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

[polygamma]

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]

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..."

 

[Ackbach]

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]

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

 

[polygamma]

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]

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."

 

[DreamWeaver]

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]

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]

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]

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]

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]

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.

View attachment 1318

 

[DreamWeaver]

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!]

 

[I like Serena]

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.

 

[polygamma]

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.

 

[alyafey22]

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 ? .

 

[I like Serena]

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 ? .

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.

 

[I like Serena]

When someone reads "It's trivial", this is what it means:

 

[Krizalid1]

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]

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]

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

 

[Deveno]

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]

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]

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]

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. :)

 

[Deveno]

Well I think it's great you CARE about these things. I personally believe mathematicians in general should strive for clarity, and if "avoidance" (by use of certain phrases, or just through too abbreviated a style) does not help, the subject as a whole suffers.

I'll try to take a look at those texts you mention, but I don't know *when*. I get distracted by other things to do...

You've brought up some good points to think about, and I'll frankly admit I'm not entirely done "digesting" them. Perhaps I will become a better writer as a result, perhaps not. I will say I have thoroughly enjoyed the exchange, discussions such as these are part and parcel of what makes these forums near and dear to MY heart.

 

[Fantini]

Well I think it's great you CARE about these things. I personally believe mathematicians in general should strive for clarity, and if "avoidance" (by use of certain phrases, or just through too abbreviated a style) does not help, the subject as a whole suffers.

You've brought up some good points to think about, and I'll frankly admit I'm not entirely done "digesting" them. Perhaps I will become a better writer as a result, perhaps not. I will say I have thoroughly enjoyed the exchange, discussions such as these are part and parcel of what makes these forums near and dear to MY heart.

I care DEEPLY at this. I remember a discussion once in the forums regarding religion and how it was a hot topic for some members. Education and teaching are two for me. My blood boils whenever I think about the current state.

Like you, I personally believe mathematicians in general should strive for clarity, but I'll throw in that I also believe 90% haven't really thought what "clarity" means. This tends to affect in bigger number the younger ones, who believe "clarity" is writing like your PhD thesis (which are mostly unreadable, consequently) as a barrage of definitions, lemmas, propositions and theorems without words connecting and full of missing steps.

I'm glad I've managed to instill doubt and reflection on you, because this opens up the possibility that one more person will contribute to a better mathematics stand. I have an admiration for your persistence and effort to help others in this forums, your thorough answers and breadth of knowledge. I am nothing less than honored to have this intellectually stimulating discussion with you and see your openness. We need more mathematicians like you. :)

I would also like for others to participate in this discussion. I very much enjoy MarkFL, Jameson, Ackbach, chisigma, Bacterius, Plato, Evgeny.Makarov, Zaid, Sudharaka, ProveIt, Chris, caffeine, anemone, soroban, Opalg, I Like Serena, HallsOfIvy, Jester, SuperSonic, Fernando Revilla and others's opinions and I wish they would come in the thread and post them. :) This is a great community and this is a topic where everyone can chime in.

(I'll stop ranting now.)

 

[Prometheus1]

Mathematics is not something that's supposed to be read like a novel. You're expected to pause, no matter how good you're. At any sufficiently advanced level I don't think it's possible to write mathematics without - reasonably, of course - leaving out some details.

 

[mathbalarka]

reasonably, of course

And that barrier of reasoning is destroyed by mathematicians, especially professional ones and when one points out that such jumping caused a miserable mistake (I have done such pranks with notes of mathematicians, yes) they simply gets either too irritated, or, as the comicstrip Fantini gave tells us, says, "yes, but if you overlook that step, you can see the proof is obvious".

 

[Fantini]

Mathematics is not something that's supposed to be read like a novel. You're expected to pause, no matter how good you're. At any sufficiently advanced level I don't think it's possible to write mathematics without - reasonably, of course - leaving out some details.

The question is not omission of details. As I have said, these can be necessary and instructive but one does not have to use expressions as "it is easy to see", "obvious" and "clearly".

If the proof is long and hard, sketch the main details. Surely you can condense something of 2 pages in about a paragraph if you point the main arguments. If you can't, I don't think you should be writing about it at all. Another common problem is what is considered "easy enough" by a textbook writer. Unfortunately, and I'm not going to single out mathematicians on this, familiarity is a problem. The longer you work on something, the more acquainted you are with a subject, the more problematic it becomes to explain things to others and keep their perspective. One has to try hard to not assume everything is easy, but most don't, hence the use of expressions we're discussing. :)

Thank you for your input.

 

[MarkFL]

When I was a student, I liked to write out my thoughts and findings when exploring a topic as I felt it would be good to have these notes for later reference and felt the act of elucidating things as clearly as possible would strengthen my own knowledge too. I took great care to try to be as clear and straightforward in my explanations, trying to imagine someone reading it who is relatively new to the subject.

However, I found that when I pulled these notes out some months or years later, there were points where I would scratch my head as I no longer could "obviously" see how I had previously "so easily" bridged a gap.

It is very hard I think for an author, who is fully immersed in a subject, to even realize what will even be obvious or easy to see for his audience. I suspect that this becomes easier to see for more seasoned writers, but in my own personal experience, I found it was a very difficult thing to do, even with the best of intentions.

 

[Deveno]

Fantini linked to two texts that are, in his opinion exemplars of what to strive for. I am in the process of reviewing them in the light of the discussion in this thread.

This is what I have found so far:

The first, an analysis text by Schröder, is a very good example, where he "foreshadows" future omissions and why he will omit them. I recall some similar wording in Spivak's Calculus text, especially regarding certain types of proof, and limits regarding infinity.

The second, a differential equations/Fourier analysis book by Kwong-Tin Tang, is not quite so good. While it doesn't possesses any egregious faults, there is a certain lack of rigor, and some "hand-waving" (perhaps because it is aimed at engineers, who may more readily accept a "heuristic" proof). The calculations are good, the prose could use some work.

*******

In mathematical writing, it is a tough question to answer: "who is this for?". In people that are part of the same field, some very abbreviated prose will work. For a general text, it is (from my vantage point) a difficult assessment to decide "where to start". And how rapidly (if at all), does one accelerate the flow of information?

On another site, I read of a poster who had a very hard time getting past the first couple of chapters in ANY mathematical text. He found it was just too much to hold in his head at one time, and the ONLY way he could progress was by making copious notes. And this is someone who has a college degree, and would have had to have taken at least some college level mathematics. Obviously, someone's definition of "clarity" doesn't match up with this reader's.

So, it seems to me that "clarity of exposition" is a TALENT, and as with all talents, we can try to cultivate it, but there may be a limit to our ability. If only understanding was easy to acquire, and easier still to pass along...(sigh).

 

[eddybob123]

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?

Maybe it's to avoid Aaronson #9. Quoted directly from his blog: But when it comes to something like P≠NP, to “motivate” your result is to insult your readers’ intelligence.

 

[Fantini]

The first, an analysis text by Schröder, is a very good example, where he "foreshadows" future omissions and why he will omit them. I recall some similar wording in Spivak's Calculus text, especially regarding certain types of proof, and limits regarding infinity.

The second, a differential equations/Fourier analysis book by Kwong-Tin Tang, is not quite so good. While it doesn't possesses any egregious faults, there is a certain lack of rigor, and some "hand-waving" (perhaps because it is aimed at engineers, who may more readily accept a "heuristic" proof). The calculations are good, the prose could use some work.

I hold Schröder's book in very high regard. The omissions are part of the pedagogy, but the amount of scaffolding and intuition he provides is very enlightening. I like how he tries to convey the big picture and lay out the mathematical landscape without forgetting to get his hands dirty in technical details. <rant> I feel many people today want to be Groethendicke, building fabulous abstract constructions but feel they are too noble to be actually able to do computations and/or technical work (e.g. what I came to know was usually hard analysis). </rant>

As for Kwong, I agree: it is hand-wavy. However there are three books and the sheer amount of mathematics one would need to know in order to prove everything rigorously AND still do the computations is just sheer madness. Just the chapter on Fourier Transform would require measure theory, plenty of real analysis and PDE. Each requires a separate book. What makes me like it is that it actually gets the hands dirty in the computations. It was tiring to look through classics (such as Arfken and Butkov) and not find the intermediate steps, which I really needed. I guess this is my pet peeve.

In mathematical writing, it is a tough question to answer: "who is this for?". In people that are part of the same field, some very abbreviated prose will work. For a general text, it is (from my vantage point) a difficult assessment to decide "where to start". And how rapidly (if at all), does one accelerate the flow of information?

On another site, I read of a poster who had a very hard time getting past the first couple of chapters in ANY mathematical text. He found it was just too much to hold in his head at one time, and the ONLY way he could progress was by making copious notes. And this is someone who has a college degree, and would have had to have taken at least some college level mathematics. Obviously, someone's definition of "clarity" doesn't match up with this reader's.

I guess how tough it is depends on what kind of book. I'd say as a rule of thumb if you are writing for undergraduate or graduate audiences, then be explicit and less skip-happy.

Another (possible) rant: I have seen some people claim certain texts were "very clear" and "direct". I think some mathematicians's definitions of clarity is not being able to read and grasp what is written with minimal effort as possible (minimal is a function of each person, but this is to say that the author has gone as far as he can to help the reader to the best of his abilities) but that the text is dry and is basically a dictionary/encyclopedia of sorts. No explanations.

So, it seems to me that "clarity of exposition" is a TALENT, and as with all talents, we can try to cultivate it, but there may be a limit to our ability. If only understanding was easy to acquire, and easier still to pass along...(sigh).

I don't like advocating the view of talent, although we know it exists. The main reason is why I dislike telling people about intelligence: it hinders more than it helps. It takes control of the outcome from you to some innate, immutable quality called "talent". Even talent has limits, and when you reach those people are bound to say "well, this is as far as I can go. Too bad." And they stop trying to improve.

Maybe it's to avoid Aaronson #9. Quoted directly from his blog: But when it comes to something like P≠NP, to “motivate” your result is to insult your readers’ intelligence.

While I agree that certain subjects for ARTICLES, which is meant to be read by PROFESSIONALS and RESEARCHERS in that area it may be insulting, I believe the majority of works that gets written is for a more diverse audience. If you're writing a calculus, or analysis, or abstract algebra, or manifold theory, or microelectronics, or fluid mechanics, or chemistry book, you should motivate what you are doing. Motivation isn't just saying "this is worth studying because human creations are fascinating and this particular subject is my favorite. Definition 1...", but at each step providing context and guidance to where you are headed, what are your goals, what is intended with certain constructions, difficulties associated with previous attempts, etc. You may argue: "Texts will become tomes!" You don't have to go so deep to present these points.

It's dangerous to quote this without the appropriate context. It is best if you show the source as well. :)

 

[Deveno]

I don't like advocating the view of talent, although we know it exists. The main reason is why I dislike telling people about intelligence: it hinders more than it helps. It takes control of the outcome from you to some innate, immutable quality called "talent". Even talent has limits, and when you reach those people are bound to say "well, this is as far as I can go. Too bad." And they stop trying to improve.

I don't, as a rule try to emphasize unduly talent. I have a great deal of respect for people who learned things "the hard way" through persistence and determination.

I guess I see "talent" as like the "base scores" when one creates a D&D character, it gives you a small advantage, but you can still follow your dreams, even if it means extra work. From personal experience, I can say that success is usually:

10% talent, 90% drive.

Intelligence is one of those tricky subjects: personally I don't think it is a monolithic characteristic, but an aggregate of characteristics. For example, there are:

Logical skills, spatial visualization, imagination, creativity, grammatical/linguistic skills, eloquence, numerical ability, memory, coherence.

Some of us may be gifted with natural abilities in some of these areas, but we owe it to ourselves and the world around us, to develop the areas we are lacking in. No human being "does it all", and we should be more forgiving of those who struggle in one area, and appreciate the gifts they have in the areas they shine in. And that carries with it, I think, a certain moral obligation to see to our own faults before judging others.

Mathematics is, after all, a form of communication, and communication isn't effective without common ground. Mathematicians are perhaps a bit more prone than some other fields to conceit: especially when confronted with the pure/applied dichotomy. This is, in my opinion, counter-productive, pure and applied mathematics should mutually enrich each other, and taking one to be "more" mathematics than the other, eventually deprives us all of what could be: understanding.