雙語暢銷書《艾倫圖靈傳》第3章:思考什麼是思考(55)
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.
馮•諾伊曼注意到，要想研究一個量子系統的狀態，比如氫原子中的電子，正需要希爾伯特的這種觀點。
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.
這種狀態，就像聲音一樣可以疊加，而且這樣的狀態有無窮多種可能，正如一個聲音可以有無數的泛音。
Hilbert space could be used to define a rigorous theory of quantum mechanics, proceeding logically from clear-cut axioms.
希爾伯特空間可以用於嚴密地定義量子力學，並進行清晰的公理化邏輯推導。
The unforeseen application of ‘Hilbert space’ was just the kind of thing that Alan would produce to support his claim for pure mathematics.
希爾伯特空間這種意外的應用，正支持了艾倫對純數學的看法。
He had seen another vindication in 1932, when the positron was discovered.
1932年，艾倫還得到了另一個支持，
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，狄拉克基於抽象數學而預言的正電子，被發現是確實存在的。
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.
數學和科學，究竟是什麼關係，這對艾倫•圖靈來說，是一個複雜，微妙，對他個人的現代思考很重要的一個問題。
