Number sense " is a short-hand for our ability to quickly understand, approximate, and manipulate numerical quantities. My hypothesis is that number sense rests on cerebral circuits that have evolved specifically for the purpose of representing basic arithmetic knowledge. Four lines of evidence suggesting that number sense constitutes a domain-specific, biologically-determined ability are reviewed: the presence of evolutionary precursors of arithmetic in animals; the early emergence of arithmetic competence in infants independently of other abilities, including language; the existence of a homology between the animal, infant, and human adult abilities for number processing ; and the existence of a dedicated cerebral substrate. In adults of all cultures, lesions to the inferior parietal region can specifically impair number sense while leaving the knowledge of other cognitive domains intact. Furthermore, this region is demonstrably activated during number processing. I postulate that higher-level cultural developments in arithmetic emerge through the establishment of linkages between this core analogical representation (the " number line ") and other verbal and visual representations of number notations. The neural and cognitive organization of those representations can explain why some mathematical concepts are intuitive, while others are so difficult to grasp. Thus, the ultimate foundations of mathematics rests on core representations that have been internalized in our brains through evolution.

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A new choice task was used to explore infants' spontaneous representations of more and less. Ten- and 12-month-old infants saw crackers placed sequentially into two containers, then were allowed to crawl and obtain the crackers from the container they chose. Infants chose the larger quantity with comparisons of 1 versus 2 and 2 versus 3, but failed with comparisons of 3 versus 4, 2 versus 4, and 3 versus 6. Success with visible arrays ruled out a motivational explanation for failure in the occluded 3-versus-6 condition. Control tasks ruled out the possibility that presentation duration guided choice, and showed that presentation complexity was not responsible for the failure with larger numbers. When crackers were different sizes, total surface area or volume determined choice. The infants' pattern of success and failure supports the hypothesis that they relied on object-file representations, comparing mental models via total volume or surface area rather than via one-to-one correspondence between object files.

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What aspects of knowledge emerge in children prior to their first contacts with the objects of their knowledge, and what aspects emerge through the shaping effects of experience with those objects? What aspects of knowledge are constant over human development from the moment that infants begin to make sense of the world, and what aspects change as children grow and learn? What aspects of knowledge are universal, and what aspects vary across people in different cultures or with different educational backgrounds? Finally, what aspects of knowledge can people change in themselves or their children with sufficient insight or effort, and what aspects are invari-

In two experiments, a manual search task explored 12- to 14-month-old infants' representations of small sets of objects. In this paradigm, patterns of searching revealed the number of objects infants represented as hidden in an opaque box. In Experiment 1, we obtained the set-size signature of object-file representations: infants succeeded at representing precisely 1, precisely 2, and precisely 3 objects in the box, but failed at representing 4 (or even that 4 is greater than 2). In Experiment 2, we showed that infants' expectations about the contents of the box were based on number of individual objects, and not on a continuous property such as total object volume. These findings support the hypothesis that infants maintained representations of individuals, that object-files were the underlying means of representing these individuals, and that object-file models can be compared via one-to-one correspondence to establish numerical equivalence.

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.

Young infants construct models of the world composed of objects tracked through time and occlusion. To date little is known about the degree to which these models are sensitive to the material makeup of the represented individuals. Two experiments probed 8-month-olds' ability to represent different kinds of entities: rigid, cohesive objects, flexible, cohesive objects, and non-rigid, non-cohesive portions of sand. In Experiment 1, infants represented an array of two rigid, cohesive objects hidden behind a single screen, but failed to represent hidden arrays of two flexible objects or two portions of sand. In Experiment 2, entities were hidden behind two screens instead of one, thereby reducing the information processing demands of the task. In that case, infants succeeded in representing arrays of both types of object stimuli, but again failed to represent the portions of sand. It is argued that (1) the processes by which infants individuate and track entities are sensitive to material kind, (2) rigid cohesive objects occupy a privileged status in this system, and (3) early knowledge about objects and substances has a quantificational aspect. q 2002 Published by Elsevier Science B.V.

cognitive architecture?

A neural network model that can simulate the learning of some simple proportional analogies is presented. These analogies include, for example, (a) red-square:red-circle # yellow-square:?, (b) apple:red # banana: ?, (c) a:b # c:?. Underlying the development of this network is a theory for how the brain learns the nature of association between pairs of concepts. Traditional Hebbian learning of associations is necessary for this process but not sufficient. This is because it simply says, for example, that the concepts "apple" and "red" have been associated, but says nothing about the nature of this relationship. The types of context-dependent interlevel connections in the network suggest a semilocal type of learning that in some manner involves association among more than two nodes or neurons at once. Such connections have been called synaptic triads, and related to potential cell responses in the prefrontal cortex. Some additional types of connections are suggested by the problem of modeling analogies. These types of connections have not yet been verified by brain imaging, but the work herein suggests that they may occur and, possibly, be made and broken quickly in the course of working memory encoding. These working memory connections are referred to as differential, delayed and anti-Hebbian connections. In these connections, one can learn transitions such as "keep red the same"; "change red to yellow"; "turn off red"; "turn on yellow," and so forth. Also, included in the network is a kind of weight transport so that, for example, red to red can be transported to a different instance of color, such as yellow to yellow. The network instantiation developed here, based on common connectionist building blocks such as associative learning, competition, and adaptive resonance...