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Writing a research paper in mathematics september 12, 1995 section 1: introduction: why bother? good mathematical writing, like good mathematics thinking, is a skill which must be practiced and developed for optimal performance. The purpose of this paper is to provide assistance for young mathematicians writing their first paper. The aim is not only to aid in the development of a well written paper, but also to help students begin to think about mathematical writing.

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The aim is not only to aid in the development of a well written paper, but also to help students begin to think about mathematical writing. I am greatly indebted to a wonderful booklet, how to write mathematics , which provided much of the substance of this essay. I will reference many direct quotations, especially from the section written by paul halmos, but i suspect that nearly everything idea in this paper has it origin in my reading of the booklet. It is available from the american mathematical society, and serious students of mathematical writing should consult this booklet themselves. Most of the other ideas originated in my own frustrations with bad mathematical writing. Although studying mathematics from bad mathematical writing is not the best way to learn good writing, it can provide excellent examples of procedures to be avoided. Thus, one activity of the active mathematical reader is to note the places at which a sample of written mathematics becomes unclear, and to avoid making the same mistakes his own writing. Mathematical communication, both written and spoken, is the filter through which your mathematical work is viewed.

Mathematical communication, both written and spoken, is the filter through which your mathematical work is viewed. If the creative aspect of mathematics is compared to the act of composing a piece of music, then the art of writing may be viewed as conducting a performance of that same piece. As a mathematician, you have the privilege of conducting a performance of your own composition! doing a good job of conducting is just as important to the listeners as composing a good piece. If you do mathematics purely for your own pleasure, then there is no reason to write about it. If you hope to share the beauty of the mathematics you have done, then it is not sufficient to simply write you must strive to write well. The next section will describe the difference between formal and informal parts of a paper, and give guidelines for each one. The essay will conclude with a section containing specific recommendations to consider as you write and rewrite the paper. Before you write: structuring the paper the purpose of nearly all writing is to communicate.

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Before you write: structuring the paper the purpose of nearly all writing is to communicate. In order to communicate well, you must consider both what you want to communicate, and to whom you hope to communicate it. This is no less true for mathematical writing than for any other form of writing. The primary goal of mathematical writing is to assert, using carefully constructed logical deductions, the truth of a mathematical statement. Careful mathematical readers do not assume that your work is well founded they must be convinced. However, convincing the reader of the simple truth of your work is not sufficient. When you write about your own mathematical research, you will have another goal, which includes these two you want your reader to appreciate the beauty of the mathematics you have done, and to understand its importance. If the whole of mathematics, or even the subfield in which you are working, is thought of as a large painting, then your research will necessarily constitute a relatively minuscule portion of the entire work.

If the whole of mathematics, or even the subfield in which you are working, is thought of as a large painting, then your research will necessarily constitute a relatively minuscule portion of the entire work. Its beauty is seen not only in the examination of the specific region which you have painted although this is important , but also by observing the way in which your own work 'fits' in the picture as a whole. These two goals to convince your reader of the truth of your deductions, and to allow your audience to see the beauty of your work in relation to the whole of mathematics will be critical as you develop the outline for your paper. At times you may think of yourself as a travel guide, leading the reader through territory charted only by you. A successful mathematical writer will lay out for her readers two logical maps, one which displays the connections between her own work and the wide world of mathematics, and another which reveals the internal logical structure of her own work. In order to advise your reader, you must first consider for yourself where your work is located on the map of mathematics. If your reader has visited nearby regions, then you would like to recall those experiences to his mind, so that he will be better able to understand what you have to add and to connect it to related mathematics.

If your reader has visited nearby regions, then you would like to recall those experiences to his mind, so that he will be better able to understand what you have to add and to connect it to related mathematics. Failure to address this very question will leave the reader feeling quite dissatisfied. To honestly and deliberately explain where your work fits into the big picture of mathematical research may require a great deal of humility. Do not fret! mathematics has been accumulating for thousands of years, based on the work of thousands or millions of practitioners.

Do not fret! mathematics has been accumulating for thousands of years, based on the work of thousands or millions of practitioners. It has been said that even the best mathematicians rarely have more than one really outstanding idea during their lifetimes. It would be truly surprising if yours were to come as a high school student! once you have considered the structure and relevance of your research, you are ready to outline your paper. The accepted format for research papers is much less rigidly defined for mathematics than for many other scientific fields. You have the latitude to develop the outline in a way which is appropriate for your work in particular. However, you will almost always include a few standard sections: background, introduction, body, and future work. The background will serve to orient your reader, providing the first idea of where you will be leading him.

The background will serve to orient your reader, providing the first idea of where you will be leading him. In the background, you will give the most explicit description of the history of your problem, although hints and references may occur elsewhere. The reader hopes to have certain questions answered in this section: why should he read this paper? what is the point of this paper? where did this problem come from? what was already known in this field? why did this author think this question was interesting? if he dislikes partial differential equations, for example, he should be warned early on that he will encounter them. If he isn't familiar with the first concepts of probability, then he should be warned in advance if your paper depends on that understanding. Remember at this point that although you may have spent hundreds of hours working on your problem, your reader wants to have all these questions answered clearly in a matter of minutes.

Remember at this point that although you may have spent hundreds of hours working on your problem, your reader wants to have all these questions answered clearly in a matter of minutes. In the second section of your paper, the introduction, you will begin to lead the reader into your work in particular, zooming in from the big picture towards your specific results. This is the place to introduce the definitions and lemmas which are standard in the field, but which your readers may not know. By the time you reach the final section, implications, you may be tired of your problem, but this section is critical to your readers.

By the time you reach the final section, implications, you may be tired of your problem, but this section is critical to your readers. You, as the world expert on the topic of your paper, are in a unique situation to direct future research in your field. S he will naturally have her/his own questions, but you, having worked on this paper, will know, better than your reader, which questions may be interesting, and which may not. If you were to continue working on this topic, what questions would you ask? also, for some papers, there may be important implications of your work. If you have worked on a mathematical model of a physical phenomenon, what are the consequences, in the physical world, of your mathematical work? these are the questions which your readers will hope to have answered in the final section of the paper. Formal and informal exposition once you have a basic outline for your paper, you should consider the formal or logical structure consisting of definitions, theorems, and proofs, and the complementary informal or introductory material consisting of motivations, analogies, examples, and metamathematical explanations. This division of the material should be conspicuously maintained in any mathematical presentation, because the nature of the subject requires above all else that the logical structure be clear. P.1 these two types of material work in parallel to enable your reader to understand your work both logically and cognitively which are often quite different how many of you believed that integrals could be calculated using antiderivatives before you could prove the fundamental theorem of calculus? since the formal structure does not depend on the informal, the author can write up the former in complete detail before adding any of the latter.

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P.1 these two types of material work in parallel to enable your reader to understand your work both logically and cognitively which are often quite different how many of you believed that integrals could be calculated using antiderivatives before you could prove the fundamental theorem of calculus? since the formal structure does not depend on the informal, the author can write up the former in complete detail before adding any of the latter. 2 thus, the next stage in the writing process may be to develop an outline of the logical structure of your paper. Several questions may help: to begin, what exactly have you proven? what are the lemmas your own or others on which these theorems stand. Which are the corollaries of these theorems? in deciding which results to call lemmas, which theorems, and which corollaries, ask yourself which are the central ideas. Which ones follow naturally from others, and which ones are the real work horses of the paper? the structure of writing requires that your hypotheses and deductions must conform to a linear order.