Is thirty-two three tens and two ones? The embedded structure of cardinal numbers

The acquisition and representation of natural numbers have been a central topic in cognitive science. However, a key question in this topic about how humans acquire the capacity to understand that numbers make ‘infinite use of finite means’ (or that numbers are generative) has been left unanswered. Here, we test the hypothesis that children’s understanding of the syntactic rules for building complex numerals—or numerical syntax—is a crucial foundation for the acquisition of number concepts. In two independent studies, we assessed children’s understanding of numerical syntax by probing their knowledge about the embedded structure of cardinal numbers using a novel task called Give-a-number Base-10 (Give-N10). In Give-N10, children were asked to give a large number of items (e.g., 32 items) from a pool that is organized in sets of ten items. Children’s knowledge about the embedded structure of numbers (e.g., knowing that thirty-two items are composed of three tens and two ones) was assessed from their ability to use those sets. Study 1 tested English-speaking 5- to 10-year-olds and revealed that children’s understanding of the embedded structure of numbers emerges relatively late in development (several months into kindergarten), beyond when they are capable of making a semantic induction over a local sequence of numbers. Moreover, performance in Give-N10 was predicted by children’s counting fluency even when school grade was controlled for, suggesting that children’s understanding of the embedded structure of numbers is founded on their earlier knowledge about the syntactic regularities in the counting sequence. In Study 2, this association was tested again in monolingual Korean kindergarteners (5-6 years), as we aimed to test the same effect in a language with a highly regular numeral system. It replicated the association between Give-N10 performance and counting fluency, and it also demonstrated that Korean-speaking children understand the embedded structure of cardinal numbers earlier in the acquisition path than English-speaking peers, suggesting that the knowledge of numerical syntax governs children’s understanding of the generative properties of numbers. Based on these observations and our theoretical analysis of the literature, we propose that the syntax for building complex numerals represents the structure of numerical thinking.

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The Natural Numbers

10.1093/oso/9780199641314.003.0010 ◽

2018 ◽

Author(s):

Øystein Linnebo

Keyword(s):

Natural Number ◽

Peano Arithmetic ◽

Number Sequence ◽

Natural Numbers ◽

Ordinal Numbers ◽

Cardinal Numbers

How are the natural numbers individuated? That is, what is our most basic way of singling out a natural number for reference in language or in thought? According to Frege and many of his followers, the natural numbers are cardinal numbers, individuated by the cardinalities of the collections that they number. Another answer regards the natural numbers as ordinal numbers, individuated by their positions in the natural number sequence. Some reasons to favor the second answer are presented. This answer is therefore developed in more detail, involving a form of abstraction on numerals. Based on this answer, a justification for the axioms of Dedekind–Peano arithmetic is developed.

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Monolingual and bilingual children’s understanding of Moore-paradox sentences

Cognition Brain Behavior An Interdisciplinary Journal ◽

10.24193/cbb.2021.25.07 ◽

2021 ◽

Vol 25 (2) ◽

pp. 129-155

Author(s):

Krisztina Bartha ◽

Keyword(s):

Theory Of Mind ◽

Cognitive Science ◽

Experimental Philosophy ◽

Overwhelming Majority ◽

The Other ◽

Bilingual Children ◽

Central Topic ◽

Experimental Findings

Research of the theory of mind (ToM) has long been a central topic in cognitive science and experimental philosophy. A preliminary example of a Moore-paradox sentence would be: It is raining, but I don’t think it is. Understanding the paradoxes in these sentences is considered part of ToM development. This study focuses on the recognition of Moore’s paradoxical sentences by monolingual and bilingual children. According to the first hypothesis, comprehension of Moore-paradoxical sentences is estimated to start at the age of 7. The second hypothesis assumes that balanced bilingual children develop the ability to understand Moore-paradoxical sentences earlier than Hungarian dominant bilinguals, and balanced bilinguals also outperform their monolingual peers. Romanian monolingual and Hungarian-Romanian bilingual children aged between 5 and 8 (N = 134) participated in the experiment. Balanced and dominant bilingual groups were established based on a questionnaire filled in by the children’s parents. During the experiment, children had to listen to a number of sentences. Each sentence that contained paradoxical statements had control sentences matching syntactically. Children had to choose the sentences they thought to be “silly”. According to the experimental findings, 5- and 6-year-old children performed poorly while the overwhelming majority of 7- and 8-year-olds could select the Moore-paradoxical sentences. There were differences between the performance of monolingual and balanced bilingual groups and between the two bilingual groups. Balanced bilinguals performed better, and their comprehension of understanding Moorean sentences developed earlier than those of the other groups.

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Philosophy of Number

10.1093/oxfordhb/9780199642342.013.039 ◽

2014 ◽

Author(s):

Marcus Giaquinto

Keyword(s):

Final Section ◽

Natural Numbers ◽

Set Size ◽

Cardinal Numbers

There are many kinds of number. This chapter concentrates on finite cardinal numbers, as they have a basic role in our thinking. Numbers cannot be seen, heard, touched, tasted, or smelled; they do not emit or reflect signals; they leave no traces. So what kind of thing are they? How can we have knowledge of them? The aim of this chapter is to present and assess the main answers to these questions – classical and neo-classical, nominalism, mentalism, fictionalism, logicism, and the set-size view. All views are disputed, including the view I will argue for, the set-size view. The final section relates the finite cardinal numbers to the natural numbers.

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A universal embedding property of the RETs

Journal of Symbolic Logic ◽

10.1017/s0022481200092227 ◽

1970 ◽

Vol 35 (1) ◽

pp. 51-59 ◽

Cited By ~ 1

Author(s):

Anil Nerode ◽

Alfred B. Manaster

Keyword(s):

Recursive Function ◽

Relational Structure ◽

Partial Ordering ◽

Proper Subset ◽

Natural Numbers ◽

Partial Recursive Function ◽

Cardinal Numbers ◽

Equivalence Type ◽

Universal Embedding

Recursive equivalence types are an effective or recursive analogue of cardinal numbers. They were introduced by Dekker in the early 1950's. The richness of various theories related to the recursive equivalence types is demonstrated in this paper by showing that the theory of any countable relational structure can be embedded in or interpreted in these theories. A more complete summary is presented in the last paragraph of this section.Let E = {0,1, 2, …} be the natural numbers. If α ⊆ E, β ⊆ E, and there is a 1-1 partial recursive function f such that the image under f of α is β, α and β are called recursively equivalent (see [3]). The recursive equivalence type or RET of α, denoted 〈α〉, is the class of all β recursively equivalent to α. Addition of RETs is defined by 〈α〉 + 〈β〉 = 〈{2x ∣ x ∈ α} ∪ 〈{2x + 1 ∣ x ∈ β}〉. The partial ordering ≤ is defined on the RETs by A ≤ B iff (EC)(A + C = B). An RET, X, is called an isol if X ≠ X + 1 or, equivalently, if no representative of X is recursively equivalent to a proper subset of itself. The isols are thus the recursive analogue of the Dedekind-finite cardinals.

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Discrete Infinity and the Syntax-semantics Interface

Revista Linguíʃtica ◽

10.31513/linguistica.2017.v13n2a14031 ◽

2017 ◽

Vol 13 (2) ◽

pp. 28

Author(s):

Uli Sauerland ◽

Pooja Paul

Keyword(s):

Cognitive Science ◽

Natural Number ◽

Lambda Calculus ◽

Lexical Item ◽

Central Feature ◽

Human Language ◽

Natural Numbers ◽

Computational System

Discrete infnity was identifed as a central feature of human language by Humboldt who famously spoke of making infnite use of fnite means. Later Chomsky refocused attention on this property starting with Chomsky (1957). In a number of works since, Chomsky has repeatedly stressed the centrality of infnity for understanding language. For example, Chomsky (2007) writes that “An I-language is a computational system that generates infnitely many internal expressions”. Chomsky also noted that the property of discrete infnity is shared by the natural numbers and language. This connection has also caught the interest of others in cognitive science (e.g. Dehaene 1999, Dehaene et al. 1999). In this squib, we want to discuss concrete reductions of discrete infnity of the natural number. Specifcally, we want to investigate the extent to which this connection is compatible with current views of the syntax-semantics interface. We argue that merge alone is not enough to derive infnity, but a minimal lexicon is necessary, as Chomsky (2007) has noted in passing. We furthermore show that Chomsky’s assertion that a single lexical item is sufcient to generate the natural numbers depends on two assumptions -- an untyped lambda calculus, and a specifc interpretation of the syntactic Merge operation.

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Frege’s Theorems on Simple Series

Essays on Frege's Basic Laws of Arithmetic ◽

10.1093/oso/9780198712084.003.0009 ◽

2019 ◽

pp. 207-234

Author(s):

William Stirton

Keyword(s):

Natural Number ◽

Natural Numbers ◽

Cardinal Numbers ◽

Points Of Interest ◽

Main Observation ◽

The Relationship

The paper discusses theorems 207, 263, 327 and 348 and the proofs thereof. With fidelity to Frege’s own terminology, we can describe all four theorems as being about simple series. An attempt is made to explain what he meant by “simple series”. The main observation concerning theorems 207 and 263 is that the proofs of these theorems, slightly modified, yield a proof that a set is denumerable if and only if it is simply infinite in Dedekind’s sense. Some philosophical points of interest in this result are noted. Attention is drawn to some technical results proven by Frege which help in understanding the relationship between his own concept of natural number and Dedekind’s. The discussion of theorems 327 and 348 concentrates on critically assessing the claim, made by Heck, that by proving these theorems in the particular way he did Frege was meaning to demonstrate that finite cardinal numbers can be made to do work normally thought of as proper to finite ordinals. While this claim is not clearly wrong, some grounds for doubt are mentioned. One moral is that we should be sceptical about claims like “every foundation of arithmetic must treat the natural numbers as either fundamentally finite cardinals or fundamentally finite ordinals”.

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A changing target language: trends in American English as viewed from the EFL perspective of China

English Today ◽

10.1017/s0266078408000370 ◽

2008 ◽

Vol 24 (4) ◽

pp. 34-41

Author(s):

Fan Xianlong

Keyword(s):

Language Learners ◽

American English ◽

Point Of View ◽

Target Language ◽

Foreign Language Learners ◽

Knowing That ◽

Changing Trends ◽

Social Events ◽

English Speaking ◽

Object Of Study

ABSTRACTChanging trends in colloquial American English from the viewpoint of a visitor and their implications for teaching of English in China. Knowing that language changes and an appreciation of current changes is of great importance for foreign-language learners as it helps enable them to have a good command of the current language so as to strengthen their ability to communicate with native speakers with facility. The reality Chinese learners of English face is, however, that they hardly have opportunities to be exposed to natural spoken forms of the target language around them, let alone access to its current changing trends. This paper aims to present such information. Based on the investigation I made among native English-speaking Americans, it tries, from a descriptive pragmatic point of view, to give an account of some salient trends of American English in daily communication. It takes everyday spoken American English as the object of study, for it is the kernel part of the language for social interaction. It is this part of the language that first undergoes changes in response to various social events, and that, having much to do with the study of language use, deserves our special attention.

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Patterns of Regularity, Stability and Interdependence in Research Related to Communities in ELT

Studies in English Language Teaching ◽

10.22158/selt.v7n2p195 ◽

2019 ◽

Vol 7 (2) ◽

pp. p195

Author(s):

Julia Posada-Ortiz

Keyword(s):

Teacher Education ◽

Preservice Teachers ◽

English Language ◽

Ways Of Knowing ◽

English Language Teaching ◽

Language Teacher Education ◽

Knowing That ◽

English Speaking ◽

The Senses ◽

Unitary Concept

This paper presents a review of studies on communities in ELT in English-speaking countries and Latin America, including Colombia. The purpose of the article is to show that it is necessary to understand the senses Language Preservice Teachers make of the concept of communities and the ways they relate to each other and their teachers. Also, there is a unitary concept of community in the policies related to English Language Teacher Education in Colombia, a naturalization of the concept of community and patterns of regularity, stability and interdependence in research related to communities in English Language Teaching that make invisible how the English Language Preservice Teachers make sense of the concept of community in their affiliations or no affiliations with particular groups. Understanding the senses the English Language Preservice Teachers make about communities might bring to the fore other ways of knowing that can contribute to the improvement of the design of teacher education programmes.

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