On July 3, 2016, I proposed what I thought was a completely original solution for Zeno’s paradox: a “minimum distance” in the real universe that cannot be subdivided:
You have probably heard of Zeno’s paradoxes, one of which is that to get anywhere, you first must travel half the distance, and then half the remaining distance, and then half the remaining distance, and so on. According to this theory, you never get there.
I have seen various solutions to this, but I am not sure anyone has proposed mine. Perhaps they have, but here it is anyway:
My proposed solution is that there is a minimum distance in the universe, which cannot be subdivided into smaller distances.
Imagine my amazement to learn that, at about the same time, a professional philosopher was bouncing the same theory off of a mathematics professor, who agreed with him!
Embedded for your listening pleasure is a discussion between philosopher Steve Patterson and Dr. Gary McGuire, the head of Mathematics and Statistics at the University College Dublin.
A little background: Patterson proposes a different description of Zeno’s paradox, in which you put together a pie by first creating half, then adding a fourth, then an eighth, and so on. Does the pie ever get completed? This is what they are referring to when they talk about the pie.
The most relevant part is at 40:27:
PATTERSON: If that’s true, then does that not mean that Zeno’s paradoxes are not solved by calculus? Because the claim is not that the runner will get ever so close to the final point, but that the runner will actually complete the race. That ultimately, the pie will ultimately be completed. Doesn’t that mean that Zeno, Zeno had a — was making a good point there?
DR. McGUIRE: Uh, yes. Yeah. Yeah. No, a really good point, I mean, I’m agreeing with you. I’m not disagreeing with you. So.
PATTERSON: So what do you think of this, this potential resolution: that the reason calculus does work in the real world is because reality is finite. It’s not infinitely divisible and therefore at some point the calculations terminate, and then, you know, you can complete the whole pie, and you would complete the race.
DR. McGUIRE: Well in the real world, as we were saying earlier, we don’t get into the infinite. So we have to approximate everything by a finite number. And so, in the real world, we would get, we wouldn’t be able to, if we were adding smaller and smaller and smaller pieces of the pie, we’d eventually have to stop somewhere. We can’t, we can’t get ever smaller and smaller and smaller pieces; we just can’t do that. So we have to stop at some smallest possible piece and then we add that in and then we finish the pie.
PATTERSON: But what about with something like distance? So could we say the same thing — that ultimately — this is what I think the resolution is to Zeno’s paradox, is that there is like a base, a base distance unit in the universe that you can’t actually divide in half. Because otherwise it seems like motion would be impossible. But if we’re — if there’s like a base unit of distance, then everything seems to resolve itself. Just like a base unit of pi.
DR. McGUIRE: I, yeah, I kind of agree — I agree with you. I think, in the real world, in practice, there is a base unit of distance, yeah.
I heard this in my car and almost involuntarily slammed on the brakes in surprise when I heard the bolded language — which, you’ll note, is the same as my own solution to Zeno’s paradox above: that “there is a minimum distance in the universe, which cannot be subdivided into smaller distances.”
Note that many commenters laughed at me in comments to the previous post — and yet this mathematics professor agrees with Patterson!
I am going to write Patterson about this rather bizarre confluence of thinking. I may have a hard time convincing him that I didn’t take my cue from him, since his podcast preceded my post by a little more than a month. But I never even heard of the guy before a couple of weeks ago.
Great minds think alike — and sometimes, so does mine!
P.S. If you listen to the whole interview, you’ll notice Patterson’s notion that numbers do not exist outside our conception, which I see as a kind of corollary to my hypothesis that — even abstractly and not purely as a “real-world” phenomenon — there is a smallest number (and a largest!) . . . but it’s beyond the limitations of humans to conceive of, or express. If that notion is right, it has real implications for the very concept of infinity.