康德与数学对象的同一性问题

Kant and the Identity of Mathematical Objects

数学对象具有自身的同一性,这是数学计算中的一个自明的预设。但数学同样也可以证明数学对象具有自身的不同一性,即可以证明1≠1。这一证明似乎对数学的基础学科地位提出了挑战。本文立足于康德哲学的基本思想,指出数学对象既在判断的层面上因从属于知性思维的同一性原理而具有质的同一性,同时也因出自先验想象力的构造而具有量的不同一性。所以,数学概念的同一性与不同一性是分属于形式逻辑与先验逻辑两个不同层面的问题,二者并不产生真正的冲突,也不会引起数学学科本身的混乱。

There is a basic presupposition in mathematics that all mathematical objects are identical. At the same time,however,mathematics can also prove that these objects are not identical,e. g. it can be proved that 1≠1. This prove seems to make a great challenge to mathematics＇ basic position in sciences.Based on Kant＇s philosophy,this article argues that mathematical objects and concepts have their quantitative non-identity because they are the result of the construction of transcendental imagination,and that they also have their qualitative identity because they must be subjected to the principle of identity of understanding when they are put into the form of judgment. The identity and non-identity of mathematical objects are two different problems that belong to the different levels of logic,i. e. of formal logic and transcendental logic respectively. So they neither conflict with each other nor arouse any chaos in mathematics itself.