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Hutto and Myin have proposed an account of radically enactive (or embodied) cognition (REC) as an explanation of cognitive phenomena, one that does not include mental representations or mental content in basic minds. Recently, Zahidi and Myin have presented an account of arithmetical cognition that is consistent with the REC view. In this paper, I first evaluate the feasibility of that account by focusing on the evolutionarily developed proto-arithmetical abilities and whether empirical data on them support the radical enactivist view. (...) |
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Linnebo in 2018 argues that abstract objects like numbers are “thin” because they are only required to be referents of singular terms in abstraction principles, such as Hume's principle. As the specification of existence claims made by analytic truths (the abstraction principles), their existence does not make any substantial demands of the world; however, as Linnebo notes, there is a potential counter-argument concerning infinite regress against introducing objects this way. Against this, he argues that vicious regress is avoided in the (...) |
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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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I identify two reasons for believing in the objectivity of mathematical knowledge: apparent objectivity and applications in science. Focusing on arithmetic, I analyze platonism and cognitive nativism in terms of explaining these two reasons. After establishing that both theories run into difficulties, I present an alternative epistemological account that combines the theoretical frameworks of enculturation and cumulative cultural evolution. I show that this account can explain why arithmetical knowledge appears to be objective and has scientific applications. Finally, I will argue (...) |
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A core commitment of Bob Hale and Crispin Wright’s neologicism is their invocation of Frege’s Constraint—roughly, the requirement that the core empirical applications for a class of numbers be “built directly into” their formal characterization. According to these neologicists, if legitimate, Frege’s Constraint adjudicates in favor of their preferred foundation—Hume’s Principle—and against alternatives, such as the Dedekind–Peano axioms. In this paper, we consider a recent argument for legitimating Frege’s Constraint due to Hale, according to which the primary empirical application of (...) |
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In our target article, we argued that the number sense represents natural and rational numbers. Here, we respond to the 26 commentaries we received, highlighting new directions for empirical and theoretical research. We discuss two background assumptions, arguments against the number sense, whether the approximate number system represents numbers or numerosities, and why the ANS represents rational numbers. |
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The world as we perceive it is structured into objects, actions and places that form parts of events. In this article, my aim is to explain why these categories are cognitively primary. From an empiricist and evolutionary standpoint, it is argued that the reduction of the complexity of sensory signals is based on the brain's capacity to identify various types of invariances that are evolutionarily relevant for the activities of the organism. The first aim of the article is to explain (...) |
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By around the age of 5½, many children in the United States judge that numbers never end, and that it is always possible to add 1 to a set. These same children also generally perform well when asked to label the quantity of a set after one object is added (e.g., judging that a set labeled “five” should now be “six”). These findings suggest that children have implicit knowledge of the “successor function”: Every natural number, n, has a successor, n (...) |
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The basic human ability to treat quantitative information can be divided into two parts. With proto-arithmetical ability, based on the core cognitive abilities for subitizing and estimation, numerosities can be treated in a limited and/or approximate manner. With arithmetical ability, numerosities are processed (counted, operated on) systematically in a discrete, linear, and unbounded manner. In this paper, I study the theory of enculturation as presented by Menary (2015) as a possible explanation of how we make the move from the proto-arithmetical (...) |
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Understanding what numbers are means knowing several things. It means knowing how counting relates to numbers (called the cardinal principle or cardinality); it means knowing that each number is generated by adding one to the previous number (called the successor function or succession), and it means knowing that all and only sets whose members can be placed in one-to-one correspondence have the same number of items (called exact equality or equinumerosity). A previous study (Sarnecka & Carey, 2008) linked children's understanding (...) |
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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 number-line estimation task has become one of the most important methods in numerical cognition research. Originally applied as a direct measure of spatial number representation, it became also informative regarding various other aspects of number processing and associated strategies. However, most of this work and associated conclusions concerns processing numbers in a symbolic format, by school children and older subjects. Symbolic number system is formally taught and trained at school, and its basic mathematical properties can easily be transferred into (...) |
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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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Research suggests that even young children engage in strategic behaviors to manipulate the impressions others form of them and that they manage their reputation in order to cooperate with others. The current study investigated whether young children also lie in order to manage their, or their group’s, reputation in front of ingroup and outgroup members. Five-year old children were randomly assigned to an individual reputation condition or a group reputation condition. Then, they played a mini dictator game in which they (...) |
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We explored the home learning environments of 173 Mexican preschool children (aged 3–6 years) in relation to their numeracy performance. Parents indicated the frequency of their formal home numeracy and literacy activities, and their academic expectations for children’s numeracy and literacy performance. Children completed measures of early numeracy skills. Mexican parent–child dyads from families with either high- or low-socioeconomic status (SES) participated. Low-SES parents (n= 99) reported higher numeracy expectations than high-SES parents (n= 74), but similar frequency of home numeracy (...) |
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Beck presents an outline of the procedure of bootstrapping of integer concepts, with the purpose of explicating the account of Carey. According to that theory, integer concepts are acquired through a process of inductive and analogous reasoning based on the object tracking system, which allows individuating objects in a parallel fashion. Discussing the bootstrapping theory, Beck dismisses what he calls the "deviant-interpretation challenge"—the possibility that the bootstrapped integer sequence does not follow a linear progression after some point—as being general to (...) |
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Do children understand how different numbers are related before they associate them with specific cardinalities? We explored how children rely on two abstract relations – contrast and entailment – to reason about the meanings of ‘unknown’ number words. Previous studies argue that, because children give variable amounts when asked to give an unknown number, all unknown numbers begin with an existential meaning akin to some. In Experiment 1, we tested an alternative hypothesis, that because numbers belong to a scale of (...) |
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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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One of the more distinctive features of Bob Hale and Crispin Wright’s neologicism about arithmetic is their invocation of Frege’s Constraint – roughly, the requirement that the core empirical applications for a class of numbers be “built directly into” their formal characterization. In particular, they maintain that, if adopted, Frege’s Constraint adjudicates in favor of their preferred foundation – Hume’s Principle – and against alternatives, such as the Dedekind-Peano axioms. In what follows we establish two main claims. First, we show (...) |
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Mental time travel refers to the ability of an organism to project herself backward and forward in time, using episodic memory and imagination to simulate past and future experiences. The evolution of mental time travel gives humans a unique capacity for prospection: the ability to pre-experience the future. Discussions of mental time travel treat it as an instance of explicit prospection. We argue that implicit simulations of past and future experience can also be used as a way of gaining information (...) |
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The commentators raised issues relevant to all three important theses of The Origin of Concepts (henceforth TOOC). Some questioned the very existence of innate representational primitives, and others questioned my claims about their richness and whether they should be thought of as concepts. Some questioned the existence of conceptual discontinuity in the course of knowledge acquisition and others argued that discontinuity is much more common than was portrayed in TOOC. Some raised issues with my characterization of Quinian bootstrapping, and others (...) |
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Memory for numbers improves with age. One source of this improvement may be learning linear spatial-numeric associations, but previous evidence for this hypothesis likely confounded memory span with quality of numerical magnitude representations and failed to distinguish spatial-numeric mappings from other numeric abilities, such as counting or number word-cardinality mapping. To obviate the influence of memory span on numerical memory, we examined 39 3- to 5-year-olds’ ability to recall one spontaneously produced number (1-20) after a delay, and the relation between (...) |
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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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The ability to improve in speed and accuracy as a result of repeating some task is an important hallmark of intelligent biological systems. Although gradual behavioral improvements from practice have been modeled in spiking neural networks, few such models have attempted to explain cognitive development of a task as complex as addition. In this work, we model the progression from a counting-based strategy for addition to a recall-based strategy. The model consists of two networks working in parallel: a slower basal (...) |
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