Hintikka on the Foundations of Mathematics: IF Logic and Uniformity Concepts

The initial goal of the present paper is to reveal a mistake committed by Hintikka in a recent paper on the foundations of mathematics. His claim that independence-friendly logic (IFL) is the real logic of mathematics is supported in that article by an argument relying on uniformity concepts taken from real analysis. I show that the central point of his argument is a simple logical mistake. Second and more generally, I conclude, based on the previous remarks and on another standard fact of IFL, that first-order logic (FOL) can adequately express uniformity concepts in real analysis, whereas IFL (understood as a non-trivial extension of FOL) cannot. This not only radically contradicts Hintikka’s particular claim in that article, but also undermines his whole enterprise of founding mathematics on his logic system.     

Foundations of Mathematics: Metaphysics, Epistemology, Structure

Since virtually every mathematical theory can be interpreted in set theory, the latter is a foundation for mathematics. Whether set theory, as opposed to any of its rivals, is the right foundation for mathematics depends on what a foundation is for. One purpose is philosophical, to provide the metaphysical basis for mathematics. Another is epistemic, to provide the basis of all mathematical knowledge. Another is to serve mathematics, by lending insight into the various fields. Another is to provide an arena for exploring relations and interactions between mathematical fields, their relative strengths, etc. Given the different goals, there is little point to determining a single foundation for all of mathematics.     

经典数学的逻辑基础

<正>经典数学理论的逻辑完全是一阶逻辑还是也需要二阶逻辑?逻辑学家对此一直有争议。这种争议大约开始于20世纪20年代,但是似乎直到现在还未尘埃落定。在普遍接受反基础主义的前提下,数学基础的研究任务不再是为数学的各个分支寻找最大程度上免于理性怀疑的基础,而是在重构数学分支的过程中给出各个数学分支间的关系,描绘出数学的大图景。在这样的背景下,数学基础的研究不     

Fregean Abstraction, Referential Indeterminacy and the Logical Foundations of Arithmetic

In Die Grundlagen der Arithmetik, Frege attempted to introduce cardinalnumbers as logical objects by means of a second-order abstraction principlewhich is now widely known as ``Hume's Principle' (HP): The number of Fsis identical with the number of Gs if and only if F and G are equinumerous.The attempt miscarried, because in its role as a contextual definition HP fails tofix uniquely the reference of the cardinality operator ``the number of Fs'. Thisproblem of referential indeterminacy is usually called ``the Julius Caesar problem'.In this paper, Frege's treatment of the problem in Grundlagen is critically assessed. In particular, I try to shed new light on it by paying special attention to the framework of his logicism in which it appears embedded. I argue, among other things, that the Caesar problem, which is supposed to stem from Frege's tentative inductive definition of the natural numbers, is only spurious, not genuine; that the genuine Caesar problem deriving from HP is a purely semantic one and that the prospects of removing it by explicitly defining cardinal numbers as objects which are not classes are presumably poor for Frege. I conclude by rejecting two closely connected theses concerning Caesar put forward by Richard Heck: (i) that Frege could not abandon Axiom V because he could not solve the Julius Caesar problem without it; (ii) that (by his own lights) his logicist programme in Grundgesetze der Arithmetik failed because he could not overcome that problem.     

转折与革命——逻辑经验主义的数学哲学:从《逻辑哲学论》到《逻辑句法》

与经验主义或实证主义之前的形式相比,逻辑和数学哲学是最具逻辑经验主义特征的。逻辑经验主义不仅因此得名,而且也为科学经验主义的成功点燃了希望。然而,经验主义者的心理主义与实证主义者的逻辑主义对数学本质的探讨始终不能达成共识。基于这种背景,维特根斯坦《逻辑哲学论》的发表对于维也纳小组的影响深远、意义重大。逻辑经验主义在20世纪20年代的数学哲学都是植根于《逻辑哲学论》的。本文在第一部分阐释了它何以成为该学派在"哲学上的转折点"。而维也纳学派从一开始就试图对《逻辑哲学论》进行革新,使之服务于其本身的纲领。本文在第二部分着重探讨了他们关于这些问题所展开的激烈讨论。这种状况一直持续到1931年卡尔纳普提出了"句法"纲领,该纲领从根本上超越了逻辑经验主义的原有框架。但由于当时欧洲逻辑经验主义者受到纳粹的迫害而四分五散,"句法"观点并未受到广泛关注。本文在第三、第四部分具体分析了逻辑经验主义如何由《逻辑哲学论》向"句法"思想转变的过程。     

Kinetic Tactile-Kinesthetic Bodies: Ontogenetical Foundations of Apprenticeship Learning

An ontogenetically-informed epistemology is necessary to understandings of apprenticeship learning. The methodology required in this enterprise is a constructive phenomenology, a phenomenology that takes into account the fact that as infants, we were apprentices of our own bodies: we all learned our bodies and learned to move ourselves. The major focus of this essay is on infant social relationships that develop on the ground of our original corporeal-kinetic apprenticeship. It shows how joint attention, imitation, and turn-taking - all richly examined areas in infant social development - are the foundation of apprenticeship learning in later adult life. The relationship between each infant capacity and later apprenticeship learning is demonstrated in examples from present-day research, specifically, research in the areas of medicine, sport, music, and tailoring, and research carried out by philosophers on apprenticeship learning.     

Non-Monotonic Reasoning from an Evolution-Theoretic Perspective: Ontic, Logical and Cognitive Foundations

In the first part I argue that normic laws are the phenomenological laws of evolutionary systems. If this is true, then intuitive human reasoning should be fit in reasoning from normic laws. In the second part I show that system P is a tool for reasoning with normic laws which satisfies two important evolutionary standards: it is probabilistically reliable, and it has rules of low complexity. In the third part I finally report results of an experimental study which demonstrate that intuitive human reasoning is in well accord with basic argument patterns of system P.     

New Foundations for Imperative Logic I: Logical Connectives,Consistency, and Quantifiers*

Imperatives cannot be true or false, so they are shunned by logicians. And yet imperatives can be combined by logical connectives: “kiss me and hug me” is the conjunction of “kiss me” with “hug me”. This example may suggest that declarative and imperative logic are isomorphic: just as the conjunction of two declaratives is true exactly if both conjuncts are true, the conjunction of two imperatives is satisfied exactly if both conjuncts are satisfied—what more is there to say? Much more, I argue. “If you love me, kiss me”, a conditional imperative, mixes a declarative antecedent (“you love me”) with an imperative consequent (“kiss me”); it is satisfied if you love and kiss me, violated if you love but don't kiss me, and avoided if you don't love me. So we need a logic of three‐valued imperatives which mixes declaratives with imperatives. I develop such a logic.     

数学日记对数学学习有效性的实验研究

本研究对高中数学学科进行实验研究,以探讨数学日记对主观数学学习效能感与客观数学成绩的影响.结果显示数学日记能显著提高学生的数学学习效能感,有助于数学学习成绩的提高,尤其对数学差生更具有显著作用;同时数学日记还能使学生对数学学习的信念与数学学习的实效协调起来,获得一致性效应.结论是数学日记能使学生形成一种有效性的学习.     

Mindful Learning: A Moderator of Gender Differences in Mathematics Performance

Past research has demonstrated that males outperform females in mathematics (Hyde, J. S., Fennema, E., & Lamon, S. J., Psychol Bull 107:139–155, 1990a). Research has also shown that encouraging mindful learning–learning information in a conditional rather than an absolute way–can increase mathematics performance in females (Ritchhart, R., & Perkins, D. N., J Social Issues 56:27–47, 2000). This paper examines the moderating role of mindful learning for gender differences, by manipulating mindful learning for females’ and males’ performance on a novel math task. The results from this study show that males performed better than females when mindful learning was not encouraged (absolute instruction), but males and females performed equally well when mindful learning was encouraged (conditional instruction). Thus we find that mindful learning moderates gender differences in math performance.     

数学学习障碍儿童问题解决的表征研究

本研究以小学四年级学生为被试,采用口语报告法探讨论了数学学习障碍儿童问题解决的表征情况。研究发现：数学学习障碍儿童问题解决的表征时间较短;数学学习障碍儿童问题解决的表征类型单一;数学学习障碍儿童问题解决的表征缺乏有效性。     

Explaining 'The Hardness of the Logical Must': Wittgenstein on Grammar, Arbitrariness and Logical Necessity

This paper explains (in Part A) Wittegnstein's understanding of the 'grammar' of our (or any) language, tracing its origins in the Tractatus's concept of logical syntax, and then examining the senses in which Wittegnstein, in his later work, viewed grammar as being 'arbitrary'. Then, armed with this understanding, it moves on (in Part B) to the task of examining how, within the framework of a Wittegnsteinian view of language, we should understand the inescapable 'compellingness' of logical necessity – what Wittegnstein calls the "hardness of the logical must". Whereas it is often thought that Wittegnstein's views on the nature of the 'grammar' of our concpets leads him towards a vitiatingly conventionalist or anti-realist understanding of necessity, in which its logical 'superhardness' becomes problematic, this paper will argue that there is actually no such tension in Wittegnstein's thought. In fact, it will be argued, an understanding of the ways in which our conceptual grammar is arbitrary casts a great deal of light on how it is that our concepts can nevertheless support a logically superhard, and normatively commanding, notion of necessity. In support of this view, I distinguish Wittegnstein's views on necessity from the 'classical' conventionalism of the Vienna Circle, and from the radical conventionalism of Michael Dummett, and defend Wittegnstein's view from a powerful recent attack from Quassim Cassam.     

数学学习困难儿童的编码加工特点:基于PASS理论的研究

同时加工和继时加工是PASS模型中的两类编码加工,二者在数学学习中发挥重要作用。使用基于PASS理论的认知评估系统（DN：CAS）单纯型数学困难、混合型数学困难和正常小学生进行两类编码加工的测评,比较研究发现,两类困难学生的两类编码加工水平均显著低于正常儿童,两类困难学生间的同时加工差异不显著,而混合型困难学生的继时加工水平显著低于单纯型困难学生。较低的同时加工水平似乎是两类数学困难的共同特征,继时加工水平的差异则似可作为单纯型与混合型困难的区分指标之一。     

小学生数学示例学习的信息加工过程实验研究

该研究试图让小学生通过示例学习的方式学习小数乘小数知识。口语报告分析及后测表明,小学四年级学生已具备示例学习的能力。实验还通过口语报告的个案分析,揭示示例学习的信息加工过程,并探讨了经过改进的口语报告分析法的应用问题。