每个理论都存在于某种物理载体之中Every Theory Is Held Inside a Physical Substrate
Naval：我那个解决芝诺悖论的办法算是泡汤了。芝诺悖论说，你要想到达某个地方，得先走完一半；而要走完一半，又得先走完四分之一；如此类推，所以你永远到不了。
绕过它的一个办法是说，即使是无限多个东西相加，它们的和也可以是有限的。你把无穷级数展开求和，我们很早就学过它会收敛。我还有一个想法是：你必须走完一个最小距离——普朗克长度——所以你最终会到达。那是一串有限的步骤。但你的意思是，我们就是不知道。
Brett：如果物理定律说我们能在一定时间内走完一米，那我们就会走完。而我们目前对物理定律的理解恰恰就是这样。所以，芝诺悖论只需一句话就能解决：我们能在这么多时间内走完这么远的距离。至于空间是否无限可分，这个说法并不涉及。
如果有人问：“空间是无限可分的吗？”我会回答：“是的。”他们可能会反问：“你怎么知道？”我会说：“广义相对论。”我怎么知道那是真的？呃，我不知道那是真的。但它是我们目前对时空最好的解释。然后他们可能会展开讨论：“嗯，如果空间无限可分，那芝诺悖论就原封不动地回来了。”我会说：“不，一个简单的实验就能驳倒它。”
所以，我们不知道实际情况如何，但如果真的存在无限多个点，我们就能穿越所有这些无限的点。芝诺悖论属于纯数学的领域。但我们并不生活在纯数学的世界里；我们生活在物理的世界里。如果物理学说我们能在有限时间内穿越无限多个点，那么无论数学怎么说，我们都会做到。
Naval：每一个数学理论都栖身于大脑或计算机这样的物理载体之中。你始终受物理定律约束，而这些纯粹、抽象的领域可能与现实没有任何对应关系。
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.