So, I’m up late after dancing and can’t sleep and I’m thinking about this great article I read today about the Humanities, and how we as a society are slowly eroding their importance by forcing them to justify themselves in economic terms.
It was really good, but there was this one epigram about how nobody was ever killed for getting Hubble’s constant wrong. Which may be true, but math and science have their own martyrs – not just Galileo’s ‘eppur si muova’ or Giordano Bruno before him. They were killed for heresy, by people who believed them to be not just wrong, but so gravely wrong their ideas were too dangerous to allow.
My favorite math martyr is Hippasus (… of Metapontum, says Wikipedia). He was killed by the Pythagoreans because he was telling people that the square root of two is irrational. The thing is, the Pythagoreans knew he was right. They knew that he was right, and that the foundation of their religion (and thus their temporal power) was flawed.
Galileo and Bruno could be accused of being misled, or incorrect. Hippasus couldn’t – the proof’s so elegant that it’s impossible to argue with.
Suppose the square root of two were rational.
Then there would be a ratio of p and q, where p and q don’t have any factors in common (i.e., they’re mutually prime)
and p over q squared is two.
so p squared over q squared is two
so p squared is two q squared.
so p squared is even.
so there’s a number r, equal to q squared, where two r equals p.
therefore four r squared is equal to two q squared.
therefore two r squared is equal to q squared.
so q squared is even.
Therefore p squared and q squared have a factor in common
And so p and q must also have a factor in common
But we defined them as mutually prime.
Driving home the other night I was trying to think of how to express the mathematical idea of elegance. I think most peopl edon’t understand how mathematicians view math, or understand how much art there is to it. This idea of elegance – where somehow the means by which you arrive at a conclusion transparently illustrates that very conclusion – it’s so appealing and not really found anywhere else.
Which is to say, that proof’s one of the most elegant ones I know.