雙語暢銷書《艾倫圖靈傳》第3章:思考什麼是思考(86)
Alan Turing had dealt the death-blow to the Hilbert programme.
He had shown that mathematics could never be exhausted by any finite set of procedures.
他已經證明，數學不可能被任何有限的程序擊敗。He had gone to the heart of the problem, and settled it with one simple, elegant observation.
他直奔問題的核心，並用一個簡明而漂亮的方法解決了。
But there was more to what he had done than a mathematical trick, or logical ingenuity.
然而，他並沒有止步於一個數學把戲，He had created something new—the idea of his machines.
接下來，他還開創了新的東西——關於機器的想法。And correspondingly, there remained a question as to whether his definition of the machine really did include everything that could possibly be counted as a 'definite method'.
對於那些有明確方法的問題，這種機器真的能夠全部解決嗎？Was this repertoire of reading, writing, erasing, moving and stopping enough?
讀出，寫入，清除，移動，停止，這套動作就夠了嗎？It was crucial that it was so, for otherwise the suspicion would always lurk that some extension of the machine's faculties would allow it to solve a greater range of problems.
這個問題是至關重要的，他有一種潛在的懷疑，那就是這種機器也許還能解決更廣泛的問題。One approach to this question led him to demonstrate that his machines could certainly compute any number normally encountered in mathematics.
他演示了他的機器能計算出任何在數學中常見的數字，He also showed that a machine could be set up to churn out every provable assertion within Hilbert's formulation of mathematics.
他還說明可以組建一個機器，快速地推導出希爾伯特數學體系中任何一個可證明的命題。But he also gave some pages of discussion that were among the most unusual ever offered in a mathematical paper, in which he justified the definition by considering what people could possibly be doing when they 'computed' a number by thinking and writing down notes on paper:
然而，他還寫下了幾頁在數學研究中很不尋常的想法，他爲了改進機器的設計，開始考慮人類如何通過思考和在紙上記錄符號來進行計算：Computing is normally done by writing certain symbols on paper.
人們通常在紙上記一些特定的符號來計算。We may suppose this paper is divided into squares like a child's arithmetic book.
我們可能把這張紙想象成，分成若干個方格，就像小孩的算術本一樣。In elementary arithmetic the two-dimensional character of the paper is sometimes used.
在基本算術中，有時會用到紙的二維特徵，But such a use is always avoidable, and I think that it will be agreed that the two-dimensional character of paper is no essential of computation.
但這是有辦法避免的，我認爲我們顯然可以同意，紙的二維特徵對於計算來說，並不是必須的。I assume then that the computation is carried out on one-dimensional paper, i.e. on a tape divided into squares.
我把計算設想成，是在一維的紙上進行的，也就是說，在分成方格的紙帶上。I shall also suppose that the number of symbols which may be printed is finite.
我還設想，可以打印的符號是有限的。If we were to allow an infinity of symbols, then there would be symbols differing to an arbitrarily small extent.
假如我們允許無限的符號，那就會存在兩個符號的差異無窮小。
