Arithmetical Set Theory

It is well known that number theory can be interpreted in the usual set theories, e.g. ZF, NF and their extensions. The problem I posed for myself was to see if, conversely, a reasonably strong set theory could be interpreted in number theory. The reason I am interested in this problem is, simply, that number theory is more basic or more concrete than set theory, and hence a more concrete foundation for mathematics. A partial solution to the problem was accomplished by WTN in [2], where it was shown that a predicative set theory could be interpreted in a natural extension of pure number theory, PN, (i.e. classical first-order Peano Arithmetic). In this paper, we go a step further by showing that a reasonably strong fragment of predicative set theory can be interpreted in PN itself. We then make an attempt to show how to develop predicative fragments of mathematics in PN.If one wishes to know what is meant by reasonably strong and fragment please read on.     

Arithmetical completeness versus relative completeness

In this paper we study the status of the arithmetical completeness of dynamic logic. We prove that for finitistic proof systems for dynamic logic results beyond arithmetical completeness are very unlikely. The role of the set of natural numbers is carefully analyzed.     

Fuzzy Logic and Arithmetical Hierarchy III

Fuzzy logic is understood as a logic with a comparative and truth-functional notion of truth. Arithmetical complexity of sets of tautologies (identically true sentences) and satisfiable sentences (sentences true in at least one interpretation) as well of sets of provable formulas of the most important systems of fuzzy predicate logic is determined or at least estimated.     

Arithmetical calculations in monolinguals and bilinguals

Summary A series of 240 arithmetical problems requiring one, two, or three operations of the same kind were presented to 32 German-Swedish bilingual and 14 German dominant subjects in the age range 14–19 years. The results provide evidence that language has an important effect on arithmetical performance, affecting latencies and error rates. Generally the bilingual subjects needed more time to perform the tasks and made more errors than the monolingual subjects. The results are tentatively explained by retrieval interference and decay of memory trace and support an interdependence hypothesis of bilingual organization in memory.This research was supported by a grant from the Royal Swedish Academy of Sciences     

发展性计算障碍儿童的数认知

发展性计算障碍作为一种特异性学习障碍,影响儿童的数学成就与数学学习观念,因而引起了多学科研究的关注和重视。文章分析了发展性计算障碍领域在数认知方面的已有研究,揭示了发展性计算障碍的数认知缺陷及内部差异,从一般认知加工障碍与数字加工模块障碍两个角度阐述了发展性计算障碍的可能成因,最后对当前研究进行了小结与展望,认为数认知缺陷的复杂关系、亚类型特征、成因之间的相互作用与机制还需进一步研究     

Fuzzy Logic and Arithmetical Hierarchy,II

A very simple many-valued predicate calculus is presented; a completeness theorem is proved and the arithmetical complexity of some notions concerning provability is determined.     

Short term Memory Impairment and Arithmetical Ability

We document the dissociation of preserved calculation skills in a patient with impaired auditory short-term memory. The patient (MRF) had a memory span of three digits. Furthermore, he showed rapid decrement in performance of single digits and letters with both auditory and visual presentation in the Brown-Peterson forgetting task. Analysis of his calculation skills revealed a normal ability to solve auditorily presented multidigit addition and subtraction problems such as 173 + 68 and to execute the Paced Auditory Serial Addition Task (Sampson, 1956, 1958; Gronwall, 1977). In addition, his performance on other tests, including arithmetic manipulation of natural numbers, decimals and fractions, approximation, magnitude, ratio, and percentage, appeared to be normal (Hitch, 1978b). It is argued that these findings require a revision of Baddeley and Hitch's (1974) concept of the function of working memory.     

工作记忆成分与儿童算术认知

根据Baddeley和Hitch(1974)提出的工作记忆多成分模型,研究者对工作记忆的各个成分与儿童算术认知之间的关系进行了广泛的探索。同语音环和视空间模板相比,中央执行在儿童算术认知加工中的作用显得更为关键。目前,探讨各种中央执行功能与儿童算术认知的关系,以及由此来解释算术学习困难儿童的认知成因是这个领域研究者最为关注的焦点。     

The Interpretability Logic of all Reasonable Arithmetical Theories

This paper is a presentation of astatus quæstionis, to wit of the problemof the interpretability logic of all reasonablearithmetical theories.We present both the arithmetical side and themodal side of the question.Dedicated to Dick de Jongh on the occasion of his 60th birthday     

Representational Structures of Arithmetical Thinking: Part I

In this paper, representational structures of arithmetical thinking, encoded in human minds, are described. On the basis of empirical research, it is possible to distinguish four types of mental number lines: the shortest mental number line, summation mental number lines, point-place mental number lines and mental lines of exact numbers. These structures may be treated as generative mechanisms of forming arithmetical representations underlying our numerical acts of reference towards cardinalities, ordinals and magnitudes. In the paper, the theoretical framework for a formal model of mental arithmetical representations is constructed. Many competitive conceptions of the mental system responsible for our arithmetical thinking may be unified within the presented framework. The paradigm underlying our research may be interpreted philosophically as a neo-Kantian approach to modeling the mind’s representational structures.     

工作记忆容量与儿童算术认知加工

工作记忆容量的差别是个体认知差异的一个重要方面。工作记忆容量在多种运算规则下与运算技能、概念性知识等因素交织在一起共同对儿童的算术认知加工产生影响,并且它是数学学习不良儿童学习困难的一个重要影响因素。同时,儿童的工作记忆容量本身也受到多种因素的制约,进而影响儿童的算术认知成绩。     

Working Memory Failures in Children with Arithmetical Difficulties

A large body of literature has examined the relationship between working memory and arithmetic achievement, but results are still ambiguous. To examine this relationship, we compared the performance of third and fifth graders with arithmetic difficulties (AD) and controls of the same age, grade, and verbal intelligence on a battery of working memory tasks, differentiating between different aspects of working memory. Children with AD scored significantly lower on active working memory tasks requiring manipulation of the to-be-recalled information (Listening Completion task, Corsi Span Backwards, Digit Backwards), but not in passive working memory tasks, requiring the recall of information in the same format in which it had been presented (Digit, Word, and Corsi Forwards Span tasks), nor in tasks involving word processing (word articulation rate, forwards and backwards word spans). A regression analysis showed that the best predictors of differences between AD children and the control group were the Corsi Span Backwards, the Listening Completion task, and the rate of articulation of pseudowords. The analysis of strategies used by children in mental calculation revealed the greater tendency of children with AD to rely on more primitive strategies: finger use never appeared as the most frequent strategy in skilled children, whereas it was the most used strategy in children with AD. Verbal and visual strategies appeared associated with successful performance in third graders, but in fifth grade, the most successful strategy was verbalization.     

儿童执行功能与算术认知策略的关系

对35名小学2年级儿童的四种执行功能与有无工作记忆负荷两种情况下算术认知策略表现之间的关系进行了探讨.结果表明,双任务协调功能与工作记忆负荷增加时放弃策略的使用次数具有显著负相关;策略转换功能越强的儿童.算术认知策略的使用越灵活;抑制功能强的儿童在提取策略的相关表现上更好;记忆更新功能与工作记忆负荷增加时策略执行的正确率具有显著正相关.这说明,执行功能与儿童算术认知策略之间存在着特定的联系.     

Arithmetical Completeness Theorem for Modal Logic $$\mathsf{}$$

We prove that for any recursively axiomatized consistent extension T of Peano Arithmetic, there exists a \(\Sigma _2\) provability predicate of T whose provability logic is precisely the modal logic \(\mathsf{K}\). For this purpose, we introduce a new bimodal logic \(\mathsf{GLK}\), and prove the Kripke completeness theorem and the uniform arithmetical completeness theorem for \(\mathsf{GLK}\).     

工作记忆广度对儿童算术认知策略的影响

选取113名小学二年级儿童为被试,探讨了工作记忆广度对儿童算术认知策略的影响。结果表明:不同工作记忆广度儿童在出声、手动、心里数数、竖式、分解、凑整、猜测和放弃策略的选择频次上具有显著差异,在算术认知策略的执行上,差异明显;随着工作记忆广度的增加,出声策略表现出了波动型的变化曲线,手动、心里数数、凑整和猜测策略具有门槛型的变化趋势,竖式、分解和放弃策略呈现出阶梯状的变化特点。这些结果说明,儿童算术认知策略表现受到其工作记忆容量明显的限制性作用。     

工作记忆中央执行对儿童算术认知策略的影响

运用“双重任务”范式,以113名小学二年级儿童为被试,考察了儿童工作记忆的中央执行对算术认知策略表现的影响。结果表明:中央执行对儿童算术认知策略的选择产生了针对性的影响,集中表现在外部策略、支持性策略和猜测放弃策略上。中央执行干扰使策略执行的正确率下降、反应时增加。在简单加法任务上,中央执行影响了外部策略和支持性策略的选择频次;在复杂任务中,这种影响体现在外部策略和放弃策略上。无论是简单任务还是复杂任务,中央执行的干扰都造成了策略整体执行效果的下降。     

From Maximal Intersubjectivity to Objectivity: An Argument from the Development of Arithmetical Cognition

Topoi - 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...     

Arithmetical Soundness and Completeness for $$\varvec{\Sigma }_{\varvec{2}}$$ Numerations

We prove that for each recursively axiomatized consistent extension T of Peano Arithmetic and \(n \ge 2\), there exists a \(\Sigma _2\) numeration \(\tau (u)\) of T such that the provability logic of the provability predicate \(\mathsf{Pr}_\tau (x)\) naturally constructed from \(\tau (u)\) is exactly \(\mathsf{K}+ \Box (\Box ^n p \rightarrow p) \rightarrow \Box p\). This settles Sacchetti’s problem affirmatively.