It's all about the deep questions.
The Racetrack Paradox
In this paradox Zeno argues that all motion is impossible, which is a pretty substantial conclusion. What kind of reasoning could possibly lead someone to this completely counterintuitive conclusion? Well here is the gist of the story that Zeno spells out:
Imagine Achilles wanted to complete a race, lets say it was one mile long for convenience. It’s clear that if he wants to complete the race, he will eventually have to make it to the half way point. After he finishes the task of traversing the first half mile, he must then reach the half way point between 1/2 a mile and 1 mile. So, he must now finish the task of getting to the 3/4 mile point. However, continuing along this line of thought, Achilles has infinitely many tasks to complete! He has to traverse 1/2 a mile, 3/4 a mile, 7/8 a mile, 15/16 a mile, etc. Each task will also take a finite number of time! Since it is logically impossible to complete a series of infinitely many tasks (each requiring a finite amount of time), Zeno tells us that Achilles can never finish the race. Furthermore, this argument can easily be generalized to anything moving from point A to point B where A and B are distinct.
The Racetrack Argument
1. If something is to move from A to B, where A and B are distinct, then it would have to complete an infinite number of tasks.
2. It is impossible for anything to complete an infinite number of tasks.
3. Therefore, nothing can move from A to B, where A and B are distinct.
So, now we should take a hard look at the premises.
One misgiving that one might have is with the word “task”. In this context, it does not mean that a person literally contemplates going to every one of these infinitely many points, one at a time. It simply means that, regardless of the person’s intentions, he/she (or it) must pass these points. Using the previous story, it is simply a fact that Achilles must pass through 1/2, 3/4, 7/8, 15/16, etc. of a mile to reach his goal. Furthermore, this whole series must be completed, because if it were not then Achilles would have stopped before he reached his goal. The one other thing that this premise assumes is that space is continuous rather than discrete. If it were discrete, then there would not be infinitely many points to “get to” before every goal. If premise two were to be accepted with no qualifications (which it won’t), then one would be forced to accept that either motion is impossible or that space is discrete. Most people would probably pick the latter. The argument for the discreteness of space would look something like this:
1. If space were not discrete, then motion would be impossible. (by The Racetrack Argument)
2. Motion is possible.
3. Therefore, space is discrete.
This is the really interesting premise. Is it impossible to complete infinitely many tasks? It seems to be perfectly possible since we do it every time we move, but that would be begging the question. One argument in support of The Racetrack argument can make is the following:
There are certain reading-lamps that have a button in the base. If the lamp is off and you press the button the lamp goes on, and if the lamp is on and you press the button the lamp goes off.
Suppose now that the lamp is off, and I succeed in pressing the button an infinite number of times, perhaps making one jab in one minute, another jab in the next half-minute, and so on, according to Russell’s recipe. After I have completed the whole infinite sequence of jabs, i.e., at the end of two minutes, is the lamp on or off? It seems impossible to answer this question. It cannot be on, because I did not ever turn it on without at once turning it off. It cannot be off, because I did in the first place turn it on, and thereafter I never turned it off without at once turning it on. But the lamp must be either on or off. This is a contradiction. (Thomas 1954; cited in Gale 1968, p. 411)
A potential response is to say that the argument is simply invalid. The supposition that the infinite series has been completed does not lead to the absurdity that the lamp is neither on nor off. Nothing follows from this supposition about the state of the lamp after the infinite series of switchings. Without any strong arguments for premise 2, however, surely it would be best to reject premise 2 in light of the unwelcoming conclusion that follows.
The standard mathematical “answer” as to why premise 2 is false is present in the mathematics of infinite series. Suppose Achilles were going at 1 mile per hour. Then he would finish the first task in 30 minutes, then the second task in 15 minutes, etc. 30+15+7.5+3.75+…=60. According to the mathematics, if one were able to finish the infinite series, then one would finish in 60 minutes. In other words, what the series approaches is 60 as one continues extending the left-hand side. Mathematics, it seems, does not adequately tell us whether completing an infinite sequence of tasks is possible or not.
It does seem, however, that premise 2 is the most likely candidate for suspicion. In the absence of sound arguments for its impossibility and basically a reductio ad absurdum for its possibility (If it were not, motion would be impossible. But, motion is possible.), one is perfectly entitled to skepticism.
There seem to be two compelling arguments for two inconsistent thesis that are closely related to this argument. Let S be the series 0, 1/2, 3/4, 7/8, etc. Let S’ be 1.
1. Passing through all the points in S is sufficient for reaching S’.
Argument: Imagine someone occupying every single point in the S series, but not any point outside it. Where would he be? He can’t be at any point in S since for every point there is a point after it. He cannot be in between any two points in S for the same reason. By assumption, he can’t be at any point external to the S series. But, these are all the possibilities. Therefore, the assumption is wrong, and he did move “outside” the series into S’.
2. Passing through all the points in S is sufficient for reaching S’.
Argument: S’ is further to the right than any member of the S series. So occupying every point in the S series does not necessitate someone going as far to the right as S’.
Clearly some clarifications of how our mathematical concepts are related to our physical conception of space are in order here.