Zenón de Elea: Filósofo griego. (c. a. C.) -Escuela eleática. – Es conocido por sus paradojas, especialmente aquellas que niegan la. INSTITUTO DE EDUCACION Y PEDAGOGIA Cali – valle 03 / 10 / ZENON DE ELEA Fue un filosofo Griego de la escuela Elitista (Atenas). Paradojas de Zenón 1. paradoja de la dicotomía(o la carrera) 2. paradja de aquiles y la tortuga. Conclusion Discípulo de parménides de Elea.

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In the fifth century B. The arguments were paradoxes for the ancient Greek philosophers.

Because many of the arguments turn crucially on the notion that space and time are infinitely divisible, Zeno was the first person to show that the concept of infinity is problematical. In the Achilles Paradox, Achilles races to catch a slower runner—for example, a tortoise that is crawling in a line away from him. The tortoise has a head start, so if Achilles hopes to overtake it, he must run at least as far as the place where the tortoise presently is, but by the time he arrives there, it will have crawled to a new place, so then Achilles must run at least to this new place, but the tortoise meanwhile will have crawled on, and so forth.

Achilles will never catch the tortoise, says Zeno. Therefore, good reasoning shows that fast runners never can catch slow ones. So much the worse for the claim that any kind of motion really occurs, Zeno says in defense of his mentor Parmenides who had argued that motion is an illusion.

Because Zeno was correct in saying Achilles needs to run at least to all those places where the tortoise once was, what is required is an analysis of Zeno’s own argument. This article explains his ten known paradoxes and considers the treatments that have been offered.

In the Achilles Paradox, Zeno assumed distances and durations can be endlessly divided into what modern mathematicians call a transfinite infinity of indivisible parts, and he assumed there are too many of these parts for the runner to complete. Aristotle ‘s treatment said Zeno should have assumed instead that there are only potential infinitiesso that at any time the hypothetical division into parts produces only a finite number of parts, and the runner has time to complete all these parts.

Aristotle’s treatment became the generally accepted solution until the late 19th century. The current standard treatment or so-called “Standard Solution” implies Zeno was correct to conclude that a runner’s path contains an actual infinity of parts at any time during the motion, but he was mistaken to assume this is too many parts. This treatment employs the mathematical apparatus of calculus which has proved its indispensability for the development of modern science.

The article ends by exploring newer treatments of the paradoxes—and related paradoxes such as Thomson’s Lamp Paradox—that were developed since the s. Zeno was born in about B. He was a friend and student of Parmenides, who was twenty-five years older and also from Elea. He was not a mathematician. Zeno is reported to have been arrested for taking weapons to rebels opposed to the tyrant who ruled Elea.

When asked about his accomplices, Zeno said he wished to whisper something privately to the tyrant. But when the tyrant came near, Zeno bit him, and would not let go until he was stabbed. It was said to be a book of paradoxes defending the philosophy of Parmenides. A thousand years after Zeno, the Greek philosophers Proclus and Simplicius commented on the book and its arguments.

They had access to some of the book, perhaps to all of it, but it has not survived. Proclus is the first person to tell us that the book contained forty arguments.

This number is confirmed by the sixth century commentator Elias, who is regarded as an independent source because he does not mention Proclus. Unfortunately, we know of no specific dates for when Zeno composed any of his paradoxes, and we know very little of how Zeno stated his own paradoxes.

We do have a direct quotation via Simplicius of the Paradox of Denseness and a partial quotation via Simplicius of the Large and Small Paradox.

In total we know of less than two hundred words that can be attributed to Zeno. Our knowledge of these two paradoxes and the other seven comes to us indirectly through paraphrases of them, and comments on them, primarily by his opponents Aristotle B. The names of the paradoxes were created by later commentators, not by Zeno. In the early fifth century B.


Reality, he said, is a seamless unity that is unchanging and can not be destroyed, so appearances of reality are deceptive.

Our ordinary observation reports are false; they do not report what is real. Although we do not know from Zeno himself whether he accepted his own paradoxical arguments or exactly what point he was making with them, according to Plato the paradoxes were designed to provide detailed, supporting arguments for Parmenides by demonstrating that our common sense confidence in the reality of motion, change, and ontological plurality that is, that there exist many thingsinvolve absurdities.

On Plato’s interpretation, it could reasonably be said that Zeno reasoned this way: His Dichotomy and Achilles paradoxes presumably demonstrate that any continuous process takes an infinite amount of time, which is paradoxical. Zeno’s Arrow and Stadium paradoxes demonstrate that the concept of discontinuous change is paradoxical. Because both continuous and discontinuous change are paradoxical, so is any change.

Eudemus, a student of Aristotle, offered another interpretation. Zeno was actually challenging the Pythagoreans and their particular brand of pluralism, not Greek common sense.

Zeno was not trying to directly support Parmenides.

Aristotle believed Paradoias Paradoxes were trivial and easily resolved, but later philosophers have not agreed on the triviality. Before Zeno, Greek thinkers favored presenting their philosophical views by writing poetry. Zeno began the grand shift away from poetry toward a prose ellea contained explicit premises and conclusions.

And he zenln the method of indirect proof in his paradoxes by temporarily assuming some thesis that he opposed and then attempting to deduce an absurd conclusion or a contradiction, thereby undermining the temporary assumption. This method of indirect proof or reductio ad absurdum probably originated with his teacher Parmenides [although this is disputed in the scholarly literature], but Zeno used it more systematically.

Any paradox can be treated by abandoning enough of its crucial assumptions. For Zeno’s it is very interesting to consider which assumptions to abandon, and why those. A paradox is an argument that reaches a contradiction by apparently legitimate steps from apparently reasonable assumptions, while the experts at the time cannot agree on the way out se the paradox, that is, agree on its resolution.

It is this latter point about disagreement among the experts that distinguishes a paradox from a mere puzzle in the ordinary sense of that term. This resolution is called the Standard Solution. It points out that, although Zeno was correct in saying that at any point or instant before reaching the goal there is always some as yet uncompleted path to cover, this does not imply that the goal is never reached.

More specifically, the Standard Solution says that for the runners in the Achilles Paradox and the Dichotomy Paradox, the runner’s path is a physical continuum that is completed by using a positive, finite speed. The details presuppose differential calculus and classical mechanics as opposed to quantum mechanics. The Standard Solution treats speed as the derivative of distance with respect to time. It assumes that physical processes are sets of point-events.

It implies that durations, distances and line segments are all linear continua composed of indivisible points, then it uses these ideas to challenge various assumptions made, and inference steps taken, by Zeno. To be very brief and anachronistic, Zeno’s mistake and Aristotle’s mistake was to fail to use calculus. More specifically, in the case of the paradoxes of motion such as the Achilles and the Dichotomy, Zeno’s mistake was not his assuming there is a completed infinity of places for the runner to go, which was what Aristotle said was Zeno’s mistake.

Instead, Zeno’s and Aristotle’s mistake was in assuming that this is too many places for the runner to go to in a finite time. Aristotle’s treatment of the paradoxes does not employ these fruitful concepts of mathematical physics. Aristotle did not believe that the use of mathematics was needed to understand the world. So, the Standard Solution is much more complicated than Aristotle’s treatment.

No single person can be credited with creating it. In ordinary discourse outside of science zwnon would never need this kind of precision, but it is needed in mathematical physics and its calculus. By “real numbers” we do not mean eoea numbers but rather decimal numbers. Calculus was invented in the late ‘s by Newton and Leibniz. Their calculus is a technique for treating continuous motion as being composed of an infinite number of infinitesimal steps.


After the acceptance of calculus, most all mathematicians and physicists believed that continuous motion should be modeled by a function which takes real numbers representing time as its argument and which gives real numbers representing spatial position as its value.

This position function should be continuous xe gap-free. In addition, the position function should be differentiable in order to make sense of speed, which is treated as the rate of change of position.

By the early 20th century most mathematicians had come to believe that, to make rigorous sense of motion, mathematics needs a fully developed set theory that rigorously defines the key concepts of real number, continuity and differentiability. Doing this requires a well defined concept of the continuum. Unfortunately Newton and Leibniz did not have a good definition of the continuum, and finding a good one required over two hundred years of work. The continuum is a very special set; it is the standard model of the real numbers.

Intuitively, a continuum is a continuous entity; it is a whole thing that has no gaps.

Distances and durations are normally considered to be real physical continua whereas treating the ocean salinity and the rod’s temperature as continua is a very useful approximation for many calculations in physics even though we know that at the atomic level the approximation breaks down.

The continuum is the mathematical line, the line of geometry, which is standardly understood to have the same structure as the real numbers in their natural order.

Zenón de Elea | Quién fue, biografía, pensamiento, arjé, paradojas

Real numbers and points on the continuum can be put into a one-to-one order-preserving correspondence. There are not enough rational numbers for this correspondence even though the rational numbers are dense, too in the sense that between any two rational numbers there is another rational number. These definitions are given in terms of the linear continuum.

Physical space is not a linear continuum because it fe three-dimensional and not linear; but it has one-dimensional subspaces such as paths of runners and orbits of planets; and these are linear continua if we use the path created by only one point on the runner and the orbit created by only one point on the planet.

Regarding timeeach point instant is assigned a real number as its time, and each instant is assigned a duration of zero. The time taken by Achilles to catch the tortoise is a temporal interval, a linear continuum of instants, according to the Standard Solution but not according to Zeno or Aristotle. The Standard Solution says that the sequence of Achilles’ goals the goals of reaching the point where the tortoise is should be abstracted from a pre-existing transfinite set, namely a linear continuum of point places along pzradojas tortoise’s path.

Zeno’s paradoxes

Aristotle’s treatment does not do this. The next section la this article presents the details of how the concepts of the Standard Solution are used to resolve each of Zeno’s Paradoxes. Of the ten known paradoxes, The Achilles attracted the most attention over the centuries.

It was generally accepted until the 19th century, but slowly lost ground to the Standard Solution. Some historians say Aristotle had no solution but only a verbal paradoja. The development of calculus was the most important step in the Standard Solution of Zeno’s paradoxes, so why did it take so long for the Standard Solution to be accepted after Newton and Leibniz developed their calculus?

The period lasted about two hundred years. There are four reasons. Today, most philosophers would not restrict meaning to empirical meaning. They believe in indivisible points even though they are not even indirectly observable. However, for an interesting exception see Dummett which contains a theory in which time is composed of overlapping intervals rather than durationless instants, and in which the endpoints of those intervals are the initiation and termination of actual physical processes.

This idea of treating time without instants develops a proposal of Russell and Whitehead.