Books Books Books

I've been reading a few interesting books lately. One is about the number Omega - Meta Math! - an extension of the ideas explored in Gödel's incompleteness theorem in the 30's, and Alan Turing's halting problem. If you're not familiar with these, they are generally taught at the Universities as interesting and quirky problems by the math and computer science departments, unsolvable problems like the logical statement "This sentence is false". The author is of the opinion that they are more than interesting curiosities, and I have to agree - in fact, I think that they are more like the electromagnetic phenomena that were a minor problem for physics a century ago.
The basic idea is this: Math is an experimental science. There will never be a mathematical Theory Of Everything (TOE), just like there will never be a TOE in Physics (the Grand Unified Theory of Everything©, also known as The Holy Grail™). The reason: A theory must be more concise than it's data. A theory is like an algorithm that allows you to generate all the data just knowing a few principles. In Math, the principles are called Axioms, and the idea is that we can prove or discover every mathematically true 'fact' by reducing it's truth to some basic axioms, and vice versa - we can construct or generate all mathematical truths given nothing but the axioms. The existence of the number Omega, however, means that this will never be the case, could never be the case, and in fact, for every mathematical truth we can reduce, there are an infinite number of truths that are irreducible. No theory could contain them. Likewise for physics.
If this doesn't make sense, think of it this way: Mathematicians and Physics have a similar aim: to describe (theorize) all of the things that are true ('facts') about math / physics. This is like asking for a map: you want one that shows you what's actually there, and where it is in relation to everything else. You don't want one with made up streets or huge missing sections. However, Gödel, Turing, Chaitin, etc., suggest that, in order to have a finished, complete map, it would have to be life-sized! (Now Google might be trying to do this, but don't go to the gas station and ask for a life-sized map). As useless as a life-sized map would be (a map that's as big as the place you're trying to represent), is how useless a life-sized TOE would be. It wouldn't even be a theory at all, technically. Now, I happen to think that this has some real deep philosophical implications, mostly about epistemology and objectivity.

The book that really makes me take a second look at objectivity is Lakoff and Nunez's book Where Mathematics Comes From. Wikipedia says that

In philosophy, an objective fact means a truth that remains true everywhere, independently of human thought or feelings.

The problem is, facts are just human thoughts. It's like saying a skrime is a kind of fish that doesn't live in water - it's out in space somewhere. Lakoff does a wonderful job of exploring the cognitive science of mathematical reasoning - showing how our conceptions about math (and the historical development of math) is an extension of our general cognitive abilities - a human property. [Think of it like this: in the universe, there is no such thing as the color blue. Our everyday experience of the color blue is a phenomena created by the particular neurological structure of our brains and eyes, etc. - the things we see do not in nature possess a quality called 'blue'. They reflect (and refract, etc), wavelengths of light that can trigger the experience of blue in us, but unfortunately for us, there are an infinite number of different combinations of wavelengths of light that can trigger the exact same experience, called metamers by vision scientists. Now think of math in the same terms - something that uniquely exists within our minds, something our mind creates as an interpretation of the experience of the world around us. So, mathematical ideas are not discovered, but created. There is no such thing as transcendental math.]

I'm really enjoying these books in combination with each other. I highly suggest both for anyone mildly interested in their subjects - from epistemology to mathematical analysis.