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Course info, instructors.
- Prof. Haynes Miller
- Dr. Nat Stapleton
- Saul Glasman
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- Mathematics
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Learning resource types, project laboratory in mathematics.
Next: Revision and Feedback »
In this section, Prof. Haynes Miller and Susan Ruff describe the criteria for good mathematical writing and the components of the writing workshop .
A central goal of the course is to teach students how to write effective, journal-style mathematics papers. Papers are a key way in which mathematicians share research findings and learn about others’ work. For each research project, each student group writes and revises a paper in the style of a professional mathematics journal paper. These research projects are perfect for helping students to learn to write as mathematicians because the students write about the new mathematics that they discover. They own it, they are committed to it, and they put a lot of effort into writing well.
Criteria for Good Writing
In the course, we help students learn to write papers that communicate clearly, follow the conventions of mathematics papers, and are mathematically engaging.
Communicating clearly is challenging for students because doing so requires writing precisely and correctly as well as anticipating readers’ needs. Although students have read textbooks and watched lectures that are worded precisely, they are often unaware of the care with which each word or piece of notation was chosen. So when students must choose the words and notation themselves, the task can be surprisingly challenging. Writing precisely is even more challenging when students write about insights they’re still developing. Even students who do a good job of writing precisely may have a different difficulty: providing sufficient groundwork for readers. When students are deeply focused on the details of their research, it can be hard for them to imagine what the reading experience may be like for someone new to that research. We can help students to communicate clearly by pointing out places within the draft at which readers may be confused by imprecise wording or by missing context.
For most students, the conventions of mathematics papers are unfamiliar because they have not read—much less written—mathematics journal papers before. The students’ first drafts often build upon their knowledge of more familiar genres: humanities papers and mathematics textbooks and lecture notes. So the text is often more verbose or explanatory than a typical paper in a mathematics journal. To help students learn the conventions of journal papers, including appropriate concision, we provide samples and individualized feedback.
Finally, a common student preconception is that mathematical writing is dry and formal, so we encourage students to write in a way that is mathematically engaging. In Spring 2013, for example, one student had to be persuaded that he did not have to use the passive voice. In reality, effective mathematics writing should be efficient and correct, but it should also provide motivation, communicate intuition, and stimulate interest.
To summarize, instruction and feedback in the course address many different aspects of successful writing:
- Precision and correctness: e.g., mathematical terminology and notation should be used correctly.
- Audience awareness: e.g., ideas should be introduced with appropriate preparation and motivation.
- Genre conventions: e.g., in most mathematics papers, the paper’s conclusion is stated in the introduction rather than in a final section titled “Conclusion.”
- Style: e.g., writing should stimulate interest.
- Other aspects of effective writing, as needed.
To help students learn to write effective mathematics papers, we provide various resources, a writing workshop, and individualized feedback on drafts.
Writing Resources
Various resources are provided to help students learn effective mathematical writing.
The following prize-winning journal article was annotated to point out various conventions and strategies of mathematical writing. (Courtesy of Mathematical Association of America. Courtesy of a Creative Commons BY-NC-SA license.)
An Annotated Journal Article (PDF)
This document introduces the structure of a paper and provides a miscellany of common mistakes to avoid.
Notes on Writing Mathematics (PDF)
LaTeX Resources
The following PDF, TeX, and Beamer samples guide students to present their work using LaTeX, a high-quality typesetting system designed for the production of technical and scientific documentation. The content in the PDF and TeX documents highlights the structure of a generic student paper.
Sample PDF Document created by pdfLaTeX (PDF)
Sample TeX Document (TEX)
Beamer template (TEX)
The following resources are provided to help students learn and use LaTeX.
LaTeX-Project. “ Obtaining LaTeX .” August 28, 2009.
Downes, Michael. “Short Math Guide for LaTeX.” (PDF) American Mathematical Society . Version 1.09. March 22, 2002.
Oetiker, Tobias, Hubert Partl, et al. “The Not So Short Introduction to LaTeX 2ε.” (PDF) Version 5.01. April 06, 2011.
Reckdahl, Keith. “Using Imported Graphics in LaTeX and pdfLaTeX.” (PDF) Version 3.0.1. January 12, 2006.
Writing Workshop
Each semester there is a writing workshop, led by the lead instructor, which features examples to stimulate discussion about how to write well. In Spring 2013, Haynes ran this workshop during the third class session and used the following slide deck, which was developed by Prof. Paul Seidel and modified with the help of Prof. Tom Mrowka and Prof. Richard Stanley.
The 18.821 Project Report (PDF)
This workshop was held before students had begun to think about the writing component of the course, and it seemed as if the students had to be reminded of the lessons of the workshop when they actually wrote their papers. In future semesters, we plan to offer the writing workshop closer to the time that students are drafting their first paper. We may also focus the examples used in the workshop on the few most important points rather than a broad coverage.
- Download video
This video features the writing workshop from Spring 2013 and includes instruction from Haynes as well as excerpts of the class discussion.
« Previous: Writing | Next: Sample Student Papers »
In this section, Prof. Haynes Miller and Susan Ruff describe how students receive feedback on their writing and what is expected from students during the revision process.
Feedback and revision are critical to students’ development as mathematical writers in the course. For each project, each student team is required to write a first draft, meet with course instructors for a debriefing meeting, make revisions, and submit a final draft. This process provides an opportunity for a mid-project check-in about the students’ writing as well as their research, and it pushes them to produce a stronger final draft than what most could have managed on their own.
In the best situations, a team’s first draft represents the students’ best efforts but is still somewhat rough; we give them lots of feedback for reworking their paper, and their final draft is substantially clearer and more rigorous, well-motivated, and technically precise. In our experience, each subsequent paper is typically better than the one before.
Instructor Feedback on Writing
After a team submits its first draft, the team’s mentor for that project, and sometimes Haynes and sometimes Susan, reads the paper and crafts feedback. First drafts typically have plenty of room for improvement. We try not to overwhelm students with a huge number of comments; commenting on everything often leads to students getting lost in the details and unable to distinguish the most important points from more trivial points. Instead, we draw attention to the most important things for the students to improve. We try to craft constructive comments so that, rather than being discouraged, students will be inspired to revise. Sometimes a second round of revision is necessary. This whole process is quite like the refereeing process for journal articles.
Debriefing Meetings
Students receive feedback on their draft at a team debriefing meeting, which usually occurs several days after the first draft is submitted. Sharing feedback via the debriefing meeting provides two key advantages:
- Clarity and emphasis via discussion. Speaking face-to-face allows us to emphasize the most important feedback; to ask students questions and understand the intentions behind their writing; and to have some back-and-forth to make sure that students understand the feedback.
- Efficiency. Reading papers and commenting on papers takes a long time. The debriefings allow us to convey some of the feedback efficiently in person rather than on paper.
Most students take the debriefing sessions very seriously. They do not see our feedback on their work beforehand, and they are naturally curious and may be somewhat anxious, especially the first time. The face-to-face interaction always helps to frame suggestions in a constructive manner, and students almost never respond defensively. They generally listen attentively and make a sincere effort to respond to our critiques.
Immediately after the debriefings, we scan the marked-up papers and send an electronic copy to the team members. The final draft is typically due a week after the debriefing, giving students time to think about research extensions of their work and to improve their writing.
Self- and Peer-Editing
One of the things we look for in papers for the course is consistency of voice and notation among sections written by different team members. We encourage the students to help each other revise.
« Previous: Revision and Feedback
To illustrate the writing and revision process for the student papers, two sample projects are presented below.
Sample Paper 1: The Dynamics of Successive Differences Over ℤ and ℝ
This project developed from the project description for Number Squares (PDF) . To view the practice presentation and final presentation from this team of students, see the Sample Student Presentations page.
The student work is courtesy of Yida Gao, Matt Redmond, and Zach Steward. Used with permission.
- First Draft of Sample Paper 1 (PDF)
- First Draft of Sample Paper 1 with Comments from Susan Ruff (PDF - 2.5MB)
- First Draft of Sample Paper 1 with Comments from Prof. Haynes Miller (PDF - 3.6MB)
- Additional Comments on Sample Paper 1 from Prof. Haynes Miller (PDF)
- Final Version of Sample Paper 1 (PDF)
Debriefing for First Draft of Sample Paper 1
This video features the debriefing meeting for the first draft of Sample Paper 1. The student team first presents their findings, and then the course instructors offer feedback and discuss the mathematics and the writing for the project.
Sample Paper 2: Tossing a Coin
This project developed from the project description for Tossing a Coin (PDF) .
The student work is courtesy of Jean Manuel Nater, Peter Wear, and Michael Cohen. Used with permission.
- First Draft of Sample Paper 2 (PDF)
- First Draft of Sample Paper 2 with Comments from Susan Ruff (PDF - 1.7MB)
- First Draft of Sample Paper 2 with Comments from Prof. Haynes Miller (PDF - 2.7MB)
- Additional Comments on Sample Paper 2 from Prof. Haynes Miller (PDF)
- Final Version of Sample Paper 2 (PDF)
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," 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. .
or structure consisting of definitions, theorems, and proofs, and the complementary or 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." (p. 2)
in the language of logic, very few actually 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 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.
by a machine (as opposed to 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. Start, as the author always starts, by putting something less than e, and then do what needs to be done--multiply by 3M2 + 7 at the right time and divide by 24 later, etc., etc.--till you end up with what you end up with. Neither arrangement is elegant, but the forward one is graspable and rememberable. (p. 43)
is bounded." What does the symbol "f" contribute to the clarity of that statement?... A showy way to say "use no superfluous letters" is to say "use no letter only once". (p. 41) is sufficiently large, then | | < e, where e is a preassigned positive number"; both disease and cure are clear. "Equivalent" is logical nonsense. (By "theorem" I mean a mathematical truth, something that has been proved. A meaningful statement can be false, but a theorem cannot; "a false theorem" is self-contradictory). As for "if...then...if...then", that is just a frequent stylistic bobble committed by quick writers and rued by slow readers. "If , then if , then ." Logically, all is well, but psychologically it is just another pebble to stumble over, unnecessarily. Usually all that is needed to avoid it is to recast the sentence, but no universally good recasting exists; what is best depends on what is important in the case at hand. It could be "If and then ", or "In the presence of , the hypothesis implies the conclusion ", or many other versions."" (p. 38-39)
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- About the Editors
A roundup of advice for writing about mathematics
April is Mathematics and Statistics Awareness Month , a time for increasing the understanding and appreciation of those fields. One way to communicate the joy and importance of math and stats? Through our writing.
Just last month, the Early Career Section of the Notices of the AMS published several articles on the theme of writing, including “Outward-Facing Mathematics: A Pitch” by Jordan Ellenberg, “To Write or Not to Write… a Book, and When?” by Joseph H. Silverman, “Preparing Your Results for Publication” by Julia Hartmann, “The Art of Writing Introductions” by John Etnyre and “Writing, and Reading, Referee Reports” by Arend Bayer.
“There’s really only one form of outward-facing math I personally know well: writing about math for general-audience publications, which I’ve been doing for more than twenty years now. And I meet a lot of graduate students and early-career mathematicians who are interested in doing it too. So let me tell you some of the lessons I’ve learned,” wrote Ellenberg. He gave this advice on getting started:
“Social media drives attention, but no one has yet figured out a great way to tweet or Snap about math. That’s why blogging is still alive for mathematicians, even as blogs have withered somewhat on the whole. I think best practice for getting started is to blog on a platform like Medium or WordPress, then use social media to bring readers to your writing. When you want to pitch a piece to a more formal publication, they’ll want to see what your writing looks like: with the blog, you’ll have something to show them.”
Etnyre’s piece focuses on writing introductions to mathematics research papers, but much of his advice is relevant for anyone who wishes to write well about mathematics.
“A common problem writers have, especially early in their career, is to overestimate what everyone will know about their work and how it fits into the research world,” Etnyre wrote. (I think this common problem often also occurs when mathematicians write about mathematics outside of the realm of their own research.) He added this advice:
“Assuming that everyone will understand the context of your work, and why it is really interesting, is not a good idea. Most work is focused on some part of a bigger program or problem, and even experts in a field might not, in any given moment, recall the subtleties and details to every interesting problem in their field. So tell them, and all the other readers who will have no chance of appreciating the context without some help from you. Explain the big picture.”
Hartmann’s piece is also intended for folks who are writing research papers but contains advice that’s useful for anyone considering writing about mathematics.
“Decide what the story is you’d like to tell. There is usually more than one way to present a result and the work that leads to it…Talk about your work to other people. Consider giving a talk at your home institution. This will force you to come up with a way to “pitch” your story. You might also receive helpful feedback on your results and comments on connections to other existing work. During the process of writing, you may find that your conception of the story has changed, and this may change the idea of how to best present it,” she wrote.
The “On writing” section for Terence Tao’s blog includes links to many pieces (written by him and others). Many of these are geared toward people who are writing research papers, but some of this advice could be helpful to folks who want their mathematics writing to take the form of blog posts, news articles and more.
Francis Su’s 2015 MAA Focus piece “Some Guidelines for Good Mathematical Writing” shares both basics and advice “toward elegance.” While the piece is written at a level that’s accessible to students, it also contains gems that will serve even experienced math writers. For instance, he advises that writers “Decide what’s important to say. Writing well does not necessarily mean writing more” and “Observe the culture. Good communication is inseparable from the culture in which it takes place.”
In “Mathematics for Human Flourishing,” Su discussed the importance of valuing public writing about mathematics:
“I would like to encourage institutions to start valuing the public writing of its faculty. More people will read these pieces than will ever read any of our research papers. Public writing is scholarly activity: it involves rigorous arguments, is subject to review process by editors, and to borrow the NSF phrase, it has broader impacts, and that impact can be measured in the digital age,” he noted.
1 Response to A roundup of advice for writing about mathematics
nice read, thank you! https://tipshire.com
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Opinions expressed on these pages were the views of the writers and did not necessarily reflect the views and opinions of the American Mathematical Society.
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How do mathematicians conduct research?
I am curious as to how mathematicians conduct research. I hope some of you can help me solve this little mystery.
To me, mathematics is a branch where you either get it or you don't. If you see the solution, then you've solved the problem, otherwise you will have to tackle it bit by bit. Exactly how this is done is elusive to me.
Unlike physicists, chemists, engineers or even sociologists, I can't see where a mathematician (other than statisticians) gather their data from. Also, unlike the other professions mentioned above, it is not apparent that mathematicians perform any experiments.
Additionally, a huge amount of work has already been laid down by other mathematicians, I wonder if there is a lot of "copy and pasting" as we see in software engineering (think of using other people's code)
So my question is, where do mathematicians get their research topics from and how do they go about conducting research? What is considered acceptable progress in mathematics?
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- 29 I like the question just fine, but: you do realize that mathematics as an academic field is not uniquely characterized by a lack of data and experiments, right? In other words, you correctly point out that theoretical mathematics is not a science . There are other non-sciences too... – Pete L. Clark Commented Dec 11, 2014 at 6:14
- 28 ff524: The NSF disagrees with me on the science thing, sometimes to the extent of putting money in my pocket. Nevertheless I think that everyone agrees that there is a sense in which (traditional, theoretical) mathematics is not a science: deductive versus inductive reasoning and all that. My point is that the OP seems to express wonderment about an academic field which lies largely outside of the scientific method. I agree and say: more amazing still, there are multiple fields like that. – Pete L. Clark Commented Dec 11, 2014 at 6:27
- 12 I have the opposite problem: I don't understand how you can call collecting some numbers from nowhere and deducing some non-sense from them without giving proper evidence (proof) as conducting research :D </sarcasm> – yo' Commented Dec 11, 2014 at 7:57
- 21 Legend has it that mathematics research consists of the following iterations coffee -> think -> coffee -> theorem -> coffee -> paper . Rinse and repeat. There may be more coffee steps involved but the general idea boils down to this (pun intended). – Marc Claesen Commented Dec 11, 2014 at 11:21
- 14 You can find a video clip on YouTube of two characters from the show "The Big Bang Theory" acting like they are "doing research", set to the song "Eye of the Tiger". The characters are playing physicists, but the clip is frighteningly accurate for what much mathematical research looks like. – Oswald Veblen Commented Dec 11, 2014 at 23:46
4 Answers 4
As far as pure mathematics, you are quite right: there are neither data nor experiments.
Drastically oversimplified, a mathematics research project goes like this:
Develop, or select from the existing literature, a mathematical statement ("conjecture") that you think will be of interest to other mathematicians, and whose truth or falsity is not known. (For example, "There are infinitely many pairs of prime numbers that differ by 2.") This is your problem .
Construct a mathematical proof (or disproof) of this statement. See below. This is the solution of the problem.
Write a paper explaining your proof, and submit it to a journal. Peer reviewers will decide whether your problem is interesting and whether your solution is logically correct. If so, it can be published, and the conjecture is now a theorem.
The following discussion will make much more sense to anyone who has tried to write mathematical proofs at any level, but I'll try an analogy. A mathematical proof is often described as a chain of logical deductions, starting from something that is known (or generally agreed) to be true, and ending with the statement you are trying to prove. Each link must be a logical consequence of the one before it.
For a very simple problem, a proof might have only one link: in that case one can often see the solution immediately. This would normally not be interesting enough to publish on its own, though mathematics papers typically contain several such results ("lemmas") used as intermediate steps on the way to something more interesting.
So one is left to, as you say, "tackle it bit by bit". You construct the chain a link at a time. Maybe you start at the beginning (something that is already known to be true) and try to build toward the statement you want to prove. Maybe you go the other way: from the desired statement, work backward toward something that is known. Maybe you try to build free-standing lengths of chain in the middle and hope that you will later manage to link them together. You need a certain amount of experience and intuition to guess which direction you should direct your chain to eventually get it where it needs to go. There are generally lots of false starts and dead ends before you complete the chain. (If, indeed, you ever do. Maybe you just get completely stuck, abandon the project, and find a new one to work on. I suspect this happens to the vast majority of mathematics research projects that are ever started.)
Of course, you want to take advantage of work already done by other people: using their theorems to justify steps in your proof. In an abstract sense, you are taking their chain and splicing it into your own. But in mathematics, as in software design, copy-and-paste is a poor methodology for code reuse. You don't repeat their proof; you just cite their paper and use their theorem. In the software analogy, you link your program against their library.
You might also find a published theorem that doesn't prove exactly the piece you need, but whose proof can be adapted. So this sometimes turns into the equivalent of copying and pasting someone else's code (giving them due credit, of course) but changing a few lines where needed. More often the changes are more extensive and your version ends up looking like a reimplementation from scratch, which now supports the necessary extra features.
"Acceptable progress" is quite subjective and usually based on how interesting or useful your theorem is, compared to the existing body of knowledge. In some cases, a theorem that looks like a very slight improvement on something previously known can be a huge breakthrough. In other cases, a theorem could have all sorts of new results, but maybe they are not useful for proving further theorems that anyone finds interesting, and so nobody cares.
Now, through this whole process, here is what an outside observer actually sees you doing:
Search for books and papers.
Stare into space for a while.
Scribble inscrutable symbols on a chalkboard. (The symbols themselves are usually meaningful to other mathematicians, but at any given moment, the context in which they make sense may exist only in your head.)
Scribble similar inscrutable symbols on paper.
Use LaTeX to produce beautifully-typeset inscrutable symbols interspersed with incomprehensible technical terms, connected by lots of "therefore"s and "hence"s.
Loop until done.
Submit said beautifully-typeset gibberish to a journal.
Apply for funding.
Attend a conference, where you speak unintelligibly about your gibberish, and listen to others do the same about theirs.
Loop until emeritus, or perhaps until dead ( in the sense of Erdős ).
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- 6 What's interesting is that sometimes in the course of proving something, you might invent an entirely new kind of mathematics, which in turn winds up being useful for other purposes. This is very loosely analogous to inventing new programming languages for the purpose of more efficiently expressing your intention and hence developing things more quickly. Many of the names of our everyday mathematical abstractions come from the names of the living, breathing people who spent their lives constructing and refining them. – Dan Bryant Commented Dec 11, 2014 at 16:21
- 15 In my own experience, it's not that common to begin with a specific problem. More often, I begin with a feeling that something I've read or heard about could be done more elegantly or more clearly. My initial goal is then just to understand better what someone else has done, but if I can really achieve a better understanding, then that often suggests improvements or generalizations of that work. Indeed, it sometimes makes such improvements obvious. If the improvement is big enough, it can constitute a paper; if not, it can sometimes become part of a paper, or of a talk. – Andreas Blass Commented Dec 11, 2014 at 21:23
- 4 Rather than starting with a conjecture (although I sometimes do that), I more often start with an idea: some specific thing that I'd like to understand. This is based on my intuition about what problems seem likely to have interesting results. As I work through the thing I am studying, I come up with specific conjectures and theorems. But the beginning of the project rarely has specific conjectures, just goals. – Oswald Veblen Commented Dec 11, 2014 at 23:31
- 6 You forgot "meet with a colleague, stare at a blackboard together and argue passionately on which definition looks the most beautiful". Pretty accurate nevertheless. – Federico Poloni Commented Dec 13, 2014 at 17:04
- 2 @Jack: The goal of pure mathematics research at any level is as I described: to be able to prove or disprove statements whose truth or falsity was not previously known. At the undergraduate level, it often begins with computations (by hand or computer) to try to evaluate whether a conjecture is plausible, and sometimes it doesn't get any further than that. There will also be a lot more interaction with an advisor. – Nate Eldredge Commented Apr 4, 2015 at 15:10
Actually, even in pure mathematics, it very often is possible to do experiments of a sort.
It's very common to come up with a hypothesis that seems plausible but you're not sure if it's true or not. If it's true, proving that is probably quite a lot of work; if it's false, proving that could be quite a lot of work, too. But, if it's true, trying to prove that it's false is a huge amount of work! Before you invest a lot of effort into trying to prove the wrong direction, it's good to gain some intuition about the situation and whether the statement seems more likely to be true or to be false. Computers can be very useful for this kind of thing: you can generate lots of examples and see if they satisfy your hypothesis. If they do, you might try to prove your hypothesis is true; if they don't, you might try to refine your hypothesis by adding more conditions to it.
See also Oswald Veblen's answer which talks about doing similar "experiments" by hand.
- 8 I "do experiments" by working out conjectures in the context of specific examples. If the conjecture works out in several examples, that makes me more confident that it may be true in general. – Oswald Veblen Commented Dec 11, 2014 at 23:42
I "gather data" and perform experiments" by working out my conjectures in the context of specific examples. If the conjecture works out in several examples, that makes me more confident that it may be true in general.
For example, suppose that I think that every topological space of a certain form has a particular property. I will start by looking at some "simple" spaces, like the real line, and see if they have the property. If they do, I may look at some more complicated space. Often, when I look at what specific attributes of the examples were necessary to show they had the property in question, it tells me what hypotheses I need to add to make my conjecture into a theorem.
This is not the same as scientific experimentation, nor the same as computer experimentation, which is also important in various areas of mathematics. But it is its own form of experimentation, nevertheless.
- 15 I think this is an important answer (especially in light of my comments above). From a philosophy of science standpoint, one must be clear that theoretical mathematics does not follow the scientific method. However, an important part of what mathematicians do in practice bears a lot of similarity to scientific experimentation. As a result, mathematical research has a similar flavor to scientific research in many respects. (There are other academic fields in which one really doesn't do experiments in any sense: philosophy, literature, law...) – Pete L. Clark Commented Dec 12, 2014 at 3:40
One point to note is that, for some questions, it is possible to do experiments to get data. Certain questions are things we now have computer programs to generate, and previously they could have been done on a far more limited scale by hand. So in some cases mathematicians do work more like experimental scientists. On the other hand, once they've found what seems to be a pattern, they change approach. Gathering further examples isn't much use (unless you then find a counter-example, but it can be encouraging) - you need to find an actual proof.
More generally, nearly every big result will come from some 'experiments': you try special cases, cases with more hypotheses, extreme cases that might result in failures...
On the 'copy-and-paste' point, mathematicians do use a lot of what other people have done (generally they must), but whereas you might copy someone's code to use it, when you cite a theorem you don't need to copy out the proof. So in terms of written space in a paper, the 'copied' section is very small. There are (fairly large) exceptions to this: fairly often a proof someone has given is very close to what you need, but not quite good enough, because you want to use it for something different to what they did. So you may end up writing out something very similar, but with your own subtle tweaks. I guess you could see this as like adjusting someone else's machine (we call things machines too, but here I mean a physical one). The difference is that generally in order to do this sort of thing you must completely understand what the machine does. Another big reason for 'copying' is that you may need (for actual theoretical reasons or for expositional ones) to build on the actual workings of the machine, not just on the output it gives.
More to the point of the question: As a mathematician, you generally read, and aim to understand, what other people have done. That gives you a bank of tools you can use - results (which you may or may or may not be completely able to prove yourself), and methods that have worked in the past. You build up an idea of things that tend to work, and how to adapt things slightly to work in similar situations. You do a fair amount of trial and error - you try something, but realise you get stuck at some point. Then you try and understand why you are stuck, and if there's a way round. You try proving the opposite to what you want, and see where you get stuck (or don't!).
Once you have a working proof, you see whether there are closely related things you can/can't prove. What happens if you remove/change a hypothesis? Also, does the reverse statement hold? If not entirely, are there some cases in which it does? Can you give examples to show your result is as good as possible? Can you combine it with other things you know about?
Another source of questions is what other people are interested in. Sometimes you know how to do something they want doing, but you didn't think of it until they asked.
One more point I'd like to make in the 'methods of proof category' is that, for me at least, there's a degree to which I work by 'feel'. You know those puzzles where all the pieces seem to be jammed in place but you're meant to take them apart (and put them back together again)? You sort of play around until you feel a bit that's looser than the rest, right? Sometimes proofs are a bit like that. When you understand something well, you can 'feel' where things are wedged tight and where they are looser.
Sometimes you also hope that lightning (inspiration) will strike. Occasionally it does.
(All of this may not exactly answer the question, but hopefully it gives some insight.)
- 3 "whereas you might copy someone's code to use it, when you cite a theorem you don't need to copy out the proof" - and when you call someone else's function, you don't need to copy the source. If you're copying the source, that's a bad sign. – user2357112 Commented Dec 11, 2014 at 8:57
- 2 @user2357112 or a sign that they don't provide a library, just an integrated implementation; or that the full library has too many requirements, or does not compile on your system. Seriously, in academic code you can usually find truly horrific things, and just copying the body of a function is one of the least abhorrent things. – Davidmh Commented Dec 11, 2014 at 9:34
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What is mathematical research like?
I'm planning on applying for a math research program over the summer, but I'm slightly nervous about it just because the name math research sounds strange to me. What does math research entail exactly? For other research like in economics, or biology one collects data and analyzes it and draws conclusions. But what do you do in math? It seems like you would sit at a desk and then just think about things that have never been thought about before. I appologize if this isn't the correct website for this question, but I think the best answers will come from here.
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- 7 $\begingroup$ Gromov said when he was young he could work on problems from early morning to midnight. $\endgroup$ – alancalvitti Commented Feb 13, 2013 at 3:19
- 5 $\begingroup$ This question has several good answers on Quora here . $\endgroup$ – David Robinson Commented Feb 13, 2013 at 8:02
- 2 $\begingroup$ You should read Théorème Vivant (Alive Theorem) by Cédric Villani, as soon as it gets translated to English (or right now if you can read French). It tells the story of a research that led to the Fields Medal, and it's fascinating. $\endgroup$ – Laurent Couvidou Commented Feb 13, 2013 at 9:48
- $\begingroup$ @LaurentCouvidou That sounds extremely fascinating, is there another book that tells a similar story? $\endgroup$ – TheHopefulActuary Commented Feb 14, 2013 at 4:51
- $\begingroup$ @Kyle Not to my knowledge. Which doesn't cover lots of math ;) $\endgroup$ – Laurent Couvidou Commented Feb 14, 2013 at 8:42
9 Answers 9
Largely (very largely, so please take everything here with a grain of salt), there are two types of mathematical research, commonly referred to as 'theorem proving/problem solving' vs. 'theory building'. Typical characteristics of theorem proving/problem solving type research is to try and tackle a famous open problem, usually stated in the form of a conjecture as to the validity of a statement or the specification of a problem. Quite often this will entail spending a lot of time learning the relevant material, analyzing particular attempts at solutions, trying to figure out why they don't work, and hopefully come up with some improvement to an existing attempt, or a whole new attempt, that has a good change of working. Very famous open-standing questions include: The Riemann hypothesis and $P\ne NP$ (which are examples of theorem proving) and the solution of the Navier-Stokes equations (an example of problem solving), all three are in the Clay's Institute millennium problems list.
Theory building is a somewhat different activity that involves the creation of new structures, or the extension of existing structures. Usually, the motivation behind the study of these new structures is coming from a desire to generalize (in order to gain better insight or be able to apply particular techniques of one area to a broader class of problems) or there might be a need to these new structures to exist, due to some application in mind. Typical activities would include a lot of reading on relevant structures, understanding their global role, figuring out what generalizations or new structures would make sense, what the aim of the new theory will be, and then a long process of proving basic structure theorems for the new structures that will necessitate tweaking the axioms. A striking example of this kind of research is Grothendieck's reformalization of modern algebraic geometry. Cantor's initial work on set theory can also be said to fall into this kind of research, and there are many other examples.
Of course, quite often a combination of the two approaches is required.
Today, research can be assisted by a computer (experimentally, computationally, and exploratory). Any mathematics research will require extensive amount of learning (both of results and of techniques) and will certainly include long hours of thinking. I find the entire process extremely creative.
I hope this helps. As should be clear, this is a rather subjective answer and I don't intend any of what I said to be taken to be said with any kind of mathematical rigor.
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- 6 $\begingroup$ Very helpful and relevant to the question! +1 $\endgroup$ – LarsH Commented Feb 13, 2013 at 16:09
"A mathematician is a blind man in a dark room looking for a black cat which isn't there." Attributed to Darwin (but I'm not convinced).
EDIT: A friend of mine found a discussion of this quote at wikiquote . It says (among other things),
The attribution to Darwin is incorrect,
In a publication of 1911 it was attributed to Lord Bowen (who died in 1894), but it was about "equity", not about mathematicians, and it was a hat instead of a cat,
It was published in 1898 as being about metaphysicians and hats,
William James, 1911, had it about philosophers,
The first reference to mathematicians seems to be in a 1948 collection of essays edited by William Schaaf.
EDIT 30 August 2016: Expanding on the last point. The collection is Mathematics, Our Great Heritage, edited by William Leonard Schaaf, published by Harper in 1948. An essay by Tomlinson Fort, Mathematics and the Sciences, appears on pages 161 to 172. A footnote states, "Address delivered at the dinner of the Southeastern Section of the Mathematical Association of America at Athens, Ga., March 29, 1940. Reprinted, by permission, from the American Mathematical Monthly, November, 1940, vol. 47, pp. 605-612." On page 163, Fort writes,
I have heard it said that Charles Darwin gave the following. (He probably never did.) "A mathematician is a blind man in a dark room looking for a black hat which isn't there."
- 3 $\begingroup$ And in the world of this quote, what is the blind man's goal? To find out (and prove) whether the cat is there? I'm having a hard time applying the quote to mathematics in a meaningful way, but maybe it's not meant to be very meaningful. $\endgroup$ – LarsH Commented Feb 13, 2013 at 16:03
- 2 $\begingroup$ I think it's meant to be funny. I don't think it stands up to a rigorous analysis. $\endgroup$ – Gerry Myerson Commented Feb 13, 2013 at 23:10
- $\begingroup$ @LarsH If anyone has a hope of finding the black cat in such a dark room, it is the Mathematician who isn't inhibited by such darkness. The main point is that the mathematician will often need to prove that something isn't the case, which is often as hopeless as trying to find said non-existent cat. If the blind man walks to every spot in the room, the cat may have simply just moved. Even if absolute confidence is reached, it may not necessarily be the case. $\endgroup$ – Display Name Commented May 27, 2014 at 0:41
- $\begingroup$ @DisplayName, can you explain why the mathematician has hope of finding the cat? As opposed to, say, a physicist, an engineer, or an animal trainer. I'm not seeing it. The mathematician is well-versed in logic and proofs, but this won't help find the cat in the complete absence of empirical data. I think we're straying from the original point, but since I don't understand the original point it's hard to get back there. Maybe the search for the meaning of this quote is like a search for a cat that doesn't exist. $\endgroup$ – LarsH Commented May 27, 2014 at 21:20
- 1 $\begingroup$ Of possible interest: quoteinvestigator.com/2015/02/15/hidden-cat (which indicates the attribution to Darwin is first found in Tomlinson's essay). $\endgroup$ – Barry Cipra Commented Aug 30, 2016 at 1:53
I guess its something like what you said, but not so much euphemic :) Mostly, researchers are dealing with problems in which there are several people working at it at the same time, so there's some kind of communication as they often work in groups. They also have to attend to conferences to get to know what's new on research world. Sometimes they are trying to "mix" different branches of mathematics in order to develop some new techniques to solve the problems.
I used to have an advisor who once explained that its contributions as a researcher involved solving problems that appeared in engineering & physics literatures but that the authors didn't had the tools and/or time and/or interest to work them out.
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Take a look at http://www.ams.org/programs/students/undergrad/emp-reu for topics near your interests. I happen to know that these are not all the REU programs coming up. there is also http://mathcs.emory.edu/~ono/REUs/ and likely others not listed. Oh, I get it, that one is already full. Anyway, these are pretty well organized. Faculty give an overall picture, students do research projects and write up both group and individual reports. Programs in other countries may or may not be this well organized.
Right, the individual sites listed should give lots of information about past year summer programs, sometimes the reports by the students.
Found this description of mathematical research at https://www.awm-math.org/noetherbrochure/Robinson82.html :
At one point, writes [Elizabeth] Scott, [Julia] Robinson was required to submit a description of what she did each day to Berkeley's personnel office. So she did: "Monday--tried to prove theorem, Tuesday--tried to prove theorem, Wednesday--tried to prove theorem, Thursday--tried to prove theorem; Friday--theorem false."
I can recommend that you read Richard Hamming's book The Art of Doing Science and Engineering: Learning to Learn . It gives many examples about research work.
I think the ultimate goal of mathematical research is to discover all possibilities. The way to do so is to think all possible thoughts, discover all possible rules, and find out all possible objects which follow the rules.
- 1 $\begingroup$ Velcome to the site! $\endgroup$ – kjetil b halvorsen Commented May 19, 2014 at 10:30
For me as an independent mathematical researcher, it includes:
1) Trying to find new, more efficient algorithms. 2) Studying data sets as projected visually through different means to see if new patterns can be made visible, and how to describe them mathematically. 3) Developing new mathematical language and improving on existing language.
It is not so different from how you describe biological or economical research, only that you try to find patterns linked to mathematical laws rather than biological or economical laws.
We know that mathematics is the subject which is not invented it's just found out from nature and as we all know that research means doing a work which is already been searched or done just to get the more accurate results and minimise some how the approximate errors .mathematics research means some how researching the nature so that we are able to do any work more logically and get the most accurate results related to anything with the help of our previous knowledge in mathematics..
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Writing a Mathematics Research Paper CURM Research Group, Fall 2014 Advisor: Dr. Doreen De Leon The following is meant as a guide to the structure and basic content of a mathematical research paper. The Structure of the Paper The basic structure of the paper is as follows: Abstract (approximately four sentences)
Mathematics papers adhere to the same standards as papers written for other classes. While it is a good idea to type your paper, you may have to leave out the formulas and insert them by hand later. It is perfectly acceptable to write formulas by hand in a math paper. Just make sure that your mathematical notation is legible.
9. Having just refereed my first paper, I'll try to say a few of meaningful things. (1) Don't obfuscate with formally correct notation where a general idea -- simply expressible in English with perhaps a few mathematical symbols -- will suffice. (2) Be consistent with notations/conventions.
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Chapter 9: Components of Your Research Paper. Chapters 4, 5, and 6 introduce you to writing mathematics, and Chapters 6, 7, and 8 instruct you in how to conduct your research in a logical fashion. Chapter 9 helps you pull it all together for the formal paper. The parts of the research paper are discussed.
2-3-4 rule: Consider splitting every sentence of more than 2 lines, every sentence with more than 3 verbs, and every paragraph with more than 4 "long" sentences. Mathspeak should be readable. BAD: Let k>0 be an integer. GOOD: Let k be a positive integer or Consider an integer k>0. BAD: Let x Î Rn be a vector.
A central goal of the course is to teach students how to write effective, journal-style mathematics papers. Papers are a key way in which mathematicians share research findings and learn about others' work. For each research project, each student group writes and revises a paper in the style of a professional mathematics journal paper.
In this note we explain the importance of clarity and give other tips for mathematical writing. Some of it is mildly opinionated, but most is just common sense and experience. 1. Be clear! This is the golden rule, really. It's absolutely paramount. Let me explain. 1.1.
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 ...
Francis Su's 2015 MAA Focus piece "Some Guidelines for Good Mathematical Writing" shares both basics and advice "toward elegance.". While the piece is written at a level that's accessible to students, it also contains gems that will serve even experienced math writers. For instance, he advises that writers "Decide what's ...
Drastically oversimplified, a mathematics research project goes like this: Develop, or select from the existing literature, a mathematical statement ("conjecture") that you think will be of interest to other mathematicians, and whose truth or falsity is not known. (For example, "There are infinitely many pairs of prime numbers that differ by 2.")
The are four general rules that must be respected: (1) zero, i.e. the vector whose coordinates are all 0, must be in the group; (2) if any element (i.e. vector) is in the group, then its negative must also be in the group; (3) if an element is in the group, then any multiple of it must also be in the group. (This applies to integer multiples ...
Mathematics research influences student learning in a number of ways: Research provides students with an understanding of what it means to do mathematics and of mathematics as a living, growing field. Writing mathematics and problem-solving become central to student's learning. Students develop mastery of mathematics topics.
The standard for citation in mathematics papers is very different than in (say) humanities. The AMS Ethical Guidelines say. The correct attribution of mathematical results is essential, both because it encourages creativity, by benefiting the creator whose career may depend on the recognition of the work and because it informs the community of when, where, and sometimes how original ideas ...
1. I think the ultimate goal of mathematical research is to discover all possibilities. The way to do so is to think all possible thoughts, discover all possible rules, and find out all possible objects which follow the rules. Share.
This guide will give you a brief overview of the parts of a mathematics research paper. Following the guide is a sample write up so you can see how one person wrote about her research experience and shared her results. A formal mathematics research paper includes a number of sections. These will be appropriate for your write-up as well.
on how to go about reading mathematics papers and gaining understanding from them. The advice is particularly aimed at inexperienced readers. A professional mathematician may read from tens to hundreds of papers every year, including pub-lished papers, manuscripts sent for refereeing by jour-nals, and draft papers written by students and col ...
In applied mathematics, we look for important connections with other disciplines that may inspire interesting and useful mathematics, and where innovative mathematical reasoning may lead to new insights and applications. Combinatorics. Computational Biology. Physical Applied Mathematics. Computational Science & Numerical Analysis.
Math papers usually don't have methodology sections and sometimes not even conclusions, unless you're doing applied math and running simulations or something. After the introduction, sometimes it's just theorem proof theorem proof etc.
In my experience, ML papers tend to have relatively poorly explained (and sometimes flimsy) math formulations, so don't be frustrated if you do not understand absolutely every equation. Most of the times in those papers what matters is the engineering/heuristic argument, and math is just the language used to describe it.
While research shows that knowledge of math facts is important, Boaler said the best way for students to know math facts is by using them regularly and developing understanding of numerical relations. Memorization, speed and test pressure can be damaging, she added. Number sense is critical.