The Origin of Concepts by Susan Carey;

Author:Susan Carey;
Language: eng
Format: epub
Publisher: OUP Premium
Published: 2009-06-15T00:00:00+00:00

Bootstrapping the Numeral List Representation of Natural Number

The output of the hypothetical bootstrapping mechanism is the numeral list representation of natural number—an ordered list of numerals such that the first one on the list represents 1 and for any word on the list that represents the cardinal value n, the next word on the list represents n + 1. The successor function is the heart of numeral list representations of integers. The numeral list representation of number is characterized by Gelman and Gallistel’s counting principles (the list is stably ordered; individuals in a given count are put in 1–1 correspondence with number words, and the cardinal value of the set is the ordinal position of the word in the count list).

The problem of how the child builds an numeral list representation decomposes into the related subproblems of learning the ordered list itself (“one, two, three, four, five, six …”), learning the meaning of each symbol on the list (e.g., “three” means three and “seven” means seven), and learning how the list itself represents number, such that the child can infer the meaning of a newly mastered numeral symbol (e.g., “eleven”) from its position in the numeral list. Rather than offering just one bootstrapping proposal, I will offer two. They differ in the planks of the process, in what aspects of core cognition are drawn upon. I offer two proposals to illustrate that it is, in fact, not difficult to imagine how to we might explain conceptual discontinuities, and to illustrate that it is possible to bring empirical data to bear on choosing among specific proposals concerning possible bootstrapping processes.

Each proposal assumes that the child first learns “one, two, three, four, five …” as a list of meaningless lexical items. This is the no numeral knower stage documented above. There is no doubt that children have the capacity to learn meaningless ordered lists of words—they learn sequences such as “eeny, meeny, miny, mo,” the alphabet, the days of the week, and so on. Indeed, nonhuman primates have this capacity, and so it is likely part of innate computational machinery (e.g., Terrace, Son, & Brannon, 2003). This step in the learning process—learning an arbitrary ordered list (“one, two, three, four, five, six …”) is a paradigmatic example of one aspect of Quinian bootstrapping: the meanings of the counting words are exhausted, initially, by their interrelations, their relative order in the list. At this point in the process, the verbal numerals are placeholders with respect to the numerical meaning they will come to have.

The two bootstrapping proposals differ in terms of the process envisioned as to how the placeholders come to be interpreted as words for natural numbers. Both proposals must capture what is known, empirically, about the earliest stages in number word learning. Both presuppose that the learning process involves combining antecedently available representations. Proposal 1 assumes that analog magnitude representations of number are the only system of core numerical cognition that is drawn upon in the process. I then present data that decisively undermine Proposal 1.

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