The Problem With Points And Lines

I was presented with an interesting question many years ago as an undergraduate. It was apparently a theory by some philosopher or other who had a claim saying why he thought math was kinda full of shit, well ok he just had an interesting paradox. I forget the name of the philosopher, but his idea went like this: Take a line (segment), it has a length correct? Well a line, we're told, is a collection of an infinite number of points crammed all together. Now, what's the length of a point? We're also told a point is dimensionless, i.e. it has zero length. Therefore, the length of a line should be zero correct? Since a line = an infinite number of points, therefore the length of a line should = 0+0+0+ ... right? Thus all lines have length zero and thus everything fucking falls apart. This, the philosopher claimed, displays that there's something wrong with the foundations of mathematics, i.e. how we treat points and lines.

Say What?

So, I mentioned before about a problem dealing with Buffy and Batman, and yes it's quite epic. The math itself isn't all that involved really ... well for me anyway. It's just the write up will be tedious, as well as the figures required, and not to mention the animations. Did I just say animations? You're damn straight, I'm pulling out all the stops ... eventually. For right now though I'm a bit too busy with other stuff to devote the time required for such a feat, and I'll probably stay this busy for a good long while. No worries though, I have plenty of things to fill the void until then. Like, for instance, explaining all my freaking steps from my last post. Those of you who are great at math and have enough background probably followed what I did ok. Those of you who are new to sigma notation and infinite series might benefit from a little explanation. There's a lot of little tricks in there that may be really useful to you. I recommend the following to those of you in Calculus 2 or a higher level probability class, ODE, PDE or any higher level class that requires nasty sigma notation funkiness (Ring Theory when you talk about polynomials for instance). Anyone else? Eh, just hope my next post has something more geared toward your level. Sorry, that just comes with the territory. Anyway, here we go:

And Beyond?

You know nothing about infinity. You probably think you can add it to your argument in order to win, or add one to it to win even more. Nope, nothing about infinity, but I know everything about it (ha!) so allow me to enlighten you. There's this statement you might have heard before, that if an infinite number of monkeys on an infinite number of typewriters type for an infinite amount of time, then at some point they will produce the complete works of Shakespeare. This is an idea that comes from probability theory, and what infinity does to it. Because when you let the time line approach infinity then the probability that anything will happen approaches 1 (i.e. 100%). Therefore we could actually reduce the goofiness down to one monkey on one typewriter. Thus if Bubbles (yes I'm naming it Bubbles) types for an infinite amount of time then at some point it'll produce Hamlet. So on a long enough time line any damn thing can happen, anything.