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 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).

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 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 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.

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, say, 10 feet. 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. Of course, simple experience tells us that Achilles will be able to overtake the tortoise, which is why this is a paradox.

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.

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.

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.

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. Aristotle solves the paradoxes by distinguishing "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").

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 construction of solutions stating that (under suitable conditions), if the distances are decreasing sufficiently rapidly, the travel time is finite (bounded by a fixed upper bound).

Using ordinary mathematics we can calculate both the time and place where Achilles overtakes the tortoise. For example, if Achilles is moving 10 metres/second and the tortoise is moving at 1 metre/second, and if the tortoise has a 100 metre head start, then Achilles will reach the tortoise's starting point in 10 seconds. The tortoise will have moved 10 metres further. Achilles would then run that 10 metres in one second, and the tortoise will have moved one metre further. Zeno argues that every time Achilles reaches the tortoise's last position, the tortoise will have advanced further - but in the next full second Achilles will run another 10 metres, passing the tortoise who will have advanced only one metre. Algebra gives us the distance and time at which Achilles would exactly match the position of the tortoise: 111 1/9 metres after running for 11 1/9 seconds. This is neither an infinite distance, nor an infinite time. While this solves the mathematics of one of the paradoxes, it does not touch the dynamics of any of the three paradoxes - namely, "How is it that motion is possible at all?"

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. Philosophers typically prefer this approach over the mathematics based approaches, since while mathematics can tell us where and when Achilles overtakes the tortoise, it does not explain how these points in space and time can ever be reached. Philosophers claim that the mathematics does not address the central point in Zeno's argument.

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. This solution is related to Heisenberg's Uncertainty Principle.

A solution to the arrow paradox is given by Rudy Rucker in Infinity and the Mind: he points out that the motion of the arrow is indeed instantaneously observable due to the small contraction in length the arrow undergoes due to the effects of special relativity.

Status of the paradoxes today

Mathematicians today tend to regard the paradoxes as resolved, but some philosophers disagree. Bertrand Russell, who was both a mathematician and a philosopher, wrote Georg Cantor invented a theory of continuity and a theory of infinity which did away with all the old paradoxes upon which philosophers had battened. ... Philosophers met the situation by not reading the authors concerned. The paradoxes certainly pose no practical difficulties.

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 there are problems in explaining how motion can happen at all.

Philosophers also point out that 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 be infinite itself, which calculus shows to be incorrect. However, Zeno's problem wasn't with any kind of infinite sum, but rather with an infinite process: how can one ever get from A to B, if an infinite number of events can be identified that need to precede the arrival at B? Philosophers claim that calculus does not resolve that question, and hence a solution to Zeno's paradoxes must be found elsewhere.

Physicists point out that in the race, after a few dozen steps, we will have to deal with dimensions where quantum mechanics can’t be disregarded. According to the uncertainty principle those distances are so small that taking a measurement would be pointless, even from a theoretical point of view: uncertainty would be too prominent.

Infinite processes remained theoretically troublesome in mathematics until the early 20th century. L. E. J. Brouwer, a Dutch mathematician 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, modern mathematics, with tools such as Kurt Gödel's proof of the logical independence of the axiom of choice and the epsilon-delta version of Weierstrass and Cauchy (or the equivalent and equally rigorous differential/infinitesimal version by Abraham Robinson), argues rigorous formulation of logic and calculus has resolved theoretical problems involving infinite processes, including Zeno's.

The quantum Zeno effect

In 1977, 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. 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.

Writings about Zeno’s paradoxes

Zeno’s paradoxes have inspired many writers

• 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.
• Paul Hornschemeier's most recent graphic novel, The Three Paradoxes, contains a comic version of Zeno presenting his three paradoxes to his fellow philosophers.
• Leslie Lamport's Specifiying Systems contains a section (9.4) introducing the character of the Zeno Specifications
• Zadie Smith references Zeno's arrow paradox, and, more briefly, Zeno's Achilles and tortoise paradox, at the end of Chapter 17 in her novel White Teeth.
• Brian Massumi shoots Zeno's "philosophical arrow" in the opening chapter of Parables for the Virtual: Movement, Affect, Sensation.
• Philip K. Dick's short science-fiction story "The Indefatiguable Frog" concerns an experiment to determine whether a frog which continually leaps half the distance to the top of a well will ever be able to get out of the well.

See also

• Zeno machine
• Supertask
• Thomson's lamp

• Balls and vase problem
• What the Tortoise Said to Achilles
• 0.999...

• Solvitur ambulando
• Incommensurable magnitudes
• Quantum Zeno effect

External links

• Stanford Encyclopedia of Philosophy: "Zeno's Paradoxes" -- by Nick Huggett.
• 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, Wolfram Demonstrations Project.
• The Dichotomy Paradox a series based solution.
• Zeno's paradoxes-wikinfo [10]
• Zeno's Paradox from PhilosophyArchive

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Zeno's paradox at PlanetMath.