Book review part 2
Feb. 5th, 2009 10:05 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
thetatianaI'm well along in the book now, though not completely finished. He's gone through again the Godel proof that shows any sufficiently powerful formal mathematical system is incomplete. I understand it better this time. For every statement in a formal system, you can map a unique number to that statement, which you can easily decode to yield the statement. So number = statement and statement = number. Cool. That's step 1.
Step 2 is that for every typographic manipulation rule of symbols in your formal system, there exists a mathematical equivalent rule for manipulating the numbers that match the statements. So there's an exact mirror of the entire formal system in mathematical properties, manipulations, etc. That is also clear.
Then step 3 is you can make a statement or formula in the mathematical system that says something about these numbers, and at the same time they're saying something about the system itself. All you have to do is come up with a statement that means (when looked at on a high level), "I am not provable inside formal system X". And it turns out that it's true! And it's not provable! In fact, there are an infinite number of true statements in any formal system that aren't provable inside that system. So much for mathematical systems mirroring all possible reasoning about numbers, and capturing all possible truths.
There's something magical about Godel's proof. It means things are much more open-ended than we ever dreamed. The universe has a way of continuing to do that to us over and over. For instance, during the 17th and 18th c. there was a thought that everything was deterministic in a strict Newtonian way. If you started the billiard balls going, they would keep on going through predetermined paths to the end. Now we think that tiny differences too small to measure in the initial conditions, and the quantum fluctuations of the vacuum, not to mention the probablistic nature of quantum mechanics means that nothing at all is predetermined, and that the future is unknowable even in theory. What an interesting universe we live in!
Step 2 is that for every typographic manipulation rule of symbols in your formal system, there exists a mathematical equivalent rule for manipulating the numbers that match the statements. So there's an exact mirror of the entire formal system in mathematical properties, manipulations, etc. That is also clear.
Then step 3 is you can make a statement or formula in the mathematical system that says something about these numbers, and at the same time they're saying something about the system itself. All you have to do is come up with a statement that means (when looked at on a high level), "I am not provable inside formal system X". And it turns out that it's true! And it's not provable! In fact, there are an infinite number of true statements in any formal system that aren't provable inside that system. So much for mathematical systems mirroring all possible reasoning about numbers, and capturing all possible truths.
There's something magical about Godel's proof. It means things are much more open-ended than we ever dreamed. The universe has a way of continuing to do that to us over and over. For instance, during the 17th and 18th c. there was a thought that everything was deterministic in a strict Newtonian way. If you started the billiard balls going, they would keep on going through predetermined paths to the end. Now we think that tiny differences too small to measure in the initial conditions, and the quantum fluctuations of the vacuum, not to mention the probablistic nature of quantum mechanics means that nothing at all is predetermined, and that the future is unknowable even in theory. What an interesting universe we live in!
