Learning math on the internet is hard. Different people use different notations and assume different backgrounds.
I encourage the poster to read and learn more about exterior algebras! Once you've seen Stoke's theorem, the Hodge decomposition, Maxwell's equations, etc. written using them you'll never go back and those old school t-shirts with "and god said..."[1] will make you cringe.
I hope/expect to see them taught in basic calculus and physics, but it takes generations for things to trickle down.
I think it was done that way once, but somewhere around seventies people dropped or moved this to advanced courses. You can't immediately solve a transformer in differential forms like the way you can by mindless symbolic manipulation over the standard formalism.
I was chagrined to find myself in this very same position. Sadly, I have not yet had the time to track down the resources necessary to "re-educate" myself. Computer science has given me a new love for math that I didn't even possess during my education, but I know of very few ways to express that enthusiasm.
Perhaps OP should have studied number theory. No number theorist has ever complained about the quality of his teacher. It's just you and all that ever was and all that ever will be. Now go discover.
Sussman's critique of mathematical notation is hollow. The point of good notation is to hide the details while conveying the high level compactly and succinctly.
Lagrange's equations written the way he derides are a perfect example. His need to think about what spaces the symbols live in based on their context in the expression is typical when translating math to code. He should be talking about about what kind of type inference engine one needs to allow the original notation and end up with his positional notation internally.
Why does he say that the arity is different for derivatives and integrals? Isn't there an arity 1 symbolic integration? If you were numerically computing an integral over a range, shouldn't you be evaluating the derivative at a point? These don't have the same arity, but at least the ideas are parallel.
I'm a math idiot, too, but I know that he's confusing indefinite and definite integration. Definite integration returns a number; indefinite integration returns a function. Differentiation does undo indefinite integration.
I have a similar, but slightly different problem. While I know that CS and programming in general can be translated into useful math knowledge, I always mistrust my math. I suspect that i don't understand the math correctly because I understand it, if that makes any sense. The weird terminologies and reputation for being way hard always erode my confidence in my understanding.
I've found that mostly it's just that mathematics is usually taught extremely poorly. Mathematicians and mathematics educators typically gloss over definitions and use terrible notation. Further compounding the problem, the mechanical aspects of mathematical computation are rarely formally defined -- talking about elementary operators and objects of a mathematical subfield is rare.
I've recently discovered abstract algebra and I love it. We need to find and celebrate good explanations of mathematics both to do better work and to give future generations the mathematical education we didn't get.
My high school math teacher refused to teach calculus. We spent year after year on algebra, trig, geometry, and did proofs. Lots and lots of proofs. He told me not to worry about when I went to college and studied calculus as I would be much better at it than the students that studied calc in high school.
Smart man, he was right. The first two weeks in my first college calc class I felt a little uneasy and understood that I was the only person in class that had not studied this stuff in high school. After those first two weeks though, it was not a problem and I swept through 6 calc classes without a sweat.
20 years later of course, I'm a math idiot again...use it or lose it ;)
I enjoyed this post a lot because I traveled the opposite path. I started out in computer science with little interest whatsoever in math all through high school.
Then when I got to college they introduced the notion of "program correctness" where you tried to prove, mathematically, that your computer programs were correct. This convinced me that computer science was simply a knock-off approximation of mathematics and I drifted away from it.
From your derivative example above I’d say it is this inexact nature of computer science that you especially like. Different strokes for different folks, I suppose.
It's not "different strokes," it's a change of perspective causing massive confusion all 'round. Computer science isn't a knock-off approximation of mathematics, it's a subfield of mathematics. Programming, which is what the post is talking about, is a third thing. The derivative is the inverse of the indefinite interval (and has the same arity), but the function he refers to computes a definite integral which is not the same thing. Etc.
Confusion about this stuff is a sign that you didn't really understand it the first time. Believe me, I've been (am still) there...
I think what he's getting at is that the math taught in high school and applied college courses isn't fully formalized, often isn't fully explained, and even (as taught) contains ambiguities. I was a math major (and took graduate level analysis, topology, and set theory as an undergrad,) but the applied math classes I took felt really rushed and confusing. I actually felt kind of lost and dumb in sophomore DiffyQ, and what's worse, I was surrounded by engineering students who were having no problems. I sometimes felt dumb studying Lebesgue integration and transfinite induction, but I never felt lost like I did in DiffyQ. It didn't feel like math at all; I loved math but hated sophomore DiffyQ.
[+] [-] onedognight|16 years ago|reply
I encourage the poster to read and learn more about exterior algebras! Once you've seen Stoke's theorem, the Hodge decomposition, Maxwell's equations, etc. written using them you'll never go back and those old school t-shirts with "and god said..."[1] will make you cringe.
I hope/expect to see them taught in basic calculus and physics, but it takes generations for things to trickle down.
[1] http://images6.cafepress.com/product/96599546v11_350x350_Fro...
[+] [-] slackenerny|16 years ago|reply
I think it was done that way once, but somewhere around seventies people dropped or moved this to advanced courses. You can't immediately solve a transformer in differential forms like the way you can by mindless symbolic manipulation over the standard formalism.
[+] [-] mcantor|16 years ago|reply
[+] [-] edw519|16 years ago|reply
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Perhaps OP should have studied number theory. No number theorist has ever complained about the quality of his teacher. It's just you and all that ever was and all that ever will be. Now go discover.
[+] [-] thunk|16 years ago|reply
http://video.google.com/videoplay?docid=-2726904509434151616...
[+] [-] onedognight|16 years ago|reply
Lagrange's equations written the way he derides are a perfect example. His need to think about what spaces the symbols live in based on their context in the expression is typical when translating math to code. He should be talking about about what kind of type inference engine one needs to allow the original notation and end up with his positional notation internally.
[+] [-] anc2020|16 years ago|reply
[+] [-] brent|16 years ago|reply
[+] [-] bkovitz|16 years ago|reply
[+] [-] sophacles|16 years ago|reply
[+] [-] silentOpen|16 years ago|reply
I've recently discovered abstract algebra and I love it. We need to find and celebrate good explanations of mathematics both to do better work and to give future generations the mathematical education we didn't get.
[+] [-] jhancock|16 years ago|reply
Smart man, he was right. The first two weeks in my first college calc class I felt a little uneasy and understood that I was the only person in class that had not studied this stuff in high school. After those first two weeks though, it was not a problem and I swept through 6 calc classes without a sweat.
20 years later of course, I'm a math idiot again...use it or lose it ;)
[+] [-] rawr|16 years ago|reply
Then when I got to college they introduced the notion of "program correctness" where you tried to prove, mathematically, that your computer programs were correct. This convinced me that computer science was simply a knock-off approximation of mathematics and I drifted away from it.
From your derivative example above I’d say it is this inexact nature of computer science that you especially like. Different strokes for different folks, I suppose.
[+] [-] arakyd|16 years ago|reply
Confusion about this stuff is a sign that you didn't really understand it the first time. Believe me, I've been (am still) there...
[+] [-] dkarl|16 years ago|reply