# The paradoxa of Zeno of Elea

In view of the apparent remoteness of mathematical abstractions, often frightening wider circles, it may be noted that even elementary mathematical training allows school disciples to see through the famous paradox of the race between Achilles and the tortoise. How could the fleet-footed hero ever catch up with and pass the slow reptile if it were given even the smallest handicap? Indeed, at his arrival at the starting point of the turtle, Achilles would find that it had moved to some further point along the race track, and this situation would be repeated in an infinite sequence. I need hardly remind you that the logical analysis of situations of this type was to play an important role in the development of mathematical concepts and methods. are an example of ancient Greek abstract reasoning that is even in contradiction to observation. How can it be true that nothing moves when our observation “confirms” (Magna Graecia, South Italy) born c. 488 BC the son of Teleutagoras was a philosopher who studied in the Eleatic School of philosophy founded by

If things have some magnitude they are infinite large

But if it exists, each thing must have some size and thickness, and part of it must be apart from the rest. And the same reasoning holds concerning the part that is in front. For that too will have size and part of it will be in front. Now it is the same thing to say this once and to keep saying it forever. For no such part of it will be last, nor will there be one part not related to another. Therefore, if there are many things, they must be both small and large; so small as not to have size, but so large as to be unlimited.
Simplicius On Aristotle's Physics

The race course (or dichotomy) paradox

For the dichotomy, Aristoteles describes Zeno's argument:

There is no motion because that which is moved must arrive at the middle of its course before it arrives at the end.

In order the traverse a line segment it is necessary to reach its midpoint. To do this one must reach the 1/4 point, to do this one must reach the 1/8 point and so on ad infinitum. Hence motion can never begin. The argument here is not answered by the well known infinite sum 1/2 + 1/4 + 1/8 + ... = 1. Zeno can argue that the sum 1/2 + 1/4 + 1/8 + ... never actually reaches 1, but more perplexing to the human mind is the attempts this sum backwards. Before traversing a unit distance we must get to the middle, but before getting to the middle we must get 1/4 of the way, but before we get 1/4 of the way we must reach 1/8 of the way etc. This argument makes us realise that we can never get started since we are trying to build up this infinite sum from the "wrong" end. Indeed this is a clever argument which still puzzles the human mind today.

Zeno bases both the dichotomy paradox and the attack on simple pluralism on the fact that once a thing is divisible, then it is infinitely divisible.

Zeno appeals here to the fact that any distance however small can be halved. It follows that there must be an infinite number of points in a line. He argues that you cannot get to the end of a racecourse because to do so you must traverse half of any give distance before you can travel the whole, and the half of that again before you can traverse it. This goes on ad infinitum so that there are an infinite number of points in any given space. However, you cannot traverse an infinite number of points in any finite time. It is assumed that finite time is composed of a finite number of instants.

The Arrow

If a stick one foot long is cut in half every day, it will still have something left after ten thousand generations ...There are times when a flying arrow is neither in motion, nor at rest, Hui Shih, Chinese Philosopher

Other paradoxes given by Zeno cause problems precisely because in these cases he considers that seemingly continuous magnitudes are made up of indivisible elements. Such a paradox is “The Arrow”:

If, says Zeno, everything is either at rest or moving when it occupies a space equal to itself, while the object moved is in the instant, the moving arrow is unmoved.

The argument rests on the fact that if in an indivisible instant of time the arrow moved, then indeed this instant of time would be divisible (for example in a smaller 'instant' of time the arrow would have moved half the distance).

The human mind, when trying to give itself an accurate account of motion, finds itself confronted with two aspects of the phenomenon. Both are inevitable but at the same time they are mutually exclusive. Either we look at the continuous flow of motion; then it will be impossible for us to think of the object in any particular position. Or we think of the object as occupying any of the positions through which its course is leading it; and while fixing our thought on that particular position we cannot help fixing the object itself and putting it at rest for one short instant.
Frankel

Vlastos points out that if we use the standard mathematical formula for velocity we have v = s/t, where s is the distance traveled and t is the time taken. If we look at the velocity at an instant we obtain v =0/0, which is meaningless. Zeno is pointing out a mathematical difficulty for ancient Greek mathematics without properly knowledge of differential calculus. If everything is at rest when it occupies a space equal to itself, and what is in flight at any given moment always occupies a space equal to itself then it cannot move. Alternatively if everything when it is behaving in a uniform manner is continually either moving or at rest, but what is moving is always in the now, then the moving arrow is motionless. The arrow in flight is at rest. We find it hard to avoid supposing that when the arrow is in flight there is a next position occupied at the next moment. The view that a finite part of time consists of a finite series of successive instants seems to be assumed in this paradox. The plausibility of the argument is dependent upon supposing that there are successive instants such that throughout an instant a moving body is where it is, though at the next instant it is somewhere else. It cannot move during the instant for that would require that the instant should have parts. Suppose we consider a period consisting of a thousand instants. The arrow is in flight throughout this period. It is never moving, but in some way, the change of position has to occur between the instants, not at any time whatever.

Consider two rows of bodies, each composed of an equal number of bodies of equal size. They pass each other as they travel with equal velocity in opposite directions. Thus, half a time is equal to the whole time. Zeno's "Stadium" paradox is a relativity gedankenexperiment made 2200 years before Albert Einstein. Consider that we have three persons A, B and C in a stadium and C in rest and the other two A, B running in opposite directions. Running in opposite directions A and B think that they approach each other with a speed which is double as large as the person C finds. Zeno thinks that we cannot have different results that depend on the observer and thus any movement is an illusion.

Achilles and the tortoise