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.

Even a ... Nevermind

 

You don't know anything about algebra. Nothing, nada, zilch, absolutely none is what you know. Ok, now that I got that obligatory math teacher berating out of my system, let me backtrack a bit and let you in on one of my big pet peeves. Like the boring goofball I am, I like to peruse the various math related internets. Most of what I find is enlightening, fun (to me at least), or at the very least correct. Every once in a while I'll stumble on a page or facebook group (*massive groan*) that touts something about paradoxes and the weak foundations of mathematics. "Oh really?" I think, sounding interesting I continue reading, thinking they must be referring to all the problems and paradoxes found in set theory (the actual foundations of math). And here I read something like: "Oh math is so crazy, through logical foundations crazy paradoxes result. Here follows a list of paradoxes proving that the foundations of math are shaky at best." Then they list some paradoxes like they promised (they're not jerks after all, just dumb), and most of them are your typical logical paradoxes, you know the ones like: "All bandits are liars," said the bandit. So on and so forth, which are all fine and dandy until they insist on adding this one: