雙語暢銷書《艾倫圖靈傳》第3章:思考什麼是思考(83)

Suppose one tried to design a 'Cantor machine' to produce this diagonAl uncomputable number.

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假設有人要設計一個康託機，來計算這個對角線上的不可計算數，

Roughly speaking, it would start with a blank tape, and write the number 1.

那大致的過程是這樣的：它從空白的紙帶開始，先寫下數字"1"，

It would then have to produce the first table, and then execute it, stopping at the first digit that it wrote, and adding on one.

產生第1個行爲表，然後運行這個表，寫下它產生的第1個數字，加1；

Then it would start again, with the number 2, produce the second table, executing it as far as the second digit, and writing this down, adding on one.

接下來，寫入數字2，產生第2個行爲表，運行，寫下它產生的第2個數字，加1；

It would have to continue doing this for ever, so that when its counter read '1000', it would produce the thousandth table, execute it as far as the thousandth digit, add on one to this and write it down.

以此類推，當它的計數器讀出"1000"，就產生第1000個行爲表，並運行它直到產生第1000個數，然後把它寫下來，加1。

One part of this process could certainly be done by one of his machines.

這個過程的一部分，確實可以用機器來做，

For the process of 'looking up the entries' in a given table, and working out what the corresponding machine would do, was itself a 'mechanical process.' A machine could do it.

在一個給定的表中查詢某一項，然後運行與之對應的機器，這是一個機械過程，機器可以做到。

There was a difficulty in that the tables were naturally thought of in two-dimensional form, but then it was only a technical matter to encode them in a form in which they could be put on a 'tape'.

有一個問題是，這個表現在是二維的，但這也很好辦，要把它編成可以放入紙帶的形式，只是個技術問題。

In fact, they could be encoded as integers, rather as Gdel had represented formulae and proofs as integers.

實際上，還可以全部用數字來表示它，哥德爾已經展示了，用數字來表示公式和證明。

Alan called them 'description numbers', so that there was a description number corresponding to each table.

艾倫稱之爲"描述數"，每個每個表都有對應的描述數。

In one way this was just a technicality, a means of putting tables on to the tape, and arranging them in an 'alphabetical order'.

總之，把這個表放入紙帶，按一定順序排列起來，只是一個技術問題。

But underneath there lay the same powerful idea that Gdel had used, that there was no essential distinction between 'numbers' and operations on numbers.

這其中也體現了哥德爾用過的強力想法，數字本身和對數字的操作，沒有實質的區別，

From a modern mathematical point of view, they were all alike symbols.

從現代數學的角度看來，它們都是符號。