Von Neumann had noticed that ‘Hilbert space’ was exactly what was needed to make precise the idea of the ‘state’ of a quantum-mechanical system, such as that of an electron in a hydrogen atom.

冯•诺伊曼注意到，要想研究一个量子系统的状态，比如氢原子中的电子，正需要希尔伯特的这种观点。

One characteristic of such ‘states’ was that they could be added like sounds, and another was that there would generally be infinitely many possible states, rather like the infinite series of harmonics above a ground.

这种状态，就像声音一样可以叠加，而且这样的状态有无穷多种可能，正如一个声音可以有无数的泛音。

Hilbert space could be used to define a rigorous theory of quantum mechanics, proceeding logically from clear-cut axioms.

希尔伯特空间可以用于严密地定义量子力学，并进行清晰的公理化逻辑推导。

The unforeseen application of ‘Hilbert space’ was just the kind of thing that Alan would produce to support his claim for pure mathematics.

希尔伯特空间这种意外的应用，正支持了艾伦对纯数学的看法。

He had seen another vindication in 1932, when the positron was discovered.

1932年，艾伦还得到了另一个支持，

For Dirac had predicted it on the basis of an abstract mathematical theory, which depended upon combining the axioms of quantum mechanics with those of special relativity.

那就是，狄拉克基于抽象数学而预言的正电子，被发现是确实存在的。

But in arguing about the relationship between mathematics and science, Alan Turing found himself tackling a perplexing, subtle, and to him personally important aspect of modern thought.

数学和科学，究竟是什么关系，这对艾伦•图灵来说，是一个复杂，微妙，对他个人的现代思考很重要的一个问题。