维特根斯坦论数学基础

Ludwig Wittgenstein on Foundations of Mathematics

维特根斯坦认为20世纪初出现的数学基础三大流派之间的不同争论,实际上都是源于对数学基础问题的认识论思考的混淆,即他们没有从语言句法角度对数学基础问题做出真正的澄清。维特根斯坦认为数学本质上并不需要任何基础,无论逻辑还是元数学都不能成为数学的基础。数学是由计算构成的自为的系统,不同的系统之间具有不同的句法规则,数学命题是一种语法规则表述,具有规范性的地位,这种规范性既不来自于逻辑公理,也不是来自于无矛盾证明或人类心灵的直觉力量,而最终在于人类的生活实践之中。

Ludwig Wittgenstein argues that the different debates among three schools of foundations of Mathematics at the beginning of 20th Century,as a matter of fact,all originate from the cognitive confusion of the foundations of Mathematics,that is to say,philosophers fail to make genuine clarifications from the point of syntax and grammar.He thinks that Mathematics does not need any foundation at all,whether it is logic or Metamathematics.Mathematics is made up of several systems which consists of different calculations.Each system has unique syntactical rules.Mathematical propositions express grammatical rules and they have normative status.This kind of normativity does not come from logical Axioms,Metamathematics or ground intuition,but comes from human practice.