While one should always study the method of a great artist, one should never imitate his manner. The manner of an artist is essentially individual, the method of an artist is absolutely universal. The first is personality, which no one should copy; the second is perfection, which all should aim at. (Oscar Wilde, A critic in Pall Mall, p. 195)

When, as a graduate student, one is starting out one’s research in a mathematical subject, one usually begins by reading the papers of the current and past leaders of the field.  Initially, one’s understanding of the subject is fairly limited, and so it is natural to view these papers as being authoritative, especially if their authors are well-known.

Eventually, though, one acquires a fair fraction of the insights and understanding conveyed by the existing literature, and is able to apply it to produce a new result or observation that goes beyond that literature (or, at least, makes explicit what was only implicitly buried in previous papers).  When the ramifications and extensions of these new advances have been explored to their natural extent, it then becomes time to write up these results as a research paper.

Of course, as your work is almost certainly based in part on the previous literature, one should cite that literature whenever appropriate, and compare and contrast your own work with that literature in an accurate, professional, and informative manner.  Also, one should try to maintain some level of notational consistency with the previous literature, such as using the same fundamental definitions and to use similar notation, so that expert readers who are already familiar with that literature can quickly get up to speed on your own work.  And if one of the arguments in your work is standard in the literature, it certainly makes sense to structure the argument in a standard fashion if possible, again to assist the experts reading your paper.

However, one should not go so far as to copy entire paragraphs or more of text from a prior paper, except when used as a direct quotation to illustrate some historical point.   First of all, if one does not properly attribute that text (e.g. “As Bourbaki [17, p. 146] writes,”, or, for that matter, the Oscar Wilde quote above), then one runs the risk of committing plagiarism.  But even if the text is properly attributed, copying the text verbatim, without updating it to reflect more recent developments (including that in the paper being written) and to add your own simplifications and insights, is a redundant waste of space and a lost opportunity to advance the subject.  If one is tempted to copy a significant portion of text from a prior reference without adding anything significantly new, one should instead simply cite the previous reference appropriately, e.g. “See [27, Section 4] for further discussion.” or “A proof can be found in [9, Lemma 2.4].” (cf. “Give appropriate amounts of detail“).

Of course, there are reasons to duplicate to some extent some discussion or argument that was present in a previous paper:

  • As mentioned earlier, one may wish to make some historical point, for instance to track the development of a mathematical idea over time.
  • If the paper is obscure and not widely available, reproducing a key argument from that paper may serve as a convenience to the reader.
  • Also, if the form of that argument can be used to motivate other arguments in your paper, then it can be worth putting in that argument so that it can be referred to later in the paper.
  • The precise result needed for your paper may differ slightly from what is already established in the literature, and so one needs to either write out a modified version of the proof, or else point to the original proof but indicate what modifications need to be made.  (The latter is suitable if the changes are particularly minor in nature.)
  • The existing paper may have an argument which can be updated, simplified, modernised, or otherwise improved thanks to more recent advances or insights in the area (including your own).  It can then be a service to the field to place an updated version of the argument in the literature (with full citations to the paper containing the original argument, of course).

However, when one is not simply quoting the prior text for historical or archival purposes, it is best to paraphrase and interpret the previous text rather than to copy that text verbatim.  This is for a number of reasons:

  • One wants to avoid conveying any impression to readers, referees, or editors of plagiarism, padding, or intellectual laziness in one’s papers.  (Note that the latter is a danger even if one is copying from one’s own work, rather than that of others.)
  • The prior work may be dated in view of more recent developments and insights, as mentioned above.
  • If you are copying or adapted a piece of text from another author that you do not fully understand yourself, then it may end up being inappropriate or incongruous for your intended purpose, and may convey the impression of superficiality or being ill-informed.  If the text becomes inaccurate due to this adaptation, then this can also cause some embarrassment and annoyance for the original author of that text.
  • Excessive use of quotation from famous mathematicians to make one’s own work look more impressive is the mathematical equivalent of name-dropping, and should be avoided.  Appeal to authority should not be the primary basis for motivating a paper; a handful of citations to demonstrate the depth of interest in the problem being studied is usually sufficient.
  • But most importantly of all, for one’s further mathematical development and career, one needs to develop one’s own consistent mathematical “voice” and style, and to avoid the impression of simply imitating the voices of other authors.  There is no need in this subject for the mathematical equivalent of a parrot, and a text which is a mix of the author’s voice and the voice of others can read very strangely.

Of course, if one is paraphrasing a previous work, one should cite that work appropriately (e.g. “The proof here is loosely based on that in [5].” or “This discussion is inspired by a related discussion in [10].”).

In some cases, the imitation of a previous author’s style and text is intended as a sign of respect or flattery for that author.  This is misguided; an author will in fact often find such mimicry to actually be somewhat offensive.  If one wants to truly respect a mathematician, then understand that mathematician’s methods, results, and exposition, and improve, update, adapt, and advance all three.  Even the greatest mathematician’s contributions should advance with the field, rather than being worshipped and preserved in some supposed state of perfection; the latter is mostly suitable only for historical purposes.

Another possible reason for copying the style of a more senior mathematician is that one does not yet have the self-confidence to write in one’s own style and voice.  While this is justifiable to some extent when one is just starting one’s career, it becomes less excusable as one continues one’s research.  If one is hesitant to state things in one’s own fashion, it is perfectly acceptable to couch such text with the appropriate caveats (e.g. “to the author’s knowledge, this observation is new” or “While Lemma 2.5 is usually phrased in a topological fashion, we found the following, more geometric, formulation to be more convenient for our applications”).    And if one does not feel confident enough in one’s understanding of a subject to explain it in any other way than copying from a previous paper, then this should be taken as a sign that one still needs to internalise the subject further.

When writing a paper with one or more coauthors, there will inevitably be distinctions in style, and so initially different sections may have sharply different tones due to their being largely written by different subsets of coauthors; but I usually find that after a few rounds of editing, the voices are harmonised into a style which is clearly derived from, but distinct from, each of the individual styles.   Ideally, one should understand and respect the underlying stylistic decisions of one’s coauthors, but at the same time be willing to take the initiative and find ways to formulate the text and arrangement to smoothly reconcile the coauthor’s preferences with one’s own; if all goes well, this can lead to a level of exposition and presentation that is superior to what each of the individual authors could separately achieve.  (Of course, if you are to perform major edits on a coauthor’s contribution, some consultation with that coauthor is presumably desirable.)  This process can be quite educational; my own writing style has definitely been influenced in a positive fashion by those of my coauthors.

Developing one’s own style is, by definition, a very personal process; while external advice or role models can certainly be of some influence, they are of limited utility after a certain point.  But finding an individual style which is comfortable and effective for both you and your readers is an important mark of one’s mathematical maturity, and is a goal that is definitely worth pursuing.

See also “Write professionally.”