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.

In non-verbal counting tasks derived from the animal literature, adult human subjects repeatedly attempted to produce target numbers of key presses at rates that made vocal or subvocal counting difficult or impossible. In a second task, they estimated the number of flashes in a rapid, randomly timed sequence. Congruent with the animal data, mean estimates in both tasks were proportional to target values, as was the variability in the estimates. Converging evidence makes it unlikely that subjects used verbal counting or time durations to perform these tasks. The results support the hypothesis that adult humans share with non-verbal animals a system for representing number by magnitudes that have scalar variability (a constant coefficient of variation). The mapping of numerical symbols to mental magnitudes provides a formal model of the underlying non-verbal meaning of the symbols (a model of numerical semantics). Animal subjects represent both number and duration by mental magnitudes (Church, 1984; Gibbon, Church, & Meck, 1984; Meck & Church, 1983). These representations are formally analogous to points on the real number line. Meck and Church (1983) proposed such a representation to account for animal non-verbal counts of objects or events (Figure 1). According to their theory each item is enumerated by an impulse of activation which is added to an accumulator. The magnitude in the accumulator at the end of the count is read into memory, where it represents the number of the counted set. The noise (trial-to-trial variability) in these remembered magnitudes is proportional to the magnitude, a property that Gibbon (1977) called scalar variability . Mathematical modeling of psychophysical data from a variety of tasks indicates that memory is the dominant source of trial-...

ebate, this approach may prove useful to probe the attentional demands of other cognitive tasks. Keywords: attention; parietal; subitizing; counting; functional magnetic resonance imaging 1. INTRODUCTION Theories of perception have proposed that human perception operates in two modes (Treisman & Gelade 1980; Eriksen & Yeh 1985). The first is assumed to be preattentive and parallel, in the sense that it can process different bits of information at the same time and before the deployment of focal attention. The second is assumed to be serial and attentive, in the sense that only the stimuli within the current focus of attention can be processed, so that multiple stimuli can only be processed by successively deploying attention towards each of them. In order to determine the experimental conditions under which preattentive parallel processes are sufficient to carry out a given task, reaction time (RT) measures have classically been used. However, chronometric measures are often ambigu

Both animals and humans represent and compare numerical quantities, but only humans have evolved multi-digit place-value number systems. This article develops a Spatial Number Network, or SpaN, model to explain how these shared numerical capabilities are computed using a spatial representation of number quantities in the Where cortical processing stream, notably the inferior parietal cortex. Multi-digit numerical representations that obey a place-value principle are proposed to arise through learned interactions between categorical language representations in the What cortical processing stream and the Where spatial representation. Learned semantic categories that symbolize separate digits, as well as place markers like `ty,' `hundred,' and `thousand,' are associated through learning with the corresponding spatial locations of the Where representation. Such What-to-Where auditory-to-visual learning generates place-value numbers as an emergent property, and may be compared with other examples of multi-modal cross-modality learning, including synesthesia. The model quantitatively simulates error rates in quantification and numerical comparison tasks, and reaction times for number priming and numerical assessment and comparison tasks. In the Where cortical process, transient responses to inputs are integrated before they activate an ordered spatial map that selectively responds to the number of events in a sequence and exhibits Weber law properties. Numerical comparison arises from activity pattern changes across the spatial map that define a `directional comparison wave.' Variants of these model mechanisms have elsewhere been used to explain data about other Where stream phenomena, such as motion perception, spatial attention, and target tracking. The model is compared wi...

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

Current theories of number processing postulate that the human abilities for arithmetic are based on cerebral circuits that are partially laid down under genetic control and later modified by schooling and education. This view predicts the existence of genetic diseases that interfere specifically with components of the number system. Here, we investigate whether Turner syndrome (TS) corresponds to this definition. TS is a genetic disorder which affects one woman in 2500 and is characterized by partial or complete absence of one X chromosome. In addition to well-characterized physical and hormonal dysfunction, TS patients exhibit cognitive deficits including dyscalculia. We tested 12 women with Turner syndrome and 13 control subjects on a cognitive battery including arithmetical tests (addition, subtraction, multiplication, division) as well as tests of the understanding of numerosity and quantity (cognitive estimation, estimation, comparison, bisection, subitizing/counting). Impairments were observed in cognitive estimation, subitizing, and calculation. We examine whether these deficits can be attributed to a single source, and discuss the possible implications of hormonal and genetic factors in the neuropsychological profile of TS patients.

Numerical abilities are thought to rest on the integration of two distinct systems, a verbal system of number words and a non-symbolic representation of approximate quantities. This view has lead to the classification of acalculias into two broad categories depending on whether the deficit affects the verbal or the quantity system. Here, we test the association of deficits predicted by this theory, and particularly the presence or absence of impairments in non-symbolic quantity processing. We describe two acalculic patients, one with a focal lesion of the left parietal lobe and Gerstmann's syndrome and another with semantic dementia with predominantly left temporal hypometabolism. As predicted by a quantity deficit, the first patient was more impaired in subtraction than in multiplication, showed a severe slowness in approximation, and exhibited associated impairments in subitizing and numerical comparison tasks, both with Arabic digits and with arrays of dots. As predicted by a verbal deficit, the second patient was more impaired in multiplication than in subtraction, had intact approximation abilities, and showed preserved processing of non-symbolic numerosities.