The Square Root of Two (part 1)

Pythagoras taught that every number can be written as a ratio. Consider a man's arm and his height. One can certainly write the ratio of the length of his arm to his height. Can't one?

Pythagoras was, of course, very familiar with the square root of 2, which is the length of the diagonal of a right triangle with sides that are both 1. But then one of his students proved that the square root of two could never be a ratio. Pythagoras figured his student was a) irrational; and b) a heretic.

I'm not sure if it's true, but they say that Pythagoras had the student murdered by his more devoted followers who would never question his teachings.

As it turns out, if one could measure the length of a man's arm perfectly, and if one could measure his height perfectly, the length of the man's arm would NOT be some perfect ratio of his height. A very small infinity of numbers can be written as perfect ratios, and a very large infinity cannot, so the length of a man's arm is never a perfect ratio to his height.

This is an important part of basic logic: one cannot say that something does not exist without a proof (like the one Pythagoras' student gave that no ratio can exist for the square root of two). Some say that anyone claiming that something does not exist has no burden of proof, those who say it does exist have all the burden. This is wrong. If no one cannot prove that something exists, and no one can prove that it does not exist, then logicians are forced into agnosticism: they must agree that no one knows if it does or does not exist.

Of course, the square root of two exists, but there is no way to write it as a ratio. It's the ratio that the student proved does not exist.

That's about all there is to say about Pythagoras and the square root of two.

Socrates used the square root of two to prove that reincarnation is true, but I'll leave that one for another day.