Value of "of course" in the mathematical literature

Question

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)

Answer 1

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..."

Answer 2

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'."

Answer 3

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.

Answer 4

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.

Answer 5

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.

Answer 6

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.

Answer 7

"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. :)

Answer 8

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.

Answer 9

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.

Answer 10

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."