociable prefrontal loci associated with component mechanisms of analogical reasoning. Cereb Cortex 20:524–533PubMedCrossRefGoogle Scholar
  16. Cohen Kadosh RC, Walsh V (2009) Numerical representation in the parietal lobes: abstract or not abstract? Behav Brain Sci 32:313–328PubMedCrossRefGoogle Scholar
  17. Cummins R (1983) The nature of psychological explanation. MIT, Cambridge, MAGoogle Scholar
  18. Danker JF, Anderson JR (2007) The roles of prefrontal and posterior parietal cortex in algebra problem solving: a case of using cognitive modeling to inform neuroimaging data. NeuroImage 35:1365–1377PubMedCrossRefGoogle Scholar
  19. De Pisapia N, Braver TS (2008) Preparation for integration: the role of anterior prefrontal cortex in working memory. NeuroReport 19:15–19PubMedCrossRefGoogle Scholar
  20. Dehaene S (2011) The number sense, 2nd edn. Oxford University Press, New YorkGoogle Scholar
  21. Dehaene S, Piazza M, Pinel P, Cohen L (2003) Three parietal circuits for number processing. Cogn Neuropsychol 20:487–506PubMedCrossRefGoogle Scholar
  22. Desai RH, Binder JR, Conant LL, Mano QR, Seidenberg MS (2011) The neural career of sensory-motor metaphors. J Cogn Neurosci 23:2376–2386PubMedCrossRefGoogle Scholar
  23. Eilenberg S, Mac Lane S (1945) Relations between homology and homotopy groups of spaces. Ann Math 46:480–509CrossRefGoogle Scholar
  24. Engel A, Burke M, Fiehler K, Bien S, Rosler F (2007) How moving objects become animated: the human mirror system assimilates non-biological movement patterns. Soc Neurosci 3:368–387CrossRefGoogle Scholar
  25. Falkenhainer B, Forbus KD, Gentner D (1989) The structure mapping engine: algorithm and examples. Artif Intell 41:1–63CrossRefGoogle Scholar
  26. Fedorenko E, Nieto-Castañón A, Kanwisher N (2012) Syntactic processing in the human brain: what we know, what we don’t know, and a suggestion for how to proceed. Brain Lang 120:187–207PubMedCrossRefGoogle Scholar
  27. Fields C (2011) Implementation of structure-mapping inference by event-file binding and action planning: a model of tool-improvisation analogies. Psychol Res 75:129–142PubMedCrossRefGoogle Scholar
  28. Fields C (2012) Motion as manipulation: implementation of force-motion analogies by event-file binding and action planning. Cogn Process 13:231–241PubMedCrossRefGoogle Scholar
  29. Fischer MH (2012) A hierarchical view of grounded, embodied, and situated numerical cognition. Cogn Process 13:161–164CrossRefGoogle Scholar
  30. Fischer MH, Brugger P (2011) When digits help digits: spatial-numerical associations point to finger counting as prime example of embodied cognition. Front Psychol Cogn 2:260. doi: 10.3389/fpsyg.2011.00260 Google Scholar
  31. Fleming TM, Beran MJ, Thompson RKR, Kleider HM, Washburn DA (2008) What meaning means for same and different: analogical reasoning in humans (Homo sapiens), chimpanzees (Pan troglodytes) and rhesus monkeys (Macaca mulatta). J Comp Psychol 122:176–185CrossRefGoogle Scholar
  32. Galton A (2006) The Church-Turing thesis: still valid after all these years? Appl Math Comput 178:93–102CrossRefGoogle Scholar
  33. Gentner D (1983) Structure-mapping: a theoretical framework for analogy. Cogn Sci 7:155–170CrossRefGoogle Scholar
  34. Gentner D (2003) Why we’re so smart. In: Gentner D, Goldin-Meadow S (eds) Language and mind: advances in the study of language and thought. MIT Press, Cambridge, pp 195–235Google Scholar
  35. Goldberg RP (1974) A survey of virtual machine research. IEEE Comput 7(6):34–45CrossRefGoogle Scholar
  36. Gottfried B (2012) Using space to represent data: diagrammatic reasoning. Cogn Process 13:371–373PubMedCrossRefGoogle Scholar
  37. Gray JR, Chabris CF, Braver TS (2003) Neural mechanisms of general fluid intelligence. Nat Neurosci 6:316–322PubMedCrossRefGoogle Scholar
  38. Green A, Fugelsang J, Kraemer D, Shamosh N, Dunbar K (2006) Frontopolar cortex mediates abstract integration in analogy. Brain Res 1096:125–137PubMedCrossRefGoogle Scholar
  39. Heyes C (2010) Where do mirror neurons come from? Neurosci Biobehav Rev 34:575–583PubMedCrossRefGoogle Scholar
  40. Heyes C (2012) Grist and mills: on the cultural origins of cultural learning. Philos Trans R Soc B 367:2181–2191CrossRefGoogle Scholar
  41. Holyoak K (2005) Analogy. In: Holyoak K, Morrison R (eds) The Cambridge handbook of thinking and reasoning. Cambridge University Press, Cambridge, pp 117–142Google Scholar
  42. Hommel B (2004) Event files: feature binding in and across perception and action. Trends Cogn Sci 8:494–500PubMedCrossRefGoogle Scholar
  43. Hopcroft JE, Ullman JD (1979) Introduction to automata, languages and computation. Addison-Wesley, BostonGoogle Scholar
  44. Kellman PJ, Massey CM, Son JY (2009) Perceptual learning modules in mathematics: enhancing students’ pattern recognition, structure extraction, and fluency. Topics Cogn Sci 2(2):1–21Google Scholar
  45. Kennedy EH, Fragaszy DM (2008) Analogical reasoning in a capuchin monkey (Cebus apella). J Comp Psychol 122:167–175PubMedCrossRefGoogle Scholar
  46. Kiefer M, Pulvermüller F (2012) Conceptual representations in mind and brain: theoretical developments, current evidence and future directions. Cortex 48:805–825PubMedCrossRefGoogle Scholar
  47. Knops A, Viarouge A, Dehaene S (2009) Dynamic representations underlying symbolic and nonsymbolic calculation: evidence from the operational momentum effect. Atten Percept Psychophys 71:803–821PubMedCrossRefGoogle Scholar
  48. Knowlton BJ, Holyoak KJ (2009) Prefrontal substrate of human relational reasoning. In: Gazzaniga MS (ed) The cognitive neurosciences. MIT Press, Cambridge, pp 1005–1017Google Scholar
  49. Kosslyn SM, Thompson WL, Ganis G (2006) The case for mental imagery. Oxford University Press, New YorkCrossRefGoogle Scholar
  50. Krawczyk DC (2012) The cognition and neuroscience of relational reasoning. Brain Res 1428:13–23PubMedCrossRefGoogle Scholar
  51. Krawczyk DC, McClelland MM, Donovan CM (2010) A hierarchy for relational reasoning in the prefrontal cortex. Cortex 47:588–597PubMedCrossRefGoogle Scholar
  52. Krueger F, Spampinato MV, Pardini M, Pajevic S, Wood JN, Weiss GH, Landgraf S, Grafman J (2008) Integral calculus problem solving: an fMRI investigation. NeuroReport 19:1095–1099PubMedCrossRefGoogle Scholar
  53. Lakoff G, Johnson M (1999) Philosophy in the flesh. Basic Books, New YorkGoogle Scholar
  54. Lakoff G, Núñez RE (2000) Where mathematics comes from: how the embodied mind brings mathematics into being. Basic Books, New YorkGoogle Scholar
  55. Landgraf S, van der Meer E, Krueger F (2010) Cognitive resource allocation for neuronal activity underlying mathematical cognition: a multi-method study. Int J Math Educ 42:579–590CrossRefGoogle Scholar
  56. Larkin J, Simon HA (1987) Why a diagram is (sometimes) worth ten thousand words. Cogn Sci 11:65–99CrossRefGoogle Scholar
  57. Mac Lane S (1972) Categories for the working mathematician. Springer, BerlinGoogle Scholar
  58. Makuuchi M, Bahlmann J, Anwander A, Friederici AD (2009) Segregating the core computational faculty of human language from working memory. Proc Natl Acad Sci USA 106:8362–8367PubMedCrossRefGoogle Scholar
  59. Markman A, Gentner D (2001) Thinking. Annu Rev Psychol 52:223–247PubMedCrossRefGoogle Scholar
  60. Marr D (1982) Vision. Freeman, New YorkGoogle Scholar
  61. Mashal N, Faust M, Hendler T, Jung-Beeman M (2007) An fMRI investigation of the neural correlates underlying the processing of novel metaphoric expressions. Brain Lang 100:115–126PubMedCrossRefGoogle Scholar
  62. McCrink K, Dehaene S, Dehaene-Lambertz G (2007) Moving along the number line: operational momentum in nonsymbolic arithmetic. Percept Psychophys 69:1324–1333PubMedCrossRefGoogle Scholar
  63. Moscovitch M (2008) The hippocampus as a “stupid”, domain-specific module: implications for theories of recent and remote memory, and of imagination. Can J Exp Psychol 62:62–79PubMedCrossRefGoogle Scholar
  64. Moulton ST, Kosslyn SM (2009) Imagining predictions: mental imagery as mental emulation. Philos Trans R Soc B 364:1273–1280CrossRefGoogle Scholar
  65. Mowat E, Davis B (2010) Interpreting embodied mathematics using network theory: implications for mathematics education. Complicity 7:1–31Google Scholar
  66. Nassi JJ, Callaway EM (2009) Parallel processing strategies of the primate visual system. Nat Rev Neurosci 10:360–372PubMedCrossRefGoogle Scholar
  67. Núñez RE, Lakoff G (2005) The cognitive foundations of mathematics: the role of conceptual metaphor. In: Campbell JID (ed) Handbook of mathematical cognition. Psychology Press, New York, pp 109–124Google Scholar
  68. Piazza M (2010) Neurocognitive start-up tools for symbolic number representations. Trends Cogn Sci 14:542–551PubMedCrossRefGoogle Scholar
  69. Preusse F, van der Meer E, Ullwer D, Brucks M, Krueger F, Wartenburger I (2010) Long-term characteristics of analogical processing in high-school students with high fluid intelligence: an fMRI study. ZDM Math Educ 42:635–647CrossRefGoogle Scholar
  70. Preusse F, van der Meer E, Deshpande G, Krueger F, Wartenburger I (2011) Fluid intelligence allows flexible recruitment of the parieto-frontal network in analogical reasoning. Front Psychol Human Neurosci 5:22. doi: 10.3389/fnhum.2011.00022 Google Scholar
  71. Qin Y, Carter C, Silk E, Stenger VA, Fissell K, Goode A, Anderson JR (2004) The change in brain activation patterns as children learn algebra equation solving. Proc Natl Acad Sci USA 101:5686–5691PubMedCrossRefGoogle Scholar
  72. Radford L (2009) Why do gestures matter? Sensuous cognition and the palpability of mathematical meanings. Educ Stud Math 70:111–126CrossRefGoogle Scholar
  73. Ranganath C (2010) A unified framework for the functional organization of the medial temporal lobes and the phenomenology of episodic memory. Hippocampus 20:1263–1290PubMedCrossRefGoogle Scholar
  74. Richards C (2002) The fundamental design variables of diagramming. In: Anderson M, Meyer B, Olivier P (eds) Diagrammatic representation and reasoning. Springer, Berlin, pp 85–102CrossRefGoogle Scholar
  75. Rosenberg-Lee M, Lovett MC, Anderson JR (2009) Neural correlates of arithmetic calculation strategies. Cogn Affect Behav Neurosci 9:270–285PubMedCrossRefGoogle Scholar
  76. Rosenberg-Lee M, Chang TT, Young CB, Wu S, Menon V (2011) Functional dissociations between four basic arithmetic operations in the human posterior parietal cortex: a cytoarchitectonic mapping study. Neuropsychologia 49:2592–2608PubMedCrossRefGoogle Scholar
  77. Santi A, Grodzinsky Y (2007) Working memory and syntax interact in Broca’s area. NeuroImage 37:8–17PubMedCrossRefGoogle Scholar
  78. Schmidt GL, Kranjec A, Cardillo ER, Chatterjee A (2010) Beyond laterality: a critical assessment of research on the neural basis of metaphor. J Int Neuropsychol Soc 16:1–5PubMedCrossRefGoogle Scholar
  79. Schubotz RI (2007) Prediction of external events with our motor system: towards a new framework. Trends Cogn Sci 11:211–218PubMedCrossRefGoogle Scholar
  80. Schubotz R, von Cramon DY (2004) Sequences of abstract nonbiological stimuli share ventral premotor cortex with action observations and imagery. J Neurosci 24:5467–5474PubMedCrossRefGoogle Scholar
  81. Searle JR (1980) Minds, brains and programs. Behav Brain Sci 3:417–458CrossRefGoogle Scholar
  82. Simons JS, Henson RNA, Gilbert SJ, Fletcher PC (2008) Separable forms of reality monitoring supported by anterior prefrontal cortex. J Cogn Neurosci 20:447–457PubMedCrossRefGoogle Scholar
  83. Stoy JE (1977) Denotational semantics: the Scott-Strachey approach to programming language theory. MIT, CambridgeGoogle Scholar
  84. Tanenbaum AS (1976) Structured computer organization. Prentice Hall, Upper Saddle RiverGoogle Scholar
  85. Turing AM (1937) On computable numbers, with an application to the Entscheidungsproblem. Proc Lond Math Soc 42:230–265CrossRefGoogle Scholar
  86. Volle E, Gilbert SJ, Benoit RG, Burgess PW (2010) Specialization of the rostral prefrontal cortex for distinct analogy processes. Cereb Cortex 20:2647–2659PubMedCrossRefGoogle Scholar
  87. Wartenburger I, Heekeren HR, Preusse F, Kramer J, van der Meer E (2009) Cerebral correlates of analogical processing and their modulation by training. NeuroImage 48:291–302PubMedCrossRefGoogle Scholar
  88. Watson C, Chatterjee A (2012) A bilateral frontoparietal network underlies visuospatial analogical reasoning. NeuroImage 59:2831–2838PubMedCrossRefGoogle Scholar
  89. Wendelken C, Nakhabenko D, Donohue SE, Carter CS, Bunge SA (2008) “Brain is to thought as stomach is to?’’ Investigating the role of rostrolateral prefrontal cortex in relational reasoning. J Cogn Neurosci 20:682–693PubMedCrossRefGoogle Scholar
  90. Wigner EP (1960) The unreasonable effectiveness of mathematics in the natural sciences. Commun Pure Appl Math 13:1–14CrossRefGoogle Scholar
  91. Wood G, Willmes K, Nuerk H-C, Fischer MH (2008) On the cognitive link between space and number: a meta-analysis of the SNARC effect. Psychol Sci Q 50:489–525Google Scholar

Copyright information

© Marta Olivetti Belardinelli and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Santa FeUSA

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