Rips et al. are correct: “Children's simple counting and enumerating does not provide rich enough constraints to formulate the right hypothesis about the natural numbers” (sect. 5.1, para. 2). However, their presentation of my position (e.g., Gallistel & Gelman Reference Gallistel, Gelman, Holyoak and Morrison2005; Gelman Reference Gelman and Medin1993; Reference Gelman2006; Gelman & Gallistel Reference Gelman and Gallistel1978; Leslie et al. Reference Leslie, Gelman and Gallistel2008) leaves out a critical feature of it – that I steadfastly have maintained that the meaning and function of counting are subservient to the arithmetic principles of ordering, and addition and subtraction.

The proposal that nonverbal counting principles facilitate learning of verbal counting does not imply that there is an immediate mapping from the nonverbal conceptual system to the verbal one. The presence of any relevant mental structure facilitates attention to, and learning about, data that share the same structure. The exact same counting principles, and their performance constraints, are shared by both the nonverbal and verbal systems. Therefore, learners have a way to collect structure-relevant data, including the fact that number words are recognized as being in their own class (Wynn Reference Wynn1992b). Similarly, beginning language learners can make sense of the fact that although the partitive follows a cardinal word, as well as quantity terms, this is but a statistical fact. The partitive occurs with other part–whole constructions such as side of, mother of, color of, and so on.

If losing one's count were the same as not being able to do addition, the Red Queen would be right. But, we know that she is not. To develop my reply to Rips et al., I start with evidence that young children can relate their use of natural numbers to arithmetic. Second, I take up the ubiquity of erroneous counts and ask: Do these reflect limits on competence alone, or are their other sources of systematic variability? Finally, I return to the question of from where the understanding of verbal counting principles and arithmetic comes.

Bullock and Gelman (Reference Bullock and Gelman1977) showed that even two-and-a-half year-old children could understand that the property of being numerically more or numerically less, defined which of two displays was the “winner.” In the transfer condition, when children first encountered the novel set sizes of 3 and 4, they responded on the basis of ordering. Further, more than 60% of the children counted and used the related cardinal values (Gelman Reference Gelman and Medin1993). This is evidence that very young children, who are still very poor counters, already can map common numerical relations to each other. Zur and Gelman (Reference Zur and Gelman2004) reported that 3- and 4-year-olds counted to check predictions about the effect of adding and subtracting items. No child ever tried to make their count be consistent with their predictions. The 4-year-olds did problems that contained values as large as 15; the maximum N for the younger group was 5. Both age groups were sensitive to the inverse relation (e.g., 5−1 followed by 5+1, or 12+3 followed by 15−3), even though these problems were not presented one after the other. Zur (Reference Zur2004) has also shown that when 4- and 5-year-olds encounter a second pair of commuted problems – that is, one followed by another, different problem that shares the property of commutativity – they are faster and more accurate than they are with pairs of problems that do not share this property. Further, this transfers to a target analogy problem.

In short, I agree with Rips et al. that the principles of arithmetic cannot be induced from the rote learning of counting or from the simple knowledge of the referents of the first few count words. It is rather the reverse: Children understand what counting is about because they can make at least implicit use of the principles of arithmetic reasoning. I think the authors seriously misestimate the age at which children have some non-trivial understanding of cardinality and the successor principle.

Sure, children take a long time to fully grasp the rules for framing the next count word in English – a non-trivial number of eighth graders have not fully mastered these skills (Harnett & Gelman Reference Hartnett and Gelman1998). When anyone replies that the next number after one quadrillion is two quadrillion, it is unlikely that they think that adding one to an unimaginably large number doubles its value. Because they do not know how to generate the word for the next number, and because, like adults, they can fail to distinguish between the word and the concept itself, they can appear to not believe that continued addition always generates a still-larger number. The large majority of Hartnett's children who were scored as “Ambiguous” regarding their understanding of the successor principle, were either concerned about the fact that they did not know the word which would result, or they would run into physical limitations. They gave answers like, “Well, the number would be make-pretend,” or “if you make up numbers, then it's alright,” or “I don't really know, because I've never really counted to where it stops.”

Rips et al. highlight the extent to which the road to counting mastery is strewn with numerous mistakes. I have shown that errors are influenced by variations in set size, density, rate of counting, time of presentation, and “touchability” of items – all variations that influence one's ability to keep entities in a display separate from each other, and/or to keep track of the counted from the to-be-counted items. So, too, does the requirement that the one-one and stable ordering outputs stay lock-stepped together. Most results bearing on these variables are reviewed in Gelman and Gallistel (Reference Gelman and Gallistel1978) and Cordes and Gelman (Reference Cordes, Gelman and Campbell2005).

No matter what, children do have to tackle the task of memorizing a sequence of terms wherein there is nothing about a given entry in the list that systematically predicts what the next one after it must be, and the next, and so on, for a very long time. Humans, unlike computers, are not good at memorizing long lists of arbitrary sounds that must be repeated in the exact same order, trial after trial. Young children also have to become fluent counters, decode task variables, and so on (Gelman & Greeno Reference Gelman, Greeno and Resnick1989). But at least they will be on a relevant learning path and be able to disambiguate other relevant data – for example, the fact that count words are in a separate class (Wynn Reference Wynn1992b) – and could use the distributional fact that number words can be followed by the partitive “of,” despite its low frequency of usage by adults (Bloom Reference Bloom2000; Bloom & Wynn Reference Bloom and Wynn1998) and its legitimate usage after many quantifiers or phrases, such as “a side of,” “the color of,” and so on. Therefore, when principles of arithmetic organize the domain, we can say that the counting principles serve to facilitate learning the verbal version of these arithmetic principles. Subsequently, more and more formal understandings of cardinality will develop, but this does not preclude an early level of understanding.