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Possible arguments:

Pro

  • For brevity for trivial deductions

Contra

  • Looks like the writer could not be bothered to write an extensive explanation
  • Definition of what is trivial to which reader and what not
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4  
I think common practice is to write "it can be shown with some algebra that …" though we spent a few years on proving that one such claim was actually completely wrong :-) –  chris May 1 at 10:29
21  
"We omit the remaining straightforward but tedious details." –  JeffE May 1 at 11:14
4  
I once had an editor directly tell me to remove that particular phrase from a paper, which has left me with a disinclination to use it further. –  Fomite May 1 at 13:55
10  
Contra #3: Condescending. –  Superbest May 2 at 7:51
2  
There's a story about a professor who got halfway through his lecture, announced "From this, we can see that obviously..." and trailed off, looking at the board uncertainly. He checked his notes, looked again, looked back at his work up to that point, and sat down to scribble on a piece of paper. 20 minutes later he looked up and shouted "Ah-HAH! I was right, it is obvious!" (The short form of this story is to use the stock phrase "rigorously and intuitively obvious to even the most casual observer", translating to "derivable by means most arcane and obscure, but take my word for it for now.") –  keshlam May 3 at 2:36

7 Answers 7

up vote 81 down vote accepted

I believe it's only appropriate in the type of writing where you are in a position to give the reader excercises. That is, in course books or other learning material. There, I think it's fine, assuming you've already given enough information for them to be able to perform such an excercise.

In papers or theses, you're actually trying to convince the reader that you're right - the reader is in the position of power, not you, as you want something from them (to accept your ideas) and not vice versa. Therefore, I'd never write such a phrase in a paper or thesis.

It's of course a viable requirement to leave out some parts of the work, either because of space constraints or because they're trivial, long but straightforward, would derail the course of the text, or something similar. If possible, such parts can be delegated to an appendix, or left out entirely. But I would accompany this with different phrasing, something like this:

The formula (7) can be obtained from (6) using straightforward application of [insert appropriate math branch here]

A detailed proof of statement (4) unfortunately exceeds the scope of this paper.

The derivation of (3) from (1) is too long to present here, but it can be worked out using a symbolic computation system such as Mathematica.

Equation (7) follows from a straightforward application of this-or-that theorem to equation (5); we refer the reader to existing literature on the topic for details of this.

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8  
+1: you said exactly what I was about to write, only better. –  Nate Eldredge May 1 at 13:35
4  
+1: "...assuming you've already given enough information for them to be able to perform such an excercise." Unfortunately, sometimes it's the most difficult and tedious exercises (the ones we most want to see done for us, even if it's only in a referenced paper) that are left as "exercises". –  Moriarty May 2 at 5:59
4  
"I found an interesting proof of this [the Fermat theorem], but the margins are too small to show it here" - Pierre Fermat –  PA6OTA May 2 at 16:47
2  
Important: actually do the details before leaving them out. Provide them in the appendix for the reviewers (and pre-print readers), too. –  Raphael May 2 at 22:07
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@Raphael: If reviewers need the details, then so do the readers. –  kundor May 4 at 17:38

I will have to advise against it, and very strongly so.

I did applied mathematics in the uni, and some of our teachers were very keen on using phrases like that. Well, I can definitely see the point in leaving exercises to the student, however we did have separate exercise sessions where everything was left as an exercise for the reader. So I have always found it odd (at times frustrating) that the course book, or lectures, by which I was supposed to get the tools to be able to solve the exercises, left some of the necessary bits as an exercise.

There are two major problems with the phrasing, in my humble opinion:

  1. It might not be as obvious or straight-forward as you (the author) might think. In some cases the exercise is practically calculations that have already been performed in the previous page or so, in that case it's understandable to omit the same calculations/derivations, but a reference to the last place these calculations, would do just as well:

    "... the rest of the derivations are analogous to..."

    or

    "... as we have seen in Theorem/Lemma X ... "

  2. That particular phrasing has a condescending/patronizing resonance in the eyes of the reader. This has been something practically all of my old classmates commented on. As a matter of fact, we still joke about the phrase "... is left an exercise to the (ambitious) reader". I am not sure what you have to gain from alienating your audience.

Lastly, regarding the use of the phrase in articles, I have to admit I don't do any work in maths anymore, but in the fields I work with (computational biology, biomedicine, bioinformatics, ...) I have never seen something like this, and I am pretty sure it would not fly well.

If the "exercise" in question is important, you include it in the paper. If you lack the space to include it, you put in supplementary and refer to it. If it is not important you do not mention it.

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I don't agree with any of these assessments. Often the omitted deductions are rudimentary, but not necessarily trivial. The reason they are omitted is generally due to lack of space, or the fact that the omitted proofs do not add anything to the exposition, not laziness. The final point doesn't even apply.

That said, stating that something "is left to the reader" is a little cheeky. You often find the equivalent, "an astute reader will be able fill in the details", which may come across as a little pretentious.

For a paper, it is okay to use such a statement, assuming that the author has actually done the proofs.

In a thesis, it is expected that the author does the work, not the reader, so I don't think it is acceptable.

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12  
I think the phrase "an astute reader will be able..." is beyond pretentious -- it effectively shifts the burden from the author to the reader, by saying that any reader who can't find a proof must just not be astute. Also I would say your third paragraph more emphatically: the author should have written complete detailed proofs of all such assertions. I think it is not sufficient for an author to believe he could probably produce such proofs, even though many authors seem to only hold themselves to that standard (which often leads to false assertions in their papers). –  Michael Zieve May 1 at 11:49
1  
Ideally "rudimentary, but not necessarily trivial" statements (that disrupt the flow of the paper) are what appendices, online supplements and potentially the ArXiv are for. If the proof is not trivial it should be explained by the author somewhere. If it is trivial or tedious then the statements in the other posts are appropriate. –  MHH May 1 at 19:45

I guess this is not the popular opinion, but I have to agree with Dave Clarke in that I do not find the phrase in the least offensive. Even in the absence of such a phrase, many papers in mathematics have terse proofs in which a lot of detail is missing, providing ample exercise in filling in the details. I don't think this is a bad thing as it forces the reader to actively engage the material (still the best way to learn it).

There is a story, doubtless embellished over the years, about G. H. Hardy saying in the middle of a lecture "It is clear that ... ," and then pausing. He went over to a corner of the board, worked for a time, and then continued with "Yes, it is clear ... ."

Having said this, I woud still prefer to see something such as "The proof is a routine computation," or, "The proof is straightforward." Nonetheless, we should not be too quick to protest an author's choice of language in these situations.

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Leaving some of the demonstration in a paper to the reader is definitely cheeky at the very least, and I for one find it quite frustrating if, for instance, I'm reading up on something and really not in the mood to have to work through it all at that point. It may seem lazy, but how fresh are you when you've got to the seventh paper of the evening?

I'd definitely suggest referring to other places if space is at a premium - to my mind, that would be the only reason for omitting something that would be important in the paper (as if it's not important, why would you mention it?) Citations would be my first port of call for this, even if it is to an elementary text book ("... for a more complete treatment of x, see Smith & Jones (2009, pp. 37-41)."), but if there is genuinely nowhere else and not enough space but it needs to be referred to, URLs are also an option. Ideally, there should be a relatively permanent place that you can keep the text, so that someone going to look for it in several years time would still have access.

At the end of the day, if it is in a work to be published in a book or journal, I'd recommend speaking to the editorial staff, as they may have existing policies for assessing the necessity of extensive material and placing it accordingly.

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I'm in philosophy, but I've only ever seen the phrase used in a scholarly work in the form of a joke, as in:

"From this theorem, it follows that God exist. The proof is left as an exercise for the reader."

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The phrase about an exercise for the reader is very passé, and can only be used in textbooks.

The normal phrase in research articles is ''we omit the details''.

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