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Auteur: Akira WATANABE

Mental representation of natural numbers and acquisition of numerals

Abstract/Résumé: In this talk, I suggest that the Merge-based conception of natural numbers advocated by Chomsky (2008) enables us to explain why full mastering of the meaning of numerals by children is a prolonged process, completed as late as at age 3½ or 4 (Carey 2009, Wynn 1992). The difficulty for children can be reduced to the unusual complexity of linguistic representations of numerals, once Chomsky's original proposal is slightly modified. [Linguistic Representations]: Chomsky (2008) suggests that repeated application of Merge to a single lexical item LI gives rise to a sequence of natural numbers as in (1). (1) 1 = one, 2 = {one}, 3 = {one, {one}}, ... Since any lexical item will do, (1) can be rendered more generally as: (1') 1 = LI, 2 = {LI}, 3 = {LI, {LI}}, ... My proposal is to shift the sequence by one unit and adopt the following: (2) 1 = {LI}, 2 = {LI, {LI}}, 3 = {LI, {LI, {LI}}}, ... In other words, one application of Merge is equated with 1, two applications with 2, and so on. The point of this modification is to match the linearized string of n LI's with the corresponding numeral list. In the case of 3, we have: (3) LI + LI + LI = One, two, three. The idea is to subject the set-theoretic object created by Merge to PF linearization, just as in the case of ordinary syntactic objects. In fact, when we count, we can insert a sequence of numerals in the DP-internal slot for a numeral as in (4). (4) We have one, two three, four, five syntacticians here. Thus, the lexical entries for numerals look like the following: (5) a. original conception (meaning): 1 = {LI}, 2 = {LI, {LI}}, 3 = {LI, {LI, {LI}}} b. linearization of the original structure (PF1): 1 = LI, 2 = LI + LI 3 = LI + LI + LI c. actual counting sequence (PF2): 1 = one, 2 = one + two 3 = one + two + three d. actual PF (PF3): 1 = one, 2 = two, 3 = three (5b) is not used as such, but helps connect (5a) to (5c). Inclusion of (5c) as a piece of PF information derives as a lexical property one of Gelman and Gallistel's (1978) counting principles, which says that the last numeral in the sequence represents the cardinality of the set counted. [Acquisition]: Compared with ordinary lexical items, which are associated with a single phonological form, entries like (5) are unusually rich. It is quite natural to expect this complexity to slow down the acquisition process. Crucially, experimental measures used to assess children's understanding of natural number concepts (Le Corre et al. 2006) all involve expressions like "three frogs". What is difficult for children is not natural number concepts but numerals with language-particular phonological forms.