Newman's lectures finished with the proof of Gdel's theorem, and thus brought Alan up to the frontiers of knowledge.
纽曼的课程就以证明哥德尔定理作为结束，因此把艾伦带到了学术界的前沿。
The third of Hilbert's questions still remained open, although it now had to be posed in terms of 'provability' rather than 'truth'.
希尔伯特的第三个问题仍然悬而未决，
Gdel's results did not rule out the possibility that there was some way of distinguishing the provable from the non-provable statements.
哥德尔的结论，并不排除存在某种方法，可以区分一个命题是否可被证明。
Perhaps the rather peculiar Gdelian assertions could somehow be separated off.
也许相当古怪的哥德尔式主张可以以某种方式被分开。
Was there a definite method, or as Newman put it, a mechanical process which could be applied to a mathematical statement, and which would come up with the answer as to whether it was provable?
正如纽曼所说，有没有一个明确的方法，可以用一个机械的过程，来判断一个数学命题是否可以证明呢？
From one point of view this was a very tall order, going to the heart of everything known about creative mathematics.
从某种角度来说，这是一个很高的要求，直奔当前数学界所有知识的核心。
Hardy, for instance, had said rather indignantly in 1928 that
比如哈代在1928年相当愤慨地说：
There is of course no such theorem, and this is very fortunate, since if there were we should have a mechanical set of rules for the solution of all mathematical problems, and our activities as mathematicians would come to an end.
很幸运，当然不存在这样的方法，否则如果存在，那我们就有了一套机械的规则，来解决所有的数学问题，而我们的数学家生涯也就走到尽头了。
There were plenty of statements about numbers which the efforts of centuries had failed either to prove or disprove.
有很多关于数字的命题，是经过了几个世纪的努力，也没有成功地证明或证伪的。
There was Fermat's so-called Last Theorem, which conjectured that there was no cube which could be expressed as the sum of two cubes, no fourth power as sum of two fourth powers, and so on.
比如费马大定理，说任意立方数都不能表示为两个立方数的和，任意四次幂也不能表示为两个四次幂的和，等等。
Another was Goldbach's conjecture, that every even number was the sum of two primes.
还有哥德巴赫猜想：任意偶数都是两个素数的和。
It was hard to believe that assertions which had resisted attack so long could in fact be decided automatically by some set of rules.
很难相信，这些顽强的命题，可以被一套规则自动证明。
Furthermore, the difficult problems which had been solved, such as Gauss's Four Square Theorem, had rarely been proved by anything like a 'mechanical set of rules', but by the exercise of creative imagination, constructing new abstract algebraic concepts.
另外，那些已经得到解决的难题，比如四平方数定理，极少有被“机械的规则”证明的，往往都是通过创造性推演，或者构建新的抽象代数概念。
As Hardy said, 'It is only the very unsophisticated outsider who imagines that mathematicians make discoveries by turning the handle of some miraculous machine.'
哈代说：“只有完全不懂数学的人，才会相信有一台超自然的机器，数学家们只要转动他的摇把，就能得到新的发现。”
On the other hand, the progress of mathematics had certainly brought more and more problems within the range of a 'mechanical' approach.
从另一方面来说，数学的发展确实给“机械方法”的问题带来了越来越多的麻烦。
Hardy might say that 'of course' this advance could never encompass the whole of mathematics, but after Gdel's theorem, nothing was 'of course' any more.
哈代也许会说，这些发展“显然”还不是整个数学，但是自从有了哥德尔的定理，没有什么东西的是“显然”的。
The question deserved a more penetrating analysis.
这个问题需要更加严格的分析。