[jey]

"There's nothing Bayesain about 2+2=4 -- that's the way the symbols are supposed to work."

How did you come to this decision that "that's the way the symbols are supposed to work"? I bet it's some sort of process of taking in information and updating your beliefs. And the idealized optimal version of that is Bayesian inference.

"Wolfram came up with a great question in NKS -- what if the universe is really discrete and not continuous?"

Sorry for the anal-retentive nitpicking, but this question isn't due to Wolfram. I'm not qualified either, but it's definitely a fascinating thing to think about. Maybe our universe is just a small computer program. http://en.wikipedia.org/wiki/Digital_physics

[DanielBMarkham]

I did not come to any decision about that's the way the symbols are supposed to work -- that's my whole point. The symbols represent the way it works. They are a self-referencing representation. There's no decision involved here at all. You can call it an axiom, but that misses the point. An "axiom" only has meaning inside certain formal symbolic systems. It. Just. Is. It exists. Different people with different numbers and operators? They'd be asking why %^ $% $#^ #$$%, for instance. But it's the same thing. The answer is in the question. It's like asking "Why is red a color?"

I didn't mean to imply Wolfram came up with computational reality, I simply mentioned that he brought it up in his book. It may turn out the integral was just a nifty little shortcut that took a lot of impossible-to-calculate math off the table for physicists. Thanks for letting me clarify that.

[jey]

"I did not come to any decision about that's the way the symbols are supposed to work -- that's my whole point. The symbols represent the way it works. They are a self-referencing representation. There's no decision involved here at all."

There's a difference here between what you believe and what is true in external reality. (Yes, I know you can spend lots of time debating whether the latter is even a coherent idea, blah blah blah, but all that stuff isn't relevant.) You had to come to a belief about what "+", "=", "2", and "4" mean. You were not born with this information embedded inside your head! :) This is true regardless of whether the symbols by themselves unambiguously and uniquely pin down the meaning of the expression. Even if a certain symbol sequence has a "unique unambiguous" interpretation, you still had to read those symbols and interpret them, and you learned something in the process -- which means you updated your beliefs about the symbols. I agree that the "real" interpretation in external reality of these symbols didn't change (if such a thing even exists), but your understanding of the symbols changed.

In general, probabilities are subjective and a property of the observer. They describe degrees of belief that the observer has in various propositions, and don't directly have anything to do with external reality. The only way they're connected to external reality is that you make observations in reality and use the gathered information to update the probabilities you assign. This means the claim that the symbols "2", "4", "=", and "+" are unambiguous is irrelevant even if it's true.

BTW - Thanks for the link on your blog to the NYT graph of the price of gas in constant dollars.