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.     

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.     

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。     

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.     

A Physical Approach to the Construction of Cognition and to Cognitive Evolution

It is shown that the method of operationaldefinition of theoretical terms applied inphysics may well support constructivist ideasin cognitive sciences when extended toobservational terms. This leads to unexpectedresults for the notion of reality, inductionand for the problem why mathematics is sosuccessful in physics.A theory of cognitive operators is proposedwhich are implemented somewhere in our brainand which transform certain states of oursensory apparatus into what we call perceptionsin the same sense as measurement devicestransform the interaction with the object intomeasurement results. Then, perceivedregularities, as well as the laws of nature wewould derive from them can be seen asinvariants of the cognitive operators concernedand are by this human specific constructsrather than ontologically independent elements.(e.g., the law of energy conservation can bederived from the homogeneity of time and bythis depends on our mental time metricgenerator). So, reality in so far it isrepresented by the laws of nature has no longeran independent ontological status. This isopposed to Campbell's `natural selectionepistemology'. From this it is shown that thereholds an incompleteness theorem for physicallaws similar to Gödels incompletenesstheorem for mathematical axioms, i.e., there isno definitive or object `theory of everything'.This constructivist approaches to cognitionwill allow a coherent and consistent model ofboth cognitive and organic evolution. Whereasthe classical view sees the two evolutionrather dichotomously (for ex.: most scientistssee cognitive evolution converging towards adefinitive world picture, whereas organicevolution obviously has no specific focus (the`pride of creation').     

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.     

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.     

数学实在论的奎因-普特南不可或缺性论证及其影响

奎因、普特南等人以数学在自然科学的不可或缺性应用为基础,为数学实在论提出了一种新的辩护。他们的辩护引发了数学实在论与唯名论对此问题的争论,由此产生了许多有价值的成果,并暴露出许多深层次的哲学问题,这对数学与科学的关系的探讨有重要的意义。     

What is Applied Mathematics?

A number of issues connected with the nature of applied mathematics are discussed. Among the claims are these: mathematics "hooks onto" the world by providing models or representations, not by describing the world; classic platonism is to be preferred to structuralism; and several issues in the philosophy of science (reality of spacetime, the quantum state) are intimately connected to the nature of applied mathematics.     

工程科学与技术哲学:从工程科学的特征看技术哲学--德国哲学家H.波塞尔专访录

技术与工程,技术哲学与工程哲学是两个既相关联又有差异的研究领域.本文就这一问题展开了对话,并试图对二者关系做一梳理和探讨.文中当代德国著名技术哲学家波塞尔教授认为,就技术哲学是与工程哲学的关系而言,尚无必要把二者划分为两个相互对立的研究领域.文中强调了技术(工程)不同于科学的特质在于其中所包含的意向或价值因素.工程中使用的是人工制品,工程哲学是否属于技术哲学,取决于有关的定义.从理论的意义上看,工程哲学已经预设了技术哲学.     

The Role of Symmetry in Mathematics

Over the past few decades the notion of symmetry has played a major role in physics and in the philosophy of physics. Philosophers have used symmetry to discuss the ontology and seeming objectivity of the laws of physics. We introduce several notions of symmetry in mathematics and explain how they can also be used in resolving different problems in the philosophy of mathematics. We use symmetry to discuss the objectivity of mathematics, the role of mathematical objects, the unreasonable effectiveness of mathematics and the relationship of mathematics to physics.     

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。     

What if Haecceity is not a Property?

In some sense, both ontological and epistemological problems related to individuation have been the focal issues in the philosophy of mathematics ever since Frege. However, such an interest becomes manifest in the rise of structuralism as one of the most promising positions in recent philosophy of mathematics. The most recent controversy between Keränen and Shapiro seems to be the culmination of this phenomenon. Rather than taking sides, in this paper, I propose to critically examine some common assumptions shared by both parties. In particular, I shall focus on their assumptions on (1) haecceity as an individual essence, (2) haecceity as a property, (3) the classification of properties, and thereby (4) the search for the principle of individuation in terms of properties. I shall argue that all these assumptions are mistaken and ungrounded from Scotus’ point of view. Further, I will fathom what consequences would follow, if we reject each of these assumptions.     

刘徽的无限思想及其解释

该文包括两方面的内容。一是从元限侵害过程、不可分量可积性、有限过程等几个方面新考察了刘徽的无限思想，力图澄清此课题的研究中存在的若干误解。二是从中国古代数学传统，刘徽的思想渊源特别是他受墨家、道家和玄学的影响等方面，对刘徽利用无限思想处理问题的方式进行解释。     

奥斯特罗格拉茨基的概率思想探赜

奥斯特罗格拉茨基是圣彼得堡概率学派的杰出代表,其对相关实际问题的应用研究推动了概率论在俄罗斯的传播和发展。他坚持认为概率论是数学分析最重要的应用分支之一:为天文学提供了大量基本数学观察方法,可确定比数学观察误差影响还要小的随机事件原因等,其社会服务功能刺激了诸如保险业等社会福利机构的产生和发展,进而促进了自然科学的相关理论发展。由于深受拉普拉斯概率思想之影响,奥斯特罗格拉茨基虽然把概率论看作研究随机现象规律的有力工具,但经常犯一些哲学观和方法论错误。     

伊斯兰精密科学对印度及其邻国的传播和影响

Ⅰ Let me first clarify that science as developed in Islamic cultural areas or societies during the Middle Ages is abbreviated here as Islamic Science. In that context, the Exact Science is confined to astronomy (including mathematical geography) and mathematics only; physics was then a part of natural philosophy, and chemistry a branch of medicine.     

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.