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世纪，人们相信几何是科学的一个分支，是这个世界的真理，而欧几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世纪，人们发现，几何系统的新发展与欧几里得产生了分歧，人们开始怀疑，宇宙是否真的是欧氏的。

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.

在抽象系统的角度上，欧几里得几何是不是完备而自洽的，是有待于探讨一下的。

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

欧几里得原理，是否是一个完备的几何理论，此时还搞不清楚。

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

那些关于点和线的概念，都是凭直觉得到的，还有些多余的假设也被用来做证明。

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.

从现代的观点看，有必要抽象化点和线的逻辑关系，用形式规则来描述它们，使它们不局限于特定的物理意义，展现抽象游戏本身的意义。

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.’

如同希尔伯特所说："我们应该同样可以用『桌子、椅子、酒杯』来描述问题，而不只是『点、线、面』"。