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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.

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closed as not a real question by Robin Chapman, Steve Huntsman, S. Carnahan, José Figueroa-O'Farrill, Loop Space Jun 21 '10 at 7:47

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

I can't see a concrete mathematical question here. It's another question of the form: "is there a connection between topic A and topic B?". There's no real answer to these, just scope for lots of discussion and argument. – Robin Chapman Jun 20 '10 at 16:45
This absolutely is a real question. 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. – DoubleJay Jun 20 '10 at 16:48
You say it's a real question; I say it's not a mathematical question - I expect we'll have to agree to differ :-) – Robin Chapman Jun 20 '10 at 16:51
In my opinion, the question is either not mathematical or needs clarification. The Anthropic Principle is not a mathematical statement, but rather (charitably, perhaps?) a statement about the physical world. How therefore could it have a mathematical consequence? Am I right in thinking that your question is more like "How can we convert the Anthropic Principle into a statement that has some mathematical language and content?" – Pete L. Clark Jun 20 '10 at 17:15
"half of all the people that have ever been born have been born"??? Maybe you mean "half of all the people that have ever been born are now alive"? or, "half the people that will ever have been born have already been born"? I've always taken the "half ... are now alive" statement, in the form, "only half the people born so far have died" as evidence that there's a 50% chance that I'm immortal. – Gerry Myerson Jun 21 '10 at 7:20

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.

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I'm getting a broken link for the Arxiv paper. Are there any alternate links? – DoubleJay Jun 20 '10 at 17:58
@DoubleJay: Seems that arXiv has connectivity problem at the moment. Try a mirror, e.g. (worked for me) rather than – Sergei Ivanov Jun 20 '10 at 18:58

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 (!).

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