Humans and other animals are able to perceive and represent a number of objects present in a scene, a core cognitive ability thought to underlie the development of mathematics. However, the perceptual mechanisms that underpin this capacity remain poorly understood. Here, we show that our visual sense of number derives from a visual system designed to efficiently encode the location of objects in scenes. Using a mathematical model, we demonstrate that an efficient but information-limited encoding of objects' locations can explain many key aspects of number psychophysics, including subitizing, Weber's law, underestimation, and effects of exposure time. In two experiments (N = 100 each), we find that this model of visual encoding captures human performance in both a change-localization task and a number estimation task. In a third experiment (N = 100), we find that individual differences in change-localization performance are highly predictive of differences in number estimation, both in terms of overall performance and inferred model parameters, with participants having numerically indistinguishable inferred information capacities across tasks. Our results therefore indicate that key psychophysical features of numerical cognition do not arise from separate modules or capacities specific to number, but rather as by-products of lower level constraints on perception. (PsycInfo Database Record (c) 2024 APA, all rights reserved).
In a large (N = 300), pre-registered experiment and data analysis model, we find that individual variation in overall performance on Raven's Progressive Matrices is substantially driven by differential strategizing in the face of difficulty. Some participants choose to spend more time on hard problems while others choose to spend less and these differences explain about 42% of the variance in overall performance. In a data analysis jointly predicting participants' reaction times and accuracy on each item, we find that the Raven's task captures at most half of participants' variation in time-controlled ability (48%) down to almost none (3%), depending on which notion of ability is assumed. Our results highlight the role that confounding factors such as motivation play in explaining individuals' differential performance in IQ testing.
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.
Cognition , Learning , Child , Humans , Mathematics
We investigate number and arithmetic learning among a Bolivian indigenous people, the Tsimane', for whom formal schooling is comparatively recent in history and variable in both extent and consistency. We first present a large-scale meta-analysis on child number development involving over 800 Tsimane' children. The results emphasize the impact of formal schooling: Children are only found to be full counters when they have attended school, suggesting the importance of cultural support for early mathematics. We then test especially remote Tsimane' communities and document the development of specialized arithmetical knowledge in the absence of direct formal education. Specifically, we describe individuals who succeed on arithmetic problems involving the number five-which has a distinct role in the local economy-even though they do not succeed on some lower numbers. Some of these participants can perform multiplication with fives at greater accuracy than addition by one. These results highlight the importance of cultural factors in early mathematics and suggest that psychological theories of number where quantities are derived from lower numbers via repeated addition (e.g., a successor function) are unlikely to explain the diversity of human mathematical ability.
Child Development , Cognition , Child , Humans , Bolivia , Indigenous Peoples , Knowledge
Words that are more surprising given context take longer to process. However, no incremental parsing algorithm has been shown to directly predict this phenomenon. In this work, we focus on a class of algorithms whose runtime does naturally scale in surprisal-those that involve repeatedly sampling from the prior. Our first contribution is to show that simple examples of such algorithms predict runtime to increase superlinearly with surprisal, and also predict variance in runtime to increase. These two predictions stand in contrast with literature on surprisal theory (Hale, 2001; Levy, 2008a) which assumes that the expected processing cost increases linearly with surprisal, and makes no prediction about variance. In the second part of this paper, we conduct an empirical study of the relationship between surprisal and reading time, using a collection of modern language models to estimate surprisal. We find that with better language models, reading time increases superlinearly in surprisal, and also that variance increases. These results are consistent with the predictions of sampling-based algorithms.
Many social and legal conflicts hinge on semantic disagreements. Understanding the origins and implications of these disagreements necessitates novel methods for identifying and quantifying variation in semantic cognition between individuals. We collected conceptual similarity ratings and feature judgements from a variety of words in two domains. We analyzed this data using a non-parametric clustering scheme, as well as an ecological statistical estimator, in order to infer the number of different variants of common concepts that exist in the population. Our results show at least ten to thirty quantifiably different variants of word meanings exist for even common nouns. Further, people are unaware of this variation, and exhibit a strong bias to erroneously believe that other people share their semantics. This highlights conceptual factors that likely interfere with productive political and social discourse.
The capacity to generate recursive sequences is a marker of rich, algorithmic cognition, and perhaps unique to humans. Yet, the precise processes driving recursive sequence generation remain mysterious. We investigated three potential cognitive mechanisms underlying recursive pattern processing: hierarchical reasoning, ordinal reasoning, and associative chaining. We developed a Bayesian mixture model to quantify the extent to which these three cognitive mechanisms contribute to adult humans' performance in a sequence generation task. We further tested whether recursive rule discovery depends upon relational information, either perceptual or semantic. We found that the presence of relational information facilitates hierarchical reasoning and drives the generation of recursive sequences across novel depths of center embedding. In the absence of relational information, the use of ordinal reasoning predominates. Our results suggest that hierarchical reasoning is an important cognitive mechanism underlying recursive pattern processing and can be deployed across embedding depths and relational domains.
Cognition , Problem Solving , Humans , Adult , Bayes Theorem , Semantics
Hierarchical cognitive mechanisms underlie sophisticated behaviors, including language, music, mathematics, tool-use, and theory of mind. The origins of hierarchical logical reasoning have long been, and continue to be, an important puzzle for cognitive science. Prior approaches to hierarchical logical reasoning have often failed to distinguish between observable hierarchical behavior and unobservable hierarchical cognitive mechanisms. Furthermore, past research has been largely methodologically restricted to passive recognition tasks as compared to active generation tasks that are stronger tests of hierarchical rules. We argue that it is necessary to implement learning studies in humans, non-human species, and machines that are analyzed with formal models comparing the contribution of different cognitive mechanisms implicated in the generation of hierarchical behavior. These studies are critical to advance theories in the domains of recursion, rule-learning, symbolic reasoning, and the potentially uniquely human cognitive origins of hierarchical logical reasoning.
Logic , Problem Solving , Humans , Learning
Standard models in quantitative linguistics assume that word usage follows a fixed frequency distribution, often Zipf's law or a close relative. This view, however, does not capture the near daily variations in topics of conversation, nor the short-term dynamics of language change. In order to understand the dynamics of human language use, we present a corpus of daily word frequency variation scraped from online news sources every 20 min for more than 2 years. We construct a simple time-varying model with a latent state, which is observed via word frequency counts. We use Bayesian techniques to infer the parameters of this model for 20,000 words, allowing us to convert complex word-frequency trajectories into low-dimensional parameters in word usage. By analyzing the inferred parameters of this model, we quantify the relative mobility and drift of words on a day-to-day basis, while accounting for sampling error. We quantify this variation and show evidence against "rich-get-richer" models of word use, which have been previously hypothesized to explain statistical patterns in language.
Linguistics , Models, Theoretical , Humans , Bayes Theorem , Language , Communication
Spatial cognition is central to human behavior, but the way people conceptualize space varies within and across groups for unknown reasons. Here, we found that adults from an indigenous Bolivian group used systematically different spatial reference frames on different axes, according to known differences in their discriminability: In both verbal and nonverbal tests, participants preferred allocentric (i.e., environment-based) space on the left-right axis, where spatial discriminations (like "b" versus "d") are notoriously difficult, but the same participants preferred egocentric (i.e., body-based) space on the front-back axis, where spatial discrimination is relatively easy. The results (i) establish a relationship between spontaneous spatial language and memory across axes within a single culture, (ii) challenge the claim that each language group has a predominant spatial reference frame at a given scale, and (iii) suggest that spatial thinking and language may both be shaped by spatial discrimination abilities, as they vary across cultures and contexts.
Children in industrialized cultures typically succeed on Give-N, a test of counting ability, by age 4. On the other hand, counting appears to be learned much later in the Tsimane', an indigenous group in the Bolivian Amazon. This study tests three hypotheses for what may cause this difference in timing: (a) Tsimane' children may be shy in providing behavioral responses to number tasks, (b) Tsimane' children may not memorize the verbal list of number words early in acquisition, and/or (c) home environments may not support mathematical learning in the same way as in US samples, leading Tsimane' children to primarily acquire mathematics through formalized schooling. Our results suggest that most of our subjects are not inhibited by shyness in responding to experimental tasks. We also find that Tsimane' children (N = 100, ages 4-11) learn the verbal list later than US children, but even upon acquiring this list, still take time to pass Give-N tasks. We find that performance in counting varies across tasks and is related to formal schooling. These results highlight the importance of formal education, including instruction in the count list, in learning the meanings of the number words.
Learning , Bolivia , Child , Child, Preschool , Humans , Mathematics
Previous findings suggest that mentally representing exact numbers larger than four depends on a verbal count routine (e.g., "one, two, three . . ."). However, these findings are controversial because they rely on comparisons across radically different languages and cultures. We tested the role of language in number concepts within a single population-the Tsimane' of Bolivia-in which knowledge of number words varies across individual adults. We used a novel data-analysis model to quantify the point at which participants (N = 30) switched from exact to approximate number representations during a simple numerical matching task. The results show that these behavioral switch points were bounded by participants' verbal count ranges; their representations of exact cardinalities were limited to the number words they knew. Beyond that range, they resorted to numerical approximation. These results resolve competing accounts of previous findings and provide unambiguous evidence that large exact number concepts are enabled by language.
Cognition , Language , Adult , Humans , Language Development , Mathematics
A major goal of linguistics and cognitive science is to understand what class of learning systems can acquire natural language. Until recently, the computational requirements of language have been used to argue that learning is impossible without a highly constrained hypothesis space. Here, we describe a learning system that is maximally unconstrained, operating over the space of all computations, and is able to acquire many of the key structures present in natural language from positive evidence alone. We demonstrate this by providing the same learning model with data from 74 distinct formal languages which have been argued to capture key features of language, have been studied in experimental work, or come from an interesting complexity class. The model is able to successfully induce the latent system generating the observed strings from small amounts of evidence in almost all cases, including for regular (e.g., an , [Formula: see text], and [Formula: see text]), context-free (e.g., [Formula: see text], and [Formula: see text]), and context-sensitive (e.g., [Formula: see text], and xx) languages, as well as for many languages studied in learning experiments. These results show that relatively small amounts of positive evidence can support learning of rich classes of generative computations over structures. The model provides an idealized learning setup upon which additional cognitive constraints and biases can be formalized.
Learning/physiology , Linguistics/methods , Humans , Language
It is popular in psychology to hypothesize that representations of exact number are innately determined-in particular, that biology has endowed humans with a system for manipulating quantities which forms the primary representational substrate for our numerical and mathematical concepts. While this perspective has been important for advancing empirical work in animal and child cognition, here we examine six natural predictions of strong numerical nativism from a multidisciplinary perspective, and find each to be at odds with evidence from anthropology and developmental science. In particular, the history of number reveals characteristics that are inconsistent with biological determinism of numerical concepts, including a lack of number systems across some human groups and remarkable variability in the form of numerical systems that do emerge. Instead, this literature highlights the importance of economic and social factors in constructing fundamentally new cognitive systems to achieve culturally specific goals. (PsycInfo Database Record (c) 2023 APA, all rights reserved).
Cognition , Mathematical Concepts , Child , Animals , Humans
We examine the conceptual development of kinship through the lens of program induction. We present a computational model for the acquisition of kinship term concepts, resulting in the first computational model of kinship learning that is closely tied to developmental phenomena. We demonstrate that our model can learn several kinship systems of varying complexity using cross-linguistic data from English, Pukapuka, Turkish, and Yanomamö. More importantly, the behavioral patterns observed in children learning kinship terms, under-extension and over-generalization, fall out naturally from our learning model. We then conducted interviews to simulate realistic learning environments and demonstrate that the characteristic-to-defining shift is a consequence of our learning model in naturalistic contexts containing abstract and concrete features. We use model simulations to understand the influence of logical simplicity and children's learning environment on the order of acquisition of kinship terms, providing novel predictions for the learning trajectories of these words. We conclude with a discussion of how this model framework generalizes beyond kinship terms, as well as a discussion of its limitations.
Language Development , Verbal Learning , Child , Concept Formation , Humans , Learning , Linguistics
The ability to represent approximate quantities appears to be phylogenetically widespread, but the selective pressures and proximate mechanisms favouring this ability remain unknown. We analysed quantity discrimination data from 672 subjects across 33 bird and mammal species, using a novel Bayesian model that combined phylogenetic regression with a model of number psychophysics and random effect components. This allowed us to combine data from 49 studies and calculate the Weber fraction (a measure of quantity representation precision) for each species. We then examined which cognitive, socioecological and biological factors were related to variance in Weber fraction. We found contributions of phylogeny to quantity discrimination performance across taxa. Of the neural, socioecological and general cognitive factors we tested, cortical neuron density and domain-general cognition were the strongest predictors of Weber fraction, controlling for phylogeny. Our study is a new demonstration of evolutionary constraints on cognition, as well as of a relation between species-specific neuron density and a particular cognitive ability. This article is part of the theme issue 'Systems neuroscience through the lens of evolutionary theory'.
Biological Evolution , Cognition , Animals , Bayes Theorem , Cognition/physiology , Humans , Mammals , Phylogeny , Psychophysics , Species Specificity
