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I am a PhD student (mid stage) and my work includes the use of mathematics. I try to learn as much as possible. I am facing a problem: I spent enough time on the mathematics required for my research, and according to my research supervisor I am good at understanding proofs, theorems, etc. at an intuitive level, but in writing I still have some work to do. Many times I try to write things and, although I think I am not that so bad, there is still a problem to be resolved.

Question: Is writing in research so very important? I have seen some research papers of some star people whose writing does seems to be very impressive to me (although I may be wrong).

Edit: Please note that I don't consider myself a star: that's why I am asking this question. I am trying my best to improve my writing skills and it has improved to some extent, but to me it appears that I may not get that much better at writing. Let me tell you a story: once during my PhD I worked with a professor other than my supervisor, and then we got some result so he ask me to write a paper. I took 4-5 days write a paper 8-9 pages long, then he told me that I wrote in a cryptic manner. (This happened in my first year.)

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    Yes writing matters a lot. If/when you become a "star" you can get away with bad writing. Until then... – user9646 Apr 10 '18 at 15:33
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    Is writing in research important? Yes. With that in mind, perhaps you have other questions? – user2768 Apr 10 '18 at 15:49
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    If you can’t clearly convey your results and the impact of then, then your efforts are basically wasted... – Jon Custer Apr 10 '18 at 15:56
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    "I took 4-5 days write a paper 8-9 pages long, then he told me that I wrote in a cryptic manner." I don't really see what you want to say here but to me it seems you want to stress that although you wrote for such a long time it was seen as cryptic? I just want to comment on that because 4-5 days for a 8-9 pages paper is REALLY short and hardly anybody would create a proper scientific paper in that short time. Writing takes time and as other stated is definitely really important in any field of research. Good luck! – Petey Pete Apr 11 '18 at 7:42
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    Because these don't seem to be mentioned elsewhere, here are a couple of well known treatments of mathematics writing: (1) How to Write Mathematics by Paul Halmos (1970; see this also); (2) Mathematical Writing by Donald E. Knuth and Tracy Larrabee and Paul M. Roberts (118 page summary/notes of a 1987 Stanford University course; errata and information about the notes) – Dave L Renfro Apr 12 '18 at 6:29
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Writing matters a lot, and a lot of the time probably most PhDs (and even postdocs) face a steep learning curve. This curve is particularly bad in a field where writing is your tool, such as Philosophy. Even when analysis takes the form of symbols, in mathematics and logic, it is essential to convey your ideas clearly. Clarity is particularly problematic for young researchers, because of our tendency to hide behind pretension. Pretension, confusion, clunkiness must be removed for your ideas to come through.

Overcoming these issues is not essential to get an inexperienced researcher to approve of your work, but any researcher worth their salt will know when what you say is bullshit, or when what you say might have value but is unclear. I say this as an experienced, but not necessarily a good, writer. My writing has often been criticised and quite rightly too. I wish I had learned to write better earlier, is my point and the advice I give.

Because writing is important, there is a second question. How to focus on improving it?

  1. Read good writing.

  2. Read a book on good writing, I recommend Joshua Schimel's Writing Science.

  3. Once you have written, wait, and then go back and read it. And then edit it. You will read it from the perspective of another person and hence realise why writing is so important and partly what you need to do to improve.

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    +1 from me because I think your writing/wording is very impressive.I can suggest an additional book: "On Writing Well: The Classic Guide to Writing Nonfiction" from William Zinsser – Amazonasmann Apr 10 '18 at 17:56
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    +1 for 3) --- I think not enough people do this. Many times after I reread a draft after letting it sit for a few weeks, I think: "What it the world what I trying to say?" It's also a great way to catch errors. – Kimball Apr 10 '18 at 20:38
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    Might be worth adding that many grad programs offer non-credit writing courses for native-English and English-as-an-additional-language speakers/writers. If it's a resource at the OP's institution it would be worth taking advantage of. My school offers an introductory course that essentially is based on your 3 suggestions. Plus it can be helpful to learn with others in a similar situation. – CuriousFindings Apr 10 '18 at 22:31
  • Is there any reason to single out mathematicians in the list "young scientists, mathematicians, and other researchers" which apparently covers the whole spectrum of academia? – user9646 Apr 11 '18 at 16:55
  • "Pretension, confusion, clunkiness". You forgot excessive wordiness. – Faheem Mitha Apr 11 '18 at 17:23
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I have not seen an answer mentioning the following aspect:

Writing grant proposals.

Being a mediocre article writer can be ok, you'll still get your publications, eventually, since mathematicians can get through bad writing with some effort. Or, they might not read your papers.

But, to stay in academia, you will eventually write research proposals, and explain your research ideas and previous results in a very clear, coherent and professional manner. This is a very fierce competition, and the audience is not experts in your field. Well, some are, and some are not, making the writing of research proposals very difficult, compared to writing a research article.

And since grants eventually will pay your salary, I'd say, it's quite important.

  • This is somewhat field/location specific. If OP does Science, it’s applicable, if they do more “pure mathematics” or technical humanities (like technical work in philosophy), they might never write a grant proposal in their career. – Dennis Apr 11 '18 at 4:25
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    @Dennis What would make you think mathematicians don’t need to write grant proposals? They do!! – Danu Apr 11 '18 at 8:36
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    @Dennis I do pure mathematics, and basically everyone I talk to in my profession does grant proposals. No grants usually means to money to do research, and you're stuck with teaching. Getting a pure teaching position still requires a solid research track record in Europe (the competition is fierce!). – Per Alexandersson Apr 11 '18 at 10:33
  • @PerAlexandersson yea, I think that’s more the “location dependent” component. Europe seems more uniformly grant-driven, whereas teaching heavy jobs seem more common in the US. – Dennis Apr 11 '18 at 13:24
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Is writing in research so very important?

Yes.

Because you share your work with the world using some symbols. These symbols, might be the definition of a particular function as well as a collection of Latin letters to explain your findings.

No matter how impressive and groundbreaking your theorem is, it is not true until you prove it with a mathematical notation. If you think in a wider scale, no matter how good your research is, it is not complete unless you carefully and technically write it down.

To put it in different words, even if you discover teleportation, without technical writing and proper explanation, it is the same thing some random person claiming that they discovered time travel.

I would refer to this excellent and short guide to gain an insight about technical writing.

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    I'd pick on the wording "it is not true until you prove it with a mathematical notation" - what I think would be better to say is that nobody cares about your theorem until you communicate it to them. But that communication doesn't necessarily have to be in the form of mathematical notation. If you somehow discovered a theorem which is true and which you can explain perfectly well in words, that's fine. I wouldn't want anyone getting the impression that you have to arbitrarily introduce some mathematical symbols into the proof before it becomes valid. – David Z Apr 11 '18 at 6:21
  • @DavidZ You are right. That is a good observation – padawan Apr 11 '18 at 6:52
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You should probably set aside a lot of time on projects for editing your writing. It is a skill that you get better at over time, especially with practice and coaching.

Why does writing matter? I've learned that writing is part of the thinking process. All of my education and work has been in my native language, but even so I found working with my university's writing center to be profoundly helpful. Often writing and editing helps you refine ideas as you go. It is while trying to explain or properly capture an idea that some of the limitations or gaps become apparent, such that you can acknowledge them and think of additional lines of argument or additional aspects of the problem.

I am guessing that few writing tutors will be able to help with the details of a mathematical proof, though they might be able to help make sure that the linear logic of an argument is apparent. ("What's your starting point? How are you defining Z? What's your next step? If the theorem requires that X and Y are satisfied, where do you address X, and where do you address Y?")

Can your professors tell you about the types of problems in your current writing? (E.g. confusing wording; ideas are too vague / not specific enough; logical jumps between steps; hard to follow the setup of the initial problem; not considering key cases; etc.) You may want to ask them if there is a class in writing proofs that you can take or audit, or if they have other resources to suggest. (Also, you say that you're using math; if your field does not have its own class in proofs, you may want to talk with the math department and arrange to attend the class they have that is meant to introduce undergraduates to proofs, even if the math content is review for you.)

Other advice: Based on some of the words you selected in the question, I am guessing you do not natively speak English. (I edited the question to help make the grammar more standard.) So if your university has a writing center or writing tutors, or if you can get your more finalized work line-edited, you should take advantage of those resources. (For instance, you could hire a graduate student who is good at writing to edit your work (expect a few rounds--they'll have questions), or arrange a time swap so you can tutor a graduate student in a required math/stats class in exchange for editing help.)

Good luck!

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It seems you are probably asking about "good English writing". I'd like to offer a slightly different perspective, from the point of view of someone who also does a fair amount of mathematics (I'm a CS theory student). I find that formally writing up proofs helps me discover minutiae that I might have missed earlier - in fact there are always some technical edge cases that go unnoticed by me on the whiteboard until I actually start writing a formal statement. So I think writing things as you go also helps solidify your math.

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Most of the answers assume it's about scientific writing, but not about writing down you proofs. (You have a bigger problem, if it's about the latter.)

While all stress that writing is important, and I agree, I'd like to put in my 2¢.

  • Writing can be learned. Look around for courses on scientific writing. There should be some. Attend them. Read other's papers, especially those praised for good writing. Read Strunk and White or Chicago manual of style or whatever.
  • You gradually get better over time. It should not be underestimated how much better at writing you get the longer you are doing your usual academic inner loop.

    Remember your advisor with god-like writing skills, who basically rewrote your first paper from scratch and it got so much better? Now look up their CV and look how long they are in academia. When you'd be in academia for the same time, you'd be comparably good.

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It's important but not as important as others here are conveying. The notion of proxy judgements is pertinent; people don't judge you (your thoughts, personality), they judge what they can see. Under this principle, badly formatted references and/or poor grammar could spell someone's doom. But as anyone doing research in STEM knows, there are tons of papers out there that are so badly written that the proxy judgement principle would hardly seem to have any force.

My personal problem with scholarly articles is the tendency towards obfuscation - hiding the intuition of ideas behind unnecessarily complex phrasing (mathematical or otherwise). Unfortunately this appears to be the norm with most such writings.

A way to make your research stand out would be to intentionally expose your ideas in very simple terms that (almost) anyone can understand and then slowly introduce the formalisms. If your ideas are good and you take this approach, you will stand out in a good light.

If your ideas are weak, then obfuscation and high prose will be essential.

Nice writing will still always be a plus, but its absence is not a show stopper.

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“Those who know that they are profound strive for clarity. Those who would like to seem profound to the crowd strive for obscurity.” ---- Nietzsche (1882) The Gay Science (The Joyful Wisdom), p. 173.

Even for mathematical research, good writing is crucial: Mathematical writing requires clear and simple presentation of results. You should be aiming to make your writing as clear and simple as possible, so as not to obscure the mathematical insights in your work. As with all academic writing, you should have a strong command of the English language (or whatever language you are writing in) and this should also be backed up with the ability to revise your notation and presentation of mathematical work to make it as simple as possible.

In mathematical research there is an additional aspect to your writing involving the presentation of mathematical results. There are many stylistic choices that must be made in presenting theorems and other results (e.g., notational choices, use of intermediate functions, lemmas, theorems, etc.), and these make an enormous difference to how easy it is for your reader to follow what you have done. Good mathematical writing requires you to present your results in a logical order, with clear notation of all objects, which is minimally taxing on the reader. This often involves choices about whether it helps to define subsidiary notation for functions/objects you will use over and over again. It also requires revising the presentation of results to get them to their simplest form.

If your work is being described as "cryptic" by other researchers, this is a sign that you are not succeeding in presenting your results in their most clear and simple form. Practice simplifying your notation and presentation until others in your field are able to read and understand your arguments with ease. Simplify, simplify, simplify!

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Yes, writing matters a lot. There are many, many papers out there and every reader will be selective in what they read. Math is no exception, even there are two counterbalancing forces that make math special. Good writing matters less in mathematics because expertise is very narrow and someone interested in a very specific question might not have much choice in what papers to consult. Good writing matters more in mathematics because mathematical texts are incredibly dense and hard to read even when executed well. In some experimental sciences, it might be possible to ignore the lousy writing and just go to table 5 with the results; in math, this is not an option.

You write that it appears to you that you may not get much better at writing, but writing is learnable. A big problem with learning to write well is that good writing is invisible. It is possible to show off one's way with words and use clever turns of phrase every now and then, but good writing in an academic context usually means getting the message across with the least amount of struggle on the side of the reader. This requires making a lot of choices that you might not notice have been made. But one can learn what these choices are. Sadly, a lot of writing advice is really bad and comes from people with no linguistic background. Excellent books on writing by people with an academic background are The Sense of Style by Steven Pinker and Style by Joseph M. Williams (there exist several versions and editions of that book).

If you are struggling a lot with writing, academic phrasebooks might be useful. They will not turn you into a good writer, but allow you to be at least okayish. In the context of mathematics, there is the excellent Writing Mathematical Papers in English by Jerzy Trzeciak, which I would recommend even to people who write well.

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