This is the name given to an interesting puzzle in game
theory. The outline was originally created by Melvin Dresher and Merrill Flood
of the RAND Corporation, and Albert W. Tucker gave it its name.
Two prisoners are
each given exactly the same information: there is enough evidence against each
of them to get them both sentenced to a gaol term. If one is prepared to give
evidence against the other, then that prisoner will get off free, while the
other prisoner serves five years. On the other hand, if each provides evidence
against the other, each will be convicted and serve four years. The dilemma is
that each prisoner knows the other prisoner has the same information, and will
act in some way: so what is the best choice to make?
The problem has a
number of practical applications: a child immunised against a certain disease
may run a small risk (let us say one chance in a million) of dying because of a
complication caused by the immunisation. On the other hand, if only half of the
population are immunised against the disease, one child in fifty will certainly
die in an epidemic. If 95% of the population are immunised, an epidemic will
not take place, as the disease will be eliminated.
To many people,
the selfish choice is best: avoid immunisation, and let others to take the
risks for you, but if nobody is immunised, then those who are without immunity
run a much greater chance of dying. Other applications are found in economics,
evolutionary biology, and even in military planning.
What
use are paradoxes?
Imagine a ball, thrown into the air, just as it reaches the
highest point. One moment it is going up, the next, it is going down. If you
take smaller and smaller fractions of time, there must be a point where the
ball is neither going up nor down. Of course, if you measure accurately enough,
there is no such moment: the ball is always either rising or falling, and its
speed is changing. So how can we ever calculate how fast it is going, or how
long it will take to get from A to B, and what are the mathematics of limits?
How can we find the mathematics to describe the path of a cricket ball?
In the late 1600s,
Isaac Newton and Wilhelm Leibniz invented the branch of mathematics called
calculus, but that was too late for the Greeks, who said there had to be a way
of dealing with limits like that, but the mathematics of very small things made
no sense to them, without calculus.
The old Greeks had
a way of testing puzzles by reasoning, and if they got a contradictory result
or a nonsense result, they called this a paradox, and knew there was a problem.
Sometimes they deliberately set up a paradox to test something, like the one
known as Achilles and the tortoise,
or Zeno’s paradox.
Invented by Zeno
of Elea, nearly 2500 years ago, one version says Achilles runs 10 times as fast
as the tortoise, but he gives the tortoise a 10-metre lead. At the start,
Achilles sprints 10 metres, but tortoise has travelled another metre, and when
Achilles travels that metre, the tortoise has gone a tenth of a metre, and so
on. Zeno said Achilles can never catch the tortoise.
He wanted to prove
that something is impossible, even though we can see it happening, from which
it follows that since we can see the impossible happening, our senses must be
faulty. In other words, his paradoxes were designed to make people think
harder.
Later, Aristotle
argued against Zeno’s assumption that space and time were infinitely divisible.
This argument made another Greek called Democritus suggest that matter was not
infinitely divisible. That gave him the idea of atoms, and all because Zeno
believed the senses could not be trusted. Even though Zeno had proved that Achilles could never catch the tortoise, we know that
in real life, he can!
Practical people
knew the tortoise was toast, and a few of them might even have had a feeling
that the tortoise would be passed at 11.111111... metres, but they had no
system of mathematics to deal with the sums, so maybe they didn’t. Anyhow, the
realm of fairy-tale mathematics, and treacherous tiny segments of time
attracted the interest of scientists in the 17th century. They needed to deal
with these ideas, and they did—but there were other paradoxes as well.
Self-reference
paradoxes
Street sign, Athens: I choose to translate Odos Epimenidou as 'Epimenides Street', but am I right?
I am easily amused, so when I saw Odos Epimenidou (Epimenides Street) in the old quarter of Athens, I laughed out loud. Epimenides of Crete is credited with one of the first self-reference paradoxes when he said, “All Cretans are liars”, meaning by implication that this statement (made by a Cretan) was necessarily untrue. But if it is untrue, then not all statements made by Cretans are liars, and so on… Stop reading now, before your brain starts hurting.
There is a more complex form of this paradox, consisting of
two sentences. “The next sentence is false” and “The previous sentence is
true”. And try this one: there are two
erors in this sentence.
No, really, stop,
before your brain hurts. You don’t need the next four paragraphs. Jump over
them!
Some words
describe themselves. “Short” is a short word, but “long” is not a long word,
“English” is an English word, but “German” is not a German word, and so on.
Bertrand Russell
sent a paradox to mathematician Gottlob Frege in 1902, and it runs something
like this: some sets, such as the set of all insects, are not members of
themselves. Other sets, such as the set of all non-insects, being not insects,
are members of themselves. If we call the set of all sets that are not members
of themselves R, we have a problem. If R is a member of itself, then by
definition it must not be a member of itself.
Or, if R is not a
member of itself, then by definition it must be a member of itself, and either
way, something is up! His letter about the paradox reached Frege as he awaited
the appearance of the second volume of a two-volume treatise on mathematics. He
added a note at the end of the second volume:
A scientist can hardly meet with anything more undesirable
than to have the foundation give way just as the work is finished. In this
position I was put by a letter from Mr Bertrand Russell just as the work was
nearly through the press.
Well, I warned you to stop…
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