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?

It's hasn't been clear to me that when you say, "or our science is incorrect", that you recognize the relativity of incorrectness. — wonderer1

Then I'll make it clear: I'm not saying that therefore all science is completely wrong and that all the facts may be utterly different than what we believe them to be.

The argument here provides a more precise account: either (7) is true or at least one of (1)-(4) is false.

Yes. How is that relevant?

So, an argument from authority then? — Janus

Yes. I defer to what physicists say about what the scientific evidence entails, as is proper.

Even worse, it seems that they are not really saying what you seem to want them to be saying. — Janus

I don’t want them to be saying anything. I’m simply reporting on what they're saying.

See, for example, Big Brain Theory: Have Cosmologists Lost Theirs?

It could be the weirdest and most embarrassing prediction in the history of cosmology, if not science.

If true, it would mean that you yourself reading this article are more likely to be some momentary fluctuation in a field of matter and energy out in space than a person with a real past born through billions of years of evolution in an orderly star-spangled cosmos. Your memories and the world you think you see around you are illusions.

This bizarre picture is the outcome of a recent series of calculations that take some of the bedrock theories and discoveries of modern cosmology to the limit. Nobody in the field believes that this is the way things really work, however. And so in the last couple of years there has been a growing stream of debate and dueling papers, replete with references to such esoteric subjects as reincarnation, multiple universes and even the death of spacetime, as cosmologists try to square the predictions of their cherished theories with their convictions that we and the universe are real.

...

Alan Guth, a cosmologist at the Massachusetts Institute of Technology who agrees this overabundance is absurd, pointed out that some calculations result in an infinite number of free-floating brains for every normal brain, making it “infinitely unlikely for us to be normal brains.” Welcome to what physicists call the Boltzmann brain problem, named after the 19th-century Austrian physicist Ludwig Boltzmann, who suggested the mechanism by which such fluctuations could happen in a gas or in the universe. Cosmologists also refer to them as “freaky observers,” in contrast to regular or “ordered” observers of the cosmos like ourselves. Cosmologists are desperate to eliminate these freaks from their theories, but so far they can’t even agree on how or even on whether they are making any progress.

A straightforward reading of this is that cosmologists accept that our best scientific models, best supported by the scientific evidence, entail that we are most likely Boltzmann brains.

Of course, like you and others they cannot accept the conclusion, and so believe that the scientific models must be wrong. Which is why, as I said earlier, either we are most likely Boltzmann brains or our science is incorrect.

But as explained by the simple argument here, avoiding this "absurd" conclusion is no easy task, as rejecting one of the premises – contrary to the evidence that favours them – simply to avoid the conclusion doesn't seem rational.

In fact I'll set out the above in a more structured format:

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 infinite¹
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 infinite¹
7. Given (5) and (6) we are infinitely more likely to be a Boltzmann brain than a normal observer¹

The current scientific evidence supports (1)-(4), and (5)-(7) are rational deductions.

If (7) is false then at least one of (1)-(4) is false. But which of (1)-(4) is it rational to reject for no other reason than that "(7) must be false!"?

¹ _{If the time after the Big Freeze is finite but sufficiently large, i.e. many orders of magnitude greater than the time required for a Boltzmann brain to form, then although we are not infinitely more likely to be Boltzmann brains, we are still more likely to be Boltzmann brains.}

In a single de Sitter universe with a cosmological constant, and starting from any finite spatial slice, the number of "normal" observers is finite and bounded by the heat death of the universe. If the universe lasts forever, the number of nucleated Boltzmann brains is, in most models, infinite; cosmologists such as Alan Guth worry that this would make it seem "infinitely unlikely for us to be normal brains". — https://en.wikipedia.org/wiki/Boltzmann_brain#In_single-universe_scenarios

The general principle is that the time between the Big Bang and the Big Freeze is finite. Therefore the number of normal observers is finite.

The time after the Big Freeze is infinite. The time required for a Boltzmann brain to form via nucleation is very large – $10^{10^{69}}$ years in fact – but still finite. Given infinite time anything that can happen within a finite time – however large – will happen an infinite number of times.

Therefore there are infinitely more Boltzmann brains – of every variety that has a non-zero probability of forming; including those that appear to themselves to be normal observers – than normal observers. Therefore any randomly selected observer is infinitely more likely to be a Boltzmann brain – even one that appears to itself to be a normal observer – than to be a normal observer.

This is the model that is best supported by the current evidence.

Less supported models are those that predict a Big Crunch or a Big Rip, each of which avoid the problem of Boltzmann brains.

Personally, I think it would be strange to argue that either the Big Crunch or the Big Rip must be correct, or that the time after the Big Freeze must be finite, or that the probability of a Boltzmann brain forming must be zero, simply because it must be that we are not most likely to be a Boltzmann brain.

...and the longer you spend on this topic, the less likely it is that you are one of them. — Banno

But still more likely than not being one.

As an explanation for our observations, yes. — wonderer1

How so? There will be Boltzmann brains that have the same observations as ordinary observers; and in fact, there will be significantly (infinitely?) more Boltzmann brains that have those same observations as ordinary observers.

or if folk prefer simple language, there are no observations that settle the issue, and hence it remains mere speculation — Banno

The same with us not being Boltzmann brains.

That is not nearly as self defeating as a scientific hypothesis. — wonderer1

You think a quantum fluctuation universe is more likely than quantum fluctuation brains with false, consistent memories?

Any method for calculating probabilities ought come to the conclusion that the most likely observer is the one we know exists - us. — Banno

If we are Boltzmann brains then a calculation that shows a Boltzmann brain to be more likely than a regular observer has satisfied the above.

Your reasoning (or Vilenkin's) seems to beg the question.

Perhaps a different line of reasoning:

Borrowing from Tryon and Vilenkin, assume that any universe is itself a quantum fluctuation. Which is more likely; that we are a decades old Boltzmann brain or that we are an ordinary brain in a 14 billion year old Boltzmann universe?

I suppose the answer to that depends on what "surrounds" this universe. If it's an infinite and eternal vacuum then presumably there are an infinite number of Boltzmann universes and an infinite number of Boltzmann brains and so it is meaningless to say that one is more likely than the other.

We're just either a Boltzmann brain or an ordinary brain in a Boltzmann universe, with no evidence or reasoning to prefer either.

Somewhat of a coincidence, but that scientist wrote a paper that continues the work of the article I posted above.

Adding to the above, there's also Is the Universe a Vacuum Fluctuation?:

The author proposes a big bang model in which our Universe is a fluctuation of the vacuum, in the sense of quantum field theory. The model predicts a Universe which is homogeneous, isotropic and closed, and consists equally of matter and anti-matter. All these predictions are supported by, or consistent with, present observations.

Although it doesn't specifically refer to Boltzmann brains, the above is analogous to a Boltzmann universe, and as quantum fluctuations giving rise to a 14 billion year old universe is exponentially less likely than a quantum fluctuation giving rise to a several decades old brain, it stands to reason that evidence of the former is also evidence of the latter.

What you put in quotes there was something that someone wrote on Wikipedia. Can you quote a physicist making such a claim? — wonderer1

Brian Greene

I am confident that I am not a Boltzmann brain. However, we want our theories to similarly concur that we are not Boltzmann brains, but so far it has proved surprisingly difficult for them to do so. So, I see Boltzmann brains as a mathematical problem that we need to solve, as opposed to an existential affront. I believe it is a problem that we will one day overcome.

Sean Carroll

Some modern cosmological models predict the appearance of Boltzmann Brains: observers who randomly fluctuate out of a thermal bath rather than naturally evolving from a low-entropy Big Bang.

...

The issue is not that the existence of such observers is ruled out by data, but that the theories that predict them are cognitively unstable: they cannot simultaneously be true and justifiably believed.

Sean Carroll (again)

We're not arguing that Boltzmann Brains exist — we're trying to avoid them.

Those are people who have considered the possibility that we are Boltzmann brains. Not people who claim what you attribute to them. I already quoted Sean Carroll on the topic and it seemed pretty clear to me that Carroll doesn't make the claim that you are attributing to him. — wonderer1

"The consensus amongst cosmologists is that some yet to be revealed error is hinted at by the surprising calculation that Boltzmann brains should vastly outnumber normal human brains."

They accept that the science entails that we are most likely Boltzmann brains. They consider this proof that something is wrong with the science.

This is consistent with what I've been saying.

Either we are most likely Boltzmann brains or our science is inaccurate.

And I want to know what justifies the assertion that we are not Boltzmann brains given that the science suggests otherwise.

As for Carroll specifically, see Why Boltzmann Brains Are Bad:

Some modern cosmological models predict the appearance of Boltzmann Brains: observers who randomly fluctuate out of a thermal bath rather than naturally evolving from a low-entropy Big Bang.

...

The issue is not that the existence of such observers is ruled out by data, but that the theories that predict them are cognitively unstable: they cannot simultaneously be true and justifiably believed.

The science says one thing; he then suggests that it's irrational to believe it, i.e. that it's rational to reject the science.

On ChatGPT, here is an example of it contradicting itself three times in a row.
https://chat.openai.com/share/96378835-0a94-43ce-a25b-f05e5646ec40
And don't ever ask it to do any engineering https://chat.openai.com/share/b5241b53-e4d8-4cab-9a81-87fa73d740ad — Lionino

Yeah, ChatGPT doesn't reason. It basically just repeats what it's read elsewhere. Sometimes it makes stuff up. I tried using it for programming once and it fabricated a function that doesn't exist.

What scientist makes the claim that we have scientific evidence that we are most likely Boltzmann brains? — wonderer1

Several are mentioned in the Wikipedia article, e.g. Boltzmann, Eddington, Feynman, Sean Carroll, and Brian Greene.

3 is a contradiction, hard to disagree with that. My point was more that you seemed to agree with Sean Caroll, because his argument was the opposite of the premise that you refuted by probability in the original argument. Sean's point is about justification. — Lionino

Carroll pointed out the paradoxical nature of this:

1. Assume that we are most likely Boltzmann brains
2. Most Boltzmann brains do not have accurate scientific knowledge
3. Therefore, we most likely do not have accurate scientific knowledge
4. Our scientific knowledge entails that we are most likely Boltzmann brains
5. Therefore, our scientific knowledge that entails that we are most likely Boltzmann brains is most likely inaccurate
6. Therefore, we are most likely not Boltzmann brains

He then uses this to reject (1).

I then simply offered an inverse of the argument:

1. Assume that we are most likely ordinary humans
2. Assume that we have accurate scientific knowledge
3. Our scientific knowledge entails that we are most likely Boltzmann brains
4. Therefore, our scientific knowledge that entails that we are most likely Boltzmann brains is most likely accurate
5. Therefore, we are most likely not ordinary humans

I then use this to reject either (1) or (2).

The "cognitive instability" applies to both sides of the issue.

That's a false dichotomy, and things aren't nearly that black or white. — wonderer1

Then to be more accurate: either we are most likely Boltzmann brains or we cannot trust the scientific evidence that entails that we are most likely Boltzmann brains.

There are areas in which I can and do trust my scientific knowledge with my life (and the lives of other people) depending on it. There are other areas of scientific knowledge in which I can't trust my knowledge to nearly the same extent. — wonderer1

Which is why I asked the question: given that we have scientific evidence that entails that we are most likely Boltzmann brains, what justifies our claim that we are not most likely Boltzmann brains?

Sorry, me no understand. — Lionino

Consider this strengthened argument:

1. If we are not Boltzmann brains then our scientific theories are true
2. One of our scientific theories is that we are most likely Boltzmann brains
3. Therefore, if we are not Boltzmann brains then the scientific theory that we are most likely Boltzmann brains is true

Do you see the problem with (3)?

But in that case, your hard-won scientific knowledge just randomly fluctuated into existence as well; you have no reason to actually think that it represents an accurate view of reality.

Also on this, the same argument I made to Banno earlier can be used.

Each of these is true (if our current theories are correct):

1. Most Boltzmann brains have inaccurate scientific knowledge
2. Most observers with accurate scientific knowledge are Boltzmann brains

His argument is if P, not Q (if we are Boltzmann brains, we cannot trust our scientific knowledge); you showed how {if not P, Q} (if we are not Boltzmann brains, we can trust our scientific knowledge) entails a contradiction. How does that show his position has the same issue? — Lionino

It's right there in that post you first responded to:

1. If we are not Boltzmann brains then we can trust our scientific knowledge
2. Our scientific knowledge strongly suggests that we are most likely Boltzmann brains
3. Therefore, if we are not Boltzmann brains then we are most likely Boltzmann brains

Obviously (3) is problematic. Therefore we must reject either (1) or (2). As I said in my above post, many prominent physicists accept that (2) is true (even if they want to avoid the implication), and so I defer to their expertise.

So, as I said earlier, either we are most likely Boltzmann brains or we cannot trust our scientific knowledge.

That just begs the question by assuming that our scientific theories show that we are most likely Boltzmann brains — Janus

I'm not assuming it. It's what physicists like Boltzmann, Eddington, Feynman, Sean Carroll, Brian Greene, and others say. I'm deferring to their expertise.

I think number 1 is upside down, what Sean Carroll seems to suggest is that: — Lionino

I was presenting the inverse of his argument to show that his position suffers from that same cognitive instability.

How about a Boltzmann universe?

Is it possible that you and your surrounding environment, including all of your purported knowledge of the past and the outside world, randomly fluctuated into existence out of a chaotic soup of particles? Sure, it’s possible. But you should never attach very high credence to the possibility. Such a scenario is cognitively unstable, in the words of David Albert. You use your hard-won scientific knowledge to put together a picture of the world, and you realize that in that picture, it is overwhelmingly likely that you have just randomly fluctuated into existence. But in that case, your hard-won scientific knowledge just randomly fluctuated into existence as well; you have no reason to actually think that it represents an accurate view of reality. It is impossible for a scenario like this to be true and at the same time for us to have good reasons to believe in it. The best response is to assign it a very low credence and move on with our lives. — Sean Carroll

1. If we are not Boltzmann brains then we can trust our scientific knowledge
2. Our scientific knowledge strongly suggests that we are most likely Boltzmann brains
3. Therefore, if we are not Boltzmann brains then we are most likely Boltzmann brains

There's certainly some "cognitive instability" in his position, too.

So it's back to what I said here. Either we are most likely Boltzmann brains or we cannot trust our scientific knowledge.

There are lots of theories outside of physics that have a lot of evidentiary support as well. — wonderer1

So which theories outside physics are evidence against the eventual heat death of the universe and/or the fluctuations/nucleations that will give rise to Boltzmann brains? And how do we determine which evidence is stronger?

Of course, one solution to all of this is to abandon scientific realism and favour instrumentalism instead.

To play chess you must accept the reality of the pieces as objects in order to move them, therefore you must accept "chess reality" to play chess. Since it may not be stated in the rules that the pieces are "objects" the acceptance is only implicit, unlike set theory in which case the rule is explicit, therefore acceptance is explicit. — Metaphysician Undercover

You can play chess without a physical board and physical pieces. You can play it with pen and paper if you like; much like we do with maths. Or, if you're very smart, you can play it in your head; again, much like we do with maths.

When playing chess in your head you're not committed to being a realist about the queen you're playing with. When using set theory you're not committed to being a realist about the mathematical objects you're using.

You just follow the rules.

p) In A, A is a member of itself/A.
q) In U, A is not a member of itself/A.

p and q are true by definition. — Philosopher19

No, it's nonsense. That's not how set theory works.

1 is a member of N and R.
A is a member of A and B.

That's it.

Just take a few actual lessons in set theory.

I don't see the relevance. You do not need to accept the premise of "mathematical objects" to play chess. You do need to accept the premise of "mathematical objects" to follow the rules of set theory. — Metaphysician Undercover

You need to accept the premise of queens and kings and pawns to play chess, but accepting this premise doesn't commit you to "chess realism".

You need to accept the premise of a murderer and a victim when playing Cluedo, but accepting this premise doesn't commit you to "Cluedo realism".

And so accepting the premise of mathematical objects when using set theory doesn't commit you to mathematical realism.

When using set theory, mathematical objects "exist" only in the sense that queens "exist" in chess and a murderer "exists" in Cluedo, i.e. not in any realist sense.

And how would you justify that claim? — Metaphysician Undercover

I already did above. The axioms of some given set theory are just rules that you must follow when using that set theory. Different set theories have different axioms and so different rules. Given that there's no connection between using some set theory and believing in the mind-independent existence of abstract mathematical objects, there's no hypocrisy in using some set theory and being a mathematical antirealist.

Your position is like arguing that it's hypocritical to play chess if I do not believe that the rules of chess correspond to some mind-independent fact about the world.

However, it's hypocrisy to say "I'm a mathematical antirealist" and then go ahead and use set theory. — Metaphysician Undercover

No it's not.