Wednesday, April 05, 2006

Ultimate Dr. Seuss Geek Topic

In The Cat In The Hat Comes Back, the Cat comes returns to visit the brother and sister during the winter-time when they are tasked with shovelling a large amount of snow. The Cat goes into the house and takes a bath, leaving a large ring around the bath tub. When the kids start to panic, the Cat says that he knows how to clean it up and takes it off of the tub and onto the wall. When the kids tell him he needs to take it off of the wall, he lifts his hat revealing a smaller Cat A who takes it off of the wall and onto the rug. When this won't do, Cat A lifts his hat to reveal Cat B who puts it on the bed covers and so on with nested Cats C, D, E,... transfering the stain onto the bedsheets, the tv,...until finally it ends up on the snow. When the kids demand that the snow be cleaned, Cat Z is revealed with "VOOM" a cleanser so powerful it will clean anything and indeed, it takes the stain out of the snow.

This is exactly what Bertrand Russell, Rudolf Carnap and the Logical Positivists hoped would happen with mathematical truth.

In the 5th century B.C., Euclid wrote the Elements, a book that took Greek knowledge of geometry and brought it together in an incredible fashion. By starting with a handful of definitions, 5 axioms, and 5 postulates, he showed how all of geometry could be derived with absolute certainty. It was one of the greatest acheivements in human thought.

But then along came non-Euclidean geometry which made use of different axioms. The question became which system is right. Surely, it had to be Euclid because the other systems were so screwy in what they said about space. If a contradiction could be shown to follow from the other geometries, then they could be rejected and Euclid wins the game. So the problem was to find a contradiction in non-Euclidean geometry.

It was a problem that had mathemticians working for centuries until Felix Klein, Eugeno Beltrami, and Henri Poincare (twice) created "relativie consistency proofs." These were ways to reduce non-Euclidean geometry to Euclidean geometry. The only way non-Euclidean geometry could have a contradiction is if Euclidean geometry had a contradiction. And surely Euclidean geometry did not entail a contradiction. Everyone believed it, but no one had ever proved it.

David Hilbert took a half-step. He was able to create a relative consistency proof that reduced Euclidean geometry to arithmetic. But did arithmetic contain a contradiction? Arithmetic could be reduced to set theory. But did set theory contain a contradiction? Each nested in the next, nested in the next, nested in the next. What we needed was meta-mathematical VOOM. That last step that would demonstrate in the bottom layer absolute consistency, that is to say, no possibility of contradiction.

Russell and Carnap thought they found it -- logic. If arithmetic could be reduced to logic itself, surely logic can't give rise to contradictions. Logic would be mathematico-philosophical VOOM!

But then came Kurt Godel. In 1931, Godel proved with absolute certainty that this consistency couldn't be demonstrated. That any logical system powerful enough to be able to prove all true sentence would also allow some false sentences to be provable, and that any logical system that proved only true sentence would miss a few.

Godel, who starved to death because he thought someone was trying to poison him, lived in a world closer to that of Dr. Seuss than the rest of us, but he proved to us with absolute certainty that it is impossible for us to be living in the world where the Cat in the Hat comes back.