The distinction between science and mathematics had only been clarified in the late nineteenth century.

数学和科学的区别，在19世纪末才澄清。

Until then it might be supposed that mathematics necessarily represented the relations of numbers and quantities appearing in the physical world, although this point of view had really been doomed as soon as such concepts as the ‘negative numbers’ were developed.

在此之前，人们往往认为，数学就是代表现实世界中的数量，然而这样的观点很快就被"负数"这样的概念终结了。

The nineteenth century, however, had seen developments in many branches of mathematics towards an abstract point of view.

十九世纪，数学的很多分支都朝着抽象观点发展，

Mathematical symbols became less and less obliged to correspond directly with physical entities.

数学符号越来越脱离物质实体。

In school algebra – eighteenth century algebra, in effect – letters would be used as symbols for numerical quantities.

学校教的代数，也就是十八世纪的代数，会用字母来代表数值。

The rules for adding and multiplying them would follow from the assumption that they were ‘really’ interpreted in terms of numbers, but it would not be necessary nor indeed always appropriate to do so.

这种规则其实也可以用数字来说明，但这既没必要，也不怎么合适。

The point of such abstraction was that it liberated algebra, and indeed all mathematics, from the traditional sphere of counting and measurement.

这种抽象的意义，就是把数学从传统的计数和测量中解脱出来。

In modern mathematics, symbols might be used according to any rules whatever, and might be interpreted in ways far more general than in terms of numerical quantities, if indeed they bore any interpretation at all.

在现代数学中，如果你愿意，符号可以遵守任何规则，它的意义不仅是数值，它还可以根本没意义。