Question

A form of the anthropic principle is as follows: "We can observe the universe only because we can exist within it in some way such that we can observe it, and it exists such that we can observe it."

What mathematical consequence does this have? I know it's broadly a problem of Bayesian probability, and we must consider all that we see from the perspective P(A|B), A = some aspect of observed reality, B = we think, therefore we are.

Can this be formulated in some useful and general way to answer questions about the universe, existential, cosmological or otherwise, or do the mathematics here give us little information?

NOTE: I know that the anthropic principle is often stated in a much more specific way and looked at from the perspective of cosmology, but that's not what I'm looking for here.

Edit: To clarify the mathematical content of this question I'll give two examples (one from a comment below).

1) I've seen claims like "the anthropic principle indicates that we most likely live at a time such that half of all people that have ever been born have been born". I want to know if a statement like this is at all reasonable or not.

2) Consider it in these (not entirely sufficient) terms: You have a vague outline of a set of prior distributions in addition to some error-prone observations whose errors depend on the prior distribution. How can you glean information about the prior distribution.

Answer 1

The question could use some clarification, but perhaps you would like to read The Height of a Giraffe for an example of a fascinating calculation based on 'anthropic reasoning'. This was one of a number of similar papers to come out a couple of years ago and there was plenty of discussion about it on physics blogs at the time.

I suppose in some sense anthropic reasoning is the cousin of 'Fermi problem' type calculations. I think there probably are interesting mathematical/logical questions in there somewhere about the validity of such estimates but I don't know what the best way to frame them is either.

Answer 2

I agree with Chapman and Clark that this isn't really a math question: anthropics is the epistemological problem about how to interpret probability theory; the theorems themselves are not in doubt.

However, since this question hasn't been closed, I might as well make my first post and answer it: if you're interested in anthropic reasoning, I recommend checking out the work of Nick Bostrom, who has written a book (Anthropic Bias: Observation Selection Effects in Science and Philosophy, available free online) and a number of papers on the subject. I hasten to mention again that these are primarily works of philosophy and not mathematics, but they do include equations where relevant.

The first example you mention is known as the doomsday argument (because if half of all observers that will ever exist have already been born and population grows exponentially, then we should expect the end of the world to come quite soon!). Bostrom discusses the doomsday argument in chapters six and seven of his book and argues that it has not been refuted (!).