Saturday, February 6, 2010

The Limits of Knowledge, Part III

(Continued from parts I and II)

Kurt Gödel (1906-1978)

I left off last time with Bertrand Russell and Alfred Whitehead trying to derive all of mathematics from a small set of logical axioms, preserving consistency and completeness in the process. A consistent mathematical system will always give the same answer to the same question. 1+1 = 3 doesn't occasionally turn out to be true. It's always false. Completeness says that if you can form a valid mathematical statement, you can derive whether it's true or false in your system.

It's easy to see why math being inconsistent would be problematic: You can never be confident about anything again. 1+1 might equal 3. The oven might start shooting out cold air. The Earth could disintegrate at any moment. Who knows? Any argument against even the craziest idea is ultimately based on logic, and that could always change.

An Incomplete Mathematics is less worrisome, but it would upend the scientific and mathematical philosophy of much of history, dating back at least to Greece. We were often seen either as being guided explicitly by divine forces, or as the only logical outcome to a specific setup of the universe. Truth was viewed as a platonic beauty: every statement is true or false. You may not be able to figure it out, but in theory it could be figured out. At the very least God must know the answer. But if all mathematics and logic are incomplete, then there are meaningful, interesting questions without answers. For the divine to be all knowing he'd have to be inconsistent. Which is arguably his prerogative, but it's a very different world. It's not that any path towards truth would be doomed to failure because of your human limitations, any path towards truth would be inherently doomed to failure. No such path could possibly exist. It's a much more pessimistic understanding of the world.

And fittingly, a very pessimistic individual proved that no formal system can be both consistent and complete. It seems common for ideas that upend fields to come from the periphery, from the marginalized groups of scientists. Einstein was unable to find work within the physics community before he revolutionized our understanding of the universe. While Kurt Gödel did his work from within the field of logic, he was inherently an outsider. One of my favorite anecdotes about Gödel revolves around his dislike of human interactions. When someone would try to schedule a meeting with him he would oblige, making explicit plans to meet at a particular location at a particular time and date. Gödel would never show up. When asked why he made all those appointments he had no intention of keeping, Gödel answered that it was the only system he'd found that stopped people from persisting in trying to meet him.

Russel had found a paradox in Frege's attempt to formalize mathematics, so he constructed elaborate rules to keep those paradoxes out. If in doing so he introduced new paradoxes, he'd invent new axioms to hide those. G
ödel proved that any attempt to remove paradoxes just added new ones. It's a tricky bit of math, but it comes down to the idea that any interesting formal system can construct a statement along the lines of "this statement is false". Any system that can't make that statement in some form is too restricted to do anything more interesting than addition in. Gödel didn't just prove paradoxes existed in some system, he proved they'll exist in any system. Mathematics is filled with infinities of paradoxes, and there's nothing you can do about it.

ödel grew less stable as he aged, eventually become extremely paranoid of both germs and poison. He refused to eat anything unless his wife tasted it first. When she was hospitalized for six months and suddenly unable to serve this role, Gödel starved to death. It was an odd end for such a monumental mathematician. Gödel's Incompleteness Theorem is one of those truths so fundamental to the universe, it's unlikely his name will ever be forgotten as long as man exists. It's not the most exciting theorem, but it conclusively set down limits to knowledge, a concept that would be extended with the advent of computers.

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