Imagine the following situation:
A runner is trying to run 1km. Before they can run 1km, then must run 0.5km. Before they can run 0.5km, they must run 0.25km. Before they can run 0.25km, they must run 0.125km and so on. This would require completing an infinite number of tasks, which Zeno claimed to be impossible.
This naturally divides the space into intervals:
0.5-1, 0.25-0.5, 0.125-0.25...
Note that Zeno assumes that an infinite number of non-overlapping space intervals can fit within a finite space. But he seems to doubt that an infinite number of non-overlapping time intervals can fit within a finite time. Why the asymmetry?
Archilles and the Tortoise Transformation Solution:
Suppose Achilles is in a foot race with a tortoise over 100m. By the time Archilles runs 100m, the tortoise has advanced a further 1m. By the time he has run the 1m, the tortoise has moved 1cm. By the time he runs the 1cm, the tortoise has moved 0.01cm, ect. Thus whenever Archilles has arrived where the tortoise was, he still has further distance to go
(This is a better solution that I edited in)
If Achilles were to reach 200m, then given the speeds, the tortoise would be at 102m. If Archilles never reaches 200m, then we can conclude that a finite amount of space can be infinitely divided into intervals. This begs the question, why isn't time infinitely divisible into intervals as well? If we don't want an assymetry between space and time, then we'll accept this, which would mean there's no problem with an infinite number of intervals or instants occuring in a finite period of time. In fact, this will occur regardless of whether the arrow is at motion or rest.
(This was the original solution that I posted, but I now have a simpler one).
Let's simplify this and imagine that the tortoise moves half as fast as Archilles. To find when they intersect absent this paradox, let t be how far the tortoise has run and a be how far Archilles has run. So if they intersect, they would intersect at:
2t=a=t+100; that is t=100m, a=200m
Just to be absolutely clear, we haven't assumed that they actually intersect, just calculated where they will intersect if they do.
We can then transform the argument about Archilles always being behind the tortoise into one about Archilles never quite reaching the 200m. This then becomes almost equivalent to the previous paradox. The difference is that the distances are ...1/8, 1/4, 1/2 instead of 1/2, 3/4, 7/8... However, this is really just the Dichotomy Paradox in reverse.
If everything when it occupies an equal space is at rest at that instant of time, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless at that instant of time and at the next instant of time but if both instants of time are taken as the same instant or continuous instant of time then it is in motion
It may be useful to abstract this out to a discussion of a mathematical function (such as ) to avoid becoming tangled in metaphysics.
Even though the arrow moves zero distance in each instant, it doesn't mean that the speed is zero as 0/0 is undefined. We simply need to define speed using limits instead.
The problem setup assumes that a time interval can be divided into instants, which is equivalent to saying that we can aggregate instants to form a proper time interval. If we don't want to create an assymetry between time and space, we need to allow that points in a space can also aggregate to from a proper space interval.
Suppose we represent spatial movement over particular periods of time as segments of a line. Whenever we choose an instant as the time period, the segment will be a point. Suppose we aggregate the spatial segments corresponding to each instant in the time period. We might naively assume that the aggregate of 0-size points must also have size of 0, but we argued in the previous section that these can aggregate to a proper space interval.
Hopefully you agree with me that these resolutions are more satisfying than just invoking geometric series. I'm still working on a write-up for the arrow paradox. I feel my current write-up is 80% there, but still has a few loose ends.