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This chapter examines Kant's much-criticized views on mathematics in general and arithmetic in particular. It makes a case for the claim that Kant's theory of arithmetic is not subject to the most familiar and forceful objection against it, namely, that his doctrine of the dependence of arithmetic on time is plainly false, or even worse, simply unintelligible. It is argued that Kant's doctrine about time and arithmetic is highly original, fully intelligible, and with qualifications due to the inherent limitations of his conceptions of arithmetic and logic, to an important extent defensible. The most philosophically striking thing about Kant's doctrine is the fact that arithmetic turns out to be a paradigm of the exact sciences (exacten Naturwissenschaften) only by virtue of its ultimately being one of the human or moral sciences (Geisteswissenschaften).

This chapter canvasses proposed connections between minds and numbers that might make knowledge of mathematics possible. The goal is to determine whether any promising leads are available in accounting for people's ability to represent math objects. It argues that we have certain primitive concepts (e.g., CAUSE in the case of physical objects, UNIQUENESS in the case of mathematical ones) and certain primitive operations (instantiation, recursion, and other procedures specialized for concept combination) that allow us to form schemas or theories for both physical and mathematical domains. We may then posit that the best of these theories are true—that they correctly describe the nature of our world—and that the objects they describe are elements of that world. Such a schema-based approach has advantages over most current theories of mathematical knowledge.

This chapter reviews the literature on infant quantification. This includes habituation studies that tested infants' discrimination of small sets, preferential looking studies that tested whether infants can detect relations between sets (e.g. equivalence), and other procedures that seemed to indicate infants performing simple calculations. For each body of evidence in support of early number concepts, evidence in support of alternative accounts is also provided. Taken together, it appears that the evidence for sensitivity to number in infancy is too weak and inconclusive to justify the strong claims based on it. In short, for every class of evidence of infant number concepts, there is a plausible counter-explanation which, in many cases, already has empirical support.

This chapter reviews the literature on early childhood number concepts, focusing on skills that do not require verbal counting, such as matching equivalent sets or ordering sets of objects, smallest to largest. Studies of young children indicate that, although basic number concepts emerge prior to formal schooling, these concepts develop gradually and with considerable limitations. For example, starting around the third birthday, children begin to match small sets of objects (e.g. two disks = two disks); however, it takes nearly two more years before children can match very disparate object sets (e.g. two flowers = two cats, but not three flowers). Verbal counting ability seems to be related to these later accomplishments. Similarly, three-year-olds can perform simple calculations using objects, but only for very small numerosities. There is a gradual progression in which children become capable of solving parallel problems with larger numerosities or multiple transformations (e.g. 3 + 2 - 1 = 4).

This chapter examines the evolution of number concept, via the ability to conceive of and use other representations of quantity. It approaches the evolution of number concept via the development of the concept in children. It finds that the child's acquisition of the concept leans heavily on the language scaffold of labelling. It considers the notion that the key in the child's construction of the number concept is the memorized set of words that constitutes the numeral list. This, in turn, raises the possibility that the presence of number concept might correlate with, and consequently be evidence for, the presence of language, provided that the presence of number in deep prehistory could be documented. It is possible that the evolutionary development of an integer concept may differ from its development in children. Hence, the chapter turns to the ethnographic and archaeological records for evidence about its evolutionary development.