Seems like you can make the argument that with an infinity of objects each surrounded by an infinite amount of separation (as why not?), the resulting density would be infinitesimal.

Suppose a B-brain occupies one cubic foot of space. Then 10 of them occupy 10 cubic feet, 100 of them occupy 100 cubic feet, etc. Density is just mass/volume. So the density for any volume of space is just one.

How would you calculate density for a infinite number of things (e.g., Boltzmann brains) in an infinitely large space?

Discussion points:

1: it would be impossible. If the space is infinite then, no matter how much space you surveyed, there would always be more space beyond it, and you would have no reason to suppose its density must be consistent with the space already surveyed.

2: How are you defining "density"? Do you mean the average number of Boltzmann brains within a given cubic area? But how could anyone possibly know the answer to that question?

3: What are the physical characteristics of a Boltzmann brain? Obviously, it won't resemble anything that we routinely understand by the term "brain". Would it resemble something like a Hoyle Black Cloud?

4: Suppose there were only one Boltzmann brain, and supposing further that its physical dimensions could be quantified, then the average density in an infinite space would be 1/∞, or pretty much 0.

It occurs to me, someone might be tempted to object to discussion point 1 because, although we posit the space to be infinitely large, we also posit the number of B Brains to be infinitely large. But, since a B brain would not be subject to the conditions which normally restrict the development of a mammal brain, it is conceivable that it would occupy an infinitely small space.

Here's a way to conceptualize the density of things in an infinite volume, granted that the things in-question are evenly distributed:

$\displaystyle D = \lim_{V \rightarrow \infty} \frac NV$

Where V is the volume you are looking at, and N is the number of the things in-question within that volume. As this limit grows, it will (over large enough growth) approach D, since the even distribution entails that you will likely increase the accuracy of your density measurement with each sample expansion; as you continue doing so indefinitely, you will eventually increase the accuracy.

2: How are you defining "density"? Do you mean the average number of Boltzmann brains within a given cubic area? — alan1000

yes

But how could anyone possibly know the answer to that question?

Well, when we explore space, we don't see any Boltzmann brains, which suggests the density is very low (or we occupy a special place).

Well, when we explore space, we don't see any Boltzmann brains — RogueAI

Cosmology at its best. A PhD thesis could be written on this observation. :roll:

Cosmology at its best. A PhD thesis could be written on this observation. — jgill

The lack of observation of any Boltzmann Brains anywhere (they're not limited to space) suggests the density of Boltzmann Brains is rather low. If it were high, we would have seen some by now, right?

Well, when we explore space, we don't see any Boltzmann brains, which suggests the density is very low (or we occupy a special place). — RogueAI

But your OP postulated there were an infinite number of things, but here you reference our world which doesn't have an infinite number of things, so this empirical evaluation doesn't help us.

Making inferences from such experiences is only works if:
A. You are not yourself a Boltzmann Brain
B. We are not inhabitants of a Boltzmann Universe

If Boltzmann Brains are vastly more likely than any other sort of brain then the fact that we have experiences is evidence for Boltzmann Brains being common, since such experiences are likely to be caused by Boltzmann Brains.

Me, I like to assume I'm at least in a Boltzmann Room, and the when I open the door I will fall out into a nebula.

Making inferences from such experiences is only works if:
A. You are not yourself a Boltzmann Brain
B. We are not inhabitants of a Boltzmann Universe — Count Timothy von Icarus

(A) yes
(B) what is a "Boltzmann Universe"?

A universe the same size as our visible universe that just happened to form from random fluctuations. The Boltzmann Brain was originally a criticism of the Boltzmann Universe, which was a popular way to explain low entropy conditions (paired with the Anthropic Principle). This view was popular before the Big Bang Theory and evidence that supported the Big Bang emerged. Theorists had tried to avoid a universe with a beginning because of intractable problems with the Cosmological Argument, but the eternal universe brings up the BB problem.

IMO the BB problem is less of an issue that the problems posed by a universe with a starting point though.

Well, when we explore space, we don't see any Boltzmann brains — RogueAI


Cosmology at its best. A PhD thesis could be written on this observation. :roll: — jgill

A hit! A palpable hit! (Hamlet)

Are Boltzman brains more likely than Boltzman anthing else? Automobiles, trees, pencils, boulders that look like octopi, buckets filled with water, buckets filled with oil, buckets filled with gravel, 55 gallon drums filled with those and other options, Empire State Buildings, bodies that look like Jean Luc Picard, bodies that look like Kirk, octopi, septapi etc., etc., etc., etc… ALL things are as likely as Boltzman brains, aren’t they? And an infinite number of all things.

How do we calculate the density of any of it?

And how do we calculate the possibility of the necessary number of particles being in the necessary positions at the same time for any of them to be formed?

In Incomplete Nature, Thomas Deacon writes

This reliably asymmetric habit of nature provides the ultimate background with respect to which an attribute of one thing can exemplify an attribute of something else. The reason is simple: since non-correlation and disorder are so highly likely, any degree of orderliness of things typically means that some external intervention has perturbed them away from this most probable state. In other words, this spontaneous relentless tendency toward messiness provides the ultimate slate for recording outside interference. If things are not in their most probable state, then something external must have done work to divert them from that state.