

The pitfalls of jargon and the damage it does to mathematics - Gelada
http://scienceblogs.com/evolutionblog/2011/05/monday_math_a_rant_about_jargo.php

======
impendia
Some thoughts, as an academic mathematician who got his Ph.D. three years ago:

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 _point_ 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.]

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.

Moreover I can think of lots of well-known books that are loaded with
exposition. _Representation Theory_ 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?)

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 _specific examples_ of what you would like to see instead.

The fact is that learning new math is just _damn hard_ , 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.

~~~
rudiger
Hehe, who here got through Rudin's _Real and Complex Analysis_?

~~~
NY_USA_Hacker
I've had no interest in the complex half, but I can claim to have done well
enough with the real half. I like it.

He does especially well with Fourier things, which is not surprising
considering some of his other work,

He gives the von Neumann proof of the Radon-Nikodym theorem. I prefer the
approaches with more steps with each step smaller, but it's great to see the
von Neumann masterstroke.

Rudin's chapter on Banach space knocks off the main results quickly and
cleanly.

His construction of Lebesgue measure early on is a bit long but surprising
and, net, shorter than the other ways to do it.

Curiously he has a nice chapter with more general results on change of
variable in multiple integration.

He does well with the various duality results, which are surprising and
important.

A good competitor, of course, is Royden, 'Real Analysis'.

------
tokenadult
The article by William P. Thurston (a Fields medalist) called "On Progress and
Proof in Mathematics

[http://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-...](http://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-00502-6/S0273-0979-1994-00502-6.pdf)

(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.

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"

[http://www.amazon.com/I-Want-Be-Mathematician-
Automathograph...](http://www.amazon.com/I-Want-Be-Mathematician-
Automathography/dp/0387960783/)

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.

~~~
lliiffee
About Halmos, that is quite scary. I happened to be reading "Finite-
Dimensional Vector Spaces" the last few days and while the material is pretty
elementary, Halmos is obviously _very_ good, and the idea that there are
people leagues ahead of him is amazing...

~~~
NY_USA_Hacker
Well, Halmos wrote the first version of 'Finite Dimensional Vector Spaces'
while he was working as an assistant to von Neumann at the Institute for
Advanced Study. So the book is doing finite dimensional linear algebra over
the real and complex numbers using the techniques of Hilbert space theory, a
von Neumann speciality. Once von Neumann had to explain what he meant by
'Hilbert space' to Hilbert!

But that Halmos was a good writer is no joke: He was one of the best of the
20th century.

~~~
lliiffee
This is slightly off-topic, but I noticed that you did work on stochastic
optimal control. Do you have any books you would recommend on the subject
(either optimal control or stochastic optimal control)? Ideally as good as
Halmos. :)

I come from a physics/cs background, and find that standard treatments of
control are very much intended for EE folks. But this seems like a historical
accident-- the techniques would seem to be very broadly useful in other
fields.

~~~
NY_USA_Hacker
If you want something written as well as Halmos, then start writing, and good
luck!

"Historical accident": Well, sure, in part nearly all the pure math
departments pushed out any such topics!

You are correct: Optimal control has been mostly in advanced parts of
electrical engineering.

So, optimal control in EE was an example doing math outside math departments.
Of math done outside math departments, control theory is relatively good
mathematically.

And, yes, there should be applications elsewhere.

For physics, yes, the deterministic theory of optimal control has been seen as
replacing the older calculus of variations which goes back to Newton.

My references are old.

More recent work on stochastic optimal control has been by R. T. Rockafellar
at University of Washington.

For

Stuart E. Dreyfus and Averill M. Law, 'The Art and Theory of Dynamic
Programming', ISBN 0-12-221860-4, Academic Press, New York.

'dynamic programming' is a big part of the discrete time versions of optimal
control. It can be stochastic or deterministic. The linear-quadratic-Gaussian
(linear 'plant' or system, quadratic cost to be minimized, and Gaussian
exogenous random variables) has 'deterministic equivalence' -- nice -- and
this book treats it. The book is a good, elementary start. Apparently Dreyfus
was a R. Bellman student.

For

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mischenko,
'The Mathematical Theory of Optimal Processes', ISBN 0-470-69381-9,
Interscience Publishers, John Wiley & Sons, New York.

it was, of course, the main source of the Pontryagin maximum principle, and,
thus, a swift kick in the back side for parts of US aerospace in the 1960s.

For

Michael Athans and Peter L. Falb, 'Optimal Control: An Introduction to the
Theory and Its Applications', McGraw-Hill Book Company, New York.

Athans was long in EE at MIT and did some military work, e.g., on parts of the
C5A airplane. Falb was at Brown's Division of Applied Mathematics.

One Athans story was the 'control' for least time to climb to, say, 100,000
feet for an F-4: Go up to a few thousand feet, go into a dive, get supersonic,
get the lower drag of a few hundred knots above Mach 1, and then with the
lower drag continue supersonic to the final altitude. I was not able to know
if that 'control' idea was just intuitive or directly from computation and the
Pontryagin maximum principle which is, after all, just a necessary condition,
i.e., local optimality.

Also in that division at Brown, see the papers of Harold Kushner. As I recall,
he wrote out a stochastic version of the Pontryagin maximum principle.

For

E. B. Dynkin and A. A. Yushkevich, 'Controlled Markov Processes', ISBN
0-387-90387-9, Springer-Verlag, Berlin.

there are connections with economic planning.

For

Dimitri P. Bertsekas and Steven E. Shreve, 'Stochastic Optimal Control: The
Discrete Time Case', ISBN 0-12-093260-1, Academic Press, New York.

the math is done carefully. So there is a lot of attention to measurability.
Part of the reason is the issue of 'measurable selection': This can be a deep
subject, but often a relatively simple way out is via regular conditional
probabilities as in

Leo Breiman, 'Probability', ISBN 0-89871-296-3, SIAM, Philadelphia.

For

David G. Luenberger, 'Optimization by Vector Space Methods', John Wiley and
Sons, Inc., New York.

this has likely the easiest mathematical treatment of deterministic optimal
control and also Kalman filtering and also the math needed for 'least action'
in physics.

For

E. B. Lee and L. Markus, 'Foundations of Optimal Control Theory', ISBN
0471-52263-5, John Wiley & Sons, New York.

this tried to be clean mathematically when it was written. At the beginning,
should know some relatively advanced results in ordinary differential
equations, e.g., as in

Earl A. Coddington and Norman Levinson, 'Theory of Ordinary Differential
Equations', McGraw-Hill, New York.

For more, you can consider non-linear filtering and connections with
mathematical finance.

The above is just from my bookshelf. Likely now a better bibliography could be
assembled via the Internet which actually is at least a little larger than my
bookshelf!

~~~
lliiffee
Thank you thank you thank you!

------
scott_s
Jargon is a _necessity_ 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.

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.

------
ubasu
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":

[PDF] [http://www.math.uh.edu/~tomforde/Books/Halmos-How-To-
Write.p...](http://www.math.uh.edu/~tomforde/Books/Halmos-How-To-Write.pdf)

<http://www.math.northwestern.edu/graduate/Forum/HALMOS.html>

collected in his Selecta of expository writing.

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.

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.

------
lurker19
Classic: Simon says how to write a paper and give a talk:

Overview page: [http://research.microsoft.com/en-
us/um/people/simonpj/papers...](http://research.microsoft.com/en-
us/um/people/simonpj/papers/giving-a-talk/giving-a-talk.htm)

PDF slideshow: [http://research.microsoft.com/en-
us/um/people/simonpj/papers...](http://research.microsoft.com/en-
us/um/people/simonpj/papers/giving-a-talk/writing-a-paper-slides.pdf)

~~~
pjscott
This is pure gold. There's a reason why so many people like reading Simon
Peyton-Jones's papers: he's _really_ good at writing them.

------
vbtemp
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.

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.

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).

------
wccrawford
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.

Why on Earth would a common joe want to read that paper anyhow? What could it
possibly do for them?

~~~
hugh3
I don't see toning down the jargon as being advantageous to the common Joe,
but it could be advantageous to mathematicians outside the extremely immediate
field. For instance I'm guessing that any mathematician will understand
"immersed surfaces in 4-manifolds" and maybe even "homology cylinder" even if
the layman won't. However, how many mathematicians know "Whitney move" and
"Whitney tower" and "Arf invariant" and "Milnor invariant" and "Sato-Levine
invariant" and so forth? Some tiny fraction. A little less jargon could make
your paper accessible to ten or a hundred times as many people.

Of course, quoting the abstract is a little unfair. The abstract needs to be
short so there's no time to explain what the hell a Sato-Levine invariant is.
Perhaps the actual paper is a lot clearer.

~~~
Someone
_I am guessing that any mathematician will understand "immersed surfaces in
4-manifolds"_?

I think you are guessing wrong. A 4-manifold, that I can more or less
understand (and that is about as far as one can get, as there are several
slightly different definitions in common use), but immersed surfaces in
manifolds? And I do not think that is because I haven't done abstract math
since graduating. One can graduate in mathematics without encountering any
manifold, for example by focussing more on applied fields such as discrete
math, analysis, or statistics.

I do not think that that abstract is necessarily bad, though. It seems to
mention sufficiently many terms to sell it to the target audience of, say, 100
people world-wide. If you do not know any of these, you should read something
more basic.

------
limmeau
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.

I don't say it's impossible, but it should make visible all the difficulties
on the way.

~~~
peterbotond
i wrote a few 'transl(iter)ation' from math/physics papers to be
understandable by a CS degree, and yes, the jargon is in the way for a CS
student. The biggest leap is to associate a greek letter with a definition. a
CS student, in my experience, wants the letter (short variable name) to be a
meaningful word (longer descriptive variable names). i am not a mathematician.
physicists, in my experience, had no problem though.

------
dkarl
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.

~~~
marshray
Couldn't they spend a little more time and write an expository background on
papers? Certainly it would be the easiest part for an expert to write.

I was reading several papers in the EE field the other day. Most of them all
spend the first page restating the problem in practical terms and then have a
substantial section reviewing the prior techniques. I found this darned
helpful, especially since the primary material was often not online.

Are they worried about wasting paper, or do they actually want to be
uninviting?

------
btilly
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.

See <http://bentilly.blogspot.com/2009/11/why-i-left-math.html> for more.

------
hsmyers
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.

~~~
lurker19
I had a stint where I bought and tried to read a few dozen cheap $8-$14 Dover
math books. There is a reason most of these books were not revised and
republished before their copyright expired and they became Doverable, and it
is not because they were perfect originally. Poor 1940s typesetting, only the
barest sketches of images, and linguistic quirks that are not present in
modern language. The recommended booms from my high school and college courses
were much better. Of course, 90% of everything is crap, so there logged be a
few gems in there.

The $3 copies of Plato et al were better than the math books.

------
cyrus_
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.

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).

------
jedbrown
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.

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]".

~~~
pas
That should be the point where to, once and for all, clearly state the
differences between analogy and the subject matter at hand.

------
br1
The saddest thing is that computer science looks up to math and emulates the
worst of it, such as cryptic, short variable names.

~~~
Wickk
What you're describing is actually bad practice. Not saying it's not a
problem, but that's not an intentional emulation.

~~~
Peaker
It depends. Variable name length is mainly related to two things:

* Scope the variable resides over

* Generality of the variable

If the scope is large (the variable is global and exposed to a lot of code),
the variable name has to be longer, so that far away readers can make sense of
it. A variable that scopes over 2 lines is allowed to be much shorter,
especially if its definition is as clear or clearer than an arbitrary name.

As for the generality, if you write a function addEmployeeSalaries, of course
that is a better name than some arbitrary short name (e.g: "addEm").

But when writing (very very) general code, it is hard to avoid situations
where the variable denotes only structure. It is just too general to find a
useful long (and specific) name.

For example, consider:

    
    
        flip f x y = f y x
        
        sequence :: Monad m => [m a] -> m [a]
    

Single-letter variable (and type variable) names, but considered perfectly
good practice.

------
sdh
I can't stand technical jargon or people who use without regard for their
audience.

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.

Legal jargon is probably the worst example.

~~~
to3m
Jargon is just a time-saving and precision measure. Try doing without it
sometime - it's painful.

It is totally unavoidable as long as people keep communicating with words. The
better solution is not to complain about it, but to ask for definitions
(recursively) - and then you can reap the time-saving and precision benefits
of the terminology, too.

------
nothis
>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.

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.

~~~
TillE
Ugh, I had a physical chemistry textbook like that. All math, very little
explanation of why or how to _apply_ the math. I remember reading one chapter
over and over again, desperately attempting to glean what shreds of meaning I
could from the bare minimum of information presented. It was like trying to
solve a puzzle.

Please, please don't write textbooks like that.

------
NY_USA_Hacker
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.

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.

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.

