Dehaene (this volume) articulates a naturalistic approach to the cognitive foundations of mathematics. Further, he argues that the ‘number line’ (analog magnitude) system of representation is the evolutionary and ontogenetic foundation of numerical concepts. Here I endorse Dehaene’s naturalistic stance and also his characterization of analog magnitude number representations. Although analog magnitude representations are part of the evolutionary foundations of numerical concepts, I argue that they are unlikely to be part of the ontogenetic foundations of the capacity to represent natural (...) number. Rather, the developmental source of explicit integer list representations of number are more likely to be systems such as the object–file representations that articulate mid–level object based attention, systems that build parallel representations of small sets of individuals. (shrink)

This article reviews current methods for evaluating models in the cognitive sciences, including theoretically based approaches, such as Bayes factors and minimum description length measures; simulation approaches, including model mimicry evaluations; and practical approaches, such as validation and generalization measures. This article argues that, although often useful in specific settings, most of these approaches are limited in their ability to give a general assessment of models. This article argues that hierarchical methods, generally, and hierarchical Bayesian methods, specifically, can provide a (...) more thorough evaluation of models in the cognitive sciences. This article presents two worked examples of hierarchical Bayesian analyses to demonstrate how the approach addresses key questions of descriptive adequacy, parameter interference, prediction, and generalization in principled and coherent ways. (shrink)

Remarkable progress in the mathematics and computer science of probability has led to a revolution in the scope of probabilistic models. In particular, ‘sophisticated’ probabilistic methods apply to structured relational systems such as graphs and grammars, of immediate relevance to the cognitive sciences. This Special Issue outlines progress in this rapidly developing field, which provides a potentially unifying perspective across a wide range of domains and levels of explanation. Here, we introduce the historical and conceptual foundations of the approach, explore (...) how the approach relates to studies of explicit probabilistic reasoning, and give a brief overview of the field as it stands today. (shrink)

I consider the role of cognitive modeling in cognitive science. Modeling, and the computers that enable it, are central to the field, but the role of modeling is often misunderstood. Models are not intended to capture fully the processes they attempt to elucidate. Rather, they are explorations of ideas about the nature of cognitive processes. In these explorations, simplification is essential—through simplification, the implications of the central ideas become more transparent. This is not to say that simplification has no downsides; (...) it does, and these are discussed. I then consider several contemporary frameworks for cognitive modeling, stressing the idea that each framework is useful in its own particular ways. Increases in computer power (by a factor of about 4 million) since 1958 have enabled new modeling paradigms to emerge, but these also depend on new ways of thinking. Will new paradigms emerge again with the next 1,000‐fold increase? (shrink)