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1900年在巴黎举行的国际数学会議上,希尔伯特(D.Hilbert)作了以数学問題为題的讲演,向数学界提出了23个問題。这篇讲演具有十分重大的意义,这不仅是因为希尔伯特恰恰在两个世紀轉折的时候提出了这些問題,更重要的是,如在他讲演的一开始所說的,“揭起包着未来的面紗,一瞥我們今后科学的进展,探索未来世紀如何发展的秘密,以及有否不可解者,追求引导这些一般化的数学思想的特殊目的是什么,在未来的世紀里,在广闊而且丰富的数学思想的各个領域里,将能发現什么新的方法和新的事实”。这篇讲演中的各个問題之間的联系不太大,問題的大小和难易也各不相同,但是可以說几乎是包含了本世紀数学界所有致力研究的課題。本文所要談的第五問題已經肯定地被解决,它已成为数学界閑談的資料。問題是这样叙述的:“試不用可微性来定义Lie羣”。首先,让我們来說明它的意义。
At the International Mathematics Conference held in Paris in 1900, D. Hilbert gave a lecture on mathematics and presented 23 questions to the mathematics community. This lecture was of great significance, not only because Hilbert had raised these issues precisely at the turning point of two centuries, but more importantly, as he said at the beginning of his speech, “lifting the bag. The veil of the future, a glimpse of our scientific progress in the future, exploring the secrets of how to develop in the coming century, and whether there are any indisputable people, what is the special purpose of pursuing these generalized mathematical ideas will be broad and in the coming century? What new methods and new facts will be found in various fields of rich mathematical thinking.” The links between the questions in this lecture are not too big. The size and difficulty of the questions are also different, but it can be said that it contains almost all the research topics of the mathematics community in this century. The fifth issue to be discussed in this article has been resolved with certainty, and it has become a chatter for the mathematics community. The question is this: “Try not to define the Lie group without subtlety.” First, let us explain its significance.