It was the logical consistency of the theory, not the experimental results, that von Neumann’s work treated.

冯•诺伊曼的工作，是基于理论的逻辑自洽性，而不是实验结果。

This suited Alan,

这正合艾伦的胃口。

who sought that kind of toughness, and it made a beautiful example of how the expansion of pure mathematics for its own sake had borne unexpected fruit in physics.

这是一个漂亮的例子，表明了纯数学的扩展如何在物理学上起到意想不到的作用。

Before the war, Hilbert had developed a certain generalisation of Euclidean geometry, which involved considering a space with infinitely many dimensions.

在战前，希尔伯特提出了一个关于欧几里得几何学的构想，他考虑了无限维度的空间。

This ‘space’ had nothing to do with physical space.

这个空间，在物理空间中是不能想象的，它更像是用音乐描绘的虚拟场景，你可以想象长笛音、小提琴音或钢琴音，它们是由很多个基音，第一泛音，第二泛音等等组成的──每种声音（理论上）是有无数个分音来使它与其它声音区分开。

It was more like an imaginary graph on which could be plotted all musical sounds, by thinking of a flute, or violin, or piano tone as made up of so much of the fundamental, so much of the first harmonic, so much of the second harmonic, and so on – each kind of sound requiring (in principle) the specification of infinitely many ingredients.

在这个空间中，一个点就好比一个声音，两个点可以相加（好比两个声音叠加），一个点能够和一个因子相乘（好比放大一个声音）注。