雙語暢銷書《艾倫圖靈傳》第3章:思考什麼是思考(58)

In the eighteenth century, it was possible to believe that geometry was a branch of science, being a system of truths about the world, which Euclid’s axioms boiled down into an essential kernel.

18世紀，人們相信幾何是科學的一個分支，是這個世界的真理，而歐幾里得公理就是它的核心。

But the nineteenth century saw the development of geometrical systems different from Euclid’s.

但到了19世紀，人們發現，幾何系統的新發展與歐幾里得產生了分歧，人們開始懷疑，宇宙是否真的是歐氏的。

It was also doubted whether the real universe was actually Euclidean. In the modern separation of mathematics from science, it became necessary to ask whether Euclidean geometry was, regarded as an abstract exercise, a complete and consistent whole.

在抽象系統的角度上，歐幾里得幾何是不是完備而自洽的，是有待於探討一下的。

It was not clear that Euclid’s axioms really did define a complete theory of geometry.

歐幾里得原理，是否是一個完備的幾何理論，此時還搞不清楚。

It might be that some extra assumption was being smuggled into proofs, because of intuitive, implicit ideas about points and lines.

那些關於點和線的概念，都是憑直覺得到的，還有些多餘的假設也被用來做證明。

From the modern point of view it was necessary to abstract the logical relationships of points and lines, to formulate them in terms of purely symbolic rules, to forget about their ‘meaning’ in terms of physical space, and to show that the resulting abstract game made sense in itself.

從現代的觀點看，有必要抽象化點和線的邏輯關係，用形式規則來描述它們，使它們不侷限於特定的物理意義，展現抽象遊戲本身的意義。

Hilbert, who was always down-to-earth, liked to say: ‘One must always be able to say tables, chairs, beer-mugs, instead of points, lines, planes.’

如同希爾伯特所說："我們應該同樣可以用『桌子、椅子、酒杯』來描述問題，而不只是『點、線、面』"。