These are old
fond paradoxes to make fools laugh i' the alehouse. - Othello, Act 1, Scene
1
Paradoxes are as old as humankind. The
ancient Greeks studied them intensely which eventually helped lead to the
discovery of irrational numbers, and paradoxes are mentioned in the Bible: "It was
one of their own prophets who said 'Cretans were never anything but liars,
dangerous animals, all greed and laziness;' and that is a true statement." (Titus
1:12-13)
Even today, we are surrounded by paradoxes such as Blackwood's
"the more terrible the prospect of thermonuclear war becomes the less likely it is to
happen," or the Moebius Strip
- a topological paradox.
For this article, we define a paradox as a statement or sentiment that is seemingly
contradictory or opposed to common sense and yet is perhaps true in fact. Another
way of thinking of a paradox is a statement that is actually self-contradictory and
hence false even though its true character is not immediately apparent.
One of the oldest paradoxes is the one cited by the Apostle Paul in his letter to
Titus (see above.) The following famous Liar Paradox
is interesting because
it cannot be true because it would make the speaker a liar and therefore what he
says is false. Neither can it be true because that would imply that Cretans are
truth-tellers, and consequently what the speaker says would be true. (For classic
Star Trek fans... "Norman, coordinate." {I, Mudd})
Self-reference would appear to be the problem with the Liar Paradox, but upon
further study, eliminating reference to oneself does not eliminate the paradox. Try
this one on for size:
If sentence A is true, then B is false, and if B is false then A must also be false. But
if sentence A is false, then B must be true, and if B is true, then A must be true.
Neither sentence talks about itself, but taken together they keep changing the
truth-value of the other, so that eventually, we are unable to say whether either
sentence is true or false.
Closely related to the Liar Paradox and the Jourdain Truth-Value Paradox are
prediction paradoxes such as the Unexpected
Exam.
Most people admit that Mike's reasoning is correct, that the exam can't be offered
on Friday, because it would not be "unexpected." Once this is admitted as sound
reasoning, the rest of Mike's reasoning seems to follow. However, even the first
step in Mike's reasoning is faulty. Suppose he has attended class every day. As he
walks to class on Friday, can he deduce correctly that there will be no test that
day? No, because if he makes such a deduction, he might walk into class and see
the unexpected examination.
The consensus among logicians is that although the professor knows he can keep
his word, there is no way that Mike can know it. Therefore, there is no way he can
make a valid deduction about the test on any day, including Friday.
The Ancient Greeks were confounded by many paradoxical situations, which, as
noted above, helped lead to the discovery of irrational numbers (numbers such as
the square root of 2 or pi). Without irrational numbers we would not have
progressed beyond elementary arithmatic and such things as geometry or calculus
would be unknown. Zeno, a 5th Century B.C. Greek philosopher, used a paradox to
argue that a person could never cross a room because to do so would require an
infinite amount of time. Zeno's Paradox, where the sum
of a set of infinite numbers can be derived to a finite total reads thus.
But obviously, it is possible to cross a room and touch the other wall and obviously
it does not take an infinite amount of time to do this. Zeno's Paradox was not
"resolved" until Newton and Liebnitz discovered the concept of the limit. "An
irresistible inference is in conflict with an inescapable fact." A Tour of the
Calculus, David Berlinski, Pantheon Books, 1995.
An irresistible inference in conflict with an inescapable fact: A marvelous modern
definition of a paradox.
Bertrand Russell, a philosopher/mathematician/political activist, changed the
direction of mathematics in the early 20th Century when he reported his famous Barber
Paradox. Russell's Paradox arises within set theory
by considering the set of all sets which are not members of themselves. Such a set
appears to be a member of itself if and only if it is not a member of itself.
The significance of Russell's paradox can be seen once it is realized that, using
classical logic, all
sentences follow from a contradiction. In the eyes of many, it therefore appeared
that no
mathematical proof could be trusted once it was discovered that the logic and set
theory apparently
underlying all of mathematics was contradictory.
Based on Russell's Paradox, metaphysicians, mathematicians and philosophers have
introduced the concept of Metalanguages to help describe such sets.
Russell's basic idea is that we can
avoid reference to S (the set of all sets which are not members of themselves) by
arranging all
sentences into a hierarchy. This hierarchy will consist of sentencesindividuals at the lowest
level, sentences about sets of individuals at the next lowest level, sentencessets of sets of
individuals at the next lowest level, etc. This hierarchy has helped bridge the gap
between mathematics, logic and philosophy in an important way.
Another of Zeno's paradoxes, the Arrow Paradox,
also
illustrates the impossibility of motion or change. This paradox is part of the work of
Werner Heisenberg, who was awarded the 1932 Nobel Prize for his work on
the eponymous Heisenberg Uncertainty Principle. Heisenberg observed that the
closer one gets to measuring the velocity of a particle of energy, the further one
gets from observing its position. The Heisenberg Uncertainty Principle prompted the
famous remark from Einstein: "I shall never believe that God plays dice with the
universe."
One of the latest and most profound prediction paradoxes is
Newcomb's Paradox. This paradox is closely related to
game theory (See my other page
on politics for more information about game and decision theory).
The paradox reveals a lot about whether or not a person believes in free will.
Reactions are almost equally divided between believers in free will, who favor taking
both boxes, and believers in determinism who favor taking only Box B. Others argue
that conditions demanded by the paradox are contradictory regardless of whether
or not the future is or isn't completely determined.
Aside from their amusing nature, paradoxes have practical applications as well.
Based on the Omnipotence Paradox
, legal scholars have
debated whether or not a constitution can be non-paradoxically amended. While
some may argue that this is an angels dancing on the head of a pin question, there
are important implications for the rights of people. For instance, can a constitution
contain a clause that prevents certain sections of the constitution from being
amended in the future? What about amending the section that deals with
amendments? For more information about self-amendment paradox, see Peter
Suber, "The Paradox of Self-Amendment in American Constitutional Law," Stanford
Literature Review, 7, 1-2 (Spring-Fall 1990) 53-78.
Political scientists have been looking at the Voter
Paradox as an example of a "government failure." Just as individual choice
sometimes fails to promote social values in desired and predictable ways, so to
does collective choice. Collective choice exercised through governmental structures
offeres at least the possibility for correcting the perceived deficiencies of individual
choice (See The
Prisoners Dilemma as an example). However, at times even government cannot
overcome these deficiencies.
For more information about the Voter Paradox and Kenneth Arrow's General
Possibility Theorem which states that any fair rule for choice will fail to
ensure a transitive social ordering of policy alternatives (in other words, cyclical
social preferences like those appearing in the Voter Paradox can arise from
any fair voting system), see William H. Riker, Liberalism Against
Populism (San Francisco: Freeman, 1982). and Kenneth Arrow,
Social Choice and Individual Values 2nd ed. (New Haven, Conn.:Yale
University Press, 1963)
The St.
Petersburg Paradox, played an important role
in the development of decision theory and the concept of the utility
function of money. While much too complex to get into in great detail here,
decision theory is the development of systemic rules for decision-making. The utility
function of money was first developed by the 18th century mathematician Daniel
Bernoulli, who proposed that the true worth of an individual's wealth is the
logarithm of the amount of money possessed.
Imagine that there is a scientific law that says "All crows are black." If only three or
four crows are observed, then the law is weakly confirmed. If millions of
crows are seen to be black, then it is strongly confirmed.
Professor Carl Hempel, who invented this paradox,
believes that a purple cow actually does slightly increase the probability that all
crows are black.
Hempel's Paradox and Nelson Goodman's "grue" paradox are examples of
confirmation paradoxes. For more information about confirmation, see Wesley C.
Salmon, "Confirmation" Scientific American May, 1973.
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