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.

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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The traditional rules of voice-leading in Western music are explicated using experimentally established perceptual principles. Six core principles are shown to account for the majority of voice-leading rules given in historical and contemporary music theory tracts. These principles are treated in a manner akin to axioms in a formal system from which the traditional rules of voice-leading are derived. Nontraditional rules arising from the derivation are shown to predict formerly unnoticed aspects of voice-leading practice. In addition to the core perceptual principles, several auxiliary principles are described. These auxiliary principles are occasionally linked to voice-leading practice and may be regarded as compositional “options ” that shape the music-making in perceptually unique ways. It is suggested that these auxiliary principles distinguish different types of part writing, such as polyphony, homophony, and close harmony. A theory is proposed to account for the aesthetic origin of voiceleading practices.

Five-month-old infants were presented with a small object, a larger object, and a background surface arranged in depth so that all were within reaching distance. Patterns of reaching for this display were observed, while spatial and kinetic properties of the display were varied. When the infants reached for the display, they did not reach primarily for the surfaces that were nearer, smaller, or presented in motion. The infants reached, instead, for groups of surfaces that formed a unit that was spatially connected and/or that moved as a whole relative to its surroundings. Infants reached for the nearer of two objects as a distinct unit when the objects were separated in depth or when one object moved relative to the other. They reached for the two objects as a single unit when the objects were adjacent or when they moved together. The reaching patterns provided evidence that the infants organized each display into the kind of units that adults call objects: manipulable units with internal coherence and external boundaries. Infants, like adults, perceived objects by detecting both the spatial arrangements and the relative movements of surfaces in the three-dimensional layout.

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.

Are abstract representations of number -- representations that are independent of the particular type of entities that are enumerated -- a product of human language or culture, or do they trace back to human infancy? To investigate these questions, four experiments investigated whether human infants discriminate between sequences of actions (jumps of a puppet) on the basis of numerosity. At 6 months, infants successfully discriminated 4- vs. 8-jump sequences, when the continuous variables of sequence duration, jump duration, jump rate, jump interval and duration and extent of motion were controlled and rhythm was eliminated. In contrast, infants failed to discriminate 2- vs. 4-jump sequences, suggesting that infants fail to form cardinal number representations of small numbers of actions. Infants also failed to discriminate between sequences of 4 vs. 6 jumps at 6 months, and succeeded at 9 months, suggesting that infants' number representations are imprecise and increase in precision with age. All of these findings agree with those of studies using visual-spatial arrays and auditory sequences, providing evidence that a single, abstract system of number representation is present and functional in infancy. Infants' Enumeration of Actions: Numerical Discrimination and its Signature Limits Recent research provides evidence that human infants discriminate between large sets of elements on the basis of numerosity, when a variety of continuous quantitative variables are controlled. For example, 6-month-old infants discriminate visual arrays of 8 vs. 16 dots when array size and density, dot size, summed area and brightness, and summed contour length are equated either during habituation or during test (e.g., Brannon, 2002;Brannon, Abbott, & Lutz, in press; Xu & Spelke, 2000; Xu...

Four experiments used a preferential looking method to investigate six-month-old infants' capacity to represent numerosity in visual-spatial displays. Building on previous findings that such infants discriminate between arrays of 8 vs. 16 discs, but not 8 vs. 12 discs (Xu & Spelke, 2000), Experiments 1 and 2 investigated whether infants' numerosity discrimination depends on the ratio of the two set sizes with even larger numerosities. Infants successfully discriminated between arrays of 16 vs. 32 discs, but not 16 vs. 24 discs, providing evidence that their discrimination shows the set-size ratio signature of numerosity discrimination in human adults, children, and many non-human animals. Experiments 3 and 4 addressed a controversy concerning infants' ability to discriminate large numerosities (observed under conditions that control for total filled area, array size and density, item size, and correlated properties such as brightness: Brannon, 2002; Xu, 2003; Xu & Spelke, 2000) vs. small numerosities (not observed under conditions that control for total contour length: Clearfield & Mix, 1999). To investigate the sources of these differing findings, Experiment 3 tested infants' large-number discrimination with controls for contour length, and Experiment 4 tested small-number discrimination with controls for total filled area. Infants successfully discriminated the large-number displays but showed no evidence of discriminating the small-number displays. These findings provide evidence that infants have robust abilities to represent large numerosities. In contrast, infants may fail to represent small numerosities in visual-spatial arrays with continuous quantity controls, consistent with the thesis that separate systems serve to represent large vs. small numerosities. A we...

Developmental research suggests that some of the mechanisms that underlie numerical cognition are present and functional in human infancy. To investigate these mechanisms and their developmental course, psychologists have turned to behavioral and electrophysiological methods using briefly presented displays. These methods, however, depend on the assumption that young infants can extract numerical information rapidly. Here we test this assumption and begin to investigate the speed of numerical processing in 5-month-old infants. Infants successfully discriminated between arrays of 4 vs. 8 dots on the basis of number when a new array appeared every 2 seconds, but not when a new array appeared every 1.0 or 1.5 seconds. These results suggest alternative interpretations of past findings, provide constraints on the design of future experiments, and introduce a new method for probing infants' enumeration process. Further experiments using this method provide initial evidence that infants' enumeration mechanism operates in parallel and yields increasingly accurate numerical representations over time, as does the enumeration mechanism used by adults in symbolic and nonsymbolic tasks. Over the past two decades, a wealth of research has focused on the nature and origins of numerical knowledge. Although reports that infants represent small numbers of objects have been interpreted in multiple ways (e.g., Carey, 2001; Clearfield & Mix, 1999; Feigenson, Carey & Spelke, 2002; Simon, 1997; Starkey & Cooper, 1980; Treiber & Wilcox, 1984; Wynn, 1992; Wynn, Bloom & Chiang, 2002), recent research provides clear evidence that infants as young as 6 months represent the approximate cardinal values of large sets of entities. In studies using a looking time method, for example, 6-month-old infant...

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