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.

Did evolution endow the human brain with a predisposition to represent and acquire knowledge about numbers? Although the parietal lobe...

Does visual imagery engage some of the same representations used in visual perception? The evidence collected by cognitive psychologists in support of this claim has been challenged by three types of alternative explanation: Tacit knowledge, according to which subjects use nonvisual representations to simulate the use of visual representations during imagery tasks, guided by their tacit knowledge of their visual systems; experimenter expectancy, according to which the data implicating shared representations for imagery and perception is an artifact of experimenter expectancies; and nonvisual spatial representation, according to which imagery representations are partially similar to visual representations in the way they code spatial relations but are not visual representations. This article reviews previously overlooked neuropsychological evidence on the relation between imagery and perception, and discusses its relative immunity to the foregoing alternative explanations. This evidence includes electrophysiological and cerebral blood flow studies localizing brain activity during imagery to cortical visual areas, and parallels between the selective effects of brain damage on visual perception and imagery. Because these findings cannot be accounted for in the same way as traditional cognitive data using the alternative explanations listed earlier, they can play a decisive role in answering the title question.

Related perceptual, motor, and cognitive performances were examined to reveal the accuracy of the properties of action spontaneously represented when mentally simulating moving one's hand. The kinematic configuration of the body represented and transformed in mental simulations was not fixed or canonical but corresponded to one's current configuration. Mental simulation time mimicked movement time for natural efficient movement from a posture midway between each of the hand's joint limits into many other postures. Equal time was required for simulated and real movements into more common, comfortable postures; shorter but proportional time was required for simulated movement than real movement into less common postures that involved longer trajectories, coordinated activity at more joints, motion near extremes of joint limits, and uncomfortable kinesthetic sensations. The findings suggest that sensorimotor structures support mental simulations of actions. Humans can envision an object, scene, or event and then inspect the mental representation in a manner that mimics or reflects real perceptual-motor performance (e.g., Craik,

We measured cerebral activation with functional magnetic resonance imaging at 3 Tesla while eight healthy volunteers performed various number processing tasks known to be dissociable in brain-lesioned patients: naming, comparing, multiplying, or subtracting single digits. The results revealed the activation of a circuit comprising bilateral intraparietal, prefrontal, and anterior cingulate components. The extension and lateralization of this circuit was modulated by task demands. The intraparietal and prefrontal activation was more important in the right hemisphere during the comparison task and in the left hemisphere during the multiplication task and was intensely bilateral during the subtraction task. Thus, partially distinct cerebral circuits with the dorsal parietal pathway underlie distinct arithmetic operations.

Is our knowledge about the appearance of objects more closely related to verbal thought or to perception? In a behavioural study using a property verification task, Kosslyn (1976) reported that there are both amodal and perceptual representations of concepts, but that amodal representations may be more easily accessed. However, Solomon (1997) argued that due to the nature of Kosslyn’s stimuli, subjects may be able to bypass semantics entirely and perform this task using differences in the strength of association between words in true trials (e.g., cat–whiskers) and those in false trials (e.g., mouse–stinger). Solomon found no evidence for amodal representations when the task materials were altered to include associated false trials (e.g., cat–litter), which require semantic processing, as opposed to associative strategies. In the current study, we used fMRI to examine the response of regions of visual association cortex while subjects performed a property verification task with either associated or unassociated false trials. We found reliable activity across subjects within the left fusiform gyrus when true trials were intermixed with associated false trials but not when true trials were intermixed with unassociated false trials. Our data support the idea that conceptual knowledge is organised visually and that it is grounded in the perceptual system. One of the leading theories of the organisation of

INTRODUCTION The goal of the present work is to examine whether the semantic representations of numbers and body parts are associated with partially distinct cortical territories. Clinical and cognitive neuropsychology studies associate semantic deficits in both domains to lesions coarsely localized to the left parietal lobe (McCarthy and Warrington, 1990). Furthermore, patients with left inferior parietal lesions often exhibit simultaneous deficits for numbers and body parts (Benton, 1992; Gerstmann, 1940). Such an association of neuropsychological deficits is however notoriously ambiguous, and has been the subject of much debate. It might suggest that there is a shared substrate for numbers and body parts in the left parietal region, perhaps based on a common functional system for spatial representation and manipulation (Gerstmann, 1940) or on the crucial role that finger counting plays in numerical development (Butterworth, 1999). However, it might also reflect the existence of dis

Mathematicians frequently evoke their “intuition” when they are able to quickly and automatically solve a problem, with little introspection into their insight. Cognitive neuroscience research shows that mathematical intuition is a valid concept that can be studied in the laboratory in reduced paradigms, and that relates to the availability of “core knowledge” associated with evolutionarily ancient and specialized cerebral subsystems. As an illustration, I discuss the case of elementary arithmetic. Intuitions of numbers and their elementary transformations by addition and subtraction are present in all human cultures. They relate to a brain system, located in the intraparietal sulcus of both hemispheres, which extracts numerosity of sets and, in educated adults, maps back and forth between numerical symbols and the corresponding quantities. This system is available to animal species and to preverbal human infants. Its neuronal organization is increasingly being uncovered, leading to a precise mathematical theory of how we perform tasks of number comparison or number naming. The next challenge will be to understand how education changes our core intuitions of number.

xxi I General

Abstract. Connectionist models based on activation spreading and attractor dynamics are functionally limited by representational and processing flexibility constraints, the ‘feature binding problem ’ and the need to balance accurately activation and inhibition. We suggest an alternative approach, in which network units are characterized by two variables: activation and phase. Whereas activation evolves according to a ‘classical ’ connectionist rule, the phase variable is characterized by a chaotic evolution. We present a model of memory retrieval with reference to the paradigmatic McClelland’s 1981 ‘Jets and Sharks ’ model. The model solves the ‘multiple reinstantiation problem’, i.e. the problem of retrieval of multiple items with overlapping features, implied by its classical predecessor. In our network, multiple pattern reinstantiation in terms of activation spreading is disambiguate through selective and differential coherence patterns. The system �exibly represents pattern similarity and feature relationships by means of graded and intermittent synchrony. The domain-general implications of this approach for connectionist ‘interactive activation models ’ and its neurophysiological plausibility are discussed.