The article is intended to provide practical help for authors of mathematical papers. It is written mainly for non-English speaking writers but should prove useful even to native speakers of English who are beginning their mathematical writing and may not yet have developed a template of the structure of mathematical discourse. The article is oriented mainly to research mathematics but applies to almost all mathematics writing, except more elementary texts. There is no intention whatsoever to impose any uniformity of mathematical style. Quite the contrary, the aim is to encourage prospective authors to write structurally correct manuscripts as expressively and flexibly as possible, but without compromising certain basic and universal rules.
Mathematics writing is different from ordinary writing and harder in addition to all the requirement of ordinary good writing, there are additional constraints and conventions in mathematics. An additional constraint is that mathematics follows much more demanding rules of logic than ordinary discourse, and your writing must follow and display this logic. However, this does not mean that there is only one right way to present a mathematical argument. Some of the additional conventions are those for defining new concepts and those for organizing the material through theorems and examples.
Communicating your ideas and knowledge through writing and other media is a very important skill to mathematicians. Mathematics often includes concepts and ideas that you cannot easily express using equations and formulae. Mathematicians must write their ideas down to add to the body of mathematical knowledge. They must communicate their thought processes to non-mathematicians, often their employers. The ability to write clearly is as important a skill to mathematicians as solving equations.
Writing mathematics is not the same as showing your work. You do not write papers to demonstrate that you have done your work, but rather to demonstrate how well you understand the ideas and concepts. A list of calculations without any context or explanation demonstrates that you have spent time doing computations, but it omits ideas. It contains no mathematics. Writing good, clear mathematical explanations will also help you improve your knowledge and understanding of the mathematical ideas and concepts you encounter. The act of writing the explanation will force you to think more carefully about what you are doing. This means clear, carefully-written mathematics will more likely be correct, and the process will help you learn and retain the concepts. This paper discusses some of the basic ideas involved in writing a mathematical paper. For more information, you can consult reference books such as Krantz [1997], Higham [1998], and Knuth [1989]. You might also consult Turner [1998] for a brief introduction to proofs [1].
A good mathematical paper has a fairly standard format. Most short papers are divided up into about a half dozen sections, which are numbered and titled. Most papers have an abstract, an introduction, a number of sections of discussion, and a list of references, but no formal table of contents or index. On occasion, papers have appendices, which give special detailed information or provide necessary general background to secondary audiences. In some fields, papers routinely have a conclusion. This section is not present simply to balance the introduction and to close the paper. Rather, the conclusion discusses the results from an overall perspective, brings together the loose ends, and makes recommendations for further research. In mathematics, these issues are almost always treated in the introduction, where they reach more readers; so a conclusion is rare [2].
A well-organized paper is easier to read than a disorganized one. Fortunately, there are some standard ways to order a mathematics essay. A math paper should be punctuated, spelled, and organized clearly, just as in papers for disciplines besides math.
The title of your paper should be informative. It should be informative without being too long. Choosing a good title in a mathematics paper is not so easy. Often a paper hinges on a concept that is defined only within the paper itself, so using the name of that concept in the title will convey no meaning at all. Generally, titles should have no more than ten words, although, admittedly, I have not followed this advice on several occasions.
The abstract is the most important section. First it identifies the subject; it repeats words and phrases from the title to corroborate a reader's first impression, and it gives details that didn't fit into the title. Then it lays out the central issues, and summarizes the discussion to come. The abstract includes no general background material. It is essentially a table of contents in a paragraph of prose. The abstract allows readers to decide quickly about reading on. While many will decide to stop there, the potentially interested will continue. The goal is not to entice all, but to inform the interested efficiently. Remember, readers are busy. They have to decide quickly whether your paper is worth their time. They have to decide whether the subject matter is of interest to them, and whether the presentation will bog them down. A well-written abstract will increase the readership.
Some writers list key words supplied by the author, usually after abstract. The number of key words is usually ten or less. Since the key words may be used in computer searches, you should try to anticipate words for which reader might search and make them specific enough to give a good indication of paper’s content.
The Introduction is the most important part of your paper. Although some mathematicians advise that the Introduction be written last, I advocate that the Introduction be written first. I find that writing the Introduction first helps me to organize my thoughts. However, I return to the Introduction many times while writing the paper, and after I finish the paper, I will read and revise the Introduction several times.
Get to the purpose of your paper as soon as possible. Don’t begin with a pile of notation. Even at the risk of being less technical, inform readers of the purpose of your paper as soon as you can. Readers want to know as soon as possible if they are interested in reading your paper or not. If you don’t immediately bring readers to the objective of your paper, you will lose readers who might be interested in your work but, being pressed for time, will move on to other papers or matters because they do not want to read further in your paper.
To state your main results precisely, considerable notation and terminology may need to be introduced. At this point, you do not want the reader to be bogged down with technical definitions and notation, and so it is therefore preferable to informally describe your results in such instances. Try to be as informative and precise as possible without drifting off into too much technical jargon.
Why are you writing this paper? The logic in climbing a mountain, «because it is there», does not apply to writing and publishing a paper. Just because you can prove a theorem does not mean that you should publish it and its proof. For example, the theorem may be of interest to no one else, the proof may involve no new ideas, or, despite a proof not being in the literature, the theorem can be easily proved by many, in particular, students.
Put your paper in an historical context. Indicate what you have done in relation to what others have done. Briefly survey the pertinent results of others to your work. On the other hand, as you place your results in an historical perspective, do not name drop. Ramanujan, J.P. Serre, and P. Deligne are common names that writers like to drop to enhance their own particular work. Readers will recognize that you are referring to these famous mathematicians in an attempt to bring attention to your work, which likely may not receive any notice otherwise [3].
Write a good introduction. Most people who read a mathematics paper will only read the introduction and skim the theorems.
The paper body discusses the various aspects of the subject individually. First, present the material in small digestible portions. Second, beware of jumping haphazardly from one detail to another, and of illogically making some details specific and others generic. Third, if possible, follow a sequential path through the subject. If such a path simply doesn't exist, then break the subject down into logical units, and present them in the order most conducive to understanding. If the units are independent, then order them according to their importance to the primary audience.
After stating what the problem is, we usually then state the answer, even before we show how we got it. Sometimes we even state the answer right along with the problem. It's uncommon, although not so uncommon as to be exceptional, to read a math paper in which the answer is left for the very end. Explaining the solution and then the answer is usually reserved for cases where the solution technique is even more interesting than the answer, or when the writers want to leave the readers in suspense. But if the solution is messy or boring, then it's typically best to hook the readers with the answer before they get bogged down in details.
Math is difficult enough that the writing around it should be simple. «Beautiful» math papers are the ones that are the easiest to read: clear explanations, uncluttered expositions on the page, well-organized presentation. For that reason, mathematical writing is not a creative endeavor the same way that, say, poetry is:
you shouldn't be spending a lot of time looking for the perfect word, but rather should be developing the clearest exposition. Unlike humanities students, mathematicians don't have to worry about over-using 'trite' phrases in mathematics. In fact, at the end of this article is a list of trite but useful advice that you may want to use in your papers, either in this class or in the future.
This guide, together with the advice, should serve as a reference while you write and will also be referred to when I comment on the drafts of the problems for your writing paper. If you can master these basic areas, your writing may not be spectacular, but it should be clear and easy to read which is the goal of mathematical writing, after all.
- First and foremost, even if your native language isn’t English, avoid poor or careless linguistic presentation. Be sensitive to the language, its idiom and
- Presentation shouldn’t be overly flowery or informal: this is not a paper in literary criticism and you are judged on your ideas and their clear presentation, not on linguistic virtuosity. The language should be clear, unambiguous and
- Avoid the sort of lifeless formalism and dry linguistic
- Occasional flash of lighthearted humor or informal lingo is fine. Mathematical style is not
- Be verbose enough to be clear – yet concise enough to privilege your core mathematical argument over its
- Not using a spell-checker is major folly. Relying totally on a spell-checker is carelessness: no spellchecker will distinguish between «some» and «same» [5].
The book of Knuth [1] contains a lot of good advice from Don Knuth’s technical writing class at Stanford University. Trzeciak [3] is also a good reference text. I’ve collected some of their best advice above, though I added a few thoughts of my own too.
As you write a mathematics paper remember that, unlike you, the reader has not been thinking intensely about the material for an extended period of time. Therefore, provide the reader with references, include useful comments, and give additional explication so that someone unfamiliar with the work can follow it.
Write a paper that you yourself would want to read. Make it accessible. Bear in mind that the referee for your paper will be a busy person who has no patience for a tract that he cannot fathom. Lay out the material so that it is rapidly apparent what your main result is, what the background for that result is, and how you are going to go about proving it. If the proof is long and complicated, then break it up into digestible pieces. Tell the reader what is going to happen before it happens. Tell the reader what has just happened before you go on to the next step. At the end of a long argument, summarize it.
Writing mathematics is not the easiest thing to do. Writing mathematics is a skill which takes practice and experience to learn. There are many resources here at Purdue Calumet which is available to you to help you with your mathematical writing. Among these are the Math Lab and the Writing Lab.
If you have not written mathematics much before, it may feel frustrating at first. But learning to write mathematics can only be done by actually doing it. It may be hard at first, but it will get easier with time and you will get better at it. Do not get discouraged! Being able to write mathematics well is a good skill to learn, and one which you will keep for a lifetime.
References
- Knuth, D.E., Larrabee, T., Roberts, P.M. (1989). Mathematical Writing. Washington: Publishing House of Mathematical Society of
- Sosinskii, B. (2004). Kak napisat matematicheskuiu statiu po-anhliiski [How to write a mathematical article in English].Moscow: Factorial Press [in Russian].
- Trzeciak, J. (1995). Writing Mathematical Papers in English. Warszawa: Publishing House of European Mathematical Society Publishing
- Savchenko, A.A. (2015). Metodicheskaia razrabotka dlia aspirantov [Methodical development for graduate students]. Moscow: Izdatelstvo MHU [in Russian].
- Baber, R.L. (2011). The language of mathematics. Utilizing math in practice. New Jersey: Publishing House «John Wiley & Sons».