Math Research Paper Conclusion Examples

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 on logic and a specific type of language, including symbols and 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 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.

Definitions

In order for the reader to understand the author’s work, definitions for terms and notations used throughout the paper must be set at the beginning of the paper. 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.

LaTeX and Wolfram

Mathematical sentences contain equations, figures, and notations that are difficult to typeset using a typical word-processing program. Both LaTeX and Wolfram have expert typesetting capabilities to assist authors in writing.

LaTeX is highly recommended for researchers whose papers constitute mathematical figures and notation. It produces professional-looking documents and authentically represents mathematical language.

Wolfram Language & System Documentation Center’s Mathematica has sophisticated and convenient mathematical typesetting technology that produces professional-looking documents.

The main differences between the two systems are due to cost and accessibility. LaTeX is freely available, whereas Wolfram is not. In addition, any updates in Mathematica will come with an additional charge. LaTeX is an open-source system, but Mathematica is closed-source.

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. Authors must apply a strict, logical construct when writing a mathematics research paper.

There are resources that provide very specific guidelines related to following sections to write and publish a mathematics research paper.

  • Concept of a math paper
  • Title, acknowledgment, and list of authors
  • Abstract
  • Introduction
  • Body of the work
  • Conclusion, appendix, and references
  • Publication of a math paper
  • Preprint archive
  • Choice of the journal, submission
  • Decision
  • Publication

 

The critical elements of a mathematics research paper are good writing and a logical construct that allows the reader to follow a clear path to the author’s conclusions.



D. J. Bernstein
Notes on writing papers

The devil's guide to conclusions

Most mathematicians and computer scientists will tell you to state your main points at the beginning of your paper, first in summarized form ("Abstract"), and then in more detail ("Section 1. Introduction"). They will tell you that this allows the busy reader to pick up your paper and immediately understand what the paper is saying, so that the uninterested reader can put your paper down after the abstract, and the partially interested reader can put your paper down after the introduction.

Is that what you want? Does it sound good to have the reader put your paper down? Wouldn't you rather have the reader absorbing your words of wisdom? Which is better for your career: minimizing the time that readers spend staring at a paper with your name on it, or maximizing the time that readers spend staring at a paper with your name on it?

Bury your conclusions

One of the easiest ways to increase the time that readers are forced to spend on your paper is to bury some of the main points of your paper in a section near the end.

Some people will tell you a simplified version of this paper-writing technique, in which you mechanically copy all of the main points of your paper into a section labelled "Conclusion" immediately before the bibliography. See, for example, E. Robert Schulman, "How to write a scientific paper", Annals of Improbable Research 2 (1996), page 8, http://www.fisheries.vims.edu/hoenig/How_to_Write_a_Scientific_Paper.htm:

Abstract

We (meaning I) present observations on the scientific publishing process which (meaning that) are important and timely in that unless I have more published papers soon, I will never get another job. These observations are consistent with the theory that it is difficult to do good science, write good scientific papers, and have enough publications to get future jobs.

...

5. Conclusions

The conclusion section is very easy to write: all you have to do is to take your abstract and change the tense from present to past. It's considered good form to mention at least one relevant theory only in the abstract and conclusion. By doing this, you don't have to say why your experiment does (or does not) agree with the theory, you merely have to state that it does (or does not).

We (meaning I) presented observations on the scientific publishing process which (meaning that) are important and timely in that unless I have more published papers soon, I will never get another job. These observations are consistent with the theory that it is difficult to do good science, write good scientific papers, and have enough publications to get future jobs.

See also Ashley C. McDowell, "How to write a philosophy paper", 2002, http://cc.kzoo.edu/~mcdowell/writing.htm:
Conclusion

After you have presented the arguments you need a conclusion. This is the last paragraph of your paper and is basically the same as your introduction, except in the past tense.

Unfortunately, this simplified technique means that the reader has absorbed everything by the end of your introduction, and can simply skip your visibly redundant "Conclusion" section. It is much better to remove some conclusions from your "Introduction" section, and some other conclusions from your "Conclusion" section, so that the reader is forced to read both sections. It is even better to spread the main points through several sections near the end (e.g., "Results" and "Analysis" and "Conclusion" and "Appendix C").

Don't be afraid to explicitly tell the reader that he will have to read more. For example, if you notice yourself saying something like

In Section 8 we compare our proposed mechanism to the previous state of the art, and show that we achieve a factor of 2 improvement in situation X.
in your introduction, replace it with
After describing our proposed mechanism we compare it to the previous state of the art.
and avoid leaking the result of the comparison.

Maybe a reader will open up your paper and, seeing or anticipating your omission of information from the "Introduction" section, will immediately flip through to find the "Conclusion" section. This is why it is important for your "Conclusion" section to be missing critical information. Omit essential terminology; omit other critical background information; omit at least one of your main points; be vague in stating other main points. But don't go overboard in removing information! Remember that the "Conclusion" section is a wonderful opportunity to add pages to your paper, adding weight to your CV, by simply repeating yourself without putting in any new scientific effort. No research required!

Defend yourself if necessary

As mentioned above, most mathematicians and computer scientists will tell you to state your main points at the beginning of your paper. See, for example, J. S. Marron, "Effective writing in mathematical statistics", 1999, http://www.unc.edu/~haipeng/teaching/marron.pdf:
2.6 Conclusion section

Here I make a controversial suggestion. Many people believe a good paper is wound up with some conclusions which highlight a few of the most important lessons of the paper. If everyone were to read every part of every paper, this would be appropriate. However, in view of the way that modern researchers approach the literature, as discussed in Section 2.1, I suggest that a summary of the main points is more effective if it is in the introduction instead. It is not so elegant, since the conclusions are not properly backed up at that point. But this does have the effect of leading those who have doubts to read further and more carefully.

See, as another example, Oded Goldreich, "How to write a paper", 2004, http://www.wisdom.weizmann.ac.il/~oded/writing.html:
4.5 Conclusions and/or suggestions for further work are not a "must"

Some people tend to think that each paper should end with conclusions and/or suggestions for further work. We strongly disagree with this opinion, and see little use in a "conclusion section" that merely re-iterates things said in the abstract and/or in the introduction. Similarly, we see no point in listing well-known open problems or re-iterating questions that were already raised in the introduction. On the other hand, we do value a conclusion section that contains high-level material that better fits after the main part of the paper (and thus is not placed in the introduction). Similarly, for raising important questions that are more appealing after reading the technical part (even if they were raised already in the technical part but not in the introduction).

To summarize: There are papers that may benefit from a conclusion section, but they are relatively few (say, less than 5% of the papers). Certainly, the inclusion of a conclusion section should not be the default.

Occasionally these people will have the nerve to criticize you for not following their advice. Here are several ways that you can fight back.

Cite the "scientific format". In some fields of science, the vast majority of papers are descriptions of scientific experiments, with each description following a rigid structure: Abstract, Introduction, Methods, Results, Conclusion/Discussion, Literature Cited. Of course, your paper isn't a description of a scientific experiment, and you probably aren't following the "Methods"/"Results" structure for the intermediate sections, but you can still claim that a "Conclusion" section is a completely standard part of the "scientific format".

Exploit the ambiguity of the word "conclusion". Explain that your paper begins with the "Introduction" and ends with the "Conclusion." Say that the "Conclusion" section obviously belongs at the end of the paper (let's not quibble about the bibliography and appendices), and that any other position for the "Conclusion" would be silly, since the word "Conclusion" means "the last part" by definition.

Point to talks. The audience of a talk has no opportunity to rewind the talk. If some of the target audience members need to understand X before they can understand Y, and need to understand Y before they can understand Z, then the speaker is forced to say Z after Y, and Y after X. If the speaker wants to say Z early in the talk for other reasons then the speaker has to repeat Z later in the talk. The end of a talk has the advantage of being able to rely on the maximum amount of preparatory material, and the further advantage of being the freshest material in the audience's mind when the talk ends. Consequently, you should easily be able to find examples of good talks in your area that end by summarizing their main points. Point to those talks, and say that you are imitating this masterful expository style in your paper.

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