Can’t we assume that the Boltzmann Brain scenario can be cognitively stable?

Question

In the Boltzmann Brain scenario, we are all brains produced by random fluctuations within a high entropy universe. The argument which I had accepted before was that our very reasoning can not be trusted in this scenario because all of our senses, experiments, etc. concerning natural laws of science were all fake memories, and therefore, weren’t actually true.

Yet a Boltzmann Brain scenario entails an infinite universe, meaning an infinite number of brains. And therefore, here would also be an infinite number of subsets of these brains. Couldn’t we argue that therefore, there is an infinite number (or at least an extremely large number) of brains which actually do have correct ideas and memories about natural laws - therefore the cognitive instability of our reasoning could be trusted? And as an infinite or huge number of such brains, it is still more likely to be one of these brains than to not be a Boltzmann Brain. Then what argument could refute the BB scenario?

Answer 1

Are Boltzmann brain scenarios cognitively stable?

Assuming that there are an exceptionally large but finite number of Boltzmann brains, each of these is true:

• Any randomly selected observer is most likely to be a Boltzmann brain
• Any randomly selected Boltzmann brain is most likely to have incorrect scientific theories
• Any randomly selected observer with correct scientific theories is most likely to be a Boltzmann brain

Although given that we cannot assume that our scientific theories are correct, the third point may not be of much use to us.

But, as per the cardinality of countable infinite sets, if there are an infinite number of Boltzmann brains then any randomly selected Boltzmann brain is equally likely to have correct scientific theories as incorrect scientific theories. This may be sufficient to be considered cognitively stable.

So prima facie we may be able to say that the Boltzmann brain scenario is cognitively stable if and only if there are an infinite number of Boltzmann brains.

Can we reject cognitively unstable Boltzmann brain scenarios?

Given that you are describing Boltzmann brain scenarios as "cognitively stable", I assume you are familiar with Sean Carroll's Why Boltzmann Brains Are Bad. In it he concludes:

The best we can do is to decline to entertain the possibility that the universe is described by a cognitively unstable theory, by setting our prior for such a possibility to zero (or at least very close to it). That is what priors are all about: setting credences for models on the basis of how simple and reasonable they seem to be before we have collected any relevant data. It seems unreasonable to grant substantial credence to the prospect that we have no right to be granting substantial credence to anything. If we discover that a certain otherwise innocuous cosmological model doesn’t allow us to have a reasonable degree of confidence in science and the empirical method, it makes sense to reject that model, if only on pragmatic grounds. This includes theories in which the universe is dominated by Boltzmann Brains and other random fluctuations. It’s not that we’ve gathered evidence against such theories by noticing that we are not BBs; it’s that we should discard such theories from consideration even before we’ve looked.

To consider this let's examine a simplified argument in favour of the Boltzmann brain scenario as best as I understand the physics (and I'm not a physicist so take the specifics with a grain of salt):

1. The universe will succumb to the Big Freeze
2. The time between the Big Bang and the Big Freeze is finite
3. The time after the Big Freeze is exceptionally¹ (perhaps infinitely) larger than the time between the Big Bang and the Big Freeze
4. The probability of a Boltzmann brain with experiences like ours forming via quantum fluctuation or nucleation within a finite time is non-zero
5. Given (1) and (2) the number of normal observers is finite
6. Given (3) and (4) the number of Boltzmann brains with experiences like ours is exceptionally (perhaps infinitely) larger than the number of normal observers
7. Given (5) and (6) we are exceptionally (perhaps infinitely) more likely to be a Boltzmann brain than a normal observer

¹ _{by "exceptionally large" I mean orders of magnitude greater than the time it would take for a Boltzmann brain to form}.

(1)-(4) are supported by our best scientific models, and (5)-(7) are rational deductions.

If (7) is false then at least one of (1)-(4) is false.

If we accept Carroll's claim that Boltzmann brain scenarios are cognitively unstable, and if we accept that we are therefore rationally justified to reject them a priori, then we are rationally justified to reject (7) a priori.

However, this then entails that we are rationally justified to reject (1), (2), (3), and/or (4) a priori. There's certainly something spurious about this.

The inverse argument then is that we are not rationally justified to reject (1), (2), (3), and/or (4) a priori, and so not rationally justified to reject (7) a priori. Therefore either we are not rationally justified to reject a cognitively unstable theory a priori or a (finite) Boltzmann brain scenario is not cognitively unstable.

Addendum

a. Even in infinite Boltzmann brain scenarios it may be rational to assert that the time between the Big Bang and now must be finite, and so that we are almost certainly Boltzmann brains with incorrect scientific theories – infinite Boltzmann brain scenarios therefore drop out of consideration.

b. It is perhaps reasonable to assert that (1) entails (2) above and so the question is whether or not we can rationally reject (1), (3), and/or (4) a priori.

c. We may be dealing with something of a lottery paradox; although we cannot rationally reject (1) a priori, cannot rationally reject (3) a priori, and cannot rationally reject (4) a priori, we may be able to rationally reject the conjunction of (1), (3), and (4) a priori.