When talking about anthropics, people often say things like "assume the universe is finite; weird things happen in infinite universes". I've myself argued that SSA breaks down when we encounter infinities; SIA breaks down sooner, when we encounter expected infinities.
You can formalise this informally with the thought that:
- In an infinite universe, anything can happen, no matter how unlikely: life must exist somewhere. So our existence doesn't tell us anything about life; its probability could be anything at all.
A superficially convincing argument; but not one you'd use for anything else. For instance, consider the following:
- In an infinite universe, anything can happen, no matter how unlikely: if gravity didn't exist, somewhere it must seem to exist by shear chance. So our observation of gravity doesn't tell us anything about gravity; its probability could be anything at all.
I've argued before that anthropic questions are pretty normal. Why would we accept the reasoning in question 1, but reject it in question 2?
We shouldn't. We can deal with questions like 2 by talking about limits of probabilities in larger and larger spaces, or by discounting distant observations (similar to sections 2.3 and 3.1 in infinite ethica). So we might define conditional probabilities like in an infinite universe in the following way:
- Let be the ratio of observers, within a large hypersphere of radius centered on location , that observe and , relative to the proportion that observes . If this tends to a limit as , independently of , then define that limit to be .
Note that this definition works just as well for "we observe the force of gravity to be blah" as with "we exist".
Now, that definition might not be ideal (in particular, "radius" is not defined for relativistic space-time). No problem: different definitions of probability are asking different questions, and can lead to different anthropic probabilities, just as in the finite case.
I'll call these class of questions "SIA-limit questions", since they are phrased as ratios of observers, and dependent on how we use limits to define probability in infinite universes. They each lead to various "SIA-limit anthropic probability theories"; in most standard situations, these should reach the same answers as each other.
Yes, it's perfectly possible to formalise informally, and I encourage people to do it more often. ↩︎
You've fallen prey to observer bias; that the existence of observers in one place and time means that observers must exist in most places and times. In other words, that the environs of any observer must represent a typical sampling of the universe, but observers will normally find themselves in environs that permit the existence of observers; which might not be a typical sampling of the universe at all.
I can illustrate this with an historical example; when it was determined in the 16th century that the stars were like our sun, and that the planets (wandering stars) were worlds like our Earth; many intellectuals adopted the view that all stars had planets and that all planets supported not just life, but sentient life. While it appears that most star systems may include planetary bodies; the second supposition is much less certain. Certainly there are no intelligent Martians, Venerians and Mercurians.
There was an interesting article "Watchers of the Infinity" in which is suggested that multiverse has coherent timelines which exist without beginning and end. Thus observer's probabilities could be calculated along such timeline in unique way (no spheres and ambiguities). But it requires that black holes don't have singularities.
A similarly odd question is how this plays with Solomonoff induction. Is a universe with infinite stuff in it of zero prior probability, because it requires infinite bits to specify where the stuff is? Quantum mechanics would say no: we can just specify a simple quantum state of the early universe, and then we're within one branch of that wavefunction. And the (quantum) information required to locate us within that wavefunction is only related to the information we actually see, i.e. finite.
I find the use of P(X|Y) with radius r centered on location l very refreshing. Though I have a different focus. Here by letting radius approaches infinity the choice of centered location becomes irrelevant. I'm more interested in what if the radius is not infinite, what if it is quite small or even approaching zero? What is the location centered on then?
I think the location outght to be centered on us, where we are at in spacetime, and the radius would represent the extend of our observations. R approaching zero could represent a state of ignorance, a prior state before making any observations about the universe. Then the location centers on the primity concept of self and now. This way one's own existence is always a prior knowledge.