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...

Three theories currently compete to explain the conceptual deficits that result from brain damage: sensory-functional theory, domain-specific theory, and conceptual structure theory. We argue that all three theories capture important aspects of conceptual deficits, and offer different insights into their origins. Conceptual topography theory (CTT) integrates these insights, beginning with A. R. Damasio’s (1989) convergence zone theory and elaborating it with the similarity-in-topography (SIT) principle. According to CTT, feature maps in sensory-motor systems represent the features of a category’s exemplars. A hierarchical system of convergence zones then conjoins these features to form both property and category representations. According to the SIT principle, the proximity of two conjunctive neurons in a convergence zone increases with the similarity of the features they conjoin. As a result, conjunctive neurons become topographically organised into local regions that represent properties and categories. Depending on the level and location of a lesion in this system, a wide variety of deficits is possible. Consistent with the literature, these deficits range from the loss of a single category to the loss of multiple categories that share sensory-motor properties.

INTRODUCTION How do we go from seeing a word to accessing its meaning? Classical models of word processing postulate that words are initially recognized in modalityspecific input lexicons before contacting a common semantic representation (Caramazza, 1996; Morton, 1979). This predicts that areas which are engaged in semantic-level processing should activate in direct correlation with the amount of semantic manipulation required by the task and do so independent of the modality of presentation of the concept (Chao et al., 2000; Perani et al., 1999; Vandenberghe et al., 1996). Here, we attempt to identify the cerebral areas engaged in the coding and internal manipulation of an abstract semantic content, the meaning of number words. Although numbers can be written in multiple notations, such as words or digits, the parietal lobes are thought to comprise a notation-independent representation of their semantic content as quantities. According to the "triple-code model" of number process

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

We report the case of a patient suffering from a severe neologistic jargon sparing number words. Neologisms resulted from pervasive phoneme substitutions with frequent preservation of the overall syllabic structure (e.g. /revolver / ® /reveltil/). Word and nonword reading, as well as picture naming, were equally affected. No significant influence of frequency, imageability, and grammatical class was found. In striking contrast with this severe speech impairment, the patient made virtually no phonological errors when reading aloud arabic or spelled-out numerals, but made frequent word selection errors (e.g. 250 ® “four hundred and sixty”). This observation indicates that during speech planning, different categories of words are processed by separable brain systems down to the level of phoneme selection, a more peripheral level than was previously assumed. Number words may be singled out during phonological processing either because they constitute a particular semantic category, or because they benefit from special brain mechanisms devoted to the production of “automatic speech”, or because they are the elementary building blocks of speech during the production of complex numerals.

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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.

The learning of mathematics is of universal interest across all OECD countries. In modern cultures, all children are expected to master up to at least 15 years of formal instruction in mathematics. Innumeracy has serious negative consequences for society. Mathematics is important not only for its own sake but also because it allows citizens to operate more effectively in their day-to-day lives. People need to balance checkbooks, make payments on items, and plan for future retirement. The topics studied by professional mathematicians extend far beyond everyday arithmetic or algebra. For example, some mathematicians devote themselves to the study of topology, others are interested in analysis or non-Euclidian geometries. The limitations of brain-imaging technologies, however, require researchers to simplify the cognitive demands facing subjects and to focus on simpler (but more common) mathematics topics. As we have seen in the chapter on reading, brain-imaging technologies can model relatively simple tasks. Thus, even though researchers who study the relationship between brain learning and mathematics hope someday to find brain activity correlates for all of mathematics, the current focus is on the learning of small whole numbers,

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.