Abstract
One of the most important questions in the young field of numerical cognition studies is how humans bridge the gap between the quantity-related content produced by our evolutionarily ancient brains and the precise numerical content associated with numeration systems like Indo-Arabic numerals. This gap problem is the main focus of this paper. The aim here is to evaluate the extent to which cultural factors can help explain how we come to think about numbers beyond the subitizing range. To do this, I summarize Clark’s (Analysis 58:7–19, 1998) notion of a difference maker in explaining complex causation that criss-crosses between mind and world and apply it to Menary’s (In: Metzinger T, Windt JM (eds) Open MIND. MIND Group, Frankfurt, 2015a) discussion of mathematical cognition as a case of enculturation. I argue that while Menary’s views on enculturation can help explain what makes the difference between numerate and anumerate cultures, it cannot help specify what makes the difference between numerate and anumerate individuals. I argue that features of enculturation do not provide an account of innovation capable of explaining how individuals manage to improve and modify the practices of their cultural niche. This is because Menary’s construal of the role of enculturation in the development of mathematical cognition focuses mostly on the inheritance and transmission of practices, not on their origins, which involve individual-level understanding, rather than population-level practices and pressures. The upshot is that culture provides the necessary background conditions against which individuals can innovate. This role is crucial in the development of numerical abilities—crucial, but explanatorily limited.
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Notes
As Núñez (2017) has argued, the literature on numerical cognition is polluted by loose and inaccurate use of important terms, especially all things ‘numerical’. In an attempt to limit my contribution to this unfortunate trend, throughout this text, I will strive to use the general expression ‘numerical cognition’ to describe only the type of cognition observed in the practice of formal arithmetic, as done by people who have mastered numeration systems like the Indo-Arabic numerals and the physical manipulation of abacuses. To reflect different stages of mastery of numeration systems—or lack thereof—I will often qualify the term ‘numerical’ to emphasize to which extent an individual can be described as being proficient in the practice of formal arithmetic. For example, ‘formal numerical cognition’ and ‘formal numerical practices’ will be used to highlight mastery of the practice of formal arithmetic using numeration systems like the Indo-Arabic numerals, while ‘rudimentary’ numerical cognition will describe individuals who are in possession of the building blocks of formal numerical cognition, including the ANS and the OFS, but who are not capable of distinguishing between collections of arbitrary sizes based on the number of items they contain. In this sense, animals and preverbal infants can be considered as having the rudiments of numerical cognition, or as being able to perform rudimentary numerical cognition, since they possess cognitive systems that react to variations of numerical information in their environment. However, they do not possess what I will often call ‘proto-numerical abilities’ nor formal numerical abilities.
This paper concerns mainly what could be considered an intermediate stage in the development of numerical cognition, which I will refer to as ‘proto-numerical’. This stage of the development of numerical cognition describes individuals that have the ability to accurately count beyond the subitizing range (i.e. beyond four objects), but whose ability to label the numerosity of collections does not include mastery of the generative syntax of a formal numeration system capable of labelling the numerosity of reasonably large collections (see Schlimm 2018 for why it would be inaccurate to speak of ‘arbitrarily’ large collections here). Thus for example, neither the Mundurucu (Pica et al. 2004) nor the Pirahã (Gordon 2004; Frank et al. 2008) would be considered as having proto-numerical skills here, since it is debatable whether the Pirahã have labels for numbers at all (Frank et al. 2008) while the Mundurucu lexicon only contains words for the first five numbers, and these are not used consistently (Pica et al. 2004). Individuals from cultures like the Oksapmin, whose body-part-based numeration system stops at 27 (Saxe 1981), could be considered as having proto-numerical abilities, as do children who have mastered the ability to use number words to accurately label the number of objects in collections containing more than four objects. These distinctions need not be considered as being set in stone, but can hopefully facilitate the discussion, as should become clear in the main text.
As for motivation behind distinguishing as I have here between ‘rudimentary’, ‘proto-’, and ‘formal’ numerical cognition, the idea is not to draw precise lines at specific numbers, but rather to reflect different stages of understanding of what numbers are. One important aspect of numbers is that they are representations of discrete quantities. While it may be argued that understanding that numbers can be extended indefinitely is a key aspect of numerical abilities, I tend to favor the opposite view, in which what matters in attributing numerical abilities is the extent to which an individual can distinguish collections based on the number of discrete objects they contain, not their understanding of the infinite nature of the natural numbers. The realization that numbers can go on without end seems like a separate stage in the development of numerical cognition (Cheung et al. 2017). As I see it, the transition from rudimentary to proto-numerical cognition is the most important developmental step, since it is this step that allows individuals to determine the precise number of items in collections beyond the subitizing range, even if this ability does not extend indefinitely, as it might for people who have mastered formal numeration systems like the Indo-Arabic numerals. This thought seems behind Crossley’s comment that “once the idea of counting has emerged, then the idea to go on counting does not seem to lie far below the surface” (Crossley 1987, p. 13). The bottom line is that if individuals can count, calculate, and trade using their symbols for discrete quantities, they should be considered as having some form of understanding of what numbers are—or at least, more so than individuals who cannot count or calculate at all—even if they can’t easily tell the difference between 1000 and 1001, for example.
Here, by saying external artefacts are widely seen as ‘constitutive’ parts of the process that allows us to bridge the gap, I am referring to the view according to which it is impossible to bridge the gap without these because they are part of what numerical cognition is. For more on what this constitutivity might mean, see Dutilh Novaes (2013) and Schlimm (2018). For an opposing view, see Pelland (2018).
Clark describes the notion of continuous reciprocal causation as applying to cases where “some system S is both continuously affecting and simultaneously being affected by activity in some other system O…we often find processes of CRC that criss-cross brain, body, and local environment” (Clark 2008, p. 24).
See Clark and Chalmers (1998) for examples of such twisted tales.
Pöyhönen (2014) also focuses on the importance of differential influence in relation to explanatory relevance and emphasizes that good explanations of extended systems are supposed to draw a line between explanatorily relevant factors and background conditions.
For example, in an environment-gene system, the gene is the difference maker: “a gene may be ‘for x’ in the simple sense that it is a feature whose presence or absence is a difference that makes a systematic (usually population-level) difference to the presence (or absence) of x” (Clark and Chalmers 1998, pp. 155, 156). This reasoning applies to programs and neurons as well: “The extension of the line on explanatory priority to the case of neural codes and programs is immediate. Here too we should say that a neural structure or process x codes for a behavioral outcome y, if against a normal ecological backdrop, it makes the difference with respect to the obtaining of y” (Clark and Chalmers 1998, p. 160).
I will use this term to describe a process by which features of an individual’s behavior are the result of social learning, as opposed to learning ‘from scratch’ (Charbonneau 2015) by interaction with the environment. Enculturation includes what Menary describes as “redeploying older neural circuits to new, culturally specific functions” (2015a, p. 2).
A label used to refer to approaches to cognition that share a rejection of most aspects of cognitivism and its emphasis on the inside of our head to explain how cognition works. The 4E here stands for Embodied, Embedded, Extended, and Enactive cognition. See Menary (2010b).
The notion of niche here is taken from a growing literature that assigns an important role to the individual organism’s ability to modify its environment in its evolution, to the extent that it modifies selective pressures on that organism. Niche construction enters organisms into feedback cycles with their environment that alter the adaptive landscape for their (and other) species. Classical examples of niches are spider webs and beaver dams, and the surprisingly-often-discussed houses of caddys fly larvae. The idea of niche construction goes back to Lewontin (1983), who Laland et al. (2014) describe as the father figure of niche construction. For humans, an especially important case of niche construction is culture: the fact that humans are able to learn from their environment and transmit what they have learned and created—including artefacts that outlast the individuals that have crafted them—means that there are selective pressures stemming from the cultural niche constructed by our ancestors. Those who adapt to their cultural surroundings will have an advantage over those who do not. A well-known example of how cultural practices impose specific selective pressures is that of dairy farming, which resulted in increased lactose tolerance in cultures where this practice was common. See Laland et al. (2014), Wheeler and Clark (2008) and Day et al. (2003).
Similarly, Ifrah talks of “that mind-boggling moment when someone first came up with the idea of counting” (Ifrah 1998, p. x).
E.g., the Mundurucu, mentioned above in note 1. See also Hurford (1987).
Once children have learned to recite number words by heart, there is a prolonged learning period where the meaning of the first few number words is acquired piecemeal, in stages, for ‘one’, ‘two’, and ‘three’. In this process, each stage lasts a few months. However, once they learn the meaning of ‘four’, children also grasp the meaning of the remaining words in their counting routine. This sudden realization is sometimes called the Induction (Rips et al. 2008a).
This discussion relates to Boden’s (2004) distinction between P-creativity and H-creativity, where the former describes an idea that is new to the individual, while H-creativity refers to ideas that are new to humanity as a whole. As Boden points out, H-creativity is simply a special case of P-creativity. This leads us to the question of whether two people having the same idea must share all or some of the cognitive resources required to entertain that idea. While this is a fascinating question, it need not be settled here, since the point I am making does not require the Induction that children undergo to bridge the gap be the same process that allowed the initial (H-creative) development of proto-numerical cognition. Rather, what is important for me here is that the H-creative development of proto-numerical cognition cannot have involved the numeral-and-expert-enriched cognitive niche in which the P-creative Induction takes place, since there were no numerals or experts to speak of in the H-creative case. This being said, it is worth mentioning that the presence of a discontinuity around the subitizing range in both the historical development of numeration systems and the ontogenetic development of proto-numerical cognition suggests that it is possible that the same process is responsible for bridging the gap in both the H-creative and P-creative case.
Gelman & Butterworth speculate that anumerate cultures do not develop words for precise quantities because “numbers are not culturally important and receive little attention in everyday life” (Gelman and Butterworth 2005, p. 9) in such cultures.
I exploit the relative humility of the footnote to speculate that the disappearance of the Yuki octonary system as a result of the serial founder effect could be an instance of lack of pressures leading to disappearance of a numerical practice: the fact that Yuki culture spread out meant that changes in their lifestyles could go against seeing precise quantification as an important practice. This, combined with the decrease in numbers of experts capable of sharing their knowledge of this practice, could have led to the disappearance of this practice. See Overmann (2015) and Foster (1944).
Similarly, De Cruz, referring to Sperber (1985), writes: “Cultural transmission, like all types of communication, is under-determined: much information is left unspecified and therefore requires a reconstruction in the mind of the recipient…each instance of cultural transmission requires a reconstruction of the concept in the individual brain” (De Cruz 2007, p. 210).
Ingold (2014) seems sympathetic to this view, insofar as he advances that an individual’s ideas are products of history, not of their own mind.
Without claiming that the origins of numerical practices are instances of the myth of the heroic inventor (see Henrich et al. 2008 for references behind this expression), it should be noted that there are some circumstances where inventions have immediate, sudden impact on their societies. Not all innovations are created equal, and there are many cases throughout the history of mathematics of such individual-led game changers. One could consider Cantor’s work on transfinite numbers an example of such innovation. In a somewhat less grandiose, more culinary context, Dominique Ansel’s invention of the cronut—a cross between a doughnut and a croissant—could perhaps also count as such a game changer. Both cases seem to illustrate that “major transitions in society need not await a series of innovations, each of small effect, but may result instead from key innovations or from coordinated flexibility in response to changing conditions” (Laland et al. 2014, p. 12).
Presumably, this denial would require coming up with criteria for when a numeration practice is truly, well, numerical. Perhaps we could want to base ourselves on the quantities reached, or the portability, ease of use, ease of comprehension, or any combination of these and other factors—not a simple task, to say the least. Moreover, this sounds like we would end up excluding some systems and including other systems of comparable expressive power, which would make hard lines somewhat arbitrary.
The difficulty of knowing where to draw the line between those that understand what natural numbers are and those that do not is well documented (e.g. Rips et al. 2008b; see also Relaford-Doyle and Núñez 2018). I will once again exploit the relative seclusion of a footnote to comment that while there is good reason to think that there have been many degrees of complexity and expressive power of representations of discrete quantities throughout history (e.g. Ifrah 1998; Chrisomalis 2010; Menninger 1969), it would appear senseless to distinguish systems based on the number of items they have developed symbols for. For example, it seems useless to distinguish between systems that have symbols for up to 27 items, and systems that have symbols for up to 29 items, say.
On the other hand, there is a good reason to distinguish between systems whose expressive power is limited by the absence of formalized generative syntax, as I have proposed here with the term ‘proto-numerical’, and systems that require understanding a generative rule for their mastery, rather than rote memorization. This relates to Hurford’s (1987) talk of numeral lexicons that lack the syntactic resources needed to express larger numbers. The mastery of the syntax of a formal system would seem to represent a separate step in the development of formal numerical cognition that requires separate cognitive processes, such as those associated with multi-digit numeral processing (e.g. Nuerk et al. 2015). Similarly, it may be useful to distinguish systems that cannot be extended indefinitely from those that can be extended indefinitely, as is the case for the Indo-Arabic numeration system. The reasons for such distinctions go beyond our present concerns, however. See Schlimm (2018) for work on these topics, as well as Beller and Bender (2008) and Overmann (2015, 2016).
This is not to imply that the processes responsible for an innovation are the same ones involved in the re-construction of the original content in a learner’s head. While I think there is a way of reading Sperber (1985) or Claidière et al. (2014) that is consistent with this claim, the point being made here does not rely on taking a side on this issue. What must be kept in mind here is that both the innovative process and the re-constructive one are black-boxed if we construe practices as essentially cultural, as Menary seems to want to do.
As an anonymous reviewer remarked, it is controversial to claim as I do here that individuals invent practices, given that many frame practices as being constituted by structures of interactions between individuals, which would seem to prevent them from being invented by a single individual. The point I am making here does not rely on rejecting this approach to practices, since even if practices are constituted by such social interactions, they are still often framed as patterns of individual performances (e.g. Rouse 2006). Thus, it seems acceptable to claim as I do here that such patterns of individual performances have origins, and that these origins are performances by individuals. I am thankful to this reviewer for bringing my attention to this issue.
References
Ansari, D. (2008). Effects of development and enculturation on number representation in the brain. Nature Reviews Neuroscience, 9, 278–291.
Basalla, G. (1988). The evolution of technology. New York: Cambridge University Press.
Beller, S., & Bender, A. (2008). The limits of counting: Numerical cognition between evolution and culture. Science, 319, 213–215.
Benacerraf, P. (1973). Mathematical truth. Journal of Philosophy, 70, 661–679.
Boden, M. (2004). The creative mind: Myths and mechanisms (2nd ed.). London: Routledge.
Butterworth, B. (1999). The mathematical brain. London: Macmillan.
Carey, S. (2009). The origin of concepts. New York: Oxford University Press.
Carey, S. (2011). Précis of the origin of concepts. Behavioral and Brain Sciences, 34(3), 113–124. https://doi.org/10.1017/S0140525X10000919.
Chalmers, D. J. (2008). Foreword. In A. Clark (Ed.), Supersizing the mind (pp. ix–xxiv). Oxford: OUP.
Charbonneau, M. (2015). All innovations are equal, but some more than others: (Re)Integrating modification processes to the origins of cumulative culture. Biological Theory, 10(4), 322–335.
Chen, W. J., & Krajbich, I. (2017). Computational modeling of epiphany learning. Proceedings of the National Academy of Sciences, 114(18), 4637–4642. https://doi.org/10.1073/pnas.1618161114.
Cheung, P., Rubenson, M., & Barner, D. (2017). To infinity and beyond: Children generalize the successor function to all possible numbers years after learning to count. Cognitive Psychology, 92, 22–36. https://doi.org/10.1016/j.cogpsych.2016.11.002.
Chrisomalis, S. (2010). Numerical notation: A comparative history. Cambridge: Cambridge University Press.
Claidière, N., Scott-Phillips, T. C., & Sperber, D. (2014). How Darwinian is cultural evolution? Philosophical Transactions of the Royal Society B: Biological Sciences. https://doi.org/10.1098/rstb.2013.0368.
Clark, A. (2008). Supersizing the mind: Embodiment, action, and cognitive extension. Oxford University Press.
Clark, A. (2013). Whatever next? Predictive brains, situated agents, and the future of cognitive science. Behavioral and Brain Sciences, 36(03), 181–204. https://doi.org/10.1017/S0140525X12047.
Clark, A., & Chalmers, D. (1998). The extended mind. Analysis, 58, 7–19.
Coolidge, F. L., & Overmann, K. A. (2012). Numerosity, abstraction, and the emergence of symbolic thinking. Current Anthropology, 53(2), 204–225.
Crossley, J. N. (1987). The emergence of number. Singapore: World Scientific.
Day, R. L., Laland, K. N., & Odling-Smee, J. (2003). Rethinking adaptation: The niche-construction perspective. Perspectives in Biology and Medicine, 46(1), 80–95.
De Cruz, H. (2007). Innate ideas as a naturalistic source of mathematical knowledge: Towards a Darwinian approach to mathematics. Ph.D. Dissertation. Brussels: Vrije Universiteit Brussel.
De Cruz, H. (2008). An extended mind perspective on natural number representation. Philosophical Psychology, 21(4), 475–490.
Dehaene, S. (1997/2011) The number sense: How the mind creates mathematics. New York: Oxford University Press.
Dehaene, S., & Cohen, L. (2007). Cultural recycling of Cortical maps. Neuron, 56(2), 384–398. https://doi.org/10.1016/j.neuron.207.10.04.
Dehaene, S., Spelke, E., Pinel, P., Stanescu, R., & Tsivkin, S. (1999). Sources of mathematical thinking: Behavioral and brain-imaging evidence. Science, 284, 970–974.
Dutilh Novaes, C. (2013). Mathematical reasoning and external symbolic systems. Logique et Analyse, 56(221), 45–65.
Everett, C. (2017). Numbers and the making of us. Cambridge, MA: Harvard University Press.
Fabry, R. E. (2016). Enriching the notion of enculturation: Cognitive integration, predictive processing, and the case of reading acquisition: A commentary on Richard Menary. In T. Metzinger & J. M. Windt (Eds.), Open MIND: Philosophy and the mind sciences in the 21 century (Vol. 2, pp. 1017–1039). Cambridge, MA: MIT Press.
Foster, G. M. (1944). A summary of Yuki culture. Berkeley, CA: University of California Press.
Frank, M., Everett, D., Fedorenko, E., & Gibson, E. (2008). Number as a cognitive technology: Evidence from Pirahã language and cognition. Cognition, 108, 819–824.
Gelman, R., & Butterworth, B. (2005). Number and language: How are they related? Trends in Cognitive Sciences, 9, 6–10.
Gelman, R., & Gallistel, C. R. (1978). The child’s understanding of number. Cambridge: Harvard University Press.
Gordon, P. (2004). Numerical cognition without words: Evidence from Amazonia. Science, 306, 496–499.
Hadamard, J. (1945). The mathematician’s mind: The psychology of invention in the mathematical field. Princeton: Princeton University Press.
Henrich, J., Boyd, R., & Richerson, P. J. (2008). Five misunderstandings about cultural evolution. Human Nature, 19, 119. https://doi.org/10.1007/s12110-008-9037-1.
Hurford, J. R. (1987). Language and number. Oxford: Basil Blackwell.
Hutchins, E. (1995). Cognition in the wild. Cambridge, MA: MIT Press.
Ifrah, G. (1998). The universal history of numbers. London: The Harvil Press.
Ingold, T. (2014). The creativity of undergoing. Pragmatics & Cognition, 22(1), 124–139.
Kahneman, D., Treisman, A., & Gibbs, B. J. (1992). The reviewing of object files: Object-specific integration of information. Cognitive Psychology, 24, 175–219.
Lakoff, G., & Núñez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books.
Laland, K. N., Boogert, N., & Evans, C. (2014). Niche construction, innovation and complexity. Environmental Innovation and Societal Transitions, 11, 71–86. https://doi.org/10.1016/j.eist.2013.08.003.
Leibovich, T., Katzin, N., Harel, M., & Henik, A. (2017). From “sense of number” to “sense of magnitude”: The role of continuous magnitudes in numerical cognition. Behavioral and Brain Sciences, 40, E164. https://doi.org/10.1017/S0140525X16000960.
Lewontin, R. C. (1983). Gene, organism and environment. In D. S. Bendall (Ed.), Evolution from molecules to men (pp. 273–285). Cambridge: Cambridge University Press.
Malafouris, L. (2010). Grasping the concept of number: How did the sapient mind move beyond approximation? In I. Morley & C. Renfrew (Eds.), The archaeology of measurement: Comprehending heaven, earth and time in ancient societies (pp. 35–42). Cambridge: Cambridge University Press.
Matsuzawa, T. (1985). Use of numbers by a chimpanzee. Nature, 315, 57–59.
Menary, R. (2007). Cognitive integration: Mind and cognition unbounde. Basingstoke: Palgrave Macmillan.
Menary, R. (Ed.). (2010a). The extended mind. Cambridge: MIT Press.
Menary, R. (2010b). Dimensions of mind. Phenomenology and the Cognitive Sciences, 9, 561–578. https://doi.org/10.1007/s1097-010-9186-7.
Menary, R. (2015a). Mathematical cognition—A case of enculturation. In T. Metzinger & J. M. Windt (Eds.), Open MIND. Frankfurt: MIND Group.
Menary, R. (2015b). What? Now. Predictive coding and enculturation—A reply to Regina E. Fabry. In T. Metzinger & J. M. Windt (Eds.), Open MIND. Frankfurt: MIND Group. https://doi.org/10.15502/9783958571198.
Menninger, K. (1969). Number words and number symbols: A cultural history of numbers. Cambridge, MA: MIT Press.
Morley, I., & Renfrew, C. (Eds.). (2010). The archaeology of measurement: Comprehending heaven, earth and time in ancient societies. Cambridge: Cambridge University Press. https://doi.org/10.1017/CBO9780511760822.
Nuerk, H.-C., Moeller, K., & Willmes, K. (2015). Multi-digit number processing—Overview, conceptual clarifications, and language influences. In R. Cohen Kadosh & A. Dowker (Eds.), The Oxford handbook of numerical cognition (pp. 106–139). Oxford: Oxford University Press.
Núñez, R. E. (2017). Is there really an evolved capacity for number? Trends in Cognitive Sciences, 21(6), 409–424.
Overmann, K. A. (2015). Numerosity structures the expression of quantity in lexical numbers and grammatical number. Current Anthropology, 56(5), 638–653.
Overmann, K. A. (2016). The role of materiality in numerical cognition. Quaternary International, 405, 42–51.
Pelland, J.-C. (2018). Which came first, the number or the numeral? In S. Bangu (Ed.), The cognitive basis of logico-mathematical knowledge (pp. 179–194). New York: Routledge.
Pepperberg, I. M., & Gordon, J. D. (2005). Number comprehension by a grey parrot (Psittacus erithacus), including a zero-like concept. Journal of Comparative Psychology, 119, 197–209.
Pica, P., Lemer, C., Izard, V., & Dehaene, S. (2004). Exact and approximate arithmetic in an Amazonian indigene group. Science, 306, 499–503.
Poincaré, H. (1910). Mathematical creation. The Monist, 20(3), 321–335.
Pöyhönen, S. (2014). Explanatory power of extended cognition. Philosophical Psychology, 27(5), 735–759. https://doi.org/10.1080/09515089.2013.766789.
Relaford-Doyle, J., & Núñez, R. E. (2018). Beyond Peano: Looking into the unnaturalness of natural numbers. In S. Bangu (Ed.), The cognitive basis of logico-mathematical knowledge (pp. 234–251). New York: Routledge.
Richerson, P., & Boyd, R. (2005). Not by genes alone: How culture transformed human evolution. Chicago: University of Chicago Press.
Rips, L. J., Asmuth, J., & Bloomfield, A. (2008a). Do children learn the integers by induction? Cognition, 106, 940–951.
Rips, L. J., Bloomfield, A., & Asmuth, J. (2008b). From numerical concepts to concepts of number. Behavioral and Brain Sciences, 31, 623–642.
Rouse, J. (2006). Practice theory. In S. Turner & M. Risjord (Eds.), Handbook of the philosophy of science: The philosophy of sociology and anthropology (Vol. 15, pp. 499–540). Dordrecht: Elsevier.
Saxe, G. (1981). Body parts as numerals: A developmental analysis of numeration among the Oksapmin in Papua New Guinea. Child Development, 52(1), 306–316. https://doi.org/10.2307/1129244.
Schlimm, D. (2018). Numbers through numerals. In S. Bangu (Ed.), The cognitive basis of logico-mathematical knowledge (pp. 195–217). New York: Routledge.
Sperber, D. (1985). Anthropology and psychology: Towards an epidemiology of representations. Man, 20, 73–89.
Sterelny, K. (2010). Minds: Extended or scaffolded? Phenomenology and the Cognitive Sciences, 9(4), 465–481. https://doi.org/10.1007/s1097-010-9174-y.
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 of the United States of America, 103, 10775–10780.
Wheeler, M., & Clark, A. (2008). Culture, embodiment and genes: Unravelling the triple helix. Philosophical Transactions of the Royal Society of London. Series B, 363, 3563–3575.
Wiese, H. (2004). Numbers, language, and the human mind. Cambridge: Cambridge University Press.
Wood, G., Willmes, K., Nuerk, H.-C., & Fischer, M. H. (2008). On the cognitive link between space and number: A meta-analysis of the SNARC effect. Psychology Science, 50(4), 489–525.
Wynn, K. (1992). Children’s acquisition of the number words and the counting system. Cognitive Psychology, 24, 220–251.
Acknowledgements
Research for this article was supported by Grant 163129 from the Fonds de recherche du Québec—Société et culture (FRQSC). Many thanks to Catarina Dutilh Novaes, Markus Pantsar, Max Jones, Regina Fabry, Richard Menary and other participants of the ESPP 2016 Symposium for helpful and engaging discussion on these topics. I also want to thank Brian Ball, Mathieu Marion, and two anonymous reviewers for helpful comments on earlier versions of this work.
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Pelland, JC. What’s new: innovation and enculturation of arithmetical practices. Synthese 197, 3797–3822 (2020). https://doi.org/10.1007/s11229-018-02060-1
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DOI: https://doi.org/10.1007/s11229-018-02060-1