A Dissociation between Symbolic Number Knowledge and Analogue Magnitude Information

Semantic understanding of numbers and related concepts can be dissociated from rote knowledge of arithmetic facts. However, distinctions among different kinds of semantic representations related to numbers have not been fully explored. Working with numbers and arithmetic requires representing semantic information that is both analogue (e.g., the approximate magnitude of a number) and symbolic (e.g., what / means). In this article, the authors describe a patient (MC) who exhibits a dissociation between tasks that require symbolic number knowledge (e.g., knowledge of arithmetic symbols including numbers, knowledge of concepts related to numbers such as rounding) and tasks that require an analogue magnitude representation (e.g., comparing size or frequency). MC is impaired on a variety of tasks that require symbolic number knowledge, but her ability to represent and process analogue magnitude information is intact. Her deficit in symbolic number knowledge extends to a variety of concepts related to numbers (e.g., decimal points, Roman numerals, what a quartet is) but not to any other semantic categories that we have tested. These findings suggest that symbolic number knowledge is a functionally independent component of the number processing system, that it is category specific, and that it is anatomically and functionally distinct from magnitude representations.     

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.     

关键词对精加工推理的影响作用初探

本实验研究在文本中突出关键词对精加工推理的产生水平和产生阶段的影响。研究采用了让被试阅读后做测试题的方法,通过分析测试题的成绩得出的结果表明：在文章中突出关键词有助于产生近推理和内涵推理,且使这两类推理更有可能在阅读过程中产生。本研究补充了McDaniel和Donnelly 1996年的研究结果。     

Use of current explanations in multicausal abductive reasoning

In multicausal abductive tasks a person must explain some findings by assembling a composite hypothesis that consists of one or more elementary hypotheses. If there are n elementary hypotheses, there can be up to 2ⁿ composite hypotheses. To constrain the search for hypotheses to explain a new observation, people sometimes use their current explanation—the previous evidence and their present composite hypothesis of that evidence; however, it is unclear when and how the current explanation is used. In addition, although a person's current explanation can narrow the search for a hypothesis, it can also blind the problem solver to alternative, possibly better, explanations. This paper describes a model of multicausal abductive reasoning that makes two predictions regarding the use of the current explanation. The first prediction is that the current explanation is not used to explain new evidence if there is a simple (i.e., nondisjunctive, concrete) hypothesis to account for that evidence. The second prediction is that the current explanation is used when attempting to discriminate among several alternative hypotheses for new evidence. These hypotheses were tested in three experiments. The results are consistent with the second prediction: the current explanation is used when discriminating among alternative hypotheses. However, the first prediction—that the current explanation is not used when a simple hypothesis can account for new data—received only limited support. Participants used the current explanation to constrain their interpretation of new data in 46.5% of all trials. This suggests that context-independent strategies compete with context-dependent ones—an interpretation that is consistent with recent work on strategy selection during problem solving.     

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。     

抽样组合中元素数量对青少年组合成绩的影响

先前抽样组合问题研究表明达到形式运算阶段青少年的抽样组合思维成绩表现并不一致,作者分析猜想可能的原因是组合元素数量增加会降低被试的抽样组合成绩。现在实验考察高中一年级学生的抽样组合思维能力,以组合问题中的总体元素数量和样本元素数量为自变量,设置了五选三、七选三和七选四的三种抽样组合问题条件。实验结果发现,随着总体元素数量和样本元素数量的增加,被试的组合成绩明显下降。这表明,青少年的抽样组合思维能力虽已获得,但随组合元素数量增加而表现出倒退,并没达到成熟的一般性;其发展水平可能存在初级水平到高级水平的区分。     

Random error in holistic evaluations and additive decompositions of multiattribute utility — an empirical comparison

This paper details the results of an empirical investigation of the random errors associated with decomposition estimates of multiattribute utility. In a riskless setting, two groups of subjects were asked to evaluate multiattribute alternatives both holistically and with the use of an additive decomposition. For one group, the alternatives were described in terms of three attributes, and for the other in terms of five. Estimates of random error associated with the various elicitations (holistic, single-attribute utility, scaling constants, or weights) were obtained using a test-retest format. It was found for both groups that the additive decomposition had significantly smaller levels of random error than the holistic evaluation. However, the number of attributes did not seem to make a significant difference to the amount of random error associated with the decomposition estimates. The levels of error found in the various elicitations were consistent with theoretical bounds that have recently been proposed in the literature. These results show that the structure imposed on the problem through decomposition results in measurable improvement in quality of the multiattribute utility judgements, and contribute to a greater understanding of the decomposition method in decision analysis.     

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.