Natural numbers serve both as finite cardinal numbers (cardinals) and finite ordinal numbers, and belong to (or embed into) other number systems. However, following Rips et al., I will consider the natural numbers only as finite cardinals.

Each natural number is associated with several concepts. There are the concepts expressed in English by the numeral determiners (e.g., “at least three,” “at most three,” and “exactly three”); call these finite-cardinality quantifier (hereafter FCQ) concepts, as their semantic values are cardinality quantifiers. There are concepts expressed by set-theoretic predicates (e.g., “three-membered”); their semantic values are properties of sets. The syntax and role in reasoning of the nouns such as “3” and “three” at least suggest that they are singular terms. If so, they express concepts such as that of the number three “itself”; I'll call these numerical-individual concepts (hereafter NI). At many places in this article, I was unsure of which of these concepts the authors were considering. Finally, metalinguistic concepts (e.g., of Arabic numerals like “3”) deserve mention.

Three further comments are pertinent in relation to this:

1. 1. In various contexts, we use phrases of the form “three Ns” to mean “exactly three Ns,” or “at least three Ns,” or even “at most three Ns.” Such default uses might create the impression that there is a “root” concept of three-ness from which we form the three FCQ concepts by some sort of supplementation. Not so! Just consider the definitions of these quantifiers in terms of the apparatus of standard first-order logic (presented in many places; e.g., Barwise & Etchemendy Reference Barwise and Etchemendy1999, pp. 366–68).

2. 2. A quotation from Bloom (in the target article's sect. 3.2.2, para. 5, referring to Bloom Reference Bloom2000, p. 215) seems to ascribe FCQ concepts to pre-linguistic infants. This would be astonishing: having FCQ concepts requires having concepts of existence, universality, identity, and some connectives, as well as being disposed to find certain inferences involving these notions compelling (see Hodes Reference Hodes2004; Peacocke Reference Peacocke1992, Ch. 1).

3. 3. Having the concept of being three-membered requires possession of the concepts of being a set and of set-membership. The latter concepts form a “local holism” (see Peacocke Reference Peacocke1992) that entered mathematics only in the late nineteenth century. These concepts are vastly more sophisticated than FCQ concepts. (They constitute a device to bring plurality within the purview of “singularist” logic [see McKay Reference McKay2006], allowing us to replace plurals – for example, “The Three Tenors” – with a singular expression – for example, “the set whose members are exactly Domingo, Pavarotti, and Carreras.”) Perhaps this emergence of set-theoretic concepts built upon some sort of “implicit conception” (see Peacocke Reference Peacocke1996; Reference Peacocke, Jackson and Smith2005) of set-hood with older rootsFootnote ¹; but such a conception would also be sophisticated. In places (e.g., sect. 2.2, para. 4), the authors seem to suggest that reputable psychologists believe children to possess these concepts (or their precursor conceptions)! I recommend that psychologists concern themselves with the concept of being three-membered only when investigating the psychology of professional mathematicians.

Perhaps young children are in a psychological state that makes the model presented in Figure 1 applicable to them. But Rips et al. seem to presuppose that this state involves concept-possession.Footnote ² This presupposition is implausible. Having concepts requires the ability to engage in conceptual thought. Very young children might be able to discriminate between situations in which there are three toys in a box and those in which there are two toys in a box; but such ability is not by itself evidence of possession of FCQ concepts. It might not even be evidence of possession of the concept expressed by the binary determiner “more/than” (as in “More sheep are in the meadow than cows are in the barn”).

In section 3.1, Rips et al. attribute to “many theorists” these theses: (i) “[O]nce children have learned language, or, at least, language-based counting, they are in a position to attain true concepts of natural numbers”; (ii) “[W]hen they are able to perform tasks such as enumerating the items in an array or carrying out simple commands (e.g., ‘Give me six balloons’),” they have acquired these concepts (sect. 3.1, para. 4). By being “in a position to attain” a concept, the authors seem to mean being exactly one cognitive step away from possessing that concept. By “true concepts of natural numbers,” they might mean the FCQ concepts, but more probably they mean the NI concepts, associated with at least some natural numbers (maybe extending as far as the person in question can easily count).

Possession of the NI concept associated with a number probably requires prior possession of the corresponding FCQ concepts. Surely the latter requires the ability to enumerate at least certain objects. Enumeration is already at least one cognitive step beyond language-based counting itself, and beyond that lies the ability to deliver the “upshot” of enumeration. (When my daughter was acquiring FCQ concepts, or at least the ability to express them, I'd give her three toys and ask, “How many?” She'd enumerate them: “One, two, three,” and then in response to my question, say “Four”!) Does this “upshot” ability indicate possession of the “exactly” FCQ concepts? The authors, in rejecting the aforementioned thesis (ii), might be answering “No,” and “No” seems plausible to me. Even if they would answer “Yes,” I think that they would agree that having lots of FCQ concepts leaves their possessors some distance from having the corresponding NI concepts. I doubt that many American middle-school algebra students possess NI concepts, even for natural numbers less than 10. Any experiments demonstrating possession of NI concepts would have to distinguish such possession from possession of corresponding metalinguistic concepts; for example, of Arabic numerals.

Whether theses (i) and (ii) refer to the FCQ or the NI concepts, Rips et al. show good sense in rejecting them. They correctly point out (in sect. 3.2.2) that a child can “arrive at something like” the generalization to which they refer as Principle (3) only after mastering advanced counting. They then address this thesis: (iii) Children simultaneously learn to do advanced counting and to correlate the numerals to which they can count with the FCQ concepts associated with those numerals. They say that it is unclear how thesis (iii) could be true (in sect. 3.2.2, para. 10), and go on to argue against it. Their argument seems beside the point. As I understood thesis (iii), the advanced counting in question is in a numeral system for the natural numbers, not one for arithmetic mod₁₀. If so, how things go for a child who learns to count cyclically (“Zero, one, …, nine, zero, one, …”) is irrelevant; it certainly is not a way of counting on which Principle (3) would be true.