The pitfalls of jargon and the damage it does to mathematics(scienceblogs.com)
scienceblogs.com
The pitfalls of jargon and the damage it does to mathematics
http://scienceblogs.com/evolutionblog/2011/05/monday_math_a_rant_about_jargo.php
44 comments
Hehe, who here got through Rudin's Real and Complex Analysis?
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'.
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'.
"Any HN'ers tried to hack their way through Baby Rudin?"
Yup. Been there; done that!
Yes, it needs some 'explanation':
Q. Why do we care about metric spaces?
A. Eventually they are important, e.g., in Hilbert and Banach spaces. For now, they are the vanilla case of a topological space.
Q. Why do we care about topological spaces?
A. Because they are the nicest way to describe continuity. So a function from one topological space to another is 'continuous' iff (from Halmos: if and only if!) for each open set in the range space its inverse image under the function is an open set in the domain space. That's a nicer, and more general, definition than converging sequences (which are not quite as powerful, but examples are a bit bizarre) or epsilon-delta arguments from limits in calculus.
Q. An 'open' set?
A. A 'topology' is defined as a collection of sets it calls 'open'. So, really the topology is defined just by some open sets. A good and easy example of an open set is an open interval, say, (0,1) on the real line. That set is 'open' in the 'usual' topology (in principle there can be others) for the real line.
Q. Why do we care about continuity?
A. As in G. Simmons, the two main pillars of the 'analysis' part of math are linearity and continuity. They are assumptions that commonly hold in practice and that give some of the most powerful and useful results. In particular, in Baby Rudin, if a real valued function of a real variable is continuous on a closed set, then the function is also uniformly continuous, is bounded, actually achieves its upper and lower bounds, and its Riemann integral exists and is finite. All nice stuff.
Continuing, all such functions form a vector space (a 'linear' space); the difference to two such functions is again such a function; and the maximum value of the difference forms a norm. So, those functions form a normed linear space. If a sequence of such functions converges in this norm, then the limit is also such a function -- i.e., a uniform limit of continuous functions is continuous. So, the space is 'complete' in this norm. So, we have the leading example of a Banach space (a complete, normed linear space). Then we can show the Hahn-Banach result and see the tasty, ice cream and cake dessert examples in Luenberger, 'Optimization by Vector Space Methods' including Kalman filtering, results for 'least action' in physics, more in deterministic optimal control, etc. So, these are some of the coming attractions of those first chapters in Baby Rudin on metric spaces, topology, open sets, continuity, etc.
Q. A 'closed' set?
A. Back to the real line where (0,1) is an open set, [0,1] is a closed set, So, it's closed and bounded. Presto: It is the leading example of a 'compact' set, which is almost as nice to work with as a finite set. In particular, any infinite subset of a compact set has a limit point in the set which is the key to uniform continuity, the existence of the Riemann integral, and more. The nicest definition of a compact set is that each open cover has a finite subcover, and this definition works also in metric spaces and topological spaces.
So here is an explanation of what the first chapters of Baby Rudin are all about. Print a copy and pin it to the wall in any math department that teaches a Baby Rudin course!
Yup. Been there; done that!
Yes, it needs some 'explanation':
Q. Why do we care about metric spaces?
A. Eventually they are important, e.g., in Hilbert and Banach spaces. For now, they are the vanilla case of a topological space.
Q. Why do we care about topological spaces?
A. Because they are the nicest way to describe continuity. So a function from one topological space to another is 'continuous' iff (from Halmos: if and only if!) for each open set in the range space its inverse image under the function is an open set in the domain space. That's a nicer, and more general, definition than converging sequences (which are not quite as powerful, but examples are a bit bizarre) or epsilon-delta arguments from limits in calculus.
Q. An 'open' set?
A. A 'topology' is defined as a collection of sets it calls 'open'. So, really the topology is defined just by some open sets. A good and easy example of an open set is an open interval, say, (0,1) on the real line. That set is 'open' in the 'usual' topology (in principle there can be others) for the real line.
Q. Why do we care about continuity?
A. As in G. Simmons, the two main pillars of the 'analysis' part of math are linearity and continuity. They are assumptions that commonly hold in practice and that give some of the most powerful and useful results. In particular, in Baby Rudin, if a real valued function of a real variable is continuous on a closed set, then the function is also uniformly continuous, is bounded, actually achieves its upper and lower bounds, and its Riemann integral exists and is finite. All nice stuff.
Continuing, all such functions form a vector space (a 'linear' space); the difference to two such functions is again such a function; and the maximum value of the difference forms a norm. So, those functions form a normed linear space. If a sequence of such functions converges in this norm, then the limit is also such a function -- i.e., a uniform limit of continuous functions is continuous. So, the space is 'complete' in this norm. So, we have the leading example of a Banach space (a complete, normed linear space). Then we can show the Hahn-Banach result and see the tasty, ice cream and cake dessert examples in Luenberger, 'Optimization by Vector Space Methods' including Kalman filtering, results for 'least action' in physics, more in deterministic optimal control, etc. So, these are some of the coming attractions of those first chapters in Baby Rudin on metric spaces, topology, open sets, continuity, etc.
Q. A 'closed' set?
A. Back to the real line where (0,1) is an open set, [0,1] is a closed set, So, it's closed and bounded. Presto: It is the leading example of a 'compact' set, which is almost as nice to work with as a finite set. In particular, any infinite subset of a compact set has a limit point in the set which is the key to uniform continuity, the existence of the Riemann integral, and more. The nicest definition of a compact set is that each open cover has a finite subcover, and this definition works also in metric spaces and topological spaces.
So here is an explanation of what the first chapters of Baby Rudin are all about. Print a copy and pin it to the wall in any math department that teaches a Baby Rudin course!
Errata: Of course, in
"In particular, in Baby Rudin, if a real valued function of a real variable is continuous on a closed set, then the function is also uniformly continuous, is bounded, actually achieves its upper and lower bounds, and its Riemann integral exists and is finite."
replace
"on a closed set"
with
"on a closed and bounded set".
"In particular, in Baby Rudin, if a real valued function of a real variable is continuous on a closed set, then the function is also uniformly continuous, is bounded, actually achieves its upper and lower bounds, and its Riemann integral exists and is finite."
replace
"on a closed set"
with
"on a closed and bounded set".
>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.
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.
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.
Please, please don't write textbooks like that.
>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.
From the perspective of a mathematician, I'd like to see textbooks strike a happy medium between page after page of explanation and absolute terseness.
When I'm taking an intermediate level course, I know that the math I encounter may rarely be useful on its own. It should be sufficient to say "you won't even think about this until you take analysis!"
I think the biggest mistake universities make is to offer one type of math instruction. I still remember taking a linear algebra class that had mathematics majors, engineers, and secondary education majors(!) all in the same class! The professor taught it in a way that I very much appreciated -- a large focus on theory and concepts, and much less focus on application & concrete problems. Of course, I and the math majors loved this, while the education majors & engineers detested it.
It might've been more difficult, but I believe that everyone would've benefited from instruction that was tailored to what people would need & could appreciate -- the engineers could learn applications of linear algebra, the secondary education majors could learn how to teach linear algebra, and the mathematicians could learn the theory behind linear algebra. Instead, we had one class that a subset was guaranteed to hate.
From the perspective of a mathematician, I'd like to see textbooks strike a happy medium between page after page of explanation and absolute terseness.
When I'm taking an intermediate level course, I know that the math I encounter may rarely be useful on its own. It should be sufficient to say "you won't even think about this until you take analysis!"
I think the biggest mistake universities make is to offer one type of math instruction. I still remember taking a linear algebra class that had mathematics majors, engineers, and secondary education majors(!) all in the same class! The professor taught it in a way that I very much appreciated -- a large focus on theory and concepts, and much less focus on application & concrete problems. Of course, I and the math majors loved this, while the education majors & engineers detested it.
It might've been more difficult, but I believe that everyone would've benefited from instruction that was tailored to what people would need & could appreciate -- the engineers could learn applications of linear algebra, the secondary education majors could learn how to teach linear algebra, and the mathematicians could learn the theory behind linear algebra. Instead, we had one class that a subset was guaranteed to hate.
Is it worth reposting the yarn from Surely You're Joking, Mr. Feynman about teaching physics in Brazil?
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.
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
http://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-...
(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...
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.
http://www.ams.org/journals/bull/1994-30-02/S0273-0979-1994-...
(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...
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.
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...
Some brilliant people are truly, honestly humble. It's possible he's better than he gives himself credit for.
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.
But that Halmos was a good writer is no joke: He was one of the best of the 20th century.
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.
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.
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!
"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!
Thank you thank you thank you!
tokenadult, you're the only other person I know who read that book. I found it dusty in a university library years ago while I was pursuing my PhD and it was an eye opening read. I'd love to buy a copy ,but it is so expensive. But thanks for reminding me of it.
It's $30 from Amazon US; that's not so bad, surely. (And yes, it's a very nice book; Halmos was an outstanding writer.)
You're right. Will order. Where did all these versions come from? I had looked years back and it was $100+. Weird.
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.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.
[PDF] http://www.math.uh.edu/~tomforde/Books/Halmos-How-To-Write.p...
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.
Classic: Simon says how to write a paper and give a talk:
Overview page: http://research.microsoft.com/en-us/um/people/simonpj/papers...
PDF slideshow: http://research.microsoft.com/en-us/um/people/simonpj/papers...
Overview page: http://research.microsoft.com/en-us/um/people/simonpj/papers...
PDF slideshow: http://research.microsoft.com/en-us/um/people/simonpj/papers...
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.
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).
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).
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?
Why on Earth would a common joe want to read that paper anyhow? What could it possibly do for them?
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.
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.
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.
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.
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.
I don't say it's impossible, but it should make visible all the difficulties on the way.
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.
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.
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?
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?
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.
See http://bentilly.blogspot.com/2009/11/why-i-left-math.html 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.
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.
The $3 copies of Plato et al were better than the math books.
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).
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.
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]".
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]".
That should be the point where to, once and for all, clearly state the differences between analogy and the subject matter at hand.
The saddest thing is that computer science looks up to math and emulates the worst of it, such as cryptic, short variable names.
I like to think of computer science as: "Math, in terms even computers can understand"
What you're describing is actually bad practice. Not saying it's not a problem, but that's not an intentional emulation.
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:
* 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]
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.
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.
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.
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.
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.
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.
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.