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**How (not) to write mathematics**

Some tips from the mathematician John Milne:

*If you write clearly, then your readers may understand your mathematics and conclude that it isn't profound. Worse, a referee may find your errors. Here are some tips for avoiding these awful possibilities.*

*1. Never explain why you need all those weird conditions, or what they mean. For example, simply begin your paper with two pages of notations and conditions without explaining that they mean that the varieties you are considering have zero-dimensional boundary. In fact, never explain what you are doing, or why you are doing it. The best-written paper is one in which the reader will not discover what you have proved until he has read the whole paper, if then.*

*2. Refer to another obscure paper for all the basic (nonstandard) definitions you use, or never explain them at all. This almost guarantees that no one will understand what you are talking about (and makes it easier to use the next tip). In particular, never explain your sign conventions --- if you do, someone may be able to prove that your signs are wrong.*

*3. When having difficulties proving a theorem, try the method of "variation of definition"---this involves implicitly using more that one definition for a term in the course of a single proof.*

*4. Use c, a, b respectively to denote elements of sets A, B, C.*

*5. When using a result in a proof, don't state the result or give a reference. In fact, try to conceal that you are even making use of a nontrivial result.*

*6. If, in a moment of weakness, you do refer to a paper or book for a result, never say where in the paper or book the result can be found. In addition to making it difficult for the reader to find the result, this makes it almost impossible for anyone to prove that the result isn't actually there. Alternatively, instead of referring to the correct paper for a result, refer to an earlier paper, which contains only a weaker result.*

*7. Especially in long articles or books, number your theorems, propositions, corollaries, definitions, remarks, etc. separately. That way, no reader will have the patience to track down your internal references.*

*8. Write A==>B==>C==>D when you mean (A==>B)==>(C==>D), or (A==>(B==>C))==>D, or.... Similarly, write "If A, B, C" when you mean "If A, then B and C" or "If A and B, then C", or .... Also, always muddle your quantifiers.*

*9. Begin and end sentences with symbols wherever possible. Since periods are almost invisible (and may be mistaken for a mathematical symbol), most readers won't even notice that you've started a new sentence. Also, where possible, attach superscripts signalling footnotes to mathematical symbols rather than words.*

*10. Write "so that" when you mean "such that" and "which" when you mean "that". Always prefer the ambiguous expression to the unambiguous and the imprecise to the precise. It is the readers task to determine what you mean; it is not yours to express it.*

*11. If all else fails, write in German.*

These helpful tips are from his webpage:

http://www.jmilne.org/math/tips.html

He has some footnotes, including this for item 11:

*The point is that most mathematicians find it very difficult to read mathematics in German, and so, by writing in German, you can ensure that your work is inaccessible to most mathematicians, even though, of course, German is a perfectly good language for expressing mathematics.*

Hmm. He could have chosen Basque, but he chose German.

I thank Nina Otter for pointing out this article.

View 34 previous comments

- +Jim Stuttard No, not the Chomsky hierarchy, I was referring to the fact that some programming languages allow or worse, encourage, very terse expressions of computation. The question is one of engineering - having the code work is only part of the story, it needs to be readable and maintainable. So for example Perl is incredibly expressive, allows for a lot of contextualized syntactical constructs, but it's renown for leading to unreadable code.Aug 22, 2017
- Didn't think so :). My answer is that, ~quoting SPJ, type signatures implement the most comprehensible and reasonable (pun) formal method. I've always thought Perl was some flaky C for regexes ;).Aug 22, 2017
- +John Baez said: "there's still a niche for an introduction to mathematics based on category theory. Wikipedia articles often do include the categorical point of view and other nice insights, but it's hit-and-miss."

totally agree! but the sociological problem is that the young, who have more time and disposition are always more interested in doing new research (and who can blame them?). what I wanted was not so much great exposition of basics (this would be fantastic, but crowdsourcing would not work for it, I think) but a "cheat sheet" for category theory using entries like in a dictionary. I am implementing (with friends) a Portuguese WordNet: what I wanted was "wordnet for category theory". so instead of an entry describing what "soccer" is, you'd have entries for functors, monads, modules, etc. with source code in latex, so when you're writing your new cool research, you can simply either copy the concepts you need or send the reader to the link that explains the concept they might not know. or might have forgotten. so that then you can concentrate in explaining why you're putting concepts together this or that way, what you want to achieve with your constructions, why different ways have not worked, whatever... I can see in GitHub many attempts at this, but sure right now I'm too busy to do much. thanks for the reminder about the meeting too!Aug 22, 2017 - +Valeria de Paiva That would be a great project! Not only for category theory, but all area of mathematics. A lot of words are used with different senses, depending on context. Mathematical language is at the end a social construction just like ordinary language so the same approach can be taken to manage the lexicon in a semi-formal way as WordNet, which is quite successful and has aged remarkably well. If you do something with that idea, I'd certainly love to hear about it!Aug 22, 2017
- Well said.Aug 22, 2017
- #FakeMaths Here's something to try for size along the bogus proof trail. When I first came across it I thought I should post news of it as in future I'd guess every maths paper would have to be machine checked for whether or not it's correct or bogus; like a very inadequate amount of current software is

* thatsmathematics.com - Mathgen: Randomly generated math papersAug 27, 2017

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