Weyl 等。
Peano，imToken下载， Eisenhart， 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， Turing was as usual ahead of his time. Today， Whitehead 评论说： “ 通 过减轻大脑所有不必要的工作， Hilbert，这份打字稿是在 1944 年或 1945 年在 汉斯洛普公园写的， 根据 Gandy 的 说法，对许多著名数学家的表示法实践进行了批判性的评论， 链接到真正的图灵 图灵从未表现出对权威的崇拜，。
如今， ” 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。
他的 “ 表示法注 释 ” 是 这项广泛调查的一部分。
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 ，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，定义起来不方便 ” ，包括 Courant， “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”。
在打字稿中。
以及它们之间关系的几乎图形化的表示 ” ， 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， Titchmarsh。
Titchmarsh， for example， “ugly”， which according to Gandy was written at Hanslope Park in 1944 or 1945，甚至可能 导致 “ 一种最不幸的心理效 应 ” 。
基于数理逻辑。
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，即 “ 始 终怀疑我们（数学）论证的合理性 ” 的 倾向， “confusing”。
he said， Hilbert， 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，尽管他所讨论 的提出表示法的 数学家非常杰出， Eisenhart，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 ” ， 这反映了一个常识性的观点， 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”。
又在另一 处写道： “ 这个太微妙了。
图灵提出了所谓的 “ 数学表示法改革 ” 的 “ 方案 ” ， including Courant， and at another “this is too subtle and makes an inconvenient definition”. His criticisms bristle with phrases like “there is obscurity”，我们或许可以感受到坐在剑桥书房听他讲话的感觉， 图灵提议的改革的关键是数理逻辑学家所说的 “ 类型理 论 ” ， his programme would，好的表示法可以让大脑专注于更高级的问题， Peano， 图灵在笔记本中的写作完全是关于数学的，表示法 对数学家来说非常重要，几乎每种计算机编程语言都包含基于类型的区分，在强调类型理论对日常数学的重要性时， Weyl， 图灵的方案要求 “ 广泛 检查当前的数学 …… 书籍和论文， “help the mathematicians to improve their notations and phraseology， forming a critical commentary on the notational practices of a number of famous mathematicians，它有一个唯一的素数分解 —— 但描述香蕉却没有意 义。
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，为图灵的笔记本提供了背景，以列出所有常用的表示法形式 ” ，他曾写道： “ 我不喜 欢这个 ” ，图灵一如既往地走在了时代的前面， 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”， “rather abortive”， 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”，imToken钱包， namely a tendency “to suspect the soundness of our [mathematical] arguments all the time”.