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  1. Single-tape and multi-tape Turing machines through the lens of the Grossone methodology.Yaroslav Sergeyev & Alfredo Garro - 2013 - Journal of Supercomputing 65 (2):645-663.
    The paper investigates how the mathematical languages used to describe and to observe automatic computations influence the accuracy of the obtained results. In particular, we focus our attention on Single and Multi-tape Turing machines which are described and observed through the lens of a new mathematical language which is strongly based on three methodological ideas borrowed from Physics and applied to Mathematics, namely: the distinction between the object (we speak here about a mathematical object) of an observation and the instrument (...)
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  • Beyond the Number Domain.Elizabeth M. Brannon Jessica F. Cantlon, Michael L. Platt - 2009 - Trends in Cognitive Sciences 13 (2):83.
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  • How Does the Mind Work? Insights from Biology.Gary Marcus - 2009 - Topics in Cognitive Science 1 (1):145-172.
    Cognitive scientists must understand not just what the mind does, but how it does what it does. In this paper, I consider four aspects of cognitive architecture: how the mind develops, the extent to which it is or is not modular, the extent to which it is or is not optimal, and the extent to which it should or should not be considered a symbol‐manipulating device (as opposed to, say, an eliminative connectionist network). In each case, I argue that insights (...)
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  • Monkeys match and tally quantities across senses.Elizabeth M. Brannon Kerry E. Jordan, Evan L. MacLean - 2008 - Cognition 108 (3):617.
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  • Sampling from the mental number line: How are approximate number system representations formed?Matthew Inglis & Camilla Gilmore - 2013 - Cognition 129 (1):63-69.
    Nonsymbolic comparison tasks are commonly used to index the acuity of an individual's Approximate Number System (ANS), a cognitive mechanism believed to be involved in the development of number skills. Here we asked whether the time that an individual spends observing numerical stimuli influences the precision of the resultant ANS representations. Contrary to standard computational models of the ANS, we found that the longer the stimulus was displayed, the more precise was the resultant representation. We propose an adaptation of the (...)
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  • Calibrating the mental number line.Véronique Izard & Stanislas Dehaene - 2008 - Cognition 106 (3):1221-1247.
    Human adults are thought to possess two dissociable systems to represent numbers: an approximate quantity system akin to a mental number line, and a verbal system capable of representing numbers exactly. Here, we study the interface between these two systems using an estimation task. Observers were asked to estimate the approximate numerosity of dot arrays. We show that, in the absence of calibration, estimates are largely inaccurate: responses increase monotonically with numerosity, but underestimate the actual numerosity. However, insertion of a (...)
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  • WEIRD languages have misled us, too.Asifa Majid & Stephen C. Levinson - 2010 - Behavioral and Brain Sciences 33 (2-3):103-103.
    The linguistic and cognitive sciences have severely underestimated the degree of linguistic diversity in the world. Part of the reason for this is that we have projected assumptions based on English and familiar languages onto the rest. We focus on some distortions this has introduced, especially in the study of semantics.
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  • The weirdest people in the world?Joseph Henrich, Steven J. Heine & Ara Norenzayan - 2010 - Behavioral and Brain Sciences 33 (2-3):61-83.
    Behavioral scientists routinely publish broad claims about human psychology and behavior in the world's top journals based on samples drawn entirely from Western, Educated, Industrialized, Rich, and Democratic (WEIRD) societies. Researchers – often implicitly – assume that either there is little variation across human populations, or that these “standard subjects” are as representative of the species as any other population. Are these assumptions justified? Here, our review of the comparative database from across the behavioral sciences suggests both that there is (...)
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  • Précis of the origin of concepts.Susan Carey - 2011 - Behavioral and Brain Sciences 34 (3):113-124.
    A theory of conceptual development must specify the innate representational primitives, must characterize the ways in which the initial state differs from the adult state, and must characterize the processes through which one is transformed into the other. The Origin of Concepts (henceforth TOOC) defends three theses. With respect to the initial state, the innate stock of primitives is not limited to sensory, perceptual, or sensorimotor representations; rather, there are also innate conceptual representations. With respect to developmental change, conceptual development (...)
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  • Quantitative Standards for Absolute Linguistic Universals.Steven T. Piantadosi & Edward Gibson - 2014 - Cognitive Science 38 (4):736-756.
    Absolute linguistic universals are often justified by cross-linguistic analysis: If all observed languages exhibit a property, the property is taken to be a likely universal, perhaps specified in the cognitive or linguistic systems of language learners and users. In many cases, these patterns are then taken to motivate linguistic theory. Here, we show that cross-linguistic analysis will very rarely be able to statistically justify absolute, inviolable patterns in language. We formalize two statistical methods—frequentist and Bayesian—and show that in both it (...)
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  • Working Memory in Nonsymbolic Approximate Arithmetic Processing: A Dual‐Task Study With Preschoolers.Iro Xenidou‐Dervou, Ernest C. D. M. Lieshout & Menno Schoot - 2014 - Cognitive Science 38 (1):101-127.
    Preschool children have been proven to possess nonsymbolic approximate arithmetic skills before learning how to manipulate symbolic math and thus before any formal math instruction. It has been assumed that nonsymbolic approximate math tasks necessitate the allocation of Working Memory (WM) resources. WM has been consistently shown to be an important predictor of children's math development and achievement. The aim of our study was to uncover the specific role of WM in nonsymbolic approximate math. For this purpose, we conducted a (...)
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  • Non-symbolic halving in an amazonian indigene group.Koleen McCrink, Elizabeth Spelke, Stanislas Dehaene & Pierre Pica - 2013 - Developmental Science 16 (3):451-462.
    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 (...)
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  • The mapping of numbers on space : Evidence for a logarithmic Intuition.Véronique Izard, Pierre Pica, Elizabeth Spelke & Stanislas Dehaene - 2008 - Médecine/Science 24 (12):1014-1016.
    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 (...)
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  • Comparing biological motion in two distinct human societies.Pierre Pica, Stuart Jackson, Randolph Blake & Nikolaus Troje - 2011 - PLoS ONE 6 (12):e28391.
    Cross cultural studies have played a pivotal role in elucidating the extent to which behavioral and mental characteristics depend on specific environmental influences. Surprisingly, little field research has been carried out on a fundamentally important perceptual ability, namely the perception of biological motion. In this report, we present details of studies carried out with the help of volunteers from the Mundurucu indigene, a group of people native to Amazonian territories in Brazil. We employed standard biological motion perception tasks inspired by (...)
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  • Numerical Architecture.Eric Mandelbaum - 2013 - Topics in Cognitive Science 5 (1):367-386.
    The idea that there is a “Number Sense” (Dehaene, 1997) or “Core Knowledge” of number ensconced in a modular processing system (Carey, 2009) has gained popularity as the study of numerical cognition has matured. However, these claims are generally made with little, if any, detailed examination of which modular properties are instantiated in numerical processing. In this article, I aim to rectify this situation by detailing the modular properties on display in numerical cognitive processing. In the process, I review literature (...)
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  • Flexible intuitions of Euclidean geometry in an Amazonian indigene group.Pierre Pica, Véronique Izard, Elizabeth Spelke & Stanislas Dehaene - 2011 - Pnas 23.
    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 (...)
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  • Where the Sidewalk Ends: The Limits of Social Constructionism.David Peterson - 2012 - Journal for the Theory of Social Behaviour 42 (4):465-484.
    The sociology of knowledge is a heterogeneous set of theories which generally focuses on the social origins of meaning. Strong arguments, epitomized by Durkheim's late work, have hypothesized that the very concepts our minds use to structure experience are constructed through social processes. This view has come under attack from theorists influenced by recent work in developmental psychology that has demonstrated some awareness of these categories in pre-socialized infants. However, further studies have shown that the innate abilities infants display differ (...)
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  • Quantity Recognition Among Speakers of an Anumeric Language.Caleb Everett & Keren Madora - 2012 - Cognitive Science 36 (1):130-141.
    Recent research has suggested that the Pirahã, an Amazonian tribe with a number-less language, are able to match quantities > 3 if the matching task does not require recall or spatial transposition. This finding contravenes previous work among the Pirahã. In this study, we re-tested the Pirahãs’ performance in the crucial one-to-one matching task utilized in the two previous studies on their numerical cognition, as well as in control tasks requiring recall and mental transposition. We also conducted a novel quantity (...)
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  • Concrete magnitudes: From numbers to time.Christine Falter, Valdas Noreika, Julian Kiverstein & Bruno Mölder - 2009 - Behavioral and Brain Sciences 32 (3-4):335-336.
    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).
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  • Generalized Quantifiers and Number Sense.Robin Clark - 2011 - Philosophy Compass 6 (9):611-621.
    Generalized quantifiers are functions from pairs of properties to truth-values; these functions can be used to interpret natural language quantifiers. The space of such functions is vast and a great deal of research has sought to find natural constraints on the functions that interpret determiners and create quantifiers. These constraints have demonstrated that quantifiers rest on number and number sense. In the first part of the paper, we turn to developing this argument. In the remainder, we report on work in (...)
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  • Indexed Natural Numbers in Mind: A Formal Model of the Basic Mature Number Competence. [REVIEW]Wojciech Krysztofiak - 2012 - Axiomathes 22 (4):433-456.
    The paper undertakes three interdisciplinary tasks. The first one consists in constructing a formal model of the basic arithmetic competence, that is, the competence sufficient for solving simple arithmetic story-tasks which do not require any mathematical mastery knowledge about laws, definitions and theorems. The second task is to present a generalized arithmetic theory, called the arithmetic of indexed numbers (INA). All models of the development of counting abilities presuppose the common assumption that our simple, folk arithmetic encoded linguistically in the (...)
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  • Interface transparency and the psychosemantics of most.Jeffrey Lidz, Paul Pietroski, Tim Hunter & Justin Halberda - 2011 - Natural Language Semantics 19 (3):227-256.
    This paper proposes an Interface Transparency Thesis concerning how linguistic meanings are related to the cognitive systems that are used to evaluate sentences for truth/falsity: a declarative sentence S is semantically associated with a canonical procedure for determining whether S is true; while this procedure need not be used as a verification strategy, competent speakers are biased towards strategies that directly reflect canonical specifications of truth conditions. Evidence in favor of this hypothesis comes from a psycholinguistic experiment examining adult judgments (...)
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  • Children’s understanding of the relationship between addition and subtraction.Camilla K. Gilmore & Elizabeth S. Spelke - 2008 - Cognition 107 (3):932-945.
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  • Preschool children master the logic of number word meanings.Jennifer S. Lipton & Elizabeth S. Spelke - 2006 - Cognition 98 (3):57-66.
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  • Observability of Turing Machines: a refinement of the theory of computation.Yaroslav Sergeyev & Alfredo Garro - 2010 - Informatica 21 (3):425–454.
    The Turing machine is one of the simple abstract computational devices that can be used to investigate the limits of computability. In this paper, they are considered from several points of view that emphasize the importance and the relativity of mathematical languages used to describe the Turing machines. A deep investigation is performed on the interrelations between mechanical computations and their mathematical descriptions emerging when a human (the researcher) starts to describe a Turing machine (the object of the study) by (...)
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  • Linguistic Determinism and the Innate Basis of Number.Stephen Laurence & Eric Margolis - 2005 - In Peter Carruthers, Stephen Laurence & Stephen P. Stich (eds.), The Innate Mind: Structure and Contents. New York, US: Oxford University Press on Demand.
    Strong nativist views about numerical concepts claim that human beings have at least some innate precise numerical representations. Weak nativist views claim only that humans, like other animals, possess an innate system for representing approximate numerical quantity. We present a new strong nativist model of the origins of numerical concepts and defend the strong nativist approach against recent cross-cultural studies that have been interpreted to show that precise numerical concepts are dependent on language and that they are restricted to speakers (...)
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  • Bootstrapping the Mind: Analogical Processes and Symbol Systems.Dedre Gentner - 2010 - Cognitive Science 34 (5):752-775.
    Human cognition is striking in its brilliance and its adaptability. How do we get that way? How do we move from the nearly helpless state of infants to the cognitive proficiency that characterizes adults? In this paper I argue, first, that analogical ability is the key factor in our prodigious capacity, and, second, that possession of a symbol system is crucial to the full expression of analogical ability.
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  • Beyond Core Knowledge: Natural Geometry.Elizabeth Spelke, Sang Ah Lee & Véronique Izard - 2010 - Cognitive Science 34 (5):863-884.
    For many centuries, philosophers and scientists have pondered the origins and nature of human intuitions about the properties of points, lines, and figures on the Euclidean plane, with most hypothesizing that a system of Euclidean concepts either is innate or is assembled by general learning processes. Recent research from cognitive and developmental psychology, cognitive anthropology, animal cognition, and cognitive neuroscience suggests a different view. Knowledge of geometry may be founded on at least two distinct, evolutionarily ancient, core cognitive systems for (...)
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  • Exact equality and successor function: Two key concepts on the path towards understanding exact numbers.Véronique Izard, Pierre Pica, Elizabeth S. Spelke & Stanislas Dehaene - 2008 - Philosophical Psychology 21 (4):491 – 505.
    Humans possess two nonverbal systems capable of representing numbers, both limited in their representational power: the first one represents numbers in an approximate fashion, and the second one conveys information about small numbers only. Conception of exact large numbers has therefore been thought to arise from the manipulation of exact numerical symbols. Here, we focus on two fundamental properties of the exact numbers as prerequisites to the concept of EXACT NUMBERS : the fact that all numbers can be generated by (...)
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  • Mathematical intuition and the cognitive roots of mathematical concepts.Giuseppe Longo & Arnaud Viarouge - 2010 - Topoi 29 (1):15-27.
    The foundation of Mathematics is both a logico-formal issue and an epistemological one. By the first, we mean the explicitation and analysis of formal proof principles, which, largely a posteriori, ground proof on general deduction rules and schemata. By the second, we mean the investigation of the constitutive genesis of concepts and structures, the aim of this paper. This “genealogy of concepts”, so dear to Riemann, Poincaré and Enriques among others, is necessary both in order to enrich the foundational analysis (...)
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  • Is linguistic determinism an empirically testable hypothesis?Helen3 De Cruz - 2009 - Logique Et Analyse 52 (208):327-341.
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  • Counting systems and the First Hilbert problem.Yaroslav Sergeyev - 2010 - Nonlinear Analysis Series A 72 (3-4):1701-1708.
    The First Hilbert problem is studied in this paper by applying two instruments: a new methodology distinguishing between mathematical objects and mathematical languages used to describe these objects; and a new numeral system allowing one to express different infinite numbers and to use these numbers for measuring infinite sets. Several counting systems are taken into consideration. It is emphasized in the paper that different mathematical languages can describe mathematical objects (in particular, sets and the number of their elements) with different (...)
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  • Numerical point of view on Calculus for functions assuming finite, infinite, and infinitesimal values over finite, infinite, and infinitesimal domains.Yaroslav Sergeyev - 2009 - Nonlinear Analysis Series A 71 (12):e1688-e1707.
    The goal of this paper consists of developing a new (more physical and numerical in comparison with standard and non-standard analysis approaches) point of view on Calculus with functions assuming infinite and infinitesimal values. It uses recently introduced infinite and infinitesimal numbers being in accordance with the principle ‘The part is less than the whole’ observed in the physical world around us. These numbers have a strong practical advantage with respect to traditional approaches: they are representable at a new kind (...)
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  • The long reach of philosophy of biology: Michael Ruse: The Oxford handbook of philosophy of biology. Oxford University Press, 2008.Matt Gers - 2011 - Biology and Philosophy 26 (3):439-447.
    The Oxford Handbook of Philosophy of Biology covers a broad range of topics in this field. It is not just a textbook focusing on evolutionary theory but encompasses ethics, social science and behaviour too. This essay outlines the scope of the work, discusses some points on methodology in the philosophy of biology, and then moves on to a more detailed analysis of cultural evolution and the applicability of a philosophy of biology toolkit to the social sciences. It is noted that (...)
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  • The meaning of 'most': Semantics, numerosity and psychology.Paul Pietroski, Jeffrey Lidz, Tim Hunter & Justin Halberda - 2009 - Mind and Language 24 (5):554-585.
    The meaning of 'most' can be described in many ways. We offer a framework for distinguishing semantic descriptions, interpreted as psychological hypotheses that go beyond claims about sentential truth conditions, and an experiment that tells against an attractive idea: 'most' is understood in terms of one-to-one correspondence. Adults evaluated 'Most of the dots are yellow', as true or false, on many trials in which yellow dots and blue dots were displayed for 200 ms. Displays manipulated the ease of using a (...)
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  • Non-symbolic arithmetic in adults and young children.Hilary Barth, Kristen La Mont, Jennifer Lipton, Stanislas Dehaene, Nancy Kanwisher & Elizabeth Spelke - 2006 - Cognition 98 (3):199-222.
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  • Where our number concepts come from.Susan Carey - 2009 - Journal of Philosophy 106 (4):220-254.
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  • Don't throw the baby out with the math water: Why discounting the developmental foundations of early numeracy is premature and unnecessary.Kevin Muldoon, Charlie Lewis & Norman Freeman - 2008 - Behavioral and Brain Sciences 31 (6):663-664.
    We see no grounds for insisting that, because the concept natural number is abstract, its foundations must be innate. It is possible to specify domain general learning processes that feed into more abstract concepts of numerical infinity. By neglecting the messiness of children's slow acquisition of arithmetical concepts, Rips et al. present an idealized, unnecessarily insular, view of number development.
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  • From numerical concepts to concepts of number.Lance J. Rips, Amber Bloomfield & Jennifer Asmuth - 2008 - Behavioral and Brain Sciences 31 (6):623-642.
    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 (...)
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  • The evolution of foresight: What is mental time travel, and is it unique to humans?Thomas Suddendorf & Michael C. Corballis - 2007 - Behavioral and Brain Sciences 30 (3):299-313.
    In a dynamic world, mechanisms allowing prediction of future situations can provide a selective advantage. We suggest that memory systems differ in the degree of flexibility they offer for anticipatory behavior and put forward a corresponding taxonomy of prospection. The adaptive advantage of any memory system can only lie in what it contributes for future survival. The most flexible is episodic memory, which we suggest is part of a more general faculty of mental time travel that allows us not only (...)
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  • Theoretical implications of the study of numbers and numerals in mundurucu.Pierre Pica & Alain Lecomte - 2008 - Philosophical Psychology 21 (4):507 – 522.
    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 (...)
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  • An extended mind perspective on natural number representation.Helen De Cruz - 2008 - Philosophical Psychology 21 (4):475 – 490.
    Experimental studies indicate that nonhuman animals and infants represent numerosities above three or four approximately and that their mental number line is logarithmic rather than linear. In contrast, human children from most cultures gradually acquire the capacity to denote exact cardinal values. To explain this difference, I take an extended mind perspective, arguing that the distinctly human ability to use external representations as a complement for internal cognitive operations enables us to represent natural numbers. Reviewing neuroscientific, developmental, and anthropological evidence, (...)
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  • Children’s understanding of the relationship between addition and subtraction.Elizabeth Spelke & Camilla Gilmore - 2008 - Cognition 107 (3):932-945.
    In learning mathematics, children must master fundamental logical relationships, including the inverse relationship between addition and subtraction. At the start of elementary school, children lack generalized understanding of this relationship in the context of exact arithmetic problems: they fail to judge, for example, that 12 + 9 - 9 yields 12. Here, we investigate whether preschool children’s approximate number knowledge nevertheless supports understanding of this relationship. Five-year-old children were more accurate on approximate large-number arithmetic problems that involved an inverse transformation (...)
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  • The wisdom of nature: an evolutionary heuristic for human enhancement.Nick Bostrom & Anders Sandberg - 2009 - In Julian Savulescu & Nick Bostrom (eds.), Human Enhancement. Oxford University Press. pp. 375--416.
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  • The universal density of measurement.Danny Fox & Martin Hackl - 2006 - Linguistics and Philosophy 29 (5):537 - 586.
    The notion of measurement plays a central role in human cognition. We measure people’s height, the weight of physical objects, the length of stretches of time, or the size of various collections of individuals. Measurements of height, weight, and the like are commonly thought of as mappings between objects and dense scales, while measurements of collections of individuals, as implemented for instance in counting, are assumed to involve discrete scales. It is also commonly assumed that natural language makes use of (...)
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  • A representational account of self-knowledge.Albert Newen & Gottfried Vosgerau - 2007 - Erkenntnis 67 (2):337 - 353.
    Self-knowledge is knowledge of one’s own states (or processes) in an indexical mode of presentation. The philosophical debate is concentrating on mental states (or processes). If we characterize self-knowledge by natural language sentences, the most adequate utterance has a structure like “I know that I am in mental state M”. This common sense characterization has to be developed into an adequate description. In this investigation we will tackle two questions: (i) What precisely is the phenomenon referred to by “self-knowledge” and (...)
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  • A Multidisciplinary Approach to Research in Small-Scale Societies: Studying Emotions and Facial Expressions in the Field.Carlos Crivelli, Sergio Jarillo & Alan J. Fridlund - 2016 - Frontiers in Psychology 7:204619.
    Although cognitive science was multidisciplinary from the start, an under-emphasis on anthropology has left the field with limited research in small scale, indigenous societies. Neglecting the anthropological perspective is risky, given that once-canonical cognitive science findings have often been shown to be artifacts of enculturation rather than cognitive universals. This imbalance has become more problematic as the increased use of Western theory-driven approaches, many of which assume human uniformity (“universality”), confronts the absence of a robust descriptive base that might provide (...)
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  • Rational Number Representation by the Approximate Number System.Chuyan Qu, Sam Clarke, Francesca Luzzi & Elizabeth Brannon - 2024 - Cognition 250 (105839):1-13.
    The approximate number system (ANS) enables organisms to represent the approximate number of items in an observed collection, quickly and independently of natural language. Recently, it has been proposed that the ANS goes beyond representing natural numbers by extracting and representing rational numbers (Clarke & Beck, 2021a). Prior work demonstrates that adults and children discriminate ratios in an approximate and ratio-dependent manner, consistent with the hallmarks of the ANS. Here, we use a well-known “connectedness illusion” to provide evidence that these (...)
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  • The Use of Local and Global Ordering Strategies in Number Line Estimation in Early Childhood.Jaccoline E. Van ’T. Noordende, M. J. M. Volman, Paul P. M. Leseman, Korbinian Moeller, Tanja Dackermann & Evelyn H. Kroesbergen - 2018 - Frontiers in Psychology 9.
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  • Young Children Intuitively Divide Before They Recognize the Division Symbol.Emily Szkudlarek, Haobai Zhang, Nicholas K. DeWind & Elizabeth M. Brannon - 2022 - Frontiers in Human Neuroscience 16.
    Children bring intuitive arithmetic knowledge to the classroom before formal instruction in mathematics begins. For example, children can use their number sense to add, subtract, compare ratios, and even perform scaling operations that increase or decrease a set of dots by a factor of 2 or 4. However, it is currently unknown whether children can engage in a true division operation before formal mathematical instruction. Here we examined the ability of 6- to 9-year-old children and college students to perform symbolic (...)
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