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Zeno's paradoxes

"Arrow paradox" redirects here. For other uses, see Arrow paradox (disambiguation).

Zeno's paradoxes are a set of philosophical problems generally thought to have been devised by Greek philosopher Zeno of Elea (c. 490–430 BC) to support Parmenides' doctrine that contrary to the evidence of one's senses, the belief in plurality and change is mistaken, and in particular that motion is nothing but an illusion. It is usually assumed, based on Plato's Parmenides (128a–d), that Zeno took on the project of creating these paradoxes because other philosophers had created paradoxes against Parmenides' view. Thus Plato has Zeno say the purpose of the paradoxes "is to show that their hypothesis that existences are many, if properly followed up, leads to still more absurd results than the hypothesis that they are one." Plato has Socrates claim that Zeno and Parmenides were essentially arguing exactly the same point. Some of Zeno's nine surviving paradoxes (preserved in Aristotle's Physics and Simplicius's commentary thereon) are essentially equivalent to one another. Aristotle offered a refutation of some of them. Three of the strongest and most famous—that of Achilles and the tortoise, the Dichotomy argument, and that of an arrow in flight—are presented in detail below.

Zeno's arguments are perhaps the first examples of a method of proof called reductio ad absurdum, also known as proof by contradiction. They are also credited as a source of the dialectic method used by Socrates. Some mathematicians and historians, such as Carl Boyer, hold that Zeno's paradoxes are simply mathematical problems, for which modern calculus provides a mathematical solution. Some philosophers, however, say that Zeno's paradoxes and their variations (see Thomson's lamp) remain relevant metaphysical problems. The origins of the paradoxes are somewhat unclear. Diogenes Laërtius, a fourth source for information about Zeno and his teachings, citing Favorinus, says that Zeno's teacher Parmenides was the first to introduce the paradox of Achilles and the tortoise. But in a later passage, Laërtius attributes the origin of the paradox to Zeno, explaining that Favorinus disagrees.

Contents

Dichotomy paradox

That which is in locomotion must arrive at the half-way stage before it arrives at the goal.

as recounted by Aristotle, Physics VI:9, 239b10

Suppose Atalanta wishes to walk to the end of a path. Before she can get there, she must get halfway there. Before she can get halfway there, she must get a quarter of the way there. Before traveling a quarter, she must travel one-eighth; before an eighth, one-sixteenth; and so on.

The dichotomy

The resulting sequence can be represented as:

{ , 1 16 , 1 8 , 1 4 , 1 2 , 1 } {\displaystyle \left\{\cdots ,{\frac {1}{16}},{\frac {1}{8}},{\frac {1}{4}},{\frac {1}{2}},1\right\}}

This description requires one to complete an infinite number of tasks, which Zeno maintains is an impossibility.

This sequence also presents a second problem in that it contains no first distance to run, for any possible (finite) first distance could be divided in half, and hence would not be first after all. Hence, the trip cannot even begin. The paradoxical conclusion then would be that travel over any finite distance can be neither completed nor begun, and so all motion must be an illusion.

This argument is called the "Dichotomy" because it involves repeatedly splitting a distance into two parts. An example with the original sense can be found in an asymptote. It is also known as the Race Course paradox.

Achilles and the tortoise

"Achilles and the Tortoise" redirects here. For other uses, see Achilles and the Tortoise (disambiguation).
Achilles and the tortoise

In a race, the quickest runner can never over­take the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead.

as recounted by Aristotle, Physics VI:9, 239b15

In the paradox of Achilles and the tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 meters, for example. Suppose that each racer starts running at some constant speed, one faster than the other. After some finite time, Achilles will have run 100 meters, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say 2 meters. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles arrives somewhere the tortoise has been, he still has some distance to go before he can even reach the tortoise. As Aristotle noted, this argument is similar to the Dichotomy. It lacks, however, the apparent conclusion of motionlessness.

Arrow paradox

The arrow
Not to be confused with other paradoxes of the same name.

If everything when it occupies an equal space is at rest at that instant of time, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless at that instant of time and at the next instant of time but if both instants of time are taken as the same instant or continuous instant of time then it is in motion.

as recounted by Aristotle, Physics VI:9, 239b5

In the arrow paradox, Zeno states that for motion to occur, an object must change the position which it occupies. He gives an example of an arrow in flight. He states that in any one (duration-less) instant of time, the arrow is neither moving to where it is, nor to where it is not. It cannot move to where it is not, because no time elapses for it to move there; it cannot move to where it is, because it is already there. In other words, at every instant of time there is no motion occurring. If everything is motionless at every instant, and time is entirely composed of instants, then motion is impossible.

Whereas the first two paradoxes divide space, this paradox starts by dividing time—and not into segments, but into points.

Paradox of place

From Aristotle:

If everything that exists has a place, place too will have a place, and so on ad infinitum.

Paradox of the grain of millet

Description of the paradox from the Routledge Dictionary of Philosophy:

The argument is that a single grain of millet makes no sound upon falling, but a thousand grains make a sound. Hence a thousand nothings become something, an absurd conclusion.

Aristotle's refutation:

Zeno is wrong in saying that there is no part of the millet that does not make a sound: for there is no reason why any such part should not in any length of time fail to move the air that the whole bushel moves in falling. In fact it does not of itself move even such a quantity of the air as it would move if this part were by itself: for no part even exists otherwise than potentially.

Description from Nick Huggett:

This is a Parmenidean argument that one cannot trust one's sense of hearing. Aristotle's response seems to be that even inaudible sounds can add to an audible sound.

The moving rows (or stadium)

The moving rows

From Aristotle:

... concerning the two rows of bodies, each row being composed of an equal number of bodies of equal size, passing each other on a race-course as they proceed with equal velocity in opposite directions, the one row originally occupying the space between the goal and the middle point of the course and the other that between the middle point and the starting-post. This...involves the conclusion that half a given time is equal to double that time.

For an expanded account of Zeno's arguments as presented by Aristotle, see Simplicius's commentary On Aristotle's Physics.[full citation needed]

Diogenes the Cynic

According to Simplicius, Diogenes the Cynic said nothing upon hearing Zeno's arguments, but stood up and walked, in order to demonstrate the falsity of Zeno's conclusions (see solvitur ambulando). To fully solve any of the paradoxes, however, one needs to show what is wrong with the argument, not just the conclusions. Through history, several solutions have been proposed, among the earliest recorded being those of Aristotle and Archimedes.

Aristotle

Aristotle (384 BC−322 BC) remarked that as the distance decreases, the time needed to cover those distances also decreases, so that the time needed also becomes increasingly small.[failed verification] Aristotle also distinguished "things infinite in respect of divisibility" (such as a unit of space that can be mentally divided into ever smaller units while remaining spatially the same) from things (or distances) that are infinite in extension ("with respect to their extremities"). Aristotle's objection to the arrow paradox was that "Time is not composed of indivisible nows any more than any other magnitude is composed of indivisibles."

Archimedes

Before 212 BC, Archimedes had developed a method to derive a finite answer for the sum of infinitely many terms that get progressively smaller. (See: Geometric series, 1/4 + 1/16 + 1/64 + 1/256 + · · ·, The Quadrature of the Parabola.) His argument, applying the method of exhaustion to prove that the infinite sum in question is equal to the area of a particular square, is largely geometric but quite rigorous. Today's analysis achieves the same result, using limits (see convergent series). These methods allow the construction of solutions based on the conditions stipulated by Zeno, i.e. the amount of time taken at each step is geometrically decreasing.

Thomas Aquinas

Thomas Aquinas, commenting on Aristotle's objection, wrote "Instants are not parts of time, for time is not made up of instants any more than a magnitude is made of points, as we have already proved. Hence it does not follow that a thing is not in motion in a given time, just because it is not in motion in any instant of that time."

Bertrand Russell

Bertrand Russell offered what is known as the "at-at theory of motion". It agrees that there can be no motion "during" a durationless instant, and contends that all that is required for motion is that the arrow be at one point at one time, at another point another time, and at appropriate points between those two points for intervening times. In this view motion is just change in position over time.

Hermann Weyl

Another proposed solution is to question one of the assumptions Zeno used in his paradoxes (particularly the Dichotomy), which is that between any two different points in space (or time), there is always another point. Without this assumption there are only a finite number of distances between two points, hence there is no infinite sequence of movements, and the paradox is resolved. According to Hermann Weyl, the assumption that space is made of finite and discrete units is subject to a further problem, given by the "tile argument" or "distance function problem". 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. Jean Paul Van Bendegem has argued that the Tile Argument can be resolved, and that discretization can therefore remove the paradox.

Henri Bergson

An alternative conclusion, proposed by Henri Bergson in his 1896 book Matter and Memory, is that, while the path is divisible, the motion is not. In this argument, instants in time and instantaneous magnitudes do not physically exist. An object in relative motion cannot have an instantaneous or determined relative position, and so cannot have its motion fractionally dissected.

Peter Lynds

In 2003, Peter Lynds put forth a very similar argument: all of Zeno's motion paradoxes are resolved by the conclusion that instants in time and instantaneous magnitudes do not physically exist. Lynds argues that an object in relative motion cannot have an instantaneous or determined relative position (for if it did, it could not be in motion), and so cannot have its motion fractionally dissected as if it does, as is assumed by the paradoxes. For more about the inability to know both speed and location, see Heisenberg uncertainty principle.

Nick Huggett

Nick Huggett argues that Zeno is assuming the conclusion when he says that objects that occupy the same space as they do at rest must be at rest.

Infinite processes remained theoretically troublesome in mathematics until the late 19th century. With the epsilon-delta definition of limit, Weierstrass and Cauchy developed a rigorous formulation of the logic and calculus involved. These works resolved the mathematics involving infinite processes.

While mathematics can calculate where and when the moving Achilles will overtake the Tortoise of Zeno's paradox, philosophers such as Kevin Brown and Francis Moorcroft claim that mathematics does not address the central point in Zeno's argument, and that solving the mathematical issues does not solve every issue the paradoxes raise.

Popular literature often misrepresents Zeno's arguments. For example, Zeno is often said to have argued that the sum of an infinite number of terms must itself be infinite–with the result that not only the time, but also the distance to be travelled, become infinite. However, none of the original ancient sources has Zeno discussing the sum of any infinite series. Simplicius has Zeno saying "it is impossible to traverse an infinite number of things in a finite time". This presents Zeno's problem not with finding the sum, but rather with finishing a task with an infinite number of steps: how can one ever get from A to B, if an infinite number of (non-instantaneous) events can be identified that need to precede the arrival at B, and one cannot reach even the beginning of a "last event"?

A humorous take is offered by Tom Stoppard in his play Jumpers (1972), in which the principal protagonist, the philosophy professor George Moore, suggests that according to Zeno's paradox, Saint Sebastian, a 3rd Century Christian saint martyred by being shot with arrows, died of fright.

Debate continues on the question of whether or not Zeno's paradoxes have been resolved. In The History of Mathematics: An Introduction (2010) Burton writes, "Although Zeno's argument confounded his contemporaries, a satisfactory explanation incorporates a now-familiar idea, the notion of a 'convergent infinite series.'".

Bertrand Russell offered a "solution" to the paradoxes based on the work of Georg Cantor, but Brown concludes "Given the history of 'final resolutions', from Aristotle onwards, it's probably foolhardy to think we've reached the end. It may be that Zeno's arguments on motion, because of their simplicity and universality, will always serve as a kind of 'Rorschach image' onto which people can project their most fundamental phenomenological concerns (if they have any)."

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Ancient Chinese philosophers from the Mohist School of Names during the Warring States period of China (479-221 BC) developed equivalents to some of Zeno's paradoxes. The scientist and historian Sir Joseph Needham, in his Science and Civilisation in China, describes an ancient Chinese paradox from the surviving Mohist School of Names book of logic which states, in the archaic ancient Chinese script, "a one-foot stick, every day take away half of it, in a myriad ages it will not be exhausted." Several other paradoxes from this philosophical school (more precisely, movement) are known, but their modern interpretation is more speculative.

Main article: Quantum Zeno effect

In 1977, physicists E. C. George Sudarshan and B. Misra discovered that the dynamical evolution (motion) of a quantum system can be hindered (or even inhibited) through observation of the system. This effect is usually called the "quantum Zeno effect" as it is strongly reminiscent of Zeno's arrow paradox. This effect was first theorized in 1958.

In the field of verification and design of timed and hybrid systems, the system behaviour is called Zeno if it includes an infinite number of discrete steps in a finite amount of time. Some formal verification techniques exclude these behaviours from analysis, if they are not equivalent to non-Zeno behaviour. In systems design these behaviours will also often be excluded from system models, since they cannot be implemented with a digital controller.

What the Tortoise Said to Achilles, written in 1895 by Lewis Carroll, was an attempt to reveal an analogous paradox in the realm of pure logic. If Carroll's argument is valid, the implication is that Zeno's paradoxes of motion are not essentially problems of space and time, but go right to the heart of reasoning itself. Douglas Hofstadter made Carroll's article a centrepiece of his book Gödel, Escher, Bach: An Eternal Golden Braid, writing many more dialogues between Achilles and the Tortoise to elucidate his arguments. Hofstadter connects Zeno's paradoxes to Gödel's incompleteness theorem in an attempt to demonstrate that the problems raised by Zeno are pervasive and manifest in formal systems theory, computing and the philosophy of mind.

  1. Parmenides 128d
  2. Parmenides 128a–b
  3. Aristotle's Physics "Physics" by Aristotle translated by R. P. Hardie and R. K. Gaye
  4. "Greek text of "Physics" by Aristotle (refer to §4 at the top of the visible screen area)". Archived from the original on 2008-05-16.
  5. ([fragment 65], Diogenes Laërtius. IX Archived 2010-12-12 at the Wayback Machine 25ff and VIII 57).
  6. Boyer, Carl (1959).The History of the Calculus and Its Conceptual Development. Dover Publications. p. 295. ISBN 978-0-486-60509-8. Retrieved2010-02-26. If the paradoxes are thus stated in the precise mathematical terminology of continuous variables (...) the seeming contradictions resolve themselves.
  7. Brown, Kevin. "Zeno and the Paradox of Motion". Reflections on Relativity. Archived from the original on 2012-12-05. Retrieved2010-06-06.
  8. Moorcroft, Francis. "Zeno's Paradox". Archived from the original on 2010-04-18.
  9. Papa-Grimaldi, Alba (1996). "Why Mathematical Solutions of Zeno's Paradoxes Miss the Point: Zeno's One and Many Relation and Parmenides' Prohibition"(PDF). The Review of Metaphysics. 50: 299–314.
  10. Diogenes Laërtius, Lives, 9.23 and 9.29.
  11. Lindberg, David (2007). The Beginnings of Western Science (2nd ed.). University of Chicago Press. p. 33. ISBN 978-0-226-48205-7.
  12. Huggett, Nick (2010). "Zeno's Paradoxes: 3.1 The Dichotomy". Stanford Encyclopedia of Philosophy. Retrieved2011-03-07.
  13. Huggett, Nick (2010). "Zeno's Paradoxes: 3.2 Achilles and the Tortoise". Stanford Encyclopedia of Philosophy. Retrieved2011-03-07.
  14. Aristotle. "Physics". The Internet Classics Archive. Zeno's reasoning, however, is fallacious, when he says that if everything when it occupies an equal space is at rest, and if that which is in locomotion is always occupying such a space at any moment, the flying arrow is therefore motionless. This is false, for time is not composed of indivisible moments any more than any other magnitude is composed of indivisibles.
  15. Laërtius, Diogenes (c. 230). "Pyrrho". Lives and Opinions of Eminent Philosophers. IX. passage 72. ISBN 1-116-71900-2.
  16. Huggett, Nick (2010). "Zeno's Paradoxes: 3.3 The Arrow". Stanford Encyclopedia of Philosophy. Retrieved2011-03-07.
  17. Aristotle Physics IV:1, 209a25
  18. The Michael Proudfoot, A.R. Lace. Routledge Dictionary of Philosophy. Routledge 2009, p. 445
  19. Aristotle Physics VII:5, 250a20
  20. Huggett, Nick, "Zeno's Paradoxes", The Stanford Encyclopedia of Philosophy (Winter 2010 Edition), Edward N. Zalta (ed.), http://plato.stanford.edu/entries/paradox-zeno/#GraMil
  21. Aristotle Physics VI:9, 239b33
  22. Aristotle. Physics 6.9
  23. Aristotle's observation that the fractional times also get shorter does not guarantee, in every case, that the task can be completed. One case in which it does not hold is that in which the fractional times decrease in a harmonic series, while the distances decrease geometrically, such as: 1/2 s for 1/2 m gain, 1/3 s for next 1/4 m gain, 1/4 s for next 1/8 m gain, 1/5 s for next 1/16 m gain, 1/6 s for next 1/32 m gain, etc. In this case, the distances form a convergent series, but the times form a divergent series, the sum of which has no limit.[original research?] Archimedes developed a more explicitly mathematical approach than Aristotle.
  24. Aristotle. Physics 6.9; 6.2, 233a21-31
  25. Aristotle. Physics. VI. Part 9 verse: 239b5. ISBN 0-585-09205-2.
  26. George B. Thomas, Calculus and Analytic Geometry, Addison Wesley, 1951
  27. Aquinas. Commentary on Aristotle's Physics, Book 6.861
  28. Huggett, Nick (1999). Space From Zeno to Einstein. ISBN 0-262-08271-3.
  29. Salmon, Wesley C. (1998). Causality and Explanation. p. 198. ISBN 978-0-19-510864-4.
  30. Van Bendegem, Jean Paul (17 March 2010). "Finitism in Geometry". Stanford Encyclopedia of Philosophy. Retrieved2012-01-03.
  31. Cohen, Marc (11 December 2000). "ATOMISM". History of Ancient Philosophy, University of Washington. Archived from the original on July 12, 2010. Retrieved2012-01-03.
  32. van Bendegem, Jean Paul (1987). "Discussion:Zeno's Paradoxes and the Tile Argument". Philosophy of Science. Belgium. 54 (2): 295–302. doi:10.1086/289379. JSTOR 187807.
  33. Bergson, Henri (1896). Matière et Mémoire [Matter and Memory](PDF). Translation 1911 by Nancy Margaret Paul & W. Scott Palmer. George Allen and Unwin. pp. 77–78 of the PDF.
  34. "Zeno's Paradoxes: A Timely Solution". January 2003.
  35. Lynds, Peter. Time and Classical and Quantum Mechanics: Indeterminacy vs. Discontinuity. Foundations of Physics Letter s (Vol. 16, Issue 4, 2003). doi:10.1023/A:1025361725408
  36. Time’s Up, Einstein, Josh McHugh, Wired Magazine, June 2005
  37. S E Robbins (2004) On time, memory and dynamic form. Consciousness and Cognition 13(4), 762-788: "Lynds, his reviewers and consultants (e.g., J.J.C. Smart) are apparently unaware of his total precedence by Bergson"
  38. Lee, Harold (1965). "Are Zeno's Paradoxes Based on a Mistake?". Mind. Oxford University Press. 74 (296): 563–570. doi:10.1093/mind/LXXIV.296.563. JSTOR 2251675.
  39. B Russell (1956) Mathematics and the metaphysicians in "The World of Mathematics" (ed. J R Newman), pp 1576-1590.
  40. Benson, Donald C. (1999).The Moment of Proof : Mathematical Epiphanies. New York: Oxford University Press. p. 14. ISBN 978-0195117219.
  41. Huggett, Nick (2010). "Zeno's Paradoxes: 5. Zeno's Influence on Philosophy". Stanford Encyclopedia of Philosophy. Retrieved2011-03-07.
  42. Burton, David, A History of Mathematics: An Introduction, McGraw Hill, 2010, ISBN 978-0-07-338315-6
  43. Russell, Bertrand (2002) [First published in 1914 by The Open Court Publishing Company]. "Lecture 6. The Problem of Infinity Considered Historically". Our Knowledge of the External World: As a Field for Scientific Method in Philosophy. Routledge. p. 169. ISBN 0-415-09605-7.
  44. "School of Names > Miscellaneous Paradoxes (Stanford Encyclopedia of Philosophy)". plato.stanford.edu. Retrieved2020-01-30.
  45. Sudarshan, E. C. G.; Misra, B. (1977). "The Zeno's paradox in quantum theory"(PDF). Journal of Mathematical Physics. 18 (4): 756–763. Bibcode:1977JMP....18..756M. doi:10.1063/1.523304. OSTI 7342282.
  46. W.M.Itano; D.J. Heinsen; J.J. Bokkinger; D.J. Wineland (1990). "Quantum Zeno effect"(PDF). Physical Review A. 41 (5): 2295–2300. Bibcode:1990PhRvA..41.2295I. doi:10.1103/PhysRevA.41.2295. PMID 9903355. Archived from the original(PDF) on 2004-07-20. Retrieved2004-07-23.
  47. Khalfin, L.A. (1958). "Contribution to the Decay Theory of a Quasi-Stationary State". Soviet Phys. JETP. 6: 1053. Bibcode:1958JETP....6.1053K.
  48. 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.". Handbook of dynamic system modeling. Chapman & Hall/CRC Computer and Information Science (hardcover ed.). Boca Raton, Florida, USA: CRC Press. pp. 15–22 to 15–23. ISBN 978-1-58488-565-8. Retrieved2010-03-05.
  49. Lamport, Leslie (2002). Specifying Systems(PDF). Microsoft Research. Addison-Wesley. p. 128. ISBN 0-321-14306-X. Retrieved2010-03-06.
  50. Zhang, Jun; Johansson, Karl; Lygeros, John; Sastry, Shankar (2001). "Zeno hybrid systems"(PDF). International Journal for Robust and Nonlinear Control. 11 (5): 435. doi:10.1002/rnc.592. Archived from the original(PDF) on August 11, 2011. Retrieved2010-02-28.
  51. Franck, Cassez; Henzinger, Thomas; Raskin, Jean-Francois (2002). "A Comparison of Control Problems for Timed and Hybrid Systems". Archived from the original on May 28, 2008. Retrieved2010-03-02.Cite journal requires |journal= ()
  52. Carroll, Lewis (1895-04-01). "What the Tortoise Said to Achilles". Mind. IV (14): 278–280. doi:10.1093/mind/IV.14.278. ISSN 0026-4423.
Wikisource has original text related to this article:

Zeno's paradoxes
Zeno s paradoxes Language Watch Edit Arrow paradox redirects here For other uses see Arrow paradox disambiguation Zeno s paradoxes are a set of philosophical problems generally thought to have been devised by Greek philosopher Zeno of Elea c 490 430 BC to support Parmenides doctrine that contrary to the evidence of one s senses the belief in plurality and change is mistaken and in particular that motion is nothing but an illusion It is usually assumed based on Plato s Parmenides 128a d that Zeno took on the project of creating these paradoxes because other philosophers had created paradoxes against Parmenides view Thus Plato has Zeno say the purpose of the paradoxes is to show that their hypothesis that existences are many if properly followed up leads to still more absurd results than the hypothesis that they are one 1 Plato has Socrates claim that Zeno and Parmenides were essentially arguing exactly the same point 2 Some of Zeno s nine surviving paradoxes preserved in Aristotle s Physics 3 4 and Simplicius s commentary thereon are essentially equivalent to one another Aristotle offered a refutation of some of them 3 Three of the strongest and most famous that of Achilles and the tortoise the Dichotomy argument and that of an arrow in flight are presented in detail below Zeno s arguments are perhaps the first examples of a method of proof called reductio ad absurdum also known as proof by contradiction They are also credited as a source of the dialectic method used by Socrates 5 Some mathematicians and historians such as Carl Boyer hold that Zeno s paradoxes are simply mathematical problems for which modern calculus provides a mathematical solution 6 Some philosophers however say that Zeno s paradoxes and their variations see Thomson s lamp remain relevant metaphysical problems 7 8 9 The origins of the paradoxes are somewhat unclear Diogenes Laertius a fourth source for information about Zeno and his teachings citing Favorinus says that Zeno s teacher Parmenides was the first to introduce the paradox of Achilles and the tortoise But in a later passage Laertius attributes the origin of the paradox to Zeno explaining that Favorinus disagrees 10 Contents 1 Paradoxes of motion 1 1 Dichotomy paradox 1 2 Achilles and the tortoise 1 3 Arrow paradox 2 Three other paradoxes as given by Aristotle 2 1 Paradox of place 2 2 Paradox of the grain of millet 2 3 The moving rows or stadium 3 Proposed solutions 3 1 Diogenes the Cynic 3 2 Aristotle 3 3 Archimedes 3 4 Thomas Aquinas 3 5 Bertrand Russell 3 6 Hermann Weyl 3 7 Henri Bergson 3 8 Peter Lynds 3 9 Nick Huggett 4 Paradoxes in modern times 5 A similar ancient Chinese philosophic consideration 6 Quantum Zeno effect 7 Zeno behaviour 8 Lewis Carroll and Douglas Hofstadter 9 See also 10 Notes 11 References 12 External linksParadoxes of motion EditDichotomy paradox Edit That which is in locomotion must arrive at the half way stage before it arrives at the goal as recounted by Aristotle Physics VI 9 239b10 Suppose Atalanta wishes to walk to the end of a path Before she can get there she must get halfway there Before she can get halfway there she must get a quarter of the way there Before traveling a quarter she must travel one eighth before an eighth one sixteenth and so on The dichotomy The resulting sequence can be represented as 1 16 1 8 1 4 1 2 1 displaystyle left cdots frac 1 16 frac 1 8 frac 1 4 frac 1 2 1 right This description requires one to complete an infinite number of tasks which Zeno maintains is an impossibility 11 This sequence also presents a second problem in that it contains no first distance to run for any possible finite first distance could be divided in half and hence would not be first after all Hence the trip cannot even begin The paradoxical conclusion then would be that travel over any finite distance can be neither completed nor begun and so all motion must be an illusion 12 This argument is called the Dichotomy because it involves repeatedly splitting a distance into two parts An example with the original sense can be found in an asymptote It is also known as the Race Course paradox Achilles and the tortoise Edit Achilles and the Tortoise redirects here For other uses see Achilles and the Tortoise disambiguation See also Infinity Zeno Achilles and the tortoise Achilles and the tortoise In a race the quickest runner can never over take the slowest since the pursuer must first reach the point whence the pursued started so that the slower must always hold a lead as recounted by Aristotle Physics VI 9 239b15 In the paradox of Achilles and the tortoise Achilles is in a footrace with the tortoise Achilles allows the tortoise a head start of 100 meters for example Suppose that each racer starts running at some constant speed one faster than the other After some finite time Achilles will have run 100 meters bringing him to the tortoise s starting point During this time the tortoise has run a much shorter distance say 2 meters It will then take Achilles some further time to run that distance by which time the tortoise will have advanced farther and then more time still to reach this third point while the tortoise moves ahead Thus whenever Achilles arrives somewhere the tortoise has been he still has some distance to go before he can even reach the tortoise As Aristotle noted this argument is similar to the Dichotomy 13 It lacks however the apparent conclusion of motionlessness Arrow paradox Edit The arrow Not to be confused with other paradoxes of the same name If everything when it occupies an equal space is at rest at that instant of time and if that which is in locomotion is always occupying such a space at any moment the flying arrow is therefore motionless at that instant of time and at the next instant of time but if both instants of time are taken as the same instant or continuous instant of time then it is in motion 14 as recounted by Aristotle Physics VI 9 239b5 In the arrow paradox Zeno states that for motion to occur an object must change the position which it occupies He gives an example of an arrow in flight He states that in any one duration less instant of time the arrow is neither moving to where it is nor to where it is not 15 It cannot move to where it is not because no time elapses for it to move there it cannot move to where it is because it is already there In other words at every instant of time there is no motion occurring If everything is motionless at every instant and time is entirely composed of instants then motion is impossible Whereas the first two paradoxes divide space this paradox starts by dividing time and not into segments but into points 16 Three other paradoxes as given by Aristotle EditParadox of place Edit From Aristotle If everything that exists has a place place too will have a place and so on ad infinitum 17 Paradox of the grain of millet Edit Description of the paradox from the Routledge Dictionary of Philosophy The argument is that a single grain of millet makes no sound upon falling but a thousand grains make a sound Hence a thousand nothings become something an absurd conclusion 18 Aristotle s refutation Zeno is wrong in saying that there is no part of the millet that does not make a sound for there is no reason why any such part should not in any length of time fail to move the air that the whole bushel moves in falling In fact it does not of itself move even such a quantity of the air as it would move if this part were by itself for no part even exists otherwise than potentially 19 Description from Nick Huggett This is a Parmenidean argument that one cannot trust one s sense of hearing Aristotle s response seems to be that even inaudible sounds can add to an audible sound 20 The moving rows or stadium Edit The moving rows From Aristotle concerning the two rows of bodies each row being composed of an equal number of bodies of equal size passing each other on a race course as they proceed with equal velocity in opposite directions the one row originally occupying the space between the goal and the middle point of the course and the other that between the middle point and the starting post This involves the conclusion that half a given time is equal to double that time 21 For an expanded account of Zeno s arguments as presented by Aristotle see Simplicius s commentary On Aristotle s Physics full citation needed Proposed solutions EditDiogenes the Cynic Edit According to Simplicius Diogenes the Cynic said nothing upon hearing Zeno s arguments but stood up and walked in order to demonstrate the falsity of Zeno s conclusions see solvitur ambulando To fully solve any of the paradoxes however one needs to show what is wrong with the argument not just the conclusions Through history several solutions have been proposed among the earliest recorded being those of Aristotle and Archimedes Aristotle Edit Aristotle 384 BC 322 BC remarked that as the distance decreases the time needed to cover those distances also decreases so that the time needed also becomes increasingly small 22 failed verification 23 Aristotle also distinguished things infinite in respect of divisibility such as a unit of space that can be mentally divided into ever smaller units while remaining spatially the same from things or distances that are infinite in extension with respect to their extremities 24 Aristotle s objection to the arrow paradox was that Time is not composed of indivisible nows any more than any other magnitude is composed of indivisibles 25 Archimedes Edit Before 212 BC Archimedes had developed a method to derive a finite answer for the sum of infinitely many terms that get progressively smaller See Geometric series 1 4 1 16 1 64 1 256 The Quadrature of the Parabola His argument applying the method of exhaustion to prove that the infinite sum in question is equal to the area of a particular square is largely geometric but quite rigorous Today s analysis achieves the same result using limits see convergent series These methods allow the construction of solutions based on the conditions stipulated by Zeno i e the amount of time taken at each step is geometrically decreasing 6 26 Thomas Aquinas Edit Thomas Aquinas commenting on Aristotle s objection wrote Instants are not parts of time for time is not made up of instants any more than a magnitude is made of points as we have already proved Hence it does not follow that a thing is not in motion in a given time just because it is not in motion in any instant of that time 27 Bertrand Russell Edit Bertrand Russell offered what is known as the at at theory of motion It agrees that there can be no motion during a durationless instant and contends that all that is required for motion is that the arrow be at one point at one time at another point another time and at appropriate points between those two points for intervening times In this view motion is just change in position over time 28 29 Hermann Weyl Edit Another proposed solution is to question one of the assumptions Zeno used in his paradoxes particularly the Dichotomy which is that between any two different points in space or time there is always another point Without this assumption there are only a finite number of distances between two points hence there is no infinite sequence of movements and the paradox is resolved According to Hermann Weyl the assumption that space is made of finite and discrete units is subject to a further problem given by the tile argument or distance function problem 30 31 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 Jean Paul Van Bendegem has argued that the Tile Argument can be resolved and that discretization can therefore remove the paradox 6 32 Henri Bergson Edit An alternative conclusion proposed by Henri Bergson in his 1896 book Matter and Memory is that while the path is divisible the motion is not 33 In this argument instants in time and instantaneous magnitudes do not physically exist An object in relative motion cannot have an instantaneous or determined relative position and so cannot have its motion fractionally dissected Peter Lynds Edit In 2003 Peter Lynds put forth a very similar argument all of Zeno s motion paradoxes are resolved by the conclusion that instants in time and instantaneous magnitudes do not physically exist 34 35 36 37 Lynds argues that an object in relative motion cannot have an instantaneous or determined relative position for if it did it could not be in motion and so cannot have its motion fractionally dissected as if it does as is assumed by the paradoxes For more about the inability to know both speed and location see Heisenberg uncertainty principle Nick Huggett Edit Nick Huggett argues that Zeno is assuming the conclusion when he says that objects that occupy the same space as they do at rest must be at rest 16 Paradoxes in modern times EditInfinite processes remained theoretically troublesome in mathematics until the late 19th century With the epsilon delta definition of limit Weierstrass and Cauchy developed a rigorous formulation of the logic and calculus involved These works resolved the mathematics involving infinite processes 38 39 While mathematics can calculate where and when the moving Achilles will overtake the Tortoise of Zeno s paradox philosophers such as Kevin Brown 7 and Francis Moorcroft 8 claim that mathematics does not address the central point in Zeno s argument and that solving the mathematical issues does not solve every issue the paradoxes raise Popular literature often misrepresents Zeno s arguments For example Zeno is often said to have argued that the sum of an infinite number of terms must itself be infinite with the result that not only the time but also the distance to be travelled become infinite 40 However none of the original ancient sources has Zeno discussing the sum of any infinite series Simplicius has Zeno saying it is impossible to traverse an infinite number of things in a finite time This presents Zeno s problem not with finding the sum but rather with finishing a task with an infinite number of steps how can one ever get from A to B if an infinite number of non instantaneous events can be identified that need to precede the arrival at B and one cannot reach even the beginning of a last event 7 8 9 41 A humorous take is offered by Tom Stoppard in his play Jumpers 1972 in which the principal protagonist the philosophy professor George Moore suggests that according to Zeno s paradox Saint Sebastian a 3rd Century Christian saint martyred by being shot with arrows died of fright Debate continues on the question of whether or not Zeno s paradoxes have been resolved In The History of Mathematics An Introduction 2010 Burton writes Although Zeno s argument confounded his contemporaries a satisfactory explanation incorporates a now familiar idea the notion of a convergent infinite series 42 Bertrand Russell offered a solution to the paradoxes based on the work of Georg Cantor 43 but Brown concludes Given the history of final resolutions from Aristotle onwards it s probably foolhardy to think we ve reached the end It may be that Zeno s arguments on motion because of their simplicity and universality will always serve as a kind of Rorschach image onto which people can project their most fundamental phenomenological concerns if they have any 7 A similar ancient Chinese philosophic consideration EditThis section does not cite any sources Please help improve this section by adding citations to reliable sources Unsourced material may be challenged and removed October 2019 Learn how and when to remove this template message Ancient Chinese philosophers from the Mohist School of Names during the Warring States period of China 479 221 BC developed equivalents to some of Zeno s paradoxes 44 The scientist and historian Sir Joseph Needham in his Science and Civilisation in China describes an ancient Chinese paradox from the surviving Mohist School of Names book of logic which states in the archaic ancient Chinese script a one foot stick every day take away half of it in a myriad ages it will not be exhausted Several other paradoxes from this philosophical school more precisely movement are known but their modern interpretation is more speculative Quantum Zeno effect EditMain article Quantum Zeno effect In 1977 45 physicists E C George Sudarshan and B Misra discovered that the dynamical evolution motion of a quantum system can be hindered or even inhibited through observation of the system 46 This effect is usually called the quantum Zeno effect as it is strongly reminiscent of Zeno s arrow paradox This effect was first theorized in 1958 47 Zeno behaviour EditIn the field of verification and design of timed and hybrid systems the system behaviour is called Zeno if it includes an infinite number of discrete steps in a finite amount of time 48 Some formal verification techniques exclude these behaviours from analysis if they are not equivalent to non Zeno behaviour 49 50 In systems design these behaviours will also often be excluded from system models since they cannot be implemented with a digital controller 51 Lewis Carroll and Douglas Hofstadter EditWhat the Tortoise Said to Achilles 52 written in 1895 by Lewis Carroll was an attempt to reveal an analogous paradox in the realm of pure logic If Carroll s argument is valid the implication is that Zeno s paradoxes of motion are not essentially problems of space and time but go right to the heart of reasoning itself Douglas Hofstadter made Carroll s article a centrepiece of his book Godel Escher Bach An Eternal Golden Braid writing many more dialogues between Achilles and the Tortoise to elucidate his arguments Hofstadter connects Zeno s paradoxes to Godel s incompleteness theorem in an attempt to demonstrate that the problems raised by Zeno are pervasive and manifest in formal systems theory computing and the philosophy of mind See also EditIncommensurable magnitudes Infinite regress Philosophy of space and time Renormalization Ross Littlewood paradox School of Names Supertask What the Tortoise Said to Achilles an allegorical dialogue on the foundations of logic by Lewis Carroll 1895 Zeno machine List of ParadoxesNotes Edit Parmenides 128d Parmenides 128a b a b Aristotle s Physics Physics by Aristotle translated by R P Hardie and R K Gaye Greek text of Physics by Aristotle refer to 4 at the top of the visible screen area Archived from the original on 2008 05 16 fragment 65 Diogenes Laertius IX Archived 2010 12 12 at the Wayback Machine 25ff and VIII 57 a b c Boyer Carl 1959 The History of the Calculus and Its Conceptual Development Dover Publications p 295 ISBN 978 0 486 60509 8 Retrieved 2010 02 26 If the paradoxes are thus stated in the precise mathematical terminology of continuous variables the seeming contradictions resolve themselves a b c d Brown Kevin Zeno and the Paradox of Motion Reflections on Relativity Archived from the original on 2012 12 05 Retrieved 2010 06 06 a b c Moorcroft Francis Zeno s Paradox Archived from the original on 2010 04 18 a b Papa Grimaldi Alba 1996 Why Mathematical Solutions of Zeno s Paradoxes Miss the Point Zeno s One and Many Relation and Parmenides Prohibition PDF The Review of Metaphysics 50 299 314 Diogenes Laertius Lives 9 23 and 9 29 Lindberg David 2007 The Beginnings of Western Science 2nd ed University of Chicago Press p 33 ISBN 978 0 226 48205 7 Huggett Nick 2010 Zeno s Paradoxes 3 1 The Dichotomy Stanford Encyclopedia of Philosophy Retrieved 2011 03 07 Huggett Nick 2010 Zeno s Paradoxes 3 2 Achilles and the Tortoise Stanford Encyclopedia of Philosophy Retrieved 2011 03 07 Aristotle Physics The Internet Classics Archive Zeno s reasoning however is fallacious when he says that if everything when it occupies an equal space is at rest and if that which is in locomotion is always occupying such a space at any moment the flying arrow is therefore motionless This is false for time is not composed of indivisible moments any more than any other magnitude is composed of indivisibles Laertius Diogenes c 230 Pyrrho Lives and Opinions of Eminent Philosophers IX passage 72 ISBN 1 116 71900 2 a b Huggett Nick 2010 Zeno s Paradoxes 3 3 The Arrow Stanford Encyclopedia of Philosophy Retrieved 2011 03 07 Aristotle Physics IV 1 209a25 The Michael Proudfoot A R Lace Routledge Dictionary of Philosophy Routledge 2009 p 445 Aristotle Physics VII 5 250a20 Huggett Nick Zeno s Paradoxes The Stanford Encyclopedia of Philosophy Winter 2010 Edition Edward N Zalta ed http plato stanford edu entries paradox zeno GraMil Aristotle Physics VI 9 239b33 Aristotle Physics 6 9 Aristotle s observation that the fractional times also get shorter does not guarantee in every case that the task can be completed One case in which it does not hold is that in which the fractional times decrease in a harmonic series while the distances decrease geometrically such as 1 2 s for 1 2 m gain 1 3 s for next 1 4 m gain 1 4 s for next 1 8 m gain 1 5 s for next 1 16 m gain 1 6 s for next 1 32 m gain etc In this case the distances form a convergent series but the times form a divergent series the sum of which has no limit original research Archimedes developed a more explicitly mathematical approach than Aristotle Aristotle Physics 6 9 6 2 233a21 31 Aristotle Physics VI Part 9 verse 239b5 ISBN 0 585 09205 2 George B Thomas Calculus and Analytic Geometry Addison Wesley 1951 Aquinas Commentary on Aristotle s Physics Book 6 861 Huggett Nick 1999 Space From Zeno to Einstein ISBN 0 262 08271 3 Salmon Wesley C 1998 Causality and Explanation p 198 ISBN 978 0 19 510864 4 Van Bendegem Jean Paul 17 March 2010 Finitism in Geometry Stanford Encyclopedia of Philosophy Retrieved 2012 01 03 Cohen Marc 11 December 2000 ATOMISM History of Ancient Philosophy University of Washington Archived from the original on July 12 2010 Retrieved 2012 01 03 van Bendegem Jean Paul 1987 Discussion Zeno s Paradoxes and the Tile Argument Philosophy of Science Belgium 54 2 295 302 doi 10 1086 289379 JSTOR 187807 Bergson Henri 1896 Matiere et Memoire Matter and Memory PDF Translation 1911 by Nancy Margaret Paul amp W Scott Palmer George Allen and Unwin pp 77 78 of the PDF Zeno s Paradoxes A Timely Solution January 2003 Lynds Peter Time and Classical and Quantum Mechanics Indeterminacy vs Discontinuity Foundations of Physics Letter s Vol 16 Issue 4 2003 doi 10 1023 A 1025361725408 Time s Up Einstein Josh McHugh Wired Magazine June 2005 S E Robbins 2004 On time memory and dynamic form Consciousness and Cognition 13 4 762 788 Lynds his reviewers and consultants e g J J C Smart are apparently unaware of his total precedence by Bergson Lee Harold 1965 Are Zeno s Paradoxes Based on a Mistake Mind Oxford University Press 74 296 563 570 doi 10 1093 mind LXXIV 296 563 JSTOR 2251675 B Russell 1956 Mathematics and the metaphysicians in The World of Mathematics ed J R Newman pp 1576 1590 Benson Donald C 1999 The Moment of Proof Mathematical Epiphanies New York Oxford University Press p 14 ISBN 978 0195117219 Huggett Nick 2010 Zeno s Paradoxes 5 Zeno s Influence on Philosophy Stanford Encyclopedia of Philosophy Retrieved 2011 03 07 Burton David A History of Mathematics An Introduction McGraw Hill 2010 ISBN 978 0 07 338315 6 Russell Bertrand 2002 First published in 1914 by The Open Court Publishing Company Lecture 6 The Problem of Infinity Considered Historically Our Knowledge of the External World As a Field for Scientific Method in Philosophy Routledge p 169 ISBN 0 415 09605 7 School of Names gt Miscellaneous Paradoxes Stanford Encyclopedia of Philosophy plato stanford edu Retrieved 2020 01 30 Sudarshan E C G Misra B 1977 The Zeno s paradox in quantum theory PDF Journal of Mathematical Physics 18 4 756 763 Bibcode 1977JMP 18 756M doi 10 1063 1 523304 OSTI 7342282 W M Itano D J Heinsen J J Bokkinger D J Wineland 1990 Quantum Zeno effect PDF Physical Review A 41 5 2295 2300 Bibcode 1990PhRvA 41 2295I doi 10 1103 PhysRevA 41 2295 PMID 9903355 Archived from the original PDF on 2004 07 20 Retrieved 2004 07 23 Khalfin L A 1958 Contribution to the Decay Theory of a Quasi Stationary State Soviet Phys JETP 6 1053 Bibcode 1958JETP 6 1053K 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 Handbook of dynamic system modeling Chapman amp Hall CRC Computer and Information Science hardcover ed Boca Raton Florida USA CRC Press pp 15 22 to 15 23 ISBN 978 1 58488 565 8 Retrieved 2010 03 05 Lamport Leslie 2002 Specifying Systems PDF Microsoft Research Addison Wesley p 128 ISBN 0 321 14306 X Retrieved 2010 03 06 Zhang Jun Johansson Karl Lygeros John Sastry Shankar 2001 Zeno hybrid systems PDF International Journal for Robust and Nonlinear Control 11 5 435 doi 10 1002 rnc 592 Archived from the original PDF on August 11 2011 Retrieved 2010 02 28 Franck Cassez Henzinger Thomas Raskin Jean Francois 2002 A Comparison of Control Problems for Timed and Hybrid Systems Archived from the original on May 28 2008 Retrieved 2010 03 02 Cite journal requires journal help Carroll Lewis 1895 04 01 What the Tortoise Said to Achilles Mind IV 14 278 280 doi 10 1093 mind IV 14 278 ISSN 0026 4423 References EditKirk G S J E Raven M Schofield 1984 The Presocratic Philosophers A Critical History with a Selection of Texts 2nd ed Cambridge University Press ISBN 0 521 27455 9 Huggett Nick 2010 Zeno s Paradoxes Stanford Encyclopedia of Philosophy Retrieved 2011 03 07 Plato 1926 Plato Cratylus Parmenides Greater Hippias Lesser Hippias H N Fowler Translator Loeb Classical Library ISBN 0 674 99185 0 Sainsbury R M 2003 Paradoxes 2nd ed Cambridge University Press ISBN 0 521 48347 6 External links EditWikisource has original text related to this article Zeno of EleaDowden Bradley Zeno s Paradoxes Entry in the Internet Encyclopedia of Philosophy Antinomy Encyclopedia of Mathematics EMS Press 2001 1994 Introduction to Mathematical Philosophy Ludwig Maximilians Universitat Munchen Silagadze Z K Zeno meets modern science Zeno s Paradox Achilles and the Tortoise by Jon McLoone Wolfram Demonstrations Project Kevin Brown on Zeno and the Paradox of Motion Palmer John 2008 Zeno of Elea Stanford Encyclopedia of Philosophy This article incorporates material from Zeno s paradox on PlanetMath which is licensed under the Creative Commons Attribution Share Alike License Grime James Zeno s Paradox Numberphile Brady Haran Archived from the original on 2018 10 03 Retrieved 2013 04 13 Retrieved from https en wikipedia org w index php title Zeno 27s paradoxes amp oldid 1052791486, wikipedia, wiki, book,

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