My mathematics teacher says that a line is an infinite sum of points. I disagree and I think that she must not have thought it through very deeply. I argue that instead that though a line can be theoretically be described as a sum of smaller lines that in no way can a line be said to be described as a continuity of points because a point is not in any way extended. If a line has an atomic unit then that unit must have the same properties as the line itself and a point has an altogether different property than a line. (That you can fit a point inside a line only shows their common property of spaciality, it does not demonstrate that a line is in any way composed of points) I hope you understand what I am saying. Do you think I am right?

I understand well what you're saying. Points have zero extension, and lining up a bunch of them won't get you beyond zero extension. It's like adding up zeros:

0+0=0

0+0+0=0

and so on. There's no reason to think that adding infinitely many zeros together would get you anything other than zero. And likewise with the lining up of points.

But when we are dealing with infinities, things are often tricky and counter-intuitive. So let's see whether we can construct an argument for your teacher's conclusion. Consider this. We begin with a line -- let's say it is 32 inches long -- and we divide it into two equal segments, these again into two equal segments, and so on. Dropping the inches, we can write this as follows:

1*32 = 2*16 = 4*8 = 8*4 = 16*2 = 32*1 = 64*1/2 = ....

Here the number before the "*" signifies the number of segments and the number after the "*" signifies the length of each segment.

Now the question is this. If we keep dividing an infinite number of times, then what is the extension of the resulting segments? In particular, is their extension zero or is it greater than zero?

Your answer is clear. The extension of these segments cannot be zero. For, it were zero, then these segments could not form the original line. Even infinitely many points of zero extension would not add up to anything of greater-than-zero extension. Therefore the extension of each of these segments must be greater than zero.

Your teacher, I imagine, might say this. Suppose that, after infinitely many divisions, the extension of the segments were indeed greater than zero. Then, lining up these infinitely many segments would get us an infinitely long line. (Multiplied by infinity, even the tiniest non-zero quantity becomes infinite.) But this is not the finite line we started with. Therefore the extension of each of these segments cannot be greater than zero. Since it cannot be smaller than zero either, it has to be zero.

Now it seems we are in real trouble. It seems that neither answer can be right. But is there any third possibility? Can we perhaps stipulate that there is some "infinitesimal" length that is greater than zero and also smaller than any finite length -- and then assert that points actually have infinitesimal extension rather than zero extension? Sure, we can say this. But even this "solution" leaves us with a difficult question. How long is a line that consists of infinitely many infinitesimal segments? Some might be tempted to answer: "32 inches." But what if our infinite division is imagined on a 17-inch long line? Are we to postulate many -- infinitely many -- different infinitesimalities, each yielding a different finite result when multiplied with infinity?

I leave things here, in this sorry state, but let me reiterate that infinities are pretty weird, and what we saw here isn't unusual at all. Thus consider the comparison of two infinities in the apparently simple question of whether there are more positive integers (1,2,3,4,...) than even positive integers (2,4,6,8,...). Here as well there seem to be compelling arguments on both sides. Those who say "yes" can say that there are additional integers, the odd ones, included in the first set and not in the second, and nothing included in the second that's not also in the first. Those who say "no" can point to a one-to-one mapping that will find a matching even positive integer for every positive integer (e.g., 1 is matched by 2, 2 is matched by 4, and so on). In this case, mathematicians have decided to take the latter argument to be decisive. But must we go that way?

One lovely book discussing many of these problems is Shaughan Lavine: Understanding the Infinite (Harvard U.P.). You will also find some fascinating thoughts on mathematical arguments in Ludwig Wittgenstein: Remarks on the Foundations of Mathematics.

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