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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.

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I guess the most widespread reason is that people tend to be lazy. –  t.b. Jul 29 '11 at 10:44
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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. –  Marek Jul 29 '11 at 10:59
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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. –  Asaf Karagila Jul 29 '11 at 10:59
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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". –  André Nicolas Jul 29 '11 at 11:28
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There is also the alternative "whose proof is left to the reader (as an exercise)." :D –  J. M. Jul 30 '11 at 5:32

9 Answers 9

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:

1) Tradition/habit.

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!

3) Immaturity/Laziness.

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.

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+1 "What am I, stupid?" - this is exactly how I feel –  oleksii Jul 29 '11 at 12:23
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What a great answer! I think there's one major take-way that can't be overemphasized: let your texts mature a little bit (a few months if you can afford it). Go back and try to eliminate all phrases conveying no information. Your goal should be to make it as easy as possible for the reader, not to make it as easy as possible for yourself. If you can't provide a reference or boil down the argument to a few sentences indicating the main ideas then this probably means that you should elaborate and think through it once again. –  t.b. Jul 29 '11 at 12:34
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Very interesting answer! As a side note to point 3), I recently read an article in which a proof started with "It is easy to see that the theorem is true if $a=0$, so suppose $a >0$." But actually the theorem was false in the case $a=0$..! (I indeed felt quite stupid when failing to see why the $a=0$ case was true, until I found a counterexample) –  Malik Younsi Jul 29 '11 at 12:56

Writing a paper is usually a long and tedious process. Some arguments seem (to the author) to be straightforward, but potentially painful to write out (e.g. due to having to introduce additional notation or concepts that are not relevant to the larger thread of the discussion at hand). At this point the author may simply state the required result, saying that it is "easy to prove", or something similar.

Ideally, such a result will indeed be easy to prove; e.g. at the level of difficulty of an exercise in a graduate text-book. Note that if your mastery of the subject is such that you are challenged by exercises in graduate-level text-books, then you may well find "easy to prove" statements hard to prove, not easy! The intended audience for such a statement is typically another expert in the field, not a beginner.

On the other hand, one reason that an argument can be hard, or at least tedious, to write down is that the author may not have at hand good tools for formalizing their intuition about the argument. In this case, rather than going to the trouble of developing these tools to formalize their intuition, they may just state the result, writing "it's easy to prove" or something similar. In my experience, this is usually why mistakes can creep in at these points --- because the difficulty in formalizing the intuition may be caused by an actual failing in the intuition!

The lesson I take from this in my own writing is that, when one thinks that a certain proof will be easy but tedious, one should examine the situation carefully, to make sure that the difficulty in writing out the complete argument is not being caused by some hidden flaw in one's intuition about the situtation.

As a reader of mathematics, especially if you are not an expert and are reading well-known papers that have been certified as correct by experts, it's probably best to presume that everything is in fact correct. However, one should expect that reading a paper and filling in all the details will be at least as demanding as reading a chapter in a graduate text-book on the same topic and doing all the exercises.

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I like Pete's answer a lot, but I feel I should add something that I don't think he mentions, which is an unapologetic word of sympathy for the phrase. Here's the thing: I don't think "easy" means "easy to construct", it means "easy to follow". Coming up with the proof might not be trivial, but the process of coming up with a proof and the proof itself are (as anyone in mathematics knows) nothing like the same thing. Which of these is important to you will of course depend hugely on your level of mathematics and your reason for reading the text.

Having gone through undergraduate mathematics relatively recently, I've found myself picking up phrases like "and trivially we see that..." from my lecturers. It's a nasty habit, but on the other hand, I know what I mean when I say it, and by extension I know what they mean when they say it. I certainly don't mean to imply that my undergraduate career has been one long triviality, or that anyone who can't follow my train of thought is an idiot. I mean that, if I wrote the proof down in front of you, and maybe you did a bit of scratching about, you would find it followed trivially from previous work or a definition. I don't need to assume you are comfortable with this previous work or definition yet, and perhaps I even expect you not to be, unless I'm trying to teach in "real time". Some things take a while to sink in, and maths is not a spectator sport.

It's actually pedagogically very reasonable, in some cases. In fact, I would argue that not omitting some proofs is pedagogically very bad practice. Here are a few reasons, all of which apply hugely to undergraduate texts, but do apply elsewhere too.

  1. Sometimes adding huge and reasonably elementary proofs distracts students from the main goal of the discussion.
  2. Sometimes the proofs aren't very enlightening. Few people read a proof thinking "I must check this to see whether it's right". We read a proof to see why it's right, in order to learn from it, for our own selfish gain. If it's right because of something we already know, it is of little to no value to someone trying to learn from a text.
  3. Anything I read in a textbook, especially a well known textbook, I (by gut instinct) assume to be important in my understanding of the content, otherwise it should have been consigned to an appendix (or the bin) for me not to read at my leisure. Eminent authors are authority figures in this way.

The example that sticks in my mind is a proof that the Ackermann function is not primitive recursive. The proof is a very simple concept and some very simple arithmetic manipulation, and your average 15-year-old could follow it without any trouble. But is it useful? Not in the slightest. It is one cute idea (namely "show that everything primitive recursive grows more slowly"), followed by several pages of very elementary mathematics which is so full of 'tricks' that it is very difficult to come up with, it is very difficult to memorise, and it takes half an hour to write down. And the tricks are so specific and elementary (things like "now replace n by the inequality n < 2n + 3") that they can't be used elsewhere. What's the value in that to someone learning recursion theory?

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Look at this cartoon in abstrusegoose.com http://goo.gl/K0bp2

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The phrase "it is easy to prove" is part of the mathematics protocol. It takes some initiation to get used to it, like reading a mathematics book does. A novel, poem, holiday brochure, chess magazine, economy book, mathematics book all need to be read but each of them has a different reading protocol. 'Simply', 'it is easy to see that', 'this trivial proof is left to the reader', and so on. - By the way the key to reading a math book is a pencil and paper. ( Use it. )

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This is a nice assortment of answers that we have going. Personally, I have tried to avoid using "it is easy to see" unless the intended reader (or, listener) should truly find it easy. I have used the phrase "a routine calculation shows that" in print, because the calculation truly was, probably more so for the prospective referee than for me. I have no qualms about using such phrases as appropriate for the audience being addressed. I agree with Pete that we should probably provide a little nudge in the correct direction of our argument whenever possible.

There is a possibly apochryphal story about Hardy (I think) who came to a certain point in a lecture and said "it is clear that," after which he paused, went over to a corner of the board and doodled for some time before resuming his lecture with the proclamation "Yes, it is clear that ... ." Hardy, I suppose, could get away with that. The rest of us might want to let the audience in on our little secret. I guess as with anything else, it's all a matter of balance, don't let the details obscure the flow of ideas.

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$@$Chris: I didn't see your answer until just now, but I like it and agree with it. About your story: I heard it about Wiener, I think, not Hardy. From what I know about the psychology of these two great mathematicians (both of whom died before I was born), judged purely as a piece of apocrypha, it fits Wiener much better than Hardy: Hardy, to all accounts, was really excellent at explaining why he knew what he knew. –  Pete L. Clark Mar 10 '12 at 21:00

I think you should be more explicit here in step two...

You should be more explicit here in step two

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It often means trivial in the algorithmic sense:

This part of the proof is tedious, but not interesting and does not necessitate insightful new ideas.

Almost always it would be better to be explicit:

E.g: It is trivial to check that the two sides are equal because the quotients of quotients are rational functions.

Obviously, this does not imply that the author would be able to correctly calculate the quotients of quotients at the first try. It is trivial in the sense that it is trivial to check 524 truly simple cases or calculate the gcd of two numbers.

Or be honest: The case of $B_n$ works exactly as the case of $A_n$, but the formulas are more complicated, so the details are not given.

Does not mean it is easy, it means that if you know the case $A_n$ and you are determined to solve the case $B_n$, the method will work.

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I know of at least one case where someone did this with the intention of intimidating the referees away from calling them on their claim. "It's easy to prove that..." i.e. "I dare you, referee, to say this isn't true! I will make you look like an idiot!"

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As alluded to in my answer, most moderately experienced referees smell blood in the water when they encounter an "it's easy to see" that is not actually easy to see. Also the refereeing process makes it much harder for the author to make the referee look like an idiot than vice versa. Too bad... –  Pete L. Clark Feb 4 at 6:17

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