雙語暢銷書《艾倫圖靈傳》第3章:思考什麼是思考(56)

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

數學和科學的區別，在19世紀末才澄清。

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，人們往往認爲，數學就是代表現實世界中的數量，然而這樣的觀點很快就被"負數"這樣的概念終結了。

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

十九世紀，數學的很多分支都朝着抽象觀點發展，

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

數學符號越來越脫離物質實體。

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

學校教的代數，也就是十八世紀的代數，會用字母來代表數值。

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.

這種規則其實也可以用數字來說明，但這既沒必要，也不怎麼合適。

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

這種抽象的意義，就是把數學從傳統的計數和測量中解脫出來。

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.

在現代數學中，如果你願意，符號可以遵守任何規則，它的意義不僅是數值，它還可以根本沒意義。