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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

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.

NY_USA_Hacker 15y ago
"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!
- NY_USA_Hacker 15y ago
  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".
rudiger 15y ago
Hehe, who here got through Rudin's Real and Complex Analysis?
- NY_USA_Hacker 15y ago
  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'.

NY_USA_Hacker 15y ago

"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!

NY_USA_Hacker 15y ago
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".

rudiger 15y ago

Hehe, who here got through Rudin's Real and Complex Analysis?

NY_USA_Hacker 15y ago
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'.