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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 the exact same point (Parmenides 128a-b).

Several of Zeno's eight surviving paradoxes (preserved in Aristotle's Physics and Simplicius's commentary thereon) are essentially equivalent to one another; and most of them were regarded, even in ancient times, as very easy to refute. 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 more 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.

Zeno's paradoxes were a major problem for ancient and medieval philosophers, who found most proposed solutions somewhat unsatisfactory. More modern solutions using calculus have generally satisfied mathematicians and engineers. Many philosophers still hesitate to say that all paradoxes are completely solved, while pointing out also that attempts to deal with the paradoxes have resulted in many intellectual discoveries. Variations on the paradoxes (see Thomson's lamp) continue to produce at least temporary puzzlement in elucidating what, if anything, is wrong with the argument.

The origins of the paradoxes are somewhat unclear. Diogenes Laertius, citing Favorinus, says that Zeno's teacher (Dr. Mears) 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 diagrees.

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 feet. 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 feet, bringing him to the tortoise's starting point. During this time, the tortoise has run a much shorter distance, for example 10 feet. It will then take Achilles some further time to run that distance, in 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.

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 fourth, he must travel one-eighth; before an eighth, one-sixteenth; and so on.

${\displaystyle H-{\frac {B}{8}}-{\frac {B}{4}}---{\frac {B}{2}}-------B}$

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.

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. However, they emphasize different points. In the Achilles and the Tortoise, the focus is that movement by multiple objects is just an illusion whereas in the Dichotomy the focus is that movement is actually impossible.

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, Zeno states that for motion to be occurring, an object must change the position which it occupies. He gives an example of an arrow in flight. He states that in any one instant of time, for the arrow to be moving it must either move to where it is, or it must move to where it is not. It cannot move to where it is not, because this is a single instant, and it cannot move to where it is because it is already there. In other words, in any instant of time there is no motion occurring, because an instant is a snapshot. Therefore, if it cannot move in a single instant it cannot move in any instant, making any motion impossible. This paradox is also known as the fletcher's paradox—a fletcher being a maker of arrows.

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

The contradiction appears, if we assume that the state of the arrow is completely characterized by its coordinate in the configurational space. No contradiction appears, if we treat the state as a point in the phase space, which includes not only coordinates, but also the momentum. More generally, for a macroscopic body, one may need to include into the phase space also the orientation and the angular momentum. The movement, as a transition of an object from one place of space to another place can be suppressed by frequent observation, this is considered as a version of the zeno effect. For macroscopic bodies, the required frequency of observations is high, far away from human abilities; therefore the term zeno effect is applied usually to quantum objects - particles, atoms. In the past century, this effect was considered nonsense, indicating non-completeness of quantum mechanics , or at least as a paradox . In recent publications, the same phenomenon is referred as effect, not paradox.

Proposed solutions

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Main article: Zeno's paradox solutions

Aristotle remarked that as the distance decreases, the time needed to cover those distances also decreases, so that the time needed also becomes increasingly small. In Physics 6.9, Aristotle solves the paradoxes by distinguishing "things infinite in respect of divisibility" from things (or distances) that are infinite in extension.

Before 212 BCE, Archimedes had developed a method to derive a finite answer for the sum of infinitely many terms that get progressively smaller. Theorems have been developed in more modern calculus to achieve the same result, but with a more rigorous proof of the method (see convergent series where the "reciprocals of powers of 2" series, equivalent to the Dichotomy Paradox, is listed as convergent). These methods allow construction of solutions stating that (under suitable conditions), if the distances are decreasing sufficiently rapidly, the travel time is finite (bounded by a certain amount).

Another proposed solution is to question the assumption inherent in Zeno's paradox, which is that between any two different points in space (or time), there is always another point. If this assumption is challenged, the infinite sequence of events is avoided, and the paradox resolved.

Yet another proposed solution, that of Peter Lynds, is to question the assumption that moving objects have exact positions at an instant and that their motion can be meaningfully dissected this way. If this assumption is challenged, motion remains continuous and the paradoxes are avoided.

Status of the paradoxes today

Mathematicians thought they had done away with Zeno's paradoxes with the invention of the calculus and methods of handling infinite sequences by Isaac Newton and Gottfried Wilhelm Leibniz in the 17th century, and then again when certain problems with their methods were resolved by the reformulation of the calculus and infinite series methods in the 19th century.. The paradoxes certainly pose no problems in Engineering, as the practical questions as to where and when events such as Achilles passing the Tortoise are satisfactorily handled by calculus.

However, some philosophers insist that the deeper metaphysical questions, as raised by Zeno's paradoxes, are not addressed by the calculus. That is, while calculus tells us where and when Achilles will overtake the Tortoise, philosophers do not see how calculus takes anything away from Zeno's reasoning that concludes that this event cannot take place in the first place. Most importantly, many philosophers do not see where, according to the calculus, Zeno's reasoning goes wrong .

Infinite processes have remained theoretically troublesome for other reasons as well. L. E. J. Brouwer, a Dutch mathematician of the 19th and 20th century, and founder of the Intuitionist school, was the most prominent of those who rejected arguments, including proofs, involving infinities. In this he followed Leopold Kronecker, an earlier 19th century mathematician.

However, some mathematical viewpoints (such as the epsilon-delta version of Weierstrass and Cauchy in the 19th century or the equivalent and equally rigorous differential/infinitesimal version by Abraham Robinson in the 20th) argue rigorous formulation of the calculus has resolved forever all problems involving infinities, including Zeno's.

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".

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."

The Moving Rows:

"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."

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

The quantum Zeno effect

In 1977, physicists studying quantum mechanics 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 (but not fundamentally related to) Zeno's arrow paradox.

This effect was first theorized in 1958.

See also

• Zeno's paradox solutions
• Zeno machine
• Supertask
• Thomson's lamp
• Balls and vase problem
• What the Tortoise Said to Achilles
• 0.999...
• Solvitur ambulando
• Incommensurable magnitudes
• Zeno effect

Footnotes

1. Aristotle's Physics "Physics" by Aristotle translated by R. P. Hardie and R. K. Gaye
2. (, Diog. IX 25ff and VIII 57)
3. Diogenes Laertius, Lives, 9.23 and 9.29.
4. "Math Forum".
5. "Zeno's Paradoxes:Archilles and the turtle". Stanford Encyclopedia of Philosophy.
6. Zeno's Paradoxes Stanford Encyclopedia of Philosophy. Dichotomy
7. Zeno's Paradoxes Stanford Encyclopedia of Philosophy. Arrow
8. B.Mielnik (1994). "The screen problem". Foundations of physics. 24 (8): 1113–1129.
9. K.Yamane, M.Ito, M.Kitano (2001). "Quantum Zeno effect in optical fibers". Optics Communications. 192 (3–6): 299–307.{{cite journal}}: CS1 maint: multiple names: authors list (link)
10. Aristotle. Physics 6.9
11. Kevin Brown, Reflections on Relativity, ; Francis Moorcroft, Zeno's Paradox, ;
12. http://www.wired.com/wired/archive/13.06/physics_pr.html
13. Kevin Brown, Reflections on Relativity, ; Francis Moorcroft, Zeno's Paradox, ; Alba Papa-Grimaldi, Why Mathematical Solutions of Zeno's Paradoxes Miss the Point: Zeno's One and Many Relation and Parmenides' Prohibition, The Review of Metaphysics, Vol. 50, 1996.
14. Kevin Brown, Reflections on Relativity, ; Francis Moorcroft, Zeno's Paradox, ; Stanford Encyclopedia of Philosophy, Zeno's Paradox, ; Alba Papa-Grimaldi, Why Mathematical Solutions of Zeno's Paradoxes Miss the Point: Zeno's One and Many Relation and Parmenides' Prohibition, The Review of Metaphysics, Vol. 50, 1996.
15. Aristotle Physics IV:1, 209a25
16. Aristotle Physics VII:5, 250a20
17. Aristotle Physics VI:9, 239b33
18. Sudarshan, E.C.G.; Misra, B. (1977), "The Zeno's paradox in quantum theory", Journal of Mathematical Physics, 18 (4): pp. 756–763 {{citation}}: |pages= has extra text (help)
19. W.M.Itano (1990). "Quantum Zeno effect" (PDF). PRA. 41: 2295–2300. doi:10.1103/PhysRevA.41.2295. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)
20. Khalfin, L.A. (1958), Soviet Phys. JETP, 6: 1053 {{citation}}: Missing or empty |title= (help)

Further references

• Chan, Wing-Tsit, (1969) A Source Book In Chinese Philosophy. Princeton University Press. ISBN 0691019649
• Kirk, 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 0521274559.
• Plato (1926) Plato: Cratylus. Parmenides. Greater Hippias. Lesser Hippias, H. N. Fowler (Translator), Loeb Classical Library. ISBN 0674991850.
• Sainsbury, R.M. (2003) Paradoxes, 2nd ed. Cambridge Univ. Press. ISBN 0521483476.

External links

• Stanford Encyclopedia of Philosophy: "Zeno's Paradoxes" -- by Nick Huggett.
• Wilkins, Geoff, "Some paradoxes - an anthology."
• Brown, Kevin, "Zeno's Paradoxes of Motion," from Reflections on Relativity at MathPages.
• Silagadze, Z . K. "Zeno meets modern science,"
• BBC article on shortest time measured as of 2004: 10 seconds.
• Blog "Strange Paths": "Modernity of Zeno's paradoxes."
• Platonic Realms: "Zeno's Paradox of the Tortoise and Achilles."
• Zeno's Paradox: Achilles and the Tortoise by Jon McLoone, The Wolfram Demonstrations Project.
• The Dichotomy Paradox a series based solution.
• Zeno's paradoxes-wikinfo

Zeno's paradox at PlanetMath.

• Paradoxes
• Supertasks
• Mathematics paradoxes