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  1. added 2019-01-19
    A New Applied Approach for Executing Computations with Infinite and Infinitesimal Quantities.Yaroslav D. Sergeyev - 2008 - Informatica 19 (4):567-596.
    A new computational methodology for executing calculations with infinite and infinitesimal quantities is described in this paper. It is based on the principle ‘The part is less than the whole’ introduced by Ancient Greeks and applied to all numbers (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). It is shown that it becomes possible to write down finite, infinite, and infinitesimal numbers by a finite number of symbols as particular cases of a unique framework. The (...)
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  2. added 2019-01-16
    What Frege Asked Alex the Parrot: Inferentialism, Number Concepts, and Animal Cognition.Erik Nelson - forthcoming - Philosophical Psychology.
    While there has been significant philosophical debate on whether nonlinguistic animals can possess conceptual capabilities, less time has been devoted to considering 'talking' animals, such as parrots. When they are discussed, their capabilities are often downplayed as mere mimicry. The most explicit philosophical example of this can be seen in Brandom's frequent comparisons of parrots and thermostats. Brandom argues that because parrots (like thermostats) cannot grasp the implicit inferential connections between concepts, their vocal articulations do not actually have any conceptual (...)
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  3. added 2018-12-17
    Numerical Infinities and Infinitesimals: Methodology, Applications, and Repercussions on Two Hilbert Problems.Yaroslav Sergeyev - 2017 - EMS Surveys in Mathematical Sciences 4 (2):219–320.
    In this survey, a recent computational methodology paying a special attention to the separation of mathematical objects from numeral systems involved in their representation is described. It has been introduced with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework in all the situations requiring these notions. The methodology does not contradict Cantor’s and non-standard analysis views and is based on the Euclid’s Common Notion no. 5 “The whole is greater than the (...)
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  4. added 2018-12-17
    The Difficulty of Prime Factorization is a Consequence of the Positional Numeral System.Yaroslav Sergeyev - 2016 - International Journal of Unconventional Computing 12 (5-6):453–463.
    The importance of the prime factorization problem is very well known (e.g., many security protocols are based on the impossibility of a fast factorization of integers on traditional computers). It is necessary from a number k to establish two primes a and b giving k = a · b. Usually, k is written in a positional numeral system. However, there exists a variety of numeral systems that can be used to represent numbers. Is it true that the prime factorization is (...)
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  5. added 2018-12-17
    The Exact (Up to Infinitesimals) Infinite Perimeter of the Koch Snowflake and its Finite Area.Yaroslav Sergeyev - 2016 - Communications in Nonlinear Science and Numerical Simulation 31 (1-3):21–29.
    The Koch snowflake is one of the first fractals that were mathematically described. It is interesting because it has an infinite perimeter in the limit but its limit area is finite. In this paper, a recently proposed computational methodology allowing one to execute numerical computations with infinities and infinitesimals is applied to study the Koch snowflake at infinity. Numerical computations with actual infinite and infinitesimal numbers can be executed on the Infinity Computer being a new supercomputer patented in USA and (...)
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  6. added 2018-12-17
    The Olympic Medals Ranks, Lexicographic Ordering and Numerical Infinities.Yaroslav Sergeyev - 2015 - The Mathematical Intelligencer 37 (2):4-8.
    Several ways used to rank countries with respect to medals won during Olympic Games are discussed. In particular, it is shown that the unofficial rank used by the Olympic Committee is the only rank that does not allow one to use a numerical counter for ranking – this rank uses the lexicographic ordering to rank countries: one gold medal is more precious than any number of silver medals and one silver medal is more precious than any number of bronze medals. (...)
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  7. added 2018-12-17
    Arithmetic of Infinity.Yaroslav D. Sergeyev - 2013 - E-book.
    This book presents a new type of arithmetic that allows one to execute arithmetical operations with infinite numbers in the same manner as we are used to do with finite ones. The problem of infinity is considered in a coherent way different from (but not contradicting to) the famous theory founded by Georg Cantor. Surprisingly, the introduced arithmetical operations result in being very simple and are obtained as immediate extensions of the usual addition, multiplication, and division of finite numbers to (...)
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  8. added 2018-08-15
    Testimony and Children’s Acquisition of Number Concepts.Helen De Cruz - 2018 - In Sorin Bangu (ed.), Naturalizing Logico-Mathematical Knowledge. Approaches from Philosophy, Psychology and Cognitive Science. London, UK: pp. 172-186.
    An enduring puzzle in philosophy and developmental psychology is how young children acquire number concepts, in particular the concept of natural number. Most solutions to this problem conceptualize young learners as lone mathematicians who individually reconstruct the successor function and other sophisticated mathematical ideas. In this chapter, I argue for a crucial role of testimony in children’s acquisition of number concepts, both in the transfer of propositional knowledge (e.g., the cardinality concept), and in knowledge-how (e.g., the counting routine).
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  9. added 2018-07-16
    Analog Mental Representation.Jacob Beck - forthcoming - WIREs Cognitive Science.
    Over the past 50 years, philosophers and psychologists have perennially argued for the existence of analog mental representations of one type or another. This study critically reviews a number of these arguments as they pertain to three different types of mental representation: perceptual representations, imagery representations, and numerosity representations. Along the way, careful consideration is given to the meaning of “analog” presupposed by these arguments for analog mental representation, and to open avenues for future research.
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  10. added 2018-01-18
    A Perceptual Account of Symbolic Reasoning.David Landy, Colin Allen & Carlos Zednik - 2014 - Frontiers in Psychology 5.
    People can be taught to manipulate symbols according to formal mathematical and logical rules. Cognitive scientists have traditionally viewed this capacity—the capacity for symbolic reasoning—as grounded in the ability to internally represent numbers, logical relationships, and mathematical rules in an abstract, amodal fashion. We present an alternative view, portraying symbolic reasoning as a special kind of embodied reasoning in which arithmetic and logical formulae, externally represented as notations, serve as targets for powerful perceptual and sensorimotor systems. Although symbolic reasoning often (...)
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  11. added 2017-10-20
    Infants, Animals, and the Origins of Number.Eric Margolis - 2017 - Behavioral and Brain Sciences 40.
    Where do human numerical abilities come from? This article is a commentary on Leibovich et al.’s “From 'sense of number' to 'sense of magnitude' —The role of continuous magnitudes in numerical cognition”. Leibovich et al. argue against nativist views of numerical development by noting limitations in newborns’ vision and limitations regarding newborns’ ability to individuate objects. I argue that these considerations do not undermine competing nativist views and that Leibovich et al.'s model itself presupposes that infant learners have numerical representations.
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  12. added 2017-09-07
    Learning the Natural Numbers as a Child.Stefan Buijsman - 2019 - Noûs 53 (1):3-22.
    How do we get out knowledge of the natural numbers? Various philosophical accounts exist, but there has been comparatively little attention to psychological data on how the learning process actually takes place. I work through the psychological literature on number acquisition with the aim of characterising the acquisition stages in formal terms. In doing so, I argue that we need a combination of current neologicist accounts and accounts such as that of Parsons. In particular, I argue that we learn the (...)
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  13. added 2017-04-18
    An Empirically Feasible Approach to the Epistemology of Arithmetic.Markus Pantsar - 2014 - Synthese 191 (17):4201-4229.
    Recent years have seen an explosion of empirical data concerning arithmetical cognition. In this paper that data is taken to be philosophically important and an outline for an empirically feasible epistemological theory of arithmetic is presented. The epistemological theory is based on the empirically well-supported hypothesis that our arithmetical ability is built on a protoarithmetical ability to categorize observations in terms of quantities that we have already as infants and share with many nonhuman animals. It is argued here that arithmetical (...)
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  14. added 2017-03-25
    The Psychology and Philosophy of Natural Numbers.Oliver R. Marshall - 2017 - Philosophia Mathematica (1):nkx002.
    ABSTRACT I argue against both neuropsychological and cognitive accounts of our grasp of numbers. I show that despite the points of divergence between these two accounts, they face analogous problems. Both presuppose too much about what they purport to explain to be informative, and also characterize our grasp of numbers in a way that is absurd in the light of what we already know from the point of view of mathematical practice. Then I offer a positive methodological proposal about the (...)
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  15. added 2017-03-25
    From Magnitude to Natural Numbers: A Developmental Neurocognitive Perspective.Roi Cohen Kadosh & Vincent Walsh - 2008 - Behavioral and Brain Sciences 31 (6):647-648.
    In their target article, Rips et al. have presented the view that there is no necessary dependency between natural numbers and internal magnitude. However, they do not give enough weight to neuroimaging and neuropsychological studies. We provide evidence demonstrating that the acquisition of natural numbers depends on magnitude representation and that natural numbers develop from a general magnitude mechanism in the parietal lobes.
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  16. added 2017-03-12
    Numerical Cognition and Mathematical Realism.Helen De Cruz - 2016 - Philosophers' Imprint 16.
    Humans and other animals have an evolved ability to detect discrete magnitudes in their environment. Does this observation support evolutionary debunking arguments against mathematical realism, as has been recently argued by Clarke-Doane, or does it bolster mathematical realism, as authors such as Joyce and Sinnott-Armstrong have assumed? To find out, we need to pay closer attention to the features of evolved numerical cognition. I provide a detailed examination of the functional properties of evolved numerical cognition, and propose that they prima (...)
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  17. added 2017-03-12
    The Innateness Hypothesis and Mathematical Concepts.Helen De Cruz & Johan De Smedt - 2010 - Topoi 29 (1):3-13.
    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 (...)
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  18. added 2017-02-14
    The Conceptual Basis of Numerical Abilities: One-to-One Correspondence Versus the Successor Relation.Lieven Decock - 2008 - Philosophical Psychology 21 (4):459 – 473.
    In recent years, neologicists have demonstrated that Hume's principle, based on the one-to-one correspondence relation, suffices to construct the natural numbers. This formal work is shown to be relevant for empirical research on mathematical cognition. I give a hypothetical account of how nonnumerate societies may acquire arithmetical knowledge on the basis of the one-to-one correspondence relation only, whereby the acquisition of number concepts need not rely on enumeration (the stable-order principle). The existing empirical data on the role of the one-to-one (...)
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  19. added 2017-02-14
    Neo-Fregeanism Naturalized: The Role of One-to-One Correspondence in Numerical Cognition.Lieven Decock - 2008 - Behavioral and Brain Sciences 31 (6):648-649.
    Rips et al. argue that the construction of math schemas roughly similar to the Dedekind/Peano axioms may be necessary for arriving at arithmetical skills. However, they neglect the neo-Fregean alternative axiomatization of arithmetic, based on Hume's principle. Frege arithmetic is arguably a more plausible start for a top-down approach in the psychological study of mathematical cognition than Peano arithmetic.
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  20. added 2017-02-13
    Commentary: A Pointer About Grasping Numbers.Martin H. Fischer, Elena Sixtus & Silke M. Göbel - 2015 - Frontiers in Psychology 6.
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  21. added 2017-02-13
    Ratio Dependence in Small Number Discrimination is Affected by the Experimental Procedure.Christian Agrillo, Laura Piffer, Angelo Bisazza & Brian Butterworth - 2015 - Frontiers in Psychology 6.
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  22. added 2017-02-13
    Language Influences Number Processing – A Quadrilingual Study.Korbinian Moeller, Samuel Shaki, Silke M. Göbel & Hans-Christoph Nuerk - 2015 - Cognition 136:150-155.
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  23. added 2017-02-13
    Two Steps to Space for Numbers.Martin H. Fischer & Samuel Shaki - 2015 - Frontiers in Psychology 6.
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  24. added 2017-02-13
    Analogue Magnitude Representations: A Philosophical Introduction.Jacob Beck - 2015 - British Journal for the Philosophy of Science 66 (4):829-855.
    Empirical discussions of mental representation appeal to a wide variety of representational kinds. Some of these kinds, such as the sentential representations underlying language use and the pictorial representations of visual imagery, are thoroughly familiar to philosophers. Others have received almost no philosophical attention at all. Included in this latter category are analogue magnitude representations, which enable a wide range of organisms to primitively represent spatial, temporal, numerical, and related magnitudes. This article aims to introduce analogue magnitude representations to a (...)
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  25. added 2017-02-13
    Spatial Coding of Ordinal Information in Short- and Long-Term Memory.Vã©Ronique Ginsburg & Wim Gevers - 2015 - Frontiers in Human Neuroscience 9.
  26. added 2017-02-13
    Identifying and Counting Objects: The Role of Sortal Concepts.Nick Leonard & Lance J. Rips - 2015 - Cognition 145:89-103.
    Sortal terms, such as table or horse, are count nouns (akin to a basic-level terms). According to some theories, the meaning of sortals provides conditions for telling objects apart (individuating objects, e.g., telling one table from a second) and for identifying objects over time (e.g., determining that a particular horse at one time is the same horse at another). A number of psychologists have proposed that sortal concepts likewise provide psychologically real conditions for individuating and identifying things. However, this paper (...)
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  27. added 2017-02-13
    Newborn Chicks Need No Number Tricks. Commentary: Number-Space Mapping in the Newborn Chick Resembles Humans' Mental Number Line.Samuel Shaki & Martin H. Fischer - 2015 - Frontiers in Human Neuroscience 9.
  28. added 2017-02-13
    How Space-Number Associations May Be Created in Preliterate Children: Six Distinct Mechanisms.Hans-Christoph Nuerk, Katarzyna Patro, Ulrike Cress, Ulrike Schild, Claudia K. Friedrich & Silke M. Göbel - 2015 - Frontiers in Psychology 6.
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  29. added 2017-02-13
    Spatial Biases During Mental Arithmetic: Evidence From Eye Movements on a Blank Screen.Matthias Hartmann, Fred W. Mast & Martin H. Fischer - 2015 - Frontiers in Psychology 6.
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  30. added 2017-02-13
    Can Statistical Learning Bootstrap the Integers?Lance J. Rips, Jennifer Asmuth & Amber Bloomfield - 2013 - Cognition 128 (3):320-330.
  31. added 2017-02-13
    Singing Numbers… in Cognitive Space — A Dual‐Task Study of the Link Between Pitch, Space, and Numbers.Martin H. Fischer, Marianna Riello, Bruno L. Giordano & Elena Rusconi - 2013 - Topics in Cognitive Science 5 (2):354-366.
    We assessed the automaticity of spatial-numerical and spatial-musical associations by testing their intentionality and load sensitivity in a dual-task paradigm. In separate sessions, 16 healthy adults performed magnitude and pitch comparisons on sung numbers with variable pitch. Stimuli and response alternatives were identical, but the relevant stimulus attribute (pitch or number) differed between tasks. Concomitant tasks required retention of either color or location information. Results show that spatial associations of both magnitude and pitch are load sensitive and that the spatial (...)
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  32. added 2017-02-13
    Optokinetic Stimulation Modulates Neglect for the Number Space: Evidence From Mental Number Interval Bisection.Konstantinos Priftis, Marco Pitteri, Francesca Meneghello, Carlo Umiltà & Marco Zorzi - 2012 - Frontiers in Human Neuroscience 6.
  33. added 2017-02-13
    Finger Counting and Numerical Cognition.Martin H. Fischer, Liane Kaufmann & Frank Domahs - 2012 - Frontiers in Psychology 3.
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  34. added 2017-02-13
    Representation of Numerical and Non-Numerical Order in Children.Ilaria Berteletti, Daniela Lucangeli & Marco Zorzi - 2012 - Cognition 124 (3):304-313.
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  35. added 2017-02-13
    Stability and Change in Markers of Core Numerical Competencies.Robert Reeve, Fiona Reynolds, Judi Humberstone & Brian Butterworth - 2012 - Journal of Experimental Psychology: General 141 (4):649-666.
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  36. added 2017-02-13
    Mapping Numerical Magnitudes Along the Right Lines: Differentiating Between Scale and Bias.Vyacheslav Karolis, Teresa Iuculano & Brian Butterworth - 2011 - Journal of Experimental Psychology: General 140 (4):693-706.
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  37. added 2017-02-13
    Numerosities and Space; Indeed a Cognitive Illusion! A Reply to de Hevia and Spelke.Titia Gebuis & Wim Gevers - 2011 - Cognition 121 (2):248-252.
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  38. added 2017-02-13
    How Number is Associated with Space?: The Role of Working Memory.Wim Fias, Jean-Philippe van Dijck & Wim Gevers - 2011 - In Stanislas Dehaene & Elizabeth Brannon (eds.), Space, Time and Number in the Brain. Oxford University Press. pp. 133-148.
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  39. added 2017-02-13
    Rebooting the Bootstrap Argument: Two Puzzles for Bootstrap Theories of Concept Development.Lance J. Rips, Susan J. Hespos & Susan Carey - 2011 - Behavioral and Brain Sciences 34 (3):145.
    The Origin of Concepts sets out an impressive defense of the view that children construct entirely new systems of concepts. We offer here two questions about this theory. First, why doesn't the bootstrapping process provide a pattern for translating between the old and new systems, contradicting their claimed incommensurability? Second, can the bootstrapping process properly distinguish meaning change from belief change?
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  40. added 2017-02-13
    Foundational Numerical Capacities and the Origins of Dyscalculia.Brian Butterworth - 2010 - Trends in Cognitive Sciences 14 (12):534-541.
  41. added 2017-02-13
    Verbal-Spatial and Visuospatial Coding of Number–Space Interactions.Wim Gevers, Seppe Santens, Elisah Dhooge, Qi Chen, Lisa Van den Bossche, Wim Fias & Tom Verguts - 2010 - Journal of Experimental Psychology: General 139 (1):180-190.
  42. added 2017-02-13
    Non-Abstractness as Mental Simulation in the Representation of Number.Andriy Myachykov, Wouter Platenburg & Martin H. Fischer - 2009 - Behavioral and Brain Sciences 32 (3-4):343 - 344.
    ion is instrumental for our understanding of how numbers are cognitively represented. We propose that the notion of abstraction becomes testable from within the framework of simulated cognition. We describe mental simulation as embodied, grounded, and situated cognition, and report evidence for number representation at each of these levels of abstraction.
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  43. added 2017-02-13
    Numbers Are Associated with Different Types of Spatial Information Depending on the Task.Jean-Philippe van Dijck, Wim Gevers & Wim Fias - 2009 - Cognition 113 (2):248-253.
  44. added 2017-02-13
    Mental Movements Without Magnitude? A Study of Spatial Biases in Symbolic Arithmetic.Michal Pinhas & Martin H. Fischer - 2008 - Cognition 109 (3):408-415.
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  45. added 2017-02-13
    Dissonances in Theories of Number Understanding.Lance J. Rips, Amber Bloomfield & Jennifer Asmuth - 2008 - Behavioral and Brain Sciences 31 (6):671-687.
    Traditional theories of how children learn the positive integers start from infants' abilities in detecting the quantity of physical objects. Our target article examined this view and found no plausible accounts of such development. Most of our commentators appear to agree that no adequate developmental theory is presently available, but they attempt to hold onto a role for early enumeration. Although some defend the traditional theories, others introduce new basic quantitative abilities, new methods of transformation, or new types of end (...)
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  46. added 2017-02-13
    The SNARC Effect Does Not Imply a Mental Number Line.Seppe Santens & Wim Gevers - 2008 - Cognition 108 (1):263-270.
    In this study, we directly contrast two approaches that have been proposed to explain the SNARC effect. The traditional direct mapping account suggests that a direct association exists between the position of a number on the mental number line and the location of the response. On the other hand, accounts are considered that propose an intermediate step in which numbers are categorized as either small or large between the number magnitude and the response representations. In a magnitude comparison task, we (...)
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  47. added 2017-02-13
    Reading Space Into Numbers: A Cross-Linguistic Comparison of the SNARC Effect.Samuel Shaki & Martin H. Fischer - 2008 - Cognition 108 (2):590-599.
    Small numbers are spontaneously associated with left space and larger numbers with right space (the SNARC effect), for example when classifying numbers by parity. This effect is often attributed to reading habits but a causal link has so far never been documented. We report that bilingual Russian-Hebrew readers show a SNARC effect after reading Cyrillic script (from left-to-right) that is significantly reduced after reading Hebrew script (from right-to-left). In contrast, they have similar SNARC effects after listening to texts in either (...)
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  48. added 2017-02-13
    Visuospatial Priming of the Mental Number Line.Ivilin Stoianov, Peter Kramer, Carlo Umiltà & Marco Zorzi - 2008 - Cognition 106 (2):770-779.
    It has been argued that numbers are spatially organized along a "mental number line" that facilitates left-hand responses to small numbers, and right-hand responses to large numbers. We hypothesized that whenever the representations of visual and numerical space are concurrently activated, interactions can occur between them, before response selection. A spatial prime is processed faster than a numerical target, and consistent with our hypothesis, we found that such a spatial prime affects non-spatial, verbal responses more when the prime follows a (...)
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  49. added 2017-02-13
    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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  50. added 2017-02-13
    A Spatial Perspective on Numerical Concepts.Martin H. Fischer & Richard A. Mills - 2008 - Behavioral and Brain Sciences 31 (6):651-652.
    The reliable covariation between numerosity and spatial extent is considered as a strong constraint for inferring the successor principle in numerical cognition. We suggest that children can derive a general number concept from the (experientially) infinite succession of spatial positions during object manipulation.
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