33

When writing a PhD thesis on mathematics one needs to quote many results by others, such as

The following theorem was proved by ABC in [1].

Theorem. Bla, bla, bla...

My question is when should one include a proof of the result in his/her PhD thesis, if it is basically the same as in the reference? I do not quite see the point of copy-pasting the proof by others. Of course if one has a completely different proof of the same result, it is probably suitable to include it.

  • A comment rather than an answer due to a physics perspective. Some factors to take into account: How well-known are the theorem and the particular proof? How fundamental are they to your thesis? Is it possible to summarise the proof (e.g. by reference to a better-known proof which was built on to do the bit you're interested in)? I assume your supervisor will read the thesis and comment (the approach will of course vary between supervisors) but they are best placed to help you answer this for your specific case. However you may as well do it especially if you cut/paste LaTeX a lot. – Chris H Dec 9 '14 at 19:36
41

As a general rule, you can cite other people's theorems without explaining their proofs, and omitting a proof is a good idea if it would be a lengthy distraction. However, there are several reasons why including such a proof could be helpful:

  1. Including it may be convenient for the reader if the proof is short. It's annoying to look up another paper and discover that you only needed a short argument that could easily have been explained in the original paper. Extracting information from a reference can be cumbersome (you have to locate exactly what you're looking for, figure out what it depends on, sort out the notation, etc.), while giving your own explanation can help readers avoid some of these difficulties.

  2. Even if the proof is not particularly short, it may serve as a warm-up for new applications of the same techniques. Reminding the reader how they work may make your paper much easier to read than if you just dive into the newest and most complicated case.

Ph.D. dissertations are something of a special case, because your advisor may encourage you to include extra details in the background sections (beyond what you might include in a published paper). This is partly a matter of demonstrating your mastery of the area and partly a matter of writing a useful survey for others. Advisors differ in how they approach this: some think it's a waste of time and it's best just to focus on writing a published paper, while others think writing a more extensive dissertation is a valuable learning exercise. This is an issue you should discuss with your advisor.

  • Thank you for your detailed answer! The information is very helpful. Usually a mathematical PhD thesis is over 100 pages long. I am curious usually what percentage of a typical thesis is "non-essential"; that is, containing definitions, results with or without proof by others, etc.? Of course the answer could be "it depends"; but I am looking for a general (unwritten?) rule. – Zuriel Dec 8 '14 at 14:47
  • Unfortunately I don't have a good feeling for this ratio. Besides including more background and more details of new results, theses can also be longer because of inefficient formatting required by the university. – Anonymous Mathematician Dec 8 '14 at 15:59
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    One of the things that commonly comes up in papers that I review is that the graduate student author has written up a version of the research for the thesis that has way too much background (such as proofs of previously published theorems.) It may be appropriate to include these in a thesis, but they need to be edited out before submitting the material for publication as a paper. – Brian Borchers Dec 8 '14 at 16:08
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    3. You need/want to explicitly use/cite elements from the proof somebody else write, e.g. like "With c as defined in the proof of..." or "with a similar argument to ...". – Raphael Dec 8 '14 at 20:06
  • @Zuriel Definitions are not "non-essential", unless the definition and associated notation are so well-known that the great majority of your readers will already know it. (For example, many papers redefine treewidth, to establish the notation that will be used.) If the definition is short, it doesn't take much space to repeat it; if it is long, you should include it because you can't expect your reder to remember all that detail. – David Richerby Dec 9 '14 at 8:06
17

This is a good question, and as Anonymous Mathematician indicates, it is well worth discussing with your advisor.

Essentially what you are asking is whether and when to include exposition in your PhD thesis. The answer is that it is rarely strictly required, but it is often expected, in many cases encouraged, and in some cases not necessary. There are a lot of nuances here and I don't foresee a comprehensive general answer being possible. (Anonymous Mathematician's answer is excellent, and I am essentially corroborating it.)

Mathematics has a proud tradition of PhD theses having significant expository content. (In my thesis, Chapter 0 is expository. It occupies about half of the thesis. This is a bit on the lengthy side, but not so unusual.)

One reason that this is done is because a PhD thesis is usually the last chance that your mentors get to lean on you and require that you show your mastery of highly difficult, technical concepts. When I am a committee member on a math PhD thesis, I generally want to see at least enough exposition to convince me that the writer has mastered the concepts, definitions and objects used in the thesis. Especially, I want to see key definitions in a lot of detail, even if they are long and taken from other sources.

Another reason this is done is that the cultural standard in mathematics is that PhD theses can be significantly more discursive than published papers. When a PhD thesis gets converted to a paper, often there is a compression of 2:1 or more in terms of the page count, and often the results that appear in the paper are stronger than what appear in the thesis. (In mathematics, I gather unlike some other fields, one most often publishes the lion's share of one's thesis work after completing the thesis, not before.) Something's gotta give, and often math papers published in the strongest journals are written so that every single page contains an important new idea or truly difficult calculation. This density of content is a point of pride of the top journals, but it can make the papers awfully difficult to read. A lot of theses are famous for being the best sources of exposition for the topics they contain.

Having said all this, it seems clear that little value is added by "copy-pasting". Taken literally: copying lengthy proofs verbatim from other sources would be plagiarism if carried too far. Most exposition in a PhD thesis is filling a gap in the literature, not reproducing it. Good exposition synthesizes several sources, offers new perspectives (including a bridge to the novel results, as AM mentions), chooses notation and hypotheses in a globally appropriate way, and so forth.

Finally: formal proof is often the least important part of good mathematical exposition. Getting the definitions and statements just right and putting them in context is more important. Most contemporary math PhD theses build on significant technical foundations, not all of which the student is expected to be personally conversant with. A PhD thesis is not supposed to be "logically self-contained" in any formal sense, only to demonstrate mastery in the eyes of the committee members and to be a useful document for the reader in the eyes of the advisor and (most importantly) the writer. If you are thinking of more or less copying a proof "for completeness", that may not be the way to go.

10

Generally, you should only need to reproduce the proof verbatim in cases where you need to dissect it, call out one part of the proof in particular, or you intend to extend it directly using similar arguments, otherwise, stating the theorem and citing a work where it is proved should be sufficient.

-5

I don't see any reason to copy a proof found somewhere else. I think its fine to say that this result and its proof can be found on this article. In my opinion, the proofs in a dissertation should be your own.

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    Since you are just stating your opinion, could you tell us a bit about your background and experience with math PhD theses? – Pete L. Clark Dec 8 '14 at 18:09
  • I don't have any. I have discussed people with PhD and they have been said that in every scientific papers you should put only new results that has not been done before. Otherwise it is plagiarism. – student Dec 8 '14 at 18:14
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    A PhD thesis is not quite a scientific paper, and the standards for how much exposition to include are variable. More to the point, there are many circumstances in which scientific and mathematical papers contain results which are not new and/or due to others. If you think that this is (inherently) plagiarism, you have a significant misunderstanding of this important concept. I recommend that you discuss this with your advisor. – Pete L. Clark Dec 8 '14 at 18:18
  • I didn't say that one's dissertation can't contain results found elsewhere. Of course you can, otherwise you can't really do mathematics. But I would not copy a proof found in elsewhere. – student Dec 8 '14 at 18:22
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    Restating somebody else's proof in your own words is not plagiarism. Nobody here (I hope) is recommending copying the proof verbatim from somebody else's paper. The OP says "basically the same", which I interpret as meaning not the same words, but the same basic mathematical argument. – Peter Shor Dec 8 '14 at 23:00

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