Is there some sort of a probability problem with a humongous but finite universe?

It’s easy to generate probabilistic paradoxes in a universe (or multiverse) with infinitely many people (e.g., if infinitely many people roll a die, equal numbers of people get 1 as get more than 1, so why think it’s more likely to get more than 1?). But what about a very large but finite universe? I used to think: “The only relevant difference is between finite and infinite. Really big but finite—no problem.” Now I am not so sure.

Paul Heyl measured the gravitational constant G as 6.670 × 10⁻¹¹ m³ kg⁻¹ s⁻², and denote the latter quantity by G₀. Consider two theories:

• H₁: The gravitational constant is between 6.665 × 10⁻¹¹ m³ kg⁻¹ s⁻² and 6.675 × 10⁻¹¹ m³ kg⁻¹ s⁻².

• H₂: The gravitational constant is between 7.676 × 10⁻¹¹ m³ kg⁻¹ s⁻² and 7.686 × 10⁻¹¹ m³ kg⁻¹ s⁻².

It seems obvious that:

1. Heyl’s measurement strongly supports H₁ but does not completely rule out H₂.

But let’s think this through. Suppose Heyl’s evidence is the proposition E which he would express as “I measured G to be G₀.” But, very plausibly, it is an essential property of a human being that they exist in a world with such-and-such a gravitational constant. One way of getting to this conclusion is to say that the forces of gravity are part of our causal history, and then to apply the essentiality of origins. Another is to say that we couldn’t have been made of completely different matter, but the forces exerted by the matter in our bodies are an essential property of that matter.

Given this essentiality of gravitational constant assumption, it follows that at least one of H₁ and H₂ is incompatible with Heyl’s existence. Now, to get (1), we need prior probabilities on which P(H₁|E) > P(H₂|E) > 0. Such prior probabilities will assign a non-zero value to H₁E and to H₂E. But at least one of these two claims is impossible since E entails Heyl’s existence, and a probability assignment that assigns a non-zero value to something impossible is screwed up, and we should be quite suspicious of what we get from it.

We might try to avoid this by using self-locating evidence. But my colleague Yoaav Isaacs has this great paper that gives a pretty strong argument that there isn’t a good way to working with self-locating evidence. So suppose we put this option aside.

Or we might make a distinction between logical impossibility and metaphysical impossibility. I find that suspicious, too.

So, what’s left? Well, here’s one remaining suggestion. Heyl’s evidence is equivalent to the proposition that Heyl measured G to be G₀, a proposition that rigidly refers to Heyl, and hence won’t be compatible with both H₁ and H₂. But we can weaken Heyl’s evidence to something that is compatible with H₁ and H₂, something purely qualitative, like:

• E_Q: A physicist named “Paul Heyl”, who married someone named “Lucy Daugherty”, and who …, measured G to be G₀.

Here, “…” is all the other purely qualitative stuff we know about Paul Heyl, so that E_Q is compatible with both H₁ and H₂.

But now here is a problem. Suppose we live in a vast but finite universe with, say, 10¹⁰¹⁰ people. In such a universe, we might well expect large numbers of people named “Paul Heyl” who satisfy all the conditions in E_Q, including the measurement of G to be G₀, even if in fact G is in the range indicated in H₁ (measurement error!). Thus, P(E_Q|H₂) is close to 1 as is P(E_Q|H₁). Granted, we do have P(E_Q|H₁) > P(E_Q|H₂) > 0. But because the two probabilities are so close to each other, the support E_Q gives to H₁ over H₂ is very slight, and hence we no longer have (1).

It follows that unless we can find some other way of solving the problem that the essentiality of the laws of nature to humans poses for Bayesian reasoning, a fair amount of fundamental physics research would be undercut by a large enough—even if finite—universe.

Of course, maybe we can find some other way of solving it. But maybe we can’t. And if we can’t, then the E_Q solution might be our best bet—and it’ll work just fine in a universe that isn’t too vast.

The smallness of the universe

[I am now thinking that the following line of argument is probably deeply spiritually mistaken. Or, maybe, the issue is this: It is the universe conceived of naturalistically that is small. But the universe that we in fact inhabit lacks that kind of smallness for it images the glory of God. So perhaps there is still an ad hominem against naturalism in the line of thought.]

Suppose I and a few other people were born and lived on the inner surface of a sphere that was a kilometer in diameter. Walk 3.14 km, and you're back where you started. And that was all. As far as I could tell, things had always been in the sphere more or less the same as far back as one could tell, with a small stable population and a simple, self-contained ecosystem. There is nothing particularly mysterious in this ecosystem, and there even is a little library containing books written by my ancestors which give complete systmes of physics, chemistry and biology that fit with all the data.

I think there is an intuition one might have after one had surveyed this meager habitat: there has to be more to reality than this. This intuition could be bolstered by arguments from design or causation, but I think it could also be a self-standing intuition: this can't be all there is.

Now our universe is bigger than that sphere. But I do not think that sheer size is what makes a difference. When I reflect on my stargazing, instead of thinking of the vastness and mystery of the physical world, I have lately been thinking about a certain kind of smallness that it has. Sure, there are untold numbers of stars arranged in untold numbers of galaxies. But even that one kilometer sphere had untold numbers of grains of dust, and that didn't make it all that large. The universe is physically large relative to us, but it exhibits a lawlike unity, and while there are many stars, they can mostly be classified into a limited number of types and subtypes. And this vast universe is, nonetheless, one that is small enough that we can have all-encompassing cosmological theories—that we can hold it all in our mind. It is true that our present theories are not good enough. But we seem to be making progress.

If this is right, then I think it is possible to have a similar kind of smallness intuition about the universe as a whole: this can't be all there is to reality. Reflecting on the universe as a whole—a whole made up of physical parts, indeed largely made up of parts like electrons, neutrinos and/or electromagnetic fields that are pretty familiar to us—can make the universe seem small to the mind's eye. Not small physically, of course. The relevant kind of conceptual smallness is compatible with the universe having infinite spatial extent. Rather, it is small in some deeper sense, despite all its wonder and glory.

I think this may be a way in which the physical universe proclaims to us that it is not all there is, that it is not self-sufficient, that while great and full of splendor, it is, after all, but an image. And unless it is seen expressly as an image, it will pale to us. For a while one might enjoy looking at the calligraphy in a manuscript. But eventually, unless one starts reading what the text says, the manuscript will probably stop being all that interesting. But once seen as pointing to something greater, indeed something infinite (and not just in the uninteresting spatial sense), then it takes on a new, and deeper splendor, one that shines through that natural splendor that was in danger of fading.