What are better words to use in an article than "obvious"?

I've heard often than it is ill-form to use the word "obvious" in a research paper. I was hoping to gather a list of less offensive words that mean generally the same thing.

For example, one that I can think of is the word "direct".

So instead of saying "...obviously follows from lemma 2.3..." you'd say "...this proof directly follows from lemma 2.3...".

Solution 1:

One thing that people (including me) actually write is "it is easy to see that". But even though I still write this sometimes, when I catch myself doing it, I don't like it.

I feel (well, maybe 90%; I am not quite as decisive on this point as the answer may otherwise suggest) that instead of pointing out that no explanation is necessary, you should either (i) include some actual explanation, however brief, or (ii) have the courage simply not to say anything like "obviously", "clearly", "it is easy to see that" if what you are asserting is actually meant to be clear.

As an example of the latter, if I am trying to show that the function $f(x) = x^3+x$ is increasing, then instead of writing

"We have $f'(x) =3x^2 +1$, which is clearly positive for all real $x$. Therefore $f$ is increasing."

I think that for almost any conceivable audience, it would be better to say

"We have $f'(x) = 3x^2+1$, which is positive for all real $x$. Therefore $f$ is increasing."

(In some contexts the word "real" would be taken as a given and could be safely suppressed, but I don't like unquantified variables. In this example I think the better question is whether it will be clear to the reader that you are invoking the corollary of the Mean Value Theorem that says that a function which a positive derivative on an interval is increasing.)

Or, in the example you've given (in which, by the way, your proposed alternative "...this proof directly follows from lemma 2.3..." is already much better than "..obviously follows from lemma 2.3..."), see if you can allow yourself to write simply "This follows from Lemma 2.3." If it were less than direct you'd be saying more about it, right?

What I have not entirely figured out is what to write when the claim you are making need not be immediately clear or obvious to the reader but should be straightforward for the reader to check if she cares to do so. In part the problem is that we are not being maximally nice to the reader by doing this -- purely insofar as the communication of the mathematics is involved it would be better to give the explanation/calculation, straightforward though it may be. But sometimes we don't condescend to explain every little thing in our mathematical writing; that's just a cultural fact which transcends good or bad mathematical writing. For this I find that something like "one can check that..." is the least obtrusive way to alert the reader that she may have to take out her pen.

Solution 2:

EDIT Just seen this on MO: https://mathoverflow.net/questions/22299/what-are-some-examples-of-colorful-language-in-serious-mathematics-papers/22455#22455

A few more to consider:

1. It is (readily) seen
2. It is an immediate consequence
3. Clearly
4. It is apparent
5. A quick\straightforward\routine\rudimentary\simple calculation\argument shows that
6. It is straightforward\routine to show that
7. Thence
8. It is evident

Solution 3:

"The proof of this fact is left as an exercise for the reader".

Solution 4:

One of my math profs liked the phrase "intuitively obvious to the most casual observer".