雙語暢銷書《艾倫圖靈傳》第3章:思考什麼是思考(60)

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.

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直到1872年的時候，德國數學家戴德金精確展示了，如何用整數的語言來定義實數，而不需要測量。

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.

這一進步，統一了數字和長度的概念，也把希爾伯特的幾何問題，轉化成了一個算術領域的問題。

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

正如希爾伯特所說，他把一切都歸約到了尚待解決的算術公理相容性的問題。

At this point, different mathematicians adopted different attitudes.

在這一點上，不同的數學家有不同的看法。

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

一種觀點認爲，討論算術公理是荒謬的，

Nothing could be more primitive than the integers.

沒有什麼比整數更原始低級了。

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.

而另一方面認爲，當然可以討論整數的基本屬性是否存在一個核心，其它問題都是由這個核心衍生來的。

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.

戴德金同樣解釋了這個問題，他在1888年做出說明：所有的算術，都是由三個概念衍生來的：首先有數字1，其次每個數字都有一個後繼，然後有一套歸納法，這使所有數字都能形式化描述。