3468

Zhang Hong and Zhuang Yan

philosophy of mathematics education

2019

35

actual infinity , real infinity（genuine infinity, true infinity） , potential infinity , bad infinity（wrong infinity） , the contradiction between finity and infinity , the infinite view of dialectical materialism , one-to-one correspondence principle , Infinite Exchange Paradox

This paper discusses the problem of finity and infinity based on the philosophical perspective of opposing idealism and receiving dialectical materialism. 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. From the perspective of philosophy and mathematics, this paper comprehensively reviews and analyzes Hegel's dialectical infinite view, introduces Engels’discussion about the infinity, expounds the mathematical concept of actual infinity and potential infinity, and points out that the concept of actual infinity is actually downright idealist, and Being is not equivalent to completion. In this paper, the authors put forward the Infinite Exchange Paradox, make a strong query to the idea of actual infinity in Hilbert Hotel Problem, and point out the irreconcilable contradiction inherent in the idea of actual infinity. At the same time, we made a detailed comparison of Hegel's infinity concept and the mathematical concept. On this basis, we put forward a new concept of infinity, that is, the infinite view of dialectical materialism: 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 one of the effective ways to solve many contradictions in mathematics practice. At the end of the paper, the authors analyze and comment on various thoughts of infinity.