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Calibrating the mental number line
, 2008
"... 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 ..."
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Cited by 18 (5 self)
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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.
Mathematical cognition
 In
, 2005
"... 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 scie ..."
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Cited by 15 (2 self)
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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 tictactoe. 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 nonhuman animal cognition suggests.
Evolutionary and developmental foundations of human knowledge: a case study of mathematics
 In M. Gazzaniga (Ed.), The cognitive neurosciences
, 2004
"... What are the brain and cognitive systems that allow humans to play baseball, compute square roots, cook soufflés, or navigate the Tokyo subways? It may seem that studies of human infants and of nonhuman animals will tell us little about these abilities, because only educated, enculturated human adu ..."
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Cited by 13 (2 self)
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What are the brain and cognitive systems that allow humans to play baseball, compute square roots, cook soufflés, or navigate the Tokyo subways? It may seem that studies of human infants and of nonhuman animals will tell us little about these abilities, because only educated, enculturated human adults engage in organized games, formal mathematics, gourmet cooking, or mapreading. In this chapter, we argue against this seemingly sensible conclusion. When human adults exhibit complex, uniquely human, culturespecific skills, they draw on a set of psychological and neural mechanisms with two distinctive properties: they evolved before humanity and thus are shared with other animals, and they emerge early in human development and thus are common to infants, children, and adults. These core knowledge systems form the building blocks for uniquely human skills. Without them we wouldn’t be able to learn about different kinds of games, mathematics, cooking, or maps. To understand what is special about human intelligence, therefore, we must study both the core knowledge systems on which it rests and the mechanisms by which these systems are orchestrated to permit new kinds of concepts and cognitive processes. What is core knowledge? A wealth of research on nonhuman primates and on human
Spontaneous Number Discrimination of MultiFormat . . .
, 2002
"... 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 ..."
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Cited by 10 (7 self)
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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 habituationdiscrimination procedure involving speech and pure tones was used with a colony of cottontop tamarins. In the habituation phase, we presented subjects with either two or threespeech 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 threetone sequences that also varied with respect to overall duration, intersyllable 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 nonhuman 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.
The cultural and evolutionary history of the real numbers
 In Evolution and Culture
, 2005
"... 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 quantityduration, length, area, volume, density, rate, intensity, and so on. ..."
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Cited by 9 (4 self)
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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 quantityduration, 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
Exact and approximate judgements of visual and auditory numerosity: An fMRI study
 Brain Research
, 2006
"... Number of text pages: 29 Number of tables: 3 Number of figures: 5 Human adults can assess the number of objects in a set (numerosity) by approximate estimation or by exact counting. There is evidence suggesting that numerosity estimation depends on a dedicated mechanism that is amodal and nonverba ..."
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Cited by 8 (2 self)
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Number of text pages: 29 Number of tables: 3 Number of figures: 5 Human adults can assess the number of objects in a set (numerosity) by approximate estimation or by exact counting. There is evidence suggesting that numerosity estimation depends on a dedicated mechanism that is amodal and nonverbal. By contrast, counting requires the coordination between the preexisting numerosity estimation abilities with language and onetoone correspondence principles. In this paper we investigate with fMRI the neural correlates of numerosity estimation and counting in human adults, using both visual and auditory stimuli. Results show that attending to approximate numerosity correlates with increased activity of a right lateralized frontoparietal cortical network, and that this activity is independent of the stimuli presentation’s modality. Counting activates additional left prefrontal, parietal, and bilateral premotor areas, again independently from stimulus modality. These results dissociate two neuronal systems that underlie different numerosity judgements. SECTION:
Nonverbal arithmetic in humans: Light from noise
"... manipulation of these nonverbal magnitude representations is sparse and lacking in depth. This study uses the analysis of variability as a tool for understanding properties of these combinatorial processes. Human subjects participated in tasks requiring responses dependent upon the addition, subtrac ..."
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Cited by 7 (1 self)
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manipulation of these nonverbal magnitude representations is sparse and lacking in depth. This study uses the analysis of variability as a tool for understanding properties of these combinatorial processes. Human subjects participated in tasks requiring responses dependent upon the addition, subtraction, or reproduction of nonverbal counts. Variance analyses revealed that the magnitude of both inputs and answer contributed to the variability in the arithmetic responses, with operand variability dominating. Other contributing factors to the observed variability and implications for logarithmic versus scalar models of magnitude representation are discussed in light of these results. Humans and other animals appear to compute descriptive statistics in a variety of domains—from language (e.g., Aslin, Saffran, & Newport, 1999), to foraging (Gallistel, 1990), to vision (e.g., Ariely, 2001) and motor skills (Trommershäuser, Maloney, & Landy, 2003). These statistics may derive from mental magnitudes representing elementary abstractions like number, duration, and distance (Gallistel, Gelman, & Cordes, 2006). These magnitude
Representations Underlying Transitive Choice in Humans and Other
"... There is strong evidence in the literature for at least three di#erent representations underlying transitive choice in various species of animals. This paper focuses primarily on understanding one of the most neglected: the productionrule model of Harris and McGonigle (1994). The productionrule mo ..."
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Cited by 2 (1 self)
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There is strong evidence in the literature for at least three di#erent representations underlying transitive choice in various species of animals. This paper focuses primarily on understanding one of the most neglected: the productionrule model of Harris and McGonigle (1994). The productionrule model has been to date the best model at accounting for the performance of squirrel monkeys (Saimiri sciureus) and human children under 7 trained on the transitive inference (TI) task (e.g. given A > B and B > C, then A > C) when presented with three items from the training set. This paper presents a new neurologicallyplausible version of this representation, the twotier model, which explains how this sort transitive performance is learned. This new model perfectly replicates the positive aspects of the productionrule model while accounting for more of the data, particularly subject's failures to learn transitive inference. This paper also discusses how the twotier model fits with other transitive inference models, and characterises how to recognise which TI representation underlies which sorts of TI performance. Of the three representations discussed, we suggest that the twotier model may be the most relevant for understanding generalpurpose primate task learning, and that it may even provide the scaffolding for the human acquisition of concrete operational thought.
A (2009) Compressed scaling of abstract numerosity representations in adult humans and monkeys
 J Cogn Neurosci
"... & There is general agreement that nonverbal animals and humans endowed with language possess an evolutionary precursor system for representing and comparing numerical values. However, whether nonverbal numerical representations in human and nonhuman primates are quantitatively similar and whether li ..."
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Cited by 1 (0 self)
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& There is general agreement that nonverbal animals and humans endowed with language possess an evolutionary precursor system for representing and comparing numerical values. However, whether nonverbal numerical representations in human and nonhuman primates are quantitatively similar and whether linear or logarithmic coding underlies such magnitude judgments in both species remain elusive. To resolve these issues, we tested the numerical discrimination performance of human subjects and two rhesus monkeys (Macaca mulatta) in an identical delayed matchtonumerosity task for a broad range of numerosities from 1 to 30. The results demonstrate a noisy nonverbal estimation system obeying Weber’s Law in both species. With average Weber fractions in the range of 0.51 and 0.60, nonverbal numerosity discriminations in humans and monkeys showed similar precision. Moreover, the detailed analysis of the performance distributions exhibited nonlinearly compressed numerosity representations in both primate species. However, the difference between linear and logarithmic scaling was less pronounced in humans. This may indicate a gradual transformation of a logarithmic to linear magnitude scale in human adults as the result of a cultural transformation process during the course of mathematical education. &