雙語暢銷書《艾倫圖靈傳》第3章:思考什麼是思考(61)

These could be written out as abstract axioms in the spirit of the ‘tables, chairs and beer-mugs’ if one so chose, and the whole theory of numbers could be constructed from them without asking what the symbols such as ‘1’ and ‘+’ were supposed to mean.

這些可以作爲抽象原理寫出來，如果你願意，你同樣可以用"桌子、椅子和酒杯"來描述，關於數字的所有理論，都可以由此推導，不需要考慮"1"和"+"這樣的符號意味着什麼。

A year later, in 1889, the Italian mathematician G. Peano gave the axioms in what became the standard form.

一年後，1889年，意大利數學家G.皮亞諾對此給出了標準化的公理。

In 1900 Hilbert greeted the new century by posing seventeen unsolved problems to the mathematical world.

1900年，希爾伯特對數學界提出23個未解決的問題，來作爲對新世紀的問候。

Of these, the second was that of proving the consistency of the ‘Peano axioms’ on which, as he had shown, the rigour of mathematics depended.

在這些問題中，第二個就是皮亞諾公理的相容性，他認爲，數學的嚴格性皆取決於此。

‘Consistency’ was the crucial word.

"相容性"是一個決定性的的詞語，

There were, for instance, theorems in arithmetic which took thousands of steps to prove – such as Gauss’s theorem that every integer could be expressed as the sum of four squares.

比如說，有的算術定理需要無數步來證明――比如拉格朗日定理：任一自然數都是四個平方數的和。

How could anyone know for sure that there was not some equally long sequence of deductions which led to a contradictory result? What was the basis for credence in such propositions about all numbers, which could never be tested out? What was it about those abstract rules of Peano’s game, which treated ‘+’ and ‘1’ as meaningless symbols, that guaranteed this freedom from contradictions? Einstein doubted the laws of motion.

誰能保證說，一直往下找，不會遇到矛盾？對於這種永遠無法驗證的事，憑什麼來做出這種保證？那麼皮亞諾的這套抽象規則，如何保證不會遇到矛盾？正如愛因斯坦質疑運動定理，

Hilbert doubted even that two and two made four – or at least said that there had to be a reason.

希爾伯特現在要質疑2+2=4，至少說，他要求一個理由。