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> if you don't do proofs you won't get any better at understanding mathematics There is a vast world of mathematical understanding available to programmers who

> if you don't do proofs you won't get any better at understanding mathematics There is a vast world of mathematical understanding available to programmers who don't like proofs. I think there is some miscommunication here about what constitutes mathematics. Mathematicians tend to think of the art of proving things, whereas engineers (and programmers, and scientists) think of equations and how to solve them. I think mathematicians actually do a disservice to engineers by trying to beat them over the head with real analysis and set theory (which is why most physics and engineering departments end up having to teach their own mathematical methods courses). The "foundational" knowledge of mathematics -- epsilons, deltas, Cantor sets, Banach spaces, etc. -- is irrelevant to almost all practitioners. Newton, Taylor, Euler, Gauss, Laplace, Bessel, Maxwell and the gang got pretty far without it. When programmers say they want to learn math, they mean they want practical knowledge: interesting functions, how to compute them, and their properties that can be applied with utility. They don't want -- or need -- to spend time poring over proofs.

j2kun 14y ago
It seems you're under the impression that real analysis is the foundation of mathematics (I certainly wouldn't consider Cantor sets or Banach spaces in there). Let's say you invent a new machine learning algorithm, how can you know how well it does? If you invent an algorithm to do anything, how can you be sure it's correct? Well you can always test it some things, but if it's really a new algorithm, then you need to prove it does what you say it does in all possible cases. It's fine and dandy if someone tells you that there is this function that you can compute and use in this way and it's the best for its purpose, but this is an extremely rare case. I believe I have mentioned this elsewhere in comments, but there is almost never a clear-cut answer in mathematics that someone can just use out of the box. It almost always involves tweaking to fit a particular application, and to really understand how to tweak an algorithm you need to understand why the algorithm works the way it does.
- jholman 14y ago
  I agree with you that if you want to prove the correctness of a new algorithm, you need to prove the correctness of that new algorithm. You're gonna want some proving skills there. How often are programmers developing new algorithms? I guess it depends on where you draw the line for "algorithm", but let me propose two possible places for the line. On the one hand, you might use existing techniques in a new function. In this case, the proving technique you need is the theory of invariants and so on, as Dijkstra ranted about. You're not gonna need a lot of practice with higher-level math proofs for that. That's even assuming you care about "proving that it does what you say it does in all possible cases", which of course we all know is done by roughly 0% of programmers, especially with emphasis on the "in all possible". On the other hand, you might make a bigger development, that requires more mathematical expertise. I suggest for your consideration that the correct description of a person doing this latter work is not "programmer", but "professor" (or grad student), and that this is relatively rare, and if this is what you meant all along, you've been using misleading language. I think you have it absolutely backwards when you say "this is an extremely rare case"; I think the case where you need real mathematical proof techniques is the extremely rare case, even in relatively math-heavy areas, e.g. numeric simulation. What you need, as I've said before, are techniques of calculation. Also, my ML knowledge is weak, but from my modest knowledge, I don't think any of it works provably. It works probably. :) Seems a weird example to choose. At this point, btw, I acknowledge that I'm thoroughly into "quibble" territory. =]
stiff 14y ago
> There is a vast world of mathematical understanding available to programmers who don't like proofs. Mathematics is a deep net of knowledge, you can not just somehow study "interesting functions" in isolation, only by systematically exploring the relevant areas and understanding the concepts you can apply it, and calculation is the easiest part most of the time. If you understand something clearly, you can prove it. If you can't prove it, you don't really understand it. It's that simple. Don't equate proofs with epsilon/deltas and Cantor sets. There are very practical theories developed formally that you can only study fruitfully if you are not afraid of proofs. Algorithms, probability theory, statistics, machine learning are all full of proofs, there is not much to calculate until a very late stage and it is very practical right from the start. A formal approach does not necessarily imply highly abstract concepts.

j2kun 14y ago

It seems you're under the impression that real analysis is the foundation of mathematics (I certainly wouldn't consider Cantor sets or Banach spaces in there). Let's say you invent a new machine learning algorithm, how can you know how well it does? If you invent an algorithm to do anything, how can you be sure it's correct? Well you can always test it some things, but if it's really a new algorithm, then you need to prove it does what you say it does in all possible cases. It's fine and dandy if someone tells you that there is this function that you can compute and use in this way and it's the best for its purpose, but this is an extremely rare case. I believe I have mentioned this elsewhere in comments, but there is almost never a clear-cut answer in mathematics that someone can just use out of the box. It almost always involves tweaking to fit a particular application, and to really understand how to tweak an algorithm you need to understand why the algorithm works the way it does.

jholman 14y ago
I agree with you that if you want to prove the correctness of a new algorithm, you need to prove the correctness of that new algorithm. You're gonna want some proving skills there. How often are programmers developing new algorithms? I guess it depends on where you draw the line for "algorithm", but let me propose two possible places for the line. On the one hand, you might use existing techniques in a new function. In this case, the proving technique you need is the theory of invariants and so on, as Dijkstra ranted about. You're not gonna need a lot of practice with higher-level math proofs for that. That's even assuming you care about "proving that it does what you say it does in all possible cases", which of course we all know is done by roughly 0% of programmers, especially with emphasis on the "in all possible". On the other hand, you might make a bigger development, that requires more mathematical expertise. I suggest for your consideration that the correct description of a person doing this latter work is not "programmer", but "professor" (or grad student), and that this is relatively rare, and if this is what you meant all along, you've been using misleading language. I think you have it absolutely backwards when you say "this is an extremely rare case"; I think the case where you need real mathematical proof techniques is the extremely rare case, even in relatively math-heavy areas, e.g. numeric simulation. What you need, as I've said before, are techniques of calculation. Also, my ML knowledge is weak, but from my modest knowledge, I don't think any of it works provably. It works probably. :) Seems a weird example to choose. At this point, btw, I acknowledge that I'm thoroughly into "quibble" territory. =]

stiff 14y ago

> There is a vast world of mathematical understanding available to programmers who don't like proofs. Mathematics is a deep net of knowledge, you can not just somehow study "interesting functions" in isolation, only by systematically exploring the relevant areas and understanding the concepts you can apply it, and calculation is the easiest part most of the time. If you understand something clearly, you can prove it. If you can't prove it, you don't really understand it. It's that simple. Don't equate proofs with epsilon/deltas and Cantor sets. There are very practical theories developed formally that you can only study fruitfully if you are not afraid of proofs. Algorithms, probability theory, statistics, machine learning are all full of proofs, there is not much to calculate until a very late stage and it is very practical right from the start. A formal approach does not necessarily imply highly abstract concepts.