"不可或缺性论证"与反实在论数学哲学

The indispensability argument for abstract mathematical entities has been an important issue in the philosophy of mathematics. The argument relies on several assumptions. Some objections have been made against these assumptions, but there are several serious defects in these objections. Ameliorating these defects leads to a new anti-realistic philosophy of mathematics, mainly: first, in mathematical applications, what really exist and can be used as tools are not abstract mathematical entities, but our inner representations that we create in imagining abstract mathematical entities; second, the thoughts that we create in imagining infinite mathematical entities are bounded by external conditions. __________ Translated from Zhexue Yanjiu 哲学研究 (Philosophical Researches), 2006, (8): 74–83     

《周易》对中国古代数学的影响

分析中国古代数学史上重要的数学著作可以看出,《周易》往往被古代数学家们视作数学发展最早的源头,而且在一些重要的数学著作中,数学家们运用《周易》中的有关概念表述数学问题,对《周易》中的数学问题及其相关问题进行深入的研究,取得了重要的数学成就。这一切足以表明《周易》对于古代数学发展具有非常重要的影响。     

The Nctm Standards and the Philosophy of Mathematics

It is argued that the philosophical and epistemological beliefs about the nature of mathematics have a significant influence on the way mathematics is taught at school. In this paper, the philosophy of mathematics of the NCTM's Standards is investigated by examining is explicit assumptions regarding the teaching and learning of school mathematics. The main conceptual tool used for this purpose is the model of two dichotomous philosophies of mathematics-absolutist versus- fallibilist and their relation to mathematics pedagogy. The main conclusion is that a fallibilist view of mathematics is assumed in the Standards and that most of its pedagogical assumptions and approaches are based on this philosophy.     

A multiple case study of teachers referring to their own religious beliefs in mathematics teaching

There is a need to integrate religious education and spiritual education across school curriculum. This paper reports one of the few empirical studies on bridging the intention-practice gap in classrooms. Six school teachers deliberately designed and implemented mathematics lessons which referred to their own religious beliefs in teaching. It unfolds teachers’ intention to enact their religious beliefs in mathematics classroom teaching. Different modes were identified. Implications to religious education in schools are offered.     

Informal versus formal mathematics

We discuss Kunen’s algorithmic implementation of a proof for the Paris–Harrington theorem, and the author’s and da Costa’s proposed “exotic” formulation for the P = NP hypothesis. Out of those two examples we ponder the relation between mathematics within an axiomatic framework, and intuitive or informal mathematics. The author is Visiting Researcher at IEA/USP, Professor of Communications, Emeritus, at the Federal University in Rio de Janeiro, and a full member of the Brazilian Academy of Philosophy.     

Self-reported mental health problems and performance in mathematics and reading in children across Europe

ABSTRACT

The study examines the associations of specific self-reported internalizing and externalizing problems and performance in mathematics and reading in school-aged children across Europe. Data were drawn from 5,842 children between 6 and 13 years of age participating in the School Children Mental Health in Europe study and a large cross-sectional survey in France. Self-reported child mental health was assessed using the Dominic Interactive, academic performance was reported by teachers. Across Europe, controlling for key sociodemographic factors associated with achievement including maternal education, performance in mathematics was more often associated with the presence of externalizing and internalizing problems as compared to performance in reading. In addition, the findings point to significant sex differences in the associations of internalizing problems and academic achievement. Considering the impact of early academic difficulties in terms of later internalizing and externalizing problems and academic attainment, school-based interventions interrupting the cycle as early as possible are warranted.     

The developmental relations between spatial cognition and mathematics in primary school children

Spatial thinking is an important predictor of mathematics. However, existing data do not determine whether all spatial sub‐domains are equally important for mathematics outcomes nor whether mathematics–spatial associations vary through development. This study addresses these questions by exploring the developmental relations between mathematics and spatial skills in children aged 6–10 years (N = 155). We extend previous findings by assessing and comparing performance across Uttal et al.'s (2013), four spatial sub‐domains. Overall spatial skills explained 5%–14% of the variation across three mathematics performance measures (standardized mathematics skills, approximate number sense and number line estimation skills), beyond other known predictors of mathematics including vocabulary and gender. Spatial scaling (extrinsic‐static sub‐domain) was a significant predictor of all mathematics outcomes, across all ages, highlighting its importance for mathematics in middle childhood. Other spatial sub‐domains were differentially associated with mathematics in a task‐ and age‐dependent manner. Mental rotation (intrinsic‐dynamic skills) was a significant predictor of mathematics at 6 and 7 years only which suggests that at approximately 8 years of age there is a transition period regarding the spatial skills that are important for mathematics. Taken together, the results support the investigation of spatial training, particularly targeting spatial scaling, as a means of improving both spatial and mathematical thinking.     

Redrawing Kant's philosophy of mathematics

This essay offers a strategic reinterpretation of Kant's philosophy of mathemat- ics in Critique of Pure Reason via a broad, empirically based reconception of Kant's conception of drawing. It begins with a general overview of Kant's philosophy of mathematics, observing how he differentiates mathematics in the Critique from both the dynamical and the philosophical. Second, it examines how a recent wave of critical analyses of Kant's constructivism takes up these issues, largely inspired by Hintikka's unorthodox conception of Kantian intuition. Third, it offers further analyses of three Kantian concepts vitally linked to that of drawing. It concludes with an etymologically based exploration of the seven clusters of meanings of the word drawing to gesture toward new possibilities for interpreting a Kantian philosophy of mathematics.     

Some Naturalistic Comments on Frege’s Philosophy of Mathematics

This paper compares Frege’s philosophy of mathematics with a naturalistic and nominalistic philosophy of mathematics developed in Ye (2010a, 2010b, 2010c, 2011), and it defends the latter against the former. The paper focuses on Frege’s account of the applicability of mathematics in the sciences and his conceptual realism. It argues that the naturalistic and nominalistic approach fares better than the Fregean approach in terms of its logical accuracy and clarity in explaining the applicability of mathematics in the sciences, its ability to reveal the real issues in explaining human epistemic and semantic access to objects, its prospect for resolving internal difficulties and developing into a full-fledged theory with rich details, as well its consistency with other areas of our scientific knowledge. Trivial criticisms such as “Frege is against naturalism here and therefore he is wrong” will be avoided as the paper tries to evaluate the two approaches on a neutral ground by focusing on meta-theoretical features such as accuracy, richness of detail, prospects for resolving internal issues, and consistency with other knowledge. The arguments in this paper apply not merely to Frege’s philosophy. They apply as well to all philosophies that accept a Fregean account of the applicability of mathematics or accept conceptual realism. Some of these philosophies profess to endorse naturalism.     

Inconsistency in mathematics and the mathematics of inconsistency

No one will dispute, looking at the history of mathematics, that there are plenty of moments where mathematics is “in trouble”, when paradoxes and inconsistencies crop up and anomalies multiply. This need not lead, however, to the view that mathematics is intrinsically inconsistent, as it is compatible with the view that these are just transient moments. Once the problems are resolved, consistency (in some sense or other) is restored. Even when one accepts this view, what remains is the question what mathematicians do during such a transient moment? This requires some method or other to reason with inconsistencies. But there is more: what if one accepts the view that mathematics is always in a phase of transience? In short, that mathematics is basically inconsistent? Do we then not need a mathematics of inconsistency? This paper wants to explore these issues, using classic examples such as infinitesimals, complex numbers, and infinity.     

How Gödel Relates Platonism to Mathematics

It is well known that Gödel takes his realistic world view as closely related to mathematics, especially to his own work in the foundations of mathematics. He reports, publicly as well as privately, that Platonism is fundamental to his major work in logic and set theory, and suggests that this philosophical position can be seen as a product of reflections on mathematics. These views of Gödel, however, are often regarded as being insufficiently formulated or argued for. In this article, the author tries to consider some points which are related to the understanding of the Gödelian mode of the interaction between mathematics and philosophy.     

The pernicious influence of mathematics upon philosophy

We shall argue that the attempt carried out by certain philosophers in this century to parrot the language, the method, and the results of mathematics has harmed philosophy. Such an attempt results from a misunderstanding of both mathematics and philosophy, and has harmed both subjects.Portions of the present text have previously appeared inThe Review of Metaphysics 44 (1990), 259–271, are reprinted with permission.     

Modernizing the philosophy of mathematics

The distinction between analytic and synthetic propositions, and with that the distinction between a priori and a posteriori truth, is being abandoned in much of analytic philosophy and the philosophy of most of the sciences. These distinctions should also be abandoned in the philosophy of mathematics. In particular, we must recognize the strong empirical component in our mathematical knowledge. The traditional distinction between logic and mathematics, on the one hand, and the natural sciences, on the other, should be dropped. Abstract mathematical objects, like transcendental numbers or Hilbert spaces, are theoretical entities on a par with electromagnetic fields or quarks. Mathematical theories are not primarily logical deductions from axioms obtained by reflection on concepts but, rather, are constructions chosen to solve some collection of problems while fitting smoothly into the other theoretical commitments of the mathematician who formulates them. In other words, a mathematical theory is a scientific theory like any other, no more certain but also no more devoid of content.     

儿童早期工作记忆与数学学习的关系

以往研究认为工作记忆对数学学习很重要,但是以往工作记忆和早期数学学习的关系研究结果并不一致。本研究运用元分析的方法回顾了33篇独立研究,含307项相关系数作为效应值,总被试量为42423名平均年龄3~8岁的儿童,旨在探讨工作记忆的3个主成分PL(语音回路)、VSSP(视觉空间画板)、CE(中央执行)与数学学习不同方面之间的关系,以及样本年龄对工作记忆和数学学习相关程度的调节作用。结果显示:工作记忆与数学学习确实存在强相关,工作记忆子成分中,CE与数学学习平均相关的程度最强。并且,CE与数学学习不同方面之间的相关程度不存在显著差异,PL、VSSP与数运算的相关程度比数概念更强,存在显著差异。样本年龄对CE、PL与数学学习之间相关关系存在显著调节作用,对VSSP与数学学习之间的相关关系不存在显著调节作用。文献梳理还发现,以往研究对数学学习的考察,大都聚焦数概念、数运算的内容,未来研究需要关注几何与空间、测量等方面内容。     

Mereology and mathematics: Christian Wolff's foundational programme

ABSTRACT

How did the traditional doctrine of parts and wholes evolve into contemporary formal mereology? This paper argues that a crucial missing link may lie in the early modern and especially Wolffian transformation of mereology into a systematic sub-discipline of ontology devoted to quantity. After some remarks on the traditional scholastic approach to parts and wholes (Sect. 1), Wolff's mature mereology is reconstructed as an attempt to provide an ontological foundation for mathematics (Sects. 2–3). On the basis of Wolff's earlier mereologies (Sect. 4), the origin of this foundational project is traced back to one of Wolff's private conversations with Leibniz (Sect. 5) and especially to the former's appropriation of the latter's notion of similarity as a means to define quantity (Sect. 6). Despite some hesitancy concerning the ultimate characterization of quantity (Sect. 7), Wolff's contribution was historically significant and influential. By developing a quantitative, extensional account of mereological relations, Wolff departed from the received doctrine and paved the way for the later revival of mereology at the intersection of ontology and mathematics.     

Effect of teachers' stereotyping on students' stereotyping of mathematics as a male domain

This multilevel analysis used data from a representative sample from Grades 6, 7, and 8 in public schools in Switzerland. The data included information on (a) 6,602 students (3,307 girls, 3,295 boys) nested within 338 classes and (b) 321 mathematics teachers of these classes. The teachers and the students tended to stereotype mathematics as a male domain, and the teachers' stereotypes significantly affected the students' stereotypes after the author controlled for achievement, interest, and self-confidence in mathematics and for school grade and schooling track.     

Alessandro Piccolomini and the certitude of mathematics

This paper offers a reconstruction of Alessandro Piccolomini's philosophy of mathematics, and reconstructs the role of Themistius and Averroes in the Renaissance debate on Aristotle's theory of proof. It also describes the interpretative context within which Piccolomini was working in order to show that he was not an isolated figure, but rather that he was fully involved in the debate on mathematics and physics of Italian Aristotelians of his time. The ideas of Lodovico Boccadiferro and Sperone Speroni will be analysed. This paper demonstrates that Piccolomini's attack on the certitude of mathematics was a product of discussions between Aristotelians.     

小学儿童数学焦虑的潜在类别转变及其父母教育卷入效应：3年纵向考察

研究采用潜在转变分析考察小学儿童数学焦虑的类别转变以及父母教育卷入在小学儿童数学焦虑类别转变中的作用。以1720名三、四年级儿童为被试, 对其数学焦虑和感知到的父母教育卷入进行3次追踪, 每次间隔1年。结果表明：(1)小学儿童数学焦虑存在低数学焦虑组、高数学评估焦虑组和高数学获得焦虑组3种不同类别; (2)随时间的推移, 高数学评估焦虑组倾向于向低数学焦虑组转变, 高数学获得焦虑组倾向于向高数学评估焦虑组转变, 而低数学焦虑组稳定性较强; (3)父亲/母亲教育卷入对儿童数学焦虑类别转变的预测作用, 因不同的数学焦虑类别而异。上述发现为深入理解数学焦虑的形成机制以及干预措施的制定提供了重要参考。     

A Naturalistic Look into Maddy's Naturalistic Philosophy of Mathematics

This paper discusses Penelope Maddy's (b.1950) naturalistic philosophy of mathematics,which is one of the most influential forms of post-Quinean naturalism in the philosophy of mathematics.Two defining features of Maddy's theory,namely the methodological autonomy of mathematics and the equivalence of Thin Realism and Arealism,are analyzed,and some criticisms of them are posed from within the naturalistic line of thought itself.In the course of this analysis and criticism,the paper will also consider Maddy's objections to the Quinean Indispensability Argument,which are the starting point of her own version of naturalism.