zeno of elea life

Zeno played a significant role in causing this progressive trend. The contemporary notion of measure (developed in the 20th century by Brouwer, Lebesgue, and others) showed how to properly define the measure function so that a line segment has nonzero measure even though (the singleton set of) any point has a zero measure. The thesis that Heraclitus influenced Parmenides faces serious chronological challenges. Erroneously thinking that contingent beings can provide “trustworthy thought and understanding” may indeed be an error of mortals. A criticism of supertasks. This objection is not decisive, however, as Palmer’s overall view does not require this emendation. Scholars are divided as to what the exact meaning of this relationship is supposed to be, leading to numerous mutually exclusive interpretative models. Zeno’s paradoxes are now generally considered to be puzzles because of the wide agreement among today’s experts that there is at least one acceptable resolution of the paradoxes. The physical objects in Newton’s classical mechanics of 1726 were interpreted by R. J. Boscovich in 1763 as being collections of point masses. There is another way out, namely, the Standard Solution that uses actual infinities, which are analyzable in terms of Cantor’s transfinite sets. Of the ten known paradoxes, The Achilles attracted the most attention over the centuries. Even if passages ascribed to these thinkers are seen to be rejecting Eleaticism, the rejection may need to be taken as directed against Melissus, not Parmenides’ himself. Palmer’s own view on Opinion is quite positive. 2. The sum of its terms d1 + d2 + d3 +… is a finite distance that Achilles can readily complete while moving at a constant speed. The sum of an infinite series of positive terms is always infinite. Navigating the Scylla and Charybdis of: a) taking the negative, yet often ambiguous and/or ambivalent, treatment of Opinion in the text seriously, while b) avoiding apparently absurd interpretative outcomes, is what makes understanding its relationship to Reality, and thus developing an acceptable interpretation of the poem overall, so very difficult. C.E. C/DK 7 then further identifies the reason mortals tend to fall into this confusion—by relying upon their senses, rather than rational accounts. Consider again our plurality of people and mountains. According to the Regressive version of the Dichotomy Paradox, the runner cannot even take a first step. The current standard treatment, the 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. Parmenides’ goddess endorses the first route, which recognizes that “what-is” is, and that it must be (it is not to not be), on the grounds that it is completely trustworthy and persuasive. Parmenides is not here denying the more limited claims by Heraclitus—that opposites can work together to produce some new harmony. But nobody in that century or the next could adequately explain what an infinitesimal was. The class of hyperreal numbers contains counterparts of the reals, but in addition it contains any number that is the sum, or difference, of both a standard real number and an infinitesimal number, such as 3 + h and 3 – 4h2. More telling, while it is still certainly possible to justify some of these properties on the grounds that thinking “what is not” is not allowed in the conception, others are far more problematic. Given the overall reconstruction of the poem as it stands, there appears to be a counter-intuitive account of “reality” offered in the central section (Reality)—one which describes some entity (or class of such) with specific predicational perfections: eternal—ungenerated, imperishable, a continuous whole, unmoving, unique, perfect, and uniform. Aristotle’s complaint can be expressed succinctly this way: Zeno was correct to suppose that at any time a runner’s path can be divided anywhere, but incorrect to suppose the path can be divided everywhere at the same time. According to the first, which is the standard interpretation, when a bushel of millet (or wheat) grains falls out of its container and crashes to the floor, it makes a sound. Simplicius says this argument is due to Zeno even though it is in Aristotle (On Generation and Corruption, 316a15-34, 316b34 and 325a8-12) and is not attributed there to Zeno, which is odd. Baird, Forrest E., and Walter Kaufmann, eds. This paradox is also called the Paradox of Denseness. Suppose there exist many things rather than, as Parmenides would say, just one thing. Physics flourished down to the time of Archelaus; ethics, as we have said, started with Socrates; while dialectic goes as far back as Zeno of Elea. Sending female to mix with male, and again in turn. It should also be taken as well-founded that the Opinion is epistemically inferior. He might have said the reason is (i) that there is no last goal in the sequence of sub-goals, or, perhaps (ii) that it would take too long to achieve all the sub-goals, or perhaps (iii) that covering all the sub-paths is too great a distance to run. Vlastos also comments that “there is nothing in our sources that states or implies that any development in Greek mathematics (as distinct from philosophical opinions about mathematics) was due to Zeno’s influence.”. Grünbaum, Adolf (1970). First, it is commonly claimed that Xenophanes was a philosophically-oriented poet, in contrast to Parmenides—a “genuine philosopher” who simply used poetry as a vehicle for communicating his thoughts. Following the arguments of Aletheia, the goddess explicitly ends her “trustworthy account and thought about reality” and commands the youth from there on, “hearing the deceptive arrangement of her words,” to learn mortal opinions (C 8.50-52). Diogenes explicitly reports that Xenophanes lived at two locations in Sicily (near Elea) and that Xenophanes even wrote a poem on the founding of Elea, as well as his native Colophon. However, the world as it appears also exists in some ontologically inferior manner. The vast majority of interpreters have followed both these moves. Rather, he seems to be claiming that even thinking of opposites requires thinking in terms of a more fundamental distinction—“what is” and “what is not”—and this inevitably leads to contradiction. This universal denigration is first introduced at C 8.34-41 on the traditional reconstruction (For a proposal to relocate these lines to Opinion, see Palmer’s 2009 discussion of “Ebert’s Proposal”). (More will be said about assumption (5) in Section 5c when we discuss supertasks.). Plato and Aristotle may have had access to the book, but Plato did not state any of the arguments, and Aristotle’s presentations of the arguments are very compressed. The period lasted about two hundred years. Purported Orphic parallels turn on Orphism’s revelatory journeys to the underworld, as well as initiations led by Night, and such influences are far more likely to have been relevantly parallel. Perhaps more telling, the apparent direct challenges and differences between these thinkers are belied by the similarities. Cantor argued that any potential infinity must be interpreted as varying over a predefined fixed set of possible values, a set that is actually infinite. Both C 9/DK 10 and C 10/DK 11 variably promise that the youth will learn about the generation/origins of the aether, along with many of its components (sun, moon, stars, and so forth). The speed during an instant or in an instant, which is what Zeno is calling for, would be 0/0 and is undefined. Interesting issues arise when we bring in Einstein’s theory of relativity and consider a bifurcated supertask. How does Zeno’s runner complete the trip if there is no final step or last member of the infinite sequence of steps (intervals and goals)? By the time Achilles reaches that location, the tortoise will have moved on to yet another location, and so on forever. Zeno is the greatest figure of the Eleatic School. The argument has been called the Paradox of Parts and Wholes, but it has no traditional name. This seems to be based primarily upon the fact that Xenophanes also wrote silloi (satirical poetry), which do not always have obvious or exclusively philosophical themes. From the House of Night—far below the center of the Earth—the Heliades would follow an ascending arc to the eastern edge of the Earth, where the sun/moon rise. Zeno’s paradoxes caused mistrust in infinites, and this mistrust has influenced the contemporary movements of constructivism, finitism, and nonstandard analysis, all of which affect the treatment of Zeno’s paradoxes. Both are critical of common mortal views, and both seem to acknowledge a distinction between mortal and divine knowledge. The context here seems to be that by learning the particular account offered in Opinion, which shares the mistakes any mortal account might possess, and/or which makes the failure of mortal accounts most evident, the deceptive account on offer is worth learning so as to best know how to avoid the mistakes other mortals make. From this standpoint, Dedekind’s 1872 axiom of continuity and his definition of real numbers as certain infinite subsets of rational numbers suggested to Cantor and then to many other mathematicians that arbitrarily large sets of rational numbers are most naturally seen to be subsets of an actually infinite set of rational numbers. U. S. A. In order to pass through these “aethereal” gates, the Heliades must persuade Justice to unlock the doors with soft words. The Standard Solution says we first should ask Zeno to be clearer about what he is dividing. The tortoise is a later commentator’s addition. Aristotle’s treatment of The Paradox of the Moving Rows is basically in agreement with the Standard Solution to that paradox–that Zeno did not appreciate the difference between speed and relative speed. That mortals erroneously believe otherwise is a result of relying on their fallible senses instead of reason. Let’s assume the object is one-dimensional, like a path. As J. O. (pp. “But do keep your thought from this way of enquiry. Having challenged this status quo, he goes on to advocate a new arrangement for the poem, moving some passages which make true cosmological claims out of. When Aristotle made this claim and used it to treat Zeno’s paradoxes, there was no better solution to the Achilles Paradox, and a better solution would not be discovered for many more centuries. Immediately following the Proem (C/DK 1), the poem moves into its central philosophical section: Reality (C. 2-C 8.49). Little is known for certain about Zeno's life. When exactly Parmenides was born is far more controversial. The suspicion that these lines might help shed light on the crucial relationship between Reality and Opinion is well-warranted. In other words, things are dense and there is no definite or fixed number of them, so they will be “unlimited.” This is a contradiction, because the plurality would be both limited and unlimited. It is not that mortals themselves are, in their cognitive failures, “backwards-turning,” in either case. It is uncontroversial that Reality is positively endorsed, and it is equally clear that Opinion is negatively presented in relation to Aletheia. Overall, the Proem has far more commonly been minimized, dismissed as irrelevant, and/or entirely ignored by ancients and moderns alike, probably because they saw no immediately obvious philosophical content or guidance for understanding the rest of the poem within it. The reason is that the runner must first reach half the distance to the goal, but when there he must still cross half the remaining distance to the goal, but having done that the runner must cover half of the new remainder, and so on. The A-D Paradox: Select Interpretative Strategies and their Difficulties, Parmenides’ Place in the Historical Narrative, Parmenides’ Influence on Select Successors, A work focused solely on explaining the logical aspects of. Furthermore, there is at least some textual evidence that might be understood to suggest Opinion should not be treated as negatively as the passages considered so far would suggest. One of the best sources in English of primary material on the Pre-Socratics. And in the middle of these is a goddess, who governs all things. That is, how to reconcile: a) the positively endorsed metaphysical arguments of Reality, which describe some unified, unchanging, motionless, and eternal “reality,” with b) the ambiguously negative (or perhaps, ambivalent) treatment of the ensuing “cosmology” in Opinion, which incorporates the very principles Reality denies. At most, one could argue that Anaximander’s views influenced and/or provided a particular target for Parmenides to reject, as some modern scholars have suggested. Point (3) is about the time it took for philosophers of science to reject the demand, favored by Ernst Mach and most Logical Positivists, that each meaningful term in science must have “empirical meaning.” This was the demand that each physical concept be separately definable with observation terms. This leaves little time in between, if any, for Parmenides to become aware of or be inspired to challenge Heracliteanism. From the perspective of the Standard Solution, the most significant lesson learned by researchers who have tried to solve Zeno’s paradoxes is that the way out requires revising many of our old theories and their concepts. Regarding the Paradox of the Grain of Millet, Aristotle said that parts need not have all the properties of the whole, and so grains need not make sounds just because bushels of grains do. It is also quite difficult to offer a convincing explanation for what possible grounds Parmenides could have for ascribing superiority to his own account of the apparent world offered in Opinion, in comparison to any other mortal offering of his time. Benacerraf suggests that an answer depends on what we ordinarily mean by the term “completing a task.” If the meaning does not require that tasks have minimum times for their completion, then maybe Russell is right that some supertasks can be completed, he says; but if a minimum time is always required, then Russell is mistaken because an infinite time would be required. These accomplishments by Cantor are why he (along with Dedekind and Weierstrass) is said by Russell to have “solved Zeno’s Paradoxes.”. See McLaughlin (1994) for how Zeno’s paradoxes may be treated using infinitesimals. Identification of divine entities certainly does not end after Parmenides either, as the systems of the Pluralists and Atomists continue to associate their fundamental “parmenidean” entities with divinity. Herodotus reports that members of the Phocaean tribe established this settlement ca. The pervasiveness of such “two-world” interpretative accounts likely says far more about Plato’s extensive influence, as well as the importance of finding some way out of the world-denying entailments, than it does about Parmenides’ own novelty. In this section, Parmenides’ positively endorsed epistemic and metaphysical claims are outlined. Whereas Anaximander envisions “justice” as a regulatory, ontological compensation arising from the competition for existential pervasiveness amongst opposites, this conception of natural balance via justice is entirely absent in Parmenides’ works. Whatever Parmenides himself held, however, it is clear that his writings did lead some to adopt this view. The proofs in Euclid’s Elements, for example, used only potentially infinite procedures. He further adds that “what is” cannot undergo psychological changes (such as pain, distress, or health) and explicitly denies the existence of void. Two Greek philosophers bore this name, Zeno of Elea and Zeno of Citium. Aristotle‘s treatment said Zeno should have assumed instead that there are only potential infinities, so 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. This is the central claim about mortal errors by the goddess, and it is undeniable that, in order for mortals to incorrectly specify the nature of Aletheia’s subject, mortals must have some conception of and familiarity with that relevant object (or type of entity). The more points there are on a line, the longer the line is. Zeno’s paradoxes are often pointed to for a case study in how a philosophical problem has been solved, even though the solution took over two thousand years to materialize. In conclusion, are there two adequate but different solutions to Zeno’s paradoxes, Aristotle’s Solution and the Standard Solution? Wisdom points out (1953, p. 23), “At the same time it became clear that [Leibniz’s and] Newton’s theory, with suitable amendments and additions, could be soundly based” provided Leibniz’s infinitesimals and Newton’s fluxions were removed. Here, Parmenides positively endorses certain epistemic guidelines for inquiry, which he then uses to argue for his famous metaphysical claims—that “what is” (whatever is referred to by the word “this”) cannot be in motion, change, come-to-be, perish, lack uniformity, and so forth. If so, assume the three objects A, B, and C are adjacent to each other in their tracks, and each A, B and C body are occupying a space that is one atom long. The reciprocal of an infinitesimal is an infinite hyperreal number. It is absurd for there to be numbers that are bigger than every integer. For instance, a fair amount has been written on the parallels between the chariot’s path and Babylonian Sun-mythology, as well as how the Proem supposedly contains Orphic and/or Shamanistic themes. For example, the infinitesimal dx is treated as being equal to zero when it is declared that x + dx = x, but is treated as not being zero when used in the denominator of the fraction [f(x + dx) – f(x)]/dx which is used in the derivative of the function f. In addition, consider the seemingly obvious Archimedean property of pairs of positive numbers: given any two positive numbers A and B, if you add enough copies of A, then you can produce a sum greater than B. 136). This controversial issue about interpreting Zeno’s purposes will not be pursued further in this article, and Plato’s classical interpretation will be assumed. As noted in the summary of the Proem above, there are two particularly difficult lines (C 1.31-32) which may be understood as suggesting some positive value for Opinion, despite its lacking in comparison to Reality. Diogenes reports that upon completion of his book, Heraclitus deposited it (apparently the only copy) in the Temple of Artemis (the Artemisium). And was he superficial or profound? An important philosophical issue is whether the paradoxes should be solved by the Standard Solution or instead by assuming that a line is not composed of points but of intervals, and whether use of infinitesimals is essential to a proper understanding of the paradoxes. This can be understood as referring directly to Heraclitus’ paradoxical aphorisms, which describe things like rivers and roads as being both simultaneously the same, but yet not (B39; B60). The usual way out of this paradox is to reject that controversial assumption. “What is” is the subject of 8.38b-41, which is uncontroversially the entity described in Aletheia, and thus necessary being on Palmer’s view. The bushel is composed of individual grains, so they, too, make an audible sound. Whereas the Milesians sought to explain cosmology and physics by identifying the arche (“origin” or “first principle”) from which all things originated (and possibly, remained constituted by), the only section of Parmenides’ poem that could provide an alternative or competing cosmological account (Opinion) is supposed to be fundamentally and deeply flawed, and offered for rejection on some grounds. He is challenging everyone’s understanding. The only ancient response to the content of the Proem is from the Pyrrhonian Skeptic Sextus Empiricus (2nd cn. (1953). A criticism of Thomson’s interpretation of his infinity machines and the supertasks involved, plus an introduction to the literature on the topic. That controversy has sparked a related discussion about whether there could be a machine that can perform an infinite number of tasks in a finite time. Sextus describes the chariot ride as a journey towards knowledge of all things, with Parmenides’ irrational desires and appetites represented as mares, and the path of the goddess upon which he travels as representative of the guidance provided by philosophical reasoning. In the brief introduction to Parmenides, the likelihood of a Xenophanean influence is stressed, the difficulty in reconciling, Cordero advances several theses in this paper. This deceptive arrangement could be understood to apply only to the goddess’ presentation of the account. Nevertheless, the vast majority of today’s practicing mathematicians routinely use nonconstructive mathematics. The Milesians tended to treat their fundamental and eternal arche as divine entities. But Zeno’s assumption that places have places was common in ancient Greece at the time, and Zeno is to be praised for showing that it is a faulty assumption. “Naming” only one opposite (for example, Light) seems to require thinking of it in terms of its opposite (for example, “Light” is “not-dark”), which is contrary to the path of only thinking of “what is,” and never “what is not” (compare Mourelatos 1979). His Life. In any case, these lines are probably best dealt with once one already has settled upon an interpretative stance for the overall poem given the rest of the evidence. Here are some of the issues. The continuum is a very special set; it is the standard model of the real numbers. Not only are the bulk of these lines (1.1-28) not quoted by any other ancient source, but their content is not even mentioned in passing. While some have attempted to claim that Opinion satisfies this on account of its dualistic nature, which is second-best to Reality’s monistic claims, this approach fails to account for how Opinion could possibly be superior to any other dualistic account. Reeve, C.D.C, and Patrick Lee Miller, eds. This paradox is generally considered to be one of Zeno’s weakest paradoxes, and it is now rarely discussed. This argument shows, he believes, that anyone who believes Achilles will succeed in catching the tortoise and who believes more generally that motion is physically possible is the victim of illusion. However, even if the Greek is read along these lines, it remains to be determined whether this value is based upon some substantial value in the account itself (there is some sense or perspective in which it is true), or merely some pragmatic and/or instructive value (for example, it is worthwhile to know what is wrong and why, so as to avoid not falling into such errors). Aristotle’s treatment by disallowing actual infinity while allowing potential infinity was clever, and it satisfied nearly all scholars for 1,500 years, being buttressed during that time by the Church’s doctrine that only God is actually infinite. In modern real analysis, a continuum is composed of points, but Aristotle, ever the advocate of common sense reasoning, claimed that a continuum cannot be composed of points. He is using this image to describe a way in which mortals should not think about things. Not only are these gates traditionally located immediately in front of the House of Night, but the mention of the chasm that lies beyond them is an apt poetical description of the completely dark House of Night. The historical record does not tell us which of these was Zeno’s real assumption, but they are all false assumptions, according to the Standard Solution. The following–once presumably safe–intuitions or assumptions must be rejected: Item (8) was undermined when it was discovered that the continuum implies the existence of fractal curves. However, closer inspection of these supposed parallels tends to undermine the thesis that Anaximander is particularly influential upon, or a specific target for, Parmenides. That is, a moral command from a fact in the world. Though lengthy quotations strongly suggest a certain internal structure, there is certainly some room for debate with respect to proper placement, in particular amongst the shorter fragments that do not share any common content/themes with the others. “Tasks and Super-Tasks,”. Times have only the values that they can in principle be measured to have; and all measurements produce rational numbers within a margin of error. Anaxagoras posits an extensive number of fundamental and eternal seeds, every kind of which is found in even the smallest portion of matter, and which give rise to objects of perception according to whichever kind of seed dominates the mixture at a particular spatio-temporal location, in accordance with the will of Nous. (2000) “Are there Really Instantaneous Velocities?”. Let’s take a short interlude and introduce Dedekind’s key, new idea that he discovered in the 1870s about the reals and their relationship to the rationals. Finally, the allegorical accounts available tend to offer little if any substantive guidance or interpretative weight for reading the poem overall. This article takes no side on this dispute and speaks of Aristotle’s “treatment.”. How Zeno’s paradoxes may be explained using a contemporary theory of Leibniz’s infinitesimals. This is key to solving the Dichotomy Paradox according to the Standard Solution. 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