Does Selflessness Save Atheism?

Boltzmann brains don't have to cause problems for mathematical universes

Just a song before I go
To whom it may concern
Traveling twice the speed of sound
It's easy to get burned

When the shows were over
We had to get back home
And when we opened up the door
I had to be alone

— Crosby, Stills, & Nash¹

Yesterday, I wrote about, among other things, Tegmark’s Mathematical Universe Hypothesis (MUH).

There are some problems with it that people like

Bentham's Bulldog

have pointed out. I’m gonna present a possible response to ~1.5 of those problems. This is all pretty shaky—I’m not too comfortable with cosmological, anthropic, or infinite philosophical reasoning.

But here goes!

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1.

MUH predicts a really huge universe, filled with lots of people.

This should be considered a point in its favor—according to the self-indication assumption, theories which predict infinite people existing are infinitely likelier than theories which don’t.

Not only that, but theories which predict really large infinities are infinitely likelier than theories that only predict smaller infinities.²

Luckily, MUH predicts a really huge infinity of people existing. Like, more than the number of numbers huge—not more than the number of natural numbers—more than the number of reals!

Huge universes like this have a problem, though. They predict the existence of lots of people, sure, but they also predict the existence of many many more Boltzmann brains.

The Laws of Thermodynamics have many weird implications, but this is undoubtedly the weirdest one. It goes like this:

• The position of particles is probabilistically determined.

• There are lots of particles all around.

• Therefore, there’s an extremely, microscopically, nearly-infinitesimally small chance of a bunch of particles spontaneously assuming the form of some sort of brain.

• If that happened, a consciousness would fizz into existence, potentially with (false) memories of having existed in the universe for a while, and then fizz right out as the particles jiggle into some other random configuration.

• Since the universe is so inconceivably massive, even things with extremely microscopic and nearly-infinitesimally small chances of happening are happening all the time!

• So lots of Boltzmann brains are fizzing into and out of existence all the time—brains which think they’re just as real as any of us, with our evolved-brains.

Oh no!

It seems like big universes would be full of Boltzmann brains—so full, in fact, that the odds that you’re not a Boltzmann brain are vanishingly small.³

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2.

Bentham thinks this is pretty compelling evidence for theism.

God wouldn’t want a bunch of consciousnesses fizzing in and out of existence at random—especially if those consciousnesses had false memories. That would be deception, which is bad, so God wouldn’t be doing it on an infinite scale.

Let’s… accept this for now.

On MUH, we don’t have a God to deal with the issue of Boltzmann brains. They really are fizzing in and out of existence constantly. So do we have to become radical skeptics? Do we need to assume that our memories are deceptions, that we’ll fizz out in the next instant?

No! I think.

Consciousness is a silly thing, but the even sillier concept is that of a “self.” I have a little write-up where I try to sow doubt over whether a distinct “self” really exists coherently:

More likely, we’re all just instantaneous consciousnesses with memories. And we care about other consciousnesses that are closely connected to ours by causality. In fact, we care about them so much that we give them the names “past me” and “future me,” give their wellbeing extra special consideration, and try to coordinate our actions and goals with theirs.

But the important thing is that we’re all just instantaneous consciousnesses with memories. That is to say: from the inside, there’s no difference between being an evolved brain or a Boltzmann one.⁴

Now, there are a lot of Boltzmann brains out there. A very large infinity.

But for every Boltzmann brain B that has the memory M, there is an evolved brain E that actually has done the things that make up M! (I’ll justify this in the next section, but for now, accept it.)

Of course, to be E would then feel exactly the same as being B. Really, there’s no difference between being E and being B. Whoever is E is also B, and vice versa. So when B fizzes out of existence, what happens to the associated consciousness? Seemingly, it just snaps onto E and keeps on living.

On MUH, this is happening constantly. Your consciousness expands to include any identical Boltzmann brains as they fizz into existence. And then when they fizz out, it snaps back to (what you consider) you. The evolved brain on Earth that’s reading this blog and loving it.

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3.

How can I say that there’s a matching evolved brain for every Boltzmann one, though? Didn’t we say that Boltzmann brains are more numerous? How can the match-up be one-to-one?

We’re gonna need some math to answer this.

Specifically, we need set theory, and extra-specifically, we need infinite set theory.⁵

A set is just a collection of objects—for instance, {0, 4, -100} or {potato, banana, my left pinky toe} or ℝ, the set of all real numbers (things like 2, π, 0.2143538796764…, etc.).

Sets have cardinalities—a count of their elements. This is denoted with vertical bars: ||. So, for example, if I have S = {my blog, Bentham’s blog, a three-legged cat named Stan}, then |S| = 3, or “the cardinality of S is three.”

Infinites sets have cardinalities too!

The natural numbers, denoted by ℕ, are all the numbers you would count with (they’re also sometimes called the counting numbers): {1, 2, 3, …}. Instead of the infinity symbol you might be used to—∞—we write the cardinality of ℕ with the Hebrew letter beth⁶ and a subscript zero:

\(|\mathbb{N}| = \beth_0\)

Interestingly, the set of all integers, which includes negative natural numbers and 0, has the same cardinality as the naturals. Even though it seems like it should be twice as large, we can draw a one-to-one mapping between the two.

Whenever sets share a cardinality, it means we can draw a one-to-one mapping between them, and vice versa.

The set of real numbers is a bit larger—its cardinality is Beth-1.

The Beth numbers have a nice relationship between one another, and to understand it we need one more set theory principle: the powerset.

A powerset of a set S is defined as the set of all subsets of S (including the empty set, ∅). For example, if we have S = {1, 2}, then P(S) = {∅, {1}, {2}, {1,2}}. The cardinality of the powerset is always 2 raised to the power of the cardinality of the set itself, i.e.: |P(S)| = 2^|S|.⁷

The Beth numbers are defined iteratively using the powerset:

\( \quad \mathcal{P}(\beth_\alpha) = \beth_{\alpha+1} \implies \beth_{\alpha+1} = 2^{\beth_\alpha} \)

So we actually know how many real numbers there are, in terms of the number of naturals!

Let’s get back to Boltzmann brains.

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4.

I claim there’s a one-to-one mapping between the set of Boltzmann brains, B, and the set of evolved brains, E.

The claim can be proven if I show that B and E have the same cardinality—that |B| = |E|.

Let’s assume the cardinality of |E| = Beth-(n), where n is at least 2.

That means there are more people than real numbers—what SIA predicts, and what the MUH provides for.

How can we go about finding the size of B?

Well, an important fact about the Beth numbers is that there’s nothing between them. There cannot possibly be a set S with Beth-(a) < |S| < Beth-(a+1).

So the cardinality of B has to be at least Beth-(n+1), or else it’s bound to be the same cardinality as E. (Recall, there are seemingly more Boltzmann brains than evolved ones, so we already have |B| ≥ |E|.)

For |B| to reach a higher Beth number than |E|, it’ll have to be at least a powerset of |E|.

On my so-called “selflessness” theory, we can rethink the set of “evolved brains” as the set of “evolved memories that correspond to a given consciousness.” And similarly for the set of Boltzmann brains.

Then E is some set of possible memories in the universe, and we need to see if B consists of all of the subsets of those memories.

Here are two possible memories that certainly exist inside E: C = “I ate a peach cobbler at 14:40 UTC on June 6, 2009” and K = “I ate a key lime pie at 14:40 UTC on June 6, 2009.”

C and K are contradictory.

Thus, a coherent memory for any given brain cannot possibly contain both C and K. So no Boltzmann brain could contain both of these memories.

Then B cannot include any subset of E which contains both C and K.

Therefore, |B| is strictly smaller than P(S), so |B| = |E|.

And we’ve proven the claim! A one-to-one mapping must exist between Boltzmann and evolved brains.

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5.

This post is long and dense enough already. I think this line of reasoning has promise in dealing with some other problems theists have with MUH.

For instance, Bentham thinks that MUH undermines induction. Now, not only do I think that would mean his anthropic argument for God undermines it too, but I think it’s possible that reasoning along lines similar to above, induction can survive.

I’ll save it for tomorrow, though.

For now, just rest easy with the confidence that your consciousness isn’t about to fizz out of existence.

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Lukethoughts

(Everyone’s hopefully-not-favorite part of the blog is back! And Lucas very nicely sent me his thoughts earlier today, so I didn’t even have to worry about being more responsible with my planning.)

• “When you are feeling off or tired or whatever, how can you distinguish if you are not yourself or having an off day versus just having an identity disorder. It seems to be way too black and white as an off day doesn’t need to be caused by something. Idk if that makes sense but I plan to expand on this —> stay -posted for subsequent thoughts on this.” (Ed. note: Ehhh… dude, what? I think Luke is asking some interesting questions that touch on psychopathology and identity, but he’ll need to be more specific and lucid before anything useful comes of it.)

• “Fuck [A]ri for texting me at 1AM and then being surprised that I’m not awake.” (Ed. note: Yeah, my bad. Yesterday’s failure’s on me.)

• “Practice does not guarantee improvement which is kind of fucked up.” (Ed. note: Nothing guarantees anything! Practice is more likely to lead to improvement than other actions, so you should still do it, and not get mad that the universe literally cannot hold any 100% probabilities.)

1. I’m stealing the starting-with-a-song thing from Bentham, but only cause it’s super cool.

2. If the idea of differently-sized infinities is new to you, just let it sit for now. We’ll get to an explanation of all that soon enough.

3. Why is this, exactly? As improbable as the creation of a Boltzmann brain is, the creation of an evolved brain is even more improbable! A Boltzmann brain just needs a brief random configuration, but an evolved brain needs lots of parts of a very complicated chain reaction to go right. It needs life to appear, and to evolve in the right way, and so on… Boltzmann brains are comparatively very easy to make.

4. Of course, we already knew this—it’s why Boltzmann brains are a problem, because they feel like real brains. But work backwards through my argument! If you think Boltzmann brains are a problem, I think you also have to accept that we can be well-modeled as simple instantaneous consciousnesses.

5. If you think you’ve got a good base of knowledge on these topics, skip to section 4.

6. This is pronounced much more like “bet” than like “beth,” but mathematicians fucking suck, so…

7. Proof is left as an exercise for the reader.

(Probably unhelpful hint: this principle is isomorphic to the sums of rows of Pascal’s triangle being powers of two.)