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

[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.