How to Read Mathematics
web.stonehill.edu
web.stonehill.edu
-Paul Halmos, inventor of "iff" and the ∎ symbol ( http://en.wikipedia.org/wiki/Paul_Halmos )
> Halmos argued that mathematics is a creative art, and that mathematicians should be seen as artists, not number crunchers. He discussed the division of the field into mathology and mathophysics, further arguing that mathematicians and painters think and work in related ways.
A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.
as did Bertrand Russell (one of my favorite quotes):
Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.
I suspect many mathematicians view themselves, at least to some extent, as creative types similar to poets.
"When I come across some academic mathematics paper, even if I'm somewhat familiar with the field, I generally find other things far more interesting, like the coffee stain on the floor. Effectively reading a mathematics paper requires a print out (not just of the paper itself but also many of the papers cited) and a pencil to work through some of the definitions and take notes for yourself, and maintaining laser-like focus for a sustained period of time."
I think that capture the gist of that article. Basically, it can be an exhausting and draining affair, especially for someone with my cognitive capacity.
The reviewer is a skilled experimenter, whom I enlisted to check my understanding of the physical processes. (By the way, this person knew what I was saying, just didn't like the way I said it. For our internal reviews, we tend to get pretty picky.)
It's definitely not for a general audience-- there'll probably be five people in the world who can use the paper directly (once the results are embedded in software, there'll probably be several hundred who will use it).
How can anybody with a grade school level education not understand that? Are you sure the reviewer couldn't turn it into a sentence? Or perhaps the reviewer didn't feel that was the proper formatting for a sentence? I'm not sure how else you would include equations in a sentence, but if my reviewer didn't understant grade school level mathematics I'd be horrified that this person was in a position to be reviewing anything at all.
When a sentence had an identifiable "other" verb-- for example "Then substituting gives x = y/2"-- everything was fine.
The reviewer actually made some incredible suggestions for restructuring the paper to bring out the important points. I'd hate for people to come away from my original comment thinking the reviewer is anything other than a very bright person who was pushing me to make the paper as clear as possible.
≠ -> !=
√ -> sqrtI know somebody is about to point out to me that I'm being "lazy" and should enjoy doing more work to learn so and so, but being asked regularly in the literature to reinvent somebody else's wheel seems to run counter to everything we do in CS (and academia in general, I would think). This article aside, I do sometimes feel that academic writing in the math/CS realm occasionally reeks a little bit of snobbery, where communication is held secondary to keeping up appearances.
... reinventing the wheel runs counter
to everything we do in CS.
You can read all you like about how to unicycle - if you don't put in the hours, you won't be able to do it.
You can read all you like about how to program - if you don't put in the hours, you won't be able to do it.
Skills require practice. These instances of "left as an exercise for the interested reader" are to help you really learn and properly understand the material. And if you don't care, don't bother. If you are unwilling to put in the time then the chances are that you would end up thinking you understand an explanation, whereas in fact you don't.
The above is only for good writing, of course. There are plenty of instances of bad writing, but that's not then a complaint about the subject, it's a complaint about the writer.
Someone once said of Feynman that his lectures were beautifully clear, and that those who listened gained real insight and understanding. Until they had to use it. Then they realised that they didn't really understand it at all.
If you want understanding, do the exercises.
So there you are, I've done as you predicted and pointed out that you're just being lazy, and that if you really want to learn then you have to put in the work. Just because you preempted it, doesn't mean it's wrong. It's right, you already know it. Complaining won't help you.
Put in the time.
1. it is doubtful that the author has actually done the work to prove his own assertion, and uses the phrase to cast his own burden on the reader,
2. the author has possibly done the work, but can't be bothered to condense it to the quality required for publication,
3. showing the work would clearly be useful for the target audience, but the author is more concerned with making the material appear difficult, or
4. the author is being ironic, because the assertion is either superfluous, outright incorrect, or known to be unprovable.
In fact, all these bad use-cases are common sources of humor among math nerds (http://www.jargon.net/jargonfile/e/exerciseleftasan.html), (http://abstrusegoose.com/12), (http://fasterdonuts.tumblr.com/post/4516834904/the-proof-is-...) ... so I think I'm being fair in calling this phrase a trope. As in, its overuse is well-known, tolerated, and occasionally ridiculed.
tldr: I am indeed complaining about bad writing, not the subject--specifically that "left as an exercise" has become a common idiom behind which bad writers in math and CS hide, and its inappropriate use is now all too common.
But it is a phrase that has its place, there are times when it's exactly right to use it. Don't reject it outright. I've used it in publication where the editor actually asked me to remove details and put the phrase in its place.
It's not always the author.
Obviously mathematicians don't leave "exercises for the reader" in research papers, but it is common in textbooks or expository writing.
> I always felt that phrase has no place in the internet age, where the concept of a "page limit" is laughable and a simple hyperlink can point me to chapters upon chapters of appendices.
The constraint is not page length, of course, but time. It is a much better use of a mathematician's time to leave easily reproducible proofs to the reader, and focus on the interesting ideas.
Anyway, doing these kinds of exercises is actually very useful in understanding new ideas.
ORLY? I'll leave this right here for you:
http://scholar.google.com/scholar?hl=en&q=left+as+an+exe...
That's >100k hits from what Google Scholar can index alone--a few are from books, but the vast majority are journal articles.
Do you really think anyone would benefit from such proofs in a paper? I would be interested if you could give an actual example of a paper where a reader might be inconvenienced by something being left as an exercise.
Modern programming languages and conventions are quite a bit better in that respect, since people design them to be maintainable. A program using one-letter variable names for everything and a set of similar macros meaning entirely different things in different files would be considered unreadable by pretty much any programmer.
IMO, the world would immensely benefit from another version of mathematical notation designed specifically for explaining things, rather than doing calculations on paper.
I don't think this is just about the math community. The problem - at least as I see it, as a student myself - is that Math is a required part of many fields of studies. Fields which not necessarily have much to do with Math. And that's a problem for many of the students. I won't say that Math is entirely unneeded and should be completely abolished, but what I am saying is that many of these students (including myself) don't want to be part of said "math community".
We don't want to have to learn the arcane notations and "math language" just to be able to use some of the tools we might indeed need. Yes, I might sound like a whiny student who's butthurt over having to actually study for something. I may be. However, seeing as /so many/ other students (who, for the most part, are competent in their own chosen field) are in danger of failing their studies or struggle heavily with their math lectures simply because they are /not/ competent at getting into the "math community" is ridiculous to say the least.
For the record, I study Informatics at a university in Germany, so the situation might be different elsewhere, but I've read about similar issues at American colleges and universities, and every now and then, related articles pop up here at HN.
Without this common language, we would have a specialized language for every field, allowing little "cross-pollination". As an example, in evolutionary biology, take the NK model of epistatic interactions which models gene interactions. It just so happens that this model is extremely similar to a thing in statistical physics called a spin-glass. Without mathematics the biologist would have to sit and work out all the details of these epistatic interactions before getting to actually do biology. So in short, mathematics saves you time by expressing common patterns or ideas.
It was Simpson's rule.
CORRECTION:
It was worse than that. It was the trapezoidal rule:
http://care.diabetesjournals.org/content/17/2/152.abstract
The author named it after themselves:
... The Tai model allows flexibility
in experimental conditions ...
http://fliptomato.wordpress.com/2007/03/19/medical-researche...
Unbelievable.
I wasn't arguing that. In fact, I said that math is lending me and students of other fields tools to use and apply. The problem lies in the expression of said tools (which is exactly what I mean with "arcance notations") and the inability of many mathematicians - no offense - to separate those two concepts (tools/mechanics and expression) from another. They assume that they are completely inseparable, which is not only part, but a major source of the problem. Any attempts to somehow make math less obscure (or even just suggest it) is met, in great parts, with conservative hostility. I've seen mathematicans claim "that I was telling them that their profession is pointless and should be abolished" when all I was saying that a majority of the expression of math is incredibly obscure/arcance to most people which are not mathematicians and mostly not suited for practical appliance (as in, appliance in fields of science other than math).
Also, it's a valid point to say that we shouldn't have special syntax/language for every appliance of math imaginable. That would indeed not only be stupid, but actually /increase/ obscurity. We should, however, stop excusing the obscurity of math with "it's shorter to write". Readability and - most importantly - comprehensibility should /always/ come before convenience. Even in math.
Also, a formula can be written once and read hundreds of thousands of times. Because of this, I consider ease of reading math notation far more important than efficiency of writing in it.
To create a new set of symbols and notation designed specifically for those who are not mathematicians is a waste of resources. Professors, whom would be responsible for teaching such a thing, would have to translate all of their concepts from "professional notation" to "novice notation" when lecturing. This involves them learning a totally new set of notation designed for no reason other than teaching. Furthermore, anyone who wanted to pursue mathematics professionally would have to then learn "professional notation".
Aside from these logistical difficulties, I fail to see how a different set of notation would be any clearer or easier to understand than the current notation, which is already widely accepted and considered useful. Essentially, you are arguing that math notation is too difficult and so it should be simplified. That is like saying Faulkner or Joyce is too difficult to read so should be rewritten for those without the ability to comprehend the source document: it sort of misses the point.
As an outsider who only ever needs to use maths as a "tool" for very specific things, maths books and general attitude by professors and teachers is just nightmarish. Somewhere behind that curtain of inaccessibility, it has to be their fault. Especially, since once you actually do understand certain mathematical concepts you suddenly realize they're easily explainable with a few words of plain text or --gasp-- a "childish" drawing to illustrate.
But let me ask you this. It's easy to cut a square into identical pieces so that all the pieces touch the center point.
In slightly more detail, the pieces are disjoint sets such that their union is the whole square. The pieces are identical except perhaps for details as to the boundaries. To say that they all "touch" the center point means that every non-zero radius disk centered at the center contains some points from each piece.
So now, how many ways can this be done? No, it's not five. And no, it's not six either.
When you start trying to work it out you find that the details matter, and they can't just be covered by a "childish" drawing to illustrate.
Details matter, and some of them are hard.
Yes, most math teaching is atrocious. We all know that. But it's not always just the teacher's fault. Sometimes it's at least partly the fault of the readers expecting everything to be made simple and immediately accessible with neither work nor effort.
An infinite number of ways this can be done.
Can you characterise the solutions? How many pieces do they contain? Some solutions have two pieces. Some have four. Are there other possibilities?
I think 8 pieces also works, but I think that may be it.
And I have, so far, 5 infinite families with, respectively, 2, 4, 8, 16 and 32 pieces.
But details matter, and you might start to question what it really means when I say "piece." Does the definition I've given really capture your intuition?
But I need to go back to writing the article.
Math is, in a very general sense, about understanding and reasoning about highly structured objects. Mathematicians have developed methods and notation to achieve that end, not to make it easily digestible by the uninitiated. Rigor is an important part of this. It is what allows mathematicians to be so sure their work is correct and it is often what makes math seem so arcane. It is not always necessary for teaching, but professional mathematicians are often the teachers and the ideas are intimately tied to their rigorous formulations in their head.
For example, continuity is a fairly intuitive concept in calculus, but the rigorous epsilon-delta definition is necessary in proofs. The basic concept, while easy to explain and understand is useless, while the precise formulation, though much more opaque, is ubiquitous in analysis simply because it is an incredible tool... for mathematicians.
An engineer likely just needs the machinery built on top of the analysis: the derivative and integral. If that's all you need, then pick a book that is focused on applications and a development of general intuition, not a book designed for mathematicians-in-the-making.
Are you aware, for example, that there are functions that are continuous at all irrational points and discontinuous at all rational points?
Next time you encounter such a 'aha erlebnis', try saying the 'few words of plain text' to someone who still has to master the subject, and _immediately_ test him or her on the subject. I predict that, oftentimes, (s)he understood nothing of what you said.
Case in point: at university, I had a prof who said that it was perfectly OK if you could not do any practice integral. Trust me, he said, if, a month from now, you do these exercises, you will not understand what is hard about them.
"What readability-per-line does mean, to the user encountering the language for the first time, is that source code will look unthreatening. So readability-per-line could be a good marketing decision, even if it is a bad design decision...The math paper is hard to read because the ideas are hard."
By the way, when you construct a proof you almost always have some leeway in how big your indirect section are. E.g. variant A: Assume X, do bits Y, Z, then contradiction. Variant B: Do bits Y', Z', assume X, then contradiction. In the second variant Y' and Z' could be useful on their own, and might be easier to understand without the inversion. For that reason, I often try to keep the contra-factual parts of the proof as small as possible. Though as with writing any code, clear writing trumps general rules.
[0] http://primes.utm.edu/notes/proofs/FermatsLittleTheorem.html
Suppose that ra and sa are the same modulo p,
then we have r = s (mod p), so the p-1 multiples
of a above are distinct and nonzero ...
Consider the (p-1) multiples of a given by:
a, 2a, 3a, ... (p-1)a. (mod p)
These are all distinct. To see this, consider
otherwise, and suppose ra=sa (mod p)
Is that what you meant?
The point is that all writing is aimed at an audience. I wouldn't expect someone with no experience of Science Fiction to be able to read "Quantum Thief," and I wouldn't expect anyone with a reading age of 6 to be able to read "Lord of the Rings." Similarly, that proof requires some degree of familiarity with the structure of proofs. This is not an especially difficult.
I've always found that indirect proofs, or proofs by contradiction, or proofs by the contrapositive, are mostly obvious as to what they are doing, although not always.
In short, I agree with what you say, don't think the problem is as bad as you are portraying, think the example you have given is not especially good, but I'd be hard pressed to find a better one.
And finally, it's possible to come up with bad writing in every context. Some proofs are badly written, badly expressed, and badly explained. I know - I've not only read many of them, I've written some as well.
No surprise there.
Addendum: Sometimes the proofs that gave me the greatest understanding were the ones that were the most badly written, forcing me to work through the material on my own terms and understand it in my own way. Perhaps well-written, well-expressed and well-explained proofs are actually a bad thing.
I spent years of my life reading mathematics, so I do not trust myself to judge how hard a piece of mathematics is for outsiders. I find the proof cited is easy to read.
About your addendum: You could have a look at Alexander Schrijver's "Combinatorial Optimization: Polyhedra and Efficiency". The interesting thing about its style is, that the author manages to make all lines require constant thought, while in most books there are really hard and really easy parts.
The only semi-implicit ways to start a proof by contradiction I can think of are the phrases "if x is..." or (less implicit) "if x were...".
Proof, Suppose that x in the middle of the interval has a derivative > 0. Then from the definition of the derivative there exists h > 0 such that x+h is in the interval and 0 < (f(x+h) - f(x))/h. Multiply by h and rearrange to see that f(x+h) > f(x) and therefore x is not where the maximum is achieved. The argument for f'(x) < 0 is similar except that f(x) is exceeded by f(x-h) instead.
Therefore if f achieves a maximum at x, then x is either a boundary point, or a spot where the derivative is 0 or undefined.
You may allow to yourself not to know with what intention author wrote the line but you should never allow yourself not to know why the author could write this line.
I must say, there's nothing like reading a math paper to remind me that there's no shortage of people in the world that are way smarter than me.