But it was only in 1872 that the German mathematician Dedekind had shown exactly how to define ‘real numbers’ in terms of the integers, in such a way that no appeal was made to the concept of measurement.

直到1872年的时候，德国数学家戴德金精确展示了，如何用整数的语言来定义实数，而不需要测量。

This step both unified the concepts of number and length, and had the effect of pushing Hilbert’s questions about geometry into the domain of the integers, or ‘arithmetic’, in its technical mathematical sense.

这一进步，统一了数字和长度的概念，也把希尔伯特的几何问题，转化成了一个算术领域的问题。

As Hilbert said, all he had done was ‘to reduce everything to the question of consistency for the arithmetical axioms, which is left unanswered.’

正如希尔伯特所说，他把一切都归约到了尚待解决的算术公理相容性的问题。

At this point, different mathematicians adopted different attitudes.

在这一点上，不同的数学家有不同的看法。

There was a point of view that it was absurd to speak of the axioms of arithmetic.

一种观点认为，讨论算术公理是荒谬的，

Nothing could be more primitive than the integers.

没有什么比整数更原始低级了。

On the other hand, it could certainly be asked whether there existed a kernel of fundamental properties of the integers, from which all the others could be derived.

而另一方面认为，当然可以讨论整数的基本属性是否存在一个核心，其它问题都是由这个核心衍生来的。

Dedekind also tackled this question, and showed in 1888 that all arithmetic could be derived from three ideas: that there is a number 1, that every number has a successor, and that a principle of induction allows the formulation of statements about all numbers.

戴德金同样解释了这个问题，他在1888年做出说明：所有的算术，都是由三个概念衍生来的：首先有数字1，其次每个数字都有一个后继，然后有一套归纳法，这使所有数字都能形式化描述。