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### Achilles and the tortoise

{{Redirect|Achilles and the Tortoise}}(File:Race between Achilles and the tortoise.gif|thumb|Distance vs. time, assuming the tortoise to run at Achilles' half speed)(File:Zeno Achilles Paradox.png|thumb|left|Achilles and the tortoise)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. Supposing that each racer starts running at some constant speed (one very fast and one very slow), then 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, 10 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.WEB, Huggett, Nick,weblink Zeno's Paradoxes: 3.2 Achilles and the Tortoise, 2010, Stanford Encyclopedia of Philosophy, 2011-03-07,

Suppose Homer wishes to walk to the end of a path. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on.ImageSize= width:800 height:100PlotArea= width:720 height:55 left:65 bottom:20AlignBars= justifyPeriod= from:0 till:100TimeAxis= orientation:horizontalScaleMajor= unit:year increment:10 start:0ScaleMinor= unit:year increment:1 start:0Colors=
id:homer value:rgb(0.4,0.8,1) # light purple
PlotData=
bar:homer fontsize:L color:homer
from:0 till:100
at:50 mark:(line,red)
at:25 mark:(line,black)
at:12.5 mark:(line,black)
at:6.25 mark:(line,black)
at:3.125 mark:(line,black)
at:1.5625 mark:(line,black)
at:0.78125 mark:(line,black)
at:0.390625 mark:(line,black)
at:0.1953125 mark:(line,black)
at:0.09765625 mark:(line,black)
{{multiple image|direction = vertical|width = 220|image1 = Zeno Dichotomy Paradox alt.png|image2 = Zeno Dichotomy Paradox.png|footer = The dichotomy, both versions}}The resulting sequence can be represented as:
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.BOOK, Lindberg, David, The Beginnings of Western Science, 2007, University of Chicago Press, 978-0-226-48205-7, 33, 2nd, This sequence also presents a second problem in that it contains no first distance to run, for any possible ((wikt:finite|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 neither be completed nor begun, and so all motion must be an illusion. An alternative conclusion, proposed by Henri Bergson, is that motion (time and distance) is not actually divisible.This argument is called the Dichotomy because it involves repeatedly splitting a distance into two parts. It contains some of the same elements as the Achilles and the Tortoise paradox, but with a more apparent conclusion of motionlessness. It is also known as the Race Course paradox. Some, like Aristotle, regard the Dichotomy as really just another version of Achilles and the Tortoise.WEB, Huggett, Nick,weblink Zeno's Paradoxes: 3.1 The Dichotomy, 2010, Stanford Encyclopedia of Philosophy, 2011-03-07,

(File:Zeno Arrow Paradox.png|thumb|The arrow)|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 timeâ€”and not into segments, but into points.WEB, Huggett, Nick,weblink Zeno's Paradoxes: 3.3 The Arrow, 2010, Stanford Encyclopedia of Philosophy, 2011-03-07,

## Three other paradoxes as given by Aristotle

From Aristotle:

### Paradox of the Grain of Millet

Description of the paradox from the Routledge Dictionary of Philosophy:Aristotle's refutation:Description from Nick Huggett:

### The Moving Rows (or Stadium)

(File:Zeno Moving Rows Paradox.png|thumb|The moving rows)From Aristotle:For an expanded account of Zeno's arguments as presented by Aristotle, see Simplicius' commentary On Aristotle's Physics.

## Proposed solutions

### 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 BCâˆ’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.Aristotle. Physics 6.9Aristotle'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. Archimedes developed a more explicitly mathematical approach than Aristotle.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. Physics 6.9; 6.2, 233a21-31Aristotle'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."BOOK, Aristotle,weblink Physics, VI, Part 9 verse: 239b5, 0-585-09205-2,

### 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."Aquinas. Commentary on Aristotle's Physics, Book 6.861

### 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 + Â· Â· Â·, 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.George B. Thomas, Calculus and Analytic Geometry, Addison Wesley, 1951

### 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.

### 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.

### Peter Lynds

Peter Lynds has argued that all of Zeno's motion paradoxes are resolved by the conclusion that instants in time and instantaneous magnitudes do not physically exist.WEB,weblink Zeno's Paradoxes: A Timely Solution, 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:1025361725408Timeâ€™s Up Einstein, Josh McHugh, Wired Magazine, June 2005Lynds 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.

### 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".WEB, Van Bendegem, Jean Paul, Finitism in Geometry,weblink Stanford Encyclopedia of Philosophy, 2012-01-03, 17 March 2010, WEB, Cohen, Marc, ATOMISM,weblink History of Ancient Philosophy, University of Washington, 2012-01-03, 11 December 2000, yes,weblink July 12, 2010, 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.JOURNAL, 187807, Discussion:Zeno's Paradoxes and the Tile Argument, Jean Paul, van Bendegem, Belgium, 1987, Philosophy of Science, 54, 2, 295â€“302, 10.1086/289379,

## The paradoxes in modern times

Infinite processes remained theoretically troublesome in mathematics until the late 19th century. The epsilon-delta version of Weierstrass and Cauchy developed a rigorous formulation of the logic and calculus involved. These works resolved the mathematics involving infinite processes.JOURNAL, Lee, Harold, Are Zeno's Paradoxes Based on a Mistake?, 2251675, 1965, Mind (journal), Mind, 74, 296, Oxford University Press, 563â€“570, 10.1093/mind/LXXIV.296.563, B Russell (1956) Mathematics and the metaphysicians in "The World of Mathematics" (ed. J R Newman), pp 1576-1590.While mathematics can calculate where and when the moving Achilles will overtake the Tortoise of Zeno's paradox, philosophers such as Kevin BrownWEB
, Kevin
, Brown
, Zeno and the Paradox of Motion
, Reflections on Relativity
, 2010-06-06
, yes
, 2012-12-05
,
,
and 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 infiniteâ€“with the result that not only the time, but also the distance to be travelled, become infinite.BOOK, Benson, Donald C., The Moment of Proof : Mathematical Epiphanies, 1999, Oxford University Press, New York, 978-0195117219, 14,weblink 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 Zenoâ€™s paradox, Saint Sebastian, a 3rd Century Christian saint supposedly martyred by being shot with arrows, died of fright. 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"?WEB, Huggett, Nick,weblink Zeno's Paradoxes: 5. Zeno's Influence on Philosophy, 2010, Stanford Encyclopedia of Philosophy, 2011-03-07, 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.'".Burton, David, A History of Mathematics: An Introduction, McGraw Hill, 2010, {{isbn|978-0-07-338315-6}}Bertrand Russell offered a "solution" to the paradoxes based on the work of Georg Cantor,BOOK, Russell, Bertrand, 2002, Our Knowledge of the External World: As a Field for Scientific Method in Philosophy, Lecture 6. The Problem of Infinity Considered Historically, Routledge, 169, First published in 1914 by The Open Court Publishing Company, 0-415-09605-7, 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)."

## A similar ancient Chinese philosophic consideration

Ancient Chinese philosophers from the Mohist School of Names during the Warring States period of China (479-221 BC) independently 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.

## Quantum Zeno effect

In 1977,JOURNAL, 1977JMP....18..756M, Sudarshan, E. C. G., E. C. G. Sudarshan, Misra, B., The Zeno's paradox in quantum theory, Journal of Mathematical Physics, 18, 4, 756â€“763, 1977, 10.1063/1.523304,weblink
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.JOURNAL,weblink PDF, W.M.Itano, D.J. Heinsen, J.J. Bokkinger, D.J. Wineland, Quantum Zeno effect, Physical Review A, PRA, 41, 5, 2295â€“2300, 1990, 10.1103/PhysRevA.41.2295, 1990PhRvA..41.2295I, 2004-07-23,weblink" title="web.archive.org/web/20040720153510weblink">weblink 2004-07-20, yes,
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.JOURNAL, Khalfin, L.A., Soviet Phys. JETP, 6, 1053, 1958, 1958JETP....6.1053K, Contribution to the Decay Theory of a Quasi-Stationary State,

## Zeno behaviour

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.BOOK, Paul A. Fishwick, Handbook of dynamic system modeling,weblink 2010-03-05, hardcover, Chapman & Hall/CRC Computer and Information Science, 1 June 2007, CRC Press, Boca Raton, Florida, USA, 978-1-58488-565-8, 15â€“22 to 15â€“23, 15.6 "Pathological Behavior Classes" in chapter 15 "Hybrid Dynamic Systems: Modeling and Execution" by Pieter J. Mosterman, The Mathworks, Inc.,
Some formal verification techniques exclude these behaviours from analysis, if they are not equivalent to non-Zeno behaviour.BOOK, Lamport, Leslie, Leslie Lamport, 2002, Specifying Systems, PDF, Addison-Wesley, 0-321-14306-X,weblink 128, 2010-03-06,

In systems design these behaviours will also often be excluded from system models, since they cannot be implemented with a digital controller.JOURNAL, Henzinger, Thomas, Franck, Cassez, Jean-Francois, Raskin,weblink A Comparison of Control Problems for Timed and Hybrid Systems, 2002, 2010-03-02, yes,weblink" title="web.archive.org/web/20080528193234weblink">weblink May 28, 2008,

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## References

{{Wikisource|Catholic Encyclopedia (1913)/Zeno of Elea|Zeno of Elea}}

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