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.     

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.     

Non-Monotonic Set Theory as a Pragmatic Foundation of Mathematics

In this paper I propose a new approach to the foundation of mathematics: non-monotonic set theory. I present two completely different methods to develop set theories based on adaptive logics. For both theories there is a finitistic non-triviality proof and both theories contain (a subtle version of) the comprehension axiom schema. The first theory contains only a maximal selection of instances of the comprehension schema that do not lead to inconsistencies. The second allows for all the instances, also the inconsistent ones, but restricts the conclusions one can draw from them in order to avoid triviality. The theories have enough expressive power to form a justification/explication for most of the established results of classical mathematics. They are therefore not limited by Gödel’s incompleteness theorems. This remarkable result is possible because of the non-recursive character of the final proofs of theorems of non-monotonic theories. I shall argue that, precisely because of the computational complexity of these final proofs, we cannot claim that non-monotonic theories are ideal foundations for mathematics. Nevertheless, thanks to their strength, first order language and the recursive dynamic (defeasible) proofs of theorems of the theory, the non-monotonic theories form (what I call) interesting pragmatic foundations.     

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.     

Bridging the Gap Between Argumentation Theory and the Philosophy of Mathematics

We argue that there are mutually beneficial connections to be made between ideas in argumentation theory and the philosophy of mathematics, and that these connections can be suggested via the process of producing computational models of theories in these domains. We discuss Lakatos’s work (Proofs and Refutations, 1976) in which he championed the informal nature of mathematics, and our computational representation of his theory. In particular, we outline our representation of Cauchy’s proof of Euler’s conjecture, in which we use work by Haggith on argumentation structures, and identify connections between these structures and Lakatos’s methods.     

Proof-analysis and Continuity

During the first phase of Greek mathematics a proof consisted in showing or making visible the truth of a statement. This was the epagogic method. This first phase was followed by an apagogic or deductive phase. During this phase visual evidence was rejected and Greek mathematics became a deductive system. Now epagoge and apagoge, apart from being distinguished, roughly according to the modern distinction between inductive and deductive procedures, were also identified on account of the conception of generality as continuity. Epistemology of mathematics today only remembers the distinction, forgetting where they agreed, in this manner not only destroying the unity of the perceptual and conceptual but also forgetting what could be gained from Aristotelian demonstrative science.     

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。     

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.     

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.     

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。     

Essential Tension: Mathematics - Physics - Philosophy

The author focuses on the tension "realism - idealism" in the philosophy of mathematics, but he does that from the perspective of a theoretical physicist. It is not only that one's standpoint in the philosophy of mathematics determines our understanding of the effectiveness of mathematics in physics, but also the fact that mathematics is so effective in physical sciences tells us something about the nature of mathematics.     

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。     

The Will to Mathematics: Minds, Morals, and Numbers

The 1990s could be called The Decade of Sociology in mathematics education. It was during those years that the sociology of mathematics became a core ingredient of discourse in mathematics education and the philosophy of mathematics and mathematics education. Unresolved questions and uncertainties have emerged out of this discourse that hinge on the key concept of social construction. More generally, what is at issue is the very idea of “the social”. Within the framework of the general problem of “the social”, we want to open a discussion of boundaries and margins in mathematics and mathematics education. By theorizing the divisions of purity and danger, we will be able to better understand the intersection of logic, mathematics, and thinking with gender, race, and class, and morals, ethics, and values in the classroom. The process of transforming the sociology of mathematics and the sociology of mind into pedagogical tools for mathematics educators and philosophers of education has already begun. One of the tasks before us is the development of a more profound and at the same time more practical grasp of “the social”. Our objective in this paper is to move ourselves and our readers in the direction of just such a grasp of the social.     

《九章算法比类大全》渊源初探

明代吴敬的<九章算法比类大全>(以下简称<大全>),除了与<九章算术>有着非常密切的关系外,也与其他算书有着重要的渊源关系.将该书与现存的1450年之前刊刻的<九章算术>外的算书进行详细比较,并分析该书与它们的渊源关系后,可以认为:<大全>受到宋元算书,特别是杨辉算书的影响较大,继承了杨辉重视介绍启蒙知识和开方术的思想,以及杨辉编写算书的"纂类"方式;吴敬至少应该听说过天元术,不过,即使他看到有关天元术的著作,也已无法理解天元术;<大全>乘除开方起例的内容与现存元末明初算书的很多内容相同,反映了这些知识从元末至明初这段时间的传播和发展.结合<大全>与<九章算术>的渊源以及<大全>对其后算书的影响的分析,还认为:明初的80年问,包含汉唐宋元一些高水平数学成就的算书在民间仍能看到,但吴敬既没有能够掌握这些知识,也没有在书中记载下他看到的算书中的高水平的数学知识.     

后现代主义的数学观及其认识

后现代主义者对数学有一种基于后现代基本立场的理解。这种理解更多地表现为一种批判性。这种批判可以促进对于数学的反思，对于抑制惟数学主义和强科学主义的思想趋向是有积极一面的。应该看到后现代主义对数学的认识大多仍停留在现代性数学的视域之内，并且其认识的片面性也是需要指出的。     

建部贤弘的数学认识论--论《大成算经》中的"三要"

“象形”、“满干”和“数”,是日本江户时代数学家建部贤弘在《大成算经》中所讨论的三个范畴 ,也是该书的纲纪 ,谓之“三要”。这些范畴来源于中国传统文化中的术数 ,语言晦涩 ,一直为日本数学史界所忽视。文章从中国数学文化传统出发 ,重新解读这些文字 ,提出一些全新的观点。认为在汉字文化圈数学家中 ,建部贤弘在中国象数学文化背景下 ,首次系统地阐述了数学科学的本质 ,讨论了数学研究对象及其存在性问题 ,并已接触到数学变量的讨论 ,同时对实数系给出了一种分类。其“三要”数理观是汉字文化圈数学认识论的突出反映 ,具有数学哲学意义。     

数学中的经验:认识论上的紧张视角

就形式数学来说,它是脱离经验,在内涵上是一种社会建构。计算机技术提高了人们的计算能力,对数学的这一社会建构平添了紧张,加强了“作为计算的数学”与“作为证明的数学”之间的不平衡性。在历史上,相似的认识论上的争论表现在计算或实践的印度数学与精神或形式的西方数学之间。我们认为,认识论上的紧张,可通过以下方式得以消解:认同数学是基于经验、可错的观点,并据此引领数学课程标准的基本理念。     

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.     

试论数学哲学的文化转向及其影响

数学文化研究是数学哲学研究与发展的一个新的方向,开辟了数学哲学研究的全新视角。本文从文化哲学、科学文化哲学以及数学哲学的后现代发展历程中梳理了数学哲学的文化转向,以及所产生的影响。     

再谈"中国古代数学中的‘黄金分割率''''"--与蒋谦、李思孟先生商榷

中国古代数学虽然有过辉煌的成绩,但中国古代数学与黄金分割无内在联系,蒋文[1]搞混了黄金分割的本质,把"河图"、"洛书"、"贾宪三角形"、"五运六气"等与没有逻辑关系的黄金分割率联系在一起,这是不合逻辑的没有根据的主观臆断,它给人造成一种错觉,似乎黄金分割率在中国古代数学中早已有之,这是不符合客观事实的误断.