So many times we have said something cannot be done and we have been proven wrong. We can arrive at points east by traveling west. We can fly in a heavier-than-air craft. Altruism can arise from competition.
That’s why it’s extra amazing to me when someone proves — really proves — something is impossible. Such is the case with one of the most brilliant insights that you might never have heard of: Godel’s Incompleteness Theorem.
In the early 1900′s, Bertrand Russell and Alfred North Whitehead laid out a way to derive all mathematical knowledge with logic — or so they thought. Their approach was to start with a few common-sense axioms and a few rules of inference and build everything up from there. This is just what you did in high school, using theorems you had proved one week to prove more theorems the next. Done right, it’s infallible.
Doesn’t it seem that if math class had gone on forever (yeah, yeah, I know…) you could, in theory, have proven every mathematical thing there is to prove?
“False!” said the young logician, Kurt Gödel. “There are some statements that are true, but which you can never prove. What’s more, I’ll show you how to construct an infinite number of such statements.”
He did this by turning Russell & Whitehead’s work on itself. He showed how to make their infallible engine of mathematical facts create statements like “This statement cannot be proven.”
That’s easy to do in English (I just did it), but imagine doing it with nothing more at your disposal than the basic rules of arithmetic and logic.
As I said, Gödel proved that such statements are both true and unprovable.
They are true because he showed how to derive them from axioms and infallible logic.
They are unprovable because, well, they say they are unprovable and we have already agreed they are true!
What’s more, Gödel continued, even if those statements were added to the system as axioms, more true-but-unprovable statements could be generated from that new system.
Gödel’s achievement was as revolutionary in math and logic as Einstein’s theory of relativity was in physics. The most brilliant minds in the world had labored for hundreds of years to create a consistent and complete formulation of mathematics, and he showed this was impossible.
Of course, this only applies where Gödel proved it applies, namely the realm of math and only for those special types of statements. Gödel’s theorem should not be construed in a metaphysical way, or for all knowledge.
If you want to read more, the best and most concise exposition I could find is this humble Web page. It ends with some fun speculation about how the theorem relates to Zen Buddhism.
If you want to read a lot more, there’s a favorably reviewed book at Amazon called Gödel’s Incompleteness Theorem: An Incomplete Guide to Its Use and Abuse.