10.23：哲学与好奇”午餐会第八期：现代数学与逻辑学中的公理—符号化方法———从欧氏几何谈起
主题:现代数学与逻辑学中的公理—符号化方法———从欧氏几何谈起
The Axiomatic-Symbolic Method in Modern Mathematics and Logic - A Discussion from Euclidean Geometry
地点: 哲学系B114
时间: 10月23日(周三) 中午12:10-13:30 （12点就可以来吃饭了）
主讲人: 钟盛阳
主要内容:
根据现有文献，欧氏几何是数学中第一个使用公理化方法建立的理论，它也是公理化方法乃至数学理论的典范。本次报告将以此理论作为具体例子，讨论建立数学理论、逻辑学理论以及其他抽象理论（包括哲学理论）的方法及其相关问题。特别地，本次报告将讨论在现代数学和逻辑学中被广泛运用的公理—符号化方法的动机、后果、好处和应用。如有时间，也将提及一些引伸自公理—符号化方法的问题，例如逻辑学的研究对象和逻辑学理论的结构；现代逻辑学对哲学的影响；数学在描述自然现象方面的作用；以及物理学的公理化与量子逻辑。
数学家和逻辑学家大量使用各种符号，原因之一是，他们所研究的理论基本上都是用公理—符号化方法建立的；而公理—符号化方法背后的动机只是为了把能说清楚的话尽可能说清楚。为了说明这一点，本次报告试图阐明：第一，为了界定一个理论中的初始（基本）概念从而保证讨论有意义，公理化方法是自然（甚至唯一）的选择。第二，在一个公理化的理论中，被用于指称初始（基本）概念的语词难免沦为在不同语境下有不同解释的符号。
According to the existing literature, Euclidean geometry is the first theory in mathematics developed using the axiomatic method. It is also the paradigm of the axiomatic method and even of mathematical theories. With this theory as a concrete example, in this talk I will discuss methods and other related issues in developing theories in mathematics, theories in logic and other abstract theories (including theories in philosophy). In particular, this talk will be about the motivation, consequences, advantages and applications of the axiomatic-symbolic method which is widely used in modern mathematics and logic. If time permits, some problems springing from the axiomatic-symbolic method will be discussed. These include the object of study in logic, the structure of a theory in logic, the influence of modern logic on philosophy, the role of mathematics in describing nature, axiomatization of physics and quantum logic.
Mathematicians and logicians use a lot of symbols, partly because most of the theories in their research are developed using the axiomatic-symbolic method. The motivation behind this method is just to express as clearly as possible what is possible to be clearly expressed. To convey this point, in this talk I will try to explain: First, in order to clarify the initial (fundamental) concepts in a theory so that discussions involving them make sense, the axiomatic method is a natural (even the unique) choice. Second, in an axiomatized theory, terms that refer to initial (fundamental) concepts may inevitably become just symbols that have different interpretations in different contexts.