Thursday, January 20, 2011

What is Truth?

Pilate said to him, "What is truth?"
- John 18:38
Here is a clever quote that ties in with yesterday's topic of paradoxes vs knowing all things..

"There is a theory which states that if ever anyone discovers exactly what the Universe is for and why it is here, it will instantly disappear and be replaced by something even more bizarre and inexplicable.
There is another theory which states that this has already happened"

- Douglas Adam, Hitchiker's Guide to the Galaxy
Over 2000 years ago, the Greek philosopher Euclid proposed five laws (Axioms) that could be used to prove "all" of geometry.

  1. Any two points define a unique straight line
  2. A straight line can be extended infinitely far
  3. A center and a radius define a unique circle
  4. All right angles are equal to one anothe.
These first four laws are simple and "obviously" true. The fifth law is not so simple:

5. The parallel postulate: If a straight line crosses two other straight lines and makes the same interior angles on the two lines, then the two lines are parallel, meaning they will never cross if infinitely extended.

These five Axioms worked very well as the basis for all of anceint Geometry. But the fifth law bothered mideval mathematicians. It is so wordy and complex compared to the others. Could it be simplified? Not really. Is it necessary? Yes, if you want to prove anything interesting. Could it be proved from the first four laws. No.

One method of proof is called "Proof by Contradiction". Geometrists said, suppose law 5 is false and the two lines do meet. Does this result in something impossible or inconsistent? To their surprise, it was consistent and created a stranger and more bizzare universe of mathematics. Today we call this non-Euclidean Geometry. The easiest example is a sphere or globe. The equator crosses all longitude lines at right angles so longitudes are "parallel" but contrary to Axiom 5 they meet at the north and south poles.
Bottom Line

It turns out that the fifth law is only true on perfectly flat surfaces and is independent of the first four laws which are "universally" true. So if asked if the parallel postulate is "true" the answer is, "it depends". This is an instance of a statement which can be true or false - both modes describes a different reality, both cases equally "real".

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