雙語暢銷書《艾倫圖靈傳》第3章:思考什麼是思考(59)

In 1899, Hilbert succeeded in finding a system of axioms which he could prove would lead to all the theorems of Euclidean geometry, without any appeal to the nature of the physical world.

1899年，希爾伯特成功地提出一個公理體系，使他可以不依靠特定的實體，而推匯出歐幾里德的所有定理。

However, his proof required the assumption that the theory of ‘real numbers’* was satisfactory.

然而，他的證明需要另外一個假設，那就是關於"實數"注的理論。

‘Real numbers’ were what to the Greek mathematicians were the measurements of lengths, infinitely subdivisible, and for most purposes it could be assumed that the use of ‘real numbers’ was solidly grounded in the nature of physical space.

對於希臘數學家們來說，"實數"是對長度的測量值，它以可無限細分，最重要的是，假設"實數"在物理空間中是固定的。

But from Hilbert’s point of view this was not good enough.

但是對於希爾伯特的觀點來說，這並不夠。

Fortunately it was possible to describe ‘real numbers’ in an essentially different way.

幸運的是，人們發現，還可以用另一種方式描述"實數"。

By the nineteenth century it was well understood that ‘real numbers’ could be represented as infinite decimals, writing the number π for instance as 3.14159265358979.… A precise meaning had been given to the idea that a ‘real number’ could be represented as accurately as desired by such a decimal – an infinite sequence of integers.

到了19世紀，人們理解了"實數"還可以表現成無限小數，比如把π寫成3.14159265358979……一個實數可以用這樣的方式精確表達――整數的無限序列。