I wrote a long response to Matthew's argument in a comment on his post, where I pointed out many problems with the argument (there are plenty). But none of your objections here are convincing to me.
The argument doesn't require the assumption that scientific evidence proves that an infinite number of people exist, so there was no need for BB to defend that. His argument for infinitely many people existing only relies on SIA.
I'm not sure what part of his argument you think is a Pascal's mugging - this doesn't make any sense to me either.
Nothing about SIA relies on a physical coin flip actually occurring. If God chose to create either 1 or 100 people, one of which was you, and you believe based on all non-anthropic reasons that there's a 50% chance of each, SIA says you should update to believing he created 100 people with 100-to-1 odds. This objection only makes sense if you reject Bayesian reasoning and don't even believe in credences, but then you're not going to be able to evaluate any claims about God's existence at all - P(God) doesn't make sense from a frequentist or propensity theory standpoint.
I guess the argument over whether goodness is a fundamental feature of the universe is just because it affects the prior probability of God? It's true that BB has made some bad arguments to that effect to try to prove that God has a high prior probability. But you don't need to believe that goodness or humanity is a fundamental feature of the universe to believe that God would create as many people as possible - all you need is total utilitarianism.
> I'm not sure what part of his argument you think is a Pascal's mugging - this doesn't make any sense to me either.
Unsubstantiated claims of super-infinity to infinitely counter a prior of nearly-infinite unlikelihood.
>you believe based on all non-anthropic reasons that there's a 50% chance of each, SIA says you should update to believing he created 100 people with 100-to-1 odds
Yeah, I'm not saying there needs to be a physical coin flip. I'm saying SIA *in the hypothetical* is telling us how to update our belief of what the result of the coin flip was. What is the equivalent of the coin flip in Matthew's analogy? I'm still not sure.
>you don't need to believe that goodness or humanity is a fundamental feature of the universe to believe that God would create as many people as possible
> Unsubstantiated claims of super-infinity to infinitely counter a prior of nearly-infinite unlikelihood.
But his claims aren't unsubstantiated - he provided arguments for them. And his argument isn't meant to counter a prior of nearly-infinite unlikelihood. No argument in the world is meant to do that. Obviously, if you have a near-zero prior in God's existence, you should conclude that it's more likely that he's made a mistake in his argument somewhere than that he really did prove God. But, unless you can identify a mistake that's totally fatal to the argument, you should still update your credence to be larger than it was before.
And I still don't see what this has to do with Pascal's mugging. Pascal's mugging is about using infinite rewards or punishments to counter extremely low credences, not infinite evidence.
> What is the equivalent of the coin flip in Matthew's analogy?
It's not an analogy, just an example of SIA in action. SIA says you should believe it's more likely that more people exist, so it favors any theory on which a large number of people exist over one where a small number exist.
In Matthew's argument, he's just using SIA directly to tell you how many people exist, and then he claims that it's more likely that that many people would exist if God exists.
> OK, but Matthew does
Yes, but it's not necessary for the argument, so it doesn't really matter.
Will you walk me through how the SIA works without some coin-flip-like assumption? The intuition pump is the coin flip because then we're just evaluating probabilities between a set of known scenarios. Without the coin flip I don't know what you really have.
I don't really understand the question because the coin flip just has nothing to do at all with SIA reasoning. SIA just says that anytime you have multiple hypotheses that predict that different numbers of people exist, and you know you're one of those people, you should multiply the probability of each hypothesis by the number of people it predicts and renormalize.
As an example, say there are two possible theories of cosmology, A and B (say physicists have eliminated all other options beyond reasonable doubt). A and B are equally well-confirmed by experiment and equally simple and elegant, such that, based only on non-anthropic considerations, they both seem equally likely. Then you discover that A predicts that twice as many people as B. SIA says you should believe A with credence of 2/3.
Okay, to be clear, BB does lean heavily into the coin toss example in the original post and in the podcast. I see that before giving the example he does formulate it once:
> Imagine thinking of your existence as being drawn randomly from the collection of possible people. If a theory predicts 10 times as many people exist, then it’s 10 times as likely you would come to exist. That’s how SIA instructs you to reason.
But then why does he give the coin toss as an example to pump our intuition? The coin toss seems to define priors for us: *all other things being equal* the universe has a 50/50 chance of being a Heads world vs a Tails world. Then we use the knowledge that I exist and the tails world has a million people to add weight to thise probabilities.
But if we take out the coin flip, we take out that prior. So it's not like we have all the possible worlds starting at equal probability: 1 human exists, 2 humans exist, 3 humans exist, ... a trillion humans exists,.... infinity humans exist. We have good reason to believe infinity humans don't exist, e.g. they take up space and we have empty space. I guess the point is that if it becomes infinity times more likely it doesn't matter how low your prior is?
I think that's where it starts to feel like Pascal's Mugging... not in that it is a reward/punishment issue, just that it plays with infinities in a way that seems... Fishy. I think the best illustration in both cases is that there are a plethora (an infinitude?) of similar arguments that can't all work.
Consider these possible worlds:
A. You exist and a teacup is in your bathroom. You can think of yourself as Plasma1.
B. You exist and 2 teacups are in your bathroom. You can think of yourself as Plasma2.
C. You exist and 3 teacups are in your bathroom. You can think of yourself as Plasma3.
...
Aleph. You exist and there are countably infinite teacups in you bathroom. You can think of yourself as PlasmaAleph
...
You exist and there are uncountably infinite teacups in your bathroom.
Now I ask you, "knowing that you exist, what is the probability of being in a world with fewer than 1000 teacups in your bathroom?"
"Imagine thinking of your existence as being drawn randomly from the collection of possible people." The chances that you're Plasma1-Plasma1000. is infinitesimal. In this case, each theory is no more likely that the others, but there are infinitely many of those theories where teacup >= 1000.
I admit to not being sure exactly where things go wrong, but that ability of infinities to smash any common sense belief seems to prove too much.
> But then why does he give the coin toss as an example to pump our intuition?
A coin toss is just a standard example to illustrate probability, since it's a clear example of a probabilistic event whose prior probability everyone agrees on.
> But if we take out the coin flip, we take out that prior.
You take out the 50-50 prior, but you still have some prior probability in any given hypothesis. SIA doesn't need the prior probabilities to be equal. If, for example, you had rolled a six-sided die that would create 10 people if it rolled a 1-5 and 20 people if it rolled a six, and you are one of the created people, SIA says you should update your probability that a six was rolled from 1/6 to 2/7.
> We have good reason to believe infinity humans don't exist, e.g. they take up space and we have empty space.
This isn't a good reason if there is an infinite amount of space, which there probably is, even just going by non-anthropic evidence.
> I guess the point is that if it becomes infinity times more likely it doesn't matter how low your prior is?
Yes, that is the argument he is making. Based on SIA, you should update your probability that infinitely many people exist to 1 unless your prior was 0.* Of course, in reality, no one should be 100% certain of SIA, so you should really only update your credence to be at least as large as your credence in SIA.
[*There are some possible caveats to this depending on how exactly you formulate SIA and how many possible people there are.]
> I think that's where it starts to feel like Pascal's Mugging...
I think the reason it feels like a Pascal's Mugging is because you're not actually 100% confident in SIA (nor should you be). SIA would only actually provide infinite evidence for a claim if you were infinitely confident in SIA. So in real life, it only provides a finite amount of evidence. If you have strong evidence against a claim that SIA claims is almost certainly true, then that also counts as evidence against SIA.
In the teacup example, I think you are misunderstanding one aspect of SIA: SIA doesn't say that you should consider yourself to be randomly chosen from all possible observers with equal probability. It says you should weight the probability of being any particular observer by the prior probability of that observer existing in the first place. SIA actually does not affect the probability at all in the teacup example because all scenarios described have the same number of observers - my probability is based purely on non-anthropic reasons.
"I’ve heard that many cosmologists say the universe is infinite."
I don't think any cosmologists today say that *our* universe is infinite. Our universe has finite size, mass, and energy; and thus a finite number of particles. Finite energy implies finite information, which quantum theory also implies.
It is still possible that our universe can do infinite computation, because the thermodynamic lower bound on energy required per bit of information produced or destroyed lessens as temperature decreases, at a rate that appears to me to provide an infinite number of computable bits. I don't know whether that relates to Matthew's argument. (Also, IIRC, the theoretical upper bound on computable bits is also the theoretical lower bound on the number needed to provide infinite bits, so the system has to be absolutely perfect to provide infinite computation. But you shouldn't trust me on this.)
Even a Multiverse is finite if there is just one starting point, and a finite number of particle interactions from start to finish, and a finite number of new universes created by each particle interaction. Speculation today about a Multiverse is based in quantum theory, which I think indeed posits a finite number of interactions and possible outcomes.
Ones who propose a Multiverse don't even necessarily imply that the Multiverse is infinite.
> Our universe has finite mass, size, and energy, and thus a finite number of particles.
None of these things are true. Most cosmologists still believe the Universe is infinite, and the only cosmological evidence we have points tentatively to that conclusion. Sometimes people refer to things like "The number of particles in the Universe," but when they do, they're talking about the observable universe, not the entire universe. No serious cosmologist believes that the observable universe is actually the entire universe - in fact, even if the universe is finite, it has to be way bigger than the observable universe to fit with cosmology.
The observable Universe has 2 meanings that I'm aware of: (a) the part of our Universe within our light-cone, or (b) the part of our Universe that isn't dark matter. Neither use posits that the rest of the Universe is infinite.
The observable universe refers to (a), not (b). Use of the phrase "observable universe" doesn't "posit" that the rest of the universe is infinite, but it implies that there is more to the Universe than the part within our light cone. It's not ruled out completely, but there is absolutely no evidence for the claim that, "Our universe has finite size, mass, and energy; and thus a finite number of particles." Nor does quantum mechanics imply that there is only a finite amount of information.
The simplest finite universe theory requires that it have positive curvature, while the simplest infinite universe theory is that it's just a regular Euclidean space on a large scale (zero curvature).
I wrote a long response to Matthew's argument in a comment on his post, where I pointed out many problems with the argument (there are plenty). But none of your objections here are convincing to me.
The argument doesn't require the assumption that scientific evidence proves that an infinite number of people exist, so there was no need for BB to defend that. His argument for infinitely many people existing only relies on SIA.
I'm not sure what part of his argument you think is a Pascal's mugging - this doesn't make any sense to me either.
Nothing about SIA relies on a physical coin flip actually occurring. If God chose to create either 1 or 100 people, one of which was you, and you believe based on all non-anthropic reasons that there's a 50% chance of each, SIA says you should update to believing he created 100 people with 100-to-1 odds. This objection only makes sense if you reject Bayesian reasoning and don't even believe in credences, but then you're not going to be able to evaluate any claims about God's existence at all - P(God) doesn't make sense from a frequentist or propensity theory standpoint.
I guess the argument over whether goodness is a fundamental feature of the universe is just because it affects the prior probability of God? It's true that BB has made some bad arguments to that effect to try to prove that God has a high prior probability. But you don't need to believe that goodness or humanity is a fundamental feature of the universe to believe that God would create as many people as possible - all you need is total utilitarianism.
> I'm not sure what part of his argument you think is a Pascal's mugging - this doesn't make any sense to me either.
Unsubstantiated claims of super-infinity to infinitely counter a prior of nearly-infinite unlikelihood.
>you believe based on all non-anthropic reasons that there's a 50% chance of each, SIA says you should update to believing he created 100 people with 100-to-1 odds
Yeah, I'm not saying there needs to be a physical coin flip. I'm saying SIA *in the hypothetical* is telling us how to update our belief of what the result of the coin flip was. What is the equivalent of the coin flip in Matthew's analogy? I'm still not sure.
>you don't need to believe that goodness or humanity is a fundamental feature of the universe to believe that God would create as many people as possible
OK, but Matthew does
> Unsubstantiated claims of super-infinity to infinitely counter a prior of nearly-infinite unlikelihood.
But his claims aren't unsubstantiated - he provided arguments for them. And his argument isn't meant to counter a prior of nearly-infinite unlikelihood. No argument in the world is meant to do that. Obviously, if you have a near-zero prior in God's existence, you should conclude that it's more likely that he's made a mistake in his argument somewhere than that he really did prove God. But, unless you can identify a mistake that's totally fatal to the argument, you should still update your credence to be larger than it was before.
And I still don't see what this has to do with Pascal's mugging. Pascal's mugging is about using infinite rewards or punishments to counter extremely low credences, not infinite evidence.
> What is the equivalent of the coin flip in Matthew's analogy?
It's not an analogy, just an example of SIA in action. SIA says you should believe it's more likely that more people exist, so it favors any theory on which a large number of people exist over one where a small number exist.
In Matthew's argument, he's just using SIA directly to tell you how many people exist, and then he claims that it's more likely that that many people would exist if God exists.
> OK, but Matthew does
Yes, but it's not necessary for the argument, so it doesn't really matter.
Will you walk me through how the SIA works without some coin-flip-like assumption? The intuition pump is the coin flip because then we're just evaluating probabilities between a set of known scenarios. Without the coin flip I don't know what you really have.
I don't really understand the question because the coin flip just has nothing to do at all with SIA reasoning. SIA just says that anytime you have multiple hypotheses that predict that different numbers of people exist, and you know you're one of those people, you should multiply the probability of each hypothesis by the number of people it predicts and renormalize.
As an example, say there are two possible theories of cosmology, A and B (say physicists have eliminated all other options beyond reasonable doubt). A and B are equally well-confirmed by experiment and equally simple and elegant, such that, based only on non-anthropic considerations, they both seem equally likely. Then you discover that A predicts that twice as many people as B. SIA says you should believe A with credence of 2/3.
Okay, to be clear, BB does lean heavily into the coin toss example in the original post and in the podcast. I see that before giving the example he does formulate it once:
> Imagine thinking of your existence as being drawn randomly from the collection of possible people. If a theory predicts 10 times as many people exist, then it’s 10 times as likely you would come to exist. That’s how SIA instructs you to reason.
But then why does he give the coin toss as an example to pump our intuition? The coin toss seems to define priors for us: *all other things being equal* the universe has a 50/50 chance of being a Heads world vs a Tails world. Then we use the knowledge that I exist and the tails world has a million people to add weight to thise probabilities.
But if we take out the coin flip, we take out that prior. So it's not like we have all the possible worlds starting at equal probability: 1 human exists, 2 humans exist, 3 humans exist, ... a trillion humans exists,.... infinity humans exist. We have good reason to believe infinity humans don't exist, e.g. they take up space and we have empty space. I guess the point is that if it becomes infinity times more likely it doesn't matter how low your prior is?
I think that's where it starts to feel like Pascal's Mugging... not in that it is a reward/punishment issue, just that it plays with infinities in a way that seems... Fishy. I think the best illustration in both cases is that there are a plethora (an infinitude?) of similar arguments that can't all work.
Consider these possible worlds:
A. You exist and a teacup is in your bathroom. You can think of yourself as Plasma1.
B. You exist and 2 teacups are in your bathroom. You can think of yourself as Plasma2.
C. You exist and 3 teacups are in your bathroom. You can think of yourself as Plasma3.
...
Aleph. You exist and there are countably infinite teacups in you bathroom. You can think of yourself as PlasmaAleph
...
You exist and there are uncountably infinite teacups in your bathroom.
Now I ask you, "knowing that you exist, what is the probability of being in a world with fewer than 1000 teacups in your bathroom?"
"Imagine thinking of your existence as being drawn randomly from the collection of possible people." The chances that you're Plasma1-Plasma1000. is infinitesimal. In this case, each theory is no more likely that the others, but there are infinitely many of those theories where teacup >= 1000.
I admit to not being sure exactly where things go wrong, but that ability of infinities to smash any common sense belief seems to prove too much.
> But then why does he give the coin toss as an example to pump our intuition?
A coin toss is just a standard example to illustrate probability, since it's a clear example of a probabilistic event whose prior probability everyone agrees on.
> But if we take out the coin flip, we take out that prior.
You take out the 50-50 prior, but you still have some prior probability in any given hypothesis. SIA doesn't need the prior probabilities to be equal. If, for example, you had rolled a six-sided die that would create 10 people if it rolled a 1-5 and 20 people if it rolled a six, and you are one of the created people, SIA says you should update your probability that a six was rolled from 1/6 to 2/7.
> We have good reason to believe infinity humans don't exist, e.g. they take up space and we have empty space.
This isn't a good reason if there is an infinite amount of space, which there probably is, even just going by non-anthropic evidence.
> I guess the point is that if it becomes infinity times more likely it doesn't matter how low your prior is?
Yes, that is the argument he is making. Based on SIA, you should update your probability that infinitely many people exist to 1 unless your prior was 0.* Of course, in reality, no one should be 100% certain of SIA, so you should really only update your credence to be at least as large as your credence in SIA.
[*There are some possible caveats to this depending on how exactly you formulate SIA and how many possible people there are.]
> I think that's where it starts to feel like Pascal's Mugging...
I think the reason it feels like a Pascal's Mugging is because you're not actually 100% confident in SIA (nor should you be). SIA would only actually provide infinite evidence for a claim if you were infinitely confident in SIA. So in real life, it only provides a finite amount of evidence. If you have strong evidence against a claim that SIA claims is almost certainly true, then that also counts as evidence against SIA.
In the teacup example, I think you are misunderstanding one aspect of SIA: SIA doesn't say that you should consider yourself to be randomly chosen from all possible observers with equal probability. It says you should weight the probability of being any particular observer by the prior probability of that observer existing in the first place. SIA actually does not affect the probability at all in the teacup example because all scenarios described have the same number of observers - my probability is based purely on non-anthropic reasons.
"I’ve heard that many cosmologists say the universe is infinite."
I don't think any cosmologists today say that *our* universe is infinite. Our universe has finite size, mass, and energy; and thus a finite number of particles. Finite energy implies finite information, which quantum theory also implies.
It is still possible that our universe can do infinite computation, because the thermodynamic lower bound on energy required per bit of information produced or destroyed lessens as temperature decreases, at a rate that appears to me to provide an infinite number of computable bits. I don't know whether that relates to Matthew's argument. (Also, IIRC, the theoretical upper bound on computable bits is also the theoretical lower bound on the number needed to provide infinite bits, so the system has to be absolutely perfect to provide infinite computation. But you shouldn't trust me on this.)
Even a Multiverse is finite if there is just one starting point, and a finite number of particle interactions from start to finish, and a finite number of new universes created by each particle interaction. Speculation today about a Multiverse is based in quantum theory, which I think indeed posits a finite number of interactions and possible outcomes.
Ones who propose a Multiverse don't even necessarily imply that the Multiverse is infinite.
> Our universe has finite mass, size, and energy, and thus a finite number of particles.
None of these things are true. Most cosmologists still believe the Universe is infinite, and the only cosmological evidence we have points tentatively to that conclusion. Sometimes people refer to things like "The number of particles in the Universe," but when they do, they're talking about the observable universe, not the entire universe. No serious cosmologist believes that the observable universe is actually the entire universe - in fact, even if the universe is finite, it has to be way bigger than the observable universe to fit with cosmology.
Can you list some references for that?
The observable Universe has 2 meanings that I'm aware of: (a) the part of our Universe within our light-cone, or (b) the part of our Universe that isn't dark matter. Neither use posits that the rest of the Universe is infinite.
The observable universe refers to (a), not (b). Use of the phrase "observable universe" doesn't "posit" that the rest of the universe is infinite, but it implies that there is more to the Universe than the part within our light cone. It's not ruled out completely, but there is absolutely no evidence for the claim that, "Our universe has finite size, mass, and energy; and thus a finite number of particles." Nor does quantum mechanics imply that there is only a finite amount of information.
The tentative evidence for an infinite universe comes from measurements of the global curvature, which are consistent with it being zero: https://www.aanda.org/articles/aa/full_html/2020/09/aa33910-18/aa33910-18.html
The simplest finite universe theory requires that it have positive curvature, while the simplest infinite universe theory is that it's just a regular Euclidean space on a large scale (zero curvature).