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。     

Arithmetization and Rigor as Beliefs in the Development of Mathematics

With the arrival of the nineteenth century, a process of change guided the treatment of three basic elements in the development of mathematics: rigour, the arithmetization and the clarification of the concept of function, categorised as the most important tool in the development of the mathematical analysis. In this paper we will show how several prominent mathematicians contributed greatly to the development of these basic elements that allowed the solid underpinning of mathematics and the consideration of mathematics as an axiomatic way of thinking in which anyone can deduce valid conclusions from certain types of premises. This nineteenth century stage shares, possibly with the Heroic Age of Ancient Greece, the most revolutionary period in all history of mathematics.     

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.     

Games as formal tools versus games as explanations in logic and science

This paper addresses the theoretical notion of a game as it arisesacross scientific inquiries, exploring its uses as a technical andformal asset in logic and science versus an explanatory mechanism. Whilegames comprise a widely used method in a broad intellectual realm(including, but not limited to, philosophy, logic, mathematics,cognitive science, artificial intelligence, computation, linguistics,physics, economics), each discipline advocates its own methodology and aunified understanding is lacking. In the first part of this paper, anumber of game theories in formal studies are critically surveyed. Inthe second part, the doctrine of games as explanations for logic isassessed, and the relevance of a conceptual analysis of games tocognition discussed. It is suggested that the notion of evolution playsa part in the game-theoretic concept of meaning.     

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.     

Mathematical Naturalism: Origins, Guises, and Prospects

During the first half of the twentieth century, mainstream answers to the foundational crisis, mainly triggered by Russell and Gödel, remained largely perfectibilist in nature. Along with a general naturalist wave in the philosophy of science, during the second half of that century, this idealist picture was finally challenged and traded in for more realist ones. Next to the necessary preliminaries, the present paper proposes a structured view of various philosophical accounts of mathematics indebted to this general idea, laying the ground for a desirable integration of their strenghts.     

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.     

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。     

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.     

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。     

中算数学机械化思想在和算中的发展——解伏题的机械化特征

和算解伏题是关于多元高次联立方程组求解问题，因其在世界数学史上首次导入了行列式算法而向为学术界所重视。文章从数学机械化这一视角对其提出新的认识，重新讨论和算解伏题的消元理论问题，认为和算解伏题是中算代数化几何与以元天术为核心的代数演算的机械化数学传统的后续发展，关孝和给出了多元高次方程组消元的一般性程序，从而构筑了和算后期计算几何发达的基础，比诸行列式理论，解伏题的数学机械化思想的价值更为重要。     

《缀术算经》:东亚数学归纳推理的典范

《缀术算经》是日本和算家建部贤弘最富创造性的一部和算著作,代表了当时日本数学发展的最高成就。文章在前人工作的基础上,从数学传播史和比较数学史的角度,将其置于整个汉字文化圈数学文化背景下,对其数学内容、思想实质与中国传统数学的关系予以深入分析,对其数学成就与建部贤弘的数学方法论予以客观公正的评价。     

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.     

东方约瑟夫问题研究选析

历史上东方有过约瑟夫问题多则，它们与西方有异。文章选述的中国和日本典型四例，在故事叙述上充满诙谐，富戏剧性。从数学要求来看，它们是问题原型的进一步发展，至今还具有足够能量向现代组合数学界挑战。     

《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.     

The Emergence of Symbolic Algebra as a Shift in Predominant Models

Historians of science find it difficult to pinpoint to an exact period in which symbolic algebra came into existence. This can be explained partly because the historical process leading to this breakthrough in mathematics has been a complex and diffuse one. On the other hand, it might also be the case that in the early twentieth century, historians of mathematics over emphasized the achievements in algebraic procedures and underestimated the conceptual changes leading to symbolic algebra. This paper attempts to provide a more precise setting for the historical context in which this decisive step to symbolic reasoning took place. For that purpose we will consider algebraic problem solving as model-based reasoning and symbolic representation as a model. This allows us to characterize the emergence of symbolic algebra as a shift from a geometrical to a symbolic mode of representation. The use of the symbolic as a model will be situated in the context of mercantilism where merchant activity of exchange has led to reciprocal relations between money and wealth.

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.     

The Heuristic Function of Mathematics in Physics and Astronomy

This paper considers the role of mathematics in the process of acquiring new knowledge in physics and astronomy. The defining of the notions of continuum and discreteness in mathematics and the natural sciences is examined. The basic forms of representing the heuristic function of mathematics at theoretical and empirical levels of knowledge are studied: deducing consequences from the axiomatic system of theory, the method of generating mathematical hypotheses, “pure” proofs for the existence of objects and processes, mathematical modelling, the formation of mathematics on the basis of internal mathematical principles and the mathematical theory of experiment.