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Using expressions like "easy to see" or "it is obvious"

Random Variable

Well-known member
MHB Math Helper
Jan 31, 2012
253
Why are such expressions used so incredibly frequently by mathematicians and aspiring mathematicians?

I think sometimes they are used to discourage questions, while other times it might just be an ego-boosting thing.

Any thoughts?
 

DreamWeaver

Well-known member
Sep 16, 2013
337
Interesting question! (Beer)

To be fair, I think it's also a bit of a meme-thing, although equally often it's as you say above. Some habits are just easy to pick up (which is "easy to see", non?) (Devil)


EDIT:

That said, it's also about relative complexity... So when posting a long, involved, intricate proof, the phrases above are just another way of saying "compared to the rest of these witterings, this 'ere part's pretty easy..."
 
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Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,197
Why are such expressions used so incredibly frequently by mathematicians and aspiring mathematicians?

I think sometimes they are used to discourage questions, while other times it might just be an ego-boosting thing.

Any thoughts?
In my opinion, they are used most frequently to hide the omission of steps. I find them rather arrogant to use, as if the fact that something is clear to one person when that one person was writing (which may or may not be true!) is automatically clear to another person of possibly differing abilities at a different time. And aren't you stupid if you can't see why it isn't obvious?

It's a theory of mine, completely unfounded on any evidence, that sometimes people use these expressions to hide slip-shod thinking. That would be an offence considerably worse than merely marginalizing someone's intelligence.
 

topsquark

Well-known member
MHB Math Helper
Aug 30, 2012
1,143
I had a grad professor at Purdue who would say "Is trivial, take two seconds" (do that in a strong Italian accent) about a problem I had spent hours on. I once had the nerve to ask if he was going to put a solution or not but, perhaps fortunately, he was still talking when I said it.

-Dan
 

Random Variable

Well-known member
MHB Math Helper
Jan 31, 2012
253
I came across a deceptively simple-looking 5 line proof of a transformation of the hypergeometric function. The author claimed that the intervening steps were obvious even to monkeys. Well, it took me several hours to figure out the intervening steps, and the proof I ended up writing was about 20 lines.
 

Ackbach

Indicium Physicus
Staff member
Jan 26, 2012
4,197
I came across a deceptively simple-looking 5 line proof of a transformation of the hypergeometric function. The author claimed that the intervening steps were obvious even to monkeys. Well, it took me several hours to figure out the intervening steps, and the proof I ended up writing was about 20 lines.
The American astronomer, Nathaniel Bowditch, when he translated Laplace's treatise Traite de mecanique celeste into English, remarked, "I never come across one of Laplace's 'Thus it plainly appears' without feeling sure that I have hours of hard work before me to fill up the chasm and find out and show how it plainly appears."
 
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DreamWeaver

Well-known member
Sep 16, 2013
337
But it goes both ways... If you write out every little step in a proof, some of those who know more with think you patronising, whereas if you right out more of a bare-bones proof, some who know a bit less will think you're inferring that they're dim (rather than unfamiliar with the idea/concept/etc in question).

It's not exactly win/win... (Headbang)
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,725
But it goes both ways... If you write out every little step in a proof, some of those who know more will think you patronising
The great mathematician John von Neumann used to write inordinately long papers, with every detail spelt out laboriously. The story goes that he did this because everything was more or less obvious to him, and he couldn't decide which steps really were obvious and which ones would need more explanation for ordinary mortals to understand them.
 

DreamWeaver

Well-known member
Sep 16, 2013
337
The great mathematician John von Neumann used to write inordinately long papers, with every detail spelt out laboriously. The story goes that he did this because everything was more or less obvious to him, and he couldn't decide which steps really were obvious and which ones would need more explanation for ordinary mortals to understand them.

(Inlove)(Inlove)(Inlove)

Classic!!!
 

HallsofIvy

Well-known member
MHB Math Helper
Jan 29, 2012
1,151
There is an old story of a professor doing a complicated derivation saying "and now it is obvious that" while he wrote a formula on a blackboard, stopping and looking at it for a few seconds saying "Now, why is that obvious?"

He sat down at his desk and spent the next five to ten minutes scribbling furiously over several sheets of paper, finally going back to the black board and announcing "Yes, it is obvious!"
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,775
I find it particularly amusing when I am reading a paper I myself wrote years ago where I state "and so it follows..." or "it is obvious that..." and no longer being entrenched in the topic, I then have to figure out all over again just why "it follows..." (Rofl)
 

Jameson

Administrator
Staff member
Jan 26, 2012
4,052
I think this totally depends on context. It's not reasonable or efficient to write out every step, but being condescending to the reader or student is never a good time. Intent is very important here.

This is a little off topic, but this reminds me of this joke I've seen a few times.

EvayE.jpg
 

DreamWeaver

Well-known member
Sep 16, 2013
337
I think this totally depends on context. It's not reasonable or efficient to write out every step, but being condescending to the reader or student is never a good time. Intent is very important here.

I quite agree, Jameson, but therein also lies a large part of the problem, since if - in good faith - mammal A writes a proof, with good intention and nothing but a well-meaning heart, there's nothing to stop mammal B transferring his or her own personal hang-ups onto kindly mammal A. End result, mammal A wonders why they bothered, and mammal B feels (unjustifiably) insulted/condescended to...

Bipeds...!! Meh! (Headbang)(Headbang)(Headbang)




EDIT:

My apologies... That should have been mammal X and mammal Y, since I assume we're having a Cartesian discussion here... (Heidy)

[I'll get my coat!]
 
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Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,910
Why are such expressions used so incredibly frequently by mathematicians and aspiring mathematicians?

I think sometimes they are used to discourage questions, while other times it might just be an ego-boosting thing.

Any thoughts?
I have learned to interpret such an expression as jargon meaning: it follows from the definition. So I look up the definition if necessary.

If it doesn't follow from the definition, I draw the conclusion that the person in question is conceited. Next time, when I see a piece of the same author, I take whatever he writes with a grain of salt.
 

Random Variable

Well-known member
MHB Math Helper
Jan 31, 2012
253
When I want to skip steps, I'll just state something as fact. And if I feel it's needed, I'll briefly state how one can derive that fact.

Every once and a while I might say something is obvious.

But with quite a few people it is like a broken record. And I know it discourages people from asking questions.
 

ZaidAlyafey

Well-known member
MHB Math Helper
Jan 17, 2013
1,667
As to me I like to write laborious proofs because I enjoy showing each step of a beautiful proof . Sometimes when I read papers I get a headache when the authors says ''it is obvious '' , ''it is clear '' especially if I understood non of it . Am I that stupid ? :confused:.
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,910
Sometimes when I read papers I get a headache when the authors says ''it is obvious '' , ''it is clear '' especially if I understood non of it . Am I that stupid ? :confused:.
That's why I always avoid using those terms. They tend to have an emotional impact that gets in the way of communicating (in both directions) what it's really about.
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,910
When someone reads "It's trivial", this is what it means:

 

Krizalid

Active member
Feb 9, 2012
118
If you see something like:

- This is trivial.
- It's obvious.
- It's clear.

and there's no other stuff in there, like an indication on how to proceed, the one who posted said it to feel great haha.
 

Evgeny.Makarov

Well-known member
MHB Math Scholar
Jan 30, 2012
2,506
I agree that the phrase "It is obvious" can be used when the author should explain the issue in detail, but does not feel like it. (Another similar phrase is "Without loss of generality", where some authors consider a special case without bothering to explain how the general case can be reduced to it.) But when used properly, it can be quite helpful. Ideally, the phrase "It is obvious" should have more or less precise meaning. For example, in the Coq proof assistant, there is a tactic (command) "trivial" that succeeds when the goal can be proved using the predefined database of lemmas that don't have the form of an implication, i.e., don't have assumptions that themselves have to be proved. There is also a tactic that applies exclusively logical (first-order) reasoning as opposed to domain-specific facts, such as those about limits in calculus. When a phrase like this is used, it puts a limit on the complexity of the proof, so the reader does not have to consider proofs that rely on specialized facts or those that are longer than just a few steps. Sometimes it helps a great deal to know that a proof of a certain statement is indeed simple.
 

Fantini

"Read Euler, read Euler." - Laplace
MHB Math Helper
Feb 29, 2012
342

It is easy to see why it is used, it's obvious.
 

Deveno

Well-known member
MHB Math Scholar
Feb 15, 2012
1,967
I tend to use the phrase: "clearly,..." or "it is easy to see..." for this purpose:

If you already know the truth of what I'm saying, you can just continue on.

If you don't, it's a subtle cue to pause, and work it out.

My intention is neither to insult/praise the intelligence or ability of the reader, nor to hide things I feel should be "spelled out", but to cut down references to more basic facts that would make a long discussion even longer.

I know full well that sometimes it may NOT be clear, it sort of goes with the territory that you cannot in advance decide the sophistication of whomever may read what you're writing.

It's, unfortunately, hit and miss...sometimes you wind up explaining it anyway. That's why a forum (post..response...further post) format is such a good idea.

In a lecture setting, one would hope a student had the bravery to say: "Wait...I don't get that". Lecturers often prove by intimidation...THEY obviously know so WHY DON'T YOU? But I suppose it's human nature to be afraid to admit one's ignorance. It's sad, really, we all start out knowing absolutely nothing at all.

How "enigmatic" should "proving" be? Gee, it depends. There's two competing forces at work: the attention span of the reader, and the need to communicate information.

Just avoiding the phrases doesn't quite work, either. One can commit the same errors of omission without calling attention to the fact, which can go unnoticed until the reader wonders...wait, what?

In a more general vein, such phrases are used by me when I feel the results used ought to be able to be worked out by the reader, in that if they can't, then there is a more serious lack of understanding at the general level than there is with some specific problem. In other words, I'm answering the wrong question entirely. Can't know that, until you get there.
 

Fantini

"Read Euler, read Euler." - Laplace
MHB Math Helper
Feb 29, 2012
342
I tend to use the phrase: "clearly,..." or "it is easy to see..." for this purpose:

If you already know the truth of what I'm saying, you can just continue on.

If you don't, it's a subtle cue to pause, and work it out.

My intention is neither to insult/praise the intelligence or ability of the reader, nor to hide things I feel should be "spelled out", but to cut down references to more basic facts that would make a long discussion even longer.

Just avoiding the phrases doesn't quite work, either. One can commit the same errors of omission without calling attention to the fact, which can go unnoticed until the reader wonders...wait, what?
If you want the reader to pause and work out something, why not just write (pause now and work this out)? Has the same effect.

As you mention, the omission could be done without any warning whatsoever. It's just as bad. It's as if some authors think: "Well, I'm lazy to write this up. I could just leave it to the reader. Oh, I know! I'll INSULT his intelligence and still leave it to him. That'll teach him to do things." Wonderful pedagogy, let's try it with all kids starting in arithmetic. I'm sure it's going to improve morale and enthusiasm. (Angry)
 

Deveno

Well-known member
MHB Math Scholar
Feb 15, 2012
1,967
It's hard to divine the intention of an author, unless you are that author.

Furthermore, some people may use such phrases without ANY of the baggage associated with them, in cases where it really IS obvious (like 2+2 = 4 obvious), just to improve narrative flow.

"Clearly" probably most fits in with this line of thought: if something previously proved and used several times is NOT clear, the pedagogical failure lies earlier on, not in the use of the word at that time.

Perhaps something like: "details left to the reader" is a more honest approach, the fact there is an omission is acknowledged, but the motive may in fact, be the same.

There is a deeper issue underneath all of this: when is "proven" proved? It seems unrealistic to reduce everything to its basic dependency on an axiomatic system; while this is, in some cases possible, it makes for dull reading.

To give an example: one might say that some specific case "follows by induction on $n$, but omit the actual inductive proof. Is this fair? How can one assess this?

Nevertheless, the objection IS a valid one. While stylistically, it might seems arrogant, I think it's just a bit presumptuous to assume it was INTENDED so. Teaching (in my humble opinion) works best as a dialogue, the instructor who cannot understand and remedy the difficulties of his/her students is not effective.

Perhaps, Fantini, if you find yourself in the position of making lecture notes, or even a textbook, you can "break with the trend" and find a novel solution to this dilemma. As for myself, I can try to be aware of such considerations when I write, but be warned: my memory isn't what it used to be, and I just might forget :p
 

Fantini

"Read Euler, read Euler." - Laplace
MHB Math Helper
Feb 29, 2012
342
It's hard to divine the intention of an author, unless you are that author.

Furthermore, some people may use such phrases without ANY of the baggage associated with them, in cases where it really IS obvious (like 2+2 = 4 obvious), just to improve narrative flow.
How does that improve narrative flow? This is one writing habit most have and when changed improves texts dramatically, in my opinion.

"Clearly" probably most fits in with this line of thought: if something previously proved and used several times is NOT clear, the pedagogical failure lies earlier on, not in the use of the word at that time.

Perhaps something like: "details left to the reader" is a more honest approach, the fact there is an omission is acknowledged, but the motive may in fact, be the same.

To give an example: one might say that some specific case "follows by induction on $n$, but omit the actual inductive proof. Is this fair? How can one assess this?
I agree with your second paragraph: "details left to the reader" is more honest. Just as we can't divine authors intentions, we can't divine their motives. Therefore, if both are good or bad, we can't tell. However, from what is written you can have an impression upon you.

An example is just when you have someone you like do something and you said something in the best interest, trying to help, but you end up offending. It was best left unsaid, even if with the greatest intentions/motives.

As for the "follows on induction on $n$", we thread on more dangerous terrain. We both know that there are induction proofs just like

$$\sum_{i=1}^n i = \frac{n(n+1)}{2}$$

and inductions where the base case is easy and the $k \to k+1$ is hell because it requires tricks and nonstandard methods. You can't possibly know unless:

1) You try it yourself, which is often hoped to be encouraged with such phrases;

2) The author tells you so. There is nothing demeaning to say "the proof follows induction at $n$. A rough sketch of important steps is..." You will have to write a couple more lines, but it should be nothing compared to the added clarity of your text plus you are giving the roadmap. Walking the road is always different.

Nevertheless, the objection IS a valid one. While stylistically, it might seems arrogant, I think it's just a bit presumptuous to assume it was INTENDED so. Teaching (in my humble opinion) works best as a dialogue, the instructor who cannot understand and remedy the difficulties of his/her students is not effective.

Perhaps, Fantini, if you find yourself in the position of making lecture notes, or even a textbook, you can "break with the trend" and find a novel solution to this dilemma. As for myself, I can try to be aware of such considerations when I write, but be warned: my memory isn't what it used to be, and I just might forget :p
I must have given the impression that I thought it was done arrogantly on purpose. I don't. However, that IS how it comes out. I don't read "obvious" and "it is easy to see" and am flattered.

I agree with you that teaching works best as a dialogue, but there's no teaching if there's no learning. Ken Robinson, a famous educator, mentions a distinction one of his friends used to make. You can be doing something but not really be achieving it. That is to say, you can be teaching, but if people aren't learning then you aren't achieving it. Textbooks authors don't get to know if there is learning going on because they aren't at their side. The dialogue is done through the text. Therefore, it is best to err on the side that it isn't clear.

These are two textbooks I own and hold in high regard as to mathematical exposition, clarity and rigor. I would certainly be happy if you could check them and see if you agree with me.

Mathematical Analysis - A Concise Introduction, Bernd Schröder

Mathematical Methods for Scientists and Engineers - Fourier Analysis, Partial Differential Equations and Calculus of Variations, Kwong-Tin Tang

Special attention goes to Schröder. At the first five chapters he is very careful to guide the reader, such as gray boxes indicating what specific theorems/computations were done at important passages, providing heuristic paragraphs motivating proofs and giving examples while leaving others explicitly for the reader (and saying so). If possible, see for yourself and comment back. :)