[This is the eighth piece in my series on the Parametric Process Assumption (PPA), a new model of anthropics. You should start here if you are just seeing this for the first time.

The previous piece is "Temporal Consistency and Texas Sharpshooting."

TLDR:

• The Fermi Paradox is easily resolved by realizing the universe is gigantic. This means that there are a huge number of possible worlds where life is very very far apart, despite the fact that it also abundant throughout the universe.
• If all you know is that you exist and that the universe is big and random, then the SSA, contrary to popular belief, is sufficient to conclude that life in the universe is abundant. The SIA, on the other hand, adds an additional bias on top.

The next piece is "Putting it all Together."]

Big and Sparse

I confess that I’ve never really understood the Fermi Paradox. Or at least, I never understood why everyone always seemed so surprised by it.

If you haven’t heard, the Fermi Paradox is the apparent contradiction that, due to the vast age of the universe and the (seemingly) high probability of alien life, we should be observing signs of intelligent alien life all around the universe, and yet we don’t. If near-lightspeed travel were possible, for instance, it would presumably only take an intelligent species a few million years to colonize the entire Milky Way. For a galaxy that’s been around more than 13 billion years, the argument goes, you’d think someone would have gotten to that by now, right?

I do personally believe our existence implies other life is likely to exist somewhere, but the problem with the Fermi Paradox is that it’s not just assuming that life is likely to exist, but that life is likely to be dense. Maybe an intelligent species could theoretically populate the entire Milky Way in a few million years, but why do we expect there to be more than one intelligent species in a given galaxy? Why do we expect there to be more than one intelligent species in the closest thousand galaxies?

I’m not the first to make this point, and I won’t be the last, but if we’re going to attempt a coherent theory of anthropics, the point needs to be made here: the universe, it turns out, is really really big. 

The observable universe - just the observable part, which may or may not make up only an infinitesimal percentage - has a radius of 46.5 billion light-years, literally millions of times more than the range at which we can supposedly find life.^[1]

Lower estimates tell us the number of galaxies in the observable universe measures a few hundred billion; let’s say for the sake of argument that the number is 200 billion. This means that the chance of intelligent life existing in any particular galaxy could be 1 in a million, and there would still be 200 thousand different planets out there just teeming with intelligent life, each one hundreds of millions of light-years away from its closest life-bearing neighbor. 

I know this is the “low math” track of this series, but I cannot resist the temptation to drive this point with just a tiny bit of math. Let’s label the physical extent of the universe X, where X is basically any measure of size, such as "number of galaxies" in our last example. Then let’s declare another variable Y, which we’ll call the “life density” of the universe. Y tells us, for one unit of size, how many times we expect life to emerge in that region. So if the universe is 100 universe units, and Y is 1/10, then on average we would expect to see life spontaneously emerge in the universe 10 times - X times Y. This value, X times Y, is what we’ll call the universe’s “life propensity.”

We can guess at lower bounds for X, but we don’t know Y at all. In the example above, X was “number of galaxies”, and I suggested the density Y could be maybe 1/1,000,000. But it could reasonably be orders of magnitude lower than that. If the actual density was 1 in a billion, there would still be, in expectation, 200 intelligent species within the observable universe; the chance of life existing somewhere, anywhere, would still be effectively 100%. 

This point is very hard to overstate: if X is large, the range of possible values for Y is massive. If we selected a value for Y “at random,”^[2] then we could wind up with virtually anything. To get a feel for it, even if we were to assume that literally nothing existed (or even could exist) outside of the observable universe, then the range of possible densities would look something like this gigantic thing:

Near the lower half of this range, even traveling many times the speed of light would not get aliens in range of humanity, even if they somehow knew which direction to go in.

And, I must remind you, this is assuming the observable universe is literally the only thing that exists. If anything exists outside the observable universe, and especially if there’s a multiverse, then the red part of this chart would go on basically forever. If you could go many times the speed of light, and even if you know which direction to go in, you might never reach another civilizaiton.

Just ask the SSA 

Nothing above is really all that new. These ideas are related to things like the Anthropic Principle, and they come up all the time, not just when talking about the Fermi paradox but also in discussions about cosmic fine-tuning. Across all kinds of contexts, the question “why do we exist at all” is very easily answered by “because the universe is big enough and random enough that life has a high chance of emerging at least somewhere.”

Now, I never know how much to claim “most people” believe something, but at least in many cases I’ve seen, the default assumption seems to be that if you want to believe in a big universe, then you really ought to adopt the SIA. But that’s not actually true. In fact, I would (and will) argue that the conclusions I drew above are actually more aligned with the SSA.

I think the disconnect comes from the fact that when people talk about the SSA or SIA, they typically focus on universes where at least one observer always exists. Sleeping Beauty must choose between one version of herself or two, the presumptuous philosopher must select between a trillion humans or a trillion trillion. For possible universes where no one might exist, however, the SSA and SIA basically take the same attitude, which is to say “that universe isn’t possible.” If there was a 20% chance Sleeping Beauty wouldn’t wake up at all, both the SSA and SIA would give that possibility 0% probability. The SSA’s famous smallness bias does not extend to the situations where nothing actually exists.

Let’s go back to playing dice. Suppose your friend Jim rolls some number of ten-sided dice, say N dice. You don’t know N, but after he’s finished rolling, he comes to you and tells you whether any of the dice came up 10. Not which die, not how many dice there were, just “yes, at least one of them was 10,” or “no, none of them were 10s.”

If getting a 10 corresponds to the observation that life exists, then 1/10 is the life density (because of the ten-sided die)^[3], and N is the size of the universe. The propensity is the expected number of times he gets a ten, and that’s N/10. Since we don’t know which roll got the 10, this is equivalent to us knowing we exist, but not our relation to other species which may also exist.

What we want to know is, once we learn that Jim has rolled at least one 10, what credence should we give to each possible value of N? As with all anthropic problems, we’d have to start with a prior over universes (like the 50% chance of heads or tails in the Sleeping Beauty problem) and then see how our anthropic assumptions distort that prior. For the sake of argument, let’s say that N can be, with equal likelihood, any value between 0 and 100:

Now, having learned that Jim has rolled a 10, let’s go through each possible value of N and update our predictions accordingly by removing the percentage of rolls which would not get 10. The first thing we do is eliminate N = 0  - no dice means no chance of rolling 10. N = 1 reduces by exactly 90%, to 1/10 of its original value, since that’s the chance of a single die succeeding. With a little calculation, we find that N = 2 is reduced by 81% of its original value, to just under 20% of where it started. At N = 10, we’re down by about 35% of our original value, and by the time we reach N = 100, while rolling a 10 still isn’t technically guaranteed, our original prior is only adjusted by 0.003%. For all intents and purposes, if you roll a hundred 10-sided dice, you can be pretty damn sure you’ll get at least one 10. Hell, even if you roll only 50 10-sided dice, you can still be pretty sure at least one will come up 10.

However, since this process didn’t actually tell you which die came up 10 (by anaolgy, which observer you are), the SSA makes no further adjustments beyond removing these impossible universes. The SSA therefore transforms the above distribution of universes into something that looks like this:

The result is that universes with low N are more or less eliminated, but high N universes - universes where the size outstrips the density by even a small margin - are left basically unaffected. The flip side of this is that if you believe and/or observe that the density of life is low (e.g. that the dice are many-sided), you must conclude that N is high.^[4]

And remember that this is the SSA, the assumption that presumably forces us to be deeply solipsistic. Even there, low density means the universe must be vast. This happens by default, simply because of the SSA’s distortions to the prior.

What about the SIA? In my opinion, it fares worse. The conventional narrative is that the SIA, by virtue of predicting larger universes, is the one that champions hugeness, and that our observations of the universe’s hugeness therefore confirm its accuracy. But what’s actually happening is that the SSA is the one which supports conventional hugeness arguments, and the SIA just throws in some weird, extra bias on top. Specifically, the SIA multiplies the likelihood of each N by about N, creating a completely unjustified assumption that N must be, a priori, quite high:

Ignore anthropics for a second and focus solely on Jim’s dice rolls: how, exactly, are we supposed to motivate a distribution of this shape in that problem? All we know is that there was at least one roll that came up 10. Past maybe 40 rolls, this event is pretty much guaranteed no matter how far above 40 you go. We must, in that situation at least, favor the SSA.

The next piece is "Putting it all Together"

1. ^{^}

   From Wikipedia, the range is apparently a few tens of thousands of light-years, assuming a radio signal directed straight at us.

2. ^{^}

   Which would obviously require a measure we don’t have.

3. ^{^}

   Technicality: I switched from a rate to probability here. I did this because the example is discrete. In the continuous case we’ll look at later (with equations), we’ll obviously use rates again

4. ^{^}

   Or I guess that God made humanity anyway.

5. ^{^}

   Actually, what happens is our prior becomes impossible to define until we gain enough evidence to condition on a finite universe.

6. ^{^}

   That is not, in my opinion, the weirdest or worst part of the AA for God, by the way. The weirdest part is the implication that the existence of infinity implies God. As I’ve been arguing, “infinity + randomness” seems a lot easier to swallow than “infinity + an intelligent entity that can nonetheless manipulate infinity at will.”

7. ^{^}

   It should technically be “life mass,” since it’s a density times a volume. But for some reason I find that phrase off-putting.

8. ^{^}

   To be a bit more precise,you can always assign arbitrary numbers to the dice and say "dice 3 rolls a 10." But what might be harder to do is come up with a numbering system that someone else would be able to identify if they didn't already have access to the dice. Even then, with dice you could