In a masterful treatise, Carey (Reference Carey2009) argues that Quinian bootstrapping drives human conceptual development by building on innate domain-specific concepts, domain-general concepts and reasoning processes (especially logical operators and analogical reasoning), and capacities to learn “placeholder” symbols and their structural relations. She proposes, in particular, that preschool children construct the system of positive integer concepts in three steps. First they learn an ordered list of meaningless number words. Then they map the first four words to sets of 1–4 numerically distinct individuals, drawing on representations from two innate but highly limited systems that together allow for “enriched parallel individuation” (EPI). Finally, they make an analogy between the sequential structure in the counting list and the numerical structure of EPI. Armed with this analogy, and with the ordering of words in the count list, children bootstrap their way over the set-size limit of EPI (3 around 4 individuals) and construct the integers.
Conspicuously absent from this account is a well-studied core system of numerical representation: the approximate number system (ANS) (Dehaene Reference Dehaene2009). Carey discusses this system, by which animals and humans of all ages, including newborn infants (Izard et al. Reference Izard, Sann, Spelke and Streri2009), represent the cardinal values of arrays of objects and events with ratio-limited precision. She argues, however, that ANS representations play no role in the construction of integer concepts, even though they later become associated with those concepts and enhance their use. This argument is crucial for Carey's bootstrapping account. If young children mapped number words both to (precise but set-size limited) representations of numerically distinct individuals and to (noisy but essentially unbounded) representations of cardinal values, they would not need the ordinal list structure, or analogical reasoning, to overcome the EPI limit of four. Instead, children could learn that “one” designates a set with a cardinal value around 1 (as given by the ANS) consisting of a single individual (as given by EPI), and that “two” and “three” designate sets with cardinal values around 2 and 3, composed by adding individuals to a set of one. Without regard to the order of these words, children could discover how these ANS and EPI representations relate to one another: progressively larger cardinal values result from progressive addition of one.
Carey presents compelling arguments that ANS representations are not the sole basis of children's learning of number words and counting, but several considerations suggest that they contribute to this process. First, children who have recently mastered counting recruit ANS representations to solve novel tasks involving number words and symbols: they perform approximate symbolic addition and place symbolic numbers on a line in ways that reveal critical signatures of the ANS (Gilmore et al. Reference Gilmore, McCarthy and Spelke2007; Siegler & Opfer Reference Siegler and Opfer2003). Second, when pre-counting children who already know number words to “three” are taught “four,” they first map the word to an ANS representation, showing more false alarms to nearby numerosities (such as five) than to distant ones (Huang et al. Reference Huang, Spelke and Snedeker2010). Third, some adults with impairments to the ANS but normal number word comprehension and verbal counting have trouble determining whether “nine” denotes a larger quantity than “seven” (e.g., Dehaene & Cohen Reference Dehaene and Cohen1997). If the ordinal structure of the count list provided critical information about numerical order independently of the ANS, then impairments to the ANS should not obscure this order.
This observation suggests a final reason to question Carey's bootstrapping theory of the development of integer concepts. Although children learn to recite ordered lists of meaningless words, and “one, two, three…” may be an example, it does not follow that children or adults can access and use the ordinal structure of such a list. Adults learn songs and poems without accessing this structure (if you can recite the United States national anthem, then consider which word comes first, “stripes” or “gleaming”? I can only answer this question by rattling off the song.) Early counting-based arithmetic strategies suggest that children who have mastered counting initially fail to access the ordinal positions of the words in their count list. A child who knows that “four” denotes the fourth word on the list should add 4+3 by starting with “four” and counting on. When children first use counting to solve such problems, however, more children start the count with “one” (Siegler & Jenkins Reference Siegler and Jenkins1989).
I suggest that Quinian bootstrapping – learning symbols as placeholders, deciphering the structural relations among them, and then using analogical reasoning to map those relations onto other conceptual domains – is probably not the source of integer concepts. Nevertheless, Carey reviews rich evidence that these concepts depend in some way on mastery of verbal counting. How else might language, and other symbol systems, support cognitive development in this domain and others? First, language may provide efficient ways to express and use concepts that children already possess (Frank et al. Reference Frank, Everett, Fedorenko and Gibson2008). Second, words and other symbols may help learners to select, from among the myriad concepts at their disposal, those that are most useful or relevant in some context (Csibra & Gergely Reference Csibra and Gergely2009; Waxman & Markow Reference Waxman and Markow1995). Third, language may serve as a medium in which information from distinct, domain-specific cognitive systems can be productively combined (Spelke Reference Spelke, Gentner and Goldin-Meadow2003b). Learning the meanings of words like “two” and “three” may be useful to children, because the meanings of these terms combine information from distinct cognitive systems. Prelinguistic infants and nonlinguistic animals possess these systems but may lack the means to combine their outputs flexibly and productively (Spelke Reference Spelke, Kanwisher and Duncan2003a). On any of these views, new concepts would arise from processes that repackage, select, or combine preexisting concepts, as envisioned by Fodor (Reference Fodor1975), rather than from the constructive processes that Carey develops from Quine's metaphors. Carey's bold and rigorously argued case for Quinian bootstrapping sets a high standard for theories and research addressing this family of Fodorian conjectures.
In a masterful treatise, Carey (Reference Carey2009) argues that Quinian bootstrapping drives human conceptual development by building on innate domain-specific concepts, domain-general concepts and reasoning processes (especially logical operators and analogical reasoning), and capacities to learn “placeholder” symbols and their structural relations. She proposes, in particular, that preschool children construct the system of positive integer concepts in three steps. First they learn an ordered list of meaningless number words. Then they map the first four words to sets of 1–4 numerically distinct individuals, drawing on representations from two innate but highly limited systems that together allow for “enriched parallel individuation” (EPI). Finally, they make an analogy between the sequential structure in the counting list and the numerical structure of EPI. Armed with this analogy, and with the ordering of words in the count list, children bootstrap their way over the set-size limit of EPI (3 around 4 individuals) and construct the integers.
Conspicuously absent from this account is a well-studied core system of numerical representation: the approximate number system (ANS) (Dehaene Reference Dehaene2009). Carey discusses this system, by which animals and humans of all ages, including newborn infants (Izard et al. Reference Izard, Sann, Spelke and Streri2009), represent the cardinal values of arrays of objects and events with ratio-limited precision. She argues, however, that ANS representations play no role in the construction of integer concepts, even though they later become associated with those concepts and enhance their use. This argument is crucial for Carey's bootstrapping account. If young children mapped number words both to (precise but set-size limited) representations of numerically distinct individuals and to (noisy but essentially unbounded) representations of cardinal values, they would not need the ordinal list structure, or analogical reasoning, to overcome the EPI limit of four. Instead, children could learn that “one” designates a set with a cardinal value around 1 (as given by the ANS) consisting of a single individual (as given by EPI), and that “two” and “three” designate sets with cardinal values around 2 and 3, composed by adding individuals to a set of one. Without regard to the order of these words, children could discover how these ANS and EPI representations relate to one another: progressively larger cardinal values result from progressive addition of one.
Carey presents compelling arguments that ANS representations are not the sole basis of children's learning of number words and counting, but several considerations suggest that they contribute to this process. First, children who have recently mastered counting recruit ANS representations to solve novel tasks involving number words and symbols: they perform approximate symbolic addition and place symbolic numbers on a line in ways that reveal critical signatures of the ANS (Gilmore et al. Reference Gilmore, McCarthy and Spelke2007; Siegler & Opfer Reference Siegler and Opfer2003). Second, when pre-counting children who already know number words to “three” are taught “four,” they first map the word to an ANS representation, showing more false alarms to nearby numerosities (such as five) than to distant ones (Huang et al. Reference Huang, Spelke and Snedeker2010). Third, some adults with impairments to the ANS but normal number word comprehension and verbal counting have trouble determining whether “nine” denotes a larger quantity than “seven” (e.g., Dehaene & Cohen Reference Dehaene and Cohen1997). If the ordinal structure of the count list provided critical information about numerical order independently of the ANS, then impairments to the ANS should not obscure this order.
This observation suggests a final reason to question Carey's bootstrapping theory of the development of integer concepts. Although children learn to recite ordered lists of meaningless words, and “one, two, three…” may be an example, it does not follow that children or adults can access and use the ordinal structure of such a list. Adults learn songs and poems without accessing this structure (if you can recite the United States national anthem, then consider which word comes first, “stripes” or “gleaming”? I can only answer this question by rattling off the song.) Early counting-based arithmetic strategies suggest that children who have mastered counting initially fail to access the ordinal positions of the words in their count list. A child who knows that “four” denotes the fourth word on the list should add 4+3 by starting with “four” and counting on. When children first use counting to solve such problems, however, more children start the count with “one” (Siegler & Jenkins Reference Siegler and Jenkins1989).
I suggest that Quinian bootstrapping – learning symbols as placeholders, deciphering the structural relations among them, and then using analogical reasoning to map those relations onto other conceptual domains – is probably not the source of integer concepts. Nevertheless, Carey reviews rich evidence that these concepts depend in some way on mastery of verbal counting. How else might language, and other symbol systems, support cognitive development in this domain and others? First, language may provide efficient ways to express and use concepts that children already possess (Frank et al. Reference Frank, Everett, Fedorenko and Gibson2008). Second, words and other symbols may help learners to select, from among the myriad concepts at their disposal, those that are most useful or relevant in some context (Csibra & Gergely Reference Csibra and Gergely2009; Waxman & Markow Reference Waxman and Markow1995). Third, language may serve as a medium in which information from distinct, domain-specific cognitive systems can be productively combined (Spelke Reference Spelke, Gentner and Goldin-Meadow2003b). Learning the meanings of words like “two” and “three” may be useful to children, because the meanings of these terms combine information from distinct cognitive systems. Prelinguistic infants and nonlinguistic animals possess these systems but may lack the means to combine their outputs flexibly and productively (Spelke Reference Spelke, Kanwisher and Duncan2003a). On any of these views, new concepts would arise from processes that repackage, select, or combine preexisting concepts, as envisioned by Fodor (Reference Fodor1975), rather than from the constructive processes that Carey develops from Quine's metaphors. Carey's bold and rigorously argued case for Quinian bootstrapping sets a high standard for theories and research addressing this family of Fodorian conjectures.