介绍图灵一篇未发表的文章：数学表示法和用语的改革。

这篇文章被收录在纪念图灵诞辰100年的论文集中，原文出自于图灵的个人数学笔记本，在伦敦附近的布莱切利园展出。

布莱切利园是二战期间盟军密码破译行动的欧洲总部，我今年五月参观布莱切利园时，看到了这个笔记本。

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引言部分翻译：

题目：数学表示法和用语的改革 - Turing

人们早就认识到数学和逻辑几乎是相同的，而且它们可能会不知不觉地融合在一起。实际上，这种融合过程并没有取得多大进展，数学从符号逻辑研究中获益甚微。造成这种情况的主要原因似乎是逻辑学家和普通数学家（mathematician-in-the-street）之间缺乏联系。符号逻辑对于大多数数学家来说是些非常难懂的术语，逻辑学家对使其更容易接受并不十分感兴趣。然而，符号逻辑似乎为数学家提供了一些小课程，这些课程可以教给他们，而数学家不需要学习太多符号逻辑。

尤其是符号逻辑似乎可以帮助数学家改进他们的表示法和用语，目前这些表示法和用语非常缺乏系统性，对有意学习者和无法表达思想的人来说都是一个明显的障碍，因为表达思想所需的表示法并不为人所知。当然，我所说的表示法不是指压力是否应该用 p 或 P 来表示这样的琐碎问题，而是指更深层次的问题，例如我们应该说“z 的函数 f(z)”还是“函数 f”。

不建议改革采取一种坚不可摧的逻辑系统的形式，让未来的所有数学都在此系统中表达，没有一个民主的数学界会支持这样的想法，而且这种想法也不可取。相反，我们必须提出一些明确的改进小建议，每个建议都有很好的论据和例子支持，每个建议都应该可以单独采纳。在这种情况下，人们可能希望其中四分之一或更多的建议得到采纳，并且所有建议的使用范围都会扩大。

尽管不希望尝试将数学置于逻辑系统的直接框架中，但在研究表示法时可能希望使用这样的系统。人们很可能会从数学教科书中选取典型的短语并分析其含义，有一个逻辑系统来以相当明确的方式表达这些含义是有用的。为此目的使用哪种系统可能并不重要，并且不同的人员可能会使用不同的系统。

具体来说，我倾向于建议以下方案：

i) 广泛检查当前的数学，物理和工程书籍以及论文，以列出所有常用的表示法形式。

ii) 检查这些表示法以发现它们的真正含义。这通常涉及各种在作者和读者之间隐含理解的陈述，也可能包括标准表示法中所涉及的表示法的等效性。

ii) 放下所需表示法最低要求，这些要求应该非常温和。 我认为被此要求涵盖的事项应包括以下内容：

a) 自由变量和约束变量应被所有人理解并得到适当遵循。

b) 应该做出某种规定以符合类型理论。假设俄罗斯世界观（Weltenscheung），我认为适用于大多数普通数学家。

c) 应当考虑推理定理，即应该认识到，各种论证形式是推理定理的一种或多种表现。因此，推理定理应该是众所周知的部分之间相互作用的规则。

d) 应该对这些符号的基本性质做出非常明确的陈述，应该没有将实数变量错当取实数值的函数的危险。

iv) 符号逻辑提出新表示法。

v) 发展遵循新规范并体现新表示法的数学相对基础部分的示例。这些示例只应在产生巨大优势的情况下纳入新表示法。应尽可能分别展示各项独立改革的效果，以促进其独立采用。

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原文：

Title : The Reform of Mathematics Notation and Phraseology  by Turing

It has long been recognised that mathematics and logic are virtually the same and they may be expected to merge imperceptibly into one another. Actually this merging process has not gone at all far, and mathematics has profited very little from researches in symbolic logic. The cried reasons for this seem to be a lack of liaison between the logician and the mathematician-in-the-street. Symbolic logic is a very alarming mouthful for most mathematicians, and the logician are not very much interested in making it more palatable. It seems however that symbolic logic has a number of small lessons for the mathematician which may ne taught without it being necessary for him to learn very much of symbolic logic.

In particular it seems that symbolic logic will help the mathematicians to improve their notation and phraseology, which are at present exceedingly unsystematic, and constitute a definite handicap both to the would-be-learner and to the writer who is unable to express ideas because the necessary notation for expressing them is not widely known. By notation I do not of course refer to such trivial questions as whether pressure should be denoted by p to P, but deeper ones such as whether we should say « the function f(z) of z » or « the function f ».

It would not be advisable to let the reform take the form of a cast-iron logical system into which all the mathematics of the future are to be expressed. No democratic mathematical community would stand for such an idea, nor would it be desirable. Instead one must put forward a number of definite small suggestions for improvement, each backed up by good argument and examples. It should be possible for each suggestion to be adopted singly. Under these circumstances one may hope that some of the suggestions will be adapted in one quarter or anther, and that the use of all will spread.

Although it is not desirable to try and put mathematics into the straight-jacket of a logical system, it may be desirable to use such a system when investigating notation. One is likely to be taking typical phrases from mathematical text-books and analyzing their meaning. It is useful to have a logical system for expressing these meanings in a fairly unambiguous way. It may not greatly matter what system is used for this purpose, and it would be quite possible for different workers to use different systems.

To be specific I am inclined to suggest the following programme :

i) An extensive examination of current mathematical and physical and engineering books and papers with a view to listing all commonly used forms of notation.

ii) Examination of these notations to discover what they really means. This will usually involve statements of various implicit understandings as between writer and readers it may also include the equivalent of the notation in question in a standard notation.

ii) Laying down a code of minimum requirements for desirable notations. These requirements should be exceedingly mild. In my opinion the points which should be covered by this code should include the following

    a) Free and bound variables should be understood by all and properly respected.

    b) Some sort of provision should be made for falling in line with the theory of types. This assumes a Russian Weltenscheung, as applies I think to the majority of mathemaicians-in-the-street.

    c) The deduction theorem should be taken account of, I.e., it should be recognized that numerous forms of argument consist in one form or another of duplications of the deduction theorem. The deduction theorem should therefore be a well known as the rule for interaction by parts.

    d) Very clear statements of the fundamental nature of the symbols should be made. There should be no danger of mistaking a real variable for a function taking real values.

iv) New notations suggested by symbolic logic.

v) Examples of the development of comparatively elementary parts of mathematics in obedience to the new code and embodying the new notations. These examples should only incorporante the new notations in cases where great advantage results. The effects of the various independent reforms should be shown separately, so far as possible; to facilitate their independent adoptions.

参考文献：

https://turingarchive.kings.cam.ac.uk/unpublished-manuscripts-and-drafts-amtc/amt-c-12