数学改革者

图灵在笔记本中的写作完全是关于数学的，对许多著名数学家的表示法实践进行了批判性的评论，包括Courant, Eisenhart, Hilbert, Peano, Titchmarsh, Weyl等。表示法对数学家来说非常重要，正如现代数理逻辑的创始人之一 Alfred North Whitehead 在其 1911 年的论文《The Symbolism of Mathematics》中所说，好的表示法“代表了对主题思想的分析，以及它们之间关系的几乎图形化的表示”。 Whitehead评论说：“通过减轻大脑所有不必要的工作，好的表示法可以让大脑专注于更高级的问题。”图灵在一份题为《数学表示法和用语改革》的战时打字稿中说，考虑不周的表示法是一种可能引发“麻烦”的“障碍”，甚至可能导致“一种最不幸的心理效应”，即“始终怀疑我们（数学）论证的合理性”的倾向。

根据Gandy的说法，这份打字稿是在 1944 年或 1945 年在汉斯洛普公园写的，为图灵的笔记本提供了背景。在打字稿中，图灵提出了所谓的“数学表示法改革”的“方案”。他说，基于数理逻辑，他的方案将“帮助数学家改进他们目前极其不系统的表示法和用语”。图灵的方案要求“广泛检查当前的数学……书籍和论文，以列出所有常用的表示法形式”，并“检查这些表示法以发现它们的真正含义”。他的“表示法注释”是这项广泛调查的一部分。

图灵提议的改革的关键是数理逻辑学家所说的“类型理论”。这反映了一个常识性的观点，即数字和香蕉是不同类型的实体：有些东西可以有意义地描述数字——例如，它有一个唯一的素数分解——但描述香蕉却没有意义。在强调类型理论对日常数学的重要性时，图灵一如既往地走在了时代的前面。如今，几乎每种计算机编程语言都包含基于类型的区分。

链接到真正的图灵

图灵从未表现出对权威的崇拜，尽管他所讨论的提出表示法的数学家非常杰出，但他在《表示法注释》中的语气却远非恭敬。他曾写道：“我不喜欢这个”，又在另一处写道：“这个太微妙了，定义起来不方便”。他的批评充斥着诸如“晦涩难懂”、“相当失败”、“丑陋”、“令人困惑”和“有点令人遗憾”之类的短语。在图灵的其他地方找不到这种坦率的直白；通过这些短语，我们或许可以感受到坐在剑桥书房听他讲话的感觉。这本破旧的笔记本向我们展示了朴素的图灵。

原文：

Mathematical reformer  

Turing’s own writings in the notebook are entirely mathematical, forming a critical commentary on the notational practices of a number of famous mathematicians, including Courant, Eisenhart, Hilbert, Peano, Titchmarsh, Weyl, and others. Notation is an important matter to mathematicians. As Alfred North Whitehead — one of the founders of modern mathematical logic — said in his 1911 essay “The Symbolism of Mathematics”, a good notation “represents an analysis of the ideas of the subject and an almost pictorial representation of their relations to each other”. “By relieving the brain of all unnecessary work”, Whitehead remarked, “a good notation sets it free to concentrate on more advanced problems”. In a wartime typescript titled “The Reform of Mathematical Notation and Phraseology” Turing said that an ill-considered notation was a “handicap” that could create “trouble”; it could even lead to “a most unfortunate psychological effect”, namely a tendency “to suspect the soundness of our [mathematical] arguments all the time”.

This typescript, which according to Gandy was written at Hanslope Park in 1944 or 1945, provides a context for Turing’s notebook. In the typescript Turing proposed what he called a “programme” for “the reform of mathematical notation”. Based on mathematical logic, his programme would, he said, “help the mathematicians to improve their notations and phraseology, which are at present exceedingly unsystematic”. Turing’s programme called for “An extensive examination of current mathematical … books and papers with a view to listing all commonly used forms of notation”, together with an “[e]xamination of these notations to discover what they really mean”. His “Notes on Notations” formed part of this extensive investigation.

Key to Turing’s proposed reforms was what mathematical logicians call the “theory of types”. This reflects the commonsensical idea that numbers and bananas, for example, are entities of different types: there are things which makes sense to say about a number — e.g. that it has a unique prime factorization — that cannot meaningfully be said of a banana. In emphasizing the importance of type theory for day-to-day mathematics, Turing was as usual ahead of his time. Today, virtually every computer programming language incorporates type-based distinctions.

Link to the real Turing

Turing never displayed much respect for status and — despite the eminence of the mathematicians whose notations he was discussing — his tone in “Notes on Notations” is far from deferential. “I don’t like this” he wrote at one point, and at another “this is too subtle and makes an inconvenient definition”. His criticisms bristle with phrases like “there is obscurity”, “rather abortive”, “ugly”, “confusing”, and “somewhat to be deplored”. There is nothing quite like this blunt candor to be found elsewhere in Turing’s writings; and with these phrases we perhaps get a sense of what it would have been like to sit in his Cambridge study listening to him. This scruffy notebook gives us the plain unvarnished Turing.

参考文献：

https://blog.oup.com/2017/02/alan-turing-lost-notebook/

Alan Turing’s lost notebook

JACK COPELAND