数学美学思想的历史演变

根据数学思想和理论形式的历史发展,将数学美学思想的历史过程概括为古希腊时期神秘主义倾向的数学美、近代形式主义倾向的数学美和现代理性主义倾向的数学美三个阶段。     

数学证明严格性之相对意义与综合评判标准

数学命题的证明在结构上必然存在某种预设性,在其论证过程中必然存在逻辑空隙。数学推理在所采用的逻辑工具和方法上并非无懈可击,加之数学形式体系的非完整性与非封闭性,这些都充分表明数学知识无论是在其原初形式上、形成过程中和构成方式三个维度上都无法获得完全意义上的可靠性。即使在元层面的证明论诉求亦复如此。数学证明的标准是一个综合评判的动态体系。     

数学名词中文化意蕴的探析——以张量概念的演变为例

数学名词与日常用语相比,具有高度抽象性的特点.数学一般以逻辑的形式呈现,常常掩盖了其丰富的文化内涵.如果我们褪去这层外壳,数学名词的文化内涵就会呈现出来.本文以广义相对论的数学基础张量分析和黎曼几何为线索,重点考察了张量概念的词源理解、形成过程和物理学隐喻,发现“球形宇宙”的隐喻原来蕴含在张量隐喻之中,希望以此表明,数学是文化进化的认知变异体和延伸,具有深刻的文化意蕴.     

《数学汇编》中的数学问题在中国——兼与古希腊数学思想比较

帕普斯的《数学汇编》是世界数学史中一部承前启后的重要数学著作,它对希腊经典数学问题的研究为后来的数学指出了方向,拓展和预见了许多数学新领域。本文分析了明末至清末近300年间传入的《数学汇编》中的典型问题,探讨了中算家对古希腊传统问题的理解、接受情况。通过比较发现其在中国和在希腊发展的差异,西方由此导致了数学的进化,而中国延续了中算以算为主和重视实用的传统,发展了不严谨的证明方法,在西学的引导下触及到了数学发展的一些理论问题,所有这些为中国进一步接受近代数学奠定了基础。     

卡拉比猜想及其证明

卡拉比猜想的证明,引出了许多重要结果,对于后来数学和物理的发展做出了基础性贡献.文章依据原始文献,详细考察了卡拉比提出猜想,丘成桐解决卡拉比猜想的工作;同时讨论了卡拉比猜想与凯勒-爱因斯坦度量的密切关系以及丘成桐和奥宾在这两个问题上的工作.     

Unification of Mathematical Theories

In this article the problem of unification of mathematical theories is discussed. We argue, that specific problems arise here, which are quite different than the problems in the case of empirical sciences. In particular, the notion of unification depends on the philosophical standpoint. We give an analysis of the notion of unification from the point of view of formalism, Gödel's platonism and Quine's realism. In particular we show, that the concept of “having the same object of study” should be made precise in the case of mathematical theories. In the appendix we give a working proposal of a certain understanding of this notion.     

工具与文化之间的数学品格——模式观的数学本体论下对数学意义的探索

数学是通过逻辑建构模式并以模式为对象进行研究的科学.这种数学模式观扬弃了数学本体论中的实在论与反实在论的极端主义哲学理解,充分肯定了数学的抽象性与客观性.在模式观的数学本体论视野下,数学兼具工具与文化两种品格.数学无形地渗透在科学的每个分支里,为其提供必要的工具;数学是理性精神的化身,深刻地影响着人们的观念、精神以及思维方式的养成.     

反证法与归谬法的现代分析

本文以现代命题逻辑为分析工具,证明了:在正命题逻辑系统的基础上,反证法的证明能力强于归谬法,它们之间相差一个双重否定律;归谬律与不矛盾律加上充分条件否定后件律相等价;反证律与不矛盾律、排中律、充分条件否定后件律加上选言推理否定肯定律相等价.     

苏颂水运仪象台的“尺寸”论证

考察从东汉至明代的现存六座历史天文文物,得出其天文尺值为1尺=24.5cm,这便是中国历代八尺高表测影和制造天文仪器的长度标准,该值从东汉至明末基本保持不变.从而把北宋苏颂水运仪象台的高度(35.65-36尺)定为873-882cm之间,宽度(21尺)为514.5cm,其他部件的尺寸也依此类推.这是对王振铎及后来国内外许多研制者把水运仪象台的高度和宽度分别定为11-12米和7米的重大校正和更改.     

Mathematical Arguments in Context

Except in very poor mathematical contexts, mathematical arguments do not stand in isolation of other mathematical arguments. Rather, they form trains of formal and informal arguments, adding up to interconnected theorems, theories and eventually entire fields. This paper critically comments on some common views on the relation between formal and informal mathematical arguments, most particularly applications of Toulmin’s argumentation model, and launches a number of alternative ideas of presentation inviting the contextualization of pieces of mathematical reasoning within encompassing bodies of explicit and implicit, formal and informal background knowledge.     

数理逻辑教学刍议——兼谈我国高校文科应设置数理逻辑基础课

高校文科逻辑教学改革工作一直是颇受逻辑界同仁关注的一个话题,关于在文科生中开数理逻辑谍程的意义及其必要性还有许多争论。数理逻辑在高校教学中要占有一席之地,似还需有一段路程娶走。还需各位同仁的共同努力。...     

Towards a Theory of Mathematical Argument

In this paper, I assume, perhaps controversially, that translation into a language of formal logic is not the method by which mathematicians assess mathematical reasoning. Instead, I argue that the actual practice of analyzing, evaluating and critiquing mathematical reasoning resembles, and perhaps equates with, the practice of informal logic or argumentation theory. It doesn’t matter whether the reasoning is a full-fledged mathematical proof or merely some non-deductive mathematical justification: in either case, the methodology of assessment overlaps to a large extent with argument assessment in non-mathematical contexts. I demonstrate this claim by considering the assessment of axiomatic or deductive proofs, probabilistic evidence, computer-aided proofs, and the acceptance of axioms. I also consider Jody Azzouni’s ‘derivation indicator’ view of proofs because it places derivations—which may be thought to invoke formal logic—at the center of mathematical justificatory practice. However, when the notion of ‘derivation’ at work in Azzouni’s view is clarified, it is seen to accord with, rather than to count against, the informal logical view I support. Finally, I pose several open questions for the development of a theory of mathematical argument.     

迈蒂的数学自然主义立场评述

迈蒂致力于提出一种基于数学实践本身的数学自然主义纲领,以改进蒯因的严重偏离数学实践的自然主义图景.然而,她将数学本体论和认识论问题都归之于方法论问题的策略引起了广泛的质疑.     

建部贤弘的数学方法论与数学思想

从数学方法论及中国宋元理学的视角,探讨了贯穿和算著作《缀术算经》始末的“缀术”方法的本质特征,以及书末“自质说”所反映的其作者建部贤弘的数学思想与方法论。经分析认为,该书自序和“自质说”中所反映出来的建部贤弘的治学思想,与当时的主流哲学即朱子学、阳明学思想有一定的联系,其思想渊源可追溯至中国的宋元理学。     

The Role of Diagrams in Mathematical Arguments

Recent accounts of the role of diagrams in mathematical reasoning take a Platonic line, according to which the proof depends on the similarity between the perceived shape of the diagram and the shape of the abstract object. This approach is unable to explain proofs which share the same diagram in spite of drawing conclusions about different figures. Saccheri’s use of the bi-rectangular isosceles quadrilateral in Euclides Vindicatus provides three such proofs. By forsaking abstract objects it is possible to give a natural explanation of Saccheri’s proofs as well as standard geometric proofs and even number-theoretic proofs.     

A Simple Identification Proof for a Mixture of Two Univariate Normal Distributions

A simple proof of the identification of a mixture of two univariate normal distributions is given. The proof is based on the equivalence of local identification with positive definiteness of the information matrix and the equivalence of the latter to a condition on the score vector that is easily checked for this model. Two extensions using the same line of proof are also given. We would like to thank Tom Wansbeek, Michel Wedel, Arie Kapteyn, and two anonymous reviewers for helpful comments on earlier versions of this paper.     

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.     

数学说明及其哲学使命

与科学说明相比,数学说明在数学哲学中长期遭到忽视。蒯因提出的不可或缺性论证、爱因斯坦-维格纳之谜以及数学哲学对数学实践的日趋重视,使数学说明重返哲学议程。数学自身内部对说明的需求和自然现象的数学说明都表明,数学不仅求真,而且寻求说明。哲学的使命就是要建立数学说明的相应标准、机制和模型,为数学决策提供建议,并推进对数学的本性、数学与世界及科学之关联的理解,为数学与科学统一的哲学解释搭建通道。     

惠施的数学思想探析

惠施是春秋战国时期诸子百家之一,提出了著名的"合同异"思想理论。为了支撑和论证"合同异"理论,惠施对自然现象和客观事物进行了深入研究,提出了"历物"十事、"尺棰"论、"镞矢"论等命题。这些命题蕴含了丰富的无穷、极限、运动、连续等数学思想。     

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。