45

I'm writing a paper in which I cite a lot of results that appear in Schikhof's Ultrametric Calculus. Some of these results are exercises in Schikhof's book. These exercises are not difficult, but are laborious. Thus, if I write the proofs, the article may extend by about two or three pages.

Should I write the proofs or simply cite them? Schikhof is a very well respected mathematician, and I have never found any errors in his book. Obviously, I have checked that the exercises are correct.

(If it were one exercise, I would write the proof in my article, as I have seen in other articles, but in my case there are about five exercises.)

TRiG
  • 121
  • 7
efs
  • 3,099
  • I forgot a detail. I have not been able to found this results proved in old papers, or in other books. – efs May 02 '17 at 14:43
  • 4
    I would cite the exercises. – Anthony Quas May 02 '17 at 14:46
  • 10
    A compromise, which I have used, is to post the solutions to the exercise on your blog or in some other public place online, and then cite the exercises and their online solutions. – Joel David Hamkins May 02 '17 at 14:59
  • If there are hints (or if you can give them near the end of your paper as spoilers), I would mention them in your papers for those readers who want to digest the material, not challenge themselves unduly. Gerhard "Unless You're Writing Another Book" Paseman, 2017.05.02. – Gerhard Paseman May 02 '17 at 15:01
  • 14
    I think this is a useful question, and do not agree with the vote to close. Authors often use exercises as part of the logical structure of their books, so in general one can't feel compelled to trust a book's theorems any more than its exercises. – R.P. May 02 '17 at 15:02
  • 2
    @René, I agree that this is a useful question; I've wondered about it myself. I do submit from a psychological point of view that, while logically speaking the proof of a theorem is no sounder than its weakest link, which may be an exercise, psychologically speaking I have a lot more confidence that a lot of people have done basic sanity checking on the statements of most of the theorems in the major textbooks than that that sanity checking has been done for most of the exercises. – LSpice May 02 '17 at 16:07
  • (I imagine I could come up with more examples of wrong exercises—as in, the statement, not just the proof, is wrong—than wrong theorems in popular textbooks.) – LSpice May 02 '17 at 16:08
  • 8
    One of my papers grew up from an exercise that turned out to be wrong. So I would be careful with relying on exercises. – Yiftach Barnea May 02 '17 at 16:31
  • 22
    I would vote for putting the proofs in the paper, perhaps as an appendix. In general I think that people worry too much about keeping papers short. There is no real loss in adding three pages. – Neil Strickland May 02 '17 at 17:06
  • 1
    I would leave the exercises out of the paper, and see what the referee thinks... – Theo Johnson-Freyd May 02 '17 at 19:08
  • i recommend proceeding exactly as you would if they did not appear as exercises. in my view, a reference is only useful for a proof, not an assertion by a reputable person. would you have cited Fermat's marginalia? (he is very reputable). I have known of erroneous assertions made informally that wound up being repeated as if true in print. so even stating that you have checked them and making the details available somehow, is much preferable to just pointing out that some authority has claimed them as true. I would also cite the exercise, but only in order to grant priority. – roy smith May 02 '17 at 21:44
  • i may be an outlier; e.g. i once declined to referee a paper because it relied on a result claimed by a famous mathematician, but whose proof had not yet appeared. i just felt i could not vouch for the correctness of the result with such a gap. that one was ok, but other famous mathematicins have made similar assertions to me that they later retracted. – roy smith May 02 '17 at 21:57
  • 1
    Given the comment from @YiftachBarnea, and the answer from Pace Nielsen, perhaps a good (soft, CW) question for MO would be, what have been some examples of faulty exercises in (university-level) Mathematics textbooks? – Gerry Myerson May 03 '17 at 23:27

3 Answers3

27

The answer is essentially given in the comments, so let me summarize:

  1. It is a frequent situation that one has to cite an exercise.

  2. It is legitimate. (Polya-Szego is cited > 1400 times according to Mathscinet)

  3. The best thing is to cite a place where the statement is proved, but if you cannot find such a place, citing an exercise is the second best choice.

  4. You can solve the exercise in your paper, or not solve (depending on the difficulty of the exercise and space limitations and other considerations).

And finally my own recommendation: When you refer to an exercise, solve it yourself, no matter whether you include a solution to your paper or not.

Similar considerations apply to handbooks, like Tables of Integrals, etc. They are essentially made for this purpose, but there are sometimes mistakes, not frequently. (Gradshtein-Ryzhik is cited > 2200 times according to Mathscinet, Abramowitz-Stegun 1740.)

Alexandre Eremenko
  • 88,753
  • 1
    I would include a solution or at least a hint, to save time to your readers. – Leo Alonso May 02 '17 at 16:59
  • 3
    An exercise rated 40 or higher in Knuth's The Art of Computer Programming would not necessarily be something that one should expect to solve in such a context... – Steve Huntsman May 02 '17 at 17:04
  • 3
    @Steve Huntsman: My principle is that one has to know and understand the proofs of EVERYTHING one uses in a paper. – Alexandre Eremenko May 02 '17 at 17:19
  • 1
    Gradshtein-Ryzhik has a citation for every formula. But often the citation is to an earlier table of integrals, which brings one no closer to an actual proof . . . – Noam D. Elkies May 02 '17 at 17:26
  • @Noam D. Elkies: yes. But people usually do not care to dig out the original source: they generally trust Gradshtein-Ryzhik. – Alexandre Eremenko May 02 '17 at 17:32
  • 14
    BTW "know and understand the proofs of EVERYTHING one uses in a paper" is a good principle but sometimes unworkable -- an important example is the classification of finite simple groups and various results that depend on it. – Noam D. Elkies May 02 '17 at 17:42
  • 1
    @Noam D. Elkies: I agree. This point was once discussed at length on MO, sorry I don't remember the exact reference. – Alexandre Eremenko May 02 '17 at 17:44
  • @SteveHuntsman: I'd certainly trust an exercise from Knuth. (And the books include solutions or pointers to solutions to most exercises, although I'm not sure if that falls off at high difficulties.) – Michael Lugo May 02 '17 at 17:49
  • 1
    Polya-Szego is not a good example for this. Sure, it has exercises. But it also has the solutions. – Gerald Edgar May 02 '17 at 21:47
  • i would cite Hironaka's resolution of singularities, but would still feel somewhat guilty about not having understood the proof. I.e. I agree with Alexandre in principle, but am not entirely true to my own principles. – roy smith May 02 '17 at 21:48
  • 1
    @AlexandreEremenko : You might be thinking about this MO question: https://mathoverflow.net/questions/23758/published-results-when-to-take-them-for-granted – Timothy Chow May 02 '17 at 21:52
  • 1
    @NoamD.Elkies "(...) is a good principle but sometimes unworkable". Well, my impression is that is not just "sometimes" but "usually". – Matemáticos Chibchas May 03 '17 at 05:14
  • @Gerald Edgar: Polya-Szego has hints and references, rather than solutions. – Alexandre Eremenko May 03 '17 at 06:15
14

I agree wholeheartedly with Alexandre's answer, but there is one other principle I'd add to his list which I believe is essential.

  1. Whether you merely cite the problem, include some small hints, provide copious hints, or give a full solution, should roughly correspond to the difficulty of the problem.

Of course, you can only know how difficult an exercise is if you have done it for yourself. Some exercises really are easy to experts in the field. Others are extremely difficult. Some are impossible.

And sometimes problems are just wrong. Indeed, one of my papers is a counter-example to the first two exercises in a well-known text.

Pace Nielsen
  • 18,047
  • 4
  • 72
  • 133
  • 4
    I once submitted a 10 page paper quoting a number of "well known" results, for which the referee requested references or hints. When no references were available, the paper swelled to over 70 pages. Fortunately all the results did survive intact. – roy smith May 02 '17 at 21:50
  • Yikes! Hopefully that upped your citation count, as others could use your paper as a reference for those well-known results. – Pace Nielsen May 02 '17 at 21:56
  • 1
    interestingly, a shorter version, using the same method to give an easier proof of an already known, and more appealing result, gets more citations. reading the long one is too big a slog i guess. i even had requests for references for one of the results from someone to whom i had already given a copy of the paper which included a proof of the desired result. but maybe that provoked a citation. – roy smith May 02 '17 at 22:14
6

Here is one additional data point. The standard reference for symmetric function is Macdonald's "Symmetric functions and Hall polynomials". Most of the content of this book is in the exercises; each section of the book contains many pages of useful results and formulas stated without proof. According to Google Scholar, this book has been cited 7735 times, and it seems likely that many (most?) of these citations are references to exercises in the book.

Dan Petersen
  • 39,284
  • 3
    Strictly speaking, Macdonald's book contains no exercises. What you're (quite reasonably) calling "exercises" are officially called Examples in the book. – Timothy Chow May 02 '17 at 21:46