Philo Toy: Is space infinitely divisible?

Posted: Fri Feb 15, 2008 5:08 am

Since we were discussing philosophy as primarily an amusement, I thought this was amusing. i.e. I couldn't decide one way or the other.

I was looking into this maybe a year ago and ran across two arguments, one for and one against, that both seem to make sense.

Space is infinitely divisible: Motion of any kind is dependent on plasticity. Plasticity depends on an object being divisible into some number of objects that make it up. But for those objects to yield motion, they too must be divisible. If you go to the bottom level, and there is no plasticity, then there is no motion. If there is no motion, objects made of objects at that level cannot exhibit motion either. That there is motion, then, implies infinite divisibility into ever smaller objects.

Space is not infinitely divisible: This argument takes Xeno's paradox at face value. Assuming space is infinitely divisble, Xeno's paradox presents a real problem. A fired arrow cannot go 1 foot because it must first go a 1/2 foot, and before this it must go a 1/4 foot, to infinity. The way out is to say that space is not infinitely divisble. Space comes in quanta. At the quantum level, there simply is no motion between points. Motion is a series of quantum leaps.

I personally don't like either argument, but I also don't see how one is more problematic than the other.

Posted: Fri Feb 15, 2008 7:30 am

I think the problem with Zeno's (it usually is written with a 'Z'; don't confuse him with the Scientology guy) paradox is as follows. If the arrow is travelling at one meter per second, then it travels one meter in one second, 1/2 meter in 1/2 second, and so on to infinity. Also, once you get to 1/infinity, there is an infinite number of such divisions, meaning you multiply the distance travelled by infinity/infinity=1.

But I think the "infinitely divisible" thing is problematic too, in light of quantum physics.

Posted: Fri Feb 15, 2008 7:52 am

Interesting stuff, Kmisho. Thanks for bringing it.

[quote= "kmisho"]Space is not infinitely divisible: This argument takes Xeno's paradox at face value. Assuming space is infinitely divisble, Xeno's paradox presents a real problem. A fired arrow cannot go 1 foot because it must first go a 1/2 foot, and before this it must go a 1/4 foot, to infinity. The way out is to say that space is not infinitely divisble. Space comes in quanta. At the quantum level, there simply is no motion between points. Motion is a series of quantum leaps. [/quote]

I am not studied on this, but I will take a crack at it.

I think the problem here is simply - and I mean simply - one of vantage point. As humans, we have a particular vantage point owing to our size. If we were dinosaurs, we'd have another - bigger - vantage point, and if we were beetles, we'd have another - smaller -vantage point.

Well, the measrement of 1 foot is a wholly human made thing and is NOT precise but to a certain point. (That is, however precisely we measure a foot, there will always be a 'deep enough down' level at which our hitherto measurement becomes imprecise).

Now, commonsensically, we all know that an arrow will pass the foot marker at some point. And Xeno would see this as well, IF HE ACKNOWLEDGED THAT THE MEASUREMENT OF 'FOOT' IS HUMAN MADE AND ONLY WORKS AT THE HUMAN VANTAGE POINT (or one similar to it; a newt, though, would have wholly different measurements owing to its size).

I am not sure it is a paradox unless you take the extreme stance that the mearurement of 'foot' has to be 100% precise all the way down. It is not, and it will not be. But, it is precise enough for us AT OUR VANTAGE POINT. If we ever needed to design a more precise measurement (for, say, measuring something in a microscope that magnifies 2,000x, then we wuld create that measurement. (And still, it would never be 100% accurate, but it would be accurate enough AT THAT VANTAGE POINT.)

So, I am not sure I see this as a paradox as much as a problem with Zeno's premise (probably unconsciously held) that in order to be unproblematic, a measurement has to be precise to the 100th percent ALL THE WAY DOWN.

Posted: Sat Mar 08, 2008 1:11 pm

In the classic sense, Xeno's paradox evaporates once you introduce modern math, where we have devised ways to handle infinite sums and limiting cases of infinitesimals.

Xeno's error is assuming that the sum of an infinite series must be infinite.

We can easily show that 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... [to infinity] = 2.

So the sum of that infinite list of progressively smaller time intervals is easily show to be quite finite, so there is no paradox, merely a lack of mathematical insight on Xeno's part. It is 'solved' by high-school math.

We should always remember that, however advanced the Greeks were in science, math, philosophy, for their time, much of their stuff is way obsolete.


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