Some thoughts, as an academic mathematician who got his Ph.D. three years ago:<p>First of all, the author is right in one aspect, in that bad writing -- and also bad speaking -- are too widely tolerated in mathematics. Most departments have a regular colloquium, where the entire <i>point</i> is to listen to a talk to mathematicians outside your specialty. Many speakers do a good job, but there are some who will start out with "Let M be a symplectic manifold, let omega be the associated bilinear form, let X be the ring of differentials, ..." [ugh.]<p>And yet I think there is more pressure to write and speak well than there is to do so poorly. Certainly as a graduate student I was pressured by my advisor to explain things clearly, to add examples and exposition to my papers, to omit technical details from talks, and in general to keep in mind the perspective of the non-expert. The author complains, "Submit a paper with two consecutive sentences of exposition and watch how quickly the referee gets on you for it." But I have not experienced this myself, nor heard this complaint from anyone until this article.<p>Moreover I can think of lots of well-known books that are loaded with exposition. <i>Representation Theory</i> by Fulton and Harris comes particularly to mind. Of course, there are many books which are notoriously terse as well. (Any HN'ers tried to hack their way through Baby Rudin?)<p>Indeed, there are many mathematicians who make quite an effort to write and speak well, and largely succeed, and the author seems unable to identify any mathematical writing which he does approve of. If we are doing a bad job, please give us <i>specific examples</i> of what you would like to see instead.<p>The fact is that learning new math is just <i>damn hard</i>, even from well-written papers. I think that we could do a much better job of explaining our work to others. But, IMHO, this article overstates the problem, and omits to propose any practical solutions.
>Sadly, the rot extends to math textbooks as well, which, with very few exceptions, are simply horrible. I mean really, really bad. It is commonly considered a great faux pas to actually explain what you're doing. You will be accused of being overly wordy if you do anything other than produce an endless sequence of definition-theorem-proof. Mathematicians too often seem to take absolute delight in being as opaque as possible. I can't tell you how many times I have heard friends and colleagues praise for their concision textbooks which, to my mind, are better described as harbingers of the apocalypse. If, as a textbook author, you place yourself in the student's shoes and try to anticipate the sorts of questions he is likely to have approaching the material for the first time, a great many of your colleagues will say that you have done it wrong.<p>As someone who only ever encounters math as a tool rather than a a great passion or source of intellectual stimulation, it makes me really happy to see this said out loud.
Jargon is a <i>necessity</i> in all fields. Otherwise we would have to use only existing words to explain all of our new concepts.. Math - and some physics - necessarily deal with things that we don't have an intuition for. I think that those subjects are going to be inherently less approachable for people outside of those sub-disciplines because their prior knowledge is much less help.<p>Now, with that said, it's probably true that some math papers could be improved with more exposition. But I think that excluding the layman from new math research is probably an inherent problem, not an accidental one. Improved exposition would be to help other mathematicians who are not in that sub-discipline to understand the broader field.
The article by William P. Thurston (a Fields medalist) called "On Progress and Proof in Mathematics<p><a href="http://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-00502-6/S0273-0979-1994-00502-6.pdf" rel="nofollow">http://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-...</a><p>(which I learned about from a comment here on HN, thanks) does a good job of demonstrating how a mathematician who makes new discoveries has to invent a new language for describing those discoveries. Then the mathematician has to relentlessly practice communicating those results first to other professional mathematicians, helping them to see the connections between their research and the new research results. Mathematicians who work hard at communicating with other mathematicians, Thurston says, can help greatly with the progress of mathematical research.<p>After edit: Paul Halmos, quite a well regarded mathematician who by his own self-evaluation was not in the same league as Fields medalists, wrote in his "automathography"<p><a href="http://www.amazon.com/I-Want-Be-Mathematician-Automathography/dp/0387960783/" rel="nofollow">http://www.amazon.com/I-Want-Be-Mathematician-Automathograph...</a><p>that he learned a lot of mathematics as he continued his career after his Ph.D. degree by "reading the first ten pages of a lot of mathematics books." Sometimes he could only get ten pages into a book by another mathematician before he was lost, but by reading dozens and dozens of books, ten pages each, he gained more conceptual foundations in more areas of mathematical research and could gradually apply what he self-learned to advance his own research. I strongly encourage students I know to follow that same strategy of reading at least the introductory portion of many books on subjects they desire to know. Don't just read what your professor assigns you to read. Go to the library and read widely. Read as far as you can before you get stuck, and then find another book and start reading it from the beginning until you get stuck again. Eventually, you will find that you can read harder books, and go farther before getting stuck.
While it is true that there are many examples of bad writing in mathematics, it is not true that unclear writing is glorified over clear exposition. To the contrary, authors like Paul Halmos are rightly celebrated as great expositors, e.g. see his books like Naive Set Theory, Finite Dimensional Vector Spaces, Measure Theory etc. and his expository writing such as "How to Write Mathematics" and "How to Talk Mathematics":<p>[PDF] <a href="http://www.math.uh.edu/~tomforde/Books/Halmos-How-To-Write.pdf" rel="nofollow">http://www.math.uh.edu/~tomforde/Books/Halmos-How-To-Write.p...</a><p><a href="http://www.math.northwestern.edu/graduate/Forum/HALMOS.html" rel="nofollow">http://www.math.northwestern.edu/graduate/Forum/HALMOS.html</a><p>collected in his Selecta of expository writing.<p>Some other books I like are Sheldon Axler's Linear Algebra Done Right, Real Mathematical Analysis by Charles Pugh, and the three books on manifolds (Topological Manifolds, Smooth Manifolds and Reimannian Manifodls) by John Lee, but there are many other well-known books that are clear in their exposition.<p>Edit: Part of the issue is also the approach of the reader, i.e. is s/he there to learn mathematics, where the goal is to explore the mathematical world, or is s/he there to use the mathematics to explore the physical world? That should guide questions like "What is the point of studying vector spaces?" The motivation to study vectors spaces (and other mathematical topics) may need to come from other courses, e.g. on mechanics, and the course/book on vector spaces would serve to give in-depth knowledge of the particular topic.
Classic: Simon says how to write a paper and give a talk:<p>Overview page:
<a href="http://research.microsoft.com/en-us/um/people/simonpj/papers/giving-a-talk/giving-a-talk.htm" rel="nofollow">http://research.microsoft.com/en-us/um/people/simonpj/papers...</a><p>PDF slideshow:
<a href="http://research.microsoft.com/en-us/um/people/simonpj/papers/giving-a-talk/writing-a-paper-slides.pdf" rel="nofollow">http://research.microsoft.com/en-us/um/people/simonpj/papers...</a>
For me, at least, once I understand it, the notation makes perfect sense and I find it far more expeditious and precise to express my idea in the domain language of the field. The problem is that getting to the point of becoming proficient in the formalism of a field can take an extremely long time (for me), and has the unfortunate side effect of making me feel like a total idiot for not understanding it sooner, once I understand it.<p>The fact is, mathematics deals with constructs with extremely specific properties. These properties can be completely and soundly stated without ambiguity quite succinctly in the formalism of the field. Once I'm comfortable with the formalism, trying to express it in English is awkward, imprecise, and generally just extremely verbose. It can then be a frustrating experience going back and convey the idea to colleagues or others in prose.<p>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 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 (more intense than programming a rather difficult problem, imo).
That paper referenced is WAY beyond most people, as the article noted. To understand what the person is going to say, you had better already know what those terms mean, and be able to explain them. If you can't, then you have no basis for the rest of the paper anyhow.<p>Why on Earth would a common joe want to read that paper anyhow? What could it possibly do for them?
I wish the author had, as a PoC, taken one of the papers in his career as a maths PhD, and rewritten it so that a non-mathematician (say, one with only a CS or physics degree) can immediately understand it.<p>I don't say it's impossible, but it should make visible all the difficulties on the way.
I'm of two minds about this. On the one hand, I think it's a misconception to think that a nothing-but-the-proofs-and-definitions approach makes it any harder to understand math. Math is just freakin' hard. Making it fluffier only makes it seem easier if you confuse page rate with learning rate. On the other hand, knowing the motivation and context for a piece of mathematics is very pleasant, makes it easier to focus and work hard on the math itself, and occasionally can be as important as knowing the math itself. My ideal math text would contain a lot of exposition at the beginning of each chapter and then the traditional dry presentation of the mathematics itself. I wouldn't want the exposition and the mathematics mixed too finely.
As long as mathematicians are dependent upon finding a small clique to work with, there are no pressures to make themselves understood by more than a small number of people who have a lot of common knowledge. Thus jargon will proliferate.<p>See <a href="http://bentilly.blogspot.com/2009/11/why-i-left-math.html" rel="nofollow">http://bentilly.blogspot.com/2009/11/why-i-left-math.html</a> for more.
One of the reasons that I prefer the reprints of much older texts by the great mathematicians is that in most cases (all?) the writing meets the conditions that are listed as desired. The previous sentence clearly doesn't---sorry! The books from Chelsea and for that matter Dover may be old, but they are accessible.
Without jargon, it would be roughly impossible to talk about compound mathematical structures. But I certainly agree that more exposition and a greater number of worked-out examples are a great help when trying to understand abstract math.<p>It would also be great to have papers written in a hyperlinked format (after so many years of the web...) so that when you click on a jargon term, you get something like a Wikipedia article about that concept with a few examples and some exposition. Wikipedia itself would be fine with me, though academics might want more official peer review (see Scholarpedia).
Often the jargon is unavoidable because "common sense" definitions are actually subtly incorrect. This does not mean that examples and motivation have no place, but there is definitely a chicken/egg problem and starting by stating definitions is an easy and expedient way to overcome it.<p>Also, the best examples usually link seemingly disparate fields. If you can't assume the reader has a background in these fields, it can be difficult to even state the link. I sometimes find myself writing "readers unfamiliar with X can think of it as [intuitive but imprecise definition]".
I can't stand technical jargon or people who use without regard for their audience.<p>The only purpose of jargon is to make a conversation exclusive to only those who know the language. If your purpose is to educate or collaborate or truly communicate, then jargon is harmful.<p>Legal jargon is probably the worst example.
In an overwhelmingly important sense, pure math has the least 'jargon' of any field. The reason is, math exposition has an absolute requirement: Each use of a word not the same as just its dictionary meaning becomes a 'term' and just MUST be defined, and 'well-defined', before it is used. So, what the article is calling 'jargon' is just such terms, but in well written pure math they all have rock solid definitions, and math is the unique field where this precision is true.<p>For the rest of the complaints about 'jargon', these are essentially just necessary given that math, due to its precision and age, is now by far the deepest field of study. So, there are a LOT of terms defined.<p>If the article wanted to claim that for each math result or paper there is a 10,000 foot intuitive view that can be explained in 90 seconds to just anyone on a street, then okay, but the article did not do this. This claim is not quite true, but it is close enough so that some such explanations, maybe with some pictures, and some connections with more elementary topics, can be useful, even in research papers. But the article didn't make this claim, either.