Zeno's paradoxes

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

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

Zeno's paradoxes are a set of problems 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 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". Plato remarked that if we are convinced by Zeno's paradoxes, we should take Parmenides' view more seriously.

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 says that Zeno's teacher, Parmenides, was "the first to use the argument known as 'Achilles and the Tortoise' ", and attributes this assertion to Favorinus. In a later statement, Laertius attributed the paradoxes to Zeno.

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 period of time to run that distance, in which period the tortoise will advance farther; and then another period of time 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.

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

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. However, they emphasise 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.

Proposed 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. Such an approach to solving the paradoxes would amount to a denial that it must take an infinite amount of time to traverse an infinite sequence of distances.

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. These methods allow construction of solutions stating that (under suitable conditions), if the distances are always decreasing, the travel time is finite (bounded by a certain amount).

Further information: Zeno's paradox solutions

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. Many philosophers, and certainly engineers, generally went along with the mathematical results.

Nevertheless, Zeno's paradoxes are still hotly debated by philosophers in academic circles. Infinite processes have remained theoretically troublesome. 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. It would be incorrect to say that a rigorous formulation of the calculus (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) has resolved forever all problems involving infinities, including Zeno's.

As a practical matter, however, no engineer has been concerned about them since knowledge of the calculus became common at engineering schools. In ordinary life, very few people have ever been much concerned.

Two 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."

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

The quantum Zeno effect

Recently, physicists studying quantum mechanics have noticed 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.

See also

• Zeno's paradox solutions
• Loki's wager
• 0.999...
• Gödel, Escher, Bach
• Sankhya Philosophy
• Solvitur ambulando
• Supertask
• What the Tortoise Said to Achilles
• Zeno effect
• Zeno machine
• Bertrand paradox
• Incommensurable magnitudes

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: 10⁻¹⁶ 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.

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Zeno's paradox at PlanetMath.