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* the most advanced math I had was Algebra 2* How could you get a BS in CS without taking calculus courses? Which school did you go to?

by p1esk 10y ago

kragen 10y ago
It's kind of a shame that so many schools push you to study calculus in order to study CS; digital computers are algebraic machines, not analytical ones, pace Babbage. Combinatorics and graph theory would be far more useful. (Although maybe this will change with machine learning.)
- tzs 10y ago
  Richard Feynman spent some time working at Thinking Machines, working on the router for the Connection Machine. From Danny Hillis' account of this [1]: By the end of that summer of 1983, Richard had completed his analysis of the behavior of the router, and much to our surprise and amusement, he presented his answer in the form of a set of partial differential equations. To a physicist this may seem natural, but to a computer designer, treating a set of boolean circuits as a continuous, differentiable system is a bit strange. Feynman's router equations were in terms of variables representing continuous quantities such as "the average number of 1 bits in a message address." I was much more accustomed to seeing analysis in terms of inductive proof and case analysis than taking the derivative of "the number of 1's" with respect to time. Our discrete analysis said we needed seven buffers per chip; Feynman's equations suggested that we only needed five. We decided to play it safe and ignore Feynman. The decision to ignore Feynman's analysis was made in September, but by next spring we were up against a wall. The chips that we had designed were slightly too big to manufacture and the only way to solve the problem was to cut the number of buffers per chip back to five. Since Feynman's equations claimed we could do this safely, his unconventional methods of analysis started looking better and better to us. We decided to go ahead and make the chips with the smaller number of buffers. Fortunately, he was right. When we put together the chips the machine worked. [1] http://longnow.org/essays/richard-feynman-connection-machine/ http://longnow.org/essays/richard-feynman-connection-machine...
- p1esk 10y ago
  I actually agree with you: basic calculus should not be studied in college. It belongs in high school, and should be a required prerequisite for college admission.
  - icantdrive55 10y ago
    In high school, I was taught math horribly. I wish high school would stick with just the basics, and work on doing it better. I spent a year in a community college making up for what I should have learned in high school; basic math up to advanced algebra. Sure I applied myself more, but the teachers, and even the text books seemed better? Once I learned the basics, it made math enjoyable, and I didn't fear courses that were heavy in math. By the way, most Medical doctors never sat in a calculus course. Here, in the U. S., there's always had two calipers of physics courses. The hard, and easy physics courses. The easy physics courses don't require calculus. They hard require calculus. Most med students too the easy courses, and aced them. It's all about the GPA when trying to pretty yourself up for med. school. I worried way too much about grades in college. I look back and wish I took the courses I was interested in. My interests are completely different as I've aged. It's tough in college because so much rides on getting into that certain graduate program, or professional school-- graduating, and getting a Job.
    - p1esk 10y ago
      I learned more advanced math in high school than I did in college (as a mech. eng. major). I wished that instead of sitting through basic calculus/lin. algebra courses again in college, they had challenged me with something more advanced.
- sn9 10y ago
  Discrete mathematics (with Rosen or Epps as the text) is usually explicitly a required course in CS programs, often a prereq for Algorithms. Calculus does usually build some mathematical maturity for those who haven't encountered it. And it's useful as an introduction to sequences and series, and for anyone interested in numerical analysis or physics simulation (e.g., computational science, modeling, game engine development, etc.). Not to mention having it is useful if you find that you'd rather do computer engineering or EE halfway through your undergrad career (though this last point is tangential at best). I do wish linear algebra was a more commonly required course in CS programs.
- fsloth 10y ago
  Computers would be darn boring without calculus. Graphics, games, audio, animation - basically anything enabling creativity on a computer needs calculus tools. The interesting bits start to happen once one has built sufficient substrate out of the discrete parts. This is my personal opinion only, of course.
  - kragen 10y ago
    What those have in common is that they're numerical, not that they require (integral and differential) calculus. It's true that in a lot of cases, deeply understanding discrete numerical algorithms is a lot easier if you can analyze the continuous versions, which of course cannot be executed directly. But you can get really far with just the discrete versions, and you can understand useful things about the continuous versions without knowing what a derivative or an integral is. And I don't just mean that you can use Unity or Pure Data to wire together pre-existing algorithms and get interesting results, although that's true too. You don't even need to understand any calculus to write a ray-tracer from scratch like http://canonical.org/~kragen/sw/aspmisc/my-very-first-raytracer http://canonical.org/~kragen/sw/aspmisc/my-very-first-raytra..., which is four pages of C. You could maybe argue that it's using square roots, and calculating square roots efficiently requires using Newton's method or something more sophisticated. But Heron of Alexandria described "Newton's" method 2000 years ago, although he hadn't generalized it to finding zeroes of arbitrary analytic functions, perhaps because he didn't have concepts of zero or functions. You could argue that it's using the pow() function, but it's using it to take the 64th power of a dot product in order to get specular reflections. People were taking integer powers of things quite a long time ago. Even using computers for really analytic things, like finding zeroes of arbitrary analytic functions, can be done with just a minimal, even intuitive, notion of continuity. Alan Kay's favorite demo of using computers to build human-comprehensible models of things is to take a video of a falling ball and then make a discrete-time model of the ball's position. A continuous-time model really does require calculus, and famously this is one of the things calculus was invented for; a discrete-time model requires the finite difference operator (and maybe its sort-of inverse, the prefix sum). Mathematics for the Million starts out with finite difference operators in its first chapter or two. You don't even need to know how to multiply and divide to compute finite differences, although a little algebra will get you a lot farther with them. A deep understanding of the umbral calculus may be inspirational and broadening in this context, and may even help you debug your programs, but you can get by without it. I agree that calculus is really powerful in extending the abilities of computers to model things, but I think you're overstating how fundamental it is.
mungoid 10y ago
Unfortunately in my case it was a private, non-profit university which had accelerated degrees (5 semesters squeezed into each year) - I'm rather dissapointed at how it turned out for the price (almost 50k!) and I wish i would have just put that money towards an actual, recognized university instead. Heck, even last year I talked to another University to look into electrical engineering and only 1, ONE, class would transfer. All others wouldnt count because since they were a private school, their curriculum was different than most. Thats not something they put in brochures.

* the most advanced math I had was Algebra 2* How could you get a BS in CS without taking calculus courses? Which school did you go to?

by p1esk 10y ago

* the most advanced math I had was Algebra 2* How could you get a BS in CS without taking calculus courses? Which school did you go to?

kragen 10y ago

It's kind of a shame that so many schools push you to study calculus in order to study CS; digital computers are algebraic machines, not analytical ones, pace Babbage. Combinatorics and graph theory would be far more useful. (Although maybe this will change with machine learning.)

tzs 10y ago
Richard Feynman spent some time working at Thinking Machines, working on the router for the Connection Machine. From Danny Hillis' account of this [1]: By the end of that summer of 1983, Richard had completed his analysis of the behavior of the router, and much to our surprise and amusement, he presented his answer in the form of a set of partial differential equations. To a physicist this may seem natural, but to a computer designer, treating a set of boolean circuits as a continuous, differentiable system is a bit strange. Feynman's router equations were in terms of variables representing continuous quantities such as "the average number of 1 bits in a message address." I was much more accustomed to seeing analysis in terms of inductive proof and case analysis than taking the derivative of "the number of 1's" with respect to time. Our discrete analysis said we needed seven buffers per chip; Feynman's equations suggested that we only needed five. We decided to play it safe and ignore Feynman. The decision to ignore Feynman's analysis was made in September, but by next spring we were up against a wall. The chips that we had designed were slightly too big to manufacture and the only way to solve the problem was to cut the number of buffers per chip back to five. Since Feynman's equations claimed we could do this safely, his unconventional methods of analysis started looking better and better to us. We decided to go ahead and make the chips with the smaller number of buffers. Fortunately, he was right. When we put together the chips the machine worked. [1] http://longnow.org/essays/richard-feynman-connection-machine/ http://longnow.org/essays/richard-feynman-connection-machine...
p1esk 10y ago
I actually agree with you: basic calculus should not be studied in college. It belongs in high school, and should be a required prerequisite for college admission.
- icantdrive55 10y ago
  In high school, I was taught math horribly. I wish high school would stick with just the basics, and work on doing it better. I spent a year in a community college making up for what I should have learned in high school; basic math up to advanced algebra. Sure I applied myself more, but the teachers, and even the text books seemed better? Once I learned the basics, it made math enjoyable, and I didn't fear courses that were heavy in math. By the way, most Medical doctors never sat in a calculus course. Here, in the U. S., there's always had two calipers of physics courses. The hard, and easy physics courses. The easy physics courses don't require calculus. They hard require calculus. Most med students too the easy courses, and aced them. It's all about the GPA when trying to pretty yourself up for med. school. I worried way too much about grades in college. I look back and wish I took the courses I was interested in. My interests are completely different as I've aged. It's tough in college because so much rides on getting into that certain graduate program, or professional school-- graduating, and getting a Job.
  - p1esk 10y ago
    I learned more advanced math in high school than I did in college (as a mech. eng. major). I wished that instead of sitting through basic calculus/lin. algebra courses again in college, they had challenged me with something more advanced.
sn9 10y ago
Discrete mathematics (with Rosen or Epps as the text) is usually explicitly a required course in CS programs, often a prereq for Algorithms. Calculus does usually build some mathematical maturity for those who haven't encountered it. And it's useful as an introduction to sequences and series, and for anyone interested in numerical analysis or physics simulation (e.g., computational science, modeling, game engine development, etc.). Not to mention having it is useful if you find that you'd rather do computer engineering or EE halfway through your undergrad career (though this last point is tangential at best). I do wish linear algebra was a more commonly required course in CS programs.
fsloth 10y ago
Computers would be darn boring without calculus. Graphics, games, audio, animation - basically anything enabling creativity on a computer needs calculus tools. The interesting bits start to happen once one has built sufficient substrate out of the discrete parts. This is my personal opinion only, of course.
- kragen 10y ago
  What those have in common is that they're numerical, not that they require (integral and differential) calculus. It's true that in a lot of cases, deeply understanding discrete numerical algorithms is a lot easier if you can analyze the continuous versions, which of course cannot be executed directly. But you can get really far with just the discrete versions, and you can understand useful things about the continuous versions without knowing what a derivative or an integral is. And I don't just mean that you can use Unity or Pure Data to wire together pre-existing algorithms and get interesting results, although that's true too. You don't even need to understand any calculus to write a ray-tracer from scratch like http://canonical.org/~kragen/sw/aspmisc/my-very-first-raytracer http://canonical.org/~kragen/sw/aspmisc/my-very-first-raytra..., which is four pages of C. You could maybe argue that it's using square roots, and calculating square roots efficiently requires using Newton's method or something more sophisticated. But Heron of Alexandria described "Newton's" method 2000 years ago, although he hadn't generalized it to finding zeroes of arbitrary analytic functions, perhaps because he didn't have concepts of zero or functions. You could argue that it's using the pow() function, but it's using it to take the 64th power of a dot product in order to get specular reflections. People were taking integer powers of things quite a long time ago. Even using computers for really analytic things, like finding zeroes of arbitrary analytic functions, can be done with just a minimal, even intuitive, notion of continuity. Alan Kay's favorite demo of using computers to build human-comprehensible models of things is to take a video of a falling ball and then make a discrete-time model of the ball's position. A continuous-time model really does require calculus, and famously this is one of the things calculus was invented for; a discrete-time model requires the finite difference operator (and maybe its sort-of inverse, the prefix sum). Mathematics for the Million starts out with finite difference operators in its first chapter or two. You don't even need to know how to multiply and divide to compute finite differences, although a little algebra will get you a lot farther with them. A deep understanding of the umbral calculus may be inspirational and broadening in this context, and may even help you debug your programs, but you can get by without it. I agree that calculus is really powerful in extending the abilities of computers to model things, but I think you're overstating how fundamental it is.

mungoid 10y ago

Unfortunately in my case it was a private, non-profit university which had accelerated degrees (5 semesters squeezed into each year) - I'm rather dissapointed at how it turned out for the price (almost 50k!) and I wish i would have just put that money towards an actual, recognized university instead. Heck, even last year I talked to another University to look into electrical engineering and only 1, ONE, class would transfer. All others wouldnt count because since they were a private school, their curriculum was different than most. Thats not something they put in brochures.