‘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 devel‐opments in arithmetic emerge through the establishment of linkages between this core analogical representation 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. (shrink)

Cognitive developmental evidence is sometimes conscripted to support ''naturalized epistemology'' arguments to the effect that a general epistemic stance leads children to build theory-like accounts of underlying properties of kinds. A review of the evidence suggests that what prompts conceptual acquisition is not a general epistemic stance but a series of category-specific intuitive principles that constitute an evolved ''natural metaphysics''. This consists in a system of categories and category-specific inferential processes founded on definite biases in prototype formation. Evidence for this (...) system provides a better understanding of the limited ''plasticity'' of ontological commitments as well as a computationally plausible account of their initial state, avoiding ambiguities about innateness. This may provide a starting point for a ''naturalized epistemology'' that takes into account evolved properties of human conceptual structures. (shrink)

Sexual selection provides an adequate partial explanation for the difference in means between the distributions, but fails to explain the difference in variance, that is, the overrepresentation of both boys with outstanding mathematical reasoning ability and boys with mental retardation. Other genetic factors are probably at work. While spatial ability is correlated with OMRA, so are other cognitive abilities. OMRA is not reducible to spatial ability; hence selection for navigational skill is unlikely to be the only mechanism by which males (...) have gained an advantage in OMRA. (shrink)

Cardinal number knowledge-understanding “two” refers to sets of two entities-is a critical piece of knowledge that predicts later mathematics achievement. Recent studies have shown that domain-general and domain-specific skills can influence children’s cardinal number learning. However, there has not yet been research investigating the influence of domain-specific quantifier knowledge on children’s cardinal number learning. The present study aimed to investigate the influence of domain-general and domain-specific skills on Mandarin Chinese-speaking children’s cardinal number learning after controlling for a number of family (...) background factors. Particular interest was paid to the question whether domain-specific quantifier knowledge was associated with cardinal number development. Specifically, we investigated 2–5-year-old Mandarin Chinese-speaking children’s understanding of cardinal number words as well as their general language, intelligence, approximate number system acuity, and knowledge of quantifiers. Children’s age, gender, parental education, and family income were also assessed and used as covariates. We found that domain-general abilities, including general language and intelligence, did not account for significant additional variance of cardinal number knowledge after controlling for the aforementioned covariates. We also found that domain-specific quantifier knowledge did not account for significant additional variance of cardinal number knowledge, whereas domain-specific ANS acuity accounted for significant additional variance of cardinal number knowledge, after controlling for the aforementioned covariates. In sum, the results suggest that domain-specific numerical skills seem to be more important for children’s development of cardinal number words than the more proximal domain-general abilities such as language abilities and intelligence. The results also highlight the significance of ANS acuity on children’s cardinal number word development. (shrink)

Six-month-old infants discriminate between large sets of objects on the basis of numerosity when other extraneous variables are controlled, provided that the sets to be discriminated differ by a large ratio (8 vs. 16 but not 8 vs. 12). The capacities to represent approximate numerosity found in adult animals and humans evidently develop in human infants prior to language and symbolic counting.

From the perspective of evolutionary theory, we believe it makes more sense to view the sex differences in spatial cognition as being an evolutionary by-product of selection for optimal rates of fetal development. Geary does not convince us that his proposed selective factors operated with “sufficient precision, economy, and efficiency.” Moreover, the archaeological evidence does not support his proposed evolutionary scenario.

The purpose of the study was to investigate how numeracy and acuity of the approximate number system (ANS) relate to the calibration and coherence of probability judgments. Based on the literature on number cognition, a first hypothesis was that those with lower numeracy would maintain a less linear use of the probability scale, contributing to overconfidence and nonlinear calibration curves. A second hypothesis was that also poorer acuity of the ANS would be associated with overconfidence and non-linearity. A third hypothesis, (...) in line with dual-systems theory (e.g., Kahneman and Frederick, 2002 ) was that people higher in numeracy should have better access to the normative probability rules, allowing them to decrease the rate of conjunction fallacies. Data from 213 participants sampled from the Swedish population showed that: (i) in line with the first hypothesis, overconfidence and the linearity of the calibration curves were related to numeracy, where people higher in numeracy were well calibrated with zero overconfidence. (ii) ANS was not associated with overconfidence and non-linearity, disconfirming the second hypothesis. (iii) The rate of conjunction fallacies was slightly, but to a statistically significant degree decreased by numeracy, but still high at all numeracy levels. An unexpected finding was that participants with better ANS acuity gave more realistic estimates of their performance relative to others. (shrink)

Contemporary research in compositional, truth-conditional semantics often takes judgments of the relative unacceptability of certain phrasal combinations as evidence for lexical semantics. For example, observing that completely full sounds perfectly natural whereas completely tall does not has been used to motivate a distinction whereby the lexical entry for full but not for tall specifies a scalar endpoint. So far, such inferences seem unobjectionable. In general, however, applying this methodology can lead to dubious conclusions. For example, observing that slightly bent is (...) natural but slightly cheap is not (that is, not without a “too cheap” interpretation) leads researchers to suggest that the interpretation of bent involves a scalar minimum but cheap does not, contra intuition—after all, one would think that what is minimally cheap is (just) free. Such claims, found in sufficient abundance, raise the question of how we can support semantic theories that posit properties of entities that those entities appear to lack. This paper argues, using theories of adjectival scale structure as a test case, that the (un)acceptability data recruited in semantic explanations reveals properties of a two-stage system of semantic interpretation that can support divergences between our semantic and metaphysical intuitions. (shrink)

It’s often claimed in the philosophical and scientific literature on temporal representation that there is no such thing as a genuine sensory system for time. In this paper, I argue for the opposite—many animals, including all mammals, possess a genuine sensory system for time based in the circadian system. In arguing for this conclusion, I develop a semantics and meta-semantics for explaining how the endogenous rhythms of the circadian system provide organisms with a direct information link to the temporal structure (...) of their environment. In doing so, I highlight the role of sensory systems in an information processing architecture. (shrink)

Glenberg's account falls short in several respects. Besides requiring clearer explication of basic concepts, his account fails to recognize the autonomous nature of perception. His account of what is remembered, and its description, is too static. His strictures against connectionist modeling might be overcome by combining the notions of psychological space and principled learning in an embodied and situated network.

Recent years have witnessed an enormous increase in behavioral and neuroimaging studies of numerical cognition. Particular interest has been devoted toward unraveling properties of the representational medium on which numbers are thought to be represented. We have argued that a correct inference concerning these properties requires distinguishing between different input modalities and different decision/output structures. To back up this claim, we have trained computational models with either symbolic or nonsymbolic input and with different task requirements, and showed that this allowed (...) for an integration of the existing data in a consistent manner. In later studies, predictions from the models were derived and tested with behavioral and neuroimaging methods. Here we present an integrative review of this work. (shrink)

The central problem in Geary's theory is how differences are conceptualized. Recent research has shown that between-sex differences on certain tasks are a consequence of averaging within sex differences. A mixture distribution models between-sex differences on several tasks well and does not appear congenial to a sexual-selection perspective.

I have provided evidence that Geary's model does not explain male dominance in spatial abilities by sexual selection. The current literature concerning the relations of nonverbal IQ to testosterone, hand preference, and right- and left-hand skill, as well as the organizing effects of testosterone on cerebral lateralization during the perinatal period, does not support Geary's arguments.

Previous research reported that college students' symbolic addition and subtraction fluency improved after training with non-symbolic, approximate addition and subtraction. These findings were widely interpreted as strong support for the hypothesis that the Approximate Number System (ANS) plays a causal role in symbolic mathematics, and that this relation holds into adulthood. Here we report four experiments that fail to find evidence for this causal relation. Experiment 1 examined whether the approximate arithmetic training effect exists within a shorter training period than (...) originally reported (2 vs 6 days of training). Experiment 2 attempted to replicate and compare the approximate arithmetic training effect to a control training condition matched in working memory load. Experiments 3 and 4 replicated the original approximate arithmetic training experiments with a larger sample size. Across all four experiments (N = 318) approximate arithmetic training was no more effective at improving the arithmetic fluency of adults than training with control tasks. Results call into question any causal relationship between approximate, non-symbolic arithmetic and precise symbolic arithmetic. (shrink)

Achievement test differences between boys and girls and between young men and young women, mostly favoring males, extend far beyond mathematics. Such pervasive differences, illustrated here, may require an explanatory theory broader than Geary's.

Although debates continue, studies of cognition in infancy suggest that knowledge begins to emerge early in life and constitutes part of humans' innate endowment. Early-developing knowledge appears to be both domain-specific and task-specific, it appears to capture fundamental constraints on ecologically important classes of entities in the child's environment, and it appears to remain central to the commonsense knowledge systems of adults.

Spatial visualization as a key variable in sex-related differences in mathematical problem solving and spatial aspects of geometry is traced to the 1960s. More recent relevant data are presented. The variability debate is traced to the latter part of the nineteenth century and an explanation for it is suggested. An idea is presented for further research to clarify sex-related brain laterality differences in solving spatial problems.

El pensamiento es incorporado por la articulación de los dos algoritmos: la estructura interna para conocer y la realidad, externa, a ser reconocida. La realidad interna está dada desde la estructura de la lengua, la cual más adelante dará lugar a un formato lógica-matemática. La realidad externa, que es también antropológica, está mediada por las áreas somatosensoriales cerebrales, que protológicamente acercan a lo distinto del humano. La filosofía y la ciencia tienen la tarea de acercarse a estas realidades para enhebrarlas, (...) de lo cual resulta un conocimiento general y una ciencia epistemológicamente bien fundada, puesto que hay una estructura ontológica y lógica que les da razón. Desde este fundamento no resultan bien libradas ni el empirismo ni es racionalismo y se abre una ventana para una metafísica menos unilateral y más rigurosa. (shrink)

Spatial and mathematical abilities may be “sex-limited” traits. A sex-limited trait has the same determinants of variation within the sexes, but the genetic or environmental effects would be differentially expressed in males and females. New advances in structural equation modeling allow means and variation to be estimated simultaneously. When these statistical methods are combined with a genetically informative research design, it should be possible to demonstrate that the genes influencing spatial and mathematical abilities are sex-limited in their expression. This approach (...) would give an empirical confirmation of Geary's evolutionary speculations. (shrink)

Many experiments with infants suggest that they possess quantitative abilities, and many experimentalists believe that these abilities set the stage for later mathematics: natural numbers and arithmetic. However, the connection between these early and later skills is far from obvious. We evaluate two possible routes to mathematics and argue that neither is sufficient: (1) We first sketch what we think is the most likely model for infant abilities in this domain, and we examine proposals for extrapolating the natural number concept (...) from these beginnings. Proposals for arriving at natural number by (empirical) induction presuppose the mathematical concepts they seek to explain. Moreover, standard experimental tests for children's understanding of number terms do not necessarily tap these concepts. (2) True concepts of number do appear, however, when children are able to understand generalizations over all numbers; for example, the principle of additive commutativity (a+b=b+a). Theories of how children learn such principles usually rely on a process of mapping from physical object groupings. But both experimental results and theoretical considerations imply that direct mapping is insufficient for acquiring these principles. We suggest instead that children may arrive at natural numbers and arithmetic in a more top-down way, by constructing mathematical schemas. (shrink)

: Fodor argues that perceptual representations are a subset of iconic representations, which are distinguished from symbolic/discursive representations. Iconic representations are nonconceptual and they do not support the abilities afforded by concepts. Iconic representations, for example, cannot support object individuation. If someone thinks that perception or some of its parts has imagistic NCC, they face the following dilemma. Either they will have to accept that this NCC does not allow for object individuation, but it represents instead conglomerations of properties and (...) at some stage of visual processing it must interface with cognition and its conceptual capacities for the visual objects to be individuated. Or, they will have to hold that the imagistic, NCC of perception, allows for object individuation. I opt for the second thesis because I think there is strong empirical evidence that objects are individuated during early vision. I also think that early vision individuates objects by means of, what I had previously called nonconceptual perceptual demonstrative reference. I argue, first, why Fodor’s view that iconic NCC does not enable object individuation is false. I also argue, contra Fodor, that early vision allows the perception of the cardinality of sets of objects. Keywords : Early Vision; Analog Representations; Object Individuation; Arithmetic Cognition; Cardinality of Sets L'individuazione di oggetti mediante il contenuto iconico: come è rappresentata la numerosità nella rappresentazione iconica? Riassunto: Per Fodor le rappresentazioni percettive sono un sottoinsieme delle rappresentazioni iconiche, distinte dalle rappresentazioni simbolico/discorsive. Le rappresentazioni iconiche sono nonconcettuali e non supportano le abilità richieste dai concetti. Le rappresentazioni iconiche, per esempio, non supportano l'individuazione di oggetti. Se si pensa che la percezione o qualche sua parte abbia un contenuto nonconcettuale come immagine, si ci si imbatte nel seguente dilemma. O si accetta che il NCC non permetta di individuare oggetti, ma che rappresenti conglomerati di proprietà e che si interfacci con la cognizione e le sue capacità concettuali, per individuare gli oggetti visivi. O si dice che l'immagine, il NCC, della percezione consenta di individuare oggetti. Propendo per la seconda tesi, poiché penso che ci sia solida evidenza empirica per cui gli oggetti vengono individuati durante le prime fasi della visione, in cui l’individuazione avviene mediante quanto definito come riferimento dimostrativo del contenuto nonconcettuale percettivo. Chiarirò le ragioni per cui la concezione di Fodor, per cui il contenuto nonconcettuale iconico non supporta l’individuazione di oggetti, è falsa. Inoltre, contra Fodor, sosterrò che le prime fasi della visione permettono la percezione della cardinalità di insiemi di oggetti. Parole chiave: Prime fasi della visione; Rappresentazioni analogiche; Individuazione di oggetti; Cognizione aritmetica; Cardinalità degli insiemi. (shrink)

Geary's is the traditional view of the sexes. Yet each part of his argument – the move from sex differences in spatial ability and social preferences to a sex difference in mathematical ability, the claim that the former are biologically primary, and the sociobiological explanation of these differences – requires considerable further work. The notion of a biologically secondary ability is itself problematic.