In the early days of developmental psychology, psychologists were in close contact with logicians and mathematicians working on the foundations of mathematics. Karl and Charlotte Bühler participated in the discussions of the Vienna Circle, which included among its members, Rudolf Carnap, Kurt Gödel, and Karl Menger. Piaget's work was heavily influenced by Bourbaki, the French structuralist school, and Piaget had many discussions with Jean Dieudonné, one of its leading members (see Piaget Reference Piaget1968; cf. Aczel Reference Aczel2007). In recent decades, developmental psychology and foundational work in mathematics have grown apart, sometimes resulting in outright antagonism (Dehaene Reference Dehaene1997). Rips et al. restore the connection between two research fields. In their top-down approach, the cognitive math principles are based on the Dedekind-Peano axioms. They argue that these axioms or mathematically equivalent minor variations (e.g., the Least Number Principle) should be studied more seriously by psychologists. My major worry is that they have missed an important new development in foundational studies in mathematics.
Rips et al. briefly consider Frege's conception of numbers as sets of all equinumerous sets of objects, and rightly conclude that this view is untenable (Frege Reference Frege1884/1974; Reference Frege1893/1967). Frege's system was flawed, because of the inconsistency of the notorious Law V. However, Crispin Wright (Reference Wright1983) pointed out that Frege's arithmetic can be derived from Hume's principle (HP). Frege (Reference Frege1893/1967) contains all the essential inferences for a valid deduction of the laws of arithmetic, and Frege's inconsistent Law V can thus be sidestepped. The second-order Peano postulates can be derived in a consistent second-order system with only one extra-logical predicate N (“is the number of”) and one non-logical axiom HP, stating that the number of Fs is equal to the number of Gs, in case there is a one-to-one correspondence between the Fs and the Gs. This result is known as Frege's theorem, and it implies that HP can serve as the basis of arithmetic (Heck Reference Heck1993; Zalta Reference Zalta and Zalta2008).
From a mathematical point of view, second-order Peano arithmetic (PA) and Frege arithmetic (FA) (second-order logic+HP) are almost equivalent (for their relative strength, see, e.g., Heck Reference Heck2000). From an epistemological or cognitive point of view, PA and FA are very different. The basic concept in PA is the successor relation. PA is thus strongly related to the ordinal conception of number or to the cognitive abilities of enumeration and counting. FA, on the other hand, is based on equinumerosity, or one-to-one correspondence. The basic cognitive ability underlying HP is the ability to judge whether the objects of two sets can be put into a one-to-one relation, or, the ability to relate every object of a set to a single object of another set. For example, a set of knives is equinumerous to a set of forks if one can form pairs of forks and knives, without remaining forks or knives. Equinumerosity judgments are thus possible without enumerating the two collections. In the philosophy of mathematics, this discovery has triggered a lively epistemological debate on the question whether HP is an a priori or conceptual truth (Boolos Reference Boolos and Heck1997; Demopoulos Reference Demopoulos1998; Heck Reference Heck2000). Neo-Fregeans claim that HP is central in mathematical knowledge.
The problem with the neo-Fregean program is that its claims about mathematical knowledge are not based on psychological evidence. Nevertheless, HP and Frege's theorem may be quite important in the empirical study of mathematical cognition. Since the 1970s, psychologists have regarded counting as the basis of numerical skills. Gelman and colleagues have argued that three principles underlie the ability to count: the one-to-one correspondence principle, the stable-order principle, and the cardinality principle (Gelman & Gallistel Reference Gelman and Gallistel1978; Gelman & Greeno Reference Gelman, Greeno and Resnick1989; Gelman & Meck Reference Gelman and Meck1983; Gelman et al. Reference Gelman, Meck and Merkin1986). These principles constitute an enumeration procedure and suffice to explain various numerical skills. However, the fact that these studies are largely based on the numerical skills of infants that have been taught to count at a very early age may have biased this research. It is perfectly possible that the ability to make one-to-one correspondence judgments (HP) is more basic and relevant than has generally been assumed. Two (arbitrarily chosen) examples should suffice to illustrate that HP is often overlooked as an explanation for certain numerical abilities.
Gordon (Reference Gordon2004) carried out several matching tasks during his stay with the Pirahã. Gordon would put a certain number of objects (1–10) below a line, and the participant had to put an equal amount of objects on the other side of the line. In general, there is a considerable decrease in performance for larger numbers, with one striking exception, namely, the line match. This would fit well with the participants having mastered a one-to-one correspondence procedure. As Gordon tampered with the matching condition, performance decreased. Gordon's results clearly indicate that non-numerates can employ a one-to-one correspondence procedure, without using enumeration (despite his claim to the contrary; Gordon Reference Gordon2004, p. 497). The uneven line match is especially noteworthy. If rectangles below the line were put in a line with unequal distances, performance was very good until four, and then dropped below 50% for five and six, and later went up again to almost correct performance for higher numbers. An explanation might be that for small numbers, the Pirahã use the more or less precise subitizing, with the effect that performance decreases rapidly above three. For larger sets, a one-to-one correspondence procedure takes over, leading to almost perfect performance. This is a hypothesis that can rather easily be tested, because one would assume that participants can be taught to use this strategy also for smaller numbers, with a resulting overall excellent performance.
Second, Jordan and Brannon (Reference Jordan and Brannon2006) have demonstrated that 7-month-old infants can already recognize cross-modal one-to-one correspondences for low numerosities. Although Brannon and Jordan explain this result as evidence for a number representational system, it can more easily be interpreted as the mastery of HP.
In conclusion, the claim made by Rips et al. that math schemas, roughly based on the Dedekind-Peano axioms, may be important in the study of (mature) mathematical cognition deserves further empirical scrutiny. However, there is a non-trivial, almost equivalent axiomatic approach that is arguably very different from a cognitive point of view. In a top-down approach towards mathematical cognition focusing on knowledge of mathematical principles, it seems more promising to start with a psychological study of one-to-one correspondence (FA or Heck's roughly equivalent system; Heck Reference Heck2000), than with Peano's mathematical induction or commutativity.
