Ontogenetic Origins of Human Integer Representations. Issue 10 (October 2019) - Carey, Susan; Barner, David
Abstract : Do children learn number words by associating them with perceptual magnitudes? Recent studies argue that approximate numerical magnitudes play a foundational role in the development of integer concepts. Against this, we argue that approximate number representations fail both empirically and in principle to provide the content required of integer concepts. Instead, we suggest that children's understanding of integer concepts proceeds in two phases. In the first phase, children learn small exact number word meanings by associating words with small sets. In the second phase, children learn the meanings of larger number words by mastering the logic of exact counting algorithms, which implement the successor function and Hume's principle (that one-to-one correspondence guarantees exact equality). In neither phase do approximate number representations play a foundational role. Highlights: A widely accepted hypothesis has emerged in cognitive science that an analog number system supplies the cognitive foundations of human arithmetic ability. We provide evidence from experimental psychology, human history, and the philosophy of mathematics that is difficult to explain on this hypothesis. We describe evidence that acquiring integer concepts depends first on capacity-limit small set representations (for numbers up to 3–4). Later learning depends crucially on mastery of exact counting algorithms, and on how these algorithms implement logical notions that are defined over words, individuals, and relations between them, but not over analog number representations. Learning to represent integers involves learning how the ordinal structure of counting encodes the successor function, and how counting implements a sequential procedure for evaluating one-to-one correspondence and the equality of sets (i.e., Hume's principle), which allows children to overcome the capacity limits of parallel computations of one-to-one correspondence that restrict small number word meanings.