Wednesday, January 19, 2011

Quotations & Paradox

For over two years now I've started with (or used) a quote in every blog post. I think it adds some variety and it's fun searching for just the right quote. So this item on the Internet at Clayton Cramer's Blog struck my funny bone:

A Profound Thought

“The problem with Internet quotations is that many are not genuine.”  - Abraham Lincoln
For some reason this reminds me of the Liar's Paradox:
In ancient Greece the fictional speaker Epimenides, a Cretan, said that "All Cretans are liars." So is Epimenides telling the truth?

Wikipedia points out that this example is not really a paradox. It is only a contradiction if you assume Epimenides is honest. If Epimenides is liar, then his statement is false and somewhere there exists an honest Cretan, just not Epimenides himself.

Greek philosopher, Eubulides of Miletus, fourth century BC, discovered a real paradox,
A man says that he is lying. Is what he says true or false?
Is the man telling the truth about lying (and thereby is lying) or lying about lying (and thereby truthful)?

Statements like this are called self-referential and are the bane of mathematical logic. Another classic is, "A barber in town shaves every man who does not shave himself. So who shaves the barber?"

My speciality while working on a mathematics PhD was Set Theory. In the early days (pre-1900's) the definition of "set" was very loose and you could use any description to describe a set. But like the paradoxes above it was easy to describe impossible or at least very strange sets. Consider the set of all sets. This uber-set would have to contain itself as a member. This gives you an infinite recursion, S is an element of S which is an element of S which is ....

To avoid this pathological set we define X, the set of all sets which DO NOT contain themselves. So is X an element of X? Think about it for a bit and you'll see that X is a paradox, it can not exist.

Bottom Line

While this may look like fun and games, paradoxes like this changed math and science in the 20th century. At the dawn of 1900 it was believed that physical and mathematical sciences were on the verge of discovering the ultimate truths of the universe. But soon Einstein's Relativity and Heisenberg's Uncertainty Principle showed that some physical "facts" are relative to the observer or unknowable.  In mathematics in 1931, Gödel's Incompleteness Theorem showed that any system of logic which was strong enough to prove itself valid, would also contain statements like "This statement is false". Thus there could be no system of logic that could prove all statements. There will always be some things which are unknowable.

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