Friday, December 10, 2010

On the Validity of 'Unknown' - Illustrated with an Example regarding the Evolution of Numbers

In our earlier posts we discussed about Science and Art, and later about evolution as a result of the prevailing uncertainty in the universe (Realizing Perfection through Uncertainty). But in this post let us polish this philosophy by taking a finer example.

Pythagoras found that the area of the two smaller sides of a triangle is equal to the area of the largest side (hypotenuse). Such a theory was perfect in their sense, because it did not pave any way for approximation. But someone asked what is the length of the side of a square which is double the area of a square of unit length. This lead to an arithmetic crisis during Pythagorean period. But ancient Egyptians had approximated the result to 7/5. But for Pythagoreans that was not enough, since they were probably addicted to the perfect result of Pythagoras theorem. Now somebody else came up to say that the side of the square is divisible by 2 if the area of the square is quadrupled. That means the side is divisible when the area is multiplied by 4, 16 and so on. But what about other squares. Of course geometrically they can be determined, since they can cut a sting of this mysterious length and preserve for future reference. But that's not what Pythagoreans were trying to do, they wanted arithmetic answers. This lead to the finding of irrational numbers. We know the result is square root of the side (2 here) which is irrational.

This example illustrates the fact that inexpressible things are not unknown. Unknown things are just those things which can not be expressed by any means. But as the means are found out, the previously inexpressible thing become quantifiable, or in general expressible. Thus in this sense there is no unknown thing.