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I've been thinking about the value of writing "of course" in mathematical papers (or its variants such as "obviously" etc.). In particular, my current train of thought is, if something is obvious, then it is obvious that it is obvious (so why include it at all?).

The example that inspired this post is: If d divides a and d divides b, then of course d also divides a+b.

Are there examples in the mathematical literature where the term "of course" is of value?

More precisely, I'm after an example (or a few), ideally by a well-known author, where "of course" or "obviously" or similar actually adds tangible value to a sentence (rather than just implying: (a) it's obvious to me, I'm so smart or (b) I can't actually be bothered working out the details)

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It is of value, I think, to separate the routine parts of an argument from the nontrivial parts. But, of course, what is routine depends heavily on one's background. –  Qiaochu Yuan Feb 23 '10 at 21:45
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When I read papers I usually pause a bit at the "obviously", "trivially" and "of course" just to make sure I've followed everything. I think the value of these phrases isn't necessarily one in terms of manipulating pieces of a logical argument, but rather a style choice to break up the flow and give readers with less background or experience a chance to think things through with something easy. –  j.c. Feb 23 '10 at 21:47
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I absolutely think this should be community wiki. My opinion is that most (possibly all) questions tagged "soft-question" should be community wiki. But it is an interesting question. –  Grétar Amazeen Feb 23 '10 at 22:29
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I'll just note: reading this post lead me to search the paper I'm in the process of revising for "Obviously" and "Of course." I didn't delete all of them though; I kept, for example, "Obviously, this is a dissatisfying state of affairs..." refering to a theorem I hadn't proved. –  Ben Webster Feb 24 '10 at 1:45
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These keywords should be red flags when you're writing: you'll want to thoroughly check any of those statements. But they have their place, especially in the more expository parts of a paper, to distinguish facts that are easily proved -- an invitation to the readers to prove it for themselves -- from facts that are not easy at all, but happen to be well known (or ought to be well-known). –  Thierry Zell Aug 24 '10 at 2:57

10 Answers 10

I don't agree that if something is obvious, then it is obvious that it is obvious. When an author declares in a mathematical exposition that a fact is obvious, or says "of course" or something with a similar meaning, then it is a signal that the reader should be able to find a very easy reason justifying the statement, rather than a complex one. This is useful information for the author to signal, and I for one as a reader have often been grateful for it.

The truly aggravating uses of these phrases occur when the author says that something is "obvious" or "clear", but it isn't really. Surely many of us have been in situations reading a paper where the author says "clearly", but after a lot of thought, it still isn't so clear. I believe that these phrases are often used because the author didn't want to work out all the details, a kind of laziness, which can also be a dangerous source of mathematical error.

I have seen papers where the author states, "It is obvious that X, and let me explain why..."

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By the way, if it's not obviously obvious when you use "obvious", then perhaps the reader could interpret the use of "obvious" as one of those infuriating uses. Eg. X is obvious if you think Y, but maybe the reader is trying to get Z to work, thinking "this is not obvious at all!". –  Douglas S. Stones Feb 23 '10 at 22:00
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Well, I think there is an entire scale of obviousness, depending on the reader and his or her background. If the author intends to write for a certain audience, presuming a certain background, then certain statements would be obvious, which might not be so clear to readers not in that intended audience. So when one reads that something is obvious, but it doesn't seem so clear, then it needn't necessarily reflect poorly on the author, but rather simply be an indicator that the reader was not in the intended audience. –  Joel David Hamkins Feb 23 '10 at 22:09
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Of course (!) it's the cases in which something is not only not obvious, but where it is not obvious that it's not obvious, that will bite you. (Sorry, I just couldn't resist this little Cheney moment.) –  Harald Hanche-Olsen Feb 23 '10 at 22:18
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Not only are there statements that are obvious but not obviously obvious, I think there are cases where it's not so obvious that it's not obvious the statement is obvious. –  JSE Aug 24 '10 at 13:22
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I propose to introduce the modality $\square\varphi$ to mean $\phi$ is obvious,' with the dual operator $\diamond\varphi$, which means $\varphi$ is plausible,' with the equivalence $\diamond\varphi\iff\neg\square\neg\varphi$. In this language, JSE's assertions are those with $\neg\square\neg\square\square\varphi$, which can be dually expressed $\diamond\square\square\varphi$. –  Joel David Hamkins Aug 24 '10 at 13:48

When I was a graduate student, a professor (who will remain nameless since I might be misquoting) said something along the lines of "If you want to see where the errors in a math paper are, just look for the places where the author states 'it is obvious that', 'clearly', or 'of course'."

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This also holds for grading proofs on exams. And it makes sense: if you the student don't know how to do a step in the proof, you might as well hope it's obvious to the grader! –  Reid Barton Feb 24 '10 at 6:14
    
That's a brilliant tactic! –  Douglas S. Stones Feb 24 '10 at 9:24
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"By a theorem of Euler..." –  Harry Gindi Feb 24 '10 at 9:48
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Was it by any chance me? Probably not, but it is something I have often said. But I didn't think it up for myself -- I got it from David Preiss. It's useful not just for evaluating the work of others, but also one's own work. That is, if you've just written a proof of something but don't feel quite secure about it, look for the bits where you didn't give full detail. –  gowers Feb 24 '10 at 12:31
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Certainly if a point is trivial, it wouldn't hurt to at least sketch an argument. –  Harry Gindi Feb 24 '10 at 17:21

I think I read in Jacobson's BAII (now available from Dover BTW), "Of course, every irreducible module is completely reducible." A sentence that has a problem if only read in English (versus math). Often, "of course," "obviously," and "it is easy to see that" are used solely for the purpose of not starting a sentence with a symbol, or keeping two symbols from being juxtaposed. And sometimes it is important to say something is obvious or a matter of course.

It is good to use the phrases frugally. And it is good to beware when an author uses them. Of course it is obvious that these turns of phrase are useful.

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@Pete Read it in English as he suggests. "Not reducible at all implies very reducible" seems like nonsense. Obviously, that's not a good translation of the technical meaning, but imagine that you flip the book open to that statement first. –  Douglas Zare Feb 24 '10 at 0:04
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But statements like that can be true in ordinary English. To wit: inflammable objects are often completely flammable. –  Qiaochu Yuan Feb 24 '10 at 0:12
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It's still a possible source of confusion worth a remark. Btw, "flammable" is a corruption of "inflammable." Clearer examples include that we park on driveways and drive on parkways, and that many words are their own opposites, like the verbs "table," "cleave," and "sanction." –  Douglas Zare Feb 24 '10 at 0:54
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I tend not to like "By definition..." because everything is follows from the definitions... –  Mariano Suárez-Alvarez Feb 24 '10 at 3:29
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@Mariano: In some cases, "by definition" serves as a useful substitute for "obviously." It tells the reader that what follows is easy to see and not to waste time trying to infer it from the technical lemmas he's just read --- simply go back to the definition. It's surprisingly easy to get so engrossed in technical lemmas that one doesn't think to look back at the definition until one is reminded to do so. –  Andreas Blass Aug 24 '10 at 19:02

Hello,

I agree with some of the comments above: "of course" is useful to point out that some step is trivial (e.g. direct consequence of the definition), as opposed to the rest of non-trivial parts of the proof. Sometimes, "of course" is useful just as an stylistic resource in the writing, to introduce and connect a sentence to the previous one. But it can be very frustrating for the reader if this step is non-trivial, even though the author claims it is.

I was curious about this question and decided to find some examples in the "mathematical literature", as the original poster suggested. I looked through "A Course in Arithmetic", by J-P. Serre (which many consider a very good writer of mathematics) and the expression "of course" appears exactly twice. In both cases, "of course" appears in a parenthetical remark:

1) (p.35) Corollary. - For two nondegenerate quadratic forms over $\mathbb{F}_q$ to be equivalent it is necessary and sufficient that they have same rank and same discriminant. (Of course the discriminant is viewed as an element of the quotient group $\mathbb{F}_q^\ast/\mathbb{F}_q^{\ast 2}$.)

2) (p.73) Let $A$ be a subset of $P$ [$P$ is the set of prime numbers]. One says that $A$ has for density a real number $k$ when the ratio $$ \left(\sum_{p\in A}\frac{1}{p^s} \right)/ \left(\log \frac{1}{s-1}\right)$$ tends to $k$ when $s\to 1$. (Of course, one then has $0 \leq k \leq 1$.)

In example (1), the way the corollary is stated, a remark is needed - but (i) it is clear from the context that this is what the author means, and (ii) it is typical in this context to consider discriminants only up to squares. Here I see this "of course" as a reminder of (ii).

Example (2) is trickier, as it is not immediately obvious that the limit of the expression as $s\to 1$ is between $0$ and $1$. But I do not interpret this "of course" as a "clearly" in this case, but rather a sort-of "do not worry, if you go back and check Cor 2 in p. 70, you can convince yourself that $0\leq k \leq 1$, and it makes sense to call this number a density".

Álvaro

PS: In "A Course in Arithmetic", the word "clearly" appears many times, while "obviously" was never used in the entire book.

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Serge Lang, also often considered a good writer, uses words like "obvious," "trivial," and "of course" extremely profusely. –  Akhil Mathew Feb 24 '10 at 11:50
    
I checked and the expression "of course" appears 61 times in Serge Lang's "Algebra" (2002, Springer, 914 pages). –  Álvaro Lozano-Robledo Feb 24 '10 at 13:28
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Serge Lang, also often considered a good writer [citation needed] –  darij grinberg Feb 24 '10 at 16:52
    
I find it hard to believe that there would be an index of "good writers". –  Douglas S. Stones Feb 26 '10 at 2:24
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I know, I should return to Axler bashing... –  darij grinberg Aug 24 '10 at 11:27

If someone said "of course the equation is well-posed because it is elliptic, etc" it would teach the reader that the statement is not only true, but that there is a well known train of thought for proving the result.

In this sense there is no claim that the proof is short or contains only simple steps, but it does say that that among a certain class of people familiar with the area that it is in fact a well-walked path.

I briefly consulted a dictionary and found these two definitions for "of course" 1. "certainly; definitely" 2. "in the usual or natural order of things"

I am suggesting that "of course" is useful in the second instance as a pedagogical indicator.

I am not claiming that this is always the usage employed in mathematical writing but it is a good one if used in the right circumstances.

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Please allow me to quote André Weil twice. First, from page 27 of The Apprenticeship of a Mathematician, concerning his first form teacher, Monsieur Collin:

'There was a strong temptation to take short-cuts, saying "it is obvious that..."; Monsieur Collin taught me never to use this word. "If it were obvious," he said, "you would not feel the need to say so; if you say so, that means it is not obvious." He is the one who taught me how to write up mathematics.'

The second quote, from page 19 of Elliptic Functions According to Eisenstein and Kronecker:

"Combining the above formulas, we get for all $n\geq 1$, the following series for $E_n(x)$: $$E_n(x) = u^{-n} \sum_{v=-N}^{+N}\epsilon_n(\zeta + v\tau)+\frac{(2\pi/iu)^n}{(n-1)!}\sum_{v=N+1}^{+\infty}\sum_{d=1}^{+\infty}d^{n-1}q^{vd}[z^d+(-1)^nz^{-d}]$$ where the double series is easily seen to be absolutely convergent provided $N$ has been taken such that $|q^{N+1}z|<1$ and $|q^{N+1}z^{-1}|<1$."

I will refrain from the obligatory of course joke and leave the moral of the story as an exercise instead.

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Doing this kind of exercise on Google books is easy and fun to do. I find that Weil uses the word "obvious" (usually as a short-cut) 10 times in the 92 pages of Elliptic Functions According to Eisenstein and Kronecker. –  John Stillwell Feb 24 '10 at 22:13
    
@John Well,as Weil's mathematical grandson,allow me to defend the great Frenchman by stating that 10 times in 92 pages still has measure 0 by the "lousy teaching" measure function I propose in my post at this thread-which isn't bad at all. –  Andrew L Aug 24 '10 at 3:20
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I don't think you can claim to be anyone's grandson without a PhD... –  Scott Morrison Aug 24 '10 at 12:28

Like Konrad Swanepoel, I have found many mistakes, especially in my own work, around "Of course" or comparable expressions, and the saying from one of my early teacher that I often quote is "If it is obvious, then it is easy to prove, so prove it."

That said, I think there are instances where "of course" adds some value to a mathematical text: namely to justify to the reader why you are not taking a seemingly shorter route. Suppose you want to prove a certain assertion, and that in your context, the proof would be easy if some groups were finite, and suppose that in the standard historical paper on the subject that you are generalizing, the groups in question are finite. Then, I think it is helpful to point out at the beginning of your proof or in a remark that "Of course, if the groups were known to be finite, we could use the strategy of...". This "of course", far from drawing attention to how smart the author is, assumes that the reader might have anticipated a seemingly shorter proof and explains why this short route was not taken.

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Many years ago, a professor of mine in a graduate algebra course wrote down a totally impenetrable statement and then added:

"Of course, it's obvious. It may not be clear that it's obvious...but it's obvious."

The words "it's obvious", etc., should be discarded just as the words "We have..." should be discarded. Also, the words "Consider the following function...." As Estermann comented..."I do not know what that means."

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Although this is my own experience, it might give an answer to your first question "if something is obvious, then it is obvious that it is obvious (so why include it at all?)." There are times when I will write, "clearly," or "of course," when I feel leaving them out might insult the reader's intelligence. If we imagine your example

If d divides a and d divides b, then of course d also divides a+b,

(probably without the italics) in an advanced context, that is, as a prelude to invoking an important theorem or stating something not so obvious that depends on that simple fact, then I might look at it that way.

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"of course", "clearly", "obviously", and other like interjections are a form of higher-order punctuation. While strictly speaking they're eliminable, good authors use them as rhythmic elements, cognitive breaths to delimit chunks of an argument. Yes they are "noise words" and phrases, but they are the ones we all know and love (in English) so probably it's best not to press this point too hard. :)

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