Quantifier expressions like “many” and “at least” are part of a rich repository of words in language representing magnitude information. The role of numerical processing in comprehending quantifiers was studied in a semantic truth value judgment task, asking adults to quickly verify sentences about visual displays using numerical or proportional quantifiers. The visual displays were composed of systematically varied proportions of yellow and blue circles. The results demonstrated that numerical estimation and numerical reference information are fundamental in encoding the meaning (...) of quantifiers in terms of response times and acceptability judgments. However, a difference emerges in the comparison strategies when a fixed external reference numerosity is used for numerical quantifiers, whereas an internal numerical criterion is invoked for proportional quantifiers. Moreover, for both quantifier types, quantifier semantics and its polarity biased the response direction. Overall, our results indicate that quantifier comprehension involves core numerical and lexical semantic properties, demonstrating integrated processing of language and numbers. (shrink)

When young children attempt to locate numbers along a number line, they show logarithmic (or other compressive) placement. For example, the distance between “5” and “10” is larger than the distance between “75” and “80.” This has often been explained by assuming that children have a logarithmically scaled mental representation of number (e.g., Berteletti, Lucangeli, Piazza, Dehaene, & Zorzi, 2010; Siegler & Opfer, 2003). However, several investigators have questioned this argument (e.g., Barth & Paladino, 2011; Cantlon, Cordes, Libertus, & Brannon, (...) 2009; Cohen & Blanc-Goldhammer, 2011). We show here that children prefer linear number lines over logarithmic lines when they do not have to deal with the meanings of individual numerals (i.e., number symbols, such as “5” or “80”). In Experiments 1 and 2, when 5- and 6- year-olds choose between number lines in a forced-choice task, they prefer linear to logarithmic and exponential displays. However, this preference does not persist when Experiment 3 presents the same lines without reference to numbers, and children simply choose which line they like best. In Experiments 4 and 5, children position beads on a number line to indicate how the integers 1 100 are arranged. The bead placement of 4- and 5-year-olds is better fit by a linear than by a logarithmic model. We argue that previous results from the number line task may depend on strategies specific to the task. (shrink)

Despite their importance in public discourse, numbers in the range of 1 million to 1 trillion are notoriously difficult to understand. We examine magnitude estimation by adult Americans when placing large numbers on a number line and when qualitatively evaluating descriptions of imaginary geopolitical scenarios. Prior theoretical conceptions predict a log-to-linear shift: People will either place numbers linearly or will place numbers according to a compressive logarithmic or power-shaped function (Barth & Paladino, ; Siegler & Opfer, ). While about half (...) of people did estimate numbers linearly over this range, nearly all the remaining participants placed 1 million approximately halfway between 1 thousand and 1 billion, but placed numbers linearly across each half, as though they believed that the number words “thousand, million, billion, trillion” constitute a uniformly spaced count list. Participants in this group also tended to be optimistic in evaluations of largely ineffective political strategies, relative to linear number-line placers. The results indicate that the surface structure of number words can heavily influence processes for dealing with numbers in this range, and it can amplify the possibility that analogous surface regularities are partially responsible for parallel phenomena in children. In addition, these results have direct implications for lawmakers and scientists hoping to communicate effectively with the public. (shrink)

How do people stretch their understanding of magnitude from the experiential range to the very large quantities and ranges important in science, geopolitics, and mathematics? This paper empirically evaluates how and whether people make use of numerical categories when estimating relative magnitudes of numbers across many orders of magnitude. We hypothesize that people use scale words—thousand, million, billion—to carve the large number line into categories, stretching linear responses across items within each category. If so, discontinuities in position and response time (...) are expected near the boundaries between categories. In contrast to previous work that suggested only that a minority of college undergraduates employed categorical boundaries, we find that discontinuities near category boundaries occur in most or all participants, but that accurate and inaccurate participants respond in opposite ways to category boundaries. Accurate participants highlight contrasts within a category, whereas inaccurate participants adjust their responses toward category centers. (shrink)

There is at present a lively debate in cognitive psychology concerning the origin of natural number concepts. At the center of this debate is the system of mental magnitudes, an innately given cognitive mechanism that represents cardinality and that performs a variety of arithmetical operations. Most participants in the debate argue that this system cannot be the sole source of natural number concepts, because they take it to represent cardinality approximately while natural number concepts are precise. In this paper, I (...) argue that the claim that mental magnitudes represent cardinality approximately overlooks the distinction between a magnitude and the increments that compose to form that magnitude. While magnitudes do indeed represent cardinality approximately, they are composed of a precise number of increments. I argue further that learning the number words and the counting routine may allow one to mark in memory the number of increments that composed to form a magnitude, thereby creating a precise representation of cardinality. (shrink)

In this paper, we present a battery of empirical findings on the relationship between cultural context and theory of mind that show great variance in the onset and character of mindreading in different cultures; discuss problems that those findings cause for the largely-nativistic outlook on mindreading dominating in the literature; and point to an alternative framework that appears to better accommodate the evident cross-cultural variance in mindreading. We first outline the theoretical frameworks that dominate in mindreading research, then present the (...) relevant empirical findings, and finally we come back to the theoretical approaches in a discussion of their explanatory potential in the face of the data presented. The theoretical frameworks discussed are the two-systems approach; performance-based approach also known as modularity-nativist approach; and the social-communicative theory also known as the systems, relational-systems, dynamic systems and developmental systems theory. The former two, which both fall within the wider modular-computational paradigm, run into a challenge with the cross-cultural data presented, and the latter - the systemic framework - seems to offer an explanatorily potent alternative. The empirical data cited in this paper comes from research on cross-cultural differences in folk psychology and theory-of-mind development; the influence of parenting practices on the development of theory of mind; the development and character of theory of mind in deaf populations; and neuroimaging research of cultural differences in mindreading. (shrink)

In spite of their practical importance, the connections between technology and mathematics have not received much scholarly attention. This article begins by outlining how the technology–mathematics relationship has developed, from the use of simple aide-mémoires for counting and arithmetic, via the use of mathematics in weaving, building and other trades, and the introduction of calculus to solve technological problems, to the modern use of computers to solve both technological and mathematical problems. Three important philosophical issues emerge from this historical résumé: (...) how mathematical knowledge depends on technology, the definition of the hybrid concept of a computation, and the usefulness of mathematics in technology. Each of these issues is briefly discussed, and it is shown that in order to analyze them, we need to combine tools and ideas from both the philosophy of technology and the philosophy of mathematics. In conclusion, it is argued that much more of interest can be found in the historically and philosophically unexplored terrains of the technology–mathematics relationship. (shrink)

Several advocates of the lively field of “metaphysics of science” have recently argued that a naturalistic metaphysics should be based solely on current science, and that it should replace more traditional, intuition-based, forms of metaphysics. The aim of the present paper is to assess that claim by examining the relations between metaphysics of science and general metaphysics. We show that the current metaphysical battlefield is richer and more complex than a simple dichotomy between “metaphysics of science” and “traditional metaphysics”, and (...) that it should instead be understood as a three dimensional “box”, with one axis distinguishing “descriptive metaphysics” from “revisionary metaphysics”, a second axis distinguishing a priori from a posteriori metaphysics, and a third axis distinguishing “commonsense metaphysics”, “traditional metaphysics” and “metaphysics of science”. We use this three-dimensional figure to shed light on the project of current metaphysics of science, and to demonstrate that, in many instances, the target of that project is not defined with enough precision and clarity. (shrink)

Much research supports the existence of an Approximate Number System (ANS) that is recruited by infants, children, adults, and non-human animals to generate coarse, non-symbolic representations of number. This system supports simple arithmetic operations such as addition, subtraction, and ordering of amounts. The current study tests whether an intuition of a more complex calculation, division, exists in an indigene group in the Amazon, the Mundurucu, whose language includes no words for large numbers. Mundurucu children were presented with a video event (...) depicting a division transformation of halving, in which pairs of objects turned into single objects, reducing the array's numerical magnitude. Then they were tested on their ability to calculate the outcome of this division transformation with other large-number arrays. The Mundurucu children effected this transformation even when non-numerical variables were controlled, performed above chance levels on the very first set of test trials, and exhibited performance similar to urban children who had access to precise number words and a surrounding symbolic culture. We conclude that a halving calculation is part of the suite of intuitive operations supported by the ANS. (shrink)

Cohen Kadosh & Walsh (CK&W) present convincing evidence indicating the existence of notation-specific numerical representations in parietal cortex. We suggest that the same conclusions can be drawn for a particular type of numerical representation: the representation of time. Notation-dependent representations need not be limited to number but may also be extended to other magnitude-related contents processed in parietal cortex (Walsh 2003).

One of the important discussions in the philosophy of mathematics, is that centered on Benacerraf’s Dilemma. Benacerraf’s dilemma challenges theorists to provide an epistemology and semantics for mathematics, based on their favourite ontology. This challenge is the point on which all philosophies of mathematics are judged, and clarifying how we might acquire mathematical knowledge is one of the main occupations of philosophers of mathematics. In this thesis I argue that this discussion has overlooked an important part of mathematics, namely mathematics (...) as it is exercised by ordinary people. I do so by looking at the different theories that have been put forward in the recent debate, and showing for each of these that they are unable to account for the mathematical practices of ordinary people. In order to show that these practices do need to be accounted for, I also argue that ordinary people are doing mathematics, i.e. that they engage in properly mathematical practices. Because these practices are properly mathematical, they should be accounted for by any philosophy of mathematics. The conclusion of my thesis, then, is that current theories fail to do something that they should do, while remaining neutral on how well they perform when it comes to accounting for the practices of professional mathematicians. (shrink)

Tutkin tässä artikkelissa Kurt Gödelin epätäydellisyysteoreemojen tulkintoja filosofiassa. Aihepiiri kattaa valtavan määrän eri tulkintoja tekoälystä fysiikkaan ja runouteen asti. Osoitan, että kriittisesti tarkasteltuna kaikki radikaalit epätäydellisyysteoreemojen sovellukset ovat virheellisiä.

What is the nature of number systems and arithmetic that we use in science for quantification, analysis, and modeling? I argue that number concepts and arithmetic are neither hardwired in the brain, nor do they exist out there in the universe. Innate subitizing and early cognitive preconditions for number— which we share with many other species—cannot provide the foundations for the precision, richness, and range of number concepts and simple arithmetic, let alone that of more complex mathematical concepts. Numbers and (...) arithmetic, and mathematics in general, have unique features—precision, objectivity, rigor, generalizability, stability, symbolizability, and applicability to the real world—that must be accounted for. They are sophisticated concepts that developed culturally only in recent human history. I suggest that numbers and arithmetic are realized through precise combinations of non-mathematical everyday cognitive mechanisms that make human imagination and abstraction possible. One such mechanism, conceptual metaphor, is a neurally instantiated inference-preserving cross-domain mapping that allows the conceptualization of abstract entities in terms of grounded bodily experience. I analyze how the inferential organization of the properties and “laws” of arithmetic emerge metaphorically from everyday meaningful actions. Numbers and arithmetic are thus—outside of natural selection—the product of the biologically constrained interaction of individuals with the appropriate cultural and historical phenotypic variation supported by language, writing systems, and education. (shrink)

Human cognition is extended and enacted. Drawing the boundaries of cognition to include the resources and attributes of the body and materiality allows an examination of how these components interact with the brain as a system, especially over cultural and evolutionary spans of time. Literacy and numeracy provide examples of multigenerational, incremental change in both psychological functioning and material forms. Though we think materiality, its central role in human cognition is often unappreciated, for reasons that include conceptual distribution over multiple (...) material forms, the unconscious transparency of cognitive activity in general, and the different temporalities of metaplastic change in neurons and cultural forms. (shrink)

In historical claims for nativism, mathematics is a paradigmatic example of innate knowledge. Claims by contemporary developmental psychologists of elementary mathematical skills in human infants are a legacy of this. However, the connection between these skills and more formal mathematical concepts and methods remains unclear. This paper assesses the current debates surrounding nativism and mathematical knowledge by teasing them apart into two distinct claims. First, in what way does the experimental evidence from infants, nonhuman animals and neuropsychology support the nativist (...) hypothesis? Second, granting that infants have some elementary mathematical skills, does this mean that such skills play an important role in the development of mathematical knowledge? (shrink)

Article Authors Metrics Comments Media Coverage Abstract Author Summary Introduction Results Discussion Supporting information Acknowledgments Author Contributions References Reader Comments (0) Media Coverage (0) Figures Abstract During language processing, humans form complex embedded representations from sequential inputs. Here, we ask whether a “geometrical language” with recursive embedding also underlies the human ability to encode sequences of spatial locations. We introduce a novel paradigm in which subjects are exposed to a sequence of spatial locations on an octagon, and are asked to (...) predict future locations. The sequences vary in complexity according to a well-defined language comprising elementary primitives and recursive rules. A detailed analysis of error patterns indicates that primitives of symmetry and rotation are spontaneously detected and used by adults, preschoolers, and adult members of an indigene group in the Amazon, the Munduruku, who have a restricted numerical and geometrical lexicon and limited access to schooling. Furthermore, subjects readily combine these geometrical primitives into hierarchically organized expressions. By evaluating a large set of such combinations, we obtained a first view of the language needed to account for the representation of visuospatial sequences in humans, and conclude that they encode visuospatial sequences by minimizing the complexity of the structured expressions that capture them. (shrink)

Kant argued that Euclidean geometry is synthesized on the basis of an a priori intuition of space. This proposal inspired much behavioral research probing whether spatial navigation in humans and animals conforms to the predictions of Euclidean geometry. However, Euclidean geometry also includes concepts that transcend the perceptible, such as objects that are infinitely small or infinitely large, or statements of necessity and impossibility. We tested the hypothesis that certain aspects of nonperceptible Euclidian geometry map onto intuitions of space that (...) are present in all humans, even in the absence of formal mathematical education. Our tests probed intuitions of points, lines, and surfaces in participants from an indigene group in the Amazon, the Mundurucu, as well as adults and age-matched children controls from the United States and France and younger US children without education in geometry. The responses of Mundurucu adults and children converged with that of mathematically educated adults and children and revealed an intuitive understanding of essential properties of Euclidean geometry. For instance, on a surface described to them as perfectly planar, the Mundurucu's estimations of the internal angles of triangles added up to ∼180 degrees, and when asked explicitly, they stated that there exists one single parallel line to any given line through a given point. These intuitions were also partially in place in the group of younger US participants. We conclude that, during childhood, humans develop geometrical intuitions that spontaneously accord with the principles of Euclidean geometry, even in the absence of training in mathematics. (shrink)

Des branches entières des mathématiques sont fondées sur des liens posés entre les nombres et l’espace : mesure de longueurs, définition de repères et de coordonnées, projection des nombres complexes sur le plan… Si les nombres complexes, comme l’utilisation de repères, sont apparus relativement récemment (vers le XVIIe siècle), la mesure des longueurs est en revanche un procédé très ancien, qui remonte au moins au 3e ou 4e millénaire av. J-C. Loin d’être fortuits, ces liens entre les nombres et l’espace (...) reflèteraient une intuition fondamentale, universelle, façonnée au cours des millénaires par la sélection naturelle, et qui aurait servi de guide et d’inspiration aux mathématiciens au fil des siècles. (shrink)

Developing earlier studies of the system of numbers in Mundurucu, this paper argues that the Mundurucu numeral system is far more complex than usually assumed. The Mundurucu numeral system provides indirect but insightful arguments for a modular approach to numbers and numerals. It is argued that distinct components must be distinguished, such as a system of representation of numbers in the format of internal magnitudes, a system of representation for individuals and sets, and one-to-one correspondences between the numerosity expressed by (...) the number and its metrics. It is shown that while many-number systems involve a compositionality of units, sets and sets composed of units, few-number languages, such as Mundurucu, do not have access to sets composed of units in the usual way. The nonconfigurational character of the Mundurucu language, which is related to a property for which we coin the term 'low compositionality power', accounts for this and explains the curious fact that Mundurucus make use of marked one-to-one correspondence strategies in order to overcome the limitations of the core system at the perceptual/motor interface of the language faculty. We develop an analysis of a particular construction, parallel numbers, which has not been studied before, elucidating the whole system. This analysis, we argue, sheds new light on classical philosophical, psychological and linguistic debates about numbers and numerals and their relation to language, and more particularly, sheds light on few-number languages. (shrink)

A crucial aspect of the human mind is the ability to project the self along the time line to past and future. It has been argued that such self-projection is essential to re-experience past experiences and predict future events. In-depth analysis of a novel paradigm investigating mental time shows that the speed of this “self-projection” in time depends logarithmically on the temporal-distance between an imagined “location” on the time line that participants were asked to imagine and the location of another (...) imagined event from the time line. This logarithmic pattern suggests that events in human cognition are spatially mapped along an imagery mental time line. We argue that the present time-line data are comparable to the spatial mapping of numbers along the mental number line and that such spatial maps are a fundamental basis for cognition. (shrink)

Characterizing different kinds of representation is of fundamental importance to cognitive science, and one traditional way of doing so is in terms of the analog–digital distinction. Indeed the distinction is often appealed to in ways both narrow and broad. In this paper I argue that the analog–digital distinction does not apply to representational schemes but only to representational systems, where a representational system is constituted by a representational scheme and its user, and that whether a representational system is analog or (...) non-analog depends on facts about that user. This aspect of the distinction has gone unnoticed, and I argue that the failure to notice it can be an impediment to scientific progress. (shrink)