Core Knowledge, Language, and Number

ABSTRACT

The natural numbers may be our simplest, most useful, and best-studied abstract concepts, but their origins are debated. I consider this debate in the context of the proposal, by Gallistel and Gelman, that natural number system is a product of cognitive evolution and the proposal, by Carey, that it is a product of human cultural history. I offer a third proposal that builds on aspects of these views but rejects one tenet that they share: the thesis that counting is central to number. I suggest that children discover the natural numbers when they learn a natural language: especially nouns, number words, and the rules that compose quantified noun phrases. This learning, in turn, depends both on cognitive systems that are innate and shared by other animals, and on our species-specific language faculty. Thus, natural number concepts are unique to humans and culturally universal, yet they are learned.

Acknowledgments

Thanks to Susan Carey, Julian Jara-Ettinger, and Josh Rule for comments on earlier versions of this manuscript, and to the members of the Carey-Spelke Seminar on Abstract Thought for useful discussions. Special thanks to Veronique Izard for invaluable insights that have influenced my thinking about number for more than ten years, and for penetrating comments, questions, and criticisms relating to the present manuscript.

Funding

Supported by a grant from NSF (DRL 1348140) and by the NSF-STC Center for Brains, Minds and Machines (CCF-1231216).

Notes

¹ In the literature on numerical development in children, the acquisition of “the successor principle” refers to a milestone in children’s mastery of counting: the point at which children understand that every word in their verbal count list refers to a cardinal value that is one larger than the value designated by the previous word. In some discussions, mastery of the successor principle of verbal counting is considered as a criterion for mastery of the natural number system (e.g., Sarnecka, 2016). Use of this criterion, however, would beg the present question. Because numerical language is learned, natural number concepts that were defined by children’s learning of number word meanings necessarily would be learned, ruling out Gelman and Gallistel’s nativist claims by definition and rendering Carey’s claims true by definition. On pain of circularity, experiments testing these theories and others require characterizations of numerical concepts that are independent of children’s learning of language.

² Placebo effects are less plausible in Park & Brannon’s experiments, because their participants showed benefits on only one of three numerical tasks.

³ There are important differences between Gallistel and Gelman’s earlier and later theories, as their thinking evolved within the growing field of numerical cognition, as well as differences between their theories and those of others who posit an innate system of natural number (Brannon & Merritt, 2011; Butterworth, 1999; Wynn, 1998). For present purposes, I ignore these differences and focus on common features of all the theories that posit innate access to the natural numbers.

⁴ Although young children’s failures in Izard’s addition and substitution tasks provides evidence that they lack full natural number concepts, success would not imply that they command these concepts fully, because Izard’s task does not test whether children have a concept ONE that separates every number from those that are closest to it (see Izard et al., 2008; and Rips et al., 2008; for discussion). I return to this question below.

⁵ For simplicity, I assume in this exposition, that children are learning a verbal counting routine. Carey’s theory applies, however, to any ordered list of symbols used in counting.

⁶ One pillar of the natural number system—the concept of a minimal unit, ONE, that distinguishes each number from its nearest neighbors—is not tested by the tasks of Izard and Jara-Ettinger: although children who pass these tasks successfully infer that the cardinal value of a set is incremented by the addition of one element, and also that the operations of addition and subtraction of one element cancel one another, these studies do not reveal whether children infer that all numbers are separated from their nearest neighbors by the same minimal unit. This gap in the evidence does not bear on Carey’s theory, because the central analogy that she posits presupposes the unit principle. After learning that one, two, and three designate sets that differ by exactly one element, Carey proposes that children infer that all the successive words in their count list will differ by exactly one element. Nothing inherent to the counting procedure licenses this inference, so Carey’s theory requires that children apply the minimal unit principle prior to this learning (see Rips et al., 2008; and Izard et al., 2008; for discussion). I return to this principle in the next section (see especially footnote 13).

⁷ For ease of exposition, my discussion centers on NPs in English. Children who learn a language that lacks the count/mass and singular/plural distinction would construct kind representations by mastering other structures, such as classifiers.

⁸ Body parts are not bounded objects, but names for body parts appear in children’s earliest vocabulary and are invoked to represent space and number in many cultures. I suggest that a core system for representing agents also serves to represent body parts as entities with distinct forms and functions, as well as whole animate beings and their actions (Spelke, forthcoming, Ch. 3). These representations, together with representations of bounded objects and of geometrical forms, support children’s early learning of nouns and noun phrases.

⁹ Fei Xu’s beautiful experiments provide much of the evidence for this claim, although she herself does not endorse it.

¹⁰ Two questions arise here on which I take no stand. First, when do children begin to map noun phrases to ANS representations? Some evidence, discussed below, suggests that this mapping occurs only after children have mapped NPs containing one, two, and three to representations of 1–3 individuals of a kind. Recent evidence suggests, however, that this mapping may begin earlier, perhaps at the start of number word learning (Barner, 2016), when children first show evidence of representing sets of objects in non-verbal tasks (Feigenson, 2011; Feigenson & Halberda, 2008). The account I offer here is compatible with either of these possibilities. Second, where does the concept SET come from? If children begin to map NPs to ANS representations at the start of number word learning, then it may come from the ANS. Alternatively, it may come from the core object system (and therefore be available in representations of objects of specific kinds), from the language faculty, or from a domain-general core cognitive system supporting logical reasoning. In the absence of relevant evidence, I leave this question open. My account requires, however, that SET is available to children at the second step described above.

¹¹ The limits on parallel representations of objects vary depending on stimulus and task factors, as well as the development of perceptual skills (e.g., Alvarez & Franconeri, 2007; Green & Bavelier, 2003). So do the limits on ANS acuity, which increases with practice and with learning of mathematics (Piazza et al., 2013). In all cases, however, parallel representations of objects and ANS representations of numerical magnitudes show limits to which the natural number system is not subject.

¹² One challenge in mastering the meanings of these quantifiers is that they specify inexact quantities, in contrast to the words for natural numbers. Children distinguish number words from these inexact quantifiers early in the process of deciphering number word meanings. Before children have mastered the meanings of three and four, they know that these words contrast in meaning with one another (a set designated by three birds should not be called four birds) but not with other quantifiers (a set designated as three birds also can be called some birds: Condry & Spelke, 2008). Children gain this knowledge before they understand that sets that can be labeled “three birds” vs. “four birds” differ in number (Condry & Spelke, 2008).

¹³ I am uncommitted as to whether a child who uses natural language rules to compose new numbers thereby comes to embrace the minimal unit principle, according to which the minimal distance between two numbers is one. Noun phrases allow for expressions such as five and a half as well as five and two more. Although young children tend to misunderstand these expressions (Gelman, 1991) and older children take a long time to develop a working understanding of fractions (Carey & Spelke, 1994), young children may be open to the possibility that there are numbers beyond those that can be generated by the operations of addition and multiplication, applied recursively to the first three numbers, and therefore that there are numbers that cannot be reached by successive addition of one. Such openness might be a virtue, from the standpoint of mathematical discovery, and it surely is achieved by adults who have discovered the rational, real, and complex numbers. Using natural language, together with core knowledge, children may discover the natural numbers, but perhaps not only the natural numbers (see also footnote 14).

¹⁴ Following Chomsky (1988), I therefore propose that children come to command the generative system of natural number by mastering the generative rules of their language. Note, however, that children may master these rules while remaining stunningly ignorant of specific facts of arithmetic, just as adults master recursive rules of multiplication without being able to determine, in specific cases, whether any given number is prime. Moreover, children may come to appreciate that natural number concepts can be generated by successive addition of one long before they realize either that repeated addition of one generates all the natural numbers or that is the only operation that does so (e.g., that successive additions of two will never generate the number twenty-three). Thus, children may implement the principle of succession without appreciating its role in defining the natural number system. I thank Veronique Izard for illuminating discussion of this point.

¹⁵ This early attempt of mine—to address the limitations of theories of numerical development based exclusively on ANS representations or object representations by drawing on both types of representations, together with the combinatorial resources of natural language—contained logical gaps, because the interface of core object and number representations to language was not clear (see Laurence and Margolis (2005) and Rips et al. (2008) for illuminating discussion). The present account fills some of these gaps but others remain: especially questions concerning the innate representations that allow learning and understanding of natural language expressions concluding quantifiers and conjunction. I hope that research will fill these gaps (see, e.g., Gleitman, 2015; Odic, Pietroski, Hunter, Lidz, & Halberda, 2014).

¹⁶ Izard (personal communication) pointed out that some languages and cultures use different number words for different kinds of things. There are vestiges of such usage in English, where it is natural to talk about a dozen eggs or two dozen donuts but odd, at least, to speak of a dozen dollars or two dozen thousand people. Language-specific restrictions on the application of number words may arise as a consequence of language learners’ conservative generalization of number words.

¹⁷ For example, if John likes the red and blue fish, does he like all the fish that are red or blue, or only the fish that are both red and blue? If John and Mary read a book, how many books were read? If John and Mary met and danced, what licenses the inference that John danced but not that he met? See footnote 15.

¹⁸ Studies of speakers of a different Amazonian language, Pirahã, have been argued to adjudicate between these two accounts, providing evidence for the innateness of natural number concepts, but their findings also are inconclusive. The Pirahã language has been described as having no words for any numbers, not even a word for one (Frank et al., 2008), and no recursive rules (Everett, 2005), but these descriptions have been disputed (see Everett, 2009; Nevins et al., 2009). Moreover, decisive tests of mastery of the logic of natural number have not yet been performed on the Pirahã. When Pirahã adults were presented with a line of objects and are asked to create, from a new collection of objects, a second line that matched it, they placed objects from the new collection opposite those in the first collection by one-to-one correspondence (Frank et al., 2008). The studies of Izard and Jara-Ettinger show, however, that this performance does not imply that the Piraha represent the numerical equality of the two sets.

Additional information

Funding

Supported by a grant from NSF (DRL 1348140) and by the NSF-STC Center for Brains, Minds and Machines (CCF-1231216).