雙語暢銷書《艾倫圖靈傳》第3章:思考什麼是思考(73)
Newman's lectures finished with the proof of Gdel's theorem, and thus brought Alan up to the frontiers of knowledge.
The third of Hilbert's questions still remained open, although it now had to be posed in terms of 'provability' rather than 'truth'.
希爾伯特的第三個問題仍然懸而未決，Gdel's results did not rule out the possibility that there was some way of distinguishing the provable from the non-provable statements.
哥德爾的結論，並不排除存在某種方法，可以區分一個命題是否可被證明。
Perhaps the rather peculiar Gdelian assertions could somehow be separated off.
也許相當古怪的哥德爾式主張可以以某種方式被分開。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?
正如紐曼所說，有沒有一個明確的方法，可以用一個機械的過程，來判斷一個數學命題是否可以證明呢？From one point of view this was a very tall order, going to the heart of everything known about creative mathematics.
從某種角度來說，這是一個很高的要求，直奔當前數學界所有知識的核心。Hardy, for instance, had said rather indignantly in 1928 that
比如哈代在1928年相當憤慨地說：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.
很幸運，當然不存在這樣的方法，否則如果存在，那我們就有了一套機械的規則，來解決所有的數學問題，而我們的數學家生涯也就走到盡頭了。There were plenty of statements about numbers which the efforts of centuries had failed either to prove or disprove.
有很多關於數字的命題，是經過了幾個世紀的努力，也沒有成功地證明或證僞的。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.
比如費馬大定理，說任意立方數都不能表示爲兩個立方數的和，任意四次冪也不能表示爲兩個四次冪的和，等等。Another was Goldbach's conjecture, that every even number was the sum of two primes.
還有哥德巴赫猜想：任意偶數都是兩個素數的和。It was hard to believe that assertions which had resisted attack so long could in fact be decided automatically by some set of rules.
很難相信，這些頑強的命題，可以被一套規則自動證明。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，那些已經得到解決的難題，比如四平方數定理，極少有被“機械的規則”證明的，往往都是通過創造性推演，或者構建新的抽象代數概念。As Hardy said, 'It is only the very unsophisticated outsider who imagines that mathematicians make discoveries by turning the handle of some miraculous machine.'
哈代說：“只有完全不懂數學的人，纔會相信有一臺超自然的機器，數學家們只要轉動他的搖把，就能得到新的發現。”On the other hand, the progress of mathematics had certainly brought more and more problems within the range of a 'mechanical' approach.
從另一方面來說，數學的發展確實給“機械方法”的問題帶來了越來越多的麻煩。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.
哈代也許會說，這些發展“顯然”還不是整個數學，但是自從有了哥德爾的定理，沒有什麼東西的是“顯然”的。The question deserved a more penetrating analysis.
這個問題需要更加嚴格的分析。
