knowledge is not truth

Indeed, as Godel demonstrated and Hosftadter rephrased in Godel,Escher,Bach, "Truth is a stronger notion than provability," meaning that there are statements that are true but whose truth (or falsehood) cannot be proved logically.



There is a constraint on this, which while I don't recall the exact verbiage, goes something along the lines of "in any sufficiently complex logical system" (where "sufficiently complex..." means anything able to handle the integers).

A really curious effect of this simple statement is that say X is a true statement in our system that cannot be proven to be true. Let's say we just assume X is true. There will yet be another statement Y that is unprovable in the system. No matter how many assumptions you make, you will either a) continue to have unprovably true statements, or b) run into an inconsistency. (I've never actually read the details of the theorem - though I imagine it's similar to the computability theorems - but I think this only applies to consistent systems.)

Many moons ago, we believed that if we knew the start state of the universe in sufficient detail and all the rules governing the system, that we could theoretically predict the future (or describe the past). Now we know it aint so.

In the past couple centuries or so we have been bombarded with a bunch of crazy ideas. Not that I'm in the mode of patronizing our forebears. Had I lived in some other time, I would probably be one of the guys who thought the promulgators of these deviant notions were complete loons.

Gauss told us that playfair's axiom (statement of euclid's 5th postulate) is different than the first four. That we can assume other things and get other, perfectly consistent geometries. (Historical comment - for centuries everyone *knew* the 5th postulate was different. They thought perhaps that it was derivable from the first four.) But..but...I can draw a line and a non-colinear point on a piece of paper and it's very obvious there's EXACTLY ONE line I can draw through the point that is parallel to the line! SEE! SEE!

Cantor told us that there are some infinities larger than infinity, i.e. there are orders of infinity. There are more real numbers than there are integers. In fact, there are more irrationals (numbers like pi and square root of 2) than there are rationals (numbers like 6/1 and 3/7 and 23/444 that can be expressed as the ratio of two integers in lowest terms). How could someone even think that infinity could be such a thing as there could exist more than one of them?

Einstein told us, among other things, that distance varies with an object's speed. But distance is distance, dangit. How can a distance not be exactly what it is? If X != Y, then either the distance is X or the distance is Y. It can't be both!

Heisenberg told us that we can't measure an object's speed and momentum simultaneously - that the act of observing changes the system.

Goedel told us that even if we knew all the rules, we still wouldn't be able to know everything.

Turing (a personal hero) did the same thing for computer problems. He says there are some computer programs, the input for which cannot be proven in advance to halt.

These last three are just nonsense. Crazy talk. All of it. Madmen's musings. These are some of the prominent loons, but there were many other loons, some of lesser, some of equally loony stature who contributed to our common loony "understanding" (if you want to all it that).

k