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*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. *Niels Bohr, Essays on Atomic Physics and Human Knowledge

**Paradox** from the Greek "*para doxa*" something contrary to opinion

Nude descending a staircase, Marcel Duchamp 1912, a scandal when first shown in the Armory show. The object at each moment is at a fixed position (according to Zeno at rest) how can it then move?

The Paradoxa of Zeno of Elea 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” Heraclitus statement “everything moves and changes”?

Zeno of Elea (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 Parmenides (the son of Peirethos born c. 520-510 in Elea) , whom he later succeeded as its head. He was a student of Parmenides and there is also a story that he was adopted by Parmenides. In Plato's Parmenides we learn that Parmenides and Zeno visited Athens and that they discussed various philosophical problems with others and among these was also the young Socrates. Zeno's students were Pericles and Socrates, Empedocles, Leucippus, Pythodoros, Kephalos and Kallias. Zeno was a teacher and one had to pay 100 minae to be his student. His philosophy of monism claimed that the many things which appear to exist are merely a single eternal reality which he called Being. Zeno died under torture when a revolt against a tyrant failed:

*But Hermippus says, that he was put into a mortar, and pounded to death. And we ourselves have written the following epigram on him:*

*Your noble wish, O Zeno, was to slay*

*A cruel tyrant, freeing Elea*

*From the harsh bonds of shameful slavery,*

*But you were disappointed; for the tyrant*

*Pounded you in a mortar. I say wrong,*

*He only crushed your body, and not you.*

*Diogenes Laertius, Life of Zeno, the Eleatic*

Zeno's principle was that "all is one" and that change or non-Being are impossible. How is this possible in a world that seems to change according to Heraclitus statement: *ta panta rhei*?

Zeno's has written a book that described forty paradoxes which unfortunately was lost.

*... a youthful effort, and it was stolen by someone, so that the author had no opportunity of considering whether to publish it or not. Its object was to defend the system of Parmenides by attacking the common conceptions of things.*

*... Zeno elaborated forty different paradoxes following from the assumption of plurality and motion, all of them apparently based on the difficulties deriving from an analysis of the continuum.*

Plato

Common believe is that if you add infinite numbers the sum increases beyond any particular limit even if the numbers added gets smaller... as long as one is able to add numbers forever. A problem is that one most be careful in infinite sums. For example the sum 1 -1 +1 -1 +1 -1... can be arranged to

(1-1) + (1-1)+ (1-1) +... = 0 + 0 + 0+... = 0. But it can also be written as 1 – (1-1+1-1+...) where according to the previous result the sum in the parenthesis is 0 so that we have a sum 1 - 0 = 1 and since both methods are expected to produce the same result we must conclude that 0 = 1.*
*Also common understanding is as described by Berkeley who wrote about Newtons calculus and the infinitesimal quantities that how it is possible to sum nothing to form something from that he called

*ghosts of quantities which disappeared*.

Aristoteles describes four of Zeno's arguments: The Dichotomy, The Achilles, The Arrow and The Stadium.

**Things without magnitude do not exist**

Today physicist say that electrons probably have no structure, are points with no magnitude. Zeno's argument why something without magnitude could not exist is:

*For if it is added to something else, it will not make it bigger, and if it is subtracted, it will not make it smaller. But if it does not make a thing bigger when added to it nor smaller when subtracted from it, then it appears obvious that what was added or subtracted was nothing.* Of course the electron has mass and charge but still seems to be strange that these properties could be attributes of points with 0 radius (if we ignore the Planck scale).

**If they are many and finite they are also infinite many**

*If there are many, they must be as many as they are and neither more nor less than that. But if they are as many as they are, they would be limited. If there are many, things that are are unlimited. For there are always others between the things that are, and again others between those, and so the things that are are unlimited*.

Simplicius *On Aristotle's Physics*

Zeno's Paradox of Plurality and Proof by Contradiction

**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.

**The Stadium**

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**

The most famous of Zeno's paradox is the Achilles paradox:

*... the slower when running will never be overtaken by the quicker; for that which is pursuing must first reach the point from which that which is fleeing started, so that the slower must necessarily always be some distance ahead.*

Most known is this paradox as the paradox that Achilles will never catch a tortoise if this moves with whatsoever speed. To do so Achilles must first reach the place from where the tortoise started. By that time, the tortoise will have got some way ahead. Achilles must then make up that, and again the tortoise will be ahead. He always coming closer to the tortoise but there is always some distance from her. The first two arguments are usually interpreted as critiques of the idea of continuous motion in infinitely divisible space and time. They differ only in that the first is expressed in terms of absolute motion, whereas the second shows that the same argument applies to relative motion.

**Even if the arrow moves it cannot hit any point!**

It is possible to extend Zeno's Arrow paradox. The probability of the arrow landing in a certain region of the target is equal to the ratio of that region's area to the total target area. As the region gets smaller the probability gets less, and there is zero probability of the arrow's point landing on any particular point of the target. Then since this is true for all points the arrow cannot hit the target.

**Comments**

*In this capricious world nothing is more capricious than posthumous fame. One of the most notable victims of posterity's lack of judgment is the Eleatic Zeno. Having invented four arguments all immeasurably subtle and profound, the grossness of subsequent philosophers pronounced him to be a mere ingenious juggler, and his arguments to be one and all sophisms. After two thousand years of continual refutation, these sophisms were reinstated, and made the foundation of a mathematical renaissance ....*

*Although they have often been dismissed as logical nonsense, many attempts have also been made to dispose of them by means of mathematical theorems, such as the theory of convergent series or the theory of sets. In the end, however, the difficulties inherent in his arguments have always come back with a vengeance, for the human mind is so constructed that it can look at a continuum in two ways that are not quite reconcilable.*

Bertrand Russel

*Mathematicians, however, ... realising that Zeno's arguments were fatal to infinitesimals, saw that they could only avoid the difficulties connected with them by once and for all banishing the idea of the infinite, even the potentially infinite, altogether from their science; thenceforth, therefore, they made no use of magnitudes increasing or decreasing ad infinitum, but contented themselves with finite magnitudes that can be made as great or as small as we please.*

Heath

Zeno's arguments against discontinuity is *the Arrow*, which focuses on the instantaneous physical properties of a moving arrow. Logic breaks down if you try to describe a whole as the sum of parts, when you look too closely at the parts. The problem was illustrated by Zeno with the following example: Take a pile of sand and remove one grain. Is it still a pile? Remove grains one at a time and at some point it's no longer a pile, but where was the transition? We can escape Zeno's paradoxes either by holding that though space and time do consist of points and instants the number of them in any finite interval is infinite or by denying that space and time consist of points at all or that there is no such things as space and time. The problem comes from trying to detect discrete changes in a continuum, when you look too closely the discrete changes disappear.

Zeno shows how sophists in ancient Greece, a kind of modern lawyers, try to convince us the opposite of what is true. For example Zeno in the Achilles paradox changes continuously the time interval at which he observes the position of Achilles and the tortoise. He reduces the time intervals so that never Achilles can reach the tortoise. If we consider simply the fact that the time T that Achilles requires to reach the position where the tortoise started and we ask where Achilles will be after a time 2T we will see that Achilles will be ahead of the tortoise and that between T and 2T he must have reached her.

Antisthenes was said that he could not change Zenos opinion and in a discussion with him he went up and down. Zeno said: could you not stay for a moment? And Antisthenes replied: “So now you say that I am moving!”

**Melissos an admiral , son of Ithagenes born 490-80 BC in Samos was also a Philosopher from the Eleatic school. He said:**

If something exists it is eternal because nothing can be produced from nothing

If something is eternal it is infinite because it cannot have an end and a beginning

If something is eternal and infinite it is one since otherwise one part could cover or limit the others

If something is eternal infinite and unique then it is also not movable since there is no place in which it could move

**Forty Centuries of Ink by David N. Carvalho :
**

"Diodorus, surnamed Siculus, a contemporary of Julius Caesar and Agustus. He published a general history in forty books, under the title 'Historical Library,' which covered a period of 1138 years. We have only a small part remaining of this vast compilation. These rescued portions we owe to Eusebius, to John Malala and other writers of the lower empire, who have cited them in the course of their works. He is the reputed author of the famous sophism against motion. 'If any body be moved, it is moved in the place where it is, or in a place where it is not, for nothing can act or suffer where it is not, and therefore there is no such thing as motion.' "

**See also:**

*Death … the most awful of evils, is nothing to us, seeing that, when we are, death is not come, and, when death is come, we are not.
*Epicurus (341-270) in his

*Letter to Menoeceus*

Laelaps the dog who never failed to catch what he was hunting once had to catch the Teumessian fox that could never be caught.

**Links**

**The Tortoise, Achilles and Thompson's Lamp**

Athens, Greece (PRWEB) August 7, 2003 -- A recent publication claims a solution to Zeno’s paradox and a scientific breakthrough. A Greek researcher indicates some contradictions in such claims and declares the argument invalid: http://science-and-research.press-world.com/v/42854.html

**A Critique of Recent Claims of a Solution to Zeno’s Paradoxes**

For more details see

**Zeno's paradoxes on Motion and Size**

**Zeno Commentary**

**Zeno and the Paradox of Motion**

**The Quantum Zeno and Anti Zeno Paradox or Heraclitus Effect (Powerpoint presentation)**

RALPH E. KENYON JR. Atomism and Infinite Divisibility

Read fragments and comments about Zeno by Aristotle and others, Greek, English and French

On Nature (Peri Physeos) by Parmenides of Elea (c. 475 BC) , **Plato’s Parmenides**

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