Naval: There goes my solution for Zeno’s paradox, which says before you can get all the way somewhere, you have to get halfway there. And before you can get halfway there, you have to get a quarter of the way there, and therefore, you’ll never get there.
One way to get past that is to say even a series of infinite things can have a finite sum. You run the infinite series and sum it, and we learn pretty early on that it converges. Another thought I had was that you have to cover a minimum distance, the Planck length, and therefore you will get there. It’s a finite series of steps. But you’re saying we just don’t know.
Brett: If the laws of physics say that we can cover one meter in a certain time period, then that’s exactly what we’ll do. And our current understanding of the laws of physics says precisely that. So Zeno’s paradox is resolved simply by saying that we can cover this space in this amount of time. It’s silent on whether or not space is infinitely divisible.
When someone asks, “Is space infinitely divisible?” Then I would say, “Yes, it is.” They might turn around and say, “How do you know?” And I would say, “General relativity.” How do I know that’s true? Well, I don’t know that it’s true. However, it is the best explanation that we presently have of space-time. And then they might get into a discussion about, “Well, if it’s infinitely divisible, then you’re presented with Zeno’s paradox all over again.” And I would say, “No, you refute that by a simple experiment.”
So we don’t know how it is, but we can travel through all of these infinite points if, in fact, there are infinite points. Zeno’s paradox is about the domain of pure mathematics. But we don’t live in a world of pure mathematics; we live in a world of physics. And if physics says that we can transverse an infinite number of points in a finite amount of time, then that’s what we’ll do regardless of the mathematics.
Naval: Every mathematical theory is held inside a physical substrate of a brain or a computer. You’re always bound by the laws of physics, and these pure, abstract domains may have no mappings to reality.