Obvious Trouble

Terms such as “clearly,” “obviously,” and “it can easily be shown that” do not belong in the mathematical literature. Authors who use them (and editors who allow them) do a great disservice to their readers.

Jokes about the mathematical abuse of these terms have been around for a long time, including the following translations of “clearly” and its kin:

Clearly: I don’t want to write down all the “in-between” steps.

Trivial: If I have to show you how to do this, you’re in the wrong class.

Obviously: I hope you weren’t sleeping when we discussed this earlier, because I refuse to repeat it.

It can easily be shown: No more than four hours are needed to prove it.

Nonetheless, many mathematicians persist in using them not only in articles but also in lectures.

Consider the following example, taken from a published paper:

     Clearly, for every k > 1, there exists a point x . . .

The word “clearly” adds nothing to help the reader understand the rest of the sentence. It is too vague to be of any use, except as a shortcut for a lazy author. What the author might have meant by “clearly” is not the same as what a reader might expect it to mean.

It should not be surprising that readers may give up or feel discouraged when something labeled “obvious” isn’t obvious to them. Indeed, they should feel insulted that an author has been so careless or unheeding as to use such language in his or her writing (or felt that the reader should undergo as least as much pain and suffering as the author did in learning the subject and coming up with the published result).

“In the best of circumstances. when you use these phrases you are endeavoring to push the reader around,” Steven G. Krantz wrote in A Primer of Mathematical Writing (AMS, 1996). “In the worst of circumstances you are throwing up a smoke screen for something that you yourself have not thought through.”

Authors who are tempted to use a term such as “clearly” must think through why they want to use it, and try to solve the problem in a different way, one that helps rather than hinders the reader.

     Obviously, the answer becomes more accurate as n approaches infinity.

The answer is simple. Avoid using terms such as “clearly,” “trivially,” and “obviously.”—I. Peterson