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Zeno's paradoxes are a set of philosophical problems generally thought to have been devised by Greek philosopher Zeno of Elea (ca. 490–430 BC) to support Parmenides's 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's 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." (Parmenides 128d). Plato has Socrates claim that Zeno and Parmenides were essentially arguing exactly the same point (Parmenides 128a–b).

Some of Zeno's nine surviving paradoxes (preserved in Aristotle's Physics[1][2] and Simplicius's commentary thereon) are essentially equivalent to one another. Aristotle offered a refutation of some of them.[1] 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.[3]

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.[4] Some philosophers, however, say that Zeno's paradoxes and their variations (see Thomson's lamp) remain relevant metaphysical problems.[5][6][7]

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 Achilles and the tortoise paradox. But in a later passage, Laertius attributes the origin of the paradox to Zeno, explaining that Favorinus disagrees.[8]

### Achilles and the tortoise

Distance vs. time, assuming the tortoise to run at Achilles' half speed

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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. If we suppose 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 reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise.[9][10]

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 Homer wants to catch a stationary bus. 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.

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The resulting sequence can be represented as:

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

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 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.[11]

There are two versions of the dichotomy paradox. In the other version, before Homer could reach the stationary bus, he must reach half of the distance to it. Before reaching the last half, he must complete the next quarter of the distance. Reaching the next quarter, he must then cover the next eighth of the distance, then the next sixteenth, and so on. There are thus an infinite number of steps that must first be accomplished before he could reach the bus. Expressed this way, the dichotomy paradox is very much analogous to that of Achilles and the tortoise.

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.[12]

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

In the arrow paradox (also known as the fletcher's 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 (durationless) instant of time, the arrow is neither moving to where it is, nor to where it is not.[13] 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.[14]

## Three other paradoxes as given by Aristotle

From Aristotle:

if everything that exists has a place, place too will have a place, and so on ad infinitum.[15]

### Paradox of the Grain of Millet

From Aristotle:

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.[16]

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, which is absurd.[17]

Description from Nick Huggett:

This 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.[18]

### The Moving Rows (or Stadium)

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.[19]

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

## Proposed solutions

### Simplicius of Cilicia

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. 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.[20][21] 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").[22] 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."[23]

#### 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."[24]

### 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.) Modern calculus achieves the same result, using more rigorous methods (see convergent series, where the "reciprocals of powers of 2" series, equivalent to the Dichotomy Paradox, is listed as convergent). 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.[4][25]

### 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 a function of position with respect to time.[26][27]

### Nick Huggett

Nick Huggett argues that Zeno is begging the question when he says that objects that occupy the same space as they do at rest must be at rest.[14]

### 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.[28][29][30] 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.

### 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. The ideas of Planck length and Planck time in modern physics place a limit on the measurement of time and space, if not on time and space themselves. 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".[31][32] 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.[4][33]

### Hans Reichenbach

Hans Reichenbach has proposed that the paradox may arise from considering space and time as separate entities. In a theory like general relativity, which presumes a single space-time continuum, the paradox may be blocked.[34]

## 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.[35][36]

While mathematics can calculate where and when the moving Achilles will overtake the Tortoise of Zeno's paradox, philosophers such as Brown and Moorcroft[5][6] 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.[37] 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"?[5][6][7][38]

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.'".[39]

Bertrand Russell offered a "solution" to the paradoxes based on modern physics,{{ safesubst:#invoke:Unsubst||date=__DATE__ |\$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }} 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)."[5]

Pat Corvini offers a solution to the paradox of Achilles and the tortoise by first distinguishing the physical world from the abstract mathematics used to describe it.[40] She claims the paradox arises from a subtle but fatal switch between the physical and abstract. Zeno's syllogism is as follows:

• P1: Achilles must first traverse an infinite number of divisions in order to reach the tortoise
• P2: it is impossible for Achilles to traverse an infinite number of divisions
• C: therefore, Achilles can never surpass the tortoise

Corvini shows that P1 is a mathematical abstraction which cannot be applied directly to P2 which is a statement regarding the physical world. The physical world requires a resolution amount used to distinguish distance while mathematics can use any resolution.

## Quantum Zeno effect

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In 1977,[41] physicists E. C. G. Sudarshan and B. Misra studying quantum mechanics discovered that the dynamical evolution (motion) of a quantum system can be hindered (or even inhibited) through observation of the system.[42] 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.[43]

## 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.[44] Some formal verification techniques exclude these behaviours from analysis, if they are not equivalent to non-Zeno behaviour.[45][46]

In systems design these behaviours will also often be excluded from system models, since they cannot be implemented with a digital controller.[47]

A simple example of a system showing Zeno behaviour is a bouncing ball coming to rest. The physics of a bouncing ball, ignoring factors other than rebound, can be mathematically analyzed to predict an infinite number of bounces.

## Notes

1. Aristotle's Physics "Physics" by Aristotle translated by R. P. Hardie and R. K. Gaye
2. ([fragment 65], Diogenes Laertius. IX 25ff and VIII 57).
3. {{#invoke:citation/CS1|citation |CitationClass=book }}
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7. Diogenes Laertius, Lives, 9.23 and 9.29.
8. Template:Cite web, mathforum.org
9. Template:Cite web
10. Template:Cite web
11. {{#invoke:citation/CS1|citation |CitationClass=book }}
12. Template:Cite web
13. Aristotle Physics IV:1, 209a25
14. Aristotle Physics VII:5, 250a20
15. The Michael Proudfoot, A.R. Lace. Routledge Dictionary of Philosophy. Routledge 2009, p. 445
16. 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
17. Aristotle Physics VI:9, 239b33
18. Aristotle. Physics 6.9
19. 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. Archimedes developed a more explicitly mathematical approach than Aristotle.
20. Aristotle. Physics 6.9; 6.2, 233a21-31
21. {{#invoke:citation/CS1|citation |CitationClass=book }}
22. Aquinas. Commentary on Aristotle's Physics, Book 6.861
23. George B. Thomas, Calculus and Analytic Geometry, Addison Wesley, 1951
24. {{#invoke:citation/CS1|citation |CitationClass=book }}
25. {{#invoke:citation/CS1|citation |CitationClass=book }}
26. Lynds, Peter. Zeno's Paradoxes: a Timely Solution
27. 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
28. Time’s Up Einstein, Josh McHugh, Wired Magazine, June 2005
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31. Hans Reichenbach (1958) The Philosophy of Space and Time. Dover
32. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
33. B Russell (1956) Mathematics and the metaphysicians in "The World of Mathematics" (ed. J R Newman), pp 1576-1590.
34. {{#invoke:citation/CS1|citation |CitationClass=book }}
35. Template:Cite web
36. Burton, David, A History of Mathematics: An Introduction, McGraw Hill, 2010, ISBN 978-0-07-338315-6
37. Template:Cite web
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43. {{#invoke:Citation/CS1|citation |CitationClass=journal }}