Zeno's Logical Paradoxes

What follows is a paper I wrote for my Ancient Philosophy class... in hindsight I think I could have done better, but such is the naure of hindsight! Criticism is warmly welcomed! Enjoy! :)

Zeno’s Logical Paradoxes

As someone immensely interested in logic and its applications, I was immediately gravitated towards the paradoxes of Zeno of Elea. I thought his conclusions were particularly interesting because of how contradictory they were to experience; by saying that “there is no motion”, “the faster will never catch the slower”, and that “the number of a thing is both finite and infinite, if not one”, he captures the attention of his audience and forces them to think and question how the universe operates.

The “Racecourse” argument against motion from fragment [10] is what captivated me the most; it seems to actually be two arguments, the first one being the equivalent of a disjunctive syllogism (the first premise could be rewritten: -AvB):

1. A->B (If there is motion, the moving object must traverse infinity in a finite time)
2. -B (The moving object cannot traverse infinity in a finite time)
:: -A (Hence motion does not exist)

and the second, an expanded version:

1. B->(A->C) (if every stretch is infinitely divisible, then for an object to
transverse some stretch, it must be able to pass an infinite number of
halfway points before reaching the end)
2. B (every stretch is infinitely divisible)
3. -C (The object is not able to pass an infinite number of halfway points)
4. B->(-A) (if every stretch is infinitely divisible, the object will not reach the end)
:: -A (the object will not reach the end, motion does not exist)

Both of these arguments are valid, however, premise 2 of the first argument and premise 3 of the second (and, by extension, premise 4) shall be negated below, leaving the argument unsound:

Suppose a sprinter is running a given length (the actual length itself does not matter) at 10 meters per second. If you cut that distance in half as Zeno would, the sprinter would still be moving at 20 half-meters per second, if cut that further by half the sprinter is moving at 40 quarter-meters per second, and further again will yield a speed of 80 eighth-meters per second. No matter how far you take this, the actual distance and time does not change, only the base unit of measurement. Even though the length of the track may be infinitely divisible, the course itself is not infinite in length.

All we have proven so far is that the sprinter can reach the end of the course, given that he/she is already moving, and that the course is not infinite in length. Zeno would argue that the sprinter could never be put into motion in the first place, which can be refuted by examining Newton’s first law of motion:

I- Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it.

“Every object in a state of uniform motion” includes objects at rest such as the sprinter in position on the starting blocks, who is at rest because the net forces acting on him are absolutely equal and opposite. Should the sprinter contract his muscles with sufficient force, he will have created a state of unequal net force, resulting in acceleration from the starting blocks and a state of motion.

The “Achilles” paradox seems strange to me in light of Zeno’s “Racecourse” paradox, in that they contradict each other. In the “Achilles” paradox Zeno asserts that “the pursuer, before he can catch the pursued, must reach the point from which the pursued started at that instant, and so the slower will always be some distance in advance of the swifter” (fragment [11], page 28), which must mean that Achilles can reach the point at which the turtle previously was. This contradicts Zeno’s argument from fragment [10], which concluded that motion does not exist. Motion cannot be both existent and no-existent, rendering at least one of these arguments false, as has already been shown of the “Racecourse”.

Suppose that Achilles is running a 100 meter course at a speed of 10 meters per second, and that the turtle starts from the middle at a speed of 1 meter per second (which is pretty fast for a turtle!). After 5 seconds, Achilles will have reached the spot from which the turtle, which is 5 meters ahead at this point, began. But before the 6th second is up, Achilles will have overtaken the turtle; there is no logical trickery here, only simple math. Is the distance between Achilles and the turtle infinitely divisible? Yes. Does this mean that the distance itself is infinite? As was shown in the “Racecourse” paradox, no.

The same criticism from physics can be applied to “The Arrow”. Zeno argues “Since a thing is at rest when it has not shifted in any degree out of place equal to its own dimensions, and since at any given instant… [the] thing is in the place it occupies at that given instant, the arrow is not moving at any time during its flight.” (fragment [12], page 29). This can be interpreted two ways: the first, that at any specific point in time, the arrow can only occupy a specific point in space, which cannot be anywhere else than where it is, and the second, that an object “is at rest when it has not shifted in any degree out of a place equal to its own dimensions”, which I take to mean that unless an object has moved in relation to itself, it has not moved at all.

The first is correct in saying that an object occupies only a single point at a given instant, yet the position of the arrow at a time already past or a time still to come may be calculated by adding the net forces acting upon the arrow in a specific vector and multiplying by the difference in time between the present and either the past or future instant; the resulting number represent the spatial coordinates that the arrow occupied or will occupy. Stated simply, if the arrow was not moving, the math behind the physics would not work and predictions could not be made.
The second interpretation would violate physics: for something to have moved in relation to itself, it would have to occupy more than one point in space. But movement is always measured in relation to something else, usually stationary and most commonly the earth.

This concept of movement in relation to something else brings us to Zeno’s fourth and final argument against motion, “The Stadium”. Zeno seems to be arguing that if two trains are traveling past one another in opposite directions, they are moving twice as fast in relation to each than to some other stationary train. If such is the case, Zeno argues, it only takes half the time to reach the end of the stationary train compared to the approaching train, since the distance traveled compared to the former is half of that of the latter with their respective points of reference. This is because the latter (the approaching train) is a moving point of reference. There is nothing mystical or paradoxical about this.

Finally we come to Zeno’s arguments against plurality (fragments [3] to [5]), which I would symbolize and work through, but it would be superfluous because one premise appears to be blatantly false, the premise being: “If there are many, the existents are infinite: for there are always other [existents] between existents, and again others between these”. This premise assumes that all things within the universe work in continuous number systems, and many do. For example there may be a specific star 432.562 light years away from earth, a block of cheese may weigh 3.2 pounds, and the average male may be 1.8 meters tall, but it would be fallacious to say that there exists within some village 208.5 people; even if a person were cut in half we would not say that he is less than one whole human. Numbers that operate in this fashion are said to be isolated, they are not separated from the next or previous number by 10 tenths of a number or 100 hundredths, rather, they separated by no less that one whole number. Thus Zeno’s argument is rather silly.

Zeno was very clever with his paradoxes, and while I would agree with him that many things are not as they seem, math does not lie or deceive. If the math from my arguments did not match up with my hypotheses I would simply concede with a shrug of my shoulders that I do not know the answer, and it may still be the case that I do not know as much as I thought I did, or that I interpreted something wrong along the way. Regardless, I think Zeno was a very important Pre-Socratic philosopher, and I value very much his way-outside-the-box thinking.