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Learning 'math' as a good ugrad or grad student is NOT easy and involves a special step often called 'mathematical maturity' where the student learns to be good

Learning 'math' as a good ugrad or grad student is NOT easy and involves a special step often called 'mathematical maturity' where the student learns to be good at reading and writing definitions, theorems, and proofs in math. This step is NOT easy to learn and in practice is nearly impossible to learn without some careful teaching. The usual place to learn the step is in a very precise course, with a lot of very well graded homework, in abstract algebra and then to strengthen the learning via very precise courses in advanced calculus and, maybe, point set topology or measure theory. Without that step, nearly no one knows how to do well writing definitions, theorems, and proofs in math. Mostly fields outside math try hard to teach math within their fields. Examples of such fields include physics, chemistry, economics, psychology, sociology, electronic engineering, and, yes, computer science, In all cases, the results are at best poor. Of course, those fields are helped by the fact that the math departments don't want to do well teaching service courses, say, in second order stationary stochastic processes for signal processing in electronic engineering with the fast Fourier transform! But you noticed the problem: The CS profs make a mess out of math. For the book of this thread from MIT, that fact is a bit surprising: For one, among the authors, at least Rivest has done some good math. For another, for each math topic in that book, MIT has some high expertise. E.g., in optimization, Bertsekas is there and long there have been others. Once the basic ugrad lessons in math are learned, one still has to learn how to communicate in less severe forms. So, the book failed there, too: Their algebra for the core steps of the simplex algorithm was too long. And, their intuitive explanation, as I objected to here, really was not correct. Sadly they just wrote their chapter on the simplex algorithm too quickly from too little study. This is especially sad since there are many highly polished presentations of the simplex algorithm. They got pushed into attention to linear programming, and the simplex algorithm, because of some important connections with some problems, e.g., flows on networks that they want to regard as part of computer science, where linear programming is a very important tool. So, really, it's a strain to call 'flows on networks' computer science. Instead, it's a topic in applied math, back to Ford and Fulkerson, yes, with some by Dantzig, but more from Cunningham and Bertsekas, in particular, part of operations research. So, CS wants to hijack flows on networks. Not really good. I omitted a more serious objection to the book: It makes a mess, a really bad mess, out of probability. Yes, in the study of the performance of algorithms, probability is important. So, once again, CS wants to borrow and use or just hijack some applied math and makes a mess. That at MIT they make such a big mess out of probability is something of a surprise. Still, in the US, the good courses in probability require both measure theory and functional analysis and are rare, even in math departments. Sad situation. In using probability for performance of algorithms, in TACP Knuth did better: He stayed with a more elementary approach to probability and made it powerful enough. E.g., in the book, I saw where they mentioned that the need for probability was that the data had a 'distribution'. Mostly that's not what have to pay attention to: E.g., don't much care about Gaussian, uniform, exponential, etc. Instead, what care about most of all is independence and, then, sometimes, lack of ties. Sad situation. For both teaching and research, CS needs to clean up its act in math. Still better, math should just open up, welcome, take in, and take over all the more important fields that are heavily math -- physics, economics, social science, electronic engineering, computing, operations research, etc. Instead of being one of the smallest departments on campus, teaching a few service courses, and otherwise struggling, math could be one of the biggest departments on campus, awash in both funding and students, and one of the most important pillars of growth in our economy. I have an example: Some years ago I took a problem in practical computing and derived some new probabilistic math for a solution. Right, I used measure theory. I wrote prototype software, ran it on some real data, etc., and wrote a paper. So that I could get the paper in front of people in computing, where the applications are, I looked for journals in computer science. From one editor in chief of one of the best journals, a chaired prof of computer science at one of the best research universities, I got back "Neither I nor anyone on my board of editors has the prerequisites to review your paper.". Later I got almost the same statement from a second such person. For a third such person, at MIT, I wrote custom tutorials for two weeks before they gave up. But in the end the editor in chief of a good Elsevier journal seems to have walked the paper around campus and gotten feedback enough to get the paper reviewed -- the editor couldn't review it! The paper was a nice contribution to computer science. Generally nice contributions will want some math. For all concerned, CS needs to learn some math. Here's what they need to do: After calculus, get a good book on abstract algebra, pay full attention to the proofs, and learn some 'mathematical maturity'. Then get a good, first book on linear algebra and learn some important material, still paying attention to the proofs, and get more maturity. Then get Halmos, 'Finite Dimensional Vector Spaces' and learn from one of the best writers of math ever and get an introduction to Hilbert space theory. Then go carefully through Rudin's 'Principles'. Then do some more in ordinary differential equations where get to use both Rudin and Halmos. Then do some optimization -- e.g., pick from linear programming and some of Bertsekas. Then learn some measure theory, functional analysis, and probability based on measure theory. Cover basic stochastic processes, e.g., Markov, Gaussian, and martingales. Then learn some mathematical statistics. THEN return to computer science.

tincholio 15y ago
I think there's validity to your argument, but you come across as a bit arrogant. A few comments to your post: >The CS profs make a mess out of math. Properly done CS _is_ maths. Maybe most of it is not in the branches you're interested in, but the core of CS is maths (think Turing, Church, Von Neumann, Codd, Hoare, Martin-Löf, etc.). >it's a strain to call 'flows on networks' computer science. I've always heard it called part of operations research, which happens to be an rather useful field to teach CS students. Yes, it is taught at probably a very basic level, but it is still useful as taught. And saying that: >So, CS wants to hijack flows on networks. Not really good. is disingenuous at best. Actual networks, like the one you're using now to post about your discontent with CS are basically run on computers, and are, not surprisingly, part of what CS people worry about. There's a whole field of very applied CS that's a direct application of some basic operations research results. >For both teaching and research, CS needs to clean up its act in math. In a general sense, yes, it would be good. Maybe you also need to read some better CS stuff? >I have an example: Some years ago I took a problem in practical computing and derived some new probabilistic math for a solution. You go on to rant how editors of CS journals weren't smart enough to understand your paper. That might well be the case, but it is obvious that you were not any smarter, sending a paper which is pure maths in nature to several CS journals, when the "problem in practical computing" was merely an application of your result, and not the main point of your paper. If you were a physicist working on some very deep results of quantum mechanics that happen to have a direct application to, say, molecular biology, you would still send it to a physics journal, not a biology journal, wouldn't you? >Here's what they need to do:... That's a good plan for learning a bunch of maths, not CS.
- NY_Entrepreneur 15y ago
  "Properly done CS _is_ maths." Okay, then you would agree with my thought on this thread that the math departments should just take over CS! The CS would get some much better math, and the math departments would get a big head and tummy ache from the much needed contact with applications. Fine with me. "is disingenuous at best." No. That chemistry uses some group theory for molecular spectroscopy and makes a mess out of it, which it has been known to do, doesn't mean that a mathematician who sits in a chair with plastic fake leather shouldn't comment. So, yes, IP routing has been based on some network shortest path algorithms. So, they used some operations research material since really it was operations research that first dug into networks that deeply -- yes, I know, there was a Nobel prize in economics for some work by Kantorovich (?) in the late 1940s on the transportation problem and, earlier, work in circuit theory as in Kirchhoff. Still, e.g., with Ford and Fulkerson and then Jack Edmonds, network shortest path and flows on networks should be attributed to operations research. If IP routing got to use it, then fine. Once I used Cunningham's version of the network simplex algorithm on a problem in allocating sales forces in a tricky situation: The problem looked like just integer linear programming, but if looked again it was just network flows where can get integer solutions for no extra effort. That doesn't mean that network flows should be part of courses in selling! But CS is not just using network flows but calling the subject part of computer science. That's hijacking! Or it justifies my "Not really good.". But, such things will happen. There is no law that says that have to be an operations research person to publish in an operations research journal or to publish about networks in a computer science journal. The NSF even works to encourage 'cross-cutting'. Since my writing is so 'succinct', maybe I wasn't clear! If CS wants to take a field like network flows for their own, then actually do a competent job of it and don't mess it up. So, when there's is need, as there is, for the simplex algorithm, and not just using it but calling it part of CS, then do a competent job with the simplex algorithm. If probabilistic analysis of the performance of algorithms is to be part of CS, then CS should do a competent job with the probability. Else, CS is diluting the quality of these subjects. I argued elsewhere on HN that CS has diluted the meaning of dynamic programming, also essentially hijacked by CS. If you want to say that CS done well is math, then they should do the math well. When I was in grad school, I took some courses in measure theory and functional analysis from a department not the pure math department. And I took some statistics. When the statistics got to sufficient statistics, which is from the Radon-Nikodym theorem of measure theory, the prof totally blew it. I just walked out of the class; mostly I knew the material at the elementary level anyway, and at the level I wanted to learn the material the course was a waste of my time. Well, in the courses in measure theory and functional analysis, the profs maintained quality only slightly short of Bourbaki. They didn't have less than a beautifully polished proof for anything. Any hint or suggestion that there was a logical gap was a very big deal in the department. When I walked out of the stat course, the department dumped on me. Then I dumped back on the mess made of the R-N theorem. The next year the stat prof was gone. That department, not pure math, took the math fully seriously: A prof who blew a good proof of one important theorem was looking for a job. The situation was SERIOUS. We weren't a pure math department, but we did the math in rock solid ways. The department also had some operations research: Similarly. The math was rock solid. E.g., at one point in the Kuhn-Tucker conditions, I objected that a proof had used but not assumed continuous differentiability. I worked up a proof with just differentiability, and my proof was in the course the next semester. So, broadly, just because we were interested in some math topics not common in pure math departments, and interested in applications, didn't mean we did the math at a lower level of quality. We were as close to Bourbaki as anyone. It can be done. CS should do it, too. Sorry 'bout that. Low quality is not good; I called out some low quality. I was fully correct to do so. The problem is not my arrogance but their low quality. "In a general sense, yes, it would be good. Maybe you also need to read some better CS stuff?" So, we are in agreement, "it would be good". For my reading, that's about me, and that's not appropriate. The issue is the book CLRS or whatever it is called. I'm not the issue. But for some "better CS stuff", you mentioned von Neumann. Right: In my favorite source of the R-N theorem, the proof is by von Neumann. In my current project, there's some math at the core, some original, and some due to von Neumann. I credit him and Kolmogorov. For the paper I sent, I sent it to an appropriate journal. The motivation for the research was a problem in computer science. The main application for the original part of the research was just to that problem. That there was some original math at the core of the paper was part of the work as in your remark that CS should be math. You are agreeing with my main point that CS should clean up its math and then calling me arrogant for making this point. Can't have it both ways. For the course of study I outlined, what important about CS does that omit! I know; I know; it omits heap sort! And AVL trees. Gee, looks like that course of study needs more, maybe a lecture, maybe even two, of an hour each? Do the math. For the CS, do that in the exercises and footnotes!

tincholio 15y ago

I think there's validity to your argument, but you come across as a bit arrogant. A few comments to your post: >The CS profs make a mess out of math. Properly done CS _is_ maths. Maybe most of it is not in the branches you're interested in, but the core of CS is maths (think Turing, Church, Von Neumann, Codd, Hoare, Martin-Löf, etc.). >it's a strain to call 'flows on networks' computer science. I've always heard it called part of operations research, which happens to be an rather useful field to teach CS students. Yes, it is taught at probably a very basic level, but it is still useful as taught. And saying that: >So, CS wants to hijack flows on networks. Not really good. is disingenuous at best. Actual networks, like the one you're using now to post about your discontent with CS are basically run on computers, and are, not surprisingly, part of what CS people worry about. There's a whole field of very applied CS that's a direct application of some basic operations research results. >For both teaching and research, CS needs to clean up its act in math. In a general sense, yes, it would be good. Maybe you also need to read some better CS stuff? >I have an example: Some years ago I took a problem in practical computing and derived some new probabilistic math for a solution. You go on to rant how editors of CS journals weren't smart enough to understand your paper. That might well be the case, but it is obvious that you were not any smarter, sending a paper which is pure maths in nature to several CS journals, when the "problem in practical computing" was merely an application of your result, and not the main point of your paper. If you were a physicist working on some very deep results of quantum mechanics that happen to have a direct application to, say, molecular biology, you would still send it to a physics journal, not a biology journal, wouldn't you? >Here's what they need to do:... That's a good plan for learning a bunch of maths, not CS.

NY_Entrepreneur 15y ago
"Properly done CS _is_ maths." Okay, then you would agree with my thought on this thread that the math departments should just take over CS! The CS would get some much better math, and the math departments would get a big head and tummy ache from the much needed contact with applications. Fine with me. "is disingenuous at best." No. That chemistry uses some group theory for molecular spectroscopy and makes a mess out of it, which it has been known to do, doesn't mean that a mathematician who sits in a chair with plastic fake leather shouldn't comment. So, yes, IP routing has been based on some network shortest path algorithms. So, they used some operations research material since really it was operations research that first dug into networks that deeply -- yes, I know, there was a Nobel prize in economics for some work by Kantorovich (?) in the late 1940s on the transportation problem and, earlier, work in circuit theory as in Kirchhoff. Still, e.g., with Ford and Fulkerson and then Jack Edmonds, network shortest path and flows on networks should be attributed to operations research. If IP routing got to use it, then fine. Once I used Cunningham's version of the network simplex algorithm on a problem in allocating sales forces in a tricky situation: The problem looked like just integer linear programming, but if looked again it was just network flows where can get integer solutions for no extra effort. That doesn't mean that network flows should be part of courses in selling! But CS is not just using network flows but calling the subject part of computer science. That's hijacking! Or it justifies my "Not really good.". But, such things will happen. There is no law that says that have to be an operations research person to publish in an operations research journal or to publish about networks in a computer science journal. The NSF even works to encourage 'cross-cutting'. Since my writing is so 'succinct', maybe I wasn't clear! If CS wants to take a field like network flows for their own, then actually do a competent job of it and don't mess it up. So, when there's is need, as there is, for the simplex algorithm, and not just using it but calling it part of CS, then do a competent job with the simplex algorithm. If probabilistic analysis of the performance of algorithms is to be part of CS, then CS should do a competent job with the probability. Else, CS is diluting the quality of these subjects. I argued elsewhere on HN that CS has diluted the meaning of dynamic programming, also essentially hijacked by CS. If you want to say that CS done well is math, then they should do the math well. When I was in grad school, I took some courses in measure theory and functional analysis from a department not the pure math department. And I took some statistics. When the statistics got to sufficient statistics, which is from the Radon-Nikodym theorem of measure theory, the prof totally blew it. I just walked out of the class; mostly I knew the material at the elementary level anyway, and at the level I wanted to learn the material the course was a waste of my time. Well, in the courses in measure theory and functional analysis, the profs maintained quality only slightly short of Bourbaki. They didn't have less than a beautifully polished proof for anything. Any hint or suggestion that there was a logical gap was a very big deal in the department. When I walked out of the stat course, the department dumped on me. Then I dumped back on the mess made of the R-N theorem. The next year the stat prof was gone. That department, not pure math, took the math fully seriously: A prof who blew a good proof of one important theorem was looking for a job. The situation was SERIOUS. We weren't a pure math department, but we did the math in rock solid ways. The department also had some operations research: Similarly. The math was rock solid. E.g., at one point in the Kuhn-Tucker conditions, I objected that a proof had used but not assumed continuous differentiability. I worked up a proof with just differentiability, and my proof was in the course the next semester. So, broadly, just because we were interested in some math topics not common in pure math departments, and interested in applications, didn't mean we did the math at a lower level of quality. We were as close to Bourbaki as anyone. It can be done. CS should do it, too. Sorry 'bout that. Low quality is not good; I called out some low quality. I was fully correct to do so. The problem is not my arrogance but their low quality. "In a general sense, yes, it would be good. Maybe you also need to read some better CS stuff?" So, we are in agreement, "it would be good". For my reading, that's about me, and that's not appropriate. The issue is the book CLRS or whatever it is called. I'm not the issue. But for some "better CS stuff", you mentioned von Neumann. Right: In my favorite source of the R-N theorem, the proof is by von Neumann. In my current project, there's some math at the core, some original, and some due to von Neumann. I credit him and Kolmogorov. For the paper I sent, I sent it to an appropriate journal. The motivation for the research was a problem in computer science. The main application for the original part of the research was just to that problem. That there was some original math at the core of the paper was part of the work as in your remark that CS should be math. You are agreeing with my main point that CS should clean up its math and then calling me arrogant for making this point. Can't have it both ways. For the course of study I outlined, what important about CS does that omit! I know; I know; it omits heap sort! And AVL trees. Gee, looks like that course of study needs more, maybe a lecture, maybe even two, of an hour each? Do the math. For the CS, do that in the exercises and footnotes!