The theorem-proof format, definitions, and logic fall under this style. Informal Exposition Informal exposition complements the formal exposition by providing the reasoning behind the theorems and proofs.
Figures, proofs, equations, and mathematical sentences do not necessarily speak for themselves within a mathematics research paper. Authors will need to demonstrate why their hypotheses and deductions are valid and how they came to prove this. Analogies and examples fall under this style. Conventions of Mathematics Clarity is essential for writing an effective mathematics research paper. This means adhering to strong rules of logic, clear definitions, theorems and equations that are physically set apart from the surrounding text, and using math symbols and notation following the conventions of mathematical language.
Each area incorporates detailed guidelines to assist the authors. Logic Logic is the framework upon which every good mathematics research paper is built. Each theorem or equation must flow logically. It is more effective to include this within the Introduction section of the paper rather than having a stand-alone section of definitions.
Theorems and Equations Theorems and equations should be physically separated from the surrounding text. They will be used as reference points throughout, so they should have a well-defined beginning and end. Math Symbols and Notations Math symbols and notations are standardized within the mathematics literature. Deviation from these standards will cause confusion amongst readers.
Therefore, the author should adhere to the guidelines for equations, units, and mathematical notation, available from various resources. Protocols for mathematics writing get very specific — fonts, punctuation, examples, footnotes, sentences, paragraphs, and the title, all have detailed constraints and conventions applied to their usage. The American Mathematical Society is a good resource for additional guidelines.
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.
This essay will begin with general ideas about mathematical writing. The purpose is to help the student develop an outline for the paper. The next section will describe the difference between "formal" and "informal" parts of a paper, and give guidelines for each one. Section four will discuss the writing of an individual proof. The essay will conclude with a section containing specific recommendations to consider as you write and rewrite the paper.
Section 2. 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. This is your first goal in mathematical writing. 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. 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.
Asking several questions may help you discern the shape and location of your work: Does your result strengthen a previous result by giving a more precise characterization of something?
Have you proved a stronger result of an old theorem by weakening the hypotheses or by strengthening the conclusions? Have you proven the equivalence of two definitions?
Is it a classification theorem of structures which were previously defined but not understood? Does is connect two previously unrelated aspects of mathematics? Does it apply a new method to an old problem? Does it provide a new proof for an old theorem? Is it a special case of a larger question? It is necessary that you explicitly consider this question of placement in the structure of mathematics, because it will linger in your readers' minds until you answer it. Failure to address this very question will leave the reader feeling quite dissatisfied.
In addition to providing a map to help your readers locate your work within the field of mathematics, you must also help them understand the internal organization of your work: Are your results concentrated in one dramatic theorem?
Or do you have several theorems which are related, but equally significant? Have you found important counterexamples? Is your research purely theoretical mathematics, in the theorem-proof sense, or does your research involve several different types of activity, for example, modeling a problem on the computer, proving a theorem, and then doing physical experiments related to your work?
Is your work a clear although small step toward the solution of a classic problem, or is it a new problem? Since your reader does not know what you will be proving until after he has read your paper, advising him beforehand about what he will read, just as the travel agent prepares his customer, will allow him to enjoy the trip more, and to understand more of the things you lead him to.
To honestly and deliberately explain where your work fits into the big picture of mathematical research may require a great deal of humility. You will likely despair that your accomplishments seem rather small.
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.
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.
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. The body, which will be made up of several sections, contains most of your work. 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. A reader who likes your paper may want to continue work in your field. 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. You should take care not to disappoint them! Section 3. 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.
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. However, few research papers actually have a linear structure, in which lemmas become more and more complicated, one on top of another, until one theorem is proven, followed by a sequence of increasingly complex corollaries. On the contrary, most proofs could be modeled with very complicated graphs, in which several basic hypotheses combine with a few well known theorems in a complex way.
There may be several seemingly independent lines of reasoning which converge at the final step.Writing to Effectively Write a Mathematics Research Paper Last updated Sep paper, Mathematics research papers are different from standard academic research papers in important paper, but writing so different that they require an entirely separate set of guidelines. Mathematical papers rely heavily on living life on the edge essay writing and a specific type of language, including symbols math regimented notation. There are two basic structures of mathematical research papers: formal and informal exposition. Structure and Style Formal Exposition The author must start with an outline that develops the math structure of the paper.
The formal structure contains the formal definitions, theorem-proof format, and rigorous logic which is the language of 'pure' mathematics. As for "if Once you have considered the structure and relevance of your research, you are ready to outline your paper. Is it a classification theorem of structures which were previously defined but not understood?
Good Writing and Logical Constructions Regardless of the document preparation system selected, publication of a mathematics paper is similar to the publication of any academic research in that it requires good writing. Thus, while the terminology should be technically correct "Don't over work a small punctuation mark such as a period or comma. They will be used as reference points throughout, so they should have a well-defined beginning and end. Have you found important counterexamples? If you do mathematics purely for your own pleasure, then there is no reason to write about it. The structure should be easily discernible by headings and punctuation.
Each hypothesis and deduction should flow in an orderly and linear fashion using formal definitions and notation. Both LaTeX and Wolfram have expert typesetting capabilities to assist authors in writing.
You, as the world expert on the topic of your paper, are in a unique situation to direct future research in your field.
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. Informal Exposition Informal exposition complements the formal exposition by providing the reasoning behind the theorems and proofs. Bad notation, on the other hand, is disastrous and may deter the reader from even reading your paper.
This means adhering to strong rules of logic, clear definitions, theorems and equations that are physically set apart from the surrounding text, and using math symbols and notation following the conventions of mathematical language. The body, which will be made up of several sections, contains most of your work. Or do you have several theorems which are related, but equally significant? Both LaTeX and Wolfram have expert typesetting capabilities to assist authors in writing. Example: "On a compact space every real-valued continuous function f is bounded.
The exact way in which this will proceed depends, of course, on the specific situation. It produces professional-looking documents and authentically represents mathematical language. Of course, all the necessary hypotheses must be included.
Bad notation, on the other hand, is disastrous and may deter the reader from even reading your paper. There are two basic structures of mathematical research papers: formal and informal exposition.
What are the lemmas your own or others on which these theorems stand. It could be "If p and q then r", or "In the presence of p, the hypothesis q implies the conclusion r", or many other versions. The body, which will be made up of several sections, contains most of your work. Such a proof is easy to write.