Zeno's paradoxes

Zeno's paradoxes

Zeno's paradoxes are a set of problems generally thought to have been devised by Zeno of Elea to support Parmenides's doctrine that "all is one" and that, contrary to the evidence of our 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 128c-d, that Zeno took on the project of creating these paradoxes because other philosophers had created paradoxes against Parmenides's view. Thus Zeno can be interpreted as saying that to assume there is plurality is even more absurd than assuming there is only "the One". (Parmenides 128d). Plato makes 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] 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.[2]

Some mathematicians, such as Carl Boyer, hold that Zeno's paradoxes are simply mathematical problems, for which modern calculus provides a mathematical solution.[3] Some philosophers, however, say that Zeno's paradoxes and their variations (see Thomson's lamp) remain relevant metaphysical problems.[4][5][6]

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 Argument. But in a later passage, Laertius attributes the origin of the paradox to Zeno, explaining that Favorinus disagrees.[7]


The Paradoxes of Motion

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.

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 metres, 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 metres, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say, 10 metres. 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.[8][9]

The dichotomy paradox

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

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.


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.

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.

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.[10]

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, with no way to establish the size of any "last" step. Expressed this way, the dichotomy paradox is very much analogous to that of Achilles and the tortoise.

The arrow paradox

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.

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.[11] 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 presented divide space, this paradox starts by dividing time—and not into segments, but into points.[12]

Three other paradoxes as given by Aristotle

Paradox of Place:

"… if everything that exists has a place, place too will have a place, and so on ad infinitum."[13]

Paradox of the Grain of Millet:

"… 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."[14]

The Moving Rows (or Stadium):

"The fourth argument is that 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."[15]

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

Proposed solutions

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 (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.[16][17] 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").[18]

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.[3][19]

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."[20] Saint 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."[21] 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.[22][23] 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.[12]

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 actually exist.[24][25][26] Lynds argues that an object in relative motion cannot have a determined relative position (for if it did, it could not be in motion), and so cannot have its motion fractionally dissected as though it does as in the paradoxes.

Another proposed solution is to question one of the assumptions of 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".[27] 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.[3][28]

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.[29]

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.[30]

While mathematics can be used to calculate where and when the moving Achilles will overtake the Tortoise of Zeno's paradox, philosophers such as Brown and Moorcroft[4][5] 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.

Zeno's arguments are often misrepresented in the popular literature. That is, 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"?[4][5][6][31]

Today there is still a debate on the question of whether or not Zeno's paradoxes have been resolved. In The History of Mathematics, Burton writes, "Although Zeno's argument confounded his contemporaries, a satisfactory explanation incorporates a now-familiar idea, the notion of a 'convergent infinite series.'"[32] Bertrand Russell offered a "solution" to the paradoxes based on modern physics[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)."[4]

The quantum Zeno effect

In 1977,[33] 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.[34] 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.[35]

Zeno behavior

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.[36] Some formal verification techniques exclude these behaviours from analysis, if they are not equivalent to non-Zeno behaviour.[37][38] In systems design these behaviours will also often be excluded from system models, since they cannot be implemented with a digital controller.[39] A simple example of a system showing Zeno behavior is a bouncing ball coming to rest. The physics of a bouncing ball can be mathematically analyzed in such a way, ignoring factors other than rebound, to predict an infinite number of bounces.

Writings about Zeno’s paradoxes

Zeno’s paradoxes have inspired many writers

  • Leo Tolstoy in War and Peace (Part 11,Chapter I) discusses the race of Achilles and the tortoise when critiquing "historical science".
  • In the dialogue "What the Tortoise Said to Achilles", Lewis Carroll describes what happens at the end of the race. The tortoise discusses with Achilles a simple deductive argument. Achilles fails in demonstrating the argument because the tortoise leads him into an infinite regression.
  • In Gödel, Escher, Bach by Douglas Hofstadter, the various chapters are separated by dialogues between Achilles and the tortoise, inspired by Lewis Carroll’s works
  • The Argentinian writer Jorge Luis Borges discusses Zeno’s paradoxes many times in his work, showing their relationship with infinity. Borges also used Zeno’s paradoxes as a metaphor for some situations described by Kafka. Jorge Luis Borges traces, in an essay entitled "Avatars of the Tortoise", the many recurrences of this paradox in works of philosophy. The successive references he traces are Agrippa the Skeptic, Thomas Aquinas, Hermann Lotze, F.H. Bradley and William James.[40]
  • In Tom Stoppard's play Jumpers, the philosopher George Moore attempts a practical disproof with bow and arrow of the Dichotomy Paradox, with disastrous consequences for the hare and the tortoise.
  • Harry Mulisch's philosophical magnum opus, De compositie van de wereld (Amsterdam, 1980) is based on Zeno's Paradoxes mostly. Along with Herakleitos' thoughts and Cusanus' coïncidentia oppositorum they constitute the foundation for his own system of the 'octave'.

See also


  1. ^ a b 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. ^ a b c Boyer, Carl (1959). The History of the Calculus and Its Conceptual Development. Dover Publications. p. 295. ISBN 9780486605098. http://books.google.com/?id=w3xKLt_da2UC&dq=zeno+calculus&q=zeno#v=snippet&q=zeno. Retrieved 26 February 2010. "If the paradoxes are thus stated in the precise mathematical terminology of continuous variables (...) the seeming contradictions resolve themselves." 
  4. ^ a b c d Brown, Kevin. "Zeno and the Paradox of Motion". Reflections on Relativity. http://www.mathpages.com/rr/s3-07/3-07.htm. Retrieved 6 June 2010. 
  5. ^ a b c Moorcroft, Francis. "Zeno's Paradox". http://www.philosophers.co.uk/cafe/paradox5.htm. 
  6. ^ 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". The Review of Metaphysics. pp. 299-314. http://www.webalice.it/keithalba/timeandreality/zeno_revmet.htm. 
  7. ^ Diogenes Laertius, Lives, 9.23 and 9.29.
  8. ^ "Math Forum". http://mathforum.org/isaac/problems/zeno1.html. , matchforum.org
  9. ^ Huggett, Nick (2010). "Zeno's Paradoxes: 3.2 Achilles and the Tortoise". Stanford Encyclopedia of Philosophy. http://plato.stanford.edu/entries/paradox-zeno/#AchTor. Retrieved 2011-03-07. 
  10. ^ Huggett, Nick (2010). "Zeno's Paradoxes: 3.1 The Dichotomy". Stanford Encyclopedia of Philosophy. http://plato.stanford.edu/entries/paradox-zeno/#Dic. Retrieved 2011-03-07. 
  11. ^ Laertius, Diogenes (about 230 CE). "Pyrrho". Lives and Opinions of Eminent Philosophers. IX. passage 72. ISBN 1116719002. http://en.wikisource.org/wiki/Lives_of_the_Eminent_Philosophers/Book_IX#Pyrrho. 
  12. ^ a b Huggett, Nick (2010). "Zeno's Paradoxes: 3.3 The Arrow". Stanford Encyclopedia of Philosophy. http://plato.stanford.edu/entries/paradox-zeno/#Arr. Retrieved 2011-03-07. 
  13. ^ Aristotle Physics IV:1, 209a25
  14. ^ Aristotle Physics VII:5, 250a20
  15. ^ Aristotle Physics VI:9, 239b33
  16. ^ Aristotle. Physics 6.9
  17. ^ 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.
  18. ^ Aristotle. Physics 6.9; 6.2, 233a21-31
  19. ^ George B. Thomas, Calculus and Analytic Geometry, Addison Wesley, 1951
  20. ^ Aristotle. Physics. VI. Part 9 verse: 239b5. ISBN 0585092052. http://classics.mit.edu/Aristotle/physics.6.vi.html. 
  21. ^ Aquinas. Commentary on Aristotle's Physics, Book 6.861
  22. ^ Huggett, Nick (1999). Space From Zeno to Einstein. ISBN 0262082713. 
  23. ^ Salmon, Wesley C. (1998). Causality and Explanation. p. 198. ISBN 9780195108644. http://books.google.com/?id=uPRbOOv1YxUC&pg=PA198&lpg=PA198&dq=at+at+theory+of+motion+russell#v=onepage&q=at%20at%20theory%20of%20motion%20russell&f=false. 
  24. ^ Lynds, Peter. Zeno's Paradoxes: a Timely Solution
  25. ^ 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
  26. ^ Ricker's critique of Lynds
  27. ^ Geometrical Finitism
  28. ^ 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. 
  29. ^ Hans Reichenbach (1958) The Philosophy of Space and Time. Dover
  30. ^ Lee, Harold (1965). "Are Zeno's Paradoxes Based on a Mistake?". Mind (Oxford University Press) 74 (296): 563–570. JSTOR 2251675. 
  31. ^ Huggett, Nick (2010). "Zeno's Paradoxes: 5. Zeno's Influence on Philosophy". Stanford Encyclopedia of Philosophy. http://plato.stanford.edu/entries/paradox-zeno/#ZenInf. Retrieved 2011-03-07. 
  32. ^ Burton, David, A History of Mathematics: An Introduction, McGraw Hill, 2010, ISBN 978-0073383156
  33. ^ Sudarshan, E. C. G.; Misra, B. (1977). "The Zeno’s paradox in quantum theory". Journal of Mathematical Physics 18 (4): 756–763. Bibcode 1977JMP....18..756M. doi:10.1063/1.523304 
  34. ^ W.M.Itano; D.J.Heinsen, J.J.Bokkinger, D.J.Wineland (1990). "Quantum Zeno effect" (PDF). PRA 41 (5): 2295–2300. Bibcode 1990PhRvA..41.2295I. doi:10.1103/PhysRevA.41.2295. http://www.boulder.nist.gov/timefreq/general/pdf/858.pdf. 
  35. ^ Khalfin, L.A. (1958). Soviet Phys. JETP 6: 1053 
  36. ^ 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 9781584885658. http://books.google.com/?id=cM-eFv1m3BoC&pg=SA15-PA22. Retrieved 5 March 2010. 
  37. ^ Lamport, Leslie (2002) (PDF). Specifying Systems. Addison-Wesley. p. 128. ISBN 0-321-14306-X. http://research.microsoft.com/en-us/um/people/lamport/tla/book-02-08-08.pdf. Retrieved 6 March 2010. 
  38. ^ Zhang, Jun; Johansson, Karl; Lygeros, John; Sastry, Shankar (2001). "Zeno hybrid systems". International Journal for Robust and Nonlinear control. http://aphrodite.s3.kth.se/~kallej/papers/zeno_ijnrc01.pdf. Retrieved 2010-02-28. 
  39. ^ Franck, Cassez; Henzinger, Thomas; Raskin, Jean-Francois (2002). A Comparison of Control Problems for Timed and Hybrid Systems. http://mtc.epfl.ch/~tah/Publications/a_comparison_of_control_problems_for_timed_and_hybrid_systems.html. Retrieved 2010-03-02. 
  40. ^ Borges, Jorge Luis (1964). Labyrinths. London: Penguin. pp. 237–243. ISBN 0811200124. 


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