12 ms·

This does not seem very informed, starting from the false premise in the question up to the random concepts listed in the answers. Calculus is actually importa

by stiff 12y ago

This does not seem very informed, starting from the false premise in the question up to the random concepts listed in the answers. Calculus is actually important for Computer Science, it's actually important for everything, it's where you learn how to handle the exponential function and the natural logarithm, how to do approximations and bounds, how to handle infinite series, etc., and those things then appear all over the place, unlike most things listed it's something that you can expect to encounter almost regardless of what domain you are interested in. I mean, the guy asks what should replace Calculus and then the first answer includes "Asymptotics", "basic limits, sequences and series", so actually calculus. In general I cringe a little every time I hear Computer Science people should focus on "discrete math", because without tools from analysis you can only solve the most trivial discrete problems. And yes, calculus by itself is hardly ever applicable in CS, yet you still have to learn it, tough luck. In general what is not stressed enough I think is that applying math is hard and you need to learn a lot of it before you have enough tools to tackle problems anywhere close to real-world complexity. The top answer also lists random concepts. I am learning probability currently, for applications in machine learning. "Discrete spaces, Bayes theorem and expected values" you can learn in a day, "Markovs, Chebyshev and Chernoff inequalities" are mostly only useful for further theoretical work, so is "the law of large numbers". What will really be useful will depend a lot on the applications, if you are a theoretical computer scientist, mastery of generating functions and transforms will be useful, and it's one of those instances where discrete problems are solved via tools from calculus/analysis. For machine learning you need to know everything about the normal distribution by heart, and this means you have to know everything about the exponential function by heart, so again back to calculus. Notions from information theory are useful, but of course none of the ones he listed. The comment "This is a must for modern programmers." sounds just comic.

anaphor 12y ago
Also I'm not sure how you can mention information theory in the context of programming and not mention PCM or the Shannon-Nyquist sampling theorem...
superuser2 12y ago
> Calculus is actually important for Computer Science, it's actually important for everything, it's where you learn how to handle the exponential function and the natural logarithm, how to do approximations and bounds, how to handle infinite series, etc., and those things then appear all over the place, It's still interesting to think about which branches of math are actually applicable to programming itself. People tend to talk about programming and math as very strongly related, and of course there is the obvious relationship that "some computer programs do particular kinds of math" like you're talking about here. But there is no (intuitive) overlap between writing, say, a web application and doing algebraic or calculus computation on paper. However, there are things like: - Set theory underpinning relational databases - Typed lambda calculus underpinning functional programming I'd be interested in other examples like this.
- ThrustVectoring 12y ago
  I have a couple simple ones off the top of my head: - You write a recursive program the same way you write an inductive proof - Abstract algebra and category theory are likely relevant, especially for metaprogramming. My math education hasn't included this, so I can't say much more. - Linear algebra is just ridiculously important - Statistics for machine learning. Also for figuring out how to combine data in a meaningful way. There are also a lot of people asking statistical questions directly, and writing programs is how you get those kinds of answers in a reasonable timeframe.
  - nmrm 12y ago
    > Abstract algebra and category theory are likely relevant, especially for metaprogramming. In general, the whole "oh yeah CS people should know some category theory and abstract algebra" is pretty hilarious. First, it's a bit like saying "oh yeah CS people need to know the undegraduate basics and also the generalization that most mathematicians don't encounter until a couple years into grad school." Second, most people who say this really mean "a conceptual grasp on different types of morphisms is useful". But that's like saying you need calculus in order to drive a car; or, in the case of categories, it's like saying you need two semesters of real analysis in order to drive a car. Why not just say "knowing about different sorts of mappings is pretty useful in functional programming"? Knowing how this generalizes to more abstract mathematical objects is totally unnecessary.
vukmir 12y ago
>Calculus is actually important for Computer Science, it's actually important for everything This. If you take a look at the MIT course "Mathematics for Computer Science"[1] you'll see that the only prerequisite for learning the math for cs is ... calculus! [1]http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/index.htm http://ocw.mit.edu/courses/electrical-engineering-and-comput...
- stepstep 12y ago
  I was a TA for that course last semester. I think the actual reason calculus is a prerequisite is just so that students will have some mathematical maturity beforehand. We didn't really teach any concepts that actually used calculus (that I remember). That said, I totally agree that calculus is good to know for a CS student.
  - stiff 12y ago
    Oh, you do use calculus, at least in the notes, actually touching on the topics I mentioned, so it's a good illustration. For example in the chapters about generating functions and on sums and asymptotics: http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/readings/MIT6_042JF10_chap09.pdf http://ocw.mit.edu/courses/electrical-engineering-and-comput... http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/readings/MIT6_042JF10_chap12.pdf http://ocw.mit.edu/courses/electrical-engineering-and-comput...
nextos 12y ago
I certainly agree. I'm trying to re-learn advanced calculus and analysis from a rigorous standpoint, as I think it is crucial for developing deep knowledge in probability theory, among other things. Slightly tangential but, while there are many lovely books for linear algebra (like Halmos, Axler or Hoffman & Kunze), as a newcomer I don't find analysis literature so exciting. The standard, Rudin, is really synthetic Bourbaki-style. I like short and precise books, but I found it really removes most intuition. Any good books you happen to like? Perhaps Pugh or Zorich?
- wyclif 12y ago
  Interested in a range of book recommendations for Calculus on up.
- stiff 12y ago
  Among the many texts almost the only one I really liked and learnt from is Courant's "Differential and Integral calculus" and the newer edition "Introduction to Calculus and Analysis". It doesn't do the typical modern division of topics and instead treats single-variable calculus, multi-variable calculus and real analysis in its two volumes in a single long sequence, but the writing style is very pleasant and the exposition very intuitive, and it includes a lot of physics applications. Hardy "A course of pure mathematics" is great too, but it is much more, well, pure, but it stays relatively intuitive and the clarity with which he writes is unparalleled, many things I first really understood from this book. Those are old texts though and notation and details of exposition differ here and there from modern standards. I have the book by Pugh, but that one is pure^2, even as far as analysis texts go, the problems are difficult and there are no solutions, so I think it would work only for people very in love with absolutely pure mathematics and most likely only in an academic setting, while I am interested in applications and self-studying. From modern texts, given your interests, I would look at "Understanding analysis" by Abbott and "Measure, Integral and Probability" by Capinski, both pleasant to read and together providing a not too steep path toward measure-theoretic probability.
  - nextos 12y ago
    Thanks for taking the time to write such a thorough reply. I find Abbott a bit imprecise sometimes, but Courant is a fantastic book. Could you also mention to some of your favorite math references, in particular those that deal with probability theory and statistics?

by stiff 12y ago

anaphor 12y ago

Also I'm not sure how you can mention information theory in the context of programming and not mention PCM or the Shannon-Nyquist sampling theorem...

superuser2 12y ago

> Calculus is actually important for Computer Science, it's actually important for everything, it's where you learn how to handle the exponential function and the natural logarithm, how to do approximations and bounds, how to handle infinite series, etc., and those things then appear all over the place, It's still interesting to think about which branches of math are actually applicable to programming itself. People tend to talk about programming and math as very strongly related, and of course there is the obvious relationship that "some computer programs do particular kinds of math" like you're talking about here. But there is no (intuitive) overlap between writing, say, a web application and doing algebraic or calculus computation on paper. However, there are things like: - Set theory underpinning relational databases - Typed lambda calculus underpinning functional programming I'd be interested in other examples like this.

ThrustVectoring 12y ago
I have a couple simple ones off the top of my head: - You write a recursive program the same way you write an inductive proof - Abstract algebra and category theory are likely relevant, especially for metaprogramming. My math education hasn't included this, so I can't say much more. - Linear algebra is just ridiculously important - Statistics for machine learning. Also for figuring out how to combine data in a meaningful way. There are also a lot of people asking statistical questions directly, and writing programs is how you get those kinds of answers in a reasonable timeframe.
- nmrm 12y ago
  > Abstract algebra and category theory are likely relevant, especially for metaprogramming. In general, the whole "oh yeah CS people should know some category theory and abstract algebra" is pretty hilarious. First, it's a bit like saying "oh yeah CS people need to know the undegraduate basics and also the generalization that most mathematicians don't encounter until a couple years into grad school." Second, most people who say this really mean "a conceptual grasp on different types of morphisms is useful". But that's like saying you need calculus in order to drive a car; or, in the case of categories, it's like saying you need two semesters of real analysis in order to drive a car. Why not just say "knowing about different sorts of mappings is pretty useful in functional programming"? Knowing how this generalizes to more abstract mathematical objects is totally unnecessary.

vukmir 12y ago

>Calculus is actually important for Computer Science, it's actually important for everything This. If you take a look at the MIT course "Mathematics for Computer Science"[1] you'll see that the only prerequisite for learning the math for cs is ... calculus! [1]http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/index.htm http://ocw.mit.edu/courses/electrical-engineering-and-comput...

stepstep 12y ago
I was a TA for that course last semester. I think the actual reason calculus is a prerequisite is just so that students will have some mathematical maturity beforehand. We didn't really teach any concepts that actually used calculus (that I remember). That said, I totally agree that calculus is good to know for a CS student.
- stiff 12y ago
  Oh, you do use calculus, at least in the notes, actually touching on the topics I mentioned, so it's a good illustration. For example in the chapters about generating functions and on sums and asymptotics: http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/readings/MIT6_042JF10_chap09.pdf http://ocw.mit.edu/courses/electrical-engineering-and-comput... http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-042j-mathematics-for-computer-science-fall-2010/readings/MIT6_042JF10_chap12.pdf http://ocw.mit.edu/courses/electrical-engineering-and-comput...

nextos 12y ago

I certainly agree. I'm trying to re-learn advanced calculus and analysis from a rigorous standpoint, as I think it is crucial for developing deep knowledge in probability theory, among other things. Slightly tangential but, while there are many lovely books for linear algebra (like Halmos, Axler or Hoffman & Kunze), as a newcomer I don't find analysis literature so exciting. The standard, Rudin, is really synthetic Bourbaki-style. I like short and precise books, but I found it really removes most intuition. Any good books you happen to like? Perhaps Pugh or Zorich?

wyclif 12y ago
Interested in a range of book recommendations for Calculus on up.
stiff 12y ago
Among the many texts almost the only one I really liked and learnt from is Courant's "Differential and Integral calculus" and the newer edition "Introduction to Calculus and Analysis". It doesn't do the typical modern division of topics and instead treats single-variable calculus, multi-variable calculus and real analysis in its two volumes in a single long sequence, but the writing style is very pleasant and the exposition very intuitive, and it includes a lot of physics applications. Hardy "A course of pure mathematics" is great too, but it is much more, well, pure, but it stays relatively intuitive and the clarity with which he writes is unparalleled, many things I first really understood from this book. Those are old texts though and notation and details of exposition differ here and there from modern standards. I have the book by Pugh, but that one is pure^2, even as far as analysis texts go, the problems are difficult and there are no solutions, so I think it would work only for people very in love with absolutely pure mathematics and most likely only in an academic setting, while I am interested in applications and self-studying. From modern texts, given your interests, I would look at "Understanding analysis" by Abbott and "Measure, Integral and Probability" by Capinski, both pleasant to read and together providing a not too steep path toward measure-theoretic probability.
- nextos 12y ago
  Thanks for taking the time to write such a thorough reply. I find Abbott a bit imprecise sometimes, but Courant is a fantastic book. Could you also mention to some of your favorite math references, in particular those that deal with probability theory and statistics?