MacTutor
S B Stechkin, How to write papers
These notes are based on a lecture given by Sergei Borisovich Stechkin at the request of the participants of his seminar at Moscow State University in April 1993. The text was prepared for publication by S B Stechkin's students A R Alimov and M I Karlov based on their notes of this lecture; they also wrote the notes.
1. Selection (what to include in the paper)
The paper is not a garbage can!Before you sit down to write an article, you need to clearly imagine what you are going to include in it. To do this, you first need to clearly understand what problem you are considering. You need a clear statement of this problem. Only after the problem is completely solved, you can write the article. The more the work is cleaned of garbage (even related to the matter), the better. The work is not a garbage can into which you can throw whatever you want. Include only the most necessary things in your work.
Don't be greedy.You shouldn't be a maniac, suffering from a delusion that everything unrelated to the matter that you haven't included in your work will be stolen from you right now. Don't be afraid. It's not scary. Don't stray from the main topic of your work, don't try to shove into it all the side issues and subtasks that you've solved. Don't be greedy. Leave something for your future (or present) children for milk. [Note. S B Stechkin called his students children, the students of his students grandchildren, etc.]
The best is the enemy of the good.Improving or generalising the results is an endless process. You need to be able to stop somewhere. Make a firm person's decision - what to include in the article, and then don't change it. The worst thing is to finish the work on the go, since this always leads to swearing and lying.
2. Title
The title is the subject, not the method.The title should be as informative as possible. Remember that most readers will only look at the title of your paper. For example, it could be the statement of the main theorem. The first thing a title says is the subject, not the method. People are interested in what you solve, and only secondarily in how you do it. So, for example, a bad title for an article would be: "Probabilistic Methods in Number Theory" - it says nothing about the subject. Try to give a classic title. Don't write: "On Chebyshev Sets" etc., since "On" is a derogatory sign showing that you haven't done much. A particularly derogatory sign is "On the Question of...".
3. Introduction
Everything you need and only what you need.Those who will read your work are busy people, they have no time. They will only read the introduction. Therefore, in the introduction, please give a precise formulation of the main result (except for the case when there is no main result). It is necessary, without using unnecessary designations, to formulate the task, the results of the work and, possibly, go over the methods.
Disrespect for ancestors is a sign of immorality.It is bad if immediately after explaining the terminology they write: "I proved such and such a theorem." It is necessary to say what problem is being considered.
Remember that mathematics is divided into three parts:
1) what was done before me;
2) what I did;
3) what will happen after me.
You must know the history of the problem - who did what before you. Before formulating your theorems, describe the results of your predecessors and indicate whose methods you used. Disrespect for ancestors is the first sign of immorality.
4. Formulation of the Theorem
The truth, the whole truth and the whole truth.The formulations of theorems should be worked on separately. The main thing you need to remember is: you need to write the truth, only the truth and the whole truth and not be ashamed to show your stupidity.
As a rule, after solving a problem, it is not immediately possible to find the correct formulation of what you have actually proved. Do not offend yourself - formulate the theorem in the most general form, otherwise someone will do it for you. At the same time, try to make the formulation shorter, "comb it", give all the notations and definitions before it. In the formulations and proofs of theorems there should not even be temporary bears. [Note. S B Stechkin called symbols that appear in the text and were not previously defined "bears". He said: "For example, you write "" and do not explain what it is - a number, a vector, an operator, or something else. If you do not care, then let's consider "" a bear."]
5. Proof
Write backwards.The proof of a theorem and the process of its presentation are reverse processes. This means that it is better to start explaining the entire course of your reasoning that took place during the proof from the end and then write it backwards. This is the main thing. Break the entire chain of proof into pieces. In each such piece: if you started some reasoning - you must definitely finish it.
The beginning of a new paragraph within the proof should always mean a new thought. But do not abuse this rule - it is not good if almost every sentence begins with a new paragraph. On the other hand, the text of the proof should not be one continuous paragraph - such a text is difficult to read.
It makes sense to present some parts of the proof as separate lemmas in two cases:
1) if these are technical things, they can be taken out so as not to interrupt the coherence of the proof;
2) if they can be of independent interest.
However, the entire proof should not be shoved into lemmas. It is not good to write, for example, like this: "the theorem is a simple consequence of lemmas 1-10." There should be something left in the proof of the theorem itself.
Try not to prove by contradiction.As is well known, a young aspiring mathematician proves every theorem by contradiction, or by very contradiction. If you can give a direct proof, then it is better not to prove by contradiction. Proving theorems by contradiction, when it can be avoided, is considered bad form.
About technology.The length of the proof depends on the order of the formulas. Example: we need to prove that . The first path: . We deduce: . Next, . From this and the previous we get . The shorter path: , which means .
The word "obviously" is a breeding ground for bedbugs.[Note. Sergei Borisovich spoke about bedbugs, meaning inaccuracies in a proof.]
It is important to understand that replacing reasoning (even simple ones) with a construction like "it is obvious that" is unpleasant for anyone reading your work, because they want to understand your train of thought, but instead you tell them: "You are an idiot. It is obvious!" All this irritates the reader, so instead of "obvious" it is better to write "understandable", "clearly", etc. However, the main trouble associated with words like "obvious" is the bugs in your reasoning: it is not at all necessary that the train of thought "obvious" to you will be obvious to you in a month. It may even be incorrect. Therefore, first carefully check this train of thought for yourself, and sometimes you may not include it in the proof. Often a person writes "obvious" when he finds it difficult to explain something. Avoid words like "obvious"!
6. Designations
It's bad when .Before writing your paper, think carefully about your notation system. When solving the problem, you may have used some of your own notations. Now, when writing, you should use completely different notations - ones that are understandable to everyone and that will make reading the text as easy as possible. Each letter in the notation should be informative. The reader, even before you define it, should have a rough idea of what it might mean: a number, a set, etc. Use standard, generally accepted notations. For example, people are used to the fact that is a small number and that for any ε there is a δ, and not vice versa. In addition, it will be unpleasant if it turns out that your lower case is larger than your upper case: .
7. Links
Make work independent.If you are not referring to some classical (in the given field) result, you should formulate it completely. The worst thing here is to refer to nothing (or to a hard-to-find source) without formulating what you are referring to there. Do not be afraid to give an obscure definition, especially if it takes up less space than the reference to it. Try, if possible, to make the work independent, so that it can be read with almost no need to look at other sources.
8. End of work
The ending should be major.Nina Karlovna Bari said that the end of any work should be major. [Note. According to S B Stechkin, N K Bari was one of those who taught him how to write papers.] That is, for example, like this: "now we can see the asymptotics", and not like: "nothing is really visible in this problem yet".
9. The process of working on a problem
[Note. This is a digression made by S B Stechkin during the lecture at the request of the audience.]
First - undress, then - look.You need to start studying a problem by understanding what you are actually solving, by discarding some conditions and connections - by "undressing" the problem. This is not easy, but it is very important. After looking at the "undressed" problem, you usually immediately understand how to work with it further.
I thought and thought all day long - no idea whatsoever.This is a normal process of working on a problem. After you have started solving a problem, after a certain period of time (the moment of satiety) it becomes disgusting to you. [Note. Sergei Borisovich said that all people have their own moment of saturation with a task; for him, it is currently one month.] Then you should stop thinking about it, write down all the results obtained by this time (even the most insignificant ones) and abandon this problem for a while and start thinking about another one. After some time, return to solving the first problem again. And do not forget to check whether when you wrote "obviously that" it is still obvious to you.
Finally, knowledge of literature is half the solution to the problem.
10. Miscellaneous
A mathematical work is not a work of art. It should be written in standard, generally accepted, plain language. It should be written as it is supposed to be written today, it should follow fashion.
Look over your work again and cross out all filler words, especially in the proof.
Remember that writing an article is a serious, responsible, and nasty business, and it will take you much more time than solving the problem itself.
I hope now you understand how to write papers!
[Note. S B Stechkin often ended his lectures with questions like: "Is it clear now?" And after the standard answer - "yes, I understand" - he related the following dialogue between a father and son: "Do you understand, Vanyushka?" - "I understand, father." - "And what about it, Vanyushka?" - "And nothing, father!"]
Last Updated June 2025