雙語暢銷書《艾倫圖靈傳》第3章:思考什麼是思考(54)

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

馮•諾伊曼的工作，是基於理論的邏輯自洽性，而不是實驗結果。

This suited Alan,

這正合艾倫的胃口。

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.

這是一個漂亮的例子，表明了純數學的擴展如何在物理學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.

在戰前，希爾伯特提出了一個關於歐幾里得幾何學的構想，他考慮了無限維度的空間。

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

這個空間，在物理空間中是不能想象的，它更像是用音樂描繪的虛擬場景，你可以想象長笛音、小提琴音或鋼琴音，它們是由很多個基音，第一泛音，第二泛音等等組成的──每種聲音（理論上）是有無數個分音來使它與其它聲音區分開。

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.

在這個空間中，一個點就好比一個聲音，兩個點可以相加（好比兩個聲音疊加），一個點能夠和一個因子相乘（好比放大一個聲音）注。