双语畅销书《艾伦图灵传》第3章:思考什么是思考(80)
But there was an approach which led to the answer by a back door route. Alan hit on the idea of the 'computable numbers'.
The crucial notion was that any 'real number' which was defined by some definite rule could be calculated by one of his machines.
这个关键的想法是，任意一个由明确规则定义的实数，都可以用一个这样的机器来计算出来。For instance, there would be a machine to calculate the decimal expansion of π, rather as he had done at school.
比如说，存在一个机器，来计算圆周率π，就像他在学校时人工算的那样。
For it would require no More than a set of rules for adding, multiplying, copying, and so forth.
因为这只需要一套加、乘、复制的规则。being an infinite decimal, the work of the machine would never end, and it would need an unlimited amount of working space on its 'tape'.
因为它是一个无限小数，所以这个机器将永远不会停止，而且它需要无限长的纸带。But it would arrive at every decimal place in some finite time, having used only a finite quantity of tape.
但在某个特定的时刻，它会处于某一个小数位，并且只用了有限的纸带。And everything about the process could be defined by a finite table, left alone to work on a blank tape.
整个计算过程都可以用行为表来规定，然后把它丢在那里，让它独自在纸带上跑来跑去This meant that he had a way of representing a number like π, an infinite decimal, by a finite table.
这就是说，他现在得到了一种方法，可以用有限的表格，来表示无限的小数，比如π。The same would be true of the square root of three, or the logarithm of seven—or any other number defined by some rule.
对于3的平方根或7的对数也可以，任何一个由规则定义的数字都可以。Such numbers he called the 'computable numbers'.
他称这样的数为“可计算数”。More precisely, the machine itself would know nothing about decimals or decimal places.
准确地说，机器本身对小数或小数位一无所知，It would simply produce a sequence of digits.
它只是产生一串数字序列。A sequence that could be produced by one of his machines, starting on a blank tape, he called a 'computable sequence'.
他的一个机器，从一条空白的纸带开始，产生这样的序列，他称为“可计算序列”。Then an infinite computable sequence, prefaced by a decimal point, would define a 'computable number' between 0 and 1.
然后，用一个以小数点作为开始的可计算序列，就可以定义一个0到1之间可计算数。It was in this more strict sense that any computable number between 0 and 1 could be defined by a finite table.
严格来说，任意0到1之间的可计算数，都可以用有限的行为表来定义。It was important to his argument that the computable numbers would always be expressed as infinite sequences of digits, even if these were all 0 after a certain point.
他的论点有一个重要之处：任何可计算数字，总是由一个无限的序列来表示，哪怕它每一位都是0。But these finite tables could now be put into something like alphabetical order, beginning at the most simple, and working through larger and larger ones.
现在我们来考虑这些行为表，从简单的开始，到越来越复杂的，They could be put in a list, or counted; and this meant that all the computable numbers could be put in a list.
它们本身也可以按某种顺序排列起来，成为一个列表。这就意味着，所有的可计算数可以构成一个列表。It was not a practical proposition actually to do it, but in principle the idea was perfectly definite, and would result in the square root of three being say 678th in order, or the logarithm of π being 9369th.
虽然在现实中不太可能真的写出这个列表，但这个想法本身是可以完美定义的，这样一来，3的平方根可能是第678个，而77的对数可能是第9369个。
