Infinities are irrelevant to the "paradox". If you
do introduce an
infinity by creating an infinite
sum, well we all know that the series
converges, and the maths is easy. But you don't need any of it.
Look at the race in ordinary finite terms. There's a start, a finish, and the space in between. The start is Achilles' line in the sand, or the arrow leaving the bow, or whatever. The finish is the tortoise, or a ribbon stretched between poles. It doesn't matter. The "race" has three components, but you don't do three things:
- leave the starting line;
- run along the racetrack;
- cross the finishing line.
These aren't three things, which you do in that order. There's only one action, the
running. This has a beginning and an ending. But they're part of the action of running.
Similarly, when you run the whole racetrack, you progressively run one-third, two-thirds, then all of the
distance. But you don't do three different things,
- run the first third;
- run the middle third;
- run the last third;
It's not an obstacle course where the three thirds are distinct phases. It's just that, whenever you run one distance, you can, if you choose, look at it and mentally
divide it into a first part, a middle part, and an end part.
You can perform this subdivision before, during, or after the event: you can glance back and think "That tree was where I was half-way", or study a map and think "There's a fire hydrant at the three-quarter point, so I'll know where I am when I get to that". Or you can do it physically instead of mentally: you can measure the distance and scratch a line in the sand or plant a golf flagpole at the half-way point.
You can divide up the distance any way you like. This doesn't split the task up into multiple tasks.
The race organizers can put banners over the route saying "one third done", "two thirds done", and "FINISH". In the middle third they might put small white poles to divide that third into four parts.
Now to return to the familiar formulation of the paradox. You first run half way. Now all you have to do is run the other half.
Or you run half way then keep on going and run another quarter of the way. You've covered three-quarters of the distance. What remains to be done? This: run the final quarter.
You don't have to get out more flags, or scratch more lines. You don't have to divide the remaining quarter into two eighths. It's only a single piece of one quarter. 1/2 + 1/4 doesn't yet add up to 1, but 1/2 + 1/4 + 1/4 does.
If you happen to be fifteen-sixteenths of the way along, you're not finished yet. The sum 1/2 + 1/4 + 1/8 + 1/16 doesn't add up to the whole distance, nor can any such sum, however fine you cut it, because you haven't counted all the parts. The remaining one-sixteenth does make it add up to the whole.
The racetrack in this case is divided not into 1/2 + 1/4 + 1/8 + 1/16 + ...; but rather into 1/2 + 1/4 + 1/8 + 1/16 + 1/16. That's five sections altogether. You only ever have finitely many pieces in the sum: the ones you've already got through, however you choose to divide them, plus the ones that are still to come, however you divide them.
This explanation was put forward by Gilbert Ryle in his book Dilemmas.