I can understand discrete space and time and if it’s really how universe works than this paradox is quickly resolved, but there is another solution which my brain can’t understand. How does continuous spacetime work and solve this paradox?
In: Physics
there is no perfect solution that fully resolves the paradox. while modern mathematical constructions of similar infinite limits provide some clarity, they dont necessarily solve the philosophical part of the paradox
often solutions involve disputing the assumptions of the paradox. as you noted, insisting that time and space is really discrete would resolve it, although this causes other issues. another dispute is that its incorrect to analyze an object at some specific exact moment in time the way that the paradox does, as doing so does not capture the nature of movement
In mathematics this Paradox can be resolved using the idea of limits.
To explain I’ll use a variation of the Paradox where a runner is trying to cross some distance. Before they cross the full distance they need to cross half of the distance. After that they cross half of the remaining distance, and so on. We can write this as 1/2 + 1/4 + 1/8… Ancient mathematicians assumed that since the expression has an infinite number of terms it must be impossible to add them all up.
In modern mathematics we can observe that as you add more and more terms the value of the sum gets closer and closer to 1. It never goes above 1 and it never stops getting closer to 1. In mathematical terms, as the number of terms goes to infinity the limit of the sum is equal to 1. So even though the series has an infinite number of terms the sum of the series is still a finite value.
This same logic applies to the Paradox of Achilles and the hare. Practically we know that Achilles will catch up to the hare after traveling a finite distance in a finite amount of time. The fact that we can describe this process using an infinite number of intermediary steps doesn’t change the final outcome. It simply demonstrates that it is possible to pass through an infinite number of steps in a finite amount of time.
The Zeno paradox, or Achilles and Tortoise of you prefer, has nothing to do with continuous space and time. It is a faulty representation of the problem. There is no paradox if you don’t take this ridiculous infinite fraction representation. What you should conclude is that this representation isn’t correct, not that Achilles can’t pass the Tortoise.
Consider the time that it takes for Achilles to get to where the Tortoise started. This is some time T. Rather than step forward in smaller and smaller time steps, step forward by T again. Achilles passes the Tortoise, problem over. Yes, there are summations of fractional times that can’t ever reach 2T, but that’s no evidence that 2T is longer than the race might reasonably last. Simply that those representations are wrong.
The Zeno paradox is based on the two assumptions: “you can cut time and space infinitely” and “sum of infinite number of values is always infinite”. You correctly found that the first doesn’t hold in discrete time and space. In the continuous time and space the second doesn’t hold.
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