Representing and Learning a Large System of Number Concepts with Latent Predicate Networks

Conventional models of exemplar or rule-based concept learning tend to focus on the acquisition of one concept at a time. They often underemphasize the fact that we learn many concepts as part of large systems rather than as isolated individuals. In such cases, the challenge of learning is not so much in providing stand-alone definitions, but in describing the richly structured relations between concepts. The natural numbers are one of the first such abstract conceptual systems children learn, serving as a serious case study in concept representation and acquisition (Carey, 2009; Fuson, 1988; Gallistel & Gelman, 2005). Even so, models of natural number learning focused on single-concept acquisition have largely ignored two challenges related to natural number’s status as a system of concepts: 1) there is an unbounded set of exact number concepts, each with distinct semantic content; and 2) people can reason flexibly about any of these concepts (even fictitious ones like eighteen-gazillion). To succeed, models must instead learn the structure of the entire infinite set of number concepts, focusing on how relationships between numbers support reference and generalization. Here, we suggest that the latent predicate network (LPN) – a probabilistic context-sensitive grammar formalism – facilitates tractable learning and reasoning for natural number concepts (Dechter, Rule, & Tenenbaum, 2015). We show how to express several key numerical relationships in our framework, and how a Bayesian learning algorithm for LPNs can model key phenomena observed in children learning to count. These results suggest that LPNs might serve as a computational mechanism by which children learn abstract numerical knowledge from utterances about number.