雙語暢銷書《艾倫圖靈傳》第3章:思考什麼是思考(95)
Who then could have seen the connection with the fate of an obscure Cambridge mathematician? Yet connection there was.
For one day Hitler was to lose the Rhineland, and it would be then, and only then, that the universal machine could emerge into the world of practical action.
因爲如果有一天，希特勒失去了萊茵區，那麼那時，就在那時，通用機器將會真正地來到這世界。The idea had come out of Alan Turing's private loss. But between the idea and its embodiment had to come the sacrifice of millions.
這成就了艾倫·圖靈，但卻帶來了數百萬的犧牲，
Nor would the sacrifices end with Hitler; there was no solution to the world's Entscheidungs problem.
而且這樣的犧牲不會因爲希特勒之死而結束。這就是這個世界的一個"判定性問題"，而且沒有方法能夠解決。* The analogy is not intended to be exact; Hilbert space and quantum mechanical 'states' differ in an essential way from anything in ordinary experience.
這個說法並不特別準確，實際上，希爾伯特空間和量子態與任何日常經驗都不相同。* The word 'group', as used in mathematics, has a technical meaning quite distinct from its use in ordinary language.
數學語言中的"羣"，與自然語言中的意思不同，It refers to the idea of a set of operations, but only when that set of operations meets certain precise conditions.
它是指遵循特定規則的一組運算。These may be illustrated by considering the rotations of a sphere.
你可以想象一個球體的旋轉，If A, B and C are three different rotations, then one can see that:
設A, B和C是三種不同的旋轉動作，那麼你可以看到：(i) there exists a rotation which exactly reverses the effect of A.
（i）存在一種旋轉，與A的效果是相反的。(ii) there exists a rotation which has exactly the same effect as performing A, and then B. Let this rotation be called 'AB'.
（ii）存在一種旋轉，與A然後B的效果是一樣的，我們把這種旋轉叫作"AB"。(iii) Then AB, followed by C, has the same effect as A, followed by BC.
（iii)AB再C，和A再BC的效果是一樣的。These are essentially the conditions required for the rotations to form a 'group'.
滿足這些規則的旋轉動作，形成了一個"羣"。Abstract group theory then arose by taking these conditions, representing them appropriately with symbols, and then abandoning the original concrete embodiment.
抽象羣論用一些符號來表示這些規則，拋棄它們的實體。The resulting theory might profitably be applied to rotations, as indeed it was, in quantum mechanics.
這樣一來，推導出的理論，不但可以應用於實際的旋轉，也可以應用於量子力學，It could also apply to the apparently unrelated field of ciphering.
還可以應用於看似不相關的密碼學領域。(Ciphers enjoy the 'group' properties: a cipher must have a well-defined decipherment operation which reverses it, and if two ciphering operations are performed in succession, the result is another cipher.)
（密碼學非常喜歡"羣"的特性：密碼必須有明確的規則來解碼，而且如果你連續對一個密碼解碼兩次，結果是你會得到另一個密碼。）
