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A fresh look at research strategies in computational cognitive science: The case of enculturated mathematical problem solving

  • Regina E. FabryEmail author
  • Markus Pantsar


Marr’s seminal distinction between computational, algorithmic, and implementational levels of analysis has inspired research in cognitive science for more than 30 years. According to a widely-used paradigm, the modelling of cognitive processes should mainly operate on the computational level and be targeted at the idealised competence, rather than the actual performance of cognisers in a specific domain. In this paper, we explore how this paradigm can be adopted and revised to understand mathematical problem solving. The computational-level approach applies methods from computational complexity theory and focuses on optimal strategies for completing cognitive tasks. However, human cognitive capacities in mathematical problem solving are essentially characterised by processes that are computationally sub-optimal, because they initially add to the computational complexity of the solutions. Yet, these solutions can be optimal for human cognisers given the acquisition and enactment of mathematical practices. Here we present diagrams and the spatial manipulation of symbols as two examples of problem solving strategies that can be computationally sub-optimal but humanly optimal. These aspects need to be taken into account when analysing competence in mathematical problem solving. Empirically informed considerations on enculturation can help identify, explore, and model the cognitive processes involved in problem solving tasks. The enculturation account of mathematical problem solving strongly suggests that computational-level analyses need to be complemented by considerations on the algorithmic and implementational levels. The emerging research strategy can help develop algorithms that model what we call enculturated cognitive optimality in an empirically plausible and ecologically valid way.


Mathematical problem solving Levels of analysis Computational modelling Competence Performance Computational complexity Enculturation Neural plasticity Embodied cognition 



We would like to thank an anonymous reviewer for very helpful comments. M.P. would like to thank the Academy of Finland and professor Gabriel Sandu for making this reseach possible as part of the project “Dependence and Independence in Logic”.


  1. Aaronson, S. (2013). Why philosophers should care about computational complexity. In B. J. Copeland, C. J. Posy, & O. Shagrir (Eds.), Computability: Turing, Gödel, Church, and beyond (pp. 261–328). Cambridge, MA: MIT Press.Google Scholar
  2. Ackerman, N. L., & Freer, C. E. (2013). A notion of a computational step for partial combinatory algebras. In T.-H. H. Chan, L. C. Lau, & L. Trevisan (Eds.), International conference on theory and applications of models of computation (pp. 133–143). Berlin: Springer.CrossRefGoogle Scholar
  3. Amalric, M., & Dehaene, S. (2016). Origins of the brain networks for advanced mathematics in expert mathematicians. Proceedings of the National Academy of Sciences, 113(18), 4909–4917.CrossRefGoogle Scholar
  4. Amalric, M., & Dehaene, S. (2018). Cortical circuits for mathematical knowledge: Evidence for a major subdivision within the brain’s semantic networks. Philosophical Transactions of the Royal Society B, 373(1740), 1–9.Google Scholar
  5. Anderson, J. R. (1990). The adaptive character of thought. Hillsdale, NJ: Lawrence Erlbaum Associates.Google Scholar
  6. Anderson, J. R. (2005). Human symbol manipulation within an integrated cognitive architecture. Cognitive Science, 29(3), 313–341.CrossRefGoogle Scholar
  7. Anderson, M. L. (2010). Neural reuse: A fundamental organizational principle of the brain. Behavioral and Brain Sciences, 33(04), 245–266.CrossRefGoogle Scholar
  8. Anderson, M. L. (2015). After phrenology: Neural reuse and the interactive brain. Cambridge, MA: MIT Press.Google Scholar
  9. Anderson, M. L. (2016). Précis of after phrenology: Neural reuse and the interactive brain. Behavioral and Brain Sciences, 39, 1–45.CrossRefGoogle Scholar
  10. Ansari, D. (2008). Effects of development and enculturation on number representation in the brain. Nature Reviews Neuroscience, 9(4), 278–291.CrossRefGoogle Scholar
  11. Ansari, D. (2016). The neural roots of mathematical expertise. Proceedings of the National Academy of Sciences of the United States of America, 113(18), 4887–4889. Scholar
  12. Arora, S., & Barak, B. (2009). Computational complexity: A modern approach. Cambridge, MA: Cambridge University Press.CrossRefGoogle Scholar
  13. Avigad, J. (2006). Mathematical method and proof. Synthese, 153(1), 105–159.CrossRefGoogle Scholar
  14. Avigad, J. (2008). Understanding proofs. In P. Mancosu (Ed.), The philosophy of mathematical practice (pp. 317–353). Oxford: Oxford University Press.CrossRefGoogle Scholar
  15. Baroody, A. J. (1984). A reexamination of mental arithmetic models and data: A reply to Ashcraft. Developmental Review, 4(2), 148–156.CrossRefGoogle Scholar
  16. Barwise, J., & Etchemendy, J. (1996). Visual information and valid reasoning. In G. Allwein & J. Barwise (Eds.), Logical reasoning with diagrams (pp. 3–25). Oxford: Oxford University Press.Google Scholar
  17. Bechtel, W., & Shagrir, O. (2015). The non-redundant contributions of Marr’s three levels of analysis for explaining information-processing mechanisms. Topics in Cognitive Science, 7(2), 312–322.CrossRefGoogle Scholar
  18. Blokpoel, M. (2017). Sculpting computational-level models. Topics in Cognitive Science, 10, 1–8.Google Scholar
  19. Blum, M. (1967). A machine-independent theory of the complexity of recursive functions. Journal of the ACM, 14(2), 322–336.CrossRefGoogle Scholar
  20. Boyer, C. B. (1985). A history of mathematics. Princeton: Princeton University Press.Google Scholar
  21. Brown, J. R. (2008). Philosophy of mathematics: A contemporary introduction to the world of proofs and pictures (2nd ed.). New York: Routledge.Google Scholar
  22. Brown, J. S., & VanLehn, K. (1980). Repair theory: A generative theory of bugs in procedural skills. Cognitive Science, 4(4), 379–426.CrossRefGoogle Scholar
  23. Carey, S. (2009). The origin of concepts. Oxford: Oxford University Press.CrossRefGoogle Scholar
  24. Carroll, W. M., & Porter, D. (1998). Alternative algorithms for whole-number operations. In M. J. Kenney & L. J. Morrow (Eds.), The teaching and learning of algorithms in school mathematics (pp. 106–114). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  25. Carter, J. (2010). Diagrams and proofs in analysis. International Studies in the Philosophy of Science, 24(1), 1–14.CrossRefGoogle Scholar
  26. Carter, J. (2017). Exploring the fruitfulness of diagrams in mathematics. Synthese. Scholar
  27. Castelli, F., Glaser, D. E., & Butterworth, B. (2006). Discrete and analogue quantity processing in the parietal lobe: A functional MRI study. Proceedings of the National Academy of Sciences of the United States of America, 103(12), 4693–4698.CrossRefGoogle Scholar
  28. Chabert, J.-L. (Ed.). (1999). A history of algorithms: From the pebble to the microchip. Heidelberg: Springer.Google Scholar
  29. Chomsky, N. (2006). Language and mind. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  30. Chomsky, N. (2015). Aspects of the theory of syntax (50th anniv). Cambridge, MA: MIT Press.Google Scholar
  31. Clark, A. (1990). Connectionism, competence, and explanation. The British Journal for the Philosophy of Science, 41(2), 195–222.CrossRefGoogle Scholar
  32. Clark, A. (1997). Being there: Putting brain, body, and world together again. Cambridge, MA: MIT Press.Google Scholar
  33. Clark, A. (2016). Surfing uncertainty: Prediction, action, and the embodied mind. Oxford, NY: Oxford University Press.CrossRefGoogle Scholar
  34. Cobb, P. (1994). Where is the mind? Constructivist and sociocultural perspectives on mathematical development. Educational Researcher, 23(7), 13–20.CrossRefGoogle Scholar
  35. Cobham, A. (1965). The intrinsic computational difficulty of functions. In Y. Bar-Hillel (Ed.), Proceedings of the 1964 congress on logic, mathematics and the methodology of science (pp. 24–30). Amsterdam: North-Holland.Google Scholar
  36. Cooper, R. P., & Peebles, D. (2017). On the relation between Marr’s levels: A response to Blokpoel. Topics in Cognitive Science, 10, 1–5.Google Scholar
  37. De Houwer, J., Vandorpe, S., & Beckers, T. (2005). Evidence for the role of higher order reasoning processes in cue competition and other learning phenomena. Learning & Behavior, 33(2), 239–249.CrossRefGoogle Scholar
  38. Dehaene, S. (2005). Evolution of human cortical circuits for reading and arithmetic: The “neuronal recycling” hypothesis. In S. Dehaene, J.-R. Duhamel, M. D. Hauser, & G. Rizzolatti (Eds.), From monkey brain to human brain: A Fyssen Foundation symposium (pp. 133–157). Cambridge, MA: MIT Press.CrossRefGoogle Scholar
  39. Dehaene, S. (2010). Reading in the brain: The new science of how we read. New York: Penguin Books.Google Scholar
  40. Dehaene, S. (2011). The number sense: How the mind creates mathematics (2nd ed.). Oxford: Oxford University Press.Google Scholar
  41. Dehaene, S., Cohen, L., Sigman, M., & Vinckier, F. (2005). The neural code for written words: A proposal. Trends in Cognitive Sciences, 9(7), 335–341. Scholar
  42. Dewey, J. (1896). The reflex arc concept in psychology. Psychological Review, 3(4), 357–370.CrossRefGoogle Scholar
  43. Donald, M. (1991). Origins of the modern mind: Three stages in the evolution of culture and cognition. Cambridge, MA: Harvard University Press.Google Scholar
  44. Duncker, K. (1945). On problem-solving. Psychological Monographs, 58(5), 1–113.CrossRefGoogle Scholar
  45. Eberhardt, F., & Danks, D. (2011). Confirmation in the cognitive sciences: The problematic case of Bayesian models. Minds and Machines, 21(3), 389–410.CrossRefGoogle Scholar
  46. Edmonds, J. (1965). Paths, trees, and flowers. Canadian Journal of Mathematics, 17(3), 449–467.CrossRefGoogle Scholar
  47. Estany, A., & Martínez, S. (2014). “Scaffolding” and “affordance” as integrative concepts in the cognitive sciences. Philosophical Psychology, 27, 98–111. Scholar
  48. Everett, C. (2017). Numbers and the making of us: Counting and the course of human cultures. Cambridge, MA: Harvard University Press.CrossRefGoogle Scholar
  49. Fabry, R. E. (2017). Predictive processing and cognitive development. In T. K. Metzinger & W. Wiese (Eds.), Philosophy and predictive processing (pp. 1–18). Frankfurt am Main: MIND Group. Scholar
  50. Fabry, R. E. (2018). Betwixt and between: The enculturated predictive processing approach to cognition. Synthese, 195(6), 2483–2518. Scholar
  51. Fabry, R. E. (2019). The cerebral, extra-cerebral bodily, and socio-cultural dimensions of enculturated arithmetical cognition. Synthese. Scholar
  52. Findlay, J. M., & Gilchrist, I. D. (2003). Active vision. Oxford: Oxford University Press.CrossRefGoogle Scholar
  53. Fitch, W. T., Hauser, M. D., & Chomsky, N. (2005). The evolution of the language faculty: Clarifications and implications. Cognition, 97(2), 179–210.CrossRefGoogle Scholar
  54. Frank, M. C., Everett, D. L., Fedorenko, E., & Gibson, E. (2008). Number as a cognitive technology: Evidence from Pirahã language and cognition. Cognition, 108(3), 819–824.CrossRefGoogle Scholar
  55. Friston, K. (2005). A theory of cortical responses. Philosophical Transactions of the Royal Society B: Biological Sciences, 360(1456), 815–836. Scholar
  56. Friston, K. (2010). The free-energy principle: A unified brain theory? Nature Reviews Neuroscience, 11(2), 127–138. Scholar
  57. Frixione, M. (2001). Tractable competence. Minds and Machines, 11(3), 379–397.CrossRefGoogle Scholar
  58. Fuson, K. C. (2003). Developing mathematical power in whole number operations. In J. Kilpatrick, G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (Vol. 1, pp. 68–94). Reston, VA: National Council of Teachers of Mathematics.Google Scholar
  59. Galotti, K. M. (1989). Approaches to studying formal and everyday reasoning. Psychological Bulletin, 105(3), 331–351.CrossRefGoogle Scholar
  60. Giaquinto, M. (2007). Visual thinking in mathematics. Oxford: Oxford University Press.CrossRefGoogle Scholar
  61. Giaquinto, M. (2008). Visualizing in mathematics. In P. Mancosu (Ed.), The philosophy of mathematical practice (pp. 22–42). Oxford: Oxford University Press.CrossRefGoogle Scholar
  62. Giaquinto, M. (2015). The epistemology of visual thinking in mathematics. In E. N. Zalta (Ed.), The Stanford encyclopedia of philosophy. Retrieved from Accessed 02 June 2019.
  63. Giardino, V. (2010). Intuition and visualization in mathematical problem solving. Topoi, 29(1), 29–39.CrossRefGoogle Scholar
  64. Giardino, V. (2014). Diagramming: Connecting cognitive systems to improve reasoning. In A. Benedek & K. Nyíri (Eds.), The power of the image: Emotion, expression, explanation (pp. 23–34). Frankfurt am Main: Peter Lang.Google Scholar
  65. Giardino, V. (2016). Behind the diagrams: Cognitive issues and open problems. In S. Krämer & C. Ljungberg (Eds.), Thinking with diagrams: The semiotic basis of human cognition (pp. 77–101). Berlin: Walter De Gruyter.Google Scholar
  66. Giardino, V. (2017). Diagrammatic reasoning in mathematics. In L. Magnani & T. Bertolotti (Eds.), Springer handbook of model-based science (pp. 499–522). Heidelberg: Springer.CrossRefGoogle Scholar
  67. Goldstone, R. L., Landy, D. H., & Son, J. Y. (2010). The education of perception. Topics in Cognitive Science, 2(2), 265–284.CrossRefGoogle Scholar
  68. Goldstone, R. L., Marghetis, T., Weitnauer, E., Ottmar, E. R., & Landy, D. (2017). Adapting perception, action, and technology for mathematical reasoning. Current Directions in Psychological Science, 26(5), 434–441.CrossRefGoogle Scholar
  69. Gordon, P. (2004). Numerical cognition without words: Evidence from Amazonia. Science, 306(5695), 496–499.CrossRefGoogle Scholar
  70. Grant, E. R., & Spivey, M. J. (2003). Eye movements and problem solving: Guiding attention guides thought. Psychological Science, 14(5), 462–466.CrossRefGoogle Scholar
  71. Griffiths, T. L., Chater, N., Kemp, C., Perfors, A., & Tenenbaum, J. B. (2010). Probabilistic models of cognition: Exploring representations and inductive biases. Trends in Cognitive Sciences, 14(8), 357–364.CrossRefGoogle Scholar
  72. Gurganus, S. P. (2007). Math instruction for students with learning problems. Boston: Pearson/Allyn and Bacon.Google Scholar
  73. Hannagan, T., Amedi, A., Cohen, L., Dehaene-Lambertz, G., & Dehaene, S. (2015). Origins of the specialization for letters and numbers in ventral occipitotemporal cortex. Trends in Cognitive Sciences, 19(7), 374–382.CrossRefGoogle Scholar
  74. Harvey, D., Van Der Hoeven, J., & Lecerf, G. (2016). Even faster integer multiplication. Journal of Complexity, 36, 1–30.CrossRefGoogle Scholar
  75. Hayes, B. (2006). Gauss’s day of reckoning. American Scientist, 94(3), 200–205.CrossRefGoogle Scholar
  76. Hayes, B. (2017). Foolproof and other mathematical meditations. Cambridge, MA: MIT Press.CrossRefGoogle Scholar
  77. Henrich, J. P. (2016). The secret of our success: How culture is driving human evolution, domesticating our species, and making us smarter. Princeton: Princeton University Press.CrossRefGoogle Scholar
  78. Heyes, C. (2012). Grist and mills: On the cultural origins of cultural learning. Philosophical Transactions of the Royal Society B: Biological Sciences, 367(1599), 2181–2191. Scholar
  79. Heyes, C. (2016). Born pupils? Natural pedagogy and cultural pedagogy. Perspectives on Psychological Science, 11(2), 280–295.CrossRefGoogle Scholar
  80. Hohwy, J. (2013). The predictive mind. Oxford: Oxford University Press.CrossRefGoogle Scholar
  81. Hubbard, E. M., Piazza, M., Pinel, P., & Dehaene, S. (2005). Interactions between number and space in parietal cortex. Nature Reviews Neuroscience, 6(6), 435–448.CrossRefGoogle Scholar
  82. Huntly, I., Kaiser, G., & Luna, E. (Eds.). (2012). International comparisons in mathematics education. London: Routledge.Google Scholar
  83. Inglis, M., & Alcock, L. (2012). Expert and novice approaches to reading mathematical proofs. Journal for Research in Mathematics Education, 43(4), 358–390.CrossRefGoogle Scholar
  84. Isaac, A. M. C., Szymanik, J., & Verbrugge, R. (2014). Logic and complexity in cognitive science. In A. Baltag & S. Smets (Eds.), Johan van Benthem on logic and information dynamics (pp. 787–824). Heidelberg: Springer.CrossRefGoogle Scholar
  85. Karatsuba, A., & Ofman, Y. (1962). Multiplication of many-digital numbers by automatic computers. In Doklady Akademii Nauk (Vol. 145, pp. 293–294).Google Scholar
  86. Koedinger, K. R., & Anderson, J. R. (1990). Abstract planning and perceptual chunks: Elements of expertise in geometry. Cognitive Science, 14(4), 511–550.CrossRefGoogle Scholar
  87. Kozen, D. C. (2012). Automata and computability. New York: Springer.Google Scholar
  88. Krämer, S. (2014a). Mathematizing power, formalization, and the diagrammatical mind or: What does “computation” mean? Philosophy & Technology, 27(3), 345–357.CrossRefGoogle Scholar
  89. Krämer, S. (2014b). Trace, writing, diagram: Reflections on spatiality, intuition, graphical practices and thinking. In A. Benedek & K. Nyíri (Eds.), The power of the image: Emotion, expression, explanation (pp. 3–22). Frankfurt am Main: Peter Lang.Google Scholar
  90. Krämer, S. (2016). Figuration, Anschauung, Erkenntnis: Grundlinien einer Diagrammatologie. Berlin: Suhrkamp Verlag.Google Scholar
  91. Lakoff, G., & Núñez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books.Google Scholar
  92. Landy, D., & Goldstone, R. L. (2007a). Formal notations are diagrams: Evidence from a production task. Memory & Cognition, 35(8), 2033–2040.CrossRefGoogle Scholar
  93. Landy, D., & Goldstone, R. L. (2007b). How abstract is symbolic thought? Journal of Experimental Psychology. Learning, Memory, and Cognition, 33(4), 720–733.CrossRefGoogle Scholar
  94. Landy, D., & Goldstone, R. L. (2010). Proximity and precedence in arithmetic. The Quarterly Journal of Experimental Psychology, 63(10), 1953–1968.CrossRefGoogle Scholar
  95. Larvor, B. (Ed.). (2016). Mathematical cultures: The London meetings 2012-2014. Basel: Birkhäuser.Google Scholar
  96. Leighton, J., & Sternberg, R. (2012). Reasoning and problem solving. In A. Weiner, F. Healy, & R. W. Proctor (Eds.), Handbook of psychology (2nd ed., pp. 623–648). New York: Wiley. Scholar
  97. Love, B. C. (2015). The algorithmic level is the bridge between computation and brain. Topics in Cognitive Science, 7(2), 230–242.CrossRefGoogle Scholar
  98. Lyons, I. M., Ansari, D., & Beilock, S. L. (2015). Qualitatively different coding of symbolic and nonsymbolic numbers in the human brain. Human Brain Mapping, 36(2), 475–488.CrossRefGoogle Scholar
  99. Mancosu, P. (Ed.). (2008). The philosophy of mathematical practice. Oxford: Oxford University Press.Google Scholar
  100. Marr, D. (1977). Artificial intelligence—A personal view. Artificial Intelligence, 9(1), 37–48.CrossRefGoogle Scholar
  101. Marr, D. (1982). Vision: A computational investigation into the human representation and processing of visual information. San Francisco: W.H. Freeman and Company.Google Scholar
  102. Maruyama, M., Pallier, C., Jobert, A., Sigman, M., & Dehaene, S. (2012). The cortical representation of simple mathematical expressions. Neuroimage, 61(4), 1444–1460.CrossRefGoogle Scholar
  103. McCandliss, B. D., Cohen, L., & Dehaene, S. (2003). The visual word form area: Expertise for reading in the fusiform gyrus. Trends in Cognitive Sciences, 7(7), 293–299. Scholar
  104. Menary, R. (2007). Cognitive integration: Mind and cognition unbounded. Basingstoke: Palgrave Macmillan.CrossRefGoogle Scholar
  105. Menary, R. (2010). Dimensions of mind. Phenomenology and the Cognitive Sciences, 9(4), 561–578. Scholar
  106. Menary, R. (2013a). Cognitive integration, enculturated cognition and the socially extended mind. Cognitive Systems Research, 25–26, 26–34. Scholar
  107. Menary, R. (2013b). The enculturated hand. In Z. Radman (Ed.), The hand, an organ of the mind: What the manual tells the mental (pp. 349–367). Cambridge, MA: MIT Press.Google Scholar
  108. Menary, R. (2014). Neural plasticity, neuronal recycling and niche construction. Mind and Language, 29(3), 286–303. Scholar
  109. Menary, R. (2015). Mathematical cognition: A case of enculturation. In T. Metzinger & J. M. Windt (Eds.), Open MIND (pp. 1–20). Frankfurt am Main: MIND Group. Scholar
  110. Merkley, R., & Ansari, D. (2016). Why numerical symbols count in the development of mathematical skills: Evidence from brain and behavior. Current Opinion in Behavioral Sciences, 10, 14–20.CrossRefGoogle Scholar
  111. Miller, K. F., Smith, C. M., Zhu, J., & Zhang, H. (1995). Preschool origins of cross-national differences in mathematical competence: The role of number-naming systems. Psychological Science, 6(1), 56–60.CrossRefGoogle Scholar
  112. Monti, M. M., Parsons, L. M., & Osherson, D. N. (2012). Thought beyond language: Neural dissociation of algebra and natural language. Psychological Science, 23(8), 914–922.CrossRefGoogle Scholar
  113. Newell, A. (1980). Physical symbol systems. Cognitive Science, 4(2), 135–183.CrossRefGoogle Scholar
  114. Newell, A. (1982). The knowledge level. Artificial Intelligence, 18(1), 87–127.CrossRefGoogle Scholar
  115. Newell, A., & Simon, H. A. (1976). Computer science as empirical inquiry (The 1975 ACM turing award lecture). Communications of the ACM, 19(3), 113–126.CrossRefGoogle Scholar
  116. Nuerk, H., Moeller, K., & Willmes, K. (2015). Multi-digit number processing: Overview, conceptual clarifications, and language influences. In R. C. Kadosh & A. Dowker (Eds.), The Oxford handbook of numerical cognition (pp. 106–139). Oxford: Oxford University Press.Google Scholar
  117. Olson, D. R. (1994). The world on paper: The conceptual and cognitive implications of writing and reading. Cambridge, Mass.: Cambridge University Press.Google Scholar
  118. Ong, W. J. (2012). Orality and literacy: The technologizing of the word. London: Routledge.Google Scholar
  119. Pantsar, M. (2014). An empirically feasible approach to the epistemology of arithmetic. Synthese, 191(17), 4201–4229.CrossRefGoogle Scholar
  120. Pantsar, M. (2015). In search of aleph-null: How infinity can be created. Synthese, 192(8), 2489–2511.CrossRefGoogle Scholar
  121. Pantsar, M. (2016). The modal status of contextually a priori arithmetical truths. In F. Boccuni & A. Sereni (Eds.), Objectivity, realism, and proof (pp. 67–79). Cham: Springer.CrossRefGoogle Scholar
  122. Pantsar, M. (2018). Early numerical cognition and mathematical processes. Theoria, 33(2), 285–304.CrossRefGoogle Scholar
  123. Pantsar, M. (under review). Cognitive complexity and mathematical problem solving. Erkenntnis.Google Scholar
  124. Pantsar, M. (in press). The enculturated move from proto-arithmetic to arithmetic. Frontiers in Psychology.Google Scholar
  125. Papadimitriou, C. H. (2003). Computational complexity. In A. Ralston, E. D. Reilly, & D. Hemmendinger (Eds.), Encyclopedia of computer science (4th ed., pp. 260–265). Chichester: Wiley.Google Scholar
  126. Peirce, C. S. (1960–1966). Collected papers of Charles Sanders Peirce. In C. Hartshorne, P. Weiss, & A. W. Burks, (Eds.), (Vol. 1–8). Cambridge, MA: Belknap Press of Harvard University Press.Google Scholar
  127. Piantadosi, S. T., Tenenbaum, J. B., & Goodman, N. D. (2016). The logical primitives of thought: Empirical foundations for compositional cognitive models. Psychological Review, 123(4), 392.CrossRefGoogle Scholar
  128. Price, C. J., & Devlin, J. T. (2003). The myth of the visual word form area. NeuroImage, 19(3), 473–481. Scholar
  129. Price, C. J., & Devlin, J. T. (2004). The pro and cons of labelling a left occipitotemporal region “the visual word form area”. NeuroImage, 22(1), 477–479.CrossRefGoogle Scholar
  130. Ramsey, W. (2017). Must cognition be representational? Synthese, 194, 4197–4214. Scholar
  131. Randolph, T. D., & Sherman, H. J. (2001). Alternative algorithms: Increasing options, reducing errors. Teaching Children Mathematics, 7(8), 480–484.Google Scholar
  132. Relaford-Doyle, J., Núñez, R., Howes, A., & Tenbrink, T. (2017). When does a ‘visual proof by induction’serve a proof-like function in mathematics? In E. Davelaar & G. Gunzelmann (Eds.), Proceedings of the 39th annual conference of the cognitive science society (pp. 1004–1009). Austin, TX: Cognitive Science Society.Google Scholar
  133. Rowlands, M. (1999). The body in mind: Understanding cognitive processes. Cambridge, MA: Cambridge University Press.CrossRefGoogle Scholar
  134. Rumelhart, D. E., Smolensky, P., McClelland, J. L., & Hinton, G. (1986). Sequential thought processes in PDP models. In J. L. McClelland & D. E. Rumelhart (Eds.), Parallel distributed processing: Explorations in the microstructures of cognition (Vol. 2, pp. 3–57). Cambridge, MA: MIT Press.Google Scholar
  135. Schneider, E., Maruyama, M., Dehaene, S., & Sigman, M. (2012). Eye gaze reveals a fast, parallel extraction of the syntax of arithmetic formulas. Cognition, 125(3), 475–490.CrossRefGoogle Scholar
  136. Schönhage, A., & Strassen, V. (1971). Schnelle Multiplikation großer Zahlen. Computing, 7(3–4), 281–292.CrossRefGoogle Scholar
  137. Shaki, S., Fischer, M. H., & Petrusic, W. M. (2009). Reading habits for both words and numbers contribute to the SNARC effect. Psychonomic Bulletin & Review, 16(2), 328–331.CrossRefGoogle Scholar
  138. Shanks, D. R. (2007). Associationism and cognition: Human contingency learning at 25. The Quarterly Journal of Experimental Psychology, 60(3), 291–309.CrossRefGoogle Scholar
  139. Shanks, D. R. (2010). Learning: From association to cognition. Annual Review of Psychology, 61, 273–301.CrossRefGoogle Scholar
  140. Shin, S.-J. (2012). The forgotten individual: Diagrammatic reasoning in mathematics. Synthese, 186(1), 149–168.CrossRefGoogle Scholar
  141. Simon, H. A. (1996). The sciences of the artificial (2nd ed.). Cambridge, MA: MIT Press.Google Scholar
  142. Spelke, E. S. (2000). Core knowledge. American Psychologist, 55(11), 1233–1243.CrossRefGoogle Scholar
  143. Sterelny, K. (2012). The evolved apprentice: How evolution made humans unique. Cambridge, MA: MIT Press.CrossRefGoogle Scholar
  144. Stjernfelt, F., & Østergaard, S. (2016). Diagrammatic problem solving. In S. Krämer & C. Ljungberg (Eds.), Thinking with diagrams: The semiotic basis of human cognition (pp. 103–119). Berlin: Walter De Gruyter.Google Scholar
  145. Szymanik, J. (2016). Quantifiers and cognition: Logical and computational perspectives. Heidelberg: Springer.CrossRefGoogle Scholar
  146. Tamburrini, G. (1997). Mechanistic theories in cognitive science: The import of Turing’s Thesis. In M. L. Dalla Chiara, K. Doets, D. Mundici, & J. van Benthem (Eds.), Logic and scientific methods: The tenth international congress of logic, methodology and philosophy of science, Florence, August 1995 (pp. 239–257). Dordrecht: Springer.Google Scholar
  147. Tang, Y., Zhang, W., Chen, K., Feng, S., Ji, Y., Shen, J., et al. (2006). Arithmetic processing in the brain shaped by cultures. Proceedings of the National Academy of Sciences, 103(28), 10775–10780.CrossRefGoogle Scholar
  148. Tappenden, J. (2005). Proof style and understanding in mathematics I: Visualization, unification and axiom choice. In P. Mancosu, K. F. Jørgensen, & S. A. Pedersen (Eds.), Visualization, explanation and reasoning styles in mathematics (pp. 147–214). Dordrecht: Springer.CrossRefGoogle Scholar
  149. Tennant, N. (1986). The withering away of formal semantics? Mind and Language, 1(4), 302–318.CrossRefGoogle Scholar
  150. Tschentscher, N., Hauk, O., Fischer, M. H., & Pulvermüller, F. (2012). You can count on the motor cortex: Finger counting habits modulate motor cortex activation evoked by numbers. Neuroimage, 59(4), 3139–3148.CrossRefGoogle Scholar
  151. Turing, A. M. (1936). On computable numbers, with an application to the Entscheidungsproblem. Proceedings of the London Mathematical Society, 42, 230–265.Google Scholar
  152. Tylén, K., Fusaroli, R., Bjørndahl, J. S., Raczaszek-Leonardi, J., ∅stergaard, S., & Stjernfelt, F. (2014). Diagrammatic reasoning: abstraction, interaction, and insight. Pragmatics & Cognition, 22(2), 264–283.CrossRefGoogle Scholar
  153. Van Rooij, I. (2008). The tractable cognition thesis. Cognitive Science, 32(6), 939–984.CrossRefGoogle Scholar
  154. Varma, S. (2014). The subjective meaning of cognitive architecture: A Marrian analysis. Frontiers in Psychology, 5, 1–9. Scholar
  155. Vogel, A. C., Petersen, S. E., & Schlaggar, B. L. (2012). The left occipitotemporal cortex does not show preferential activity for words. Cerebral Cortex, 22(12), 2715–2732.CrossRefGoogle Scholar
  156. Vogel, A. C., Petersen, S. E., & Schlaggar, B. L. (2014). The VWFA: It’s not just for words anymore. Frontiers in Human Neuroscience. Scholar
  157. Walsh, M. M., & Lovett, M. C. (2016). The cognitive science approach to learning and memory. In S. E. F. Chipman (Ed.), The Oxford handbook of cognitive science (pp. 211–230). Oxford: Oxford University Press.Google Scholar
  158. Wood, D., Bruner, J. S., & Ross, G. (1976). The role of tutoring in problem solving. Journal of Child Psychology and Psychiatry, 17(2), 89–100.CrossRefGoogle Scholar
  159. Zednik, C., & Jäkel, F. (2016). Bayesian reverse-engineering considered as a research strategy for cognitive science. Synthese, 193(12), 3951–3985.CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Philosophy IIRuhr University BochumBochumGermany
  2. 2.Department of Philosophy, History, and Art StudiesUniversity of HelsinkiHelsinkiFinland

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