The main source of our knowledge of Zeno comes from the dialogue Parmenides written by Plato. Zeno was a pupil and friend of the philosopher Parmenides and studied with him in Elea. The Eleatic School , one of the leading pre-Socratic schools of Greek philosophy, had been founded by Parmenides in Elea in southern Italy.
His philosophy of monism claimed that the many things which appear to exist are merely a single eternal reality which he called Being. His principle was that "all is one" and that change or non-Being are impossible. Certainly Zeno was greatly influenced by the arguments of Parmenides and Plato tells us that the two philosophers visited Athens together in around BC. Despite Plato 's description of the visit of Zeno and Parmenides to Athens, it is far from universally accepted that the visit did indeed take place.
However, Plato tells us that Socrates , who was then young, met Zeno and Parmenides on their visit to Athens and discussed philosophy with them. Given the best estimates of the dates of birth of these three philosophers, Socrates would be about 20 , Zeno about 40 , and Parmenides about 65 years of age at the time, so Plato 's claim is certainly possible. Zeno had already written a work on philosophy before his visit to Athens and Plato reports that Zeno's book meant that he had achieved a certain fame in Athens before his visit there.
Unfortunately no work by Zeno has survived, but there is very little evidence to suggest that he wrote more than one book. The book Zeno wrote before his visit to Athens was his famous work which, according to Proclus , contained forty paradoxes concerning the continuum. Four of the paradoxes, which we shall discuss in detail below, were to have a profound influence on the development of mathematics. Diogenes Laertius [ 10 ] gives further details of Zeno's life which are generally thought to be unreliable.
Zeno returned to Elea after the visit to Athens and Diogenes Laertius claims that he met his death in a heroic attempt to remove a tyrant from the city of Elea.
The stories of his heroic deeds and torture at the hands of the tyrant may well be pure inventions. Diogenes Laertius also writes about Zeno's cosmology and again there is no supporting evidence regarding this, but we shall give some indication below of the details. Zeno's book of forty paradoxes was, according to Plato [ 8 ] But it is impossible for S to reach an unlimited number of half way points within a limited amount of time.
Therefore, it is impossible for S to traverse the stadium or, indeed, for S to move at all; in general, it is impossible to move from one place to another. Simplicius adds the identification of the slowest runner as the tortoise in Ph. Aristotle remarks that this argument is merely a variation on the Dichotomy, with the difference that it does not depend on dividing in half the distance taken Ph. Whether this is actually the case is debatable.
During the time it takes Achilles to reach the point from which the tortoise started t 0 , the tortoise will have progressed some distance d 1 beyond that point, namely to t 1 , as follows:. Therefore, the slowest runner in the race, the tortoise, will never be overtaken by the fastest runner, Achilles.
Epiphanius, Against the Heretics 3. Thus, according to Aristotle, the moving arrow A is actually standing still. The argument for this conclusion seems to be as follows: What moves is always, throughout the duration of its motion, in the now, that is to say, in one instant of time after another.
So, throughout its flight, A is in one instant of time after another. So A is resting at t. Thus A is resting at every instant of its flight, and this amounts to the moving arrow always being motionless or standing still. This description suggests a final position as represented in Diagram 2.
Apparently, Zeno somehow meant to infer from the fact that the leading B moves past two A s in the same time it moves past all four C s that half the time is equal to its double. The challenge is to develop from this less than startling fact anything more than a facile appearance of paradox.
Since it is stressed that all the bodies are of the same size and that the moving bodies move at the same speed, Zeno would appear to have relied on some such postulate as that a body in motion proceeding at constant speed will move past bodies of the same size in the same amount of time. He could have argued that in the time it takes all the C s to move past all the B s, the leading B moves past two A s or goes two lengths, and the leading B also moves past four C s or goes four lengths.
According to the postulate, then, the time the leading B travels must be the same as half the time it travels. Unfortunately, the evidence for this particular paradox does not enable us to determine just how Zeno may in fact have argued. Aristotle also gestures toward two additional ingenious arguments by Zeno, versions of which were also known to Simplicius.
The version of this argument known to Simplicius represents Zeno as engaged in a fictional argument with Protagoras, wherein he makes the point that if a large number of millet seeds makes a sound for example, when poured out in a heap , then one seed or even one ten-thousandth of a seed should also make its own sound for example, in that process Simp.
The evidence nonetheless suggests that Zeno anticipated reasoning related to that of the sorites paradox, apparently invented more than a century later. Eudemus fr. Zeno would appear to have argued as follows. Everything that is is in something, namely a place. If a place is something, then it too must be in something, namely some further place. If this second place is something, it must be in yet another place; and the same reasoning applies to this and each successive place ad infinitum.
Thus, if there is such a thing as place, there must be limitless places everywhere, which is absurd. Therefore, there is no such thing as place. This argument could well have formed part of a more elaborate argument against the view that there are many things, such as that if there are many things, they must be somewhere, i.
This is, however, only speculation. After the portion of the exchange between Socrates and Zeno quoted above sect. Socrates virtually accuses Zeno of having plotted with Parmenides to conceal the fundamental identity of their conclusions. With so many readers of Plato accustomed to taking Socrates as his mouthpiece in the dialogues, it is not surprising that this passage has served as the foundation for the common view of Zeno as Parmenidean legatee and defender, by his own special means, of Eleatic orthodoxy.
Zeno this time replies that Socrates has not altogether grasped the truth about his book. First, he says, the book had nothing like the pretensions Socrates has ascribed to it Prm. Zeno is made to explain his actual motivation as follows:. For not only does Parmenides end up examining the relation of his One to other things, which would have been impossible if his doctrine entailed their non-existence, but the relation other things have to the One actually proves responsible in a way for their existence.
Zeno cannot be supposing that his arguments against plurality entailed the doctrine of Parmenides when that doctrine is represented in this same dialogue by Parmenides himself as something altogether more involved than the simple thesis that only one thing exists. What Plato actually suggests is that Zeno aimed to show those whose superficial understanding of Parmenides had led them to charge him with flying in the face of common sense, that common sense views concerning unity and plurality are themselves riddled with latent contradictions.
Many men had mocked Parmenides: Zeno mocked the mockers. His logoi were designed to reveal the inanities and ineptitudes inherent in the ordinary belief in a plural world; he wanted to startle, to amaze, to disconcert. However, whether the historical Zeno was actually involved in anything like the dialectical context Plato envisages for him must remain uncertain. The more mature Zeno seems a little embarrassed by the combative manner evident in the arguments of his younger days, as well he might since that spirit would have come to be seen as typical of the eristic controversialists who sprang up in the sophistic era.
In the Alcibiades , Socrates reports that Pythodorus and Callias each paid Zeno a hundred minae to become clever and skilled in argument Alc. Teaching for payment is of course one hallmark of the professional educators who styled themselves experts in wisdom.
Precisely what Aristotle meant by this remains a matter of speculation, given that Aristotle also attributes the invention of dialectic to Socrates Arist. For Aristotle, then, Zeno was a controversialist and paradox-monger, whose arguments were nevertheless both sophisticated enough to qualify him as the inventor of dialectic and were important for forcing clarification of concepts fundamental to natural science.
Should we then think of Zeno as a sophist? The skill Plutarch attributes to Zeno, still evident in the fragmentary remains of his arguments, is just the kind of skill in argument manifested in a great deal of sophistic practice. His apparent demonstrations of how the common-sense view is fraught with contradiction made him an influential precursor of sophistic antilogic and eristic disputation.
It is not surprising that someone like Isocrates should have viewed Zeno as a sophist to be classed with Protagoras and Gorgias. While he perhaps does not fit exactly into any of these categories, still his development of sophisticated methods of argumentation to produce apparent proofs of the evidently false conclusions that motion is impossible and that there are not in fact many things made it quite natural for Plato, Aristotle, Isocrates, and others to refer to him under all these labels.
Several of the paradoxes involve no specifically mathematical notions at all. The Achilles is perhaps the best example since it employs only very ordinary notions, such as getting to where another has started from.
The other extant arguments for the most part deploy similarly prosaic notions: being somewhere or being in a place, being in motion, moving past something else, getting halfway there, being of some size, having parts, being one, being like, being the same, and so on. Where Zeno seems to have leapt ahead of earlier thinkers is in deploying specifically quantitative concepts, most notably quantitative concepts of limit peras and the lack of limit to apeiron.
Earlier Greek thinkers had tended to speak of limitedness and unlimitedness in ways suggesting a qualitative rather than a quantitative notion. They had an immediate impact on Greek physical theory.
His arguments, perhaps more than anything else, forced the Greek natural philosophers to develop properly physical theories of composition as opposed to the essentially chemical theories of earlier thinkers such as Empedocles. That mathematicians and physicists have worked ever since to develop responses to the more ingenious of his paradoxes is remarkable, though perhaps not surprising, for immunity to his paradoxes might be taken as a condition upon the adequacy of our most basic physical concepts.
He may even have offered his collection of paradoxes to provoke deeper consideration of the adequacy of theretofore unexamined notions. References in this bibliography to items prior to are more selective than those to more recent items. For a nearly exhaustive and annotated listing of Zenonian scholarship down to , consult L.
Paquet, M. Roussel, and Y. The long standard collection of the fragments of the Presocratics and sophists, together with testimonia pertaining to their lives and thought, has been:. For the English reader, the fragments and testimonia of the Presocratics and sophists are now most usefully presented in:. Life and Writings 2. The Extant Paradoxes 2. Is this not what you say? For if there were many things, they would incur impossibilities. So is this what your arguments intend, nothing other than to maintain forcibly, contrary to everything normally said, that there are not many things?
And do you think that each of your arguments is a proof of this very point, so that you consider yourself to be furnishing just as many proofs that there are not many things as the arguments you have written? Is this what you say, or do I not understand correctly? Bibliography Further Reading References in this bibliography to items prior to are more selective than those to more recent items. Comprehensive accounts of Zeno and his arguments may be found in: Barnes, J.
Chapters 12 and Caveing, M. Vrin, Ferber, R. Guthrie, W. Kirk, G. Raven, and M. Makin, S. Craig ed. McKirahan, R. Long, ed. Sainsbury, R. Vlastos, G. Edwards ed. When the tournament ends, Grand Zeno himself appears at the ring and declares that there will be an even greater tournament to come with all twelve universes competing against each other - an event that would soon be known as the Tournament of Power.
When Zeno first reveals himself to Universes 6 and 7, Goku immediately introduces himself, shaking hands with the god of the multiverse, much to Lord Beerus' concern. Zeno takes a liking to Goku, as the two share the same childlike outlook, seeing a potential friend in the Saiyan warrior.
Some time later, Zeno contacts Beerus to bring Goku to the god-king's palace. Beerus does as requested and Goku appears before Grand Zeno, who gives him a special button that can transport Zeno to wherever Goku is; an advantage that soon thereafter saves Goku's life. When battling Goku Black in Future Trunks' alternate timeline , Goku and Vegeta are on the verge of losing the fight along with the future world.
Goku Black and the evil Kai of the tenth universe, who are technically the same being, merge bodies and become absolutely unstoppable. Just when all hope seems lost, Goku presses the button Zeno gave him, and a second later Zeno appears and helps Goku in his fight by obliterating the entire alternate future timeline, including Goku Black.
Since this was an alternate version of Zeno from a now-defunct timeline, Goku asked if he wanted to tag along back to the present rather than live in an endless void. Future Zeno agreed, and from that point on the entirety of the Dragon Ball multiverse was run by two Zenos instead of one.
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