Rips et al. rely on a notion of “deriving” concepts, as when they say that, “understanding the natural number concept may allow us to avoid trying to derive it from unwieldy raw material from which no such derivation is possible” (target article, sect. 6, para. 3). But, prima facie, derivation requires both (1) starting points, and (2) something like a logic that permits introduction of defined notions. The possibilities for licensing particular derivations may be limited by either (1) or (2). It is a familiar point that a chain of reasoning not licensed by one system (e.g., a propositional calculus) may be licensed by another (e.g., a predicate calculus). Analogous points are relevant when reflecting on the capacity of human learners to “derive” the concept of natural number from the starting points suggested by developmental psychologists (object tracking, approximate number representations, etc.). Rips et al. suggest, in effect, that these “premises” are inadequate. What concerns us is the need of a plausible description of the relevant “background logic” that is somehow implemented by human psychology.

Our point is not merely that Rips et al. do not provide such a description, but that (so far as we know) nobody knows how to characterize the logical space of cognitive change – be it human concept acquisition or animal learning. In one sense, delineating this space is a major focus of ongoing work throughout cognitive science (e.g., debates concerning eliminative connectionism). However, not even optimists can think we have a remotely adequate description of the mental logic governing potential derivations of new concepts from available cognitive resources. Of course, in the absence of an understood mental logic, it is especially hard to know whether the resources currently posited by developmental psychologists are adequate starting points for human learners.

At the beginning of their “Concluding comments” (sect. 6), Rips et al say that “Thanks to analytic work by Dedekind (Reference Dedekind1888/1963), Frege (Reference Frege1884/1974), and others, we have a firm idea about the constituents of the natural number concept.” The target article ends with the suggestion referred to earlier, that “understanding the natural number concept may allow us to avoid trying to derive it from unwieldy raw material from which no such derivation is possible.” These statements may be too bold (current work in philosophical logic suggests disagreement), and a remark about the analytic work may be informative.

Given a consistent fragment of Frege's second-order logic, the Dedekind axioms for arithmetic follow from (i) the general principle that (the extensions of) two concepts are equinumerous if and only if they correspond one-to-one, and (ii) suitable definitions, in notation made available by the logic, of notions such as “zero” and “precedes.” In this restricted sense, arithmetic follows from a single non-logical principle concerning an equivalence of equinumerosity and one-to-one correspondence (see Demopoulos Reference Demopoulos1994; Zalta Reference Zalta2003). Trivially, however, there can be no derivation of Dedekind's axioms without a proof system and a way of encoding the requisite definitions. Correspondingly, much of Frege's accomplishment lies with his invention of a logic in which proofs by mathematical induction could be conducted, and the axioms could be explicitly represented. The background logic is needed to define/abstract Frege's official concept of natural number. And this is no surprise: Definition requires a language; abstraction requires apparatus.

In terms of cognition, there is an analogous point. It is not yet useful to be told that certain “raw materials” are not themselves “rich enough” to support abstraction of numerical concepts. No starting points are rich enough without the logical tools that enable abstraction. The interesting claim in this vicinity is presumably that the “raw materials” discussed in the article are not rich enough, even given plausible assumptions about the cognitive apparatus independently available for purposes of abstraction. But defending this claim would require an argument – not provided by Rips et al. – concerning the relevant cognitive apparatus, about which very little is known. Our claim is not that psychologists should posit unconscious knowledge of points (i) and (ii) referred to earlier and of Frege's logic; we have no firm views on this score. But because understanding a natural language plausibly requires derivational capacities of the kind reflected in Frege's logic, it seems rash to assume that humans do not have capacities that would let us derive natural arithmetic from analogs of (i) and (ii).

As Frege also stressed, although derivations can reveal logical relations among concepts, this does not yet tell us anything about the actual constituents (if any) of the relevant concepts. However, one can speculate that human learners have the representational powers required for reasoning along Fregean lines. This might involve the abilities to: determine equinumerosity via computing one-to-one correspondence (as Rips et al. note, infants do); represent an initial number, which need not be zero, in terms of prior notions (perhaps a singular/plural distinction); represent a relation of “precedence” that has a definable transitive closure; and manipulate these representations in ways corresponding to a consistent but suitably powerful fragment of Frege's (second-order) logic. In this way, one might abstract a Fregean notion of natural number and go on to derive natural analogs of the Dedekind/Peano axioms. We do not suggest that this must be the way human children acquire their concept of natural number. But such scenarios do not strike us as impossible or even especially implausible, pace Rips et al., given the alternatives.

To repeat, our point is neither that Rips et al. have it wrong, nor that the developmental psychologists have it right. But we want to caution against concluding, in the absence of empirically motivated assumptions about the abstractive/logical capacities of children, that the “raw materials” currently posited by the psychologists are inadequate starting points. One can agree that derivations are not yet to be had, while welcoming new premises.