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。     

无穷小量的命运及对数学发展动力的思考

近代力学的需要催生了无穷小量.历史上的数学家对无穷小量作了各种解读,试图论证它存在的合法性,以使微积分严密化,都没有成功.在19世纪末的数学公理化运动中,极限取代了无穷小量成为微积分的基础.无穷小量的曲折历史使人们认识到:数学有效性并不必须由其真理性来保证,数学家的信仰是数学发展的精神动力.近代数学家经历了从信仰上帝到信仰自然再到信仰数学内部的逻辑美的过程,数学家的任务也经历了从解决上帝、自然和数学三者之间的矛盾到解决自然和数学两者之间的矛盾,再到解决数学内部的矛盾过程.哥德尔不完备性定理抽走了数学家的逻辑美信仰,数学界出现了信仰危机.     

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.     

Why Do Informal Proofs Conform to Formal Norms?

Kant discovered a philosophical problem with mathematical proof. Despite being a priori, its methodology involves more than analytic truth. But what else is involved? This problem is widely taken to have been solved by Frege’s extension of logic beyond its restricted (and largely Aristotelian) form. Nevertheless, a successor problem remains: both traditional and contemporary (classical) mathematical proofs, although conforming to the norms of contemporary (classical) logic, never were, and still aren’t, executed by mathematicians in a way that transparently reveals why these proofs—written in the vernacular to this very day—succeed in conforming to those norms.     

从在国际顶尖基础数学期刊发表论文情况看我国数学近几十年来的发展

为了通过中国大陆地区在基础数学顶尖科研期刊发表论文的情况,了解我国数学的发展和现状,本文选取了10个国际最权威的基础数学期刊,分为两类,对中国数学家在这些刊物发表论文的情况进行分析。统计结果表明我国在第一类期刊发表的论文数量很少,这表明我国数学的研究水平还有待提高,原创性研究方面还有待努力;但是近30年来,我国在第二类期刊发表的论文数量显著地逐年增加,这显示我国数学研究团队的数量和实力都有了很好的发展。     

On the Persuasiveness of Visual Arguments in Mathematics

Two experiments are reported which investigate the factors that influence how persuaded mathematicians are by visual arguments. We demonstrate that if a visual argument is accompanied by a passage of text which describes the image, both research-active mathematicians and successful undergraduate mathematics students perceive it to be significantly more persuasive than if no text is given. We suggest that mathematicians’ epistemological concerns about supporting a claim using visual images are less prominent when the image is described in words. Finally we suggest that empirical studies can make a useful contribution to our understanding of mathematical practice.     

Systems and Beliefs

Systems thinking provides insights into how ideas interact and change, and constructivism is an example of this type of systemic approach. In the 1970s constructivism emphasised the development of mathematical and scientific ideas in children. Recently constructivist ideas are applied much more generally. Here I use this approach to consider beliefs and their role in conflicts and the conditions needed for reconciliation. If we look at Reality in terms of how we construct it as a human cognitive process, we recognise two things. First, that we cannot go beyond our senses and thoughts to what exists independently of us, and second, if we construct what we know we have to take responsibility for this. This inevitably focuses our thinking on the relation we have with the physical and social world, we are a part of the universe rather than apart from it. This paper argues that accepting and understanding these limits of human knowing together with our interconnectedness provide opportunities to understand conflicting positions. To resolve conflict, people with opposing viewpoints have to be prepared to understand each other. That is a challenge because our own reality plays a vital role in our lives, for everything from personal survival to social support.     

宋元数学人才群体之探索

宋元两代是我国古代数学发展的一个重要时期，涌现了大批量有成就的数学家。该文综合研究了宋元数学人才群体的结构、特点，以及在数学发展所起的重要作用，阐述了这均元时期人才政策为数学人才群体的形成所提供的良好环境，并分析了宋元空才群体的数学研究之特色。     

Memory and Geometry in Bruno: Some Analogies

In 1588 the Italian philosopher Giordano Bruno wrote a treatise against the mathematicians and philosophers of his time (Articuli centum et sexaginta adversus huius tempestatis mathematicos atque philosophos), which he dedicated to the emperor Rudolph II. The ‘oddities’ thus presented to the emperor, as an alternative to sixteenth-century mathematics, have been studied from both a mathematical and a philosophical point of view. In addition to the philosophical approach, this article indicates analogies between the Nolan’s geometry and his art of memory. Bearing in mind that Bruno was a teacher in the ars memoriae, the manner in which mnemonic aspects are woven into his mathematical thinking is brought out.     

清末数学教育对中国数学家的职业化影响

数学教育是决定数学研究能否持续性发展的重要因素，对数学的专业化及数学家的职业化都起着举足轻重的作用。     

Embedding Quantum Mechanics into a Broader Noncontextual Theory

Scholars concerned with the foundations of quantum mechanics (QM) usually think that contextuality (hence nonobjectivity of physical properties, which implies numerous problems and paradoxes) is an unavoidable feature of QM which directly follows from the mathematical apparatus of QM. Based on some previous papers on this issue, we criticize this view and supply a new informal presentation of the extended semantic realism (ESR) model which embodies the formalism of QM into a broader mathematical formalism and reinterprets quantum probabilities as conditional on detection rather than absolute. Because of this reinterpretation a hidden variables theory can be constructed which justifies the assumptions introduced in the ESR model and proves its objectivity. When applied to special cases the ESR model settles long-standing conflicts (it reconciles Bell’s inequalities with QM), provides a general framework in which previous results obtained by other authors (as local interpretations of the GHZ experiment) are recovered and explained, and supports an interpretation of quantum logic which avoids the introduction of the problematic notion of quantum truth.     

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.     

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。     

科学主义在17世纪的萌生

无论是培根的知识统一、笛卡尔的普遍数学,还是霍布斯的物理主义,都源于一种对绝对确定性的寻求,并试图用一种“科学方法”使知识成为普遍的和永恒的绝对真理,于是,“科学方法”被赋予某种向政治、经济、文学等领域转移的能力,甚至获得了某种可以支配其他事物的地位,这是科学主义在17世纪萌生的重要原因。     

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.     

《抱朴子》的科学思想管窥

晋代著名科学家葛洪在<抱朴子>一书中提出了值得重视的科学思想,"至明极聪"而不可"毕见尽闻"的科学认知论揭示了世界的无限性以及感官认识的有限性,为科学探索活动提供了不竭的精神动力;"形神相卫"的心身关系论揭示了精神与躯体、养神与养身相互依存的密切关系;"人所好恶,各各不同"的人格差异论为人们科学地识别和使用人才提供了诸多启示.     

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.     

Medieval Representations of Change and Their Early Modern Application

The article investigates the role of symbolic means of knowledge representation in concept development using the historical example of medieval diagrams of change employed in early modern work on the motion of fall. The parallel cases of Galileo Galilei, Thomas Harriot, and René Descartes and Isaac Beeckman are discussed. It is argued that the similarities concerning the achievements as well as the shortcomings of their respective work on the motion of fall can to a large extent be attributed to their shared use of means of knowledge representation handed down from antiquity and the Middle Ages. While the interpretation of medieval diagrams was unproblematic in the scholastic context from which they arose, in the early modern context, which was characterized by the confluence of natural philosophy and practical mathematics, it became ambiguous. It was the early modern mathematicians’ work within this contradictory framework that brought about a new conceptualization of motion which, in particular, eventually led to an infinitesimal concept of velocity. In this process, the diagrams themselves remained largely unchanged and thus functioned as a catalyst for concept development.     

Poincaré on Generalizations and Facts: Construction or Translation?

Much of the focus on Poincaré’s philosophy of science has been on the notion of convention, a crucial concept that has become distinctive of his position. However, other notions have received much less attention. That is the case of verifiable hypotheses. This kind of hypotheses seems to be constituted from the generalization of several observable facts. So, in order to understand what these hypotheses are, we need to know what a fact to Poincaré is. He divides facts into brute and scientific facts. The characterization of this duality is not trivial at all, and leads us to the following questions that we will discuss in this paper: (1) which the part of construction that exists in a scientific fact and which the part of translation, that is, what remains from the brute fact in the scientific one?; and (2) when we conceive a generalized hypothesis, are we supposed to do it from scientific or from brute facts? The clarification of these questions could lead to distinguish the part of construction and the part of translation in the first steps of science, which is essential to get a better understanding of Poincaré’s conception of science.