Reference to numbers in natural language

A common view is that natural language treats numbers as abstract objects, with expressions like the number of planets, eight, as well as the number eight acting as referential terms referring to numbers. In this paper I will argue that this view about reference to numbers in natural language is fundamentally mistaken. A more thorough look at natural language reveals a very different view of the ontological status of natural numbers. On this view, numbers are not primarily treated abstract objects, but rather ‘aspects’ of pluralities of ordinary objects, namely number tropes, a view that in fact appears to have been the Aristotelian view of numbers. Natural language moreover provides support for another view of the ontological status of numbers, on which natural numbers do not act as entities, but rather have the status of plural properties, the meaning of numerals when acting like adjectives. This view matches contemporary approaches in the philosophy of mathematics of what Dummett called the Adjectival Strategy, the view on which number terms in arithmetical sentences are not terms referring to numbers, but rather make contributions to generalizations about ordinary (and possible) objects. It is only with complex expressions somewhat at the periphery of language such as the number eight that reference to pure numbers is permitted.     

Language-specific effects on number computation in toddlers: A European cross-linguistic cartography

A fundamental question in developmental science is how brains with and without language compute numbers. Measuring young children's verbal reactions in Spain and Finland, we show that, although there is a general arithmetic ability for small numbers that is shared by monkeys and preverbal infants, the development of such initial knowledge in humans follows specific performance patterns, depending on what language the children speak (here, Spanish and Finnish). Together with our previous data collected in France and in England, these new results contribute to a European cross-linguistic cartography of the relationships between language and number development.     

Exceptional lexical skills but executive language deficits in school starters and young adults with Turners syndrome: implications for X chromosome effects on brain function

TS school starters had enhanced receptive and expressive language on standardised assessment (CELF-P) and enhanced rhyme judgements, spoonerisms, and lexical decision, indicating enhanced phonological skills and word representations. There was marginal but consistent advantage across lexico-semantic tasks. On executive tasks, speeded naming of numbers was impaired but not pictures. Young TS adults had enhanced naming and receptive vocabulary, indicating enhanced semantic skills. There were consistent deficits in executive language: phonemic oral fluency, rhyme fluency, speeded naming of pictures, numbers and colours; sentence completion requiring supression of prepotent responses.Haploinsufficiency of X-chromosome drives mechanisms that affect the anatomical and neurochemical development of the brain, resulting in enhanced temporal lobe aspects of language. These strengths co-exist with impaired development of frontal lobe executive language systems. This means not only that these elements of language can decouple in development but that their very independence is driven by mechanisms linked to the X-chromosome.     

Language and number: a bilingual training study

Three experiments investigated the role of a specific language in human representations of number. Russian-English bilingual college students were taught new numerical operations (Experiment 1), new arithmetic equations (Experiments 1 and 2), or new geographical or historical facts involving numerical or non-numerical information (Experiment 3). After learning a set of items in each of their two languages, subjects were tested for knowledge of those items, and new items, in both languages. In all the studies, subjects retrieved information about exact numbers more effectively in the language of training, and they solved trained problems more effectively than untrained problems. In contrast, subjects retrieved information about approximate numbers and non-numerical facts with equal efficiency in their two languages, and their training on approximate number facts generalized to new facts of the same type. These findings suggest that a specific, natural language contributes to the representation of large, exact numbers but not to the approximate number representations that humans share with other mammals. Language appears to play a role in learning about exact numbers in a variety of contexts, a finding with implications for practice in bilingual education. The findings prompt more general speculations about the role of language in the development of specifically human cognitive abilities.     

Visual field asymmetries in numerical size comparisons of digits, words, and signs

Visual field asymmetries were examined in American Sign Language-English bilinguals for speeded numerical size judgments of pairs of digits, number words, and number signs. Physical size of the number pairs was either congruent or incongruent with their numerical size. The results revealed a greater left visual field (LVF) interference for numbers represented as digits and a greater right visual field (RVF) interference for numbers represented as words or signs. Subjects' performance on number words and signs was also influenced by their skill in English and ASL: interference was greater in the RVF in the subjects' better language but was greater in the LVF for the less skilled language. These findings suggest that lateralization of numerical size judgments is moderated by the mode of number presentation and by prior language experience.     

The processing of two-digit numbers in bilinguals

We explored possible between-language influences when bilinguals processed two-digit numbers. Spanish/English bilinguals and German/English bilinguals performed a number comparison task with Arabic digits and verbal numbers in their first language (L1) and second language (L2) while the unit-decade compatibility was manipulated. The two bilingual groups showed regular compatibility effect with Arabic digits. In L1, Spanish/English bilinguals showed reverse compatibility effect, while German/English bilinguals showed regular compatibility effect. However, both groups of bilinguals presented reverse compatibility effect in English (L2), which suggested that the bilinguals' L1 did not determine the processing of number words in their L2. The results indicated that bilinguals processed two-digit number words selectively in their L1 and L2 and that they did not transcode L2 numbers into Arabic format.     

语言转换对非熟练双语者不同认知控制成分的即时影响

语言转换情境是否影响非熟练双语者的认知控制存在争议。实验1和实验2分别采用不同的实验范式探讨语言转换对非熟练双语者的认知控制是否会有即时的影响, 以及对不同认知控制成分的影响是否会有不同。结果表明, 语言转换情境能够即时地促进反应抑制, 阻碍干扰抑制, 但对认知灵活性没有影响, 同时说明了反应抑制、干扰抑制和认知灵活性属于3种不同的认知成分, 其内在作用机制不同。此外, 研究结果还进一步说明了双语者的认知控制优势是双语者长期双语使用的结果, 为双语优势的内在机制提供了实验支持。     

Speaking two languages with different number naming systems: What implications for magnitude judgments in bilinguals at different stages of language acquisition?

Differences between languages in terms of number naming systems may lead to performance differences in number processing. The current study focused on differences concerning the order of decades and units in two-digit number words (i.e., unit-decade order in German but decade-unit order in French) and how they affect number magnitude judgments. Participants performed basic numerical tasks, namely two-digit number magnitude judgments, and we used the compatibility effect (Nuerk et al. in Cognition 82(1):B25–B33, 2001) as a hallmark of language influence on numbers. In the first part we aimed to understand the influence of language on compatibility effects in adults coming from German or French monolingual and German–French bilingual groups (Experiment 1). The second part examined how this language influence develops at different stages of language acquisition in individuals with increasing bilingual proficiency (Experiment 2). Language systematically influenced magnitude judgments such that: (a) The spoken language(s) modulated magnitude judgments presented as Arabic digits, and (b) bilinguals’ progressive language mastery impacted magnitude judgments presented as number words. Taken together, the current results suggest that the order of decades and units in verbal numbers may qualitatively influence magnitude judgments in bilinguals and monolinguals, providing new insights into how number processing can be influenced by language(s).     

Parental language mixing in bilingual families

An important question about early bilingualism that concerns both parents and researchers is the degree to which one language may interfere with another. This question rests on an implicit assumption that learning more than one language must always produce confusion and/or interference between (or among) the languages. Although many naturalistic studies have addressed this issue, no firm answers are yet available from the conflicting results obtained. Several factors appear to be responsible for the contradictory evidence, including the small numbers of subjects in each study, the large number of different language combinations of varying similarity that have been examined, and the variety of linguistic input situations (e.g., sequential bilingualism, language separation between home and school, parent/language separation), that have been observed. There is a consensus that children's language mixing can be eliminated if parents adhere strictly to the principle of one parent/one language enunciated by Grammont (Ronjat, 1913). However, this claim has not been directly examined. The primary goal of the present study is to characterize the linguistic input available to a child growing up with two languages. Analyses of the data show that a large proportion of parents, even those firmly committed to maintaining a strict separation of language by parent, model linguistically mixed utterances for their children. This finding suggests that children's early language mixing does not reflect interlinguistic confusion. Rather, it suggests that the child is formulating hypotheses about language based on the data available, i.e., that using the language of both father and mother in a single utterance is acceptable.     

线性时态逻辑的决策问题复杂性

不同的时态逻辑能够适应不同的推理任务。为了符合应用,关于时间的模型从离散的自然数和整数,延伸到稠密的线性实数,甚至扩展到区间代数和树代数。如果简单的时态连接词的表达力已经足够,就只需使用这些简单的时态连接词来构造的时态逻辑。在能够承担降低运算速度的风险下,我们可以为实现更强的表达力而使用更多的连接词,也可以加上度量信息或者固定点。作者近期提出了一个令人惊讶的结论:建立在实数时间上的具有足够表达力的语言和基于自然数离散时间流的传统简单算子,它们推理的计算复杂性是一样的。在这篇论文中,作者试图对建立在标准时态连接词和线性时间流的普通类上的时态逻辑中所有决策问题的计算复杂性作新的说明。尤其是,文中指出,所有标准逻辑在PSPACE中都存在决策问题。     

Iconic and non-iconic stages in number development: the role of language

Is language the key to number? This article argues that the human language faculty provides the cognitive equipment that enables humans to develop a systematic number concept. Importantly, the number concept is based on non-iconic representations that involve relations between relations: relations between numbers are linked with relations between objects. In contrast to this, language-independent numerosity concepts provide only iconic representations. The pattern of forming relations between relations lies at the heart of our language faculty, suggesting that it is language that enables humans to make the step from these iconic representations, which we share with other species, to a generalized concept of number.     

Relating numeric cognition and language processing: Do numbers and words share a common representational platform?

Numerical processing and language processing are both grounded in space. In the present study we investigated whether these are fully independent phenomena, or whether they share a common basis. If number processing activates spatial dimensions that are also relevant for understanding words, then we can expect that processing numbers may influence subsequent lexical access to words. Specifically, if high numbers relate to upper space, then they can be expected to facilitate understanding of words such as bird that are having referents typically found in the upper vertical space. The opposite should hold for low numbers. These should facilitate the understanding of words such as ground referring to entities with referents in the lower vertical space. Indeed, in two experiments we found evidence for such an interaction between number and word processing. By eliminating a contribution of linguistic factors gained from additional investigations on large text corpora, this strongly suggests that understanding numbers and language is based on similar modal representations in the brain. The implications of these findings for a broader perspective on grounded cognition will be discussed.     

English palindromes: A distributional model

A list of English palindromes that approaches comprehensiveness is given. A stochastic model for the distribution of palindromes within the language is proposed. The model relates frequencies of palindromes and of heteropalindromes (Jones, 1980) as a function of word length, and it is shown to predict accurately the actual numbers of palindromes that occur.     

Number processing of Arabic and Hebrew bilinguals: Evidence supporting the distance effect

In the current study, a direct assessment of the effect of language lexical‐syntactic structure and magnitude semantic access on numerical processing was made by contrasting the performance of Arabic/Hebrew bilinguals in a digital (Hindi‐digits/Arabic‐digits) and verbal numerical comparison task (Arabic, an inverted language: Units‐Decades, Hebrew, a non‐inverted language: Decades‐Units). Our data revealed in the digital presentation format (Experiment 1) a regular distance effect in Arabic language‐Hindi digits and Hebrew language‐Arabic digits, characterized by an inverse relation between reaction times and numerical distance with no difference in the mean reaction times of participants in Arabic‐L1 and Hebrew‐L2. This indicates that both lexical digits of two‐digit numbers in L1 and L2 are similarly processed and semantically accessed. However in the verbal presentation format (Experiment 2) a similar pattern of distance effect was found, but the mean reaction times in Arabic were lower than in Hebrew in each numerical distance. This indicates that the processing of two‐digit number words in L1 and L2 is semantically accessed and determined by the syntactic structure of each language.     

Hobson’s Conception of Definable Numbers

In this paper, I explore an intriguing view of definable numbers proposed by a Cambridge mathematician Ernest Hobson, and his solution to the paradoxes of definability. Reflecting on König’s paradox and Richard’s paradox, Hobson argues that an unacceptable consequence of the paradoxes of definability is that there are numbers that are inherently incapable of finite definition. Contrast to other interpreters, Hobson analyses the problem of the paradoxes of definability lies in a dichotomy between finitely definable numbers and not finitely definable numbers. To bypass this predicament, Hobson proposes a language dependent analysis of definable numbers, where the diagonal argument is employed as a means to generate more and more definable numbers. This paper examines Hobson’s work in its historical context, and articulates his argument in detail. It concludes with a remark on Hobson’s analysis of definability and Alan Turing’s analysis of computability.     

New perspectives on the expression of quantity

The expression of quantity is central to many acts of communication, both formal and casual. Expressions of quantity, whether in numbers and percentages or in language, are used to convey information about frequency, certainty, risk, and degrees of association. It has typically been supposed that the many linguistic expressions that convey quantities are merely vague ways of indicating information that would be better conveyed by numbers and percentages, whenever possible. However, we show that such a view is too simplistic and is misleading. Language expressions can be more informative than numerical information alone. Language expressions carry built-in perspectives that affect the inferences and decisions made by listeners and readers. We also show that information presented through numbers (e.g., the depiction of fat content of foods) is not necessarily neutral, but also induces perspective. Failure to recognize effects of perspective may have led to some false conclusions about how people attribute causation.     

Language statistics explain the spatial–numerical association of response codes

The spatial–numerical association of response codes (SNARC) has shown that parity judgments with participants’ left hands yield faster response times (RTs) for smaller numbers than for larger numbers, with the opposite result for right-hand responses. These findings have been explained by participants perceptually simulating magnitude on a mental number line. In three RT experiments, we showed that the SNARC effect can also be explained by language statistics. Participants made parity judgments of number words (Exp. 1) and Arabic numerals (Exp. 2). Linguistic frequencies of the number words and numbers mirrored the SNARC effect, explaining aspects of processing that a perceptual simulation account could not. In Experiment 3, we investigated whether high- and low-frequency nonnumerical words would also elicit a SNARC-like effect. Again, RTs were faster for high-frequency words for left-hand responses, with the opposite result for right-hand responses. These results demonstrate that what has only been attributed to perceptual simulation should also be attributed to language statistics.     

Memory for numbers in monolinguals and bilinguals

Forty-six subjects were assigned to two groups according to a reaction time technique: one monolingual group of 14 subjects with German as the dominant language and one group of 32 German-Swedish bilingual subjects balanced in skills. The subjects were given short-term memory tasks on numbers presented aurally and a visually presented prose text involving both short-term and long-term memory processes. The results provide evidence that two language systems actively used in daily communication affect both speed and memory span for digits: when compared with monolingual subjects, balanced bilinguals read more slowly and showed a higher error rate in the recall of two-digit numbers. The results support a hypothesis of interdependent bilingual storage.