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  1. Predicative Frege Arithmetic and ‘Everyday’ Mathematics.Richard Heck - 2014 - Philosophia Mathematica 22 (3):279-307.
    The primary purpose of this note is to demonstrate that predicative Frege arithmetic naturally interprets certain weak but non-trivial arithmetical theories. It will take almost as long to explain what this means and why it matters as it will to prove the results.
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  • Bootstrapping in a language of thought: A formal model of numerical concept learning.Steven T. Piantadosi, Joshua B. Tenenbaum & Noah D. Goodman - 2012 - Cognition 123 (2):199-217.
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  • Counterfactuals as Strict Conditionals.Andrea Iacona - 2015 - Disputatio 7 (41):165-191.
    This paper defends the thesis that counterfactuals are strict conditionals. Its purpose is to show that there is a coherent view according to which counterfactuals are strict conditionals whose antecedent is stated elliptically. Section 1 introduces the view. Section 2 outlines a response to the main argument against the thesis that counterfactuals are strict conditionals. Section 3 compares the view with a proposal due to Aqvist, which may be regarded as its direct predecessor. Sections 4 and 5 explain how the (...)
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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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  • Mathematical symbols as epistemic actions.Johan De Smedt & Helen De Cruz - 2013 - Synthese 190 (1):3-19.
    Recent experimental evidence from developmental psychology and cognitive neuroscience indicates that humans are equipped with unlearned elementary mathematical skills. However, formal mathematics has properties that cannot be reduced to these elementary cognitive capacities. The question then arises how human beings cognitively deal with more advanced mathematical ideas. This paper draws on the extended mind thesis to suggest that mathematical symbols enable us to delegate some mathematical operations to the external environment. In this view, mathematical symbols are not only used to (...)
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  • Numbers and Arithmetic: Neither Hardwired Nor Out There.Rafael Núñez - 2009 - Biological Theory 4 (1):68-83.
    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 (...)
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  • The innateness hypothesis and mathematical concepts.Helen3 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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  • Number Concepts: An Interdisciplinary Inquiry.Richard Samuels & Eric Snyder - 2024 - Cambridge University Press.
    This Element, written for researchers and students in philosophy and the behavioral sciences, reviews and critically assesses extant work on number concepts in developmental psychology and cognitive science. It has four main aims. First, it characterizes the core commitments of mainstream number cognition research, including the commitment to representationalism, the hypothesis that there exist certain number-specific cognitive systems, and the key milestones in the development of number cognition. Second, it provides a taxonomy of influential views within mainstream number cognition research, (...)
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  • Situated Counting.Peter Gärdenfors & Paula Quinon - 2020 - Review of Philosophy and Psychology 12 (4):721-744.
    We present a model of how counting is learned based on the ability to perform a series of specific steps. The steps require conceptual knowledge of three components: numerosity as a property of collections; numerals; and one-to-one mappings between numerals and collections. We argue that establishing one-to-one mappings is the central feature of counting. In the literature, the so-called cardinality principle has been in focus when studying the development of counting. We submit that identifying the procedural ability to count with (...)
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  • Animal Cognition, Species Invariantism, and Mathematical Realism.Helen De Cruz - 2019 - In Andrew Aberdein & Matthew Inglis (eds.), Advances in Experimental Philosophy of Logic and Mathematics. London: Bloomsbury Academic. pp. 39-61.
    What can we infer from numerical cognition about mathematical realism? In this paper, I will consider one aspect of numerical cognition that has received little attention in the literature: the remarkable similarities of numerical cognitive capacities across many animal species. This Invariantism in Numerical Cognition (INC) indicates that mathematics and morality are disanalogous in an important respect: proto-moral beliefs differ substantially between animal species, whereas proto-mathematical beliefs (at least in the animals studied) seem to show more similarities. This makes moral (...)
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  • Thinking Materially: Cognition as Extended and Enacted.Karenleigh A. Overmann - 2017 - Journal of Cognition and Culture 17 (3-4):354-373.
    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 (...)
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  • Testimony and Children’s Acquisition of Number Concepts.Helen De Cruz - 2018 - In Sorin Bangu (ed.), Naturalizing Logico-Mathematical Knowledge: Approaches From Psychology and Cognitive Science. New York: Routledge. 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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  • 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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  • Learning to represent exact numbers.Barbara W. Sarnecka - 2015 - Synthese 198 (Suppl 5):1001-1018.
    This article focuses on how young children acquire concepts for exact, cardinal numbers. I believe that exact numbers are a conceptual structure that was invented by people, and that most children acquire gradually, over a period of months or years during early childhood. This article reviews studies that explore children’s number knowledge at various points during this acquisition process. Most of these studies were done in my own lab, and assume the theoretical framework proposed by Carey. In this framework, the (...)
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  • Rebooting the bootstrap argument: Two puzzles for bootstrap theories of concept development.Lance J. Rips & Susan J. Hespos - 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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  • Does learning to count involve a semantic induction?Kathryn Davidson, Kortney Eng & David Barner - 2012 - Cognition 123 (1):162-173.
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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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  • Cognitive Structuralism: Explaining the Regularity of the Natural Numbers Progression.Paula Quinon - 2022 - Review of Philosophy and Psychology 13 (1):127-149.
    According to one of the most powerful paradigms explaining the meaning of the concept of natural number, natural numbers get a large part of their conceptual content from core cognitive abilities. Carey’s bootstrapping provides a model of the role of core cognition in the creation of mature mathematical concepts. In this paper, I conduct conceptual analyses of various theories within this paradigm, concluding that the theories based on the ability to subitize (i.e., to assess anexactquantity of the elements in a (...)
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  • On the Philosophical Significance of Frege’s Constraint.Andrea Sereni - 2019 - Philosophia Mathematica 27 (2):244–275.
    Foundational projects disagree on whether pure and applied mathematics should be explained together. Proponents of unified accounts like neologicists defend Frege’s Constraint (FC), a principle demanding that an explanation of applicability be provided by mathematical definitions. I reconsider the philosophical import of FC, arguing that usual conceptions are biased by ontological assumptions. I explore more reasonable weaker variants — Moderate and Modest FC — arguing against common opinion that ante rem structuralism (and other) views can meet them. I dispel doubts (...)
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  • Why Theories of Concepts Should Not Ignore the Problem of Acquisition.Susan Carey - 2015 - Disputatio 7 (41):113-163.
    Why Theories of Concepts Should Not Ignore the Problem of Acquisition.
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  • Neurophilosophy of Number.Hourya Benis Sinaceur - 2017 - International Studies in the Philosophy of Science 31 (1):1-25.
    Neurosciences and cognitive sciences provide us with myriad empirical findings that shed light on hypothesised primitive numerical processes in the brain and in the mind. Yet, the hypotheses on which the experiments are based, and hence the results, depend strongly on sophisticated abstract models used to describe and explain neural data or cognitive representations that supposedly are the empirical roots of primary arithmetical activity. I will question the foundational role of such models. I will even cast doubt upon the search (...)
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  • The influence of cardiorespiratory fitness on strategic, behavioral, and electrophysiological indices of arithmetic cognition in preadolescent children.R. Davis Moore, Eric S. Drollette, Mark R. Scudder, Aashiv Bharij & Charles H. Hillman - 2014 - Frontiers in Human Neuroscience 8.
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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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  • The Computational Origin of Representation.Steven T. Piantadosi - 2020 - Minds and Machines 31 (1):1-58.
    Each of our theories of mental representation provides some insight into how the mind works. However, these insights often seem incompatible, as the debates between symbolic, dynamical, emergentist, sub-symbolic, and grounded approaches to cognition attest. Mental representations—whatever they are—must share many features with each of our theories of representation, and yet there are few hypotheses about how a synthesis could be possible. Here, I develop a theory of the underpinnings of symbolic cognition that shows how sub-symbolic dynamics may give rise (...)
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  • Can statistical learning bootstrap the integers?Lance J. Rips, Jennifer Asmuth & Amber Bloomfield - 2013 - Cognition 128 (3):320-330.
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  • Mental Magnitudes and Increments of Mental Magnitudes.Matthew Katz - 2013 - Review of Philosophy and Psychology 4 (4):675-703.
    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 (...)
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  • Acquiring mathematical concepts: The viability of hypothesis testing.Stefan Buijsman - 2021 - Mind and Language 36 (1):48-61.
    Can concepts be acquired by testing hypotheses about these concepts? Fodor famously argued that this is not possible. Testing the correct hypothesis would require already possessing the concept. I argue that this does not generally hold for mathematical concepts. I discuss specific, empirically motivated, hypotheses for number concepts that can be tested without needing to possess the relevant number concepts. I also argue that one can test hypotheses about the identity conditions of other mathematical concepts, and then fix the application (...)
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  • What’s new: innovation and enculturation of arithmetical practices.Jean-Charles Pelland - 2020 - Synthese 197 (9):3797-3822.
    One of the most important questions in the young field of numerical cognition studies is how humans bridge the gap between the quantity-related content produced by our evolutionarily ancient brains and the precise numerical content associated with numeration systems like Indo-Arabic numerals. This gap problem is the main focus of this paper. The aim here is to evaluate the extent to which cultural factors can help explain how we come to think about numbers beyond the subitizing range. To do this, (...)
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  • Bootstrapping Concepts via Hybridization: A Step-by-step Guide.Matteo De Benedetto & Nina Poth - forthcoming - Review of Philosophy and Psychology.
    Carey’s (2009) account of bootstrapping in developmental psychology has been criticized out of a lack of theoretical precision and because of its alleged circularity (Rips et al. 2013, Cognition 128 (3): 320–330; Fodor 2010, Times Literary Supplement, 7–8; Rey 2014, Mind & Language 29 (2): 109–132). In this paper, we respond to these criticisms by connecting the debate on bootstrapping with recent accounts of conceptual creativity in philosophy of science. Specifically, we build on Nersessian’s (2010) hybrid-models-based theory of scientific conceptual (...)
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  • Children's Understanding of the Natural Numbers’ Structure.Jennifer Asmuth, Emily M. Morson & Lance J. Rips - 2018 - Cognitive Science 42 (6):1945-1973.
    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, (...)
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  • Let us redeploy attention to sensorimotor experience.Nicolas Michaux, Mauro Pesenti, Arnaud Badets, Samuel Di Luca & Michael Andres - 2010 - Behavioral and Brain Sciences 33 (4):283-284.
    With his massive redeployment hypothesis (MRH), Anderson claims that novel cognitive functions are likely to rely on pre-existing circuits already possessing suitable resources. Here, we put forward recent findings from studies in numerical cognition in order to show that the role of sensorimotor experience in the ontogenetical development of a new function has been largely underestimated in Anderson's proposal.
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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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  • Grey parrot number acquisition: The inference of cardinal value from ordinal position on the numeral list.Irene M. Pepperberg & Susan Carey - 2012 - Cognition 125 (2):219-232.
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  • Is thirty-two three tens and two ones? The embedded structure of cardinal numbers.Diego Guerrero, Jihyun Hwang, Brynn Boutin, Tom Roeper & Joonkoo Park - 2020 - Cognition 203 (C):104331.
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  • Conceptual and procedural distinctions between fractions and decimals: A cross-national comparison.Hee Seung Lee, Melissa DeWolf, Miriam Bassok & Keith J. Holyoak - 2016 - Cognition 147 (C):57-69.
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  • Squeezing, striking, and vocalizing: Is number representation fundamentally spatial?Rafael Núñez, D. Doan & Anastasia Nikoulina - 2011 - Cognition 120 (2):225-235.
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