936:: "A version of nonstandard analysis, Internal Set Theory, has been used to provide a resolution of Zeno's paradoxes of motion. This resolution is inadequate because the application of Internal Set Theory to the paradoxes requires a model of the world that is not in accordance with either experience or intuition. A model of standard mathematics in which the ordinary real numbers are defined in terms of rational intervals does provide a formalism for understanding the paradoxes. This model suggests that in discussing motion, only intervals, rather than instants, of time are meaningful. The approach presented here reconciles resolutions of the paradoxes based on considering a finite number of acts with those based on analysis of the full infinite set Zeno seems to require. The paper concludes with a brief discussion of the classical and quantum mechanics of performing an infinite number of acts in a finite time."
1168:"According to this, the length of the hypotenuse of a right angled triangle in discretized space is always equal to the length of one of the two sides, in contradiction to geometry." -this sentence seems problematic, because it relates space with a distinstive attribute (discretized) to geometry without an an attribute. i cannot propose a solution since the meaning of "discretized" is unbeknowst to me, but i can propose an initial direction of reseach that is try replacing "geometry" with " euclidian geometry" as a formal (linguistical) corrction and check the meaning and reltiobn between "discretionalized" and "euclidian".
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becomes proven enough to be accepted as not just a theory but rather the actual dscription of reality (if and when that happens) it will remind us that zeno was wrong in his assumption about infinite divisibility and that can be taken as a simple cause of his paradoxes. (if i take correctly a paradox takes a seemingly correct line of arguments and points to a result with it that is obviously contradicting reality, saying either the line of thought is not as correct as we think and then lets find the mistake in it, or reality is not what it is belived and then lets find the mistake in the belief about it.)
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1215:. This makes more sense, as the figure illustrates how each of the successive stages in the race takes less time and brings Achilles closer to the tortoise until the tortoise is overtaken. The exposition is immediately followed by a discussion of the associated convergent infinite series. In the English Knowledge, the infinite-series discussion is in a different article,
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The resolution of the paradoxes comes from understanding that if someone can direct your attention to an irrelevant part of a system, while you think you are looking at the whole system, they can convince you of foolish things about the whole system. So if you look at shorter and shorter segments of
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also im missing from the article a simple worded paragraph that would tell that the zeno paradoxes result in an so to say impossibility of the event observed in reality because the question is framed to look at a time frame that by its definition excludes the event of achilles reaching the turtle or
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also theres this italian guy, carlo rovelli who deserves mention as the author of some books proposing that the infinite division used in the zeno paradoxes actually is in contradiction with the physical nature of the world. its not that he talks about zeno, its that when his quantum physics theory
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One risk of being a debunker or skeptic is that you might not be skeptical about your own skepticism. Your explanation is great until you unequivocally declare "nothing to do with infinities". I think the history of math will bear out that the notion of limits is a very old problem (not initially
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This list is intended to collect references thought to be relevant for the article. Delete entries only when they are blatantly and obviously inappropriate. In general, we want not only to collect useful references, but also be able to check new additions against previous discussions that lead to
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Is the image worth preserving? I propose to delete it. It adds nothing to the text. It merely illustrates what is obvious anyhow, that
Achilles, running at 10 m/s, will overtake anything running more slowly including, for example, a squirrel running at 5 m/s. (Squirrels can go that fast, though
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After the relevant entries in this encyclopedia, the place to begin any further investigation is Salmon (2001), which contains some of the most important articles on Zeno up to 1970, and an impressively comprehensive bibliography of works in
English in the Twentieth Century
1219:. The figure should be here, if anywhere, though the argument in this section supposes the tortoise's speed to be 0.1 m/s. Modifying the graph to suppose that the tortoise is going 0.1 or 0.2 m/s is impractical, since the line for the tortoise would be nearly horizontal.
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time (of a runner's or a Hare's motion), because a questioner like Zeno led you down that path, you discover the "paradox" that your inquiry doesn't tell you anything accurate about the characteristics of the overall motion. Or as
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Philip K. Dick's short science-fiction story "The
Indefatigable Frog" concerns an experiment to determine whether a frog which continually leaps half the distance to the top of a well will ever be able to get out of the
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Allama Iqbal's book The
Reconstruction of Religious Thought in Islam discusses the paradox in Lecture II The Philosophical Test of the Revelations of Religious Experience, and suggests that motion is not continuous but
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Paul A. Fishwick, ed. (1 June 2007). "15.6 "Pathological
Behavior Classes" in chapter 15 "Hybrid Dynamic Systems: Modeling and Execution" by Pieter J. Mosterman, The Mathworks, Inc.".
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solved by Newton and
Leibnitz when they formulated calculus) and Zeno was a key motivator by framing it as a paradox. But what do I know? I'm not a historian of science.
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Hornschemeier's most recent graphic novel, The Three Paradoxes, contains a comic version of Zeno presenting his three paradoxes to his fellow philosophers.
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Zadie Smith references Zeno's arrow paradox, and, more briefly, Zeno's Achilles and tortoise paradox, at the end of Chapter 17 in her novel White Teeth.
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