雙語暢銷書《艾倫圖靈傳》第3章:思考什麼是思考(57)

Quantum mechanics presented a fine example of where the expansion and liberation of mathematics for its own sake had paid off in physics.

量子力學是一個很好的例子，說明數學的擴展成功地應用於物理學。

It had proved necessary to create a theory not of numbers and quantities, but of ‘states’ – and ‘Hilbert space’ offered exactly the right symbolism for these.

它證明了，建立一個不是由數值組成的，而是由形式組成的理論，是非常必要的，希爾伯特空間就是代表。

Another related development in pure mathematics, which quantum physicists was now busy exploiting, was that of the ‘abstract group’.

另外一個量子物理學家們正忙於研究的純數學問題，就是"抽象羣"的發展。

It had come about through mathematicians putting the idea of ‘operation’ into a symbolic form, and treating the result as an abstract exercise.

數學家們形式化地描述"運算"，把運算的結果也看成抽象的。

On the other hand, the movement towards abstraction had created something of a crisis within pure mathematics.

但在另一方面，面向抽象的發展，也給純數學內部帶來了一些危機。

If it was to be thought of as a game, following arbitrary rules to govern the play of symbols, what had happened to the sense of absolute truth? In March 1933 Alan acquired Bertrand Russell’s Introduction to Mathematical Philosophy, which addressed itself to this central question.

它被當成一種遊戲，按照隨意的規則來玩弄符號，那麼數學的實在感跑到哪裏去了？1933年3月，艾倫讀了伯特蘭•羅素的《數學哲學引論》，這本書就試圖解決這個關鍵問題。

The crisis had first appeared in the study of geometry.

危機首先出現在幾何研究中。