The Conceptual Basis of Numerical Abilities: One-to-One Correspondence Versus the Successor Relation
2008; Taylor & Francis; Volume: 21; Issue: 4 Linguagem: Inglês
10.1080/09515080802285255
ISSN1465-394X
Autores Tópico(s)Child and Animal Learning Development
ResumoAbstract In recent years, neologicists have demonstrated that Hume's principle, based on the one-to-one correspondence relation, suffices to construct the natural numbers. This formal work is shown to be relevant for empirical research on mathematical cognition. I give a hypothetical account of how nonnumerate societies may acquire arithmetical knowledge on the basis of the one-to-one correspondence relation only, whereby the acquisition of number concepts need not rely on enumeration (the stable-order principle). The existing empirical data on the role of the one-to-one correspondence relation for numerical abilities is assessed and additional empirical tests are proposed. In the final part, it is argued that the fact that the successor relation and the one-to-one correspondence relation can play independent roles in number concept acquisition may be a complication for testing the Whorfian hypothesis. Keywords: EnumerationHume's PrincipleNumerical ConceptsWhorfian hypothesis Acknowledgements I would like to thank Brian Butterworth, Helen De Cruz, Pierre Pica, the members of the research section Epistemology and Ontology at the Vrije Universiteit Amsterdam, and two anonymous referees for valuable comments. Notes Notes [1] Rips, Asmuth, and Bloomfield (2006 Rips, L, Asmuth, J and Bloomfield, A. 2006. Giving the boot to the bootstrap: How not to learn the natural numbers. Cognition, 101: B51–B60. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) argue that learning the natural numbers hinges on Peano's axioms. They formulate this as an alternative to Carey's (2001 Carey, S. 2001. Cognitive foundations of arithmetic: Evolution and ontogenesis. Mind & Language, 16: 37–55. [Crossref], [Web of Science ®] , [Google Scholar]) bootstrap mechanism. Remarkably, they let the natural numbers start with zero (p. B59). [2] Frege disagrees with Cantor's ordinal conception of number, and his comments are most illuminating; see Frege (1978 Frege, G. 1978. Foundations of arithmetic, Oxford: Blackwell. [Google Scholar], pp. 97–98). [3] Basic Law V states that two predicates F and G have the same extension (or determine the same set), in case ∀x: Fx ↔ Gx. [4] Piaget borrowed the basic mathematical concepts ISOMORPHISM and ORDER (and a third concept NEIGHBOURHOOD, which is more relevant for geometry than for arithmetic) from the Bourbaki group (Piaget, 1968 Piaget, J. 1968. Le structuralisme, Paris: PUF. [Google Scholar], pp. 22–23). In view of the contemporary wide gap, or even antagonism, between foundational work in mathematics and the empirical study of mathematical cognition, it is quite remarkable that Piaget was in close contact with Bourbaki, especially with Jean Dieudonné (Aczel, 2007 Aczel, A. 2007. The artist and the mathematician, London: High Stakes. [Google Scholar], p. 162). Piaget (1968 Piaget, J. 1968. Le structuralisme, Paris: PUF. [Google Scholar], p. 24) already mentioned the possible role of categories in mathematical cognition. [5] Gelman and Butterworth (2005 Gelman, R and Butterworth, B. 2005. Number and language: How are they related?. Trends in Cognitive Sciences, 9: 6–10. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar], p. 8) correctly ascribe this view to John Locke (An Essay Concerning Human Understanding, Bk II, ch. 16). [6] Turconi, Campbell, & Seron (2006 Turconi, E, Campbell, J and Seron, X. 2006. Numerical order and quantity processing in number comparison. Cognition, 98: 273–285. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) argue that order and quantity processing may involve different cognitive mechanisms. In number comparisons, quantity judgments give rise to the standard distance effect, while order judgments cause a reverse order effect. It is argued that quantity judgments basically involve the “number sense,” rather than order (or enumeration). [7] Of course, this would require a concept of ORDER or an ordering algorithm. This concept should be distinguished from the concept of a SUCCESSOR RELATION or ENUMERATION and is a much weaker concept. Enumeration requires at least an initial element and some form of “accumulation,” in addition to the concept of ORDER. There is evidence that the concept of ORDER, and the ability of making of transitive order inferences, is innate already in birds (see, e.g., Paz-y-Miño, Bond, Kamil, & Balda, 2004 Paz-y-Miño, G, Bond, A, Kamil, A and Balda, R. 2004. Pinyon jays use transitive inference to predict social dominance. Nature, 430: 778–781. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]). [8] Gelman and Butterworth (2005 Gelman, R and Butterworth, B. 2005. Number and language: How are they related?. Trends in Cognitive Sciences, 9: 6–10. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar], p. 7) argue that the word ‘two’ cannot refer to a particular set of two object files, e.g., my pair of shoes. They point out that there is a difference between the particular set and the property twoness, and a difference between the particular set and another set of two objects. However, it is possible to regard the concept TWO as a compound concept EQUINUMEROUS TO MY PAIR OF SHOES. Similarly, the concept of a DOG can be regarded as a compound concept WALKS AND BARKS LIKE ROVER. In both cases, a unary predicate is obtained by relativizing a binary predicate to named object (or set). [9] Brannon (2002 Brannon, E. 2002. The development of ordinal numerical knowledge in infancy. Cognition, 83: 223–240. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) argues that 11-month-old-infants develop a capacity for nonnumerical ordinal judgments, before developing a capacity for ordinal numerical judgments. This capacity is most probably related to the approximate “number sense.” [10] In Lipton and Spelke (2004 Lipton, J and Spelke, E. 2004. Preschool children master the logic of number word meanings. Cognition, 98: B57–B66. [Crossref], [Web of Science ®] , [Google Scholar]), experiments with preschool children show that these children have mastered the logic of number-changing operations for numbers well beyond their counting range. It would be worthwhile to investigate to what extent they have mastered the one-to-one correspondence principle for large cardinal values. Mastery of the one-to-one correspondence principle would sufficiently explain their numerical abilities. [11] In Jordan and Brannon (2006 Hume, D. 1978. A treatise of human nature, Oxford: Oxford University Press. [Google Scholar]) it is argued that seven-month-old infants preferentially attend to visual displays of adult humans that numerically match the number of adult humans they hear speaking. This is taken as evidence for a multisensory representation of numbers, but a more modest interpretation would be that the experiments point at an ability of making cross-modal one-to-one correspondence judgments. [12] On the basis of the available evidence, it is not likely that this is an innate ability. Gelman and Gallistel (2004 Gelman, R and Gallistel, CR. 2004. Language on the origin of numerical concepts. Science, 306: 441–443. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar], p. 442) conclude that nonnumerate subjects “gave evidence of being indifferent to exact numerical equality.” [13] The words for ‘five’ and ‘hand’ are related in many languages, see for example Harris (1987 Harris, JW. 1987. Australian Aboriginal and Islander mathematics. Australian Aboriginal Studies, 2: 29–37. [Google Scholar]). [14] Interestingly, Butterworth (2005 Butterworth, B. 2005. The development of arithmetical abilities. Journal of Child Psychology and Psychiatry, 46: 3–18. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar], p. 7) clearly distinguishes the one-to-one correspondence principle from the stable-order principle and moreover indicates that the two principles appear independently and that learning the correct order of the counting sequence is difficult. At the same time, he rightly points out that “[o]ne of the earliest and perhaps the most important contact between the child's sense of number and the conceptual tools provided by culture is counting” (p. 6). [15] An anonymous referee pointed out that as an empirical, psychological matter of fact, it may be the case that many, most, or all one-to-one correspondence judgments invoke an enumeration procedure (for going through all the pairs). At first blush, there are several reasons to doubt this: one may forget the initial element, the accumulation of the number of pairs may be superfluous, or one may even chunk the items in groups of pairs that are judged separately in one glance. However this may be, the objection is grist for the mill, as it highlights the need for further empirical research. [16] Rouillon (2006 Rouillon, A. 2006. Au Gravettien, dan la grotte Cosquier (Marseille, Bouches-du-Rhône): L’Homme a-t-il compté sur ses doigts?. L’Antropologie, 110: 500–509. [Crossref] , [Google Scholar]) demonstrated that the late Paleolithic “negative hand” cave paintings at Grottoe Cosquer, which are made by blowing paint around one or more fingers of one hand, were always in a fixed order. The five configurations were “thumb extended,” “thumb and index extended,” “thumb, index, and middle finger extended,” “thumb, index, middle and ring finger extended,” and “all fingers extended.” This is taken as evidence that an enumeration procedure was already used 27,000 years ago. [17] This may also be the result of the difficulty the participants experienced in drawing straight lines, as Gordon points out. [18] A similar distinction is made in Gordon (2004 Gordon, P. 2004. Numerical cognition without words: Evidence from Amazonia. Science, 306(5695): 496–499. [Crossref], [PubMed], [Web of Science ®] , [Google Scholar], p. 496). I will use the distinction uncritically, although some may doubt whether the distinction is real (e.g., Pica, private communication). This controversy is partly due to the fact that no very precise definition of the Whorfian hypothesis is uncontroversial. For a recent discussion of the Whorfian hypothesis in cognitive science, see Gentner and Goldin-Meadow (2003 Gentner, D and Goldin-Meadow, S. 2003. Language in mind, Cambridge, MA: Bradford. [Crossref] , [Google Scholar]). [19] This “external” view on mathematical cognition has become popular recently; see, for example, Clark (2000 Clark, A. 2000. Mindware, Oxford: Oxford University Press. [Google Scholar], p. 146) and De Cruz (2006 De Cruz, H. 2006. Why are some numerical concepts more successful than others? An evolutionary perspective on the history of number concepts. Evolution and Human Behavior, 27: 306–323. [Crossref], [Web of Science ®] , [Google Scholar]). [20] One may doubt whether this would still count as a “concept.” Characterizing the precise nature of concepts is a difficult methodological issue in psychology and an important ontological problem in philosophy. I am not committed to particular proposal and prefer to be liberal. [21] Jordan and Brannon (2006 Hume, D. 1978. A treatise of human nature, Oxford: Oxford University Press. [Google Scholar]) demonstrate that seven-month-old infants can already recognize cross-modal one-to-one correspondence judgments for low numerosities. This would support the claim that one-to-one correspondence judgments belong to the innate numerical abilities.
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