Successor function

Abstract: Like most cognitive scientists, I take concepts to be mental symbols. Mental symbols are not all concepts, as there are also sensory representations, motor representations, and perceptual representations. From the perspective of cognitive science, a theory of concepts must specify their format, what computations they enter into, what determines their content, and how they differ from other types of mental symbols. In a recent book (Carey, in press) I present case studies of the acquisition of several important domains of conceptual representations, arguing that the details of the acquisition process adjudicate among rival theories of concepts within cognitive science. The case studies bear on the existence and nature of innate concepts, and on the existence and nature of discontinuities in development. Human conceptual development involves the construction of representational resources that go beyond those from which they are built in theoretically interesting ways. As Fodor (1975, 1980) has forcefully argued, characterizing these discontinuities and explaining how they are possible is a formidable challenge to cognitive science. Meeting this challenge informs our theories of the human conceptual system. Here I illustrate the lessons I draw from these case studies by touching on one of them: accounting for the origin of concepts of natural number. Explaining the human capacity for representing natural numbers has been a project in philosophy for centuries (e.g., Mill, 1874) and in psychology since its emergence as a scientific discipline during the past century (e.g., Piaget, 1952). As natural number is the backbone of all of arithmetic, an understanding how representations of natural number arise provides a good start on a theory of the human capacity for mathematics. Accounting for the origin of any conceptual requires specifying the innate building blocks from which the representations are built, and specifying the learning mechanisms that accomplish the feat. Two distinct research programs should be, but often are not, distinguished. In one, the logical program, the building blocks are conceived of as logically necessary prerequisites for the capacity in question. In the case of natural number representations these might include the capacity for carrying out recursive computations, the capacity to represent sets, and various logical capacities, such as those captured in second order predicate calculus. Sometimes arguments within the logical program seek to specify some necessary computational ability that other animals lack (e.g., see Hauser, Chomsky & Fitch’s, 2002, proposal that non-humans lack the capacity for recursion) that might explain why only humans have the target conceptual ability. A full account of the building blocks for some representational capacity within the logical program must include all of the necessary ones. A second research program, the ontogenetic program, conceives of the building blocks as specific representational systems out of which the target representational capacity is actually built in the course of ontogenesis or historical development. In the case of number representations these would be the innate representations with numerical content (if any). These two projects are interrelated, but clearly distinct. The first (characterizing the logical prerequisites for natural number) leads to analyses like those that attempt to derive the Peano-Dedekind axioms from Zermelo-Fraenkel set theory or Frege’s proof that attempts to derive these axioms from second order logic and the principle that if two sets can be put in 1-1 correspondence they have the same cardinal value. Such analyses seek to uncover the structure of the concept of natural number, and certainly involve representational capacities drawn upon in mature mathematical thought, but nobody would suppose that in ontogenesis or historical development people construct the concept of natural number by recapitulating such proofs (see Feferman, this volume). The ontogenetic project (characterizing the actual representational systems from which natural number is built) requires empirical studies of infants, non-human animals, young children, and historical records, discovering systems of symbols (both mental and public) that are actually created and used in thought, specifying their format and the computations they support. If such empirically attested representational systems do not have the power to represent the target concept (in this case natural number), the project then becomes one of characterizing successive representational systems that are constructed in the process of arriving at the target, and characterizing the learning mechanisms involved in the construction of each. The ontogenetic project does not reduce to the logical one. There is no presumption that the concept of natural number can be defined in terms of the earliest representations with numerical content alone. On the other hand, the ontogenetic project depends upon the logical one for a characterization of the target concept at issue (e.g., what I mean here by the concept of natural number is characterized by the Peano-Dedekind axioms), as well as for a characterization of the logical resources drawn upon in the construction process. Many recent papers illustrate the failure to distinguish the two projects. To take just one example at random: Leslie, Gelman & Gallistel (2008) argue that natural number is built from an innate system of representations that provides symbols that approximate the cardinal values of sets (analog magnitude representations, see below), plus an innate representation with the content “one,” and an innate capacity to represent the successor function. The first of these, analog magnitude number representations, is a well-studied and well-characterized system of representation found throughout the animal kingdom, as well as in human infants, children and adults. The second, an innate symbol for “one,” is a posited innate representation for which these authors offer no evidence. The third is a logical prerequisite for a concept of natural number; clearly we must have the capacity to implement the successor function somehow if we are to represent natural number. There may be representations with the content “one” early in development, and as well as systems of representation that implement the successor function, but in the ontogenetic research tradition, one must actually characterize them (format, computational role), and provide evidence for their existence. Thus, one might (and probably would) agree that representations of “one” and the successor function are logical prerequisites for representations of natural number, and in the logical tradition one could explore how representations of number could be built from such primitives (a fairly easy task, obviously). But the ontogenetic program requires providing evidence for such putative developmental primitives. Here I report on the current state of the art in accounting for the acquisition of concepts of natural number from the ontogenetic research tradition. I argue for four theses: 1) There are three distinct systems of innate representations with numerical content. 2) None contains any symbols for natural numbers. 3) The ontogenetically earliest representational systems that includes symbols for even a subset of the natural numbers is the count list, when deployed in a way that satisfies the “counting principles” described by Gelman and Gallistel (1978). 3) The learning mechanisms that accomplish the construction of the numeral list representation of natural number include, but are not exhausted by, a form of bootstrapping described by Quine (1960, 1969, 1974), among others, called here “Quinian bootstrapping.”

Abstract: Humans possess two nonverbal systems capable of representing numbers, both limited in their representational power: the first one represents numbers in an approximate fashion, and the second one co...