论哲学无限与数学无限的异同点(下)

Discuss the Similarities and Differences between Philosophical Infinity and Mathematical Infinity

众所周知,"有限无限问题"是数学的基本问题,也是哲学的基本问题。本文基于以下哲学观点开展讨论:反对唯心主义,接受辩证唯物主义,在这样的大前提下讨论"有限无限问题"。本文从哲学和数学角度全面回顾、剖析了黑格尔的辩证无限观,同时介绍了恩格斯关于无限的论述,并对数学中的实无限、潜无限观进行了分析,指出实无限观实际上是彻头彻尾的唯心主义的无限观。与此同时,作者对黑格尔的无限观和数学中的两种无限观进行了详细的分析比较,在此基础上作者提出了"辩证唯物主义无限观",摒弃潜无限、实无限观中错误的一面,积极吸取两者正确的一面,即既要承认无限的客观存在性和可认识性,又要承认无限过程的不可完成性,这是解决数学实践中诸多矛盾的唯一正确途径。

It is well known that the problem of finity and infinity is the basic problem of mathematics, and it is also the basic problem of Philosophy. This paper discusses the problem of finity and infinity based on the philosophical perspectives of opposing Idealism and receiving Dialectical Materialism. From the perspective of philosophy and mathematics, this article comprehensively reviewed and analyzed Hegel＇s dialectical infinite view, and introduced Engels＇ discussion about the infinity. Then we expounded the mathematical concept of actual infinity and potential infinity, which has pointed out that the concept of actual infinity was actually downright idealist and anti-Marxism. At the same time, we made a detailed comparison of Hegel＇s infinity concept and the mathematical concept. On this basis, we proposed the Marxism infinity concept： abandoning the wrong aspects of the potential infinity and actual infinity, while actively absorbing correct aspects of both. It is necessary to recognize the existence and knowability of infinite objectivity, and also to admit the non completion of infinity process, which is the only correct way to solve the contradictions in mathematical practice.