Motion in Greek mathematics. 

This was easy for Geometry and Number Theory, but for motion there were extreme difficulties. The problem is with the notion of instantaneous velocity, which in the mathematical model was connected with the notion of infinity . Is there an infinite, or an infinitesimally small magnitude? In particular Zeno of Elea (c.-450), a student of Parmenides, who believed that reality is unchanging and sense impressions merely illusions, showed that current ideas on motion required careful criticism to avoid logical paradoxes.

 Zeno established many of the rules of inference of logical argument, and pioneered 'Proof by Contradiction'. His works survive only in the commentary of Aristotle (-384-- -322), who opposed the philosophy of Parmenides.

 The following is Aristotle's criticsm of Zeno, and our only source for Zeno's arguments.

The text

(From Aristotle 'Physics' translated in I. Thomas, 'Greek Mathematical Works') 

 Zeno has four arguments about motion which present difficulties to those who try to resolve them. The first is that which says there is no motion because the object moved must arrive at the middle before it arrives at the end, concerning which we have already treated.

 Zeno's argument makes a false assumption in not allowing the possibility of passing through or touching an infinite number of positions one by one in a limited time. For there are two senses in which length and time, and, generally, any continuum, are said to be infinite, either in respect of division or of extension. So where the infinite is infinite in respect of quantity, it is not possible to make in a limited time an infinite number of contacts, but it is possible where the infinite is infinite in respect of division; for the time also is infinite in this respect. And so it is possible to pass through an infinite number of positions in a time which is in this sense infinite, but not in a time which is finite, and to make an infinite number of contacts because its moments are infinite, not finite.

 The second argument is the so-called Achilles; this asserts that the slowest will never be overtaken by the quickest ; for that which is pursuing must first reach the point from which the fleeing object started, so that the slower must necessarily always be some distance ahead. This is the same reasoning as that of the Dichotomy , the only difference being that when the magnitude which is successively added is divided it is not necessarily bisected. The argument leads to the conclusion that the slower will never be overtaken, and it is for the same reason as in the Dichotomy (for in both by dividing the distance in some way it is concluded that the goal will not be reached; but in this a dramatic effect is produced by saying that not even the swiftest will be successful in its pursuit of the slowest) and so the solution must necessarily be the same. The claim that the one in front is not overtaken is false ; for when in front he is not indeed overtaken, but he will nevertheless be overtaken if he give his pursuer a finite distance to go through. 

T'hese are two of the argmnents, and the third is the one just mentioned that the flying arrow is at rest. This conclusion follows from the assumption that time Is composed of instants; for if this is not granted the reasoning does not follow.

 The fourth is that about the two rows of equal bodies moving past each other in the stadium with equal velocities in opposite directions, the one row starting from the end of the stadium, the. other from the middle. This, he thinks, leads to the conclusion that half a given time is equal to its double. The fallacy lies in assuming that a body takes an equal time to pass with equal speed a body in motion and a body of equal size at rest; but this is untrue. For example, let AA be stationary bodies of equal size, let BB be the bodies equal in number and size that start from the middle, and let GG be the bodies equal in number and size that start from the end, having a speed equal to that of the Bs. In consequence, the first B and the first G move past each other and come simultaneously to the end. It follows that G has passed all the bodies it is moving past, though B has passed only half the bodies it is moving past, so that B has taken half the time taken by G. For each takes an equal time in passing each body. And it follows that at the same moment the first B has passed all the GG; for the first G and the first B will be simultaneously at opposite ends of the A s , since both take an equal time in passing the A s. Such is his argument, and it comes about from the aforementioned fallacy. 

Explanation 

• Premise 1: Space and time are infinitely divisible.

 
1. Paradox 1 Dichotomy: motion is impossible, because before an object can travel any given distance, it must first travel half the distance; but before it does this it must first travel half of this, and so on.
2. Paradox 2 Achilles: Achilles is racing a tortoise, who starts ahead. Before Achilles can pass the tortoise, he must first reach the point P₁ where the tortoise started. Say he does this when the tortoise is at P₂. Before Achilles passes the tortoise, he must first reach P₂, and so on.
Premise 2: Space and time are made up of indivisible atoms.
 
 
1. Paradox 3 The arrow: Motion is impossible, because at any atomic instant, the arrow is at rest in space; if not, space would be infinitely divisible. At the next instant, it is somewhere else at rest. So it is always at rest.
2. Paradox 4 The Stadium: Two chariots A and B race round the stadium at the same speed but in opposite directions. A third chariot G is at rest. Suppose at some atomic instant, B racing left passes a unit length of G. Then in the same time, A and B pass two unit lengths of each other. But then they pass one unit in half the time, which is indivisible. 

 What is needed here is a notion of a sum of an infinite series. In the end, Aristotle failed to convincingly prove the properties of infinite sets . So he banned the notion of "completed infinity" from mathematics, proposing instead that mathematics should only deal with "potential infinity". For example, any line can be doubled; given any set of points, another can always be found. This related also to Geometry: if space is atomic, a circle is a regular polygon of infinitely many sides.

Exercise 

 What did Euclid actually prove in IX, 20?

 Why do modern Number Theory books claim he proved that there are infinitely many prime numbers?