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，希尔伯特成功地提出一个公理体系，使他可以不依靠特定的实体，而推导出欧几里德的所有定理。

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

然而，他的证明需要另外一个假设，那就是关于"实数"注的理论。

‘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.

对于希腊数学家们来说，"实数"是对长度的测量值，它以可无限细分，最重要的是，假设"实数"在物理空间中是固定的。

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

但是对于希尔伯特的观点来说，这并不够。

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

幸运的是，人们发现，还可以用另一种方式描述"实数"。

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世纪，人们理解了"实数"还可以表现成无限小数，比如把π写成3.14159265358979……一个实数可以用这样的方式精确表达――整数的无限序列。