Paradox

For other uses, see Paradox (disambiguation).

Robert Boyle's self-flowing flask fills itself in this diagram, but perpetual motion machines cannot exist (according to our present understanding of physics).

The term paradox comes from the Greek para ("contrary to") and doxa ("opinion"). Thus a paradox was originally something that was contrary to received or common opinion. From that, the term came to be used for something that was contrary to or contradicted common sense. Today the term is usually used for something that is an apparently true statement or group of statements that leads to a contradiction or a situation which defies intuition.

Sometimes either the statements in question do not really imply the contradiction and the puzzling result is not really a contradiction, or the premises themselves are not all really true or cannot all be true together. Zeno's paradoxes of motion were such. The recognition of ambiguities, equivocations, and unstated assumptions underlying known paradoxes has sometimes led to significant advances in science, philosophy and mathematics.

The word paradox is often used interchangeably and wrongly with contradiction; but where a contradiction by definition cannot be true, many paradoxes do allow for resolution, though many remain unresolved or only contentiously resolved, such as Curry's paradox. Still more casually, the term is sometimes used for situations that are merely surprising, albeit in a distinctly "logical" manner, such as the Birthday Paradox. This is also the usage in economics, where a paradox is an unintuitive outcome of economic theory.

All that siad, however, there are some genuine paradoxes, statements that, if true, are false, and if false, are true.

Zeno's Paradoxes

The ancient Greek philosopher Parmenides of Elea (c. 515-c.450 B.C.E.) had held that motion is an illusion and that existence is one indivisible whole. His student and follower, Zeno of Elea (490-430 B.C.E.)—Zeno was regarded by Aristotle as being the founder of dialectic—produced a number of paradoxes that purport to demonstrate that space, time, and especially motion are inherently contradictory, and thus cannot exist.

• The millet seed. If a single millet seed is dropped to the ground, it does not make a sound that can be heard. A bushel of millet seeds is merely an aggregate of many millet seeds, but it seems to make a sound. It is a paradox that 10,000 dropped seeds should make a sound when a single dropped seed does not, since any number of multiplications of nothing (or zero) should not produce anything other than nothing.

• Falsity of the premise "There are many things." (a) "If there are many things, they must be both great and small; so small as to have no size, so great as to be infinite." The next part of the argument employs the principle of division: anything that has size can be divided into two things, each with size, so the process has no termination. (b) If there is a plurality, the finite things must be both finite and infinite in number. It must be finite because a plurality requires a finite and definite number; infinite because two or more things require boundaries or distinguishing marks, but here again a progression toward infinity areises when those things are broken into smaller things. (c) "If there are many things, they must be both like and unlike." The argument for that claim is not given. (From Hondereich, p. 922)

• Divisibility of a line. A line is either divisible or indivisible. If divisible, it is divisible into either a finite or an infinite number of parts, each of which either has or lacks magnitude. If finite but lacking magnitude, then we cannot reconstruct the line because any number of parts of zero magnitude is still zero. If it is an infinit enumber of parts of zero magnitude, the same result (because an infinite multiplication of zero is still zero). but if the line is divisible into an infinite number of parts of any magnitude, then this would constitute a much longer line than the original (because an infinite multiplication of any magnitude greater than zero yields an infinite magnitude). Thus, lines are not divisible.

• The arrow. The flying arrow cannot really be flying because if it is moving, it must either be moving in a place where it is, or in a place where it is not. But if it is in the place that it is (i.e. a place exactly equal to its length), then it is at rest, and if it is moving into a place where it is not, this cannot be because it cannot be where it is not.

• Achilles and the tortise. If the tortise starts the race somewhere in front of Achilles, then Achilles, in an effort to overtake the tortise, must first cover the distance between his starting point and the starting point of the tortise, but in that time the tortise will hav emoved forward. So Achilles must then cover that distance. But the tortise will have moved on, and so on, with an infinite iteration of this necessity. Thus Achilles cannot catch the tortise so long as the tortise remains in motion.

Examples

Sometimes supernatural or science fiction themes are held to be impossible due to resultant paradoxial conditions. The theme of time travel has staged many popular paradoxes arising from the traveler interfering with the past. Suppose Jones, who was born in 1950, travels back in time to 1901 and kills his own grandfather. It follows that neither his father nor he himself will be born; but then he would not have existed to travel back in time and kill his own grandfather; but then his grandfather would not have died and Jones himself would have lived; etc. This is known as the Grandfather paradox.

Paradoxes that arise from apparently intelligible uses of language are often of interest to logicians and philosophers. This sentence is false is an example of the famous liar paradox: it is a sentence which cannot be consistently interpreted as true or false, because if it is false it must be true, and if it is true it must be false. Therefore, it can be concluded the sentence is both true and false. Russell's paradox, which shows that the notion of the set of all those sets that do not contain themselves leads to a contradiction, was instrumental in the development of modern logic and set theory.

For more examples see List of paradoxes.

Etymology

The etymology of paradox can be traced back to at least Plato's Parmenides, where Zeno of Elea, who lived from 490-430 B.C.E., used the word "paradoxon" to describe some of his seminal philosophic ideas.

Zeno sought to illustrate that equal absurdities followed logically from the denial of Parmenides' views. There were apparently 40 'paradoxes of plurality' and other paradoxes that Zeno used to attack the Greek understanding of the physical world. In fact, Zeno's paradoxes of multiplicity and motion, which revealed problems in the Greek idea of space and time, were resolvable only using mathematics discovered in the 19th century.

It is unknown if incarnations of paradox were used before Zeno of Elea. Later and more frequent usage of the word has been traced to the early Renaissance. Early forms of the word appeared in the late Latin paradoxum and the related Greek παράδοξος paradoxos, which means "contrary to expectation", or "incredible". The word is a fusion of the preposition para, meaning "against" or "beyond", and the noun stem doxa, meaning "belief" or "opinion".

Common themes

Common themes in paradoxes include direct and indirect self-reference, infinity, circular definitions, and confusion of levels of reasoning. Paradoxes which are not based on a hidden error generally happen at the fringes of context or language, and require extending the context or language to lose their paradox quality.

In moral philosophy, paradox plays a central role in ethics debates. For instance, it may be considered that an ethical admonition to "love thy neighbour" is not just in contrast with, but in contradiction to an armed neighbour actively trying to kill you: if he or she succeeds, you will not be able to love him or her. But to preemptively attack them or restrain them is not usually understood as loving. This might be termed an ethical dilemma. Another example is the conflict between an injunction not to steal and one to care for a family that you cannot afford to feed without stolen money.

Types of paradoxes

W. V. Quine (1962) distinguished between three classes of paradoxes.

• A veridical paradox produces a result that appears absurd but is demonstrated to be true nevertheless. Thus, the paradox of Frederic's birthday in The Pirates of Penzance establishes the surprising fact that a person may be more than nine years old on his ninth birthday. Likewise, Arrow's impossibility theorem involves behaviour of voting systems that is surprising but all too true.
• A falsidical paradox establishes a result that not only appears false but actually is false; there is a fallacy in the supposed demonstration. The various invalid proofs (e.g. that 1 = 2) are classic examples, generally relying on a hidden division by zero. Another example would be the Horse paradox.
• A paradox which is in neither class may be an antinomy, which reaches a self-contradictory result by properly applying accepted ways of reasoning. For example, the Grelling-Nelson paradox points out genuine problems in our understanding of the ideas of truth and description.

See also

• List of paradoxes
• Impossible object
• Logical fallacy
• Puzzle

References

ISBN links support NWE through referral fees

• R. M. Sainsbury (1988). Paradoxes. Cambridge.
• W. V. Quine (1962). "Paradox". Scientific American, April 1962, pp. 84–96.
• Michael Clarke (2002). Paradoxes from A to Z. London: Routledge.

External links

• Open Directory Project: Paradoxes
• Definability paradoxes
• Insolubles (at the Stanford Encyclopedia of Philosophy)
• http://ideas.repec.org/p/wpa/wuwpmi/0405008.html
• http://www.ritsumei.ac.jp/~akitaoka/index-e.html
• http://www.torinfo.com/illusion/directory.html
• http://www.iep.utm.edu/p/par-russ.htm
• http://www.mathpages.com/rr/s3-07/3-07.htm

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