Structure and Style Formal Exposition The author must start with an outline that develops the logical structure of the paper. Each hypothesis and deduction should flow in an orderly and linear fashion using formal definitions and notation. The author should not repeat a proof or substitute words or phrases that differ from the definitions already established within the paper.
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. 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. It goes without saying that any assertion should follow the lemmas and theorems on which it depends. However, there may be many linear orders which satisfy this requirement. In view of this difficulty, it is your responsibility to, first, understand this structure, and, second, to arrange the necessarily linear structure of your writing to reflect the structure of the work as well as possible.
The exact way in which this will proceed depends, of course, on the specific situation. One technique to assist you in revealing the complex logical structure of your paper is a proper naming of results. By naming your results appropriately lemmas as underpinnings, theorems as the real substance, and corollaries as the finishing work , you will create a certain sense of parallelness among your lemmas, and help your reader to appreciate, without having struggled through the research with you, which are the really critical ideas, and which they can skim through more quickly.
Another technique for developing a concise logical outline stems from a warning by Paul Halmos, in HTWM, never to repeat a proof: If several steps in the proof of Theorem 2 bear a very close resemblance to parts of the proof of Theorem 1, that's a signal that something may be less than completely understood. Other symptoms of the same disease are: 'by the same technique or method, or device, or trick as in the proof of Theorem When that happens the chances are very good that there is a lemma that is worth finding, formulating, and proving, a lemma from which both Theorem 1 and Theorem 2 are more easily and more clearly deduced.
Now that we have discussed the formal structure, we turn to the informal structure. The formal structure contains the formal definitions, theorem-proof format, and rigorous logic which is the language of 'pure' mathematics. The informal structure complements the formal and runs in parallel. It uses less rigorous, but no less accurate! For although mathematicians write in the language of logic, very few actually think in the language of logic although we do think logically , and so to understand your work, they will be immensely aided by subtle demonstration of why something is true, and how you came to prove such a theorem.
Outlining, before you write, what you hope to communicate in these informal sections will, most likely, lead to more effective communication. Before you begin to write, you must also consider notation. The selection of notation is a critical part of writing a research paper. In effect, you are inventing a language which your readers must learn in order to understand your paper.
Good notation firstly allows the reader to forget that he is learning a new language, and secondly provides a framework in which the essentials of your proof are clearly understood. Bad notation, on the other hand, is disastrous and may deter the reader from even reading your paper. In most cases, it is wise to follow convention. Using epsilon for a prime integer, or x f for a function, is certainly possible, but almost never a good idea.
Section 4: Writing a Proof The first step in writing a good proof comes with the statement of the theorem. A well-worded theorem will make writing the proof much easier. The statement of the theorem should, first of all, contain exactly the right hypotheses.
Of course, all the necessary hypotheses must be included. On the other hand, extraneous assumptions will simply distract from the point of the theorem, and should be eliminated when possible. When writing a proof, as when writing an entire paper, you must put down, in a linear order, a set of hypotheses and deductions which are probably not linear in form.
I suggest that, before you write you map out the hypotheses and the deductions, and attempt to order the statements in a way which will cause the least confusion to the reader. This is the traditional backward proof-writing of classical analysis.
It has the advantage of being easily verifiable by a machine as opposed to understandable by a human being , and it has the dubious advantage that something at the end comes out to be less than e. The way to make the human reader's task less demanding is obvious: write the proof forward. Neither arrangement is elegant, but the forward one is graspable and rememberable. Avoid unnecessary notation. Consider: a proof that consists of a long chain of expressions separated by equal signs.
Such a proof is easy to write. The author starts from the first equation, makes a natural substitution to get the second, collects terms, permutes, inserts and immediately cancels an inspired factor, and by steps such as these proceeds till he gets the last equation.
This is, once again, coding, and the reader is forced not only to learn as he goes, but, at the same time, to decode as he goes. The double effort is needless. By spending another ten minutes writing a carefully worded paragraph, the author can save each of his readers half an hour and a lot of confusion. The paragraph should be a recipe for action, to replace the unhelpful code that merely reports the results of the act and leaves the reader to guess how they were obtained. The paragraph would say something like this: "For the proof, first substitute p for q, the collect terms, permute the factors, and, finally, insert and cancel a factor r.
Specific Recommendations As in any form of communication, there are certain stylistic practice which will make your writing more or less understandable. These may be best checked and corrected after writing the first draft. Many of these ideas are from HTWM, and are more fully justified there.
Notation that hasn't been used in several pages or even paragraphs should carry a reference or a reminder of the meaning. The structure should be easily discernible by headings and punctuation.
There should be a clear definition of the problem at hand all the way through. The title is the first contact that readers will have with your paper.
It must communicate something of the substances to the experts in your field as well as to the novices who will be interested. Thus, while the terminology should be technically correct "Don't over work a small punctuation mark such as a period or comma.How to Effectively Write a Mathematics Research Paper Last updated Sep college app essay prompts 2012 nfl, Mathematics research papers are different from standard academic research papers in important ways, but not so different that they require an entirely separate set of guidelines. Mathematical papers rely heavily paper logic how a specific type of language, math symbols and regimented notation. There are two basic structures write mathematical research papers: formal and informal exposition.
.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. Doing a good job of conducting is just as important to the listeners as composing a good piece. Notation that hasn't been used in several pages or even paragraphs should carry a reference or a reminder of the meaning.
If he dislikes partial differential equations, for example, he should be warned early on that he will encounter them. Which are the corollaries of these theorems? On the other hand, extraneous assumptions will simply distract from the point of the theorem, and should be eliminated when possible. Break it up, but not too small; use prose, but not too much. Once you have considered the structure and relevance of your research, you are ready to outline your paper.
This is the place to introduce the definitions and lemmas which are standard in the field, but which your readers may not know. What are the lemmas your own or others on which these theorems stand.
The structure should be easily discernible by headings and punctuation. Theorems and Equations Theorems and equations should be physically separated from the surrounding text. Many of these ideas are from HTWM, and are more fully justified there.