The Psychological Scaffolding of Number

Theories of early numeracy assume that the logical concepts which enable formal definitions of numbers (e.g., cardinality, one-to-one correspondence, successor function) play key roles in ontogenesis, but empirical support is lacking. These concepts may not be suitable targets for theory because such definitions are reconstructions of numbers in formal language, but not explanatory of what numbers are. We propose that there should be a correspondence between those formal ideas that are generative for the natural numbers and those realized in children’s early learning. Building on our account of foundations of arithmetic (Grice et al., 2024, Psychological Review), we give a new explanatory construction of the natural numbers and a theory of early numeracy analogous to it. We show that numbers can be derived from linear order, monotonicity and convexity – principles of perceptual organization that are evolutionarily based. A key step in our construction is the ℕ-chain, the limit of finite linear orders, which we show is equivalent to the successor function but preferable as a target for psychological theory because it can be approximated in finite terms. In our theory, building blocks of early numeracy are linear orders - the number word sequence, object play schemas - and how they are related in the count routine by structure mapping (order embeddings). The number word sequence develops gradually, giving a representation analogous to an ℕ-chain when the child realizes that numbers can be generated without limit. Ordinality is prior to cardinality as the basis for numeracy, one-to-one correspondence results from convexity, and the natural numbers and arithmetic emerge jointly as a common structure. Our theory reconciles constructivist and nativist accounts because it shows that numeracy is both culturally mediated and biologically-based, although ‘core number systems’ do not play a role. The dawning of numeracy in a child’s mind is both a reinvention and rediscovery of mathematics itself.