Is there any good reason to write "it is easy to see"? [closed]

Question

Many academic papers, particularly in mathematics and similar fields, use the phrase "it is easy to see that..." (e.g. in a mathematical proof). I never understood why this sentence is used. Such a sentence is inaccurate at best, since it is not easy for everyone to see; maybe it is easy for the author or some of the readers, but there are certainly readers to whom it is not easy. Authors often try to shorten their paper as much as possible, so it is not clear why they would lengthen their paper by an inaccurate sentence.

Is there any good reason for an author to use this phrase?

Answer 1

Mathematical papers, especially, and many others, are not written for the complete novice with no understanding of the field. One tries to tailor the explanation of a new concept to the general level of understanding of the audience and tries to avoid being overly pedantic giving every detail and argument back to Euclid.

College lecturers often use such phrases, knowing the level that their students should have attained and, thus, permitting the lecture to flow more smoothly and be less boring.

But writers do the same thing, envisioning the reader of the paper, or perhaps just the reviewers. If I say such a thing and the reviewer calls me on it, then I know I have to say more. But I trust that the reviewer can validly stand in place of the audience and will be able to easily fill such gaps.

When such a phrase is used (we can conclude X) it implies that X flows from the previous statement(s) and isn't just an unsupported statement inserted into the flow. It indicates a hopefully simple, but unstated, argument.

The alternative of including the complete argument is longer, more pedantic, more boring papers.

The other alternative of starting every sentence with "Therefore,..." is also stilted and, eventually, boring. Some variation of phrasing makes the reading more pleasant. Writers, in general, like to have a repertoire of, more or less, equivalent phrases to help the flow.

On the other hand, it has happened that the statement is used incorrectly when it takes a lot of argument and deep insight to go from point A to point B. Sometimes the author just doesn't notice the width of the gap. But, for the most part, it seems to work out.

Answer 2

Several times during seminars I heard the following exchange, which sounds like a joke but actually isn't:

Audience member: Why is X true?

Speaker: Oh, it's obvious.

Audience member: OK, thanks! [sits down satisfied]

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Especially in mathematics, it is often helpful to let the Reader know what level of difficulty and/or complexity each step is. The approach you (or at least I) would take differs significantly depending on this.

If a step is supposed to be easy and it's not quite my field, I'd take note that this sort of transition is standard in these parts and move on. Conversely, if the step is supposed to be easy and it is my field and it doesn't seem easy at all, then it's a strong indication that either the author is wrong (rare) or that my understanding is insufficient and I should think about this transition until it seems easy (not just until I find any way to justify it).

If a step is supposed to be moderate in difficulty, I might or might not work out the details myslef as an exercise, but I will also be aware that some amount of work goes into making it possible - helpful when evaluating if an argument is plausible and also in identifying where the content of the paper really is (especially in papers that are heavy on definitions, it's often a nontrivial task to figure out which transitions are easy but unfamiliar and which are responsible for the actual progress).

Finally, if a step is said to be difficult then there obviously should be a reference or a proof. If it's a proof, then more likely than not this is the point where the progress is being made in the paper, so if I'm reading the paper this is the part I would study in the most detail. If it's a reference, I would make a mental note that it's a potentially strong tool to keep in my arsenal - also, I would know better than to attempt to reproduce the result myself.

Note that it's not always all that easy to determine which is which without the Author explicitly making a judgement. The short phrase "By Thm. C in [42] we have X" could expand into either of "It is easy to see that X (see Thm C in [42] for details)" and "Because of the deep theorem of Smith (Thm C in [42]) we have X".

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Having said all of the above, I want to add that personally, I dislike the phrase "It is easy to see". I understand the sentiment, but if the paper is read by anyone other than the experts in the field (maybe undergrads or experts in another field) then chances are it's not going to be easy to see to all the Readers, and the Readers who don't find it easy might feel bad about it in one way or another. If I'm already taking the time to explain myself for not explaining a transition I try to give some more details: either a super-short sketch of a proof, or a phrase like "It follows by an application of standard techniques that ..." or "A simple but mundane computation shows that...", etc.

Answer 3

One common way that I interpret "It is easy to see X" is as an ever so slight shortening of "it easily follows that X" (which I prefer). This itself is a shortened version of "The preceding assertion implies X", or perhaps a pluralized version such as "The preceding assertions (or definitions, or assertions and definitions, or...) imply X".

There is information contained here, namely a logical implication, that would be lost if one simply wrote "X".

Answer 4

There are no good reasons for phrases like that one. If something is "easy to see" or is "obvious", then there is no point in emphasizing such truism.

However, if no additional knowledge in needed and the following steps are quite straightforward, then one can use the advice form the Mathematical writing article in The Princeton companion to applied mathematics:

The question of how much detail to give is related to the question of how formal to be, but it is not the same question. It is true that there is a tendency in informal mathematical writing to leave out details, but with even the most formal writing a decision has to be made about how much detail to give; it is just that in formal writing one probably wants to signal more carefully when details have been left out. This can be done in various ways. One can use expressions such as “It is an easy exercise to check that...,” or “The second case is similar,” which basically say to the reader, “I have decided not to spell out this part of the argument.” One can also give small hints, such as “By compactness,” or “An obvious inductive argument now shows that...,” or “Interchanging the order of summation and simplifying, we obtain....”

If you do decide to leave out detail, it is a good idea to signal to the reader how difficult it would be to put that detail in. A mistake that some writers make is to give references to other papers for arguments that can easily be worked out by the reader, without saying that the particular result that is needed is easy. This is straight- forwardly misleading; it suggests that the best thing to do is to go and look up the other paper when in fact the best thing to do is to work out the argument for oneself.

Also from How to read and understand a paper in the same book:

In mathematical writing certain standard phrases are used that have particular meanings. “It follows that” or “it is easy to see that” mean that the next statement can be proved without using any new ideas and that giving the details would clutter the text. The detail may, however, be tedious. The shorter “hence,” “therefore,” or “so” imply a more straightforward conclusion. “It can be shown that” again implies that details are not felt to be worth including but is noncommittal about the difficulty of the proof.

Answer 5

"It is easy to see" should be used sparingly. If one can say how it is easy to see why the statement is true, in a similar number of words, then this strategy is prefered. Alternative, but more specific expressions, than "it is easy to see" include

• After routine algebra, ...
• By theorem X, ...
• By definition of Y, ...

Note the phrase "It is easy to see that" is actually longer than all of the above.

Sometimes the reason why it is easy to see something cannot be explained in so few words. In which case, if it is truly easy to see for the target audience, using this phrase is useful to signal to the reader that if they don't follow, its probably some trivial thing that they've messed up in their head, rather than a need to think deeply about the statement. But even then, the phrase should only be used when it can't be made more specific.

Answer 6

My personal opinion is that "it is easy to see" should really mean that it is easy for a reader who is the target audience to see, and can be used when that part of the proof is not an important piece, such as if it is a routine or tedious but uninteresting calculation. For example, it is better to write "it is easy to see that algorithm A runs in O(n^3) time" if it is just a bunch of for-loops that obviously take O(n^3) time, than to give a long and tedious proof just for the sake of formal rigour.

That said, it does happen that sometimes what one thinks is "easy to see" is not so easy for others to see, or perhaps even oneself after a few months of not looking at it. Hence having reviewers helps.

Answer 7

Papers are not written with exhaustive detail. There is always a balance between explicitly stating your reasoning and sticking to the topic at hand. Sometimes there are points that don't take great insight to understand, and would be easy (though perhaps lengthy) for the reader to derive on their own. This is the first function of "it is easy to see".

Another aspect is managing tone and audience expectation. Papers are written for an expert audience with the premise of pushing the bleeding edge of human knowledge. If the author of a paper states something obvious or well known, the reader may be confused: If it is emphasized in a modern paper, could it be that the author is not just a fool restating things everybody knows, but actually means to state something different? The competent reader could be confused by what the author meant to accomplish in stating something already well known or "easy to see". So the author preempts this, by acknowledging the point as something well-known and not novel, but mentioned for clarity and/or as a reminder, and not something to be taken as a novel or interesting claim. This is the second function of "it is easy to see".

Some people like to criticize statements like "it can be shown" ("then why don't you show it?" - because it would be a distracting digression) or "it's obvious" ("then why say it?" - to show that the author is aware of it). They assume it is an exercise in arrogance on the part of the author. However, written communication is not just the text. There is also subtext and context. Well-structured text has a central point and a coherent tone. Going on every possible tangent, digressing into topics of wildly different level, does not necessarily serve these qualities. As such, phrases like "it is easy to see" can serve an important point in enabling effective communication of novel findings to an advanced, technical audience that has less time available to read than there are papers of interest.

Answer 8

Yes. It specifies a particular level of difficulty (not too easy and not too hard), thus managing the reader's expectations and directing their focus.

The phrase will not be used for completely trivial deductions that can be done in half a second. If the previous sentence concluded that 2x=2, nobody would write "it is therefore easy to see that x=1". It's easy to see that it's easy to see, so there's no value in pointing out that it's easy to see.

Likewise, if the deduction is difficult and requires hours to figure out, nobody would write "it's easy to see", because it's not, and saying it is will confuse the reader.

It's the middle ground where the deduction can take a few seconds or maybe a few minutes, where the phrase is useful. A priori, the reader does not know the difficulty of the deduction - is this a half-second thing and I'm too dense to figure it out? Is this a difficult deduction and the author is remiss for neglecting a proof, or maybe he's just making stuff up?

The statement "It's easy to see" signals that we're in the middle ground - No, you're not stupid for failing to recognize this immediately (if it were that easy, I wouldn't say anything). But yes, I'm confident that if you spend a minute you'll figure it out, so there's no need to encumber the paper with all the details.

So much power in such a simple phrase.

Answer 9

The author most often writes this to justify that he does not include a proof, that you can do yourself in a reasonable time. This assumes that you have the basic knowlege that is required in your field to understand the average paper.

If you think you're missing some parts and cannot see why it is easy, you usually can look up the "easy to see" (after eight pages of calculations) proofs in other manuscripts, text books or in the internet.

If they would include this proofs, the reviewers and editors would ask them, why they proof something, that is known from text books or "easy" to see. A paper should contribute something new and should not (and cannot) proof any details that it builds up on.

For more stuff that is not well-known or more complicated, the authors should provide a citation and write "the derivation of the formula and the proof can be found in [42]".

But when you cite a text book, then people can ask why do you cite text book A and not text book B? The correct citation would be the original paper, but it will usually be much harder to understand the topic in the original paper than in a text book.

And there are of course some hand-waving, when the author knows something to be true, but would need more time to proof it himself and does not consider it to be worth the effort.

Answer 10

"You'll notice that ...."

First person always reads smoother, commands attention, and prevents overwtiting.

That said, you'll want to be clear who your audience is at the outset of the publication. State it explicitly who should be reading the text, and the assumed knowledge level of the subject matter.

Answer 11

Yes, though it is sometimes abused.

It's an assertion that if you think something is true you've probably got it right. It also sets the tone of what you're about to have to think through. It's common in other walks of life the other way round: if you have something that you are asserting that is non-trivial and is going to take some working though, you give your audience a warning.

In maths, perhaps arrogantly (though unfortunately in my experience accurately), it is assumed that the statements in papers, talks, books, etc are involved. Thus instead of caveating each sentence with "this is hard", the few that are easy are caveated instead (simply as a way to reduce verbosity).

The problem comes when the caveat is incorrect or is only correct after something highly non-trivial has become internalised. In this case it alienates those readers. As I imagine you can guess it not only hampers progress, it can also be seen as a bit of a middle finger. "If you don't get this you must not be cut out to be doing this sort of thing". Perhaps in some cases this is how its meant or just laziness. However its not always easy to know what will help your readers most and "nothing deep happens here" is far more likely.

I don't like using "trivial" or "clear" because of the potential to be misread, but I have often wanted to express that sentiment.

Answer 12

Yes, there are good reasons for the general meaning of phrases like this, but "It is easy to see that..." is a very poor choice of phrase for this meaning. Others have already suggested better phrases.

As Buffy expresses, it's not expected for advanced papers to reference, let alone prove every conclusion used. The problem here is the passive voice, and, more importantly:

It is easy to see. What is easy to see?

I see two general phrase choices:

1. Longer. If you want sentence flow to present a long thought, "From Y, we observe that with/from [few steps or concepts], X.", which carries the meaning more explicitly.

2. Shorter. What is being said is, "Y. Y => X." (Y being the conclusion the reader is expected to know, from the audience the text is written for.)

This way, you are at least specifying the subject of your sentence.

Now, if Y is something you learn in high school, you would look silly specifying it; you would look as if you're proud that you can still do high school math. You write papers for your peers.

It's a safe bet that one or more of your peers won't agree Y is obvious, but perhaps will say nothing in review to not "look dumb", when in fact s/he is your peer, just working in a different field.

Widening the audience slightly from that seems a reasonable place to be.