Alan Turing had dealt the death-blow to the Hilbert programme.
艾伦·图灵，给了希尔伯特计划致命一击，
He had shown that mathematics could never be exhausted by any finite set of procedures.
他已经证明，数学不可能被任何有限的程序击败。
He had gone to the heart of the problem, and settled it with one simple, elegant observation.
他直奔问题的核心，并用一个简明而漂亮的方法解决了。
But there was more to what he had done than a mathematical trick, or logical ingenuity.
然而，他并没有止步于一个数学把戏，
He had created something new—the idea of his machines.
接下来，他还开创了新的东西——关于机器的想法。
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'.
对于那些有明确方法的问题，这种机器真的能够全部解决吗？
Was this repertoire of reading, writing, erasing, moving and stopping enough?
读出，写入，清除，移动，停止，这套动作就够了吗？
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.
这个问题是至关重要的，他有一种潜在的怀疑，那就是这种机器也许还能解决更广泛的问题。
One approach to this question led him to demonstrate that his machines could certainly compute any number normally encountered in mathematics.
他演示了他的机器能计算出任何在数学中常见的数字，
He also showed that a machine could be set up to churn out every provable assertion within Hilbert's formulation of mathematics.
他还说明可以组建一个机器，快速地推导出希尔伯特数学体系中任何一个可证明的命题。
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，他还写下了几页在数学研究中很不寻常的想法，他为了改进机器的设计，开始考虑人类如何通过思考和在纸上记录符号来进行计算：
Computing is normally done by writing certain symbols on paper.
人们通常在纸上记一些特定的符号来计算。
We may suppose this paper is divided into squares like a child's arithmetic book.
我们可能把这张纸想象成，分成若干个方格，就像小孩的算术本一样。
In elementary arithmetic the two-dimensional character of the paper is sometimes used.
在基本算术中，有时会用到纸的二维特征，
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.
但这是有办法避免的，我认为我们显然可以同意，纸的二维特征对于计算来说，并不是必须的。
I assume then that the computation is carried out on one-dimensional paper, i.e. on a tape divided into squares.
我把计算设想成，是在一维的纸上进行的，也就是说，在分成方格的纸带上。
I shall also suppose that the number of symbols which may be printed is finite.
我还设想，可以打印的符号是有限的。
If we were to allow an infinity of symbols, then there would be symbols differing to an arbitrarily small extent.
假如我们允许无限的符号，那就会存在两个符号的差异无穷小。