In their article, Rips et al. argue that inductive accounts of the development of the natural numbers suffer from an irreparable defect: they cannot explain how children arrive at the intuition that there are infinitely many numbers. Moreover, the authors argue that these accounts confuse cardinalities with natural numbers. Indeed, according to Rips et al., the natural numbers cannot be acquired from representations of cardinalities, because natural numbers are not individuated by their reference to cardinalities but, rather, by the system of rules that govern their behavior (e.g., the Dedekind-Peano axioms). Rips et al.'s solution to these problems is two-fold: (1) They postulate that human minds comprise two autonomous systems of numerical reasoning – one dedicated to the representation of cardinalities, the other dedicated to the representation of the natural numbers; and (2) they argue that the development of the system that represents the natural numbers is governed by a schema of innate constraints.
In my view, Rips et al.'s theory must answer multiple challenges. First, it does not seem to satisfy the “no competing inference” criterion rightly imposed by the authors themselves on theories of learning. As far as I can tell, the initial state they propose (in sect. 5.2) is potentially consistent with many more systems than just the natural numbers.
Second, because it postulates that the cognitive system for the natural numbers is completely independent from the system dedicated to representations of cardinalities, Rips et al.'s account predicts that one could grasp the natural numbers without having first developed a system for representing cardinalities. This seems to be wrong both historically and developmentally. Each of the histories of number notations I have read reports that, in culture after culture, the first numerical symbols were “external sets” that were connected to sets in the world via one-to-one correspondence (Dantzig Reference Dantzig1967; Hurford Reference Hurford1987; Ifrah Reference Ifrah1985; Kaplan Reference Kaplan1999). None of these report cultures in which the meaning of the first symbols was determined by inferential rules.
As Rips et al. point out, little work has been done on children's grasp of infinity, and even less has been done on their grasp of rules that apply over the entire infinite domain. However, available research on these issues (including the research reviewed in their article) suggests that children learn symbols for cardinalities long before their use of symbols shows evidence of being governed – explicitly or implicitly – by rules such as the Dedekind-Peano axioms or the rules of arithmetic. By now, multiple studies have shown that, by age 4, many children have learned how to use counting to represent cardinalities (e.g. Le Corre et al. Reference Le Corre, Van de Walle, Brannon and Carey2006; Wynn Reference Wynn1990; Reference Wynn1992b). In contrast, one of the rare studies on children's understanding of the successor principle showed that it is not until about 3 years later that the majority of children understand that there is no largest number (Hartnett & Gelman Reference Hartnett and Gelman1998). Similarly, the study of children's understanding of the commutativity of addition cited in Rips et al.'s article suggests that many kindergarteners do not yet understand that addition is commutative. Finally, Hughes (Reference Hughes1986) reports that children understand statements like “four dogs plus three dogs is seven dogs” before they understand statements about the numbers themselves (e.g., “four plus three is seven”). In sum, it seems that the normal sequence of acquisition of number symbols is the same in childhood as in history; that is, in both cases, the meaning of number symbols is determined by their relation to cardinalities of sets long before the symbols are of the kinds of formal systems Rips et al. have in mind.
Furthermore, I have recently shown that analog magnitudes play a causal role in the acquisition of knowledge of a basic inferential relation between large number words (beyond “four”), namely, numerical order (e.g., knowledge that “ten fish” is more than “six fish”). That is, I have found that children must map these number words onto analog magnitudes (Le Corre, under review) to be able to order them. These mappings are formed many months after children have learned how to use counting to determine the numerical size of a set. Therefore, children spend a period knowing how counting works but not knowing how to order large number words relative to each other. For example, some children know that correctly counting a collection of fish as “one, two, three, four, five, six” results in collection called “six fish” and that correctly counting a collection as “one, two, three, four, five, six, seven, eight, nine, ten” results in a collection called “ten fish,” but they do not know that the expression “ten fish” denotes more fish than the expression “six fish.” It is only when they can verbally estimate the numerical size of large sets without counting – a sign that they have mapped large number words to large analog magnitudes – that they can order expressions containing large number words relative to each other. Given that (1) analog magnitudes are representations of cardinalities, and that (2) knowledge of the order of linguistic expressions could be the antecedent of knowledge of the order of numerical symbols (e.g., 6<10), this suggests that knowledge of formal relations between numbers could be constructed out of representations of cardinalities.
In sum, it seems that Rips et al.'s solution to the challenge they have posed is not the right one, because their natural numbers are not the natural ones – they are not the ones that came to mind first to inventors of numerical notations, nor are they the ones that children acquire first. Of course, I have not answered Rips et al.'s challenge; we still do not know how children could induce an infinite system from a finite learning set. Yet, I suggest that the consistent appearance of symbols for cardinalities prior to more formal systems is not an accident, but rather, reveals how the former is a scaffold for the construction of the latter. I think simple learning considerations argue for the same point. Indeed, if quantificational uses of numbers were not relevant to the acquisition of the natural numbers, how would children ever learn that symbols like the number words or Arabic digits are symbols for the natural numbers?
