" Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis? "
> 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.
While it may be a bit pretentious to quote oneself, I'll go ahead and do just that from another post similar to this topic:
"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.
I just had a reviewer criticize a report I wrote for containing "sentence fragments". The paper says things like, "Then x = y/2", which my reviewer interpreted as a one-word sentence ("Then"), followed by an equation. It made me realize how unnatural it is for some people to actively read mathematics-- for example, turning "=" into the verb "equals".
What kind of reviewer, for what kind of paper? Any paper in which I wrote "Then x = y/2" would almost certainly be a math (or computer science) paper, so the reviewer would know what I was saying. If it was for a more general audience, I would probably find a better way to state my point.
It's about modeling specific physical processes in buildings. It's computer science in the sense that the paper focuses on efficient implementation. However, the audience is the indoor environment community.
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).
That's a very simple example.
Then x
equals
y
divided by
2
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.
I agree “Then x = y/2” is fine. However, your reasoning why this is OK (“=” is “equal”) is incomplete. Consider “Then the answer = y/2.”, which is grammatically correct but improper.
My secret for reading difficult math and physics texts: write out, by hand, every equation as you encounter it. Firstly, it slows me down so I'm not tempted to skim it like a novel. More importantly, my standards for what I write are much higher than my standards for what I read; if I see something I'm not completely convinced of, I may shrug and move on, but I'm not willing to write something down unless I really understand it.
This is what I also do, any step in a derivation I would glance over when read must be fully understood and proved whenever I write it down. Every non-trivial result in my math books is written down so I can fully comprehend what's going on.
Ah. Well, its not like your missing or unable to follow a fundamental concept, you just use different symbols. A bit of practice with reading things the "math way" and you'll be fine.
The most irritating trope of math and CS literature has to be whenever something interesting or useful is "left as an exercise to the reader." 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. Calling something "trivial" is swallowable, and perhaps this article's example of "it easily follows that..." is the mathematician's secret handshake that the steps in between are boring, but pointing out an interesting conclusion and then saying the reader deserves an exercise to actually make sense of it, I mean... isn't the point of academic writing to communicate research and results already performed? Wouldn't it be better to show the work for your assertion so that others could critique it or offer even better solutions?
I 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.
The most irritating thing to me is when someone says they want to understand something, but then they are unwilling to put in any real effort. It is a misapprehension, misunderstanding and misrepresentation to say:
... reinventing the wheel runs counter
to everything we do in CS.
You can read all you like about how to juggle - 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 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.
You must have only seen this phrase in places where it was used in the bright, sparkling, professorial manner assumed by your comment. I take no offense when it appears in a textbook, where the intent is to have me practice a skill. I do object to its appearance in a journal article, however; that is a context where you are supposed to communicate findings as clearly and concisely as possible. In particular, I've come across its use more and more often where one of the following likely applies:
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.
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.
Yes, it's used too often. Yes, it's used sometimes when it's inappropriate. Yes, sometimes I suspect that the author doesn't really know the answer. I'm less sure that the author uses it it all seriousness when the know the result is unproven or unprovable.
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.
> I mean... isn't the point of academic writing to communicate research and results already performed?
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.
I looked at the first few pages and saw only a few math papers, and in those the exercises were uninteresting details (e.g. just computation, or the second of two analogous cases where only the first is proved in the paper).
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.
I completely agree. This phrase appears in even some of the most hallowed tomes of computer science, such as Introduction to Algorithms. The textbook puts important theorems and proofs in the exercises section, and does not even provide solutions online (at least for the later sections). Keep in mind that this book is used as a reference manual for many implementations of algorithms and data structures.
I remember back in school that doing the math was not really a problem, but actually reading the textbooks was. I was the same way with programming when I started, where I could write but not read the language, but I'm suprised that I got through 4 years of university math without really being able to read the proofs that explained it.
That's natural, since mathematical notation is heavily slanted towards writing on paper and symbol manipulation. Use of one-letter variable names and highly context-sensitive notation reduces the number of pen strokes, but doesn't help the reader.
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.
The problem is that the mathematical notation, while not perfect, works really well for those adept with its use. It's an efficient shorthand. For explaining things in depth we have pictures and and natural language. I fail to see how creating a new set of symbols would do anything but hinder communication within the math community. If you have any deeper argument to support your position, I'd be interested to hear it.
>I fail to see how creating a new set of symbols would do anything but hinder communication within the math community.
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.
As a math major, my view is probably biased, but I feel like one of the reasons mathematics is so widely used and applied in other fields such as physics, CS, and even biology these days is that they express or model, to a high degree of accuracy, patterns that recur in all these fields. The mathematical notation is the most general and abstract form of expressing these patterns. My feeling is that to not learn the language of math dooms one to reinvent the wheel because they don't realize that the problem she is working on has been solved before in a more general case by some one else. In those cases where the problem hasn't been solved before, the solution leads to new areas or notation which can show up in unexpected ways in different branches of science.
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.
There was an example not so long ago of a paper published in, I think, biology, where the author had invented a wonderful new technique for more accurate estimates of the area under a curve.
It was Simpson's rule.
CORRECTION:
It was worse than that. It was the trapezoidal rule:
>So in short, mathematics saves you time by expressing common patterns or ideas.
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.
You speak about "math community" as if is some exclusive, elite club. In reality, anyone who needs to read a formula or some proof is part of that group.
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.
I disagree. By "math community" I refer specifically to those people who create new mathematics, i.e. mathematicians. That is not at all as inclusive as you argue. Math notation comes from mathematicians and is designed to facilitate communication between people of that profession. The notation is very functional and efficient for its designed task.
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.
That's a very elegant way of saying: "Communication in math is just as fucked up as you'd think."
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.
I'm working on something at the moment to try to explain exactly why what you've said here is, to a large extent, simply wrong. It's too long to include here, and it's not yet ready to "publish". You're about two weeks too early.
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.
Yes, but there's more that you can say. In particular, there is more than one infinite family. How many are there? And do all solutions belong to an infinite family? Or are there isolates/sporadics?
Can you characterise the solutions? How many pieces do they contain? Some solutions have two pieces. Some have four. Are there other possibilities?
An infinite family seems to refer to an infinite collection of solutions that are all related in some sense. An isolate or sporadic solution is a solution that is unique and unrelated to other solutions. They are both terms used to categorize solutions.
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?
Most math concepts -are- fairly easy to explain and, from the sounds of it, you don't actually want a math book. You want a book that gives you the executive summary or a run-down of specific tools for specific situations. The flaw is not in the books or the teaching, it is what you are expecting from mathematicians.
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.
Which comes back to the problems with the OPs argument. Sure, you can explain continuity in a few simple sentences but sooner or later you will start to hit corner cases and you find that those few simple sentences are really quite ambiguous. It's much like using pseudocode to describe an algorithm. I've found that the best mathematical texts provide both - a 'morally correct' definition in plain english to describe the intention and a mathematical description which defines the precise meaning.
I have my doubts about that claim. In most cases, the 'you' that thinks something 'is easily explainable with a few words of plain text' often is a different 'you' from the one that struggled grasping the concept. That struggle is what made you think the concept is 'easy'.
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.
I was just looking at this the other day. It reminded me of this:
"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."
The big ones: Watch out for proof by contrapositive and contradiction. Until you really get used to proving things by assuming bits of them are false it throws you off to see it used in textbooks and papers because most papers do not tell you they are using them.
Look at this proof of Fermat's Little Theorem [0], can you spot the indirect bit of the proof?
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.
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 ...
More completely, I expect you want them to say:
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)
... and so on.
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.
Yes, I just went for something really basic to give an example of a semi-implicit indirect proof.
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.
To find the maximum of a function on a closed interval it suffices to look for places where the derivative is 0, undefined, or at the boundary of the interval.
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.
Line by line carefully. Never skip anything in hopes it will clear up later.
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.
The best way I've found to read math texts is to take it concept by concept. Line by line is too mechanical. Approach each concept and get a "gist" for what is trying to be accomplished. Once you understand where they are starting from for a given concept, where they are going with it, and have a rough idea of how they'll get there, then you read the details. The details are essentially meaningless if you don't know the bigger picture.
You can skip on first reading to get an overview of where the author wants to take you, but don't expect it will clear up later on its own. Read and re-read. Fill in all the little gaps, especially when a sentence begins with "Clearly" or "As we can easily see".
"As we can easily see" == doesn't seem implausible after you read the paper countless times, re-work all the derivations from scratch at least 7 times, read the cited docs, get out your old analysis texts to look up some theorem you'd forgotten existed, and sacrifice a chicken.
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.
-Paul Halmos, inventor of "iff" and the ∎ symbol ( http://en.wikipedia.org/wiki/Paul_Halmos )