Bootstrapping in a language of thought: A formal model of numerical concept learning

Abstract

In acquiring number words, children exhibit a qualitative leap in which they transition from understanding a few number words, to possessing a recursive system of interrelated numerical concepts. We present a computational framework for understanding this inductive leap as the consequence of statistical inference over a sufficiently powerful representational system. We provide an implemented model that is powerful enough to learn number word meanings and other related conceptual systems from naturalistic data. The model shows that bootstrapping can be made computationally and philosophically well-founded as a theory of number learning. Our approach demonstrates how learners may combine core cognitive operations to build sophisticated representations during the course of development, and how this process explains observed developmental patterns in number word learning.

Why is number word learning hard? Evidence from bilingual learners

“…Specifically, on the assumption that delays between subset knower level stages are driven by the problem of mapping numerals to set sizes, it remains unknown why this 6 Consistent with this, computational models of the counting stages suggest that, to the extent that learning the cardinal principle is a process of inference to the best explanation where the successor principle is contained in a child's initial hypothesis space, children may indeed skip steps given sufficient robust training with, e.g., one and two, but not three or four (Piantadosi, et al, 2012). mapping problem might be so difficult.…”

“…Also, the stage-like nature of children's learning has been corroborated by meta-analyses and computational models (Lee & Sarnecka, 2010Sarnecka & Lee, 2009), which can model the delays between stages using a small number of parameters (Piantadosi, et al, 2012).…”

“…Up until this point, children are generally referred to as "subset-knowers" since they have only acquired the meanings for a subset of their number words. However, at the next stage children appear to recognize that the counting procedure can be used to enumerate sets and, more specifically, that the last word in a count routine labels the cardinality of the entire set (for discussion of knower levels, see Wynn, 1990Wynn, , 1992Le Corre et al, 2006;Lee & Sarnecka, 2011;Le Corre & Carey, 2007;Piantadosi, Tenenbaum, & Goodman, 2012;Sarnecka & Carey, 2008;Davidson, Eng, & Barner, 2012). At this stage, children are generally referred to as Cardinal Principle knowers ("CP-knowers").…”

To infinity and beyond: Children generalize the successor function to all possible numbers years after learning to count

“…According to some previous studies, children have knowledge of the successor function early in childhood -either innately (Leslie, Gallistel, & Gelman, 2007) or when they become Cardinal Principle knowers (CPknowers) at around 3½ to 4 years of age (Carey, 2004(Carey, , 2009Piantadosi, Tenenbaum, & Goodman, 2012;Sarnecka & Carey, 2008). Other studies have suggested that this knowledge emerges sometime after children learn the cardinal principle (Davidson, Eng, & Barner, 2012), providing evidence that children only have item-based knowledge of the successor function when they become CP-knowers (see also Wagner, et al, 2015).…”

Natural language and set-theoretic conception of natural number

“…The dominant view in developmental psychology as reviewed by Rips et al (2008), however, is that the natural number system is constructed out of other number-related concepts in the course of children's cognitive development (see Carey 2009;Feigenson et al 2004;Le Corre et al 2006;Piantadosi et al 2012;Sarnecka to appear;and Wynn 1992, among many others). The major difficulty for the nativist position is the fact that acquisition of number words (or numerals) by children takes a long time, completed only at the age of three and a half to four years.…”