Of course. But the numbering doesn’t really help; lets assume the boxes are numbered in powers of two, in meters, starting at 2^0 = 1. After opening ten boxes, the next box is 2^10 m. Let me define a new unit, M = 2^10 m. Then the situation is the same as before.

Of course, m and M are distinct – but we have no way of preferring one of them, apart from arbitrary choices.

If the physical laws we discover require the universe to be finite or infinite, well, then we know. Even if they don’t, they can still inform our choice of prior.

That used to be my argument. Turns out it’s pretty circular – it depends on the physical laws being the same everywhere – including in the unobserved portion of the universe. Getting that sort of result, all the way out to infinity, is at least as hard as showing an infinite universe in the first place.

LP, I pretty much agree with your analysis. I’d say that “the universe is infinite” is half of a scientific statement – it’s disprovable if it’s wrong. But if it’s right, then there does not exist an experiment that can distinguish between it and its inverse “the universe is finite”.

James, if the universe doesn’t exist, it would still be nice to know whether it’s an infinite or a finite universe that doesn’t exist.

Are you a Catholic Jew or a Protestant Jew?

You folks can philosophize all you like about the scientific method, but among actual cosmologists (at least non-fringe ones) there’s no debate about the finite-ness of the universe. The Big Bang theory requires a finite universe that grew from a point singularity, at a finite rate, a finite time ago; and that theory has been so incredibly strongly confirmed by observation (like COBE) that it’s nearly universally (sic) considered correct.

There were, in the past few decades, questions of the age of the universe, and whether it will keep expanding forever (meaning time is potentially infinite); both of those seem to have been mostly settled by now.

Hmm. “Scientific” here means, “in accordance with the principles of the scientific method.” The scientific method is fairly well known: observe some phenomena, form a hypothesis about them, do some experiments to validate or invalidate the hypothesis, repeat. But actually, experiments are generally designed to invalidate the hypothesis, because ‘positive’ results might just mean that we haven’t checked in the one area of the universe that would invalidate the hypothesis.

So here’s the problem: let’s say we form the hypothesis, “The universe is finite.” To verify this, we need to find a boundary somewhere, so we do experiments for a thousand years to look for one, but don’t find one. Does this invalidate the hypothesis? No, because maybe we haven’t looked in the right place yet. So instead, we form the hypothesis, “The universe is infinite.” Again, to verify this, we need to see if there are boundaries to the universe anywhere. We search for a million years and don’t find anything — does this validate the hypothesis? No, for the same reason. And a hypothesis that can’t be validated or invalidated by evidence isn’t scientific, because it’s not subject to the scientific method.

Nick: While there are physical laws in the universe, we don’t know any. We only know models that tend to accurately describe observed phenomena. Since the models shift over time (sometimes dramatically), does this mean we ought to change our views on the size of the universe, along with these changes in model? If so, this was also my point in my original comment: pick whatever makes the math (modeling) work out better.

More than that, you’re neglecting arguments based on physical law. If the physical laws we discover require the universe to be finite or infinite, well, then we know. Even if they don’t, they can still inform our choice of prior. (And, IIRC, the best arguments for the infinity of the universe are based on physical law as opposed to just observing the minimum size of the universe.)

There seems to be something fundamentally flawed in your argument if one is deeming evidence irrelevant. If one person judges the universe to be infinite and the other says “no, the evidence you used to make that determination is irrelevant and I’m not going to take it into account,” then why is it surprising that they have different answers?

“This is not so much a Jaynesian Mind Projection Fallacy, as an argument claiming there are no consistent priors for deciding the question “is the universe infinite?”. ”

There is something wrong with your claim on a deeper level. When we can’t even decide what would consitute proof of an infinite universe (perhaps impossible ?), then it seems especially silly to decide the question from the bayesian point of view. In your example, any evidence that you are taking as evidence for the universe being infinite is also evidence against your assumption of the distribution of the size of the universe in the case that it is finite.

Here the two duplicates have the same evidence – simply the second duplicate judges some of the evidence irrelevant (the extra boxes already opened). The question is whether two situations – (situation A) and (situation B = situation A + one extra box outside situation A known to contain a black ball) – should be treated the differently. In the absence of some extra structure differentiating A and B, I don’t see why this should be.

In fact, I can make the two situations precisely equivalent by adding an infinity of opened boxes containing black balls at the beginning (no longer equivalent with the size of the universe). Now the situations facing the duplicates are precisely the same, so they should come up with the same prior (at different times) and hence disagree (note: this is only true if the probability of the universe being infinite – of there being no white ball – is taken as non-zero).

This is not so much a Jaynesian Mind Projection Fallacy, as an argument claiming there are no consistent priors for deciding the question “is the universe infinite?”.

Stuart, you are making many assumptions that you are not making clear to me, but the comments of this blog post are not the right place for long explanations of your analysis.

Can you give links to relevant websites? I’m not particularly enamoured with the results of my analysis, and would be quite keen to see it shown wrong.
I made many assumptions about how the box model relates to the “size of the universe” model, yes. There are many ways of doing that, but I don’t think that really matters.
Assumptions for the box model itself: two isomorphic situations should be given identical priors, and Initial Situation is isomorphic to Initial Situation + one extra box known to contain a black ball.