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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...".

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    $\begingroup$ Years ago I went to a writing class (different subject) and was told whenever you want to say "it is obvious" you should read it as "you dummy" and see if it still seems reasonable. $\endgroup$ – Ross Millikan Aug 23 '13 at 23:53
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    $\begingroup$ Try english.stackexchange.com. I don't see how this is directly related to mathematics. $\endgroup$ – bubba Aug 24 '13 at 0:32
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    $\begingroup$ I asked this over at mathoverflow: ref.. $\endgroup$ – Douglas S. Stones Aug 24 '13 at 0:34
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    $\begingroup$ @bubba: I disagree with your recommendation. The OP is asking about writing styles in mathematical research papers. Of course (!) if you write a mathematical research paper in English then you will need to pay attention to the rules and the art of good English writing, but mathematical writing is for a particular community which has particular needs and expectations. It's better to ask advice about how to write a math research paper well from people who (have a strong command of English and English writing and) have written math research papers than English experts. $\endgroup$ – Pete L. Clark Aug 24 '13 at 0:47
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    $\begingroup$ "As even a complete fool can plainly see ..."; "It is almost painfully self-evident that ...". "That <statement> is true is so apparent, that to grace it with the proof it clearly does not deserve would be a pitiable sacrifice to the deity representing wasted effort." $\endgroup$ – Kaz Aug 24 '13 at 3:19

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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.

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    $\begingroup$ This example is perhaps borderline, but I think that for a sufficiently unpractised audience the clearly does serve a useful purpose. Even at that level arguments and calculations are often presented with steps that require a little thought or filling in, and the clearly, if used correctly, tells the student that this really isn’t such a case. $\endgroup$ – Brian M. Scott Aug 24 '13 at 0:21
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    $\begingroup$ I actually think it clarifies things sometimes to say "it is easily seen that" or "clearly" because it helps keep things in perspective. I know which parts are at least supposed to be easy, once I understand the material sufficiently well. The structure of the argument is more clear. $\endgroup$ – littleO Aug 24 '13 at 0:22
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    $\begingroup$ I think that a research paper should be written in as informal a style as the editors will permit; this is more likely to improve than to detract from its readability. $\endgroup$ – Brian M. Scott Aug 24 '13 at 0:27
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    $\begingroup$ As a shining example, Serre is generally held to be one of the greatest mathematical expositors (in particular, by me). He is certainly writing for audience who does not understand things as well as he does most of the time. But one reads him and develops a lot of faith that everything necessary to understand what he's saying is included in his writing. I have never seen him use "clearly": he doesn't need to. $\endgroup$ – Pete L. Clark Aug 24 '13 at 0:35
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    $\begingroup$ @Brian: I will only add that I admire Serre's writing style most of all because I feel like he "lets me in" to what's actually going on more than almost any other leading research mathematician I've ever read. I own and read his Collected Works for no better reason than he wrote them. So " I’ve never read Serre, nor am I likely to" makes me a little sad. I do recommend him to you most heartily. $\endgroup$ – Pete L. Clark Aug 24 '13 at 0:50
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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
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    $\begingroup$ When refereeing papers, the places where authors make mistakes usually follow right after expressions like "clearly" and "it is apparent". $\endgroup$ – Ryan Budney Aug 24 '13 at 0:05
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    $\begingroup$ Please don't use the word 'thence' in the year 2013. Or any year after. $\endgroup$ – Potato Aug 24 '13 at 0:13
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    $\begingroup$ @Potato: Thence is part of my native language: I use it naturally and have done so for most of my life. $\endgroup$ – Brian M. Scott Aug 24 '13 at 0:16
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    $\begingroup$ @Potato, I think you mean "Or any year thereafter." $\endgroup$ – Joel Reyes Noche Aug 24 '13 at 0:43
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    $\begingroup$ It may be routine to show that X although X is far from obvious. For example, it is routine to show that $2^{24} = 16777216$, but I wouldn't say it's obvious. $\endgroup$ – Michael Hardy Aug 24 '13 at 2:01
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"The proof of this fact is left as an exercise for the reader".

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    $\begingroup$ To me that seems potentially more obnoxious than "obviously" (which I think can be obnoxious, though mathematical readers tend to eventually get used to it). Is it the task of a writer of a research paper to be setting exercises for the reader? $\endgroup$ – Pete L. Clark Aug 24 '13 at 0:13
  • $\begingroup$ Well, I have read at least one research paper which contained something that sounded like that... $\endgroup$ – Marra Aug 24 '13 at 0:31
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    $\begingroup$ I've read more than one research paper for which some of the claimed results were false; I've read dozens (or hundreds?) in which I wished the authors had expressed themselves differently. It is not necessarily the case that every mathematical practice that one encounters is even acceptable, let alone worthy of emulation. $\endgroup$ – Pete L. Clark Aug 24 '13 at 0:39
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    $\begingroup$ I've probably used "left to the reader" in numerous papers, but "left as an exercise" only once, as far as I can remember. That was in connection with a tangential piece of information whose proof I thought (and still think) readers would really enjoy finding on their own. $\endgroup$ – Andreas Blass Aug 24 '13 at 1:49
  • $\begingroup$ I think the only obnoxious thing in my phrase would be the 'exercise' word, which can be substituted by anything more adequate and polite. $\endgroup$ – Marra Aug 24 '13 at 11:10
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One of my math profs liked the phrase "intuitively obvious to the most casual observer".

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  • $\begingroup$ Googling I see references back to the 90s. Alas, I believe this was at K-State in 81-83. Ancient history. :) $\endgroup$ – sfjac Aug 24 '13 at 2:55
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    $\begingroup$ One of the references continued "but we are not casual observers, so we shall prove it". Nice. $\endgroup$ – sfjac Aug 24 '13 at 3:00
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    $\begingroup$ I prefer "Even the smallest-headed of dolts would realise that..." $\endgroup$ – Anthony Carapetis Aug 24 '13 at 3:24
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"Clearer" fits nicely. Also, "apparent", or any other word that implies "short". These words imply the proof is straightforward.

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Statements like "obvious", "clear", and "trivial" imply personal opinion or value judgements.

Instead, I prefer "immediate", "straightforward", "directly follows from". These are relatively objective and based on the number of steps required.

If the above do not apply to the statement in question, then I certainly don't think it should be labeled "obvious"...and perhaps it is worth further explanation as well.

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    $\begingroup$ Isn't saying something is "immediate" or "straightforward" also a personal value judgment? What is straightforward to you may not be straightforward for a first year undergraduate. $\endgroup$ – Potato Aug 24 '13 at 0:14
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    $\begingroup$ I agree with @Potato. Nevertheless they are somehow less ad hominem than the alternatives presented above. $\endgroup$ – Pete L. Clark Aug 24 '13 at 0:15
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    $\begingroup$ @Potato: well it has the advantage that it tells the reader the author believes it's not a sophisticated argument. It has the same disadvantage in that it does not communicate what that argument is. $\endgroup$ – Ryan Budney Aug 24 '13 at 0:25
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    $\begingroup$ @Pete: Ad hominem really isn’t correct; (potentially) condescending comes a lot closer. My own judgement of them is similar but not identical: in a scale of decreasing potentially condescension I think that I’d make them $$\text{obvious}>\text{trivial}>>\text{clear}>(>)\text{ immediate, straightforward, etc.}\;.$$ $\endgroup$ – Brian M. Scott Aug 24 '13 at 0:25
  • $\begingroup$ @Brian: Yes, you're right. I was trying to quickly indicate my opinion, but I agree that you've said it better. $\endgroup$ – Pete L. Clark Aug 24 '13 at 0:29
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My current favorite version of this is from a paper of Phragmen: "On deduira sans peine...:" one may (or will) deduce painlessly. (He's being nice, the proposition is fairly obvious.)

He contrasts this with "sans trop de peine:" without too much pain. (He's being generous, the proposition may be quite hard.)

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    $\begingroup$ I think this nicely points up the one good justification for using these sorts of qualifying phrases: not just alerting the reader to an omitted argument, nor (worse) justifying the writer’s decision to omit it, but rather for informing the reader about the argument being omitted. As a reader it’s very nice to know whether an omitted step is likely to be “immediate” (OK, I’ll think about it for a moment till I get it), “a lengthy but routine calculation” (OK, I’ll probably skip it, unless I’m killing time on a long flight), or even “a simple but enlightening exercise” (hmm, lemme try it!). $\endgroup$ – Peter LeFanu Lumsdaine Aug 24 '13 at 16:03
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Corollary: ...

Otherwise why not take a look at a couple of maths books/papers and see what the Authors have written.

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Sometimes "canonical" works. For example "the canonical map" is much better than "the obvious map".

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    $\begingroup$ I think the question refers to "obvious" in the sense of "easy to verify", not "natural". $\endgroup$ – augurar Aug 24 '13 at 1:00
  • $\begingroup$ Is that the "obvious" interpretation of the question? I don't see the question being limited to that sense, in any case. $\endgroup$ – Andrew Aug 24 '13 at 1:25
  • $\begingroup$ More importantly, I don't think bad usage of "obvious" is limited to that sense. $\endgroup$ – Andrew Aug 24 '13 at 3:12
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Don’t try to find a synonym, that doesn’t solve the problem that “obvious” has. The problem is that when you feel the need to point out that a connection is obvious, it probably isn’t. Don’t say it’s obvious, make it obvious. Or, if it’s really obvious, just state it. No need for a word such as “clearly” or “evident”.

Ironically, this seems rather obvious to me but all the other answers missed it, and instead suggest synonyms which, as I’ve said, suffer from exactly the same problem.

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    $\begingroup$ Did you see Pete L. Clark’s earlier answer? It has essentially the same main point: “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.” $\endgroup$ – Peter LeFanu Lumsdaine Aug 24 '13 at 14:47
  • $\begingroup$ @Peter Hm. I completely missed that, no idea how that’s possible. $\endgroup$ – Konrad Rudolph Aug 24 '13 at 15:19
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How about "self-evident"? In certain contexts the ever popular "trivial" could work.

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    $\begingroup$ "Self-evident" is a technical term in epistemology. Obviousness is subjective; self-evidence is not. It is not self-evident that JFK was killed on a Friday, because I can understand what that statement says without knowing whether it's true. That I am conscious is self-evident, because I need to be aware that I am conscious in order to understand what that statement says. Most statements can be understood without knowing whether they're true; self-evident statements can be understood only by being aware of the facts that they assert. I think some self-evident things are far from obvious, $\endgroup$ – Michael Hardy Aug 24 '13 at 3:26
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    $\begingroup$ .... for example: Mathematics consists of more than just the technical skills that it relies on. I takes a lot of work just to understand what that statement says, so it's far from obvious to one who hasn't studied. But one can reach the point of understanding it only by a course of thinking in which one observes that it is true. It is a self-evident statement, but not at all obvious except to people with a lot of experience. $\endgroup$ – Michael Hardy Aug 24 '13 at 3:28
  • $\begingroup$ +1. Who is the audience. How much do you want to insult them. How intelligent do you think they are? How do you hope to help them? $\endgroup$ – dardisco Aug 24 '13 at 7:03
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evidently, visibly, naturally, undeniably..

Words like trivially, and obviously sound disrespectful, it is as if the author is mocking the reader. Also they sound 'empty' and many authors use these words to make up for the incompleteness in their work.

Mathematics is about deduction, not intuition. So any word that does not imply to bring in intuition to reason can be thought of as a good word. :)

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