Thanks a ton. No blog or site to plug? C’mon, we need more guys like you to help out chumps like me

Austin - Ahh yeah I guess I hadn’t explicitly reasoned through the application stuff being left-associative. I think that makes it much more clear. Your explanation is proably a hell of a lot better than mine. I tend to say in a page what can be said in a sentence.

Rez - Yeah, reading about Javascript’s functional-abilities is what prompted me to try this stuff out in AS3. C and C++ in particular allow you to have pointers to functions. They can be REALLY useful, but still tend to be pretty limited compared to these modern day languages which treat functions as “first class citizens”. As well, there’s support for ‘variadic functions’ in those languages, kind of like the ‘rest’ operator in AS3. In C++, in particular, these variadic functions are considered really bad form because of the lack of type safety, amoung other things–they’re a massive haven for runtime issues. In AS3, a runtime error nets you an accurate explanation and a line number of where it occurred. In C/C++, a runtime error nets you nightmares for the next ten years.

The Fraction class could be subclassed to a ComplexNumber representation without much trouble. These classes would give the most accurate floating point estimates of their numerical values. Taking this to the extreme, however, and allowing the introduction of irrational constants would require a means of representing things like “π^2+3*e÷4″ as an associated cluster of numerical objects, whose reduce() methods would be monstrous, at which point any programmer would say, “Forget it, let’s use floating point.”

I think that hesitation, by the way, is largely due to Flash developers’ tendency to stick with graphical development environments whenever possible. ABCDump runs in a shell, right? That can be intimidating.

However, what’s really astonishing is this idea of chaining variables to an event handler. This gives us the power to associate new variables with objects, essentially giving those objects new properties and functions (accessible via the closure) without bothering to redefine them (by subclassing them, for instance). It is in some sense a powerful form of composition.

All lambda calculus functions (in the simple lambda calculus, of course) only take one argument, ever. For the identity which just gives you back whatever you use, you can define it as so:

(\x. x)

When applied to any argument, this just returns the argument.
How do we get functions that take multiple arguments? Return other functions!

This takes two parameters:

(\x. (\y. y x))

So we can apply it to two parameters and it will apply y to x. It’s worth noting this works because of the associativity of function application in a functional language, that is:

f x y

Is actually

(f x) y

Since function application is left-associative. Also, types are right associative, so the type

Int -> Int -> Int -> Int

Is actually

Int -> (Int -> (Int -> Int))

The reason types are right-associative is really because of currying.

I think when you explain it this way, partial-application and currying make a bit more sense and may make things a bit more lucid without complicating it too much. Your explanation is perfectly valid, however.

Zack - Should be fixed now… I had changed the capitialization on one of my directories and didn’t go around updating links. Unfortunately the example is REALLY REALLY BAD. I had just suggested the FFT to a guy who asked about beat detection on Newgrounds, and someone else basically said “No that’s dumb and useless” and I just couldn’t let it slide, so I made a quick demo in an hour or two. THEN when I threw this blog up, I decided to slap up a bunch of posts I’d made on Newgrounds as ‘filler’ until I got around to updating regularly (which I haven’t yet and probably never will). Pretty good story, huh? At any rate, this is one of the articles I like seeing the least, just because the results were so bad–just good enough to prove my point to some douchebag on a Flash forum who probably never read the post anyway, but not much else. The link I gave at the end of the article is a more serious attempt by someone else.

Also, sorry about the weird spacing in the code–it shows up aligned properly in my IDE. I know it never translates anywhere else properly, but I’ve been too lazy to fix it, heh. Like I said, filler!