Confessions of a Math Idiot

http://funcall.blogspot.com/2009/07/confessions-of-math-idiot.html

by blasdel • 17 years ago

72 44 17 years ago

funcall.blogspot.com

rawr• 17 years ago

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.

arakyd2• 17 years ago

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...

scott_s2• 17 years ago

Regarding where CS stands in relation to math: http://news.ycombinator.com/item?id=690798

tetha• 17 years ago

There is even more confusion out there. Think of abstract algebraic structures.

Who needs groups? No one. Until one realizes that certain simple equation transformation rules are not based on natural numbers, or real numbers but on groups. Once something like that clicks in your brain, you can suddenly solve weird equations with sets and symmetric differences, or bitstrings and xor's or other pretty akward things which looked really scary earlier.

Who needs rings and semirings? No one, until one realized that a certain algorithm requires a structure... and this structure is a semiring! Thus, if one can prove that two operations and a set form a semiring, you can apply this algorithm without any effort, because it will just work! :)

Or, even more. Who needs the theory of katamorphisms, Anamorphisms and such? No one. Until one realizes how beautiful recursive datastructures are and how easy it is to program them once you understood the idea behind them (Check http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.41.1... for a nice paper about this)

Who needs integration (besides those numerical people)? No one, until one realizes that any simulation, most iterations and such are discrete intergrations in very, very akward and convoluted algebraic structures.

Ach, I somehow wish I studied math before studying computer science by now.

dkarl• 17 years ago

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.

bkovitz7• 17 years ago

My experience with math in school is that it was taught as if it were modern art: a bunch of meaningless procedures (up through calculus) and meaningless puzzles (beyond calculus).

If it were taught as providing some insight or serving some purpose, it might be a lot easier to learn and retain.

mcantor• 17 years ago

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.

sophacles2• 17 years ago

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.

silentOpen2• 17 years ago

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.

bkovitz7• 17 years ago

What do you love about abstract algebra?

I'm asking because I found myself deeply disappointed with it when I took it in school. It seemed to me just a haphazard collection of definitions and puzzles. Why, for example, does anyone care how many non-abelian groups of order 18 there are? I found myself unable to retain much of anything, due to the lack of any coherent explanation of "What is the point?"

In computers, by contrast, no one hides the point. Searching, sorting, bits and bytes, languages that make it easier to express things without making mistakes—there's no mystery about the purpose or importance of anything.

I wonder, if someone showed me what is attractive about abstract algebra, maybe I would find myself just as addicted to math as to programming.

brent• 17 years ago

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.

bkovitz7• 17 years ago

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.

likpok• 17 years ago

Sort of, for well-defined functions (for some definition of well-defined). I remember seeing an example in analysis which showed that it was possible to get some nonintuitive behavior in functions.

ggchappell• 17 years ago

> ... I know that he's confusing indefinite and definite integration.

Yes he is. The real solution is to banish the term "indefinite integral" from the language, and substitute "antiderivative". We should also not use the integral sign for antiderivatives, but unfortunately there is no well known alternative. :-(

Using the same terminology and symbols for antiderivatives as for (definite) integrals is confusing, as here. It also makes the Fundamental Theorem of Calculus -- a very profound fact if there ever was one -- appear to be something trivial about getting rid of limits on an integral sign. But consider: one can compute a definite integral using an antiderivative ... whodathunkit?

thunk3• 17 years ago

See also Sussman's talk on how terrible math notation is:

http://video.google.com/videoplay?docid=-2726904509434151616...

anc2020• 17 years ago

Thanks for the link, its a shame Google video makes text unreadable though.

bkovitz7• 17 years ago

Try putting it on Full Screen and hit Pause as necessary. I was able to read most of the text, though it wasn't always easy.

onedognight3• 17 years ago

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.

thunk3• 17 years ago

Maybe, but a notation arrived at by an accretion of historical accident is not the way to introduce this stuff to beginners, inertia be damned.

Edit: Notation

slackenerny3• 17 years ago

type inference

Guy Steele's ‘Fortress’ language is headed in that direction.

davidmathers2• 17 years ago

Sussman's critique of mathematical notation is hollow.

I don't do enough math to have a useful opinion, but it strikes me that something as simple as 1 + 1 = 2 is ambiguous (does f = x+x or x+1?) and using unambiguous notation would be tedious and useless.

Unless you're trying to turn it into a program. I sometimes think of programming as the opposite of math.

tome• 17 years ago

Actually, the notation is confusing even to professional mathmaticians. The notation used in differential geometry is the archetype of this phenomenon; it's sometimes incredibly hard to work out what the simple-looking notation means.

In fact, differential geometry has been described as the study of objects "invariant under a change of notation" playing on the lack of clarity in this field, along with the standard description of a tensor as an object "invariant under a change of coordinates".

See, for example, the introduction to this piece: http://www.math.jhu.edu/~yrubinst/RiemGeom/Thesis-pp-23-30-a...

onedognight3• 17 years ago

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.

[1] http://images6.cafepress.com/product/96599546v11_350x350_Fro...

slackenerny3• 17 years ago

see them taught

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.

edw519• 17 years ago

When I first saw this, it was the beginning of the beginning:

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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.

jhancock• 17 years ago

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 ;)

Confessions of a Math Idiot

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