However, it explains the historical background of the concept of infinity how controversial the concept was in detail. — Corvus

It was controversial when they didn't know better. It's not controversial now because they know better. Those opposed to set theory now are, for the most part, non-mathematicians who don't know better but think they do.

Let mathematicians argue about set theory. Anyone else just isn't equipped to understand the matter.

And the next step is to agree that there is something fishy here. Which is what I am saying. It’s incomplete. — Banno

See the argument here.

If we can dismiss the claim that we are most likely Boltzmann brains a priori then we can dismiss the possibility of heat death a priori, or we can dismiss the possibility of quantum fluctuations a priori, even though we have a posteriori evidence in favour of them.

So sure, something is fishy here, but there appear to be no good solutions.

How do you get from

"given that our scientific theories entail the eventual formation of an exceptionally large number of Boltzmann brains with experiences like ours"

to

"it is exceptionally probable that the Big Freeze has happened" — creativesoul

Because we ought reason as if we are randomly selected from the set of all possible observers with experiences like ours, and the set of all possible observers with experiences like ours contains exceptionally more post-Big Freeze Boltzmann brains than pre-Big Freeze normal observers.

That's not the only possibility entailed by our scientific theories. — creativesoul

It's what cosmologists say is the most likely consequence of our best scientific theories. This is precisely why the Boltzmann brain problem is seen as a problem. It's not just some crazy hypothesis by some crazy philosopher.

Seems to me the difference between ↪Michael and others here is that he is pretty convinced by the Boltzmann discussion, while the others are more comfortable acknowledging that it is interesting but very far form conclusive. — Banno

You seem to misunderstand what I am saying. I don't believe that I am a Boltzmann brain or that I am most likely a Boltzmann brain.

I am simply explaining that, as per the words of the cosmologists who know better than me, our best scientific models entail that we are most likely Boltzmann brains.

Therefore, either we are most likely Boltzmann brains or our best scientific models are mistaken.

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Like Tones' you refuse to stick to mathematics, committing the folly Banno pointed to, a pretense of mathematics. Until you define and demonstrate how the distinction between extensional and intensional is relevant to a discussion of mathematical values, your reference to physical objects is completely irrelevant. — Metaphysician Undercover

It's not maths, as both you and Tones have clearly demonstrated, by needing to refer to physical objects rather than mathematical values to support your claims of "identical". — Metaphysician Undercover

It's an analogy to explain to you the mistake you're making.

a. 1 + 1 is identical to 3 - 1.

Under an intensional reading (a) is false because adding one to one isn't identical to subtracting 1 from 3.

Under an extensional reading (a) is true because the value returned by adding one to one is identical to the value returned by subtracting 1 from 3.

Compare with:

b. The President of the United States is identical to the husband of Jill Biden.

Under an intensional reading (b) is false because being the President of the United States isn't identical to being the husband of Jill Biden.

Under an extensional reading (b) is true because the person who is the President of the United States is identical to the person who is the husband of Jill Biden.

There are four ifs there, so, giving equal likelihood to each, we end up with a 6,25% chance of being in an infinite multiverse in which every possibility is realized. — Lionino

That's not how it works.

We are likely not in one, but if we are, then the same thing that applies to Boltzmann brains applies here: we have no reason to believe in any reasoning we do — which bears no weight on whether it is true or not, but still. — Lionino

With Boltzmann brains there are a finite number of brains, with more of them having incorrect scientific theories than correct scientific theories, and so if we are Boltzmann brains then we are more likely to have incorrect scientific theories than correct scientific theories.

In an infinite multiverse there are not more universes in which we have incorrect scientific theories than correct scientific theories. Given the nature of infinity, there are an equal number of each universe, and so we are equally likely to have incorrect scientific theories as correct scientific theories.

Besides, this line of reasoning does not depend on scientific evidence at all. It is a priori reasoning: is it more rational to believe that an infinite multiverse in which every metaphysical possibility is realized is less probable?

Sorry Michael, I cannot follow you. You've strayed from mathematics, just like Tones did with the example of Twain=Clemens. Your example, like Tones' appears to be completely irrelevant. To me, you've changed the subject and I cannot follow the terms of the change. If you want to continue this course, please demonstrate how it is relevant to mathematics. However, in the meantime I ask that you consider the following — Metaphysician Undercover

Well, I can't explaining the mistake you're making in any simpler terms, so if you don't understand that then I can't help you further.

When we recognize that the value produced by carrying out the procedure on the right side is "equal" to the value produced by carrying out the procedure on the left side, we implicitly acknowledge with the use of "value", that this is something within the mind, dependent on that mental activity of carrying out the procedure. If we use use "identical", instead of "equal" it is implied that what is really a value (something mind dependent) is an object with an identity. This is why Platonism is implied when we replace "equal value" with "identical value". It is implied that the value is an object with an identity. — Metaphysician Undercover

You really read too much into words. There's just no substantial metaphysical implications in saying that the value returned by one operation is identical to the value returned by some other operation. It's just language and just maths. We don't need to believe in the mind-independent existence of abstract entities.

The values returned are the same. What is represent by the right and left sides is not the value itself, but the operation. Therefore the "=" signifies an equality between two operations, it does not signify "the same". — Metaphysician Undercover

You're conflating an extensional and intensional reading. To hopefully make the distinction clear, consider the below:

1. The President of the United States is identical to the husband of Jill Biden.

Under an intensional reading (1) is false because being the President of the United States isn't identical to being the husband of Jill Biden.

Under an extensional reading (1) is true because the person referred to by the term "the President of the United States" is the person referred to by the term "the husband of Jill Biden".

The intensional reading of "1 + 1" is the operation, the extensional reading is the value returned by that operation. Under that extensional reading, 1 + 1 = 3 - 1 where the "=" symbol is used to mean "is identical to".

It is irrelevant to the rest of the post, which demonstrates that "the value" of the right side, and of the left side is only produced by carrying out the procedure to its correct conclusion. — Metaphysician Undercover

Yes, and the values returned by both sides are identical.

No that is clearly not the case, because these two procedures are completely different. They are said to result in the same value, 2, but the operations represented do not have the same value, nor are they identical. — Metaphysician Undercover

Operations don't have a value. Operations return a value. The value returned by the operation of adding 1 to 1 is identical to the value returned by the operation of subtracting 1 from 3.

But moving on from Boltzmann brains, although slightly related, there is perhaps something else to consider.

If there is an infinite multiverse and if every metaphysical possibility is realized in some universe then it would seem to follow that there are an infinite number of universes in which solipsism¹ is correct, an infinite number of universes in which idealism¹ is correct, an infinite number of universes in which common sense external world realism¹ is correct, an infinite number of universes in which monotheism¹ is correct, an infinite number of universes in which polytheism¹ is correct, an infinite number of universes in which atheism¹ is correct, and so on.

¹ _{When considered as making a limited claim only about the nature of the universe in which they are an inhabitant.}

And given the cardinality of infinite sets, the probability that we are in one type of universe rather than some other empirically indistinguishable (to us) universe is equal.

So if it is rational to believe in an infinite multiverse in which every metaphysical possibility is realized, and if your metaphysics is a metaphysical possibility, then it is no less rational to believe in it than in any other.

Which would make all of these arguments rather pointless. You'd be as equally likely to be right as wrong, and either way entirely by accident.

As for whether or not it is rational to believe in such a multiverse:

Is it more rational to believe that this is necessarily the only universe? Is it more rational to believe that there are necessarily a finite number of universes? Is it more rational to believe that even an infinite multiverse necessarily only realizes some subset of all metaphysical possibilities? Is it more rational to believe that an infinite multiverse in which every metaphysical possibility is realized is less probable?

But before that singularity there was no time axis on which a previous event caused the big bang event, was it? — Quk

If time is a dimension of spacetime then it makes no sense to talk about "before" the singularity.

It is simply the case that an initial singularity of infinite temperature and density rapidly expanded.

It expanded presumably because given its inherent nature it is unstable and the probability that it will expand is non-zero (and without time there is no distinction between an instant and eternity).

And although it may be tempting to say that time is required for change, it is perhaps more accurate to say that time is change, assuming that time is in fact a dimension of spacetime.

You think there was spacetime before the Big Bang? — Quk

The Big Bang is the rapid expansion from an initial singularity; that singularity being something like an infinitely compressed spacetime.

Although if time is a dimension of this expanding spacetime then it might not make much sense to talk about before the Big Bang, as time (like length, width, and height) began with the Big Bang.

What's the alternative? An infinite past? That has its own problems. If the past is infinite then as of now an infinite period of time has completed, which seems nonsensical.

So I think that the past must be finite. I'm unsure if that entails that something came from nothing or if something "already" existed when time started – but then how did time start?

Right, A=B means that the value of A is equal to the value of B. This does not mean that A is identical to B, so the "=" signifies a relationship of equality, it does not signify a relationship of identity. — Metaphysician Undercover

The value represented by the symbol "A" is identical to the value represented by the symbol "B".

Two dollar bills are non-identical, but equal value. — Metaphysician Undercover

They are of identical value.

But this creates a procedural problem in practice. Let's take the example "1+1=2". The value represented by "1+1" would be exactly the same, identical, to the value represented by "2". The problem is that "1+1"contains the representation of an operation, and "2" does not. And in order that an operation can fulfill what is intended by the operator, the operation must have a very special type of value. Because it is necessary to recognize this special type of value, that signified by the operator, it is impossible that "1+1" signifies the exact same value as "2", because there is no operation represented by "2". In other words the value represented by "1+1" consists of an operation, and the value represented by "2" does not, therefore they are not representations of the exact same value. — Metaphysician Undercover

Given that 1 + 1 = 3 - 1, the value given by the procedure "add 1 to 1" is identical to the value given by the procedure "subtract 1 from 3" – that value being 2.

It's not the case that there are two equal but non-identical values of 2.

Is there something that you disagree with in Carroll's conclusion? — wonderer1

I'm not denying that there are scientific models that avoid the Boltzmann brain problem. I'm simply explaining that, as per the words of cosmologists like Carroll, the current leading scientific model entails that we are most likely Boltzmann brains.

I'm then questioning his suggestion that we can dismiss the conclusion that we are most likely Boltzmann brains a priori, as that then entails that we can dismiss some empirically well-supported scientific model a priori. For example, as per @RogueAI's comment above, one supposed solution is to dismiss (4) a priori. Is that really rational?

The probability of our being a normal observer is 100%. Here we are.

... The Big Freeze has not happened. — creativesoul

You're begging the question.

The Boltzmann brain problem is that given that our scientific theories entail the eventual formation of an exceptionally large number of Boltzmann brains with experiences like ours, it is exceptionally probable that the Big Freeze has happened and that we are Boltzmann brains having the illusory experience of being normal observers before the Big Freeze.

Science seems to be self-defeating re: Boltzmann Brains: our best theories imply we're probably BB's, but that's "cognitively unstable" (aka "I really don't want to believe that"), so we're probably not BB's and we can't trust our best theories.

Just ditch this idea that minds can come from mindless stuff. It just creates problems. You're not a Boltzmann Brain — RogueAI

So, regarding the argument here, your claim is that we can dismiss (4) a priori?

I get that, but if we are BBs then our scientific theories are incorrect; this is straightforward paradox, it has something in common with the "Liar' sentence.

If our scientific theories are correct, we are most likely to be Boltzmann brains.
If we are Boltzmann brains our scientific theories are incorrect.

Do you not see the problem? — Janus

If we are Boltzmann brains then our scientific theories are almost certainly incorrect.

Yes, I see the problem. But still, as I said:

If our scientific models entail that we are most likely Boltzmann brains, and if our scientific models are correct, then we are most likely Boltzmann brains.

If our scientific models entail that we are most likely Boltzmann brains, and if we are not most likely Boltzmann brains, then our scientific models are incorrect.

How is it in conflict? — Luke

Sorry, it was special relativity, not general relativity:

B-theorists typically emphasize how special relativity eliminates the past/present/future distinction from physical models of space and time. Thus what seems like an awkward way to express facts about time in ordinary English is actually much closer to the way we express facts about time in physics.

For example, see the conventionality of simultaneity.

it presupposes the experience of time that we have. — Luke

Well, yes. I think it self-evident that I experience the passage of time. I want a theory of time that can account for that.

There is no similar study we can do to see how damage/changes to “unmoving space(time)” affects physical objects. — Luke

There's perhaps gravity and any curvature of space(time) in general.

That’s not something I believe, and I doubt it takes into account the facts as we best understand them. — Luke

It's something I believe. I'm unconvinced that physics alone can explain the hard problem of consciousness.

But I am arguing against your view. — Luke

Again, I'm spit balling, not trying to argue for it. It was really just an off-hand remark, not something I intended to lead to a rigorous discussion.

You're just doubling down and are still ignoring the fact that if we are BBs our scientific models are incorrect. — Janus

I'm not ignoring it because I've never disputed it. If we are Boltzmann brains then our scientific models are almost certainly incorrect.

This doesn't refute what I said above:

If our scientific models entail that we are most likely Boltzmann brains, and if our scientific models are correct, then we are most likely Boltzmann brains. This is a straightforward modus ponens.

If our scientific models entail that we are most likely Boltzmann brains, and if we are not most likely Boltzmann brains, then our scientific models are incorrect. This is a straightforward modus tollens.

No, you have it backwards, if we are BBs our scientific models are necessarily incorrect (assuming that it would even be possible for BBs to have scientific models, which is extremely questionable), as I already explained. — Janus

1. Our scientific models tell us that we are most likely Boltzmann brains.
2. If what our scientific models tell us is true then we are most likely Boltzmann brains.
3. If we are not most likely Boltzmann brains then what our scientific models tell us is false.

(2) and (3) seem quite straightforward.

Compare with:

1. John tells us that it is raining.
2. If what John says is true then it is raining.
3. If it is not raining then what John says is false.

But since it is our science that (purportedly) tells us that we are Boltzman brains and that hence all our science is wrong, why would it be rational to believe such a self-eliminating conclusion? It is precisely this problem that you have so far completely failed to address. — Janus

I'm not saying that it is rational to believe that we are Boltzmann brains. I am simply explaining that our best scientific models seem to entail that we are most likely Boltzmann brains. Therefore, either our scientific models are correct and we are most likely Boltzmann brains or we are not most likely Boltzmann brains and our scientific models are incorrect.

I'm then questioning the extent to which it is rational to reject some scientific model a priori when it is supported a posteriori.

You are simply ignoring my argument. — Luke

I'm simply presenting an alternative view, I'm not trying to argue against your view.

The traditional view is a presentist one, where 3D objects move over time. — Luke

Yes, and this is apparently in conflict with general relativity (and time reversibility?). So I'm offering a hypothetical solution that might resolve the conflict between this and our everyday experience of the (one-way) passage of time.

If consciousness is underpinned by physical stuff, then how can it move when none of the physical stuff does? — Luke

I don't know. Much like I don't know how, according to physical presentism, physical objects can move through the unmoving space(time) that underpins them. Again, I'm spit balling. I don't have some consistent and complete mathematical model at hand.

Why should you be able to treat consciousness as a presentist object in an otherwise universe? — Luke

Because, as above, it may resolve the conflict between general relativity and our everyday experience of the passage of time. If dualism is correct then the physical and the mental need not necessarily behave according to the same laws.

Well, I would say that the traditional view is that physical objects supervene on and move through static space(time). I’m just pushing this up a level and suggesting that consciousness supervenes on and moves through the time dimension of static 4D physical objects.

But again, I’m just spit balling.

The idea is that consciousness is something that supervenes on the physical, much like any traditional dualism, and so is not separable from it. The 4D object doesn’t move as you mean it but this 0D consciousness travels through the time dimension creating the illusion of physical movement.

So time flows for consciousness but not for the physical host. At the very least it could tentatively explain the current conflict between general relativity seeming to imply physical eternalism and our everyday experience seeming to imply presentism.

Of course this would seem to entail determinism, unless something like the many worlds hypothesis is true and consciousness can choose which “world branch” of the 4D host it travels down.

The consciousness of what? If you mean the consciousness of the 4D host, then that is extended across time and doesn’t move. If you mean the consciousness of a 3D part of that 4D host, then all the 3D parts are different and none of them moves. — Luke

My random idea is that the physical host is something like a tunnel and consciousness the occupant. The tunnel is fixed in time and space with consciousness travelling through it.

As you follow Sean Carroll, see here:

In brief, the BB problem arises if our universe (1) lasts forever (or at least an extraordinarily long time, much longer than 10¹⁰⁶⁶ years), and (2) undergoes random fluctuations that could potentially create conscious observers. If the rate of fluctuations times the lifetime of the universe is sufficiently large, we would expect a “typical” observer to be such a fluctuation, rather than one of the ordinary observers (OOs) that arise through traditional thermodynamic evolution in the wake of a low-entropy Big Bang. We humans here on Earth have a strong belief that we are OOs, not BBs, so there is apparently something fishy about a cosmological model that predicts that almost all observers are BBs.

This mildly diverting observation becomes more pressing if we notice that the current best-fit model for cosmology – denoted ΛCDM, where Λ stands for the cosmological constant (vacuum energy) and CDM for “cold dark matter” – is arguably a theory that satisfies both conditions (1) and (2).

...

It is therefore reasonable to worry that BBs will be produced in the eventual future, and dominate the number of intelligent observers in the universe. Note that this conclusion doesn’t involve speculative ideas such as eternal inflation, the cosmological multiverse, or the string theory landscape – it refers to ordinary ΛCDM, the best-fit model constructed by cosmologists to describe the universe we live in today.

...

I will argue that cosmologies dominated by BBs should be rejected, not because I have empirical evidence that I am not one and I should be, but because such models are cognitively unstable.

...

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.

So, it seems to be exactly what I said above. The best evidence supports (1)-(4), and (7) follows. And his argument is that because (7) is just silly, we must reject (1), (2), (3), and/or (4) despite the evidence in their favour. It's a conceded dogma.

I'm asking if that's rational.

If it's rational to reject (7) a priori then it's rational to reject (1), (2), (3), and/or (4) a priori. If it's not rational to reject (1), (2), (3), and/or (4) a priori then it's not rational to reject (7) a priori.

We have a choice to make.

Just spit balling but how about:

Physical objects are 4D objects extended in space and time as per eternalism.
Consciousness is a non-physical 0D "object" bound to some physical object.

Time doesn't flow but consciousness travels through (its physical host's) time.

Notice "two things". Equality deals with two things, identity only involves one thing. — Metaphysician Undercover

In the context of maths, when we say that A = B we are saying that the value of A is equal to the value of B. The value of A is equal to the value of B if and only if A and B have the same value.

A non-identical but equal value makes no sense.

We’re not saying that the symbol “A” is identical to the symbol “B”. This is where I think you are misunderstanding.

I think that is what bothered Wittgenstein about mathematics — Metaphysician Undercover

Was that early or also late Wittgenstein? Because I suspect late Wittgenstein wouldn't have read any metaphysics into set theory. It's just a useful language game we play, not something that entails the realist existence of abstract mathematical objects.

Regarding the "=" sign, it was invented in 1557 by Robert Recorde:

And to avoid the tedious repetition of these words: "is equal to" I will set as I do often in work use, a pair of parallels, or duplicate lines of one [the same] length, thus: =, because no 2 things can be more equal.

People can produce whatever axioms they like, but if they are not useful they will not be used, nor become conventional. — Metaphysician Undercover

Yes, that's precisely right, and is why your talk of axioms being "false" is nonsense. Axioms aren't truth-apt; they're just either useful for their purpose or not. And given that the axioms of ZFC are the most prominently used, it stands to reason that they are considered to be the most useful.

That's all there is to say about them.

Nah, it's a matter of my understanding of the strength of the evidence. — wonderer1

So you're a cosmologist who understands the sigma level of each of (1), (2), (3), and (4)? I wasn't aware.

No. I am claiming 1-4 are insufficiently justified given the present state of scientific knowledge and my ability to distinguish well evidenced science from highly speculative science. — wonderer1

You're not just saying that.

If (1)-(4) are true then (7) is true. You're saying that (7) is false. Therefore you're saying that (1), (2), (3), and/or (4) is false.

You're dismissing some outcome on purely theoretical grounds irrespective of the strength of its evidence. I’m asking if that's rational.

And yet there's the argument here.

You're claiming that the "absurdity" of (7) is sufficient justification to reject the evidence that suggests that (1), (2), (3), and/or (4) is true. Is that really rational?

It must be that the universe won't succumb to the Big Freeze, because if it will then we are most likely Boltzmann brains!

It must be that the time between the Big Bang and the Big Freeze is infinite, because if it isn't then we are most likely Boltzmann brains!

It must be that the time after the Big Freeze is finite (and sufficiently small), because if it isn't then we are most likely Boltzmann brains!

It must be that the probability of a Boltzmann brain with experiences like ours forming via quantum fluctuation or nucleation within a finite time is zero, because if it isn't then we are most likely Boltzmann brains!

You're welcome to do it. But then you leave room for sceptics, anti-realists, idealists, and solipsists to dogmatically reject whatever scientific evidence supports common-sense non-sceptical external world realism. You've set the precedent.

Yes, and as I've shown over and over again, that definition of "=" is not representative of how "=" is actually used in mathematics. Therefore it is a false definition, designed for some other purpose, foreign to mathematics. — Metaphysician Undercover

You're putting the cart before the horse. It's not that we use maths and then retroactively describe what the symbols mean and infer the axioms; it's that we define what the symbols mean, prescribe the axioms, and then use them.

Those are the people who say "=" signifies identity in mathematics. They claim to be doing mathematics when they say that "1=1" means that what left 1 signifies is the same as what the right 1 signifies. But that's obviously not mathematics. In mathematics, the left side of the equation always signifies something different from the right side, or else the equation would be useless.

It's one thing for non-mathematicians, who don't know any better, to think that what they are doing is mathematics, when it's not. But it's truly shameful when mathematicians claim to be doing mathematics when what they are doing is not mathematical. As I explained already, that's how they come up with false axioms. — Metaphysician Undercover

The symbol "=" is defined in ZFC by saying that "A = B" is true if and only if A is B.

They could have used the symbol "#" instead, but they decided on "=".

Whether (1) is true is unknown. As far as I know, the universe as we know it might end with a false vacuum decay tomorrow. — wonderer1

Yes, there are 4 major predictions: Big Freeze, Big Rip, Big Crunch, and vacuum instability.

With Big Freeze being considered to have the most evidential support.

Is the "absurdity" of (7) sufficient justification to reject the evidence that suggests that the Big Freeze is most likely?