I'm sure many friends and family members are tired of hearing me start a sentence with "I read this article in the New Yorker..." or "I heard this really interesting piece on NPR..." but I really did just read a very interesting article in the New Yorker. Granted, it was published in February 2005 and I just got around to reading it, but.... I digress.
I'm always interested in physics even though I can't understand it. I had the most perfect notes in physics class in college, but I could not pass the exams to save my life. This article, called "Time Bandits" by Jim Holt, was mostly about the relationship that Einstein had with Gödel (who proved two incompleteness theorems, neither of which I can comprehend). But the real genius of the article is that it simplifies what Einstein did when he came up with the theory of relativity (E=mc2 was nearly an afterthought of 3 other papers published in the months before September 1905).
He decided that absolute time didn't exist:
"Einstein, however, realized that our idea of time is something we abstract from out experience with rhythmic phenomena: heartbeats, planetary rotations, the ticking of clocks. Time judgments always come down to judgments of simultaneity... Working from his two basic principles, Einstein proved that whether an observer deems two events to be happening 'at the same time' depends on his state of motion. In other words, there is no universal now."
Sometimes I think I forget that there isn't a universal now.
Everything really is relative. And to try to live as if there is absolute
time is just begging for more hardship than is really necessary for one
person.
As a side bar, this is the part of the article that reminds me why I couldn't handle philosophy or mathematics:
"But Gödel's self-referential formula comments on its provability, not on its
truthfulness. Could it by lying? No, because if it were, that would
mean it could be proved, which would make it true. So, in asserting that
it cannot be proved, it has to be telling the truth."
Wait, what? What time is it? Right, it's absolutely past my bedtime....
2 comments:
An excellent post Katy. I love this stuff.
Maybe I can give you a brief upshot of why Godel's incompleteness theorems are so cool. "Incompleteness of what?" you might ask. Well, a formal system. Formal systems are a lot like a system of logic. You've got certain "axioms" that act like rules. And everything else is a direct result of those rules.
Godel's first theorem proved something huge. Basically, in any formal system that is complicated enough to include arithmetic, you can always construct some "Godel sentence" that is true, but not provable within the system.
What does that mean? Basically it means that truth outruns provability. In any reasonably complicated system, you're inevitably going to run across a sentence that you can't prove. In fact, no one can. The theory is "incomplete", since not all of the true sentences are a direct result of the axioms (rules).
So even if you set up a really nice perfect system of rules, you're going to have situations in that system where the rules don't apply.
Godel's second incompleteness theorem is pretty cool too. It's about consistency. Consistency means there's no contradictions. The theory "makes sense".
Godel proved that in any system (which includes arithmetic and some other stuff), if the system says of itself that it is consistent, then it's not. Put more simplistically, if the theory says it makes sense, then it doesn't. (Weird huh?)
A lot of this relies on the concept of self-reference. A Godel sentence (let's call it "G") says something like, "G is not provable". (See how it references itself?) So, if you found a proof of G, then G would thereby be false. But if you can't prove G, G must be true (since that's what it says!).
The same line of reasoning can be found in an ancient paradox, called the Liar paradox: "This sentence is false". Notice, that if it is false, then what it says is true. And if it's true then what it says is false. This has caused a lot of problems for logic!
Kind of like when you were a kid and someone said, "It's opposite day!" Well, if it was opposite day, then only the opposite statement would be true. But if it wasn't opposite day, then what they said was false. As you can see, I think too hard.
Sorry for rambling on... but ya know, my geek-life is filled with such things.
Holy crap, I'm going to have to read this again when I'm more drunk...
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