Although animals of many species have been shown to discriminate between visual-spatial arrays or auditory-temporal sequences on the basis of numerosity, most of the evidence for numerosity discrimination comes from experiments involving extensive laboratory training. Under these conditions, animals' discrimination of two numerosities depends on their ratio and is independent of their absolute value. It is an open question whether any untrained nonhuman animal spontaneously represents number in this way as do human children and adults. Here we present the results of habituation-discrimination experiments on cotton-top tamarin monkeys (Saguinus oedipus) that provide evidence for numerosity discrimination in the absence of training. Presented with auditory stimuli (speech syllables) controlled for the continuous variables of sequence duration, item duration, inter-stimulus interval, and overall energy, tamarins readily discriminated sequences of 4 vs 8, 4 vs 6, and 8 vs 12 syllables. In contrast, tamarins failed to discriminate sequences of 4 vs 5 and 8 vs 10 syllables, providing evidence that their numerosity discrimination is approximate and shows the set-size ratio signature of numerosity discrimination in humans and trained non-human animals. These results provide strong support for the hypothesis that representations of large, approximate numerosity are evolutionarily ancient and spontaneously available to non-human animals.

Human adults are thought to possess two dissociable systems to represent numbers: an approximate quantity system akin to a mental number line, and a verbal system capable of representing numbers exactly. Here, we study the interface between these two systems using an estimation task. Observers were asked to estimate the approximate numerosity of dot arrays. We show that, in the absence of calibration, estimates are largely inaccurate: responses increase monotonically with numerosity, but underestimate the actual numerosity. However, insertion of a few inducer trials, in which participants are explicitly (and sometimes misleadingly) told that a given display contains 30 dots, is sufficient to calibrate their estimates on the whole range of stimuli. Based on these empirical results, we develop a model of the mapping between the numerical symbols and the representations of numerosity on the number line.

Studies using operant training have demonstrated that laboratory animals can discriminate the number of objects or events based on either auditory or visual stimuli, as well as the integration of both auditory and visual modalities. To date, studies of spontaneous number discrimination in untrained animals have been restricted to the visual modality, leaving open the question of whether such capacities generalize to other modalities such as audition. To explore the capacity to spontaneously discriminate number based on auditory stimuli, and to assess the abstractness of the representation underlying this capacity, a habituation-discrimination procedure involving speech and pure tones was used with a colony of cotton-top tamarins. In the habituation phase, we presented subjects with either two- or three-speech syllable sequences that varied with respect to overall duration, intersyllable duration, and pitch. In the test phase, we presented subjects with a counterbalanced order of either two- or three-tone sequences that also varied with respect to overall duration, inter-syllable duration, and pitch. The proportion of looking responses to test stimuli differing in number was significantly greater than to test stimuli consisting of the same number. Combined with earlier work, these results show that at least one non-human primate species can spontaneously discriminate number in both the visual and auditory domain, indicating that this capacity is not tied to a particular modality, and within a modality, can accommodate differences in format.

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...

Mathematics is a system for representing and reasoning about quantities, with arithmetic as its foundation. Its deep interest for our understanding of the psychological foundations of scientific thought comes from what Eugene Wigner called the unreasonable efficacy of mathematics in the natural sciences. From a formalist perspective, arithmetic is a symbolic game, like tic-tac-toe. Its rules are more complicated, but not a great deal more complicated. Mathematics is the study of the properties of this game and of the systems that may be constructed on the foundation that it provides. Why should this symbolic game be so powerful and resourceful when it comes to building models of the physical world? And on what psychological foundations does the human mastery of this game rest? The first question is metaphysical—why is the world the way it is? We do not treat it, because it lies beyond the realm of experimental behavioral science. We review the answers to the second question that experimental research on human and non-human animal cognition suggests.

ous (uncountable) quantities is the system of real numbers. It includes the irrational numbers, like 2, and the transcendental numbers, like p. It is used by modern humans to represent many distinct systems of continuous quantity--duration, length, area, volume, density, rate, intensity, and so on. Because the system of real numbers is isomorphic to a system of magnitudes, the terms real number and magnitude are used interchangeably. Thus, when we refer to "mental magnitudes" we are referring to a real number system in the brain. Like the culturally specified real number system, the real number system in the brain is used to represent both continuous quantity and numerosity. Magnitudes and real numbers have the property that there is no way to pick out a successor, the next number in the sequence. Given a line of some length, there is no procedure whereby one could pick out the next longer line. Similarly, given a real number, like, say, 2, there is no procedure that picks out the next

Do children draw upon abstract representations of number when they perform approximate arithmetic operations? In this study, kindergarten children viewed animations suggesting addition of a sequence of sounds to an array of dots, and they compared the sum to a second dot array that differed from the sum by 1 of 3 ratios. Children performed this task successfully with all the signatures of adults ’ nonsymbolic number representations: accuracy modulated by the ratio of the sum and the comparison quantity, equal performance for within- and cross-modality tasks and for addition and comparison tasks, and performance superior to that of a matched subtraction task. The findings provide clear evidence for nonsymbolic numerical operations on abstract numerical quantities in children who have not yet been taught formal arithmetic.

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...

Human newborns discriminate languages from different rhythmic classes, fail to discriminate languages from the same rhythmic class, and fail to discriminate languages when the utterances are played backwards. Recent evidence showing that cotton-top tamarins discriminate Dutch from Japanese, but not when utterances are played backwards, is compatible with the hypothesis that rhythm discrimination is based on a general perceptual mechanism inherited from a primate ancestor. The present study further explores the rhythm hypothesis for language discrimination by testing languages from the same and different rhythmic class. We find that tamarins discriminate Polish from Japanese (different rhythmic classes), fail to discriminate English and Dutch (same rhythmic class), and fail to discriminate backwards utterances from different and same rhythmic classes. These results provide further evidence that language discrimination in tamarins is facilitated by rhythmic differences between languages, and suggest that, in humans, this mechanism is unlikely to have evolved specifically for language. Processing a spoken language requires perceptual mechanisms that operate on the incoming signal and extract information relevant for understanding the linguistic content of the utterance. Human infants begin the language-learning process with sensitivities to many

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