Thursday, March 19, 2009

The End of Elegance

I think there are three breakthroughs in mathematics that have really shook the foundations of the field, the first is the discovery of irrational numbers (formerly accredited to Hippaus, a member of Pythagoras's school around 500 BC but Indian mathematicans are now thought to have been earlier), the second, relates to the work of Cantor in the 19th Century in showing that infinity comes in different sizes, and, thirdly, Godel's incompleteness theorem in the first part of the 20th century, which demonstrated that attempts to form systems of axioms that are entirely logically consistent are doomed. These amazing feats of insight, intellectual rigour and imagination, initially triggered rejection and a strong counter reaction from their peers. In the case of Hippaus, legend has it, that this discovery cost him his life, as Pythagoras's followers incensed with his proof that the square root of 2 is irrational threw him into the sea ! After time, these ideas were accepted, incoporated into the mathematical mainstream and built on by thinkers who followed in the wake of these tidal waves.

Lets address the first intellectual tsunami; the discovery of irrational numbers. Why was this so important ? This discovery was important because it challenged a central notion of the type of mathematics that Pythagoras and his followers were seeking to establish. Pythagoras viewed mathematics as sacred and capable of explaining the deepest ideas and describing the natural world around them. For the school of Pythagoras, shapes and numbers were elegant expressions of profound ideas. In this intellectual climate, they assumed that numbers could always be expressed in terms of simple ratios of integers (e.g. 1/7), which they understood in geometric terms - a feature of Greek mathematics that makes it hard for modern reader to appreciate their arguments directly.

What did Happaus show ? We don't have access to the original proof but we can assume that his proof followed this type of argument:

If the SQRT (2) is rational, than is follows:

SQRT (2) = a/b where a and b are integers.

It also follows:

2 = a2/b2, which can easily be turned around to 2b2 = a2

We know that 2 times any number will result in an even number and that square root of any even number results in an even number, therefore, "a" must be an even number. If "a" is an even number than we can express it as 2r, where r is another integer, and we can re-arrange the equation above to:

2b2 = 4r2, which can be simplified to b2 = 2r2.

Using exactly the same argument as the one above, we can say that "b" must also be even. Now, we have a contradiction, because any ratio of two even integers can be reduced to a ratio involving an even and a odd number (e.g. 2/8 = 1/4). Therefore, it is not possible for the square root of 2 to be expressed as a ratio of two integers. In fact, this intriguing qauntity can not be directly calculated but only approximated.

This is still a somewhat shocking result. A physical representation of the square root of 2 can be easily visualised by constructing a right angle triangle with two sides the length of 1 m. The hypotenuse of the triangle must be the square root of 2 (using Pythagoras's famous theorem) .... we can see it, we can easily esimate the length using a ruler, how can it be that we can not calculate it ? This is exactly the intellectual dilemna that haunted Pythagoras, disturbed many mathematicians since the Greeks (notably Newton) and still causes one to shake your head and muse that God must be playing some elaborate joke on us. The later discovery of the irrational nature of pi and e, and Cantor's discovery that that there are many more irrational numbers than rational on the number line, just serves to deepen the shock. The type of elegance visualised by the early Greeks was over. No wonder they metaphorically shot the messenger by throwing him into the sea.

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