Take this hypothesis: 2+2 = 4. There are an infinite number of things 2+2 could equal, but I, like Einstein, without any empirical testing of my mathematical hypothesis, believe it to be 4. If you believed the sum to be 5, again without any testing, I don't see how you or I have any different number of bits dedicated to our answers. Your bits, however, are wrong.
The apples example is from "How to Convince me that 2 + 2 = 3": http://www.overcomingbias.com/2007/09/how-to-convince.html
Interesting essay on Bayesian reasoning: http://yudkowsky.net/bayes/technical.html
"But from a Bayesian perspective, you need an amount of evidence roughly equivalent to the complexity of the hypothesis"
But what is the "complexity of the hypothesis"? Without a proper definition, there is not much left to the article, or is there?
2+2 = 4. Thus it is written.
Maybe I am missing something.
All science is provisional, this much is true. Math, however, is a formal symbolic system for representing things in reality. 2 + 2 = 4 not because of some inner truth in math but because when we observe nature and combine 2 things and 2 things we have what we call 4 things. We could change the symbols around all day and they would still work. So math is just a generic way of talking about that which we can observe.
The interesting thing happens when our symbolic system escapes that which can be observed, or when it is incomplete, say in the case of negative numbers (then rational, the imaginary, then irrational, etc.) At this point the exercise becomes one of either bringing the system of symbols to some application that has observable impact (applied physics) or changing the symbolic tools. There's nothing Bayesain about 2+2=4 -- that's the way the symbols are supposed to work.
Now whenever we get "stuck" we have to go back and check out symbolic systems. Just like geeks build O/S as a hobby or college experiment, I imagine physics and mathematicians build calculi, or systems of symbols and rules for working with them. Wolfram came up with a great question in NKS -- what if the universe is really discrete and not continuous? In other words, when Newton created the integral he might have taken math down a path that ends up breaking when you try to put a GUT together. I think that's a helluva question, but it's above my pay grade.
There was a book George Gamov wrote: 1-2-3-Infinity about the way various counting systems and numbers play together. Go read it -- it's better than this blog article.
It all depends on the axioms you use, iirc. I can think of cases where 2+2 does not equal 4. For example (very roughly -- this example isn't really 100% accurate, but you get the idea), 2 protons plus 2 protons != 4 protons. in fact, you get the helium nucleus, I believe, 2 protons plus 2 neutrons. before you say this is an esoteric example, i might point out that this is the reaction that powers the sun and you wouldn't be alive right now without it. (for the actual reaction see here: http://en.wikipedia.org/wiki/Proton-proton_chain_reaction)
Someone with more maths training might correct me here, but 2+2=4 is true because you're using the usual axioms you learned in school. you might also say it's obvious that 3 times 3 = 9, or that ab = ba, but there is a lot of interesting math around anti-commutative algebras where ab=-ba, for example. it's all about the axioms you use.
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
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.
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of course, it turns out that GR is wrong, at least at small-scales. while special relativity is compatible with quantum field theory (indeed, SR was the motivation for moving from quantum mechanics to QFT, as quantum mechanics does not respect SR), general relativity is not.
