Quantum mechanics presented a fine example of where the expansion and liberation of mathematics for its own sake had paid off in physics.

量子力学是一个很好的例子，说明数学的扩展成功地应用于物理学。

It had proved necessary to create a theory not of numbers and quantities, but of ‘states’ – and ‘Hilbert space’ offered exactly the right symbolism for these.

它证明了，建立一个不是由数值组成的，而是由形式组成的理论，是非常必要的，希尔伯特空间就是代表。

Another related development in pure mathematics, which quantum physicists was now busy exploiting, was that of the ‘abstract group’.

另外一个量子物理学家们正忙于研究的纯数学问题，就是"抽象群"的发展。

It had come about through mathematicians putting the idea of ‘operation’ into a symbolic form, and treating the result as an abstract exercise.

数学家们形式化地描述"运算"，把运算的结果也看成抽象的。

On the other hand, the movement towards abstraction had created something of a crisis within pure mathematics.

但在另一方面，面向抽象的发展，也给纯数学内部带来了一些危机。

If it was to be thought of as a game, following arbitrary rules to govern the play of symbols, what had happened to the sense of absolute truth? In March 1933 Alan acquired Bertrand Russell’s Introduction to Mathematical Philosophy, which addressed itself to this central question.

它被当成一种游戏，按照随意的规则来玩弄符号，那么数学的实在感跑到哪里去了？1933年3月，艾伦读了伯特兰•罗素的《数学哲学引论》，这本书就试图解决这个关键问题。

The crisis had first appeared in the study of geometry.

危机首先出现在几何研究中。