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Math is not generally what I am doing, but I have to read some literature and articles in dynamic systems and complexity theory. What I noticed is that authors tend to use (quite frequently) the phrase "it is easy to see/prove/verify/..." in the manuscripts. But to me, it is usually not easy at all; maybe because I haven't spent much time in the field, maybe not.

My question is: why do people use the phrase "it is easy" in their proofs?

P.S. I hope this question is not too subjective, and has some value for the community.

  • 83
    I guess the most widespread reason is that people tend to be lazy.2011-07-29
  • 3
    I think Theo nailed it. But even when you are lazy, I think it should be discouraged to use such expressions as "it is easy to prove that" or "this is trivial". It gives a bit of a smug vibe.2011-07-29
  • 25
    I believe the true meaning is actually "[I think but haven't really checked that] it is easy [for me] to see/prove/verify/..." with at least one pair of brackets removed.2011-07-29
  • 26
    Another reason, I guess, is that when the proposition which is "easy" is not part of your main result, and you are trying to keep the paper short; or that it is a nice corollary which you found which has no real importance. Sometimes however these things are indeed very trivial for the intended reader, for example - proving that embedding is a total order with respect to linear orders. It is not a "trivial" theorem for the beginning mathematician but it is for someone reading a paper titled "Forcing over long well orders" (just making up a title) which assumes some knowledge in set theory.2011-07-29
  • 0
    I think this is a soft-question type of question, and I'm not quite sure about the math-history too. Also, this might be suited for community wiki - but I am not sure as well.2011-07-29
  • 2
    Maybe you want to have a look at the comments [in this thread](http://math.stackexchange.com/q/38620/) for some related thoughts/advice. @Asaf: I did flag it for CW since it is unlikely that this question has a definitive answer..2011-07-29
  • 13
    It is true that we overuse "it is easy to show," "obviously," "clearly," and their ilk. However, such terms are hard to avoid when one wants to describe the full logic of an argument without verifying every detail. The main issue is inaccurate use of the term. It should mean roughly "it is (or should be) easy for *you*" and not "it is easy for *me*".2011-07-29
  • 9
    There is also the alternative "whose proof is left to the reader (as an exercise)." :D2011-07-30
  • 0
    A related question on MathOverflow: http://mathoverflow.net/questions/16193/value-of-of-course-in-the-mathematical-literature2011-07-30
  • 2
    @J.M. I find "left as an exercise" more honest (in some cases) :-)2011-07-31
  • 9
    I took a class on proposal writing where we were taught that every time you want to write "obviously" you should read it as "you dummy" and see if you still want to say it. Maybe "It is easy to prove" fits that category.2011-07-31
  • 0
    @Asaf, embedding does not seem to be a total order with respect to linear orders, since neither $\omega$ nor $\omega^*$ embed into the other. Have I misunderstood your remark?2011-10-20
  • 0
    @JDH: While you are correct, I believe I may have meant well orders. However that was not the point of the remark, the point was that while it is not easy to see that any two well orders are comparable in embedding, it is easy to see that if you are familiar with the basic theorems about well orderings, and whilst writing a paper in set theory which deals with an even more advanced material, the writer may assume that the fact two well orderings are comparable in embedding is indeed trivial to the reader.2011-10-20
  • 0
    @oleksii Some authors do that. Some others do something similar, but give a hint. For example, they can say "It is easy to see $P$ is true. (Note that $N$ implies $F$)" The remark inside the parenthesis is usually enlightning and helps seeing what he/she is stating.2012-03-04
  • 1
    In [Concerts](http://blog.plover.com/math/concerts.html) I said "Obviously, no more than six concerts are required. (I have a new contribution to the long-debated meaning of the mathematical jargon term 'obviously': if my six-year-old daughter could figure out the answer, so can you.)" Unfortunately Ms. 6 is now 10½ so no longer represents the same standard of obviousness.2014-12-31
  • 0
    Perhaps the proof is too annoying to typeset?2015-03-29

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