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.
Use of “it is easy to see that” is common and traditional in mathematical writing, but it is not exactly a proud tradition: really good mathematical exposition uses this and similar locutions very sparingly.
To be more specific, I think it is bad writing to say “It is easy to see that X is true” and say no more about how to prove X. If this occurs in formal mathematical writing and all else is as it should be, then no information is being conveyed. In other words, what other reason is there to assert that X is true and say no more about the proof except that the author expects the reader to be able to supply the details unassisted? If you are skipping the proof for any other reason, you had better say something!
[And of course some of the harm is psychological. If you’re carefully reading a text or paper that asserts X is true and says nothing else about it, you know you need to stop and think about how to prove X. If “it is easy to see X” then after every minute or so part of your brain will quit thinking about how to prove X and think, “They said it was easy to see this, and I can’t see it at all. What am I, stupid?” I definitely remember thinking this way when I started out reading “serious” math books.]
When I referee papers I often suggest that authors suppress their “it is easy to see that”‘s. As others have said in the comments, as a careful, skeptical reader, you also need to stop and be sure that indeed you can supply the proof yourself, and it is notorious among mathematicians that such phrases are likely places to find gaps in mathematical arguments. But it is just as easy — in fact, easier — to have a gap in the argument where you don’t have any text at all, so writing “it is easy to see that” is not really the guilty party but rather a possible piece of incriminating evidence.
So if it’s not so good to write this, why do people write this way? And they certainly do: I happened to be editing my commutative algebra notes when I read this question, so out of curiosity I searched for “easy” and found about ten instances of “it is easy to see that” in 265 pages of notes. About half of them I simply took out. The other half I thought were okay because I didn’t just say “it is easy to see that”: I went on to explain why it was easy! So having caught myself doing what I said not to, I can reflect on some causes:
I have read “it is easy to see that” thousands of times, so it is in my vocabulary whether I like it or not. Most mathematicians know that they have funny phrases which appear ubiquitously in mathematical writing but not in the rest of their lives: one of the very first questions I answered on this site was about the meaning and use of “in the sequel”. In the year or so since then I have observed it in my own writing: it just fits in there. You have to really actively dislike some of these standard locutions in order to avoid writing them yourself. For instance, I have more than a thousand pages of mathematical writings available online and I challenge anyone to find “by inspection” anywhere in these. “By inspection” is the deformed cousin of “it is easy to see that”: whereas at least it is easy to see what “it is easy to see that” means, even the meaning of “by inspection” is obscure.
2) A conflation of formal writing and informal writing / speaking / teaching.
The way you speak mathematics to someone else is very different from the way you write it: it is much more temporal. If you are teaching someone new mathematics then most often they cannot verify / process / understand every single mathematical statement you make, in real time, so they have to make choices about exactly what to think about as you’re talking to them. In spoken conversation it’s extremely useful to say “this is easy”: by saying it, you’re cueing the listener that it’s safe to direct her attention elsewhere. Also, because when you talk — or write informally — you don’t give anywhere near as complete information as you do in formal mathematical writing, commentary on what you’re skipping becomes more important. For instance, in an intermediate level graduate course I may prove approximately 2/3 of the theorems I state in class. If I’m skipping something, it’s probably because it’s too easy or too hard. I had better say which it is!
Certainly when you’re reading your own writing and you find “it is easy to see that”, you need to stop short and make sure you know exactly what you omitted. If it’s not easy for you to see what you wrote it was easy to see, you may have a serious problem: indeed, you may be papering over a gap in your argument. To do this intentionally is a sign of great mathematical immaturity — someone who hides (in plain sight!) what they don’t know in this way is not going to make it very far in this profession — but even doing it unintentionally is something that most mathematicians largely grow out of with experience.