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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.
这些可以作为抽象原理写出来，如果你愿意，你同样可以用"桌子、椅子和酒杯"来描述，关于数字的所有理论，都可以由此推导，不需要考虑"1"和"+"这样的符号意味着什么。
A year later, in 1889, the Italian mathematician G. Peano gave the axioms in what became the standard form.
一年后，1889年，意大利数学家G.皮亚诺对此给出了标准化的公理。
In 1900 Hilbert greeted the new century by posing seventeen unsolved problems to the mathematical world.
1900年，希尔伯特对数学界提出23个未解决的问题，来作为对新世纪的问候。
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.
在这些问题中，第二个就是皮亚诺公理的相容性，他认为，数学的严格性皆取决于此。
‘Consistency’ was the crucial word.
"相容性"是一个决定性的的词语，
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.
比如说，有的算术定理需要无数步来证明――比如拉格朗日定理：任一自然数都是四个平方数的和。
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.
谁能保证说，一直往下找，不会遇到矛盾？对于这种永远无法验证的事，凭什么来做出这种保证？那么皮亚诺的这套抽象规则，如何保证不会遇到矛盾？正如爱因斯坦质疑运动定理，
Hilbert doubted even that two and two made four – or at least said that there had to be a reason.
希尔伯特现在要质疑2+2=4，至少说，他要求一个理由。
