雙語暢銷書《艾倫圖靈傳》第3章:思考什麼是思考(71)
The Hilbert programme was essentially an extension of the work on which he had started in the 1890s.
It did not attempt to answer the question which Frege and Russell had tackled, that of what mathematics really was.
它並不急於解答弗雷格和羅素的問題，也就是數學到底是什麼。In that respect it was less philosophical, less ambitious.
一方面，這樣就不那麼哲學化，不那麼讓人吃力。
On the other hand, it was more far-reaching in that it asked profound and difficult questions about the systems such as Russell produced.
另一方面，羅素的那個艱難的困境實際上很難解決。In fact Hilbert posed the question as to what were, in principle, the limitations of a scheme such as that of Principia Mathematica.
希爾伯特提出的問題主要是，在原則上，《數學原理》的限制是什麼。Was there a way of finding out what could, and what could not, be proved within such a theory?
有沒有一種方法，來判斷什麼可以被這套理論證明，而什麼不可以。Hilbert's approach was called the formalist approach, because it treated mathematics as if a game, a matter of form.
希爾伯特的方法，叫作形式主義，它把數學看成一套形式規則。The allowable steps of proof were to be considered like the allowable moves in a game of chess, with the axioms as the starting position of the game.
允許的證明步驟，就好比國際象棋中允許的走法，而公理就好比是開局時的擺法。In this analogy, 'playing chess' corresponded to 'doing mathematics', but statements about chess (such as 'two knights cannot force checkmate') would correspond to statements about the scope of mathematics.
在這個類比中，"下國際象棋"就相當於"做數學"，只是把國際象棋的命題（比如『兩個馬將不死對方』）換成數學命題。And it was with such statements that the Hilbert programme was concerned.
希爾伯特計劃，就是考慮這樣的命題。At that 1928 congress, Hilbert made his questions quite precise.
在1928年的大會上，希爾伯特明確提出了他的問題。First, was mathematics complete, in the technical sense that every statement (such as 'every integer is the sum of four squares') could either be proved, or disproved.
第一，數學是完備的嗎？是不是每個命題（比如『任意自然數都是四個平方數的和』）都能證明或證僞。Second, was mathematics consistent, in the sense that the statement '2 + 2 = 5' could never be arrived at by a sequence of valid steps of proof.
第二，數學是相容的嗎？也就是說，用符合邏輯的步驟和順序，永遠不會推出矛盾的命題，比如2+2 = 5。And thirdly, was mathematics decidable? By this he meant, did there exist a definite method which could, in principle, be applied to any assertion, and which was guaranteed to produce a correct decision as to whether that assertion was true.
第三，數學是可判定的嗎？他的意思是，是否存在一個機械式的方法，可以應用於任何命題，然後自動給出該命題的真假。In 1928, none of these questions was answered.
在1928年，這些問題都不能得到解答。But it was Hilbert's opinion that the answer would be 'yes' in each case.
但希爾伯特的觀點是，每個回答都將會是“是”。In 1900 Hilbert had declared 'that every definite mathematical problem must necessarily be susceptible of an exact settlement … in mathematics there is no ignorabimus';
早在1900年，希爾伯特宣佈"所有數學問題都是有解的……沒有數學照耀不到的角落"。and when he retired in 1930 he went further:
當他1930年退休時，他研究得更深入了：In an effort to give an example of an unsolvable problem, the philosopher Comte once said that science would never succeed in ascertaining the secret of the chemical composition of the bodies of the universe.
舉一個不可解問題的例子來說，哲學家孔德曾經認爲，科學永遠無法給出宇宙的化學成分。A few years later this problem was solved.…
但沒過幾年，這個問題就被解決了……The true reason, according to my thinking, why Comte could not find an unsolvable problem lies in the fact that there is no such thing as an unsolvable problem.
在我看來，孔德找到一個不可解的問題的真正原因在於，這種不可解的問題壓根就不存在。It was a view more positive than the Positivists.
這個觀點，比實證主義者還要激進。But at the very same meeting, a young Czech mathematician, Kurt Gdel, announced results which dealt it a serious blow.
但就在這同一個大會上，一個年輕的捷克數學家，柯特·哥德爾的宣佈，給了他當頭一擊。Gdel was able to show that arithmetic must be incomplete: that there existed assertions which could neither be proved nor disproved.
哥德爾能夠證明，算術一定是不完備的：存在既不能證明，也不能證僞的命題。He started with Peano's axioms for the integers, but enlarged through a simple theory of types, so that the system was able to represent sets of integers, sets of sets of integers, and so on.
他從皮亞諾的整數公理開始，經過集合層次理論的拓展，使這個系統可以代表整數的集合、整數的集合的集合，等等。However, his argument would apply to any formal mathematical system rich enough to include the theory of numbers, and the details of the axioms were not crucial.
總之，他的論點可以應用到任何涵蓋了算術公理的形式系統，與其公理本身的內容無關。
