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Who then could have seen the connection with the fate of an obscure Cambridge mathematician? Yet connection there was.
谁能看到一位无名的剑桥数学家的命运转折？是的，这里就是转折。
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.
因为如果有一天，希特勒失去了莱茵区，那么那时，就在那时，通用机器将会真正地来到这世界。
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.
这成就了艾伦·图灵，但却带来了数百万的牺牲，
Nor would the sacrifices end with Hitler; there was no solution to the world's Entscheidungs problem.
而且这样的牺牲不会因为希特勒之死而结束。这就是这个世界的一个"判定性问题"，而且没有方法能够解决。
* The analogy is not intended to be exact; Hilbert space and quantum mechanical 'states' differ in an essential way from anything in ordinary experience.
这个说法并不特别准确，实际上，希尔伯特空间和量子态与任何日常经验都不相同。
* The word 'group', as used in mathematics, has a technical meaning quite distinct from its use in ordinary language.
数学语言中的"群"，与自然语言中的意思不同，
It refers to the idea of a set of operations, but only when that set of operations meets certain precise conditions.
它是指遵循特定规则的一组运算。
These may be illustrated by considering the rotations of a sphere.
你可以想象一个球体的旋转，
If A, B and C are three different rotations, then one can see that:
设A, B和C是三种不同的旋转动作，那么你可以看到：
(i) there exists a rotation which exactly reverses the effect of A.
（i）存在一种旋转，与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）存在一种旋转，与A然后B的效果是一样的，我们把这种旋转叫作"AB"。
(iii) Then AB, followed by C, has the same effect as A, followed by BC.
（iii)AB再C，和A再BC的效果是一样的。
These are essentially the conditions required for the rotations to form a 'group'.
满足这些规则的旋转动作，形成了一个"群"。
Abstract group theory then arose by taking these conditions, representing them appropriately with symbols, and then abandoning the original concrete embodiment.
抽象群论用一些符号来表示这些规则，抛弃它们的实体。
The resulting theory might profitably be applied to rotations, as indeed it was, in quantum mechanics.
这样一来，推导出的理论，不但可以应用于实际的旋转，也可以应用于量子力学，
It could also apply to the apparently unrelated field of ciphering.
还可以应用于看似不相关的密码学领域。
(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.)
（密码学非常喜欢"群"的特性：密码必须有明确的规则来解码，而且如果你连续对一个密码解码两次，结果是你会得到另一个密码。）
