Objects and Processes in Mathematical Practice

In this paper it is argued that the fundamental difference of the formal and the informal position in the philosophy of mathematics results from the collision of an object and a process centric perspective towards mathematics. This collision can be overcome by means of dialectical analysis, which shows that both perspectives essentially depend on each other. This is illustrated by the example of mathematical proof and its formal and informal nature. A short overview of the employed materialist dialectical approach is given that rationalises mathematical development as a process of model production. It aims at placing more emphasis on the application aspects of mathematical results. Moreover, it is shown how such production realises subjective capacities as well as objective conditions, where the latter are mediated by mathematical formalism. The approach is further sustained by Polanyi’s theory of problem solving and Stegmaier’s philosophy of orientation. In particular, the tool and application perspective illuminates which role computer-based proofs can play in mathematics.     

Mathematical Models and Reality: A Constructivist Perspective

To explore the relation between mathematical models and reality, four different domains of reality are distinguished: observer-independent reality (to which there is no direct access), personal reality, social reality and mathematical/formal reality. The concepts of personal and social reality are strongly inspired by constructivist ideas. Mathematical reality is social as well, but constructed as an autonomous system in order to make absolute agreement possible. The essential problem of mathematical modelling is that within mathematics there is agreement about ‘truth’, but the assignment of mathematics to informal reality is not itself formally analysable, and it is dependent on social and personal construction processes. On these levels, absolute agreement cannot be expected. Starting from this point of view, repercussion of mathematical on social and personal reality, the historical development of mathematical modelling, and the role, use and interpretation of mathematical models in scientific practice are discussed.     

The Surveyability of Long Proofs

The specific characteristics of mathematical argumentation all depend on the centrality that writing has in the practice of mathematics, but blindness to this fact is near universal. What follows concerns just one of those characteristics, justification by proof. There is a prevalent view that long proofs pose a problem for the thesis that mathematical knowledge is justified by proof. I argue that there is no such problem: in fact, virtually all the justifications of mathematical knowledge are ‘long proofs’, but because these real justifications are distributed in the written archive of mathematics, proofs remain surveyable, hence good.     

On the Foundations of Constructive Mathematics – Especially in Relation to the Theory of Continuous Functions

We discuss the foundations of constructive mathematics, including recursive mathematics and intuitionism, in relation to classical mathematics. There are connections with the foundations of physics, due to the way in which the different branches of mathematics reflect reality. Many different axioms and their interrelationship are discussed. We show that there is a fundamental problem in BISH (Bishop’s school of constructive mathematics) with regard to its current definition of ‘continuous function’. This problem is closely related to the definition in BISH of ‘locally compact’. Possible approaches to this problem are discussed. Topology seems to be a key to understanding many issues. We offer several new simplifying axioms, which can form bridges between the various branches of constructive mathematics and classical mathematics (‘reuniting the antipodes’). We give a simplification of basic intuitionistic theory, especially with regard to so-called ‘bar induction’. We then plead for a limited number of axiomatic systems, which differentiate between the various branches of mathematics. Finally, in the appendix we offer BISH an elegant topological definition of ‘locally compact’, which unlike the current definition is equivalent to the usual classical and/or intuitionistic definition in classical and intuitionistic mathematics, respectively.     

Mathematical,Philosophical and Semantic Considerations on Infinity (I): General Concepts

In the Reality we know, we cannot say if something is infinite whether we are doing Physics, Biology, Sociology or Economics. This means we have to be careful using this concept. Infinite structures do not exist in the physical world as far as we know. So what do mathematicians mean when they assert the existence of ω (the mathematical symbol for the set of all integers)? There is no universally accepted philosophy of mathematics but the most common belief is that mathematics touches on another worldly absolute truth. Many mathematicians believe that mathematics involves a special perception of an idealized world of absolute truth. This comes in part from the recognition that our knowledge of the physical world is imperfect and falls short of what we can apprehend with mathematical thinking. The objective of this paper is to present an epistemological rather than an historical vision of the mathematical concept of infinity that examines the dialectic between the actual and potential infinity.     

历史上的数学学派—理论初析

数学学派是数学史研究中的一个重要方面，学派研究为研究数学史提供了一种新的线索和角度。该文系统探讨了数学学派的起泊，定义，类型，它的形成条件及特征，功能与影响等，并对数学学派的衰落原因进行了分析，为数学学派提供了一个系统的理论概括。     

中国筹算和《九章算术》中的程序思维

本文以程序思维为主线，系统地研究了中国传统数学的筹算制及其学术的最重要成果《九章算术》。并指出：中国传统数学沿着《九章算术》所开辟出的方向，走上了一条不甚博大但不乏精深的程序化数学的发展之路。     

The Future of Mathematics in Economics: A Philosophically Grounded Proposal

The use of mathematics in economics has been widely discussed. The philosophical discussion on what mathematics is remains unsettled on why it can be applied to the study of the real world. We propose to get back to some philosophical conceptions that lead to a language-like role for the mathematical analysis of economic phenomena and present some problems of interest that can be better examined in this light. Category theory provides the appropriate tools for these analytical approach.     

道教思想与数学思想的互动

道教思想与数学思想的互动曾产生过积极效应,二者之间存在着深厚的历史渊源。秦九韶本着"数与道非二本"的观点,在"大衍求一术"中将道教思想与数学思想的互动体现得十分充分。在中国传统科学中,天文历法与数学的紧密结合本身也体现了这两个领域之间需要思想互动。二者的互动研究对于开辟从文化史研究数学和从科学史研究中国古代哲学的新视角富有启迪。     

从“古为今用”到“丝路精神”:吴文俊数学史观的形成与演变

吴文俊的数学史观,来自他对中国数学史的独创性研究。在"古为今用"思想的引领下,开辟了数学机械化的新领域,让中国古代数学为世界数学作出新贡献;"古证复原"原则的确立,开启了中国数学史研究的新时代;"两种数学主流"思想的提出,确立了中国传统数学在世界数学发展史上的地位。更为重要的是,2002年,吴文俊指出"丝路精神"的核心价值是"知识交流与文化融合"。因此,"古为今用""古证复原""两种主流"和"丝路精神"构成了吴文俊数学史观的核心要素,是指引新时代中国数学史研究的伟大旗帜。     

论数学中的虚构主义

数学虚构主义是"数学实体不存在、数学中不存在真理、数学在世界的说明和科学事业中可有可无"的一种典型的当代数学反实在论的解释。通过对数学虚构主义的批判性分析,得出"数学虚构主义的反实在论规划整体上并不成功","数学在世界的说明和科学中是至关重要,而非可有可无"、"数学实体确实不存在"和"数学中存在真理,但其本质需进一步研究"的结论。     

"数学常青"——从第22届国际科学史大会看数学史研究的特点与走向

通过对第22届国际科学史大会上数学史的特邀报告、主题讨论会和论文报告的全面分析，我们看到传统内史型的数学史研究依然具有重要的生命力，但新的“热点”更应值得关注，比如，“多元文化之间数学知识的交流与传播”就已成为新的主流。为了应对“全球化和文化多样性”带来的挑战，这次国际科学史大会上展现出的新思想、新视角和新进路，对于中国数学史是十分重要的。     

Top-Down and Bottom-Up Philosophy of Mathematics

The philosophy of mathematics of the last few decades is commonly distinguished into mainstream and maverick, to which a ‘third way’ has been recently added, the philosophy of mathematical practice. In this paper the limitations of these trends in the philosophy of mathematics are pointed out, and it is argued that they are due to the fact that all of them are based on a top-down approach, that is, an approach which explains the nature of mathematics in terms of some general unproven assumption. As an alternative, a bottom-up approach is proposed, which explains the nature of mathematics in terms of the activity of real individuals and interactions between them. This involves distinguishing between mathematics as a discipline and the mathematics embodied in organisms as a result of biological evolution, which however, while being distinguished, are not opposed. Moreover, it requires a view of mathematical proof, mathematical definition and mathematical objects which is alternative to the top-down approach.     

《墨经》数学今释

《墨经》是墨子著作的一部分，其中也有墨家弟子增补的内容。《墨经》包含丰富的数学、物理学和逻辑学知识，是研究先秦科学技术史的宝贵文献，在中国和世界学术史一上占有重要的地位。由于《墨经》文字古奥简晦，现传本又多有衍脱讹误，因此如何理解各条经文，历来众说纷纭，分歧很大，争论很多。本文对《墨经》中涉及数学概念和数学命题的经文作了初步的归纳和整理，并对较重要的或在认识上存在较大分歧的数学定义和命题作了新的诠释，最后对《墨经》数学的特点和意义进行了简要的评述，以期为深入探讨《墨经》数学和墨子数学思想提供一个新的基础。     

切比雪夫的概率思想及其数学文化背景

切比雪夫不等式和切比雪夫大数定律是概率论极限理论的基础,其创立是概率论成为严密数学分支的标志.大多有关研究成果都侧重于切比雪夫及其后继者的贡献,本文将重点考察切比雪夫概率思想的创新点及其数学文化背景,尤其是法国数学文化对切比雪夫概率思想形成的深刻影响.另外,还探讨了切比雪夫不等式优先发现权问题.     

Mathematics and Argumentation

Some authors have begun to appeal directly to studies of argumentation in their analyses of mathematical practice. These include researchers from an impressively diverse range of disciplines: not only philosophy of mathematics and argumentation theory, but also psychology, education, and computer science. This introduction provides some background to their work.     

数学发现和论证的两种原型:古希腊与古中国

萨博论点认为希腊数学证明源于埃利亚学派,但没有涉及与之相关的社会文化原因。把视野扩大到社会文化背景,可以发现,古希腊重甲步兵的出现以及其社会地位的提高,创造了以对抗或竞赛为象征的民族精神,从而既摆脱了怀疑主义哲学的束缚,又使得批判思维统治社会成为可能,这才是埃利亚学派及其数学产生的社会原因。另一方面,结合中国传统社会的考察,分析《九章算术》源于日常实践的问题以及复杂的理论回答,可以发现,中国古代需要大量人力完成的农业技术发展和日常生活中的实用技术发展,都得益于其算法的实用性,中国传统数学由此而成为东方数学的最高形式。两种数学实践的原型代表着两种重要的数学价值观,即确定性与实用性。     

民族主义与东亚数学编史问题

分析东亚民族主义产生的背景与东亚民族主义史学观的历史根源,论述民族主义史学观支配下东亚数学国别史编纂的研究取向及其影响。认为,这些国别史都是从民族本位出发,强调本国数学的独立性与主体性,重视本民族数学的独特性与数学成就上的差别,而轻忽东亚数学的整体性与同质性。以西方数学作为参照标准,较强的辉格史倾向在为民族科学文化先进性进行辩护方面发挥重要作用,重视数学史内史研究而忽视外史研究的研究范式也为其所需要。欲消除民族主义数学史观的消极影响,就应该树立“东亚数学一体化”的观点,超越民族,从儒家文化、汉字文化的整体视角来审视东亚数学,并且对东亚数学持连续发展的观点。只有把中国、日本与韩国(还包括越南)数学作为一个整体考察,才有可能全面认识东亚数学思想、数学精神与数学知识体系。     

20世纪50年代中苏数学交流的特点及其对中国数学发展的影响

20世纪 5 0年代是苏联数学发展的强盛时期 ,也是中国全面开展向苏联学习的时期。从中国数学发展的角度看 ,中苏数学交流对于中国现代数学的发展产生了很大的促进作用 ,为建立中国自己的数学研究基础和体系打下了良好的基础。从中国和国际数学发展的背景出发 ,对一些原始档案材料和文献进行分析和研究 ,指出 2 0世纪 5 0年代中苏数学交流的方式、内容和结果。这一时期数学交流在中国表现的主要特点即学习苏联对数学研究进行整体规划 ,有计划地重点发展数学应用和应用数学 ,继续发展我国在国际数学界的强项专业以及重视数学史研究。并进一步探讨苏联数学在 2 0世纪 5 0年代对中国现代数学发展的影响。     

Coherence of Our Best Scientific Theories

Putnam in Realism in mathematics and Elsewhere, Cambridge University Press, Cambridge (1975) infers from the success of a scientific theory to its approximate truth and the reference of its key term. Laudan in Philos Sci 49:19–49 (1981) objects that some past theories were successful, and yet their key terms did not refer, so they were not even approximately true. Kitcher in The advancement of science, Oxford University Press, New York (1993) replies that the past theories are approximately true because their working posits are true, although their idle posits are false. In contrast, I argue that successful theories which cohere with each other are approximately true, and that their key terms refer. My position is immune to Laudan’s counterexamples to Putnam’s inference and yields a solution to a problem with Kitcher’s position.