Achille’s paradox… why can’t Achilles just reach the tortoise?
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Because we are arbitrarily changing the definition of time flow, rigging it to suit our needs. In the Achilles or Zeno’s paradox, we’re running time through something like a 1/2^x function (don’t let the math throw you, it just simply means that each step of the “catch up game”, we add less time… first we add half a second to our timeline, then a quarter of a second, then an eighth of a second, then a 1/16th, etc)…essentially we rig it so that we never tack on enough time at the end to allow Achilles to pass. It’s almost like taking a photo of a ball in the air and saying “why is it that the ball never reaches the other side of the frame, no matter how long the photo sits on the table?” The answer, again, is that we are rigging the passage of time…in the case of a photo, we are freeze framing time; in the case of the paradox, we are simply taking exponentially smaller increments of time up up to hard limit (called “asymptote” in math) that we, by definition and design, will never reach. We are saying, “let’s figure out a way to keep adding on smaller and smaller increments of time to the end of our time total, in such a way that we’ll get closer and closer to a particular number (the point where Achilles would catch up with tortoise) but never reach it”. This is easily done mathematically, but it’s just gaming the system. It’s just a trick. It’s literally the same trick as a freeze frame, just playing with the numbers to make it seem like time is actually moving a very teensy bit, but never enough to pass a certain point. In reality though, time does not flow in exponentially decreasing units, it flows evenly at one second per second, right past the arbitrary limit that would make achilles never pass the tortoise.
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