How to Effectively Write a Mathematics Research Paper
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
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 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.
Related: Do you have questions on language, grammar, or manuscript drafting? Get personalized answers on the FREE Q&A Forum!
Logic is the framework upon which every good mathematics research paper is built. Each theorem or equation must flow logically.
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
- Body of the work
- Conclusion, appendix, and references
- Publication of a math paper
- Preprint archive
- Choice of the journal, submission
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.
Good advice. For me, writing an essay on mathematics was very difficult. I did not have enough time and knowledge to write a quality essay. I worked a lot in the library and read many articles on the Internet. I studied information about essay writing. But I couldn’t finish the essay in full. I had to look for professional writers on the subject of mathematics. He helped me finish a few paragraphs. The work was delivered on time and on an excellent assessment.
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How to write a Mathematics Research Paper
Mathematics research papers writing are very different from standard research paper writing . But not so unique that it requires a different set of guidelines.
While writing a mathematics research paper, the writer must follow the logic and a language that specify the research paper completely and includes symbols and regimented notation.
Also, having clarity in the paper is very effective for mathematics research papers. This means the writer must use strong logic rules, clear definitions, theorems and equations that surrounds text and maths symbols.
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Logic is the core framework found in our research paper in mathematics. Each equation we write has a core presence of theorem and flows logically.
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- Theorems and Equations
Theorems and equations must separate from the surrounding text. It is used as a reference point for a well-defined beginning.
- Math Symbols and Notations
Math symbols and notations are differentiated within mathematics literature. The deviation may cause confusion and our writers follow the guidelines for equations, units, and mathematical notation.
What you will get when you Order Mathematics Research Paper
Mathematics paper publication is similar to any academic research publication that requires good writing. Our mathematics writers apply strict and logical theorem when writing mathematics research papers.
Below are a few essentials found in our paper:
- Title, acknowledgement, and list of authors
- Body of the work
- Conclusion, appendix, and references
- Publication of a math paper
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All these are critical elements of mathematics research and are good for readers to follow a clear path to the conclusion and result section.
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Igor Pak's blog
- About Igor Pak
How to write math papers clearly
Writing a mathematical paper is both an act of recording mathematical content and a means of communication of one’s work. In contrast with other types of writing, the style of math papers is incredibly rigid and resistant to even modest innovation. As a result, both goals suffer, sometimes immeasurably. The clarity suffers the most, which affects everyone in the field.
Over the years, I have been giving advice to my students and postdocs on how to write clearly. I collected them all in these notes . Please consider reading them and passing them to your students and colleagues.
Below I include one subsection dealing with different reference styles and what each version really means. This is somewhat subjective, of course. Enjoy!
**** 4.2. How to cite a single paper. The citation rules are almost as complicated as Chinese honorifics , with an added disadvantage of never being discussed anywhere. Below we go through the (incomplete) list of possible ways in the decreasing level of citation importance and/or proof reliability.
(1) “ Roth proved Murakami’s conjecture in [Roth].” Clear.
(2) “ Roth proved Murakami’s conjecture [Roth].” Roth proved the conjecture, possibly in a different paper, but this is likely a definitive version of the proof.
(3) “ Roth proved Murakami’s conjecture, see [Roth].” Roth proved the conjecture, but [Roth] can be anything from the original paper to the followup, to some kind of survey Roth wrote. Very occasionally you have “ see [Melville]”, but that usually means that Roth’s proof is unpublished or otherwise unavailable (say, it was given at a lecture, and Roth can’t be bothered to write it up), and Melville was the first to publish Roth’s proof, possibly without permission, but with attribution and perhaps filling some minor gaps.
(4) “ Roth proved Murakami’s conjecture [Roth], see also [Woolf].” Apparently Woolf also made an important contribution, perhaps extending it to greater generality, or fixing some major gaps or errors in [Roth].
(5) “ Roth proved Murakami’s conjecture in [Roth] ( see also [Woolf]).” Looks like [Woolf] has a complete proof of Roth, possibly fixing some minor errors in [Roth].
(6) “ Roth proved Murakami’s conjecture ( see [Woolf]).” Here [Woolf] is a definitive version of the proof, e.g. the standard monograph on the subject.
(7) “ Roth proved Murakami’s conjecture, see e.g. [Faulkner, Fitzgerald, Frost].” The result is important enough to be cited and its validity confirmed in several books/surveys. If there ever was a controversy whether Roth’s argument is an actual proof, it was resolved in Roth’s favor. Still, the original proof may have been too long, incomplete or simply presented in an old fashioned way, or published in an inaccessible conference proceedings, so here are sources with a better or more recent exposition. Or, more likely, the author was too lazy to look for the right reference, so overcompensated with three random textbooks on the subject.
(8) “ Roth proved Murakami’s conjecture ( see e.g. [Faulkner, Fitzgerald, Frost]).” The result is probably classical or at least very well known. Here are books/surveys which all probably have statements and/or proofs. Neither the author nor the reader will ever bother to check.
(9) “ Roth proved Murakami’s conjecture. 7 Footnote 7: See [Mailer].” Most likely, the author never actually read [Mailer], nor has access to that paper. Or, perhaps, [Mailer] states that Roth proved the conjecture, but includes neither a proof nor a reference. The author cannot verify the claim independently and is visibly annoyed by the ambiguity, but felt obliged to credit Roth for the benefit of the reader, or to avoid the wrath of Roth.
(10) “ Roth proved Murakami’s conjecture. 7 Footnote 7: Love letter from H. Fielding to J. Austen, dated December 16, 1975. ” This means that the letter likely exists and contains the whole proof or at least an outline of the proof. The author may or may not have seen it. Googling will probably either turn up the letter or a public discussion about what’s in it, and why it is not available.
(11) “ Roth proved Murakami’s conjecture. 7 Footnote 7: Personal communication. ” This means Roth has sent the author an email (or said over beer), claiming to have a proof. Or perhaps Roth’s student accidentally mentioned this while answering a question after the talk. The proof may or may not be correct and the paper may or may not be forthcoming.
(12) “ Roth claims to have proved Murakami’s conjecture in [Roth].” Paper [Roth] has a well known gap which was never fixed even though Roth insists on it to be fixable; the author would rather avoid going on record about this, but anything is possible after some wine at a banquet. Another possibility is that [Roth] is completely erroneous as explained elsewhere, but Roth’s work is too famous not to be mentioned; in that case there is often a followup sentence clarifying the matter, sometimes in parentheses as in “( see, however, [Atwood])”. Or, perhaps, [Roth] is a 3 page note published in Doklady Acad. Sci. USSR back in the 1970s, containing a very brief outline of the proof, and despite considerable effort nobody has yet to give a complete proof of its Lemma 2; there wouldn’t be any followup to this sentence then, but the author would be happy to clarify things by email.
UPDATE 1. (Nov 1, 2017): There is now a video of the MSRI talk I gave based on the article.
UPDATE 2. (Mar 13, 2018): The paper was published in the Journal of Humanistic Mathematics . Apparently it’s now number 5 on “ Most Popular Papers ” list. Number 1 is “My Sets and Sexuality”, of course.
UPDATE 3. (March 4, 2021): I wrote a followup paper and a blog post titled “ How to tell a good mathematical story “, with a somewhat different emphasis.
I realize this is mostly humor, but I would disagree in several respects.
First, I think it’s generally bad form to use citations as objects of prepositions, so I would rarely, if ever, write your example #1, preferring greatly to write #2 instead in nearly all circumstances.
Second, your examples #3, #4, #7 are classic examples of “run-on sentences”, which both Wikipedia and I agree should be avoided. It is a common stylistic error.
Third, style guides such as the Chicago Manual will tell you that you always need a comma after “e.g.”.
Thanks, Jeffrey. You are correct on the substance in all three instances. However, this is no humor and I personally dislike all three style rules which you mention, as I believe that they diminish clarity of the math writing for both native and (especially) non-native English speakers. In fact, given the precise manner in which we in math sciences need to communicate references, having more choice is good! So I don’t feel bad at all about abandoning century old rules which were designed for different types of communication. I explain it all in the notes I linked in the post.
@quasihuman (timestamps got reordered for some reason)
About 1) – there is nothing to fix, in my opinion. The clauses and connections between them are clear. Also, I am against semicolons (this is explained in the writeup).
About 2) – Right. I am perfectly comfortable if copy editors correct me. It’s their job. But my job is to communicate as clearly as possible in the version that’s free on the web. As I explain in the writeup, clarity is not a tradeoff, it’s *everything*. We have reached the point when over 90% of the world mathematicians are not native English speakers, yet English is the language we all use to communicate math. If some pedants are annoyed by the simplified and substandard writing style while still understand it clearly, it’s fine, really (cf. Simple English Wikipedia). This is because the clarity of understanding of people with poor English improves even by just a little then, which is all I want.
In fact, if I thought that writing “ze rezults of zis teorem R enaf 2 show…” would improve the clarity of my math papers, I totally would use this style. I don’t believe it does though, not on any level. One reason is the modern technology – Google Translate wouldn’t get that, and I know many are using it to help with translation. I only recommend breaking standard rules about commas in a constructive way to signify the clauses. I feel this is a very minor grammar violation for good purpose.
About completely restructuring sentences. I am not a fan. Sometimes this works well. But sometimes this results in a lengthy cumbersome construction communicating exactly the same thing, but whose only advantage is compliance with grammar rules. The result is less clarity. Remember, it’s people who are reading your paper, not some kind of compilers. You brain won’t stop working just because of some missing comma or a period.
1) Numbers 3, 4, and 7 can all be fixed easily; put in a semicolon in place of the comma.
2) I have to disagree with the amount of encouragement you give for breaking grammar rules. The minor reason for the disagreement is that you annoy copy editors as well as grammar pedants like myself. More importantly, almost every time I am tempted to break a grammar rule for the sake of clarity, I can make the writing even more clear by completely restructuring the sentence or paragraph in the right way. Breaking a grammar rule should be the last resort when, after much effort, you find no other way of making the writing clear.
There are citation typesetting styles that distinguish between textual (\citet) and parenthetical (\citep) references. For example, one could say, “Roth  proved Murakami’s conjecture.” Being in the 21st century, the “Roth ” would be a hyperlink.
I very much enjoy this grammar book http://www.lel.ed.ac.uk/~gpullum/SIEG/ You may too. Its point of view is that grammar shouldn’t be prescriptive but descriptive. That is, it should observe empirically what facilitates communication, and then describe it. (And they basically say that Strunk is an idiot.)
@rgrig — Cool. Thanks for the link!
- July 25, 2017 at 1:35 pm Writing math papers - Skip Garibaldi
- October 26, 2022 at 10:23 pm How to start a paper? | Igor Pak's blog
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