Achilles and the Tortoise

Miscellaneous History
3

I guess a restatement of the paradox would be:

  1. The limit of an infinite series is never actually reached by any finite partial sum. [Laws of Math]

  2. Any measurable action we perform on this Earth can be subdivided [conceptually] into an infinity of partial actions, each also measurable, whose sum [in the sense of the sum of an infinite series] is the measure of the original action. [For it is valid to create a model of the universe that makes sense to us, and this model is perfectly sensible].

  3. But in the real world, only half our model is applicable. For “subdivision” in the sense of 2 is real, but “addition” in the sense of 2 isn’t.

This is because subdivision in our minds is possible, being merely a concept, and this subdivision accurately reflects reality.

But summation happens in the real world, where it is not possible to perform an infinite summation, neither by humans nor by nature. Nature is not a cosmic glue that melds all the infinite partial actions into one. Therefore no summation is possible. Therefore the tortoise never catches the hare.

So what’s the flaw? The flaw is in assuming his model makes sense. Slicing something up into a finite number of pieces makes sense to us, and we justified to feel we know the laws that apply to finite slicing, because we see countless examples of it every day of our lives.

But we have never seen anything divided into an infinity of pieces. Therefore we have no basis for assuming we know the laws that apply to infinite slicing in the real world [even granting that such a thing is possible]. So we have no basis for the assertion that nature is not a glue that melds infinitely many objects together. On the contrary, we see that it is.

EDIT: Taking it a step further, when we slice up something like a foot race into infinite pieces, implicit in the very act of slicing is the assumption that the sum of the slices is the original object. Otherwise we would not be justified in slicing it in the first place.

For example, when we physically slice a salami, we can argue along the lines of Zeno’s paradox thusly:

The whole is equal to the sum of its parts. Therefore all these slices of salami are equal to the original salami. But you can make thirty sanwhiches, say, out of those slices, but you cannot make even one sandwhich out of the original salami, because it is too big to fit into a sandwhich.