Erkenntnistheorie der Zahldefinition und philosophische Grundlegung der Arithmetik unter Bezugnahme auf einen Vergleich von Gottlob Freges Logizismus und platonischer Philosophie (Syrian, Theon von Smyrna u.a.)

The epistomology of the definition of number and the philosophical foundation of arithmetic based on a comparison between Gottlob Frege's logicism and Platonic philosophy (Syrianus, Theo Smyrnaeus, and others). The intention of this article is to provide arithmetic with a logically and methodologically valid definition of number for construing a consistent philosophical foundation of arithmetic. The – surely astonishing – main thesis is that instead of the modern and contemporary attempts, especially in Gottlob Frege's Foundations of Arithmetic, such a definition is found in the arithmetic in Euclid's Elements. To draw this conclusion a profound reflection on the role of epistemology for the foundation of mathematics, especially for the method of definition of number, is indispensable; a reflection not to be found in the contemporary debate (the predominate ‘pragmaticformalism’ in current mathematics just shirks from trying to solve the epistemological problems raised by the debate between logicism, intuitionism, and formalism). Frege's definition of number, ‘The number of the concept F is the extension of the concept ‘numerically equal to the concept F”, which is still substantial for contemporary mathematics, does not fulfil the requirements of logical and methodological correctness because the definiens in a double way (in the concepts ‘extension of a concept’ and ‘numerically equal’) implicitly presupposes the definiendum, i.e. number itself. Number itself, on the contrary, is defined adequately by Euclid as ‘multitude composed of units’, a definition which is even, though never mentioned, an implicit presupposition of the modern concept ofset. But Frege rejects this definition and construes his own - for epistemological reasons: Frege's definition exactly fits the needs of modern epistemology, namely that for to know something like the number of a concept one must become conscious of a multitude of acts of producing units of ‘given’ representations under the condition of a 1:1 relationship to obtain between the acts of counting and the counted ‘objects’. According to this view, which has existed at least since the Renaissance stoicism and is maintained not only by Frege but also by Descartes, Kant, Husserl, Dummett, and others, there is no such thing as a number of pure units itself because the intellect or pure reason, by itself empty, must become conscious of different units of representation in order to know a multitude, a condition not fulfilled by Euclid's conception. As this is Frege's main reason to reject Euclid's definition of number (others are discussed in detail), the paper shows that the epistemological reflection in Neoplatonic mathematical philosophy, which agrees with Euclid's definition of number, provides a consistent basement for it. Therefore it is not progress in the history of science which hasled to the a poretic contemporary state of affairs but an arbitrary change of epistemology in early modern times, which is of great influence even today. This revised version was published online in August 2006 with corrections to the Cover Date.     

幼儿对数的认知及其策略

该研究探查幼儿对基数、数序、运算和解应用题的认知发展过程及其认知策略。着重探查：(1)不同认知任务对幼儿数认知发展的影响;(2)幼儿对基数和数序两者认知发展的顺序;(3)从幼儿主动解决问题的策略探究其认知发展水平。该研究采用定性和定量相结合的研究方法。被试为4、5、6岁城市幼儿园儿童,共92人,男女约各半。全部实验以个别方式进行。主要研究结果表明：(1)幼儿对基数、数序、运算和解应用题的认知成绩均有随年龄发展的趋势,但快速发展的年龄阶段因任务的难度而异;(2)幼儿对基数和数序的认知在4—5岁显示出不同步的发展,对基数的认知成绩优于对数序的认知,而到6岁两者具有同步发展的趋势;(3)幼儿解决问题的策略水平随年龄发展,显示了由外化水平的智力活动向完全内化的智力操作的发展过程,并具有明显的层次性。该研究结果为幼儿数能力的培养和促进提供参考依据。     

The associations between space and order in numerical and non-numerical sequences

Spatial-numerical associations have been found across different studies, yet the basis for these associations remains debated. The current study employed an order judgment task to adjudicate between two competing accounts of such associations, namely the Mental Number Line (MNL) and Working Memory (WM) models. On this task, participants judged whether number pairs were in ascending or descending order. Whereas the MNL model predicts that ascending and descending orders should map onto opposite sides of space, the WM model predicts no such mapping. Moreover, we compared the spatial-order mapping for numerical and non-numerical sequences because the WM model predicts no difference in mapping. Across two experiments, we found consistent spatial mappings for numerical order along both horizontal and vertical axes, consistent with a MNL model. In contrast, we found no consistent mappings for letter sequences. These findings are discussed in the context of conflicting extant data related to these two models.     

To drive or not to drive: Neuropsychological assessment for driver's license among stroke patients

Seventy-two stroke patients, 43 with right hemisphere (RHD) and 29 with left hemisphere damage (LHD), and 7 coronary infarct controls with no evidence of cerebral damage, were neuropsychologically tested as part of an assessment program for driver's license. Mean age in the group was 53 years. Stroke patients were tested on average 4 months post injury. The groups did not differ on major demographic variables except that RHD patients were more often hemiplegic than LHD patients. The test battery was factor analyzed into 4 valid principal components: (I) visual perception , (II) spatial attention , (III) visuospatial processing , and (IV) language/praxis. The presence of hemianopia (factor I) excludes driving. In addition, measures of neglect and reduced speed of mental processing from factor II, III and IV, were found to be the most discriminating variables when classifying patients for driving. Even though neglect was more frequently observed among RHD than LHD patients, the two hemisphere groups did not differ significantly in number of patients denied driving, 58% RHD compared to 41% LHD patients. The need for comprehensive neuropsychological assessment is underlined.     

汉字加工的基本单元:来自笔画数效应和部件数效应的证据

用命名作业和真假字判断作业考察了笔画数和部件数对汉字加工的影响,发现,当部件数保持恒定时,笔画数少的汉字较笔画数多的汉字加工快,这种笔画数效应和字频之间不存在交互作用;当笔画数被控制时,部件数效应也存在,但和字频有交互作用,表现在低频少部件汉字的加工要快于低频多部件汉字的加工,而高频汉字中的部件数效应不显著。实验还发现独体字（只有一个部件）的加工和合体字（部件数大于等于二）的加工有差异。作者认为,汉字的加工要经过笔画、部件和整字三个层次,其中单位部件的笔画数和部件数影响着汉字加工时间     

Pigeons' preference for free choice: number of keys versus key area.

In concurrent-chains schedules, pigeons prefer terminal links that provide two keys correlated with reinforcers (free choice) over those that provide only one key (forced choice), terminal-link reinforcement rates being equal. With same-size keys, free choice provides a larger area available for pecking. Preferences were examined using terminal links that differed in key number only (one or two) or key size only (small and medium or medium and large), or that equated the area of the two free-choice keys with that of the forced-choice key. Medium (standard) keys were typically preferred to small keys, but indifference was typically obtained between medium and large keys. The size preference usually overrode free-choice preference with one medium key pitted against two small keys, but free-choice preference was reliably observed with one large key pitted against two medium keys. In other words, preferences were a joint function of key number and key area, implying that free-choice preference is not reducible to preference for larger key areas. Free-choice preference requires separate keys rather than larger areas; the relevant behavioral units are the discriminated operants correlated with each terminal-link key rather than classes defined by topographical features such as area or perimeter.     

仪表数量与信号间隔时距对监视作业效绩的影响

用信号察觉反应时与信号误报率作指标 ,对监视作业中监视仪表数量与信号间隔时距对监视作业效绩的影响问题进行研究。结果表明 ,监视作业的效绩会随仪表增多和信号间隔时距增大而呈下降趋势。当监视仪表达到 8个时 ,监视作业的效绩会因负荷过重而明显下降。信号出现间隔时距增长时 ,会因警戒水平下降使监视仪表作业的效绩明显降低。     

Delay and number of food reinforcers: Effects on choice and latencies

Pigeons were given a choice between two identical-duration situations (terminal links of chain schedules). One terminal link of the choice pair provided two food deliveries, and the other provided five. The exact times of these food deliveries differed between the terminal links and were varied over conditions. A single response during the initial link gave immediate access to the corresponding terminal link. Forced trials, during which only one of the initial-link keys was lighted, were interspersed with choice trials during which both initial-link keys were lighted. Choice tended to favor whichever terminal link was correlated with the higher sum of the immediacies (i.e., the sum of the reciprocals of the delays to each of the reinforcers following the choice, with all delays measured from the choice). Latencies on forced trials and on choice trials also were related (negatively) to the sum of the immediacies. This correlation among response measures (choice and latencies) suggests that both measures are manifestations of the effect of conditioned reinforcement on response tendencies.     

Noniterative estimation and the choice of the number of factors in exploratory factor analysis

Based on the usual factor analysis model, this paper investigates the relationship between improper solutions and the number of factors, and discusses the properties of the noniterative estimation method of Ihara and Kano in exploratory factor analysis. The consistency of the Ihara and Kano estimator is shown to hold even for an overestimated number of factors, which provides a theoretical basis for the rare occurrence of improper solutions and for a new method of choosing the number of factors. The comparative study of their estimator and that based on maximum likelihood is carried out by a Monte Carlo experiment.The author would like to express his thanks to Masashi Okamoto and Masamori Ihara for helpful comments and to the editor and referees for critically reading the earlier versions and making many valuable suggestions. He also thanks Shigeo Aki for his comments on physical random numbers.