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Switch to: References
Citations of:
Exact and Approximate Arithmetic in an Amazonian Indigene Group
Science 306 (5695):499-503 (2004)
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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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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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Why would we want to develop artificial human-like arithmetical intelligence, when computers already outperform humans in arithmetical calculations? Aside from arithmetic consisting of much more than mere calculations, one suggested reason is that AI research can help us explain the development of human arithmetical cognition. Here I argue that this question needs to be studied already in the context of basic, non-symbolic, numerical cognition. Analyzing recent machine learning research on artificial neural networks, I show how AI studies could potentially shed (...) |
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One main challenge of non-platonist philosophy of mathematics is to account for the apparent objectivity of mathematical knowledge. Cole and Feferman have proposed accounts that aim to explain objectivity through the intersubjectivity of mathematical knowledge. In this paper, focusing on arithmetic, I will argue that these accounts as such cannot explain the apparent objectivity of mathematical knowledge. However, with support from recent progress in the empirical study of the development of arithmetical cognition, a stronger argument can be provided. I will (...) |
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There is a noticeable gap between results of cognitive neuroscientific research into basic mathematical abilities and philosophical and empirical investigations of mathematics as a distinct intellectual activity. The paper explores the relevance of a Wittgensteinian framework for dealing with this discrepancy. |
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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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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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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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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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A new computational methodology allowing one to work in a new way with infinities and infinitesimals is presented in this paper. The new approach, among other things, gives the possibility to calculate the number of elements of certain infinite sets, avoids indeterminate forms and various kinds of divergences. This methodology has been used by the author as a starting point in developing a new kind of computer – the Infinity Computer – able to execute computations and to store in its (...) |
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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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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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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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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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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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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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This paper considers non-standard analysis and a recently introduced computational methodology based on the notion of ①. The latter approach was developed with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework and in all the situations requiring these notions. Non-standard analysis is a classical purely symbolic technique that works with ultrafilters, external and internal sets, standard and non-standard numbers, etc. In its turn, the ①-based methodology does not use any of these (...) |
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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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In updating our beliefs on the basis of our background attitudes and evidence we frequently employ objects in our environment to represent pertinent information. For example, we may write our premises and lemmas on a whiteboard to aid in a proof or move the beads of an abacus to assist in a calculation. In both cases, we generate extramental (that is, occurring outside of the mind) representational states, and, at least in the case of the abacus, we operate over these (...) |
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