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Professors or not, this is a quite mediocre discussion. For one, "discrete" and "continuous" mathematics are not disjoint things, and the "discrete" math that y

by stiff 12y ago

Professors or not, this is a quite mediocre discussion. For one, "discrete" and "continuous" mathematics are not disjoint things, and the "discrete" math that you can do without touching "continuous mathematics" is for the most part too simplistic to serve as a model of any real world phenomena, probably including most practically useful algorithms. Almost any "discrete" problem at the real-world level of complexity requires tools from real analysis, and in fact more often than not from complex analysis. For example, the "Concrete mathematics" book by Knuth that is praised so much is mainly a book about generating functions in the form of power series of a complex variable. Despite the problem being discrete, you only get to get answers by setting up those power series with the discrete coefficients in front of the complex powers, and then obtain answers by differentiation and whatnot. You do not get any "mathematical thinking" without understanding calculus, this is non-sense. The theory of real variables is not only one of the most ubiquitously practical one, but it is also the model for many other theories. Modern probability theory is largely advanced calculus in a particular setting, for example. That is not to say that calculus is all there is. You just need to know a whole lot of math to apply it fruitfully in modern complicated settings, and calculus is some of the math that you need to know if you want to work on machine learning or robotics. If you want to work on webapps on the other hand, you do not need almost any math at all, but then you also do not need a university to learn in in the first place.

sanxiyn 12y ago
How does, say, encoding and decoding error-correcting codes require analysis in any way? I protest that formal power series and formal derivative should not be counted under calculus or continuous math. You can define and prove properties of formal derivatives without any reference to continuum whatsoever. Math has unity, real numbers are great, but I think people are overselling continuous-discrete connections. Dividing line between continuous math and discrete math is one of the most clear boundary in math, and there are huge amounts of useful discrete math which does not require continuous math. If you work on machine learning, learn continuous math. On the other hand, if you work on programming language, learn logic and discrete math instead of continuous math. If you work on web applications, well, good luck. :)
- ColinWright 12y ago
  My PhD is in discrete math and graph theory, and yet I find calculus and probability to be indispensable tools. I devise algorithms that work on images and streams of text, and yet I use power series and vector spaces over the complex numbers to analyze them. Writing code to implement error-correcting codes doesn't use continuous math, but some error-correcting codes are best understood as working in polynomial rings over finite fields, and many of the ideas and tools are directly linked to similar structures in infinite continuous spaces. Limiting yourself to learning only about discrete math that does not require any continuous math is like limiting yourself to using only one style of programming, or one language, or one operating system, or one hardware platform. If you don't mind being so limited, fine. If you want to understand things in greater depth and thus give yourself more opportunities and have more tools in your arsenal, learn both. They support each other in ways you can't imagine without actually learning them.
  - sanxiyn 12y ago
    I am not saying connection does not exist (it certainly does), but I am saying connection is oversold. They support each other, but they don't support each other that much. While I don't have PhD in math, I did learn calculus and probability. I am still not convinced that they should come before logic.
    - ColinWright 12y ago
      See, that's the thing. People who don't have all the tools to hand think that the connections are over-rated. People who do have more of the tools to hand think that it's much more important than is realized by people who don't have all the tools to hand. It's the blub[0] paradox all over again. I would suggest that you haven't (yet) learned enough to realize that the connections are much deeper, stronger, and important than you realize. It may well be that you don't need to know that the derivative of 3x^2 is 6x, but the influence on working with discrete math of a deeper understanding of continuous math is real, but subtle and hard to explain. Like Spock said when McCoy asked what it was like to be dead: McCoy: Come on Spock this is me, McCoy!! You really have been where no one has been before, can't you tell me what it felt like? Spock: It would be impossible to discuss the subject without a common frame of reference. McCoy: You're joking - you mean I have to die before we can discuss your insight on death? We can't all learn everything. I'm not advocating that everyone should get a PhD in cross-disciplinary math subjects, just as I'm not advocating that everyone should become fluent in Japanese, Russian, and Finnish, or that everyone should become fluent in Scheme, Haskell, and Erlang. What I am saying is that I, who do have a PhD in math, believe that the connections are more extensive and useful than you realize. [0] http://www.paulgraham.com/avg.html http://www.paulgraham.com/avg.html

by stiff 12y ago

sanxiyn 12y ago

How does, say, encoding and decoding error-correcting codes require analysis in any way? I protest that formal power series and formal derivative should not be counted under calculus or continuous math. You can define and prove properties of formal derivatives without any reference to continuum whatsoever. Math has unity, real numbers are great, but I think people are overselling continuous-discrete connections. Dividing line between continuous math and discrete math is one of the most clear boundary in math, and there are huge amounts of useful discrete math which does not require continuous math. If you work on machine learning, learn continuous math. On the other hand, if you work on programming language, learn logic and discrete math instead of continuous math. If you work on web applications, well, good luck. :)

ColinWright 12y ago
My PhD is in discrete math and graph theory, and yet I find calculus and probability to be indispensable tools. I devise algorithms that work on images and streams of text, and yet I use power series and vector spaces over the complex numbers to analyze them. Writing code to implement error-correcting codes doesn't use continuous math, but some error-correcting codes are best understood as working in polynomial rings over finite fields, and many of the ideas and tools are directly linked to similar structures in infinite continuous spaces. Limiting yourself to learning only about discrete math that does not require any continuous math is like limiting yourself to using only one style of programming, or one language, or one operating system, or one hardware platform. If you don't mind being so limited, fine. If you want to understand things in greater depth and thus give yourself more opportunities and have more tools in your arsenal, learn both. They support each other in ways you can't imagine without actually learning them.
- sanxiyn 12y ago
  I am not saying connection does not exist (it certainly does), but I am saying connection is oversold. They support each other, but they don't support each other that much. While I don't have PhD in math, I did learn calculus and probability. I am still not convinced that they should come before logic.
  - ColinWright 12y ago
    See, that's the thing. People who don't have all the tools to hand think that the connections are over-rated. People who do have more of the tools to hand think that it's much more important than is realized by people who don't have all the tools to hand. It's the blub[0] paradox all over again. I would suggest that you haven't (yet) learned enough to realize that the connections are much deeper, stronger, and important than you realize. It may well be that you don't need to know that the derivative of 3x^2 is 6x, but the influence on working with discrete math of a deeper understanding of continuous math is real, but subtle and hard to explain. Like Spock said when McCoy asked what it was like to be dead: McCoy: Come on Spock this is me, McCoy!! You really have been where no one has been before, can't you tell me what it felt like? Spock: It would be impossible to discuss the subject without a common frame of reference. McCoy: You're joking - you mean I have to die before we can discuss your insight on death? We can't all learn everything. I'm not advocating that everyone should get a PhD in cross-disciplinary math subjects, just as I'm not advocating that everyone should become fluent in Japanese, Russian, and Finnish, or that everyone should become fluent in Scheme, Haskell, and Erlang. What I am saying is that I, who do have a PhD in math, believe that the connections are more extensive and useful than you realize. [0] http://www.paulgraham.com/avg.html http://www.paulgraham.com/avg.html