Rips et al. are right about the crevasse, with psychological research on children's quantitative representations is on one side and principles of mathematics or the other. But if children's quantitative representations are mappings from number-words to numbers and their successors (“one” represents one, “two” represents two), this could lead to any of the infinite number sequences in mathematics: for example, mod983. Rips et al.'s negative argument is that “children cannot bootstrap their way from these beginnings to true math concepts” (sect. 5, para. 1), where true math includes the concept of natural number. On the other side of the crevasse, principles in the philosophy of mathematics are available. But so is the stark warning – a child sitting by a heap of peas, picking them up one by one uttering a number-word, will have access to “no bridge which leads across from the kindergarten numbers” (Frege Reference Frege1979, p. 276). Rips et al.'s positive argument is that the sole way to bridge the crevasse is by means of principles.
I applaud Rips et al.'s main argument, which I support through (1) a complementary account of the nature of mathematical principles. But I also have a couple of reservations. Their bridge specifically includes principles of mathematical induction (MI) with reference to my Piaget-inspired study (Smith Reference Smith2002). My reservations are about (2) the authors' interpretation of MI, and (3) reasons in rule learning.
1. Support for the argument
Regarding the authors' discussion of math principles, first, the following characteristics are worth bearing in mind:
a. Mathematical principles are normative: “Mathematics forms a network of norms [in that] if calculation reveals a causal connexion to you, then you are not calculating” (Wittgenstein Reference Wittgenstein1978, pp. 431, 425; cf. Piaget Reference Piaget2006, p. 8). Note that this distinction between causality and normativity is exclusive.
b. Mathematical principles are modal (i.e., necessary truths): Their instantiation in experience provides occasions for their recognition. Their correct understanding depends on normative demonstration, not merely verification (Leibniz Reference Leibniz1996, p. 85; Piaget Reference Piaget2006, p. 7).
c. Mathematical principles are constitutive: Any principle has a two-part formation covering origins and constitution. Both are required because pseudo-rationality has an empirical origin without a normatively valid constitution (Kant Reference Kant1787/1933, p. B116; Piaget Reference Piaget2006, p. 9). For Kant, mathematical principles are not innate (see Kant Reference Kant1790/2002, sects. 8–221). Instead, they are constructed, making their debut in experience with their derivation being synthetic a priori (Kant 1787/1933, pp. B1, B15, B747).
This trinity supports Rips et al.'s critique of research in the psychology of mathematics. Contra (a), that mathematical principles are normative, the track-record in this research shows multiple traces of norm-denial and norm-reduction to causes, but almost nothing of norm-laden minds (Smith 2006). Contra (b), that mathematical principles are modal, the focus of this research is on truth-functionality to the exclusion of modal necessitation (Smith Reference Smith2002). Contra (c), that mathematical principles are constitutive, it deals with origins to the exclusion of constitution (Smith, in press a). That is why principles are required to bridge the crevasse from the other side.
2. Interpreting mathematical induction (MI)
Rips et al.'s positive recommendations specifically refer to MI principles, in relation to which they note my (Smith Reference Smith2002) study in section 5.3.3. However, their discussion is open to challenge in two respects:
First, Rips et al.'s definition of MI is a modernized restatement of Poincaré's definition. The open question is its interpretation. The authors interpret MI through logicism, in that MI is really logical deduction (their argument is in Rips & Asmuth [Reference Rips, Asmuth, Feeney and Heit2007]). Yet Poincaré's contrary interpretation was for constructivism (Poincaré Reference Poincaré1905, p. 15; 1952, p. 52), under which MI is really synthetic a priori in its generation of novelty (Poincaré Reference Poincaré1952, pp. 50, 162, 194). Under logicism, logical deduction and empirical induction are necessitating and universalizing respectively; for Poincaré (Reference Poincaré1905, pp. 13–14), MI combines both in a unique form of reasoning. Crucially, Piaget (Reference Piaget1942; see Smith Reference Smith2002, Ch. 3.1) was critical of logicism, and his constructivism aimed to recast Poincaré's position, notably in Inhelder and Piaget's (Reference Inhelder, Piaget, Gréco, Inhelder, Matalon and Piaget1963) seminal study, otherwise neglected until my replication. That is why I (Smith Reference Smith2002, p. 5) quoted Poincaré's definition with some sympathy for his interpretation (cf. Smith Reference Smith1999). Rips et al. state that this definition conflates universal generalization [UG] and MI. I doubt this, but reckon they should take this matter up with Poincaré and Piaget.
Next, my (2002) study was about the formation of MI in children aged 5–7 years who were investigated twice. My operationalization required serial, equal additions to two containers – in Study I, both were initially empty; in Study II, their initial content was unequal. The questions were threefold: the base equality/inequality through serial additions, universality about number, and necessity about number (Smith Reference Smith2002, p. 58). This operationalization matches Poincaré's and Rips et al.'s criteria for MI. My evidence (significant in non-parametric and logistic analyses) showed these children to be successful with the first two questions, and promising as to modality (pp. 64–66, 147). Rips et al. challenge this in view of the purported conflation of MI and UG. I disagree in seeing an early formation of MI in 5–7-year-olds. (More on this, below, with regard to the second reservation.)
3. Reasons and rule-learning
Principles are rules about which four assumptions are commonly made in psychology: (i) mathematical rules are clear and exact, (ii) their applications are well-defined, (iii) their learning is typically through social experiences that (iv) are causal mechanisms responsible for rule-learning. Rips et al. seem to be committed to this quartet, but all four are contradicted in Wittgenstein's (1978) analysis of the rule-following paradox. A companion argument from Piaget's developmental epistemology leads to the same conclusion: rule-meaning is dependent on rule-use, not the other way round (Smith, in press b). Further, rule-use is regulated by reasons whose role is “to introduce new necessities into systems where they were merely implicit or remained unacknowledged” (Piaget Reference Piaget2006, p. 8). Reasons have origins inclusive of pseudo-necessities; they may also amount to valid necessitations.
Either way, reasons are investigable as normative facts (Smith, in press a). Returning to Rips et al.'s strictures and using my (2002, Ch. 6.3.3) study, examples include:
pseudo-necessities: “more there because you have not put the right number in”; “because it wasn't fair at the start”
recursion over number: “because if there was the same in there before and you just add another one in each box, there would be the same again”
necessitation about number: “because that's the way it has to be. From the beginning I started adding more, so there must be more.”
MI starts somewhere. Well, here is a primitive version in children aged 5–7 years. Isn't rule-learning a fallible process of construction for us all!
In conclusion, I think Rips et al. will welcome my reiteration of the normative, modal, and constitutive nature of mathematical principles as complementary to their critique of math psychology, and on reflection see promise in my two reservations as contributions to Piaget's developmental epistemology directed on an empirical account of normativity.
