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.

Hey, his argument seems sound at first, in fact when my roommate landed it on me I didn't have a response other than "Um right, well that's neat and all, but we all know lines have length." Which is true in a physical (non mathematical) sense because you can take a fucking ruler and measure every line out there. So, the conclusion is definitely false, meaning that something's wrong, either geometry's foundation is sketchy or there's some bad math going on in his argument. If you've read my previous entry on infinity then you can probably guess where I'm going with this. Two problems come to my mind immediately. One, we know that if a line is cut into two pieces then the length of the line is the lengths of the two pieces added together. Can we assume this is the same for points? In other words, we know the property is true for line segments, that doesn't mean it's true for points at all, and just assuming so is a big no no. Second, remember me saying that when infinity's involved that anything can happen? When an infinite number of objects cram together there's no telling what happens without giving it some serious thought. Okay, these two problems are actually fairly interrelated, it comes down to not really knowing what infinity can do to a system, or just assuming that an infinite number of things can be treated as a finite number of things. So, I'm going to show you how it's possible for an infinite number of dimensionless objects to cram together to give us a line.

Ok, take a line, and for simplicity take a line with length 1. Now split it into n equal pieces, so each piece has length 1/n, right? Think about it for a second, 1 = 1/n + 1/n + ... + 1/n (n times). Alright,so we're adding 1/n up n times, makes sense. Let's factor out 1/n to get 1 = (1/n)(1 + 1 + 1 + ... + 1). And what happens when we have 1 added up n times? We get 1 = (1/n)n, which we all know is true algebraically. Does everyone agree so far? So this works for any n, now keeping that in mind I want to take the limit of both sides as n goes to infinity.

Now what is this saying? Well the limit of 1/n as n goes to infinity is zero (basic calculus), and n as n goes to infinity better be infinity right? So we basically have, as n goes to infinity, zero times infinity, what is that? If you said zero then you're wrong, or right; same if you said infinity, or some constant. Thing is it's undefined, that's because using calculus you can use simple examples to show that there are cases where they're all true. Take these,

This is something fairly basic you learn in Calculus 1, that zero times infinity is actually an indeterminate form of the form infinity/infinity, and it can actually be a constant, zero, or infinity (which is why it's undefined, if it were defined they'd all go to the same number every single time). I submit that above I've shown that it's possible to take an infinite number of points and cram them together to get something positive. How exactly? Well splitting up the line into n pieces each of length 1/n, what happens when we take n to infinity? The number of segments approaches infinity and the length of each segment approaches zero (1/infinity). Does that sound familiar? As n approaches infinity, the segments approach points, i.e. you're shrinking the pieces down to points. And when this happens you get the total length of the line equaling zero times infinity, which can totally equal a positive quantity.

Now why did I do this roundabout mess? I needed to use only steps that I knew were true. For instance, I don't know how to equate the length of a line in terms of its points (as I mentioned above), but I know how to do it in terms of line segments. So splitting the line into smaller lines I know how to deal with. Further, after that I apply some calculus, which we can all agree calculus is valid correct? The beauty of limits is that we don't care about what happens when it finally gets to infinity, just when it gets extremely close. So if what we found is true for all n, then it'll still be true when we rock out close to infinity. So the philosopher just assumed that adding an infinite number of points is just adding up zeroes, but that's not what you're doing. No, you're adding up an infinite number of things and multiplying by something that goes to zero. Our philosopher was distributing that zero before it actually got there, that's all. Typical newbie mistake, we all make them sometimes.

This should remind you of the development of integrals in Calculus 1, which is the same exact concept just with area instead of length. Oh, and you'll probably need to know some calculus to follow this argument. Sorry, I should really say that kinda stuff at the beginning eh?