数学的对象是什么

Objects of mathematics

所有的科学是研究范式的科学,而数学是研究范式的范式的科学,数学的对象是所有可能的结构。集合和类之间的区别必须被理解为实结构和潜结构之间的区别,这种区别是一个结构的两个不同方面之间的区别:结构的纯关系概念和结构的实在性概念,这一理解使真正的极大的集合论域的产生成为可能。亚里士多德把潜无穷和实无穷区别开来,得出结论:一般不存在实无穷的对象,无穷是没有结束的、非确定的、没有实现的,并且是不能完善和实在化的东西。数学的无穷概念明显不等于不可完全性概念,集合是确定的、似对象的、实在的和具有完全性的全体,一个集合是多样性的统一。

All sciences are the study of patterns, and mathematics is the science of pattern of patterns. The objects of mathematics are all the possible structures. The distinction between sets and classes should be understood as between actual and potential structures, and this understanding makes possible a truly maximal conception of the set-theoretical universe. Aristotle distinguished between the potential infinite and the actual infinite, and concluded there is in general no actual infinite object because the infinite is the unfinished, the indefinite, the unrealizable, which cannot be completed and actualized. The mathematical infinite is obviously not identical to the incompletable, and a set is a definite, object-like, actual, completed totality, thus can be seen as the unity of many.