This commentary pertains to section 3 of the target article, where Rips et al. criticize the induction model. I argue that Rips et al.'s critique fails in two ways. First, it overstates what the induction is supposed to explain. The induction is about the acquisition of natural-number concepts, not the concept NATURAL NUMBER itself. Second, Rips et al. fail to grasp the magnitude of children's achievement in learning the cardinal meanings of a finite set of number words (e.g., the words “one” through “nine”).

Forget NATURAL NUMBER, the induction is about SEVEN

The first issue to be addressed is one of terminology. There is a difference between natural-number concepts (e.g., SEVEN) and the concept NATURAL NUMBER itself. SEVEN is a mental symbol for a specific cardinality; NATURAL NUMBER is part of meta-numerical knowledge – knowledge about numbers.

When Rips et al. use the term “natural-number concepts” interchangeably with the terms “the natural-number concept” and “the concept NATURAL NUMBER,” they blur the distinction between number concepts and meta-numerical knowledge. Similarly, when they argue that the induction model is wrong because it does not explain how children learn that numbers go on forever, or that numbers can be added in any order, what they are really saying is that the induction model does not explain the acquisition of meta-numerical knowledge.

However, the induction model was never meant to explain the acquisition of meta-numerical knowledge. The induction model explains how children acquire natural-number concepts: mental symbols for cardinalities such as 7. Specifically, we are interested in cardinalities that are (a) bigger than 4 (and thus too big for each individual to be represented in parallel), and (b) exact (and thus too fine-grained to be represented by the analog magnitude number system).

The successor rule – no more, no less

Rips et al.'s conflation of numbers with meta-numerical concepts leads them to misunderstand the claims of the induction model. For example, Rips et al. read s(n)=n+1 as a function over the natural numbers, and argue that a child who has the concept s(n)=n+1 must ipso facto have the concept NATURAL NUMBER. Partly this is a problem with notation: We (induction theorists) have used s(n)=n+1 as shorthand for something much wordier. The following is what I think the child really learns.

Precursors: The child first learns the words for those cardinalities that are small enough to be represented directly, as sets of individuals. (These are learned one at a time, in order, over a period of 1–2 years.).

Induction: Then, all at once, the child learns how to assign cardinal meanings to the rest of the number words. To continue with Rips et al.'s example, a child who knows the list up to “nine” induces the following:

The child does not need the concept NATURAL NUMBER in order to make this induction, and the successor rule itself is represented only implicitly. The key fact, however, is that the child learns the higher number-word meanings all at once. That's how we know the child is applying a rule.

Rips et al. really seem to miss the import of what the child achieves when he or she learns this rule, even for a finite set of number words. A child who can represent the meaning of “seven” via this rule has acquired a way of mentally representing an exact, large quantity. This is information that simply was not representable until a set of symbols (the number words) were borrowed from outside. This is the really amazing thing about number-concept development, and it happens very early on – when the child knows only a few number words.

Even “one”-knowers won't accept a modular system

Finally, why do Rips et al. believe that a child who has learned the cardinal meanings of “one” through “nine” doesn't have natural-number concepts? One big reason seems to be that (according to Rips et al.) this child's knowledge state is equally compatible with either the natural-number system, or an alternative system such as Successor-Mod-10. A closer look at children's behavior, however, shows us that this is not true.

The original basis for the induction model was the finding that, from the time children learn what “one” means, they treat the number words as unique labels for specific cardinalities (Wynn Reference Wynn1992b). This is the very definition of a “one”-knower – it is a child who, when asked for “one” object, always gives only 1 and never more than that. Similarly, when a “one”-knower is asked to label a set of 1, he or she always says “one” and never other number words.

These facts mean that modular systems are off the table as a possibility by the time the child learns “one.” In order for Mod₋₁₀ or any other modular system to work, children would have to accept either multiple number words for the same cardinality, or multiple cardinalities for the same number word. In fact, however, even “one”-knowers do not accept the word “eleven” for a set of 1, or the word “one” for a set of 11.

This basic finding has been replicated many times over, not only for “one”-knowers, but also for “two”-knowers, “three”-knowers, and “four”-knowers as well (e.g., Condry & Spelke Reference Condry and Spelke2008; Le Corre & Carey Reference Le Corre and Carey2007; Le Corre et al. Reference Le Corre, Van de Walle, Brannon and Carey2006; Sarnecka & Carey, in press; Sarnecka & Gelman Reference Sarnecka and Gelman2004; Sarnecka et al. Reference Sarnecka, Kamenskaya, Yamana, Ogura and Yudovina2007). Moreover, one study (Sarnecka & Gelman Reference Sarnecka and Gelman2004) found that children apply this constraint even to number words whose meanings they do not yet know: “Two”- and “three”-knowers treat “five” and “six” as mutually exclusive cardinalities, even though they cannot identify the actual cardinalities 5 or 6.

Thus, Rips et al.'s critique does not hold up well to scrutiny. Where it is true (e.g., in pointing out that the induction does not yield the concept NATURAL NUMBER), it is irrelevant. And where it is relevant (e.g., in the claim that children's early number knowledge is consistent with modular systems), it is false.