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The study of how children learn numbers has yielded one of the most productive research programs in cognitive development, spanning empirical and computational methods, as well as nativist and empiricist philosophies. This paper provides a tutorial on how to think computationally about learning models in a domain like number, where learners take finite data and go far beyond what they directly observe or perceive. To illustrate, this paper then outlines a model which acquires a counting procedure using observations of sets and words, extending the proposal of Piantadosi et al. (2012). This new version of the model responds to several critiques of the original work and outlines an approach which is likely appropriate for acquiring further aspects of mathematics.

Zipf famously stated that, if natural language lexicons are structured for efficient communication, the words that are used the most frequently should require the least effort. This observation explains the famous finding that the most frequent words in a language tend to be short. A related prediction is that, even within words of the same length, the most frequent word forms should be the ones that are easiest to produce and understand. Using orthographics as a proxy for phonetics, we test this hypothesis using corpora of 96 languages from Wikipedia. We find that, across a variety of languages and language families and controlling for length, the most frequent forms in a language tend to be more orthographically well-formed and have more orthographic neighbors than less frequent forms. We interpret this result as evidence that lexicons are structured by language usage pressures to facilitate efficient communication.

Although the mapping between form and meaning is often regarded as arbitrary, there are in fact well-known constraints on words which are the result of functional pressures associated with language use and its acquisition. In particular, languages have been shown to encode meaning distinctions in their sound properties, which may be important for language learning. Here, we investigate the relationship between semantic distance and phonological distance in the large-scale structure of the lexicon. We show evidence in 100 languages from a diverse array of language families that more semantically similar word pairs are also more phonologically similar. This suggests that there is an important statistical trend for lexicons to have semantically similar words be phonologically similar as well, possibly for functional reasons associated with language learning.

We show that children in the Tsimane', a farming-foraging group in the Bolivian rain-forest, learn number words along a similar developmental trajectory to children from industrialized countries. Tsimane' children successively acquire the first three or four number words before fully learning how counting works. However, their learning is substantially delayed relative to children from the United States, Russia, and Japan. The presence of a similar developmental trajectory likely indicates that the incremental stages of numerical knowledge--but not their timing--reflect a fundamental property of number concept acquisition which is relatively independent of language, culture, age, and early education.

Infants must learn about many cognitive domains (e.g., language, music) from auditory statistics, yet capacity limits on their cognitive resources restrict the quantity that they can encode. Previous research has established that infants can attend to only a subset of available acoustic input. Yet few previous studies have directly examined infant auditory attention, and none have directly tested theorized mechanisms of attentional selection based on stimulus complexity. This work utilizes model-based behavioral methods that were recently developed to examine visual attention in infants (e.g., Kidd, Piantadosi, & Aslin, 2012). The present results demonstrate that 7- to 8-month-old infants selectively attend to nonsocial auditory stimuli that are intermediately predictable/complex with respect to their current implicit beliefs and expectations. These findings provide evidence of a broad principle of infant attention across modalities and suggest that sound-to-sound transitional statistics heavily influence the allocation of auditory attention in human infants.

A major open question in natural language research is the role of communicative efficiency in the origin and on-line processing of language structures. Here, we use word pairs like "chimp/chimpanzee", which differ in length but have nearly identical meanings, to investigate the communicative properties of lexical systems and the communicative pressures on language users. If language is designed to be information-theoretically optimal, then shorter words should convey less information than their longer counterparts, when controlling for meaning. Consistent with this prediction, a corpus analysis revealed that the short form of our meaning-matched pairs occurs in more predictive contexts than the longer form. Second, a behavioral study showed that language users choose the short form more often in predictive contexts, suggesting that tendencies to be information-theoretically efficient manifest in explicit behavioral choices. Our findings, which demonstrate the prominent role of communicative efficiency in the structure of the lexicon, complement and extend the results of Piantadosi, Tily, and Gibson (2011), who showed that word length is better correlated with Shannon information content than with frequency. Crucially, we show that this effect arises at least in part from active speaker choice. (Contains 2 figures.)

In acquiring number words, children exhibit a qualitative leap in which they transition from understanding a few number words, to possessing a rich 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. (Contains 6 figures and 2 tables.)

We present a general information-theoretic argument that all efficient communication systems will be ambiguous, assuming that context is informative about meaning. We also argue that ambiguity allows for greater ease of processing by permitting efficient linguistic units to be re-used. We test predictions of this theory in English, German, and Dutch. Our results and theoretical analysis suggest that ambiguity is a functional property of language that allows for greater communicative efficiency. This provides theoretical and empirical arguments against recent suggestions that core features of linguistic systems are not designed for communication. (Contains 3 figures.)