>Second, even if you think moral realism is probably false, you shouldn’t be super confident in its falsity. A ton of incredibly smart people are moral realists including most philosophers, and about a third of philosophers take goodness to be fundamental. You should, therefore, be no more than 90% confident that they’re all wrong.
It seems like something weird is going on here.
Suppose Muslims believe that the Qur'an is the exact, uncreated and eternal word of God. Suppose Islam also teaches that God had a 100% probability of sharing this word with Muhammad and preserving it for all time.
Under this possibly hypothetical version of Islam (I'm not actually sure how accurate it is), there's a 100% chance Muhammad would relay all the words in the Qur'an in its present form in their precise, current order. Under the vast majority of non-Islamic views, by contrast, even if Muhammad had some high probability of coming up with *something* as his false prophetic text, the probability that it's the exact Qur'an we have today is near-infinitesimal. So the likelihood ratio of this version of Islam over common-sense alternatives is massive, way over a googol at the very least.
Now, there are millions of really smart Muslims who (again hypothetically) adhere to this theological view and who've thought a lot about it. You might want to say that Islam therefore should get a prior of at least 10^-50, if not significantly higher. But this is of course going to be massively outweighed by the ridiculously huge Bayes factor. So on this view of the epistemology of disagreement, we should consider Islam near-certain not due to any miracles, but merely because the Qur'an contains a lot of words in a fixed order!
I think it is true of moral realism. Simply because you think otherwise doesn't mean others are obliged to think accordingly.
What if someone like me thinks it is true of moral realism, and in fact that moral realism is (if anything) less plausible than various forms of theism and various mainstream religions?
As an anti-realist, I'm not sure I agree. At least the religious view I described is unambiguously coherent, or is so if we imagine the hypothetical Islam variant to make modest claims about God's attributes (so we don't necessarily have to deal with, e.g., the problem of evil or the precise definition of God's omnipotence). Whereas I'm not even sure I can make any sense of the realist's view of normative facts, moral properties, stance-independent reasons, etc., at all.
I also personally suspect that, if we could somehow randomly generate a super-intelligent mind from the space of all possible super-intelligent minds, the probability that it would arrive at moral realism as a remotely plausible possibility after reflection is near zero - maybe even lower than the prior of the Qur'an, because the space could be *that* large. However, making this intuition rigorous is difficult.
Why not, though? The nature of the outside view update is the same regardless of how much *you in particular* think the position is crazy. The whole point of such outside view reasoning is that we outsource the measure of craziness of a position to the wisdom of a crowd of the smart people and if there are a lot of such people who hold this position then we assume that its not that crazy.
Whether or not smart people think something isn't a good reason to update that much in favor of it not being super implausible. What matters is why they think what they think. If one is a member of the field of experts in question and has knowledge of the field's methods, presumptions, and so on, then one may be in a good position to not put much stock in what lots of "smart people" think. I think I'm in just such a situation when it comes to moral realism: I take the popularity of moral realism to be less evidence that realism isn't an absurd position (it is), and far better evidence of other things: (1) strong selection effects for people disposed towards realism in analytic philosophy (2) weird and persistent biases among analytic philosophers (3) problems with the methods and presuppositions of analytic philosophy.
For comparison, if lots of smart people endorsed astrology or homeopathy, I would not be very inclined to endorse either.
I do not particularly favor outside view updates as well. They can be applied as a cheap heuristic when you do not want to spend much effort on researching some topic, so you just look at the consensus of the experts and default to it.
However, indeed, as you've said:
"If one is a member of the field of experts in question and has knowledge of the field's methods, presumptions, and so on, then one may be in a good position to not put much stock in what lots of "smart people" think."
But Bentham's Bulldog seems to accept outside view updates in principle, therefore I'm curious how he justifies discarding outside view update in this particular imaginary scenario.
What seems to be doing the work here is the fact that our having the *exact text* of the Quran is astronomically unlikely if God didn't inspire it. But our having the exact text of the Quran is also incredibly unlikely if God did inspire it: the hypothesis that God inspired it doesn't predict that the 374th word is whatever the 374th word is, etc. What predicts the exact text is: [the Quran is divinely inspired & the exact text is what it is]. But ofc, [the Quran is not divinely inspired & the exact text is what it is] also predicts the exact text. Or, to put it another way, [the Quran is divinely inspired & the exact text is what it is] takes a hit to its prior probability exactly proportional to the gain in predictive power it has over just [the Quran is divinely inspired].
I just wanted to leave a response to this objection that I thought of: it's trading on an ambiguity between two different but related claims. Define the following:
B = "Billions of smart Muslims believe Muhammad was the messenger of God, and that the Qur'an, *whatever* it says, is eternal the word of God"
B* = "Billions of smart Muslims believe Muhammad was the messenger of God, and the word of God in particular is '____'," where "____" should be replaced by the entire text of the Qur'an
Q = "The Qur'an states that ____," where "____" should be replaced as before
Given that Q is our evidence, the likelihood of B* is actually high regardless of whether Islam is true, because if billions of Muslims believe the Qur'an states that ____, then the Qur'an probably does state that. So the likelihood ratio of B* is indifferent, about 1. On the other hand, the likelihood of B is low regardless of whether Islam is true, since it's always hard to predict the exact corpus of the Qur'an without conditionalizing on more precise evidence of its contents. So the likelihood ratio of B* should also be about 1. Either way, the likelihood ratio tells us little.
In terms of specifically your example the question one has to ask is whether when looking at Qur'an we observe an extremely rare event "This exact combination of letters is observed" or a less much rare event "any combination of letters fitting the description of some prophetic text is observed".
Little known fact about philosophy but the SIA actually only applies to situations involving coin flips. Hope this helps clear up most of the confusion.
I admire your composure, civility, and friendliness in response to Eneasz's confused, condescending, dogmatic article. Reading the back-and-forth, I get the impression that he doesn't understand your argument, and he doesn't want to understand it. The discussion of the "mugging" is revelatory, though not in the way he intended it: I think the misfiring sequence looks something like this:
(have a 99.9999% prior on materialism etc.) + (to act practically on strong beliefs, one must have a system to defend against "muggings," which are literally big-number attacks and sociologically any argument that violates "ground beliefs:" ) -> (two habits of thought: taboo on ultrabig numbers and a willingness to round 99.99999% credences to axioms) -> (any argument involving big numbers and lots of inferential backtracking away from Yudkowskian dicta sets off these immune systems) -> (inflammatory response in the logic tissues)
I appreciate the rebuttal and I'm linking it at the end of my OP now, ty! I don't find it persuasive, and it seems we're just talking past each other at this point.
I think one major disagreement is that I accept SIA but I believe it provides 0 evidence rather than infinite evidence. If you accept infinite universes across infinite multiverses, then it follows that if ANY discrete thing exists (whether it be a particle or an insanely complex non-repeating pattern or a thing in between or things OOMs more small/fundamental or more big/complex than these examples) then yes, infinites of that discrete thing exist out to as many Beths as you care for. And this is true of all the things, there's no reason to limit these discrete things to just humans. This doesn't actually give evidence for anything. It's just another way of saying "yes I think there are infinite multiverses" or "no I don't think that" and neither statement is useful for anything at all.
All the actual work is being done by assuming goodness is fundamental and a god exists. If someone doesn't already share those assumptions the argument has no persuasive power. If someone does already share those assumptions, the argument is unnecessary.
I don't think we have any way to resolve this disagreement with words.
I say "your god" rather than "God" to clarify that we're talking about your beliefs rather than a thing that exists in reality.
I did use quotation marks for a summary rather than a direct quote, which I apologize for! Old, bad habit. I'm trying to revert to 'single-quote' when it's not an actual quote, but I slip up sometimes. I hope from context it's clear it's not a direct quote ^^;
> You should, therefore, be no more than 90% confident that they’re all wrong.
Pshhh. :) I have a lot of practice being confident that mystics are wrong, I'm not about to throw in the towel now!
If you get an update by a factor of N to a theory that says there are more N times more people--which is what SIA says--you'll get an infinite update in favor of infinite people.
God is good if he exists by definition. So the things that follow from God's existence as a good being and exist serve to probabilistically evidence his existence.
> If you get an update by a factor of N to a theory that says there are more N times more people--which is what SIA says--you'll get an infinite update in favor of infinite people.
Conditional upon evidence that there actually are infinite people. SIA cannot provide that evidence by itself.
> God is good if he exists by definition.
People have tried to define god into existence for ages. (Hello ontological argument). If you're going to define your god into existence you don't need all these extra math steps, you can just define him as existing. I have saved you many steps!
I'm geniunely curious how can SIA-followers deal with its craziness regarding infinities. Mathew bites the bullet all the way, you seem to have some way around it. Could you explain it?
Let's go back to Sleeping Beauty paradox. According to SIA, P(Heads|Awake) = 1/3 because there are twice as many possible awakenings on Tails. I suppose you agree with it.
Now, consider a modification of Sleeping Beauty where after the last awakening on Tails the coin is tossed once more. If it's Heads the experiment end, but if it Tails, the Beauty gets as many extra awakenings as she already had and then the loop continues until on one of the tosses of the coin comes Heads. So the possible coin sequence of the experiment are H, TH, TTH, TTTH, TTTTH... and the Beaty, correspondingly, is having 1, 2, 4, 8, 16... awakenings.
What should be the Beauty's credence that the first coin came Heads conditionally on being awakened in such experiment? Here I expect SIA followers to say that it's literally zero. After all, there are potentially infinite awakenings otherwise. Do you agree? If so why does the same principle not apply to make you absolutely certain that infinite people exist in our reality? If not, what is your reasoning for disagreement?
If that's how the math comes out then I agree because it's a hypothetical, and according to the hypothetical that's the answer. I don't agree with absolute certainty that this is the case in our universe because I'm not that certain that our universe contains complete infinities. But more to the point - so what if it did? If the universe is infinite in this way then infinite people exist and......... ? This provides no evidence of any other claims.
> I don't agree with absolute certainty that this is the case in our universe because I'm not that certain that our universe contains complete infinities.
Well, it seems that you have to be absolutely certain that our universe *doesn't contain* complete infinities, in order not to agree. If there is the most tiny but still non-zero probability that there are infinite people in the universe, SIA makes you update to full certanity that it's indeed the case, solely based on the fact of you existence. The only reason SIA-follower can not be absolutely certain that infinite people exist is to have completely trapped prior.
> so what if it did? If the universe is infinite in this way then infinite people exist and......... ?
Personally, I'm mostly worried about the rationality of SIA reasoning. For me the fact that just from entertaining the thought of infinite people existing you have to believe with absolute certanity that infinite people indeed exist, immediately sounds as wrong way to reason - definitely not how cognition engines produce accurate maps of reality, therefore discrediting SIA beyond any reasonable doubt. I wonder how people who accept SIA deal with this conundrum.
How can you accept SIA but think it doesn’t provide any evidence? Isn’t the whole point of anthropic reasoning about what the fact of your existence gives you evidence for, and the point of SIA that your existence gives you evidence that more people exist?
The hypothetical is solid, but granting that a hypothetical is solid doesn't mean it actually translates into reality. You need some actual evidence to think the hypothetical is more than just a fun hypothetical for that. Because as I said earlier, I have just as much evidence that I'm a brain in a vat, or a boltzman brain, or an em in an ancestor simulation. SIA gives 0 physical evidence that it's more likely than those, or this is all a dream, or whatever.
Maybe I didn't bother to critique it enough since I'm already at the "infinity everything exists is plausible" conclusion and didn't need to be convinced of it with further arguments.
A crisper answer for why R^4 can only have aleph_0 people, I think, is that people must take up space (have nonempty topological interior, which they don't share). Any open set in R^4 will contain a rational point. There are aleph_0 many rational points in R^4. (You could have continuum-many people if you they could have literally 0 volume, but that's not allowed in our world afaict.)
>Fourth, even if one isn’t initially inclined to accept moral non-natural realism, they should think it’s likely because of its wonderful explanatory power.
I both reject non-naturalist realism and reject that it has literally any explanatory power at all, given that I don't even think it's intelligible (FWIW, that also means I can't directly assign a probability to bit being true, since I don't think it the claim that non-naturalist moral realism is true is a proposition).
Even if others don't accept that it's unintelligible, they at least aren't obliged to agree that it's an especially good explanation for anything. Whether or not it serves as a "good explanation" will itself turn on contentious claims they may be free to reject.
I appreciate the rebuttal and I'm linking it at the end of my OP now, ty! I don't find it persuasive, and it seems we're just talking past each other at this point.
I think one major disagreement is that I accept SIA but I believe it provides 0 evidence rather than infinite evidence. If you accept infinite universes across infinite multiverses, then it follows that if ANY discrete thing exists (whether it be a particle or an insanely complex non-repeating pattern or a thing in between or things OOMs more small/fundamental or more big/complex than these examples) then yes, infinites of that discrete thing exist out to as many Beths as you care for. And this is true of all the things, there's no reason to limit these discrete things to just humans. This doesn't actually give evidence for anything. It's just another way of saying "yes I think there are infinite multiverses" or "no I don't think that" and neither statement is useful for anything at all.
All the actual work is being done by assuming goodness is fundamental and a god exists. If someone doesn't already share those assumptions the argument has no persuasive power. If someone does already share those assumptions, the argument is unnecessary.
I don't think we have any way to resolve this disagreement with words.
I say "your god" rather than "God" to clarify that we're talking about your beliefs rather than a thing that exists in reality.
I did use quotation marks for a summary rather than a direct quote, which I apologize for! Old, bad habit. I'm trying to revert to 'single-quote' when it's not an actual quote, but I slip up sometimes. I hope from context it's clear it's not a direct quote ^^;
> You should, therefore, be no more than 90% confident that they’re all wrong.
Pshhh. :) I have a lot of practice being confident that mystics are wrong, I'm not about to throw in the towel now!
The self-indication assumption says from your existence you get evidence that there are more people. By that principle, you'll get infinitely strong evidence there are infinite people. I agree other conscious beings should also reason about the self-indication assumption in the same way.
A Bayesian should reason in terms of probabilities. If you're, say, 5% confident that God exists, because this is so much more strongly predicted, you should now be a theist.
From the existence of a rock with a chip you gain infinitely strong evidence there are infinite rocks with a chip. You don't need to limit this to minds.
Why would anyone be 5% confident that your god exists? Are you 5% confident the gravity goblin exists? Such a creature is already more likely than your god, since we have evidence of gravity. Also there's infinite rocks that are attracted to each other, thus providing infinite evidence for the goblin.
No, it only applies to minds. The core intuition behind SIA--that if there are more people the odds are higher that I'd be one of them--doesn't apply to the other objects, because if there are more objects, while the odds any particular object would exist are higher, on account of there being more of them, the odds that I'd witness that one is lower. Thus, taking into account the specific evidence--my table exists and I witness it--we don't get any update in favor of more chairs existing, except insofar as it means more people will have my precise experience of seeing a chair. In contrast, your existence is different--if I exist, I'd necessarily know about it, while there being more chairs makes it less likely I'd see any particular chair.
Why have a 5% credence prior to seeing this argument? Well, because God is simple, lacks arbitrary limits, has a higher prior than naturalism, explains why anything exists, explains the physical world, laws, complex structures forming, fine-tuning, the laws applying to the stuff that exists, consciousness, psychophysical harmony, structures (brains) that produce consciousness arising, and about a dozen other things. In short, you believe in God for the same reason you believe in dark matter--it explains a lot https://benthams.substack.com/p/god-best-explains-the-world
There is no reason to privilege minds aside from really liking minds and thus being sure they MUST have cosmic importance of some sort. That is not a valid argument. The odds that you personally witness something don't matter.
>God is simple
nope
>lacks arbitrary limits
haha, lacks all sorts of things XD
>has a higher prior than naturalism
oof, no
>explains why anything exists
no
>explains the physical world
me: --quickly checks all explanations about the physical world-- Hm, another no
>laws
...? no
>complex structures forming
definitely not
>fine-tuning
no
>the laws applying to the stuff that exists
Ah! oh wait, you said this one already, bonus-no.
>consciousness
lol no
>psychophysical harmony
I don't know what this, so hey, maybe this one! But given the track record so far, I'm not going to take your word for it
>structures (brains) that produce consciousness arising
are we just double-counting No's from earlier in this list for the fun of it?
>and about a dozen other things.
hmmmmmm....
OK, I'm just being cheeky there, which is a sign I should bow out. I've had this debate at least a hundred times (maybe hundreds? definitely in the triple digits tho) back in the late 90s through late 00s. (I started out on the theist side but was won over fairly quickly, so the majority of the time was just ever greater refinement into epsilon with each new argument). I'm not the person to go through all of this with you with, but hopefully someone will take up the mantle. It can be a lot of fun! Remember that no serious person is going to start with anything close to 5%. They're either already a theist, or already disillusioned, if they've spent any serious amount of time on this.
Some people are undecided. But again, arguments are for the purpose of convincing the undecided. You don't use arguments to convince people who are, based on philosophical confusion, almost certain you're wrong.
I'd be happy to have you on the YouTube channel to discuss any of these points.
Hm. I'll think on it. I'd want to really start at something basic, like "how can you define goodness in a way that it can even conceivably be thought of fundamental to reality and existing without a valuer." If your god's first attribute is that he's the incarnation of this fundamental aspect of the universe, you first gotta give people a reason to think that's a sane sentence to even utter.
For quick clarification - eg: you know god either made 1 thing or 100 thing. In a sample of the universe you observe at least 1 thing. Odds are higher he made 100 thing, as that was more likely to result in finding at least one thing in a random sample. This thing doesn't need to be a mind. Also note that "god made X thing" is an assumption of the hypothetical, and thus cannot be used to provide evidence for what was already assumed.
Imagine that we don't live in a universe anything like ours. It's one of those universes that isn't "finely tuned," so there's no opportunity for anything like complex life. There is no multiverse or anything like it, no parallel universes - it's all that exists. However, it's sufficiently vast and lasts sufficiently long, that for just a fraction of a second, there's a Boltzmann brain, and it thinks, "Hm, I exist, ergo infinitely more people must exist," even though all its sense data is "there is literally nothing in the entire universe except you," before it dies and all consciousness ceases to exist.
Apparently the Boltzmann brain was completely, 100% rational, given the SIA. There's a reason, however, that people are expected to not merely guess populations, but instead perform censuses.
> The claim is that if there are two theories, and one of them says that N times more people exist than another, your existence confirms that theory by a factor of N. This is the self-indication assumption—it’s overwhelmingly plausible—and Eneasz accepts it! But if this is true, your existence gives you infinitely strong evidence that infinite people exist. <
Isn't this a reductio ad absurdum of the SIA? There never have been and never will have been an infinite number of people. If SIA tells you that there are (or will be) an infinite number of people, then it must be wrong.
Alternatively, doesn't the combination of SIA and the fact that there have only (and forever will have only) been finitely many people prove that an all-good God does not exist? After all, if an all-good God existed then an infinite number of people would exist, and they don't.
Of course part of the standard beliefs about God is omnipotence, so God could (presumably) create an infinite number of people, thus validating the SIA. But we have no evidence that God has done such a thing -- none of the Holy Books claims such a thing has happened!
> Finally, because God is good, and would want to make either all possible people or some huge portion of them <
This looks like another weak link in your argument. What is your evidence that God would want to make so many people? AFAIK he only made two to four people (Adam and Eve definitely, maybe Lilith, and maybe Jesus). God specifically told A&E to be fruitful and multiply. He didn't say "But don't sweat it. I'll be making infinitely many more just like you in any case." Kinda seems like our (finite!) reproduction is part of His plan. Your claim here might actually be heretical, and believing heretics is definitely not what God wants us to do. :-)
OTOH, if that claim is false then it saves the SIA and God from the devastating arguments you (presumably unintentionally) provided.
How do you know there aren't infinite people? They might exist in the multiverse or in our infinitely big universe.
I'm not a Christian, I don't think the Bible ever says the number of people God makes. But if God exists, he'd have a moral reason to make all possible people if it's good to do so (which it is), so all possible people would exist.
> Isn't this a reductio ad absurdum of the SIA? There never have been and never will have been an infinite number of people. If SIA tells you that there are (or will be) an infinite number of people, then it must be wrong.
How do you save SIA from this reducto ad absurdum, though? Whatever evidence you get of infinite people not being able to exist it can't possibly outweight the infinitely strong anthropical update in favor of infinite people that SIA provides. You need to be absolutely sure that infinite people can't possibly exist, to rescue SIA, and you can't arrive to this view just by mere updating on available finite evidence.
So a world with more people in it is more likely to contain me than a world with fewer people. Doesn't equating the ratio of population between those worlds with the ratio of the likelihood of being in each world (given that I exist) rely on the probability of each person being independent? That is, a world with 100 times more people is presented as being 100 times more likely, but I am much less likely to exist in a world that never contained my parents than a world that did. Since my probability and my parents' probabilities are not independent, I would think that a world with 100 times more people would be more likely but something less than 100 times more likely.
Yes, this is correct. What matters isn't the raw number of people but the share of people that you might currently be where you discount people if you're less likely to be them. So, if theory 1 says there are 10 people and theory 2 says there are 2, theory 1 is 5x better, because it makes it 5x likelier that there'd be someone like me. But if theory 1 says there are 10 people named John, and I'm Matthew, and theory 2 says there are 2 Matthews, theory 2 is better.
I don't think your argument that a physical universe cannot contain more than aleph0 observers works, as it seems to be top strong. It would rule out that physical reality could be continuous. Suppose that we have a rock which is continuos. It would contain beth1 non-overlapping parts. This certainly seems possible. But for each non-overlapping part, it would require beth1 sentences to describe it's spacial relation to each other non-overlapping parts, which cannot be done with only aleph0 possible sentences. So your argument would rule out space being continuous, it seems.
It also rules out universal composition for a universe which is infinitely large. If the universe is infinitely large, and there is not only a finite bounded region which contains stuff, there will be at least aleph0 things (for example rocks) in that universe. But on universal composition, the number of composite objects consisting of those rocks would then be the power set of the set of those rocks, which would at least have cardinality beth1. Now, the spacial relations of all these composites cannot be described using aleph0 sentences, meaning you should rule out the possibility of infinite space and universal composition.
But this seems like far too strong conclusions from an argument based on symbol manipulation, if you will. Though you may of course just bite the bullet and say that the positions I outlined can be dismissed out of hand.
You can't have more than aleph null observers of more than arbitrarily small size or less than some arbitrarily large size I think. If you have people made of every subset of some number of points, then a finite sentence can't pick out one particular one (over the other arbitrarily large number of subsets).
But why does a sentence have to be able to pick out one over the other for them to actually exist? We can perfectly well decribe a scenario where beth1 (or even beth2) observers, all of which are distinct and all of which are both larger than point-size and smaller than some finite size. So while we can't list the whole set of those observers using a sentence, we can still describe a scenario where they exist. That surely seems good enough?
Yes, so if any observer can be described in finite characters there can only be aleph null. That will be true in typical infinite worlds, but by having weird things like infinitely small people, it won't apply.
So, take a brain and one of its neurons. The brain is conscious both with and without the neuron, and the neuron contains aleph0 non-overlapping parts (though we should surely also think that a continuous neuron containing beth1 non-overlapping parts is possible). Now, on univeralism, for each part of the neuron, there will be a mereological fusion of that part and the rest of the brain, and each of these will be conscious (and so observers). And since the neuron has aleph0 non-overlapping parts, and since the set of parts is the powersets of the non-overlapping parts, it will have beth1 parts, and there will thus be beth1 observers there. Now each of these will be roughly the size of a brain, so there is no shenanigans with infinitely small people. But according to your argument we can be sure that such a scenario is impossible, because we don't have enough sentences to list out all the observers. But that just doesn't seem to be a reasonable criterion for possibility.
After all, I believe you think that the real numbers exist (or are in some sense "real"). We also cannot list out all of these. And what about the powerset of the real numbers, and the powerset of the powerset etc. I take it you think all these exist in some sense. But if those things can exist without us being able to describe them, why the exception with physical stuff/observers?
But most of those things can't be picked out by a sentence! If there are infinite members, the idea that there's 1, 4, 5, 9...and so on, picking out numbers forever, can't be picked out by any sentence of finite length.
Also, even if you can get Beth 1, that's not enough (I don't think Beth 2 is enough, but def not Beth 1).
Sure, I am also not objecting to the argument as a whole - especially if you think that there are more than beth2 possible people, then it really begins to be difficult to see how to get that in a physical world. I just have a gripe with the specific argument you gave, as I don't think it is sufficient.
But, I mean, most real numbers can't be picked out by a sentence either, right? You can't write down pi in a single sentence. You can of course use the sign π, but that just denotes pi. If we allow that, then we can do the following for the observers:
Denote the brain minus the neuron with "B". We can then say one of the observers is {B}. Another of the observers is {B, a}, where a is a non-overlapping part of the neuron, and another is {B, a, b} etc.
Doing this, we can pick out some of the observers, just like we can pick out some of the real numbers.
On eternalism plus mereological universalism plus the continuity of time, it's easy to get Beth-2 objects. If you choose a particle and an arbitrary collection of moments at which it exists, take the mereological sum of the time-slices of the particle at those moments. Voila, Beth-2 choices are possible. Although this observation is more nitpicking than anything and is probably not the kind of thing BB had in mind.
Why stop at Beth 2? No master why you think you should stop there, you have infinite evidence that Beth 3 is more likely! And again with Beth 4…
This leads to a contradiction.
SIA: N times more people in a hypothesis means that that hypothesis is preferred by a factor of N. (I buy it for finite N)
SIA to infinity: if hypothesis A has a larger infinity of people than hypothesis B, A is infinitely more likely than B. (This is the bad axiom).
Probability axiom: probabilities are real numbers in [0,1] (Claude says that infinitesimals can’t save us either)
Probability axiom: the sum of all mutually exclusive and exhaustive possibilities = 1. If the number of people is card1, then the number of people is not card2, and the number of people must have some value.
Fact about infinity: there’s no largest infinity
Therefore we need a function f that maps infinite cardinals to probabilities such that f(card1)/f(card2) = card1/card2 that satisfies the requirements above. The math here is difficult set theory, but both ChatGPT4o and Claude 3.5 Sonnet were confident that such a function cannot exist.
There's no biggest infinite set! But maybe you should think this shows it's not a set--it's absolute infinite, the size of the collection of truths. That's what wins the arms race.
Alternatively, you can think what matters isn't absolute numbers but the share of possible people that exist.
But then there’s certainly a size even bigger than the collection of truths. Your reasoning for why the collection of truths is the biggest is certain to be false.
You’re just stringing together words at this point. Your view of anthropic lead to a direct logical contradiction by saying that the number of people in existence has no size, (or is bigger than itself).
Not sure why you give this contradiction so little credence, and heavily unintuitive unrestricted SIA so much.
There are technically only continuum many hyperreals on the most common construction of "the" hyperreals. Although I think you can instead construct them in less typical ways with larger ultraproduct to get an arbitrarily large number of them.
>Second, even if you think moral realism is probably false, you shouldn’t be super confident in its falsity. A ton of incredibly smart people are moral realists including most philosophers, and about a third of philosophers take goodness to be fundamental. You should, therefore, be no more than 90% confident that they’re all wrong.
It seems like something weird is going on here.
Suppose Muslims believe that the Qur'an is the exact, uncreated and eternal word of God. Suppose Islam also teaches that God had a 100% probability of sharing this word with Muhammad and preserving it for all time.
Under this possibly hypothetical version of Islam (I'm not actually sure how accurate it is), there's a 100% chance Muhammad would relay all the words in the Qur'an in its present form in their precise, current order. Under the vast majority of non-Islamic views, by contrast, even if Muhammad had some high probability of coming up with *something* as his false prophetic text, the probability that it's the exact Qur'an we have today is near-infinitesimal. So the likelihood ratio of this version of Islam over common-sense alternatives is massive, way over a googol at the very least.
Now, there are millions of really smart Muslims who (again hypothetically) adhere to this theological view and who've thought a lot about it. You might want to say that Islam therefore should get a prior of at least 10^-50, if not significantly higher. But this is of course going to be massively outweighed by the ridiculously huge Bayes factor. So on this view of the epistemology of disagreement, we should consider Islam near-certain not due to any miracles, but merely because the Qur'an contains a lot of words in a fixed order!
But I think that view is crazy enough that you should have a prior of near zero in it and not defer. The same isn't true of moral realism.
I think it is true of moral realism. Simply because you think otherwise doesn't mean others are obliged to think accordingly.
What if someone like me thinks it is true of moral realism, and in fact that moral realism is (if anything) less plausible than various forms of theism and various mainstream religions?
As an anti-realist, I'm not sure I agree. At least the religious view I described is unambiguously coherent, or is so if we imagine the hypothetical Islam variant to make modest claims about God's attributes (so we don't necessarily have to deal with, e.g., the problem of evil or the precise definition of God's omnipotence). Whereas I'm not even sure I can make any sense of the realist's view of normative facts, moral properties, stance-independent reasons, etc., at all.
I also personally suspect that, if we could somehow randomly generate a super-intelligent mind from the space of all possible super-intelligent minds, the probability that it would arrive at moral realism as a remotely plausible possibility after reflection is near zero - maybe even lower than the prior of the Qur'an, because the space could be *that* large. However, making this intuition rigorous is difficult.
Why not, though? The nature of the outside view update is the same regardless of how much *you in particular* think the position is crazy. The whole point of such outside view reasoning is that we outsource the measure of craziness of a position to the wisdom of a crowd of the smart people and if there are a lot of such people who hold this position then we assume that its not that crazy.
Whether or not smart people think something isn't a good reason to update that much in favor of it not being super implausible. What matters is why they think what they think. If one is a member of the field of experts in question and has knowledge of the field's methods, presumptions, and so on, then one may be in a good position to not put much stock in what lots of "smart people" think. I think I'm in just such a situation when it comes to moral realism: I take the popularity of moral realism to be less evidence that realism isn't an absurd position (it is), and far better evidence of other things: (1) strong selection effects for people disposed towards realism in analytic philosophy (2) weird and persistent biases among analytic philosophers (3) problems with the methods and presuppositions of analytic philosophy.
For comparison, if lots of smart people endorsed astrology or homeopathy, I would not be very inclined to endorse either.
I do not particularly favor outside view updates as well. They can be applied as a cheap heuristic when you do not want to spend much effort on researching some topic, so you just look at the consensus of the experts and default to it.
However, indeed, as you've said:
"If one is a member of the field of experts in question and has knowledge of the field's methods, presumptions, and so on, then one may be in a good position to not put much stock in what lots of "smart people" think."
But Bentham's Bulldog seems to accept outside view updates in principle, therefore I'm curious how he justifies discarding outside view update in this particular imaginary scenario.
What seems to be doing the work here is the fact that our having the *exact text* of the Quran is astronomically unlikely if God didn't inspire it. But our having the exact text of the Quran is also incredibly unlikely if God did inspire it: the hypothesis that God inspired it doesn't predict that the 374th word is whatever the 374th word is, etc. What predicts the exact text is: [the Quran is divinely inspired & the exact text is what it is]. But ofc, [the Quran is not divinely inspired & the exact text is what it is] also predicts the exact text. Or, to put it another way, [the Quran is divinely inspired & the exact text is what it is] takes a hit to its prior probability exactly proportional to the gain in predictive power it has over just [the Quran is divinely inspired].
I was essentially trading on an ambiguity as mentioned here: https://open.substack.com/pub/benthams/p/eneasz-brodski-makes-silly-mistakes?r=2py43&utm_campaign=comment-list-share-cta&utm_medium=web&comments=true&commentId=65339766
I just wanted to leave a response to this objection that I thought of: it's trading on an ambiguity between two different but related claims. Define the following:
B = "Billions of smart Muslims believe Muhammad was the messenger of God, and that the Qur'an, *whatever* it says, is eternal the word of God"
B* = "Billions of smart Muslims believe Muhammad was the messenger of God, and the word of God in particular is '____'," where "____" should be replaced by the entire text of the Qur'an
Q = "The Qur'an states that ____," where "____" should be replaced as before
Given that Q is our evidence, the likelihood of B* is actually high regardless of whether Islam is true, because if billions of Muslims believe the Qur'an states that ____, then the Qur'an probably does state that. So the likelihood ratio of B* is indifferent, about 1. On the other hand, the likelihood of B is low regardless of whether Islam is true, since it's always hard to predict the exact corpus of the Qur'an without conditionalizing on more precise evidence of its contents. So the likelihood ratio of B* should also be about 1. Either way, the likelihood ratio tells us little.
Great observation! This is indeed a very curious property of probability theory that a lot of people are confused about. I talk about it here: https://www.lesswrong.com/posts/bHzcKEEHL854KsWGM/another-non-anthropic-paradox-the-unsurprising-rareness-of
In terms of specifically your example the question one has to ask is whether when looking at Qur'an we observe an extremely rare event "This exact combination of letters is observed" or a less much rare event "any combination of letters fitting the description of some prophetic text is observed".
Little known fact about philosophy but the SIA actually only applies to situations involving coin flips. Hope this helps clear up most of the confusion.
I admire your composure, civility, and friendliness in response to Eneasz's confused, condescending, dogmatic article. Reading the back-and-forth, I get the impression that he doesn't understand your argument, and he doesn't want to understand it. The discussion of the "mugging" is revelatory, though not in the way he intended it: I think the misfiring sequence looks something like this:
(have a 99.9999% prior on materialism etc.) + (to act practically on strong beliefs, one must have a system to defend against "muggings," which are literally big-number attacks and sociologically any argument that violates "ground beliefs:" ) -> (two habits of thought: taboo on ultrabig numbers and a willingness to round 99.99999% credences to axioms) -> (any argument involving big numbers and lots of inferential backtracking away from Yudkowskian dicta sets off these immune systems) -> (inflammatory response in the logic tissues)
Anyway, kudos to you for taking it well.
This is a great post title. :)
I appreciate the rebuttal and I'm linking it at the end of my OP now, ty! I don't find it persuasive, and it seems we're just talking past each other at this point.
I think one major disagreement is that I accept SIA but I believe it provides 0 evidence rather than infinite evidence. If you accept infinite universes across infinite multiverses, then it follows that if ANY discrete thing exists (whether it be a particle or an insanely complex non-repeating pattern or a thing in between or things OOMs more small/fundamental or more big/complex than these examples) then yes, infinites of that discrete thing exist out to as many Beths as you care for. And this is true of all the things, there's no reason to limit these discrete things to just humans. This doesn't actually give evidence for anything. It's just another way of saying "yes I think there are infinite multiverses" or "no I don't think that" and neither statement is useful for anything at all.
All the actual work is being done by assuming goodness is fundamental and a god exists. If someone doesn't already share those assumptions the argument has no persuasive power. If someone does already share those assumptions, the argument is unnecessary.
I don't think we have any way to resolve this disagreement with words.
I say "your god" rather than "God" to clarify that we're talking about your beliefs rather than a thing that exists in reality.
I did use quotation marks for a summary rather than a direct quote, which I apologize for! Old, bad habit. I'm trying to revert to 'single-quote' when it's not an actual quote, but I slip up sometimes. I hope from context it's clear it's not a direct quote ^^;
> You should, therefore, be no more than 90% confident that they’re all wrong.
Pshhh. :) I have a lot of practice being confident that mystics are wrong, I'm not about to throw in the towel now!
To put it more plainly:
> From this, we learn that the number of people that exist is the most there could be
No we don't
> because God is good
No he isn't (because he doesn't exist)
If you get an update by a factor of N to a theory that says there are more N times more people--which is what SIA says--you'll get an infinite update in favor of infinite people.
God is good if he exists by definition. So the things that follow from God's existence as a good being and exist serve to probabilistically evidence his existence.
> If you get an update by a factor of N to a theory that says there are more N times more people--which is what SIA says--you'll get an infinite update in favor of infinite people.
Conditional upon evidence that there actually are infinite people. SIA cannot provide that evidence by itself.
> God is good if he exists by definition.
People have tried to define god into existence for ages. (Hello ontological argument). If you're going to define your god into existence you don't need all these extra math steps, you can just define him as existing. I have saved you many steps!
I'm geniunely curious how can SIA-followers deal with its craziness regarding infinities. Mathew bites the bullet all the way, you seem to have some way around it. Could you explain it?
Let's go back to Sleeping Beauty paradox. According to SIA, P(Heads|Awake) = 1/3 because there are twice as many possible awakenings on Tails. I suppose you agree with it.
Now, consider a modification of Sleeping Beauty where after the last awakening on Tails the coin is tossed once more. If it's Heads the experiment end, but if it Tails, the Beauty gets as many extra awakenings as she already had and then the loop continues until on one of the tosses of the coin comes Heads. So the possible coin sequence of the experiment are H, TH, TTH, TTTH, TTTTH... and the Beaty, correspondingly, is having 1, 2, 4, 8, 16... awakenings.
What should be the Beauty's credence that the first coin came Heads conditionally on being awakened in such experiment? Here I expect SIA followers to say that it's literally zero. After all, there are potentially infinite awakenings otherwise. Do you agree? If so why does the same principle not apply to make you absolutely certain that infinite people exist in our reality? If not, what is your reasoning for disagreement?
If that's how the math comes out then I agree because it's a hypothetical, and according to the hypothetical that's the answer. I don't agree with absolute certainty that this is the case in our universe because I'm not that certain that our universe contains complete infinities. But more to the point - so what if it did? If the universe is infinite in this way then infinite people exist and......... ? This provides no evidence of any other claims.
> I don't agree with absolute certainty that this is the case in our universe because I'm not that certain that our universe contains complete infinities.
Well, it seems that you have to be absolutely certain that our universe *doesn't contain* complete infinities, in order not to agree. If there is the most tiny but still non-zero probability that there are infinite people in the universe, SIA makes you update to full certanity that it's indeed the case, solely based on the fact of you existence. The only reason SIA-follower can not be absolutely certain that infinite people exist is to have completely trapped prior.
> so what if it did? If the universe is infinite in this way then infinite people exist and......... ?
Personally, I'm mostly worried about the rationality of SIA reasoning. For me the fact that just from entertaining the thought of infinite people existing you have to believe with absolute certanity that infinite people indeed exist, immediately sounds as wrong way to reason - definitely not how cognition engines produce accurate maps of reality, therefore discrediting SIA beyond any reasonable doubt. I wonder how people who accept SIA deal with this conundrum.
How can you accept SIA but think it doesn’t provide any evidence? Isn’t the whole point of anthropic reasoning about what the fact of your existence gives you evidence for, and the point of SIA that your existence gives you evidence that more people exist?
The hypothetical is solid, but granting that a hypothetical is solid doesn't mean it actually translates into reality. You need some actual evidence to think the hypothetical is more than just a fun hypothetical for that. Because as I said earlier, I have just as much evidence that I'm a brain in a vat, or a boltzman brain, or an em in an ancestor simulation. SIA gives 0 physical evidence that it's more likely than those, or this is all a dream, or whatever.
Maybe I didn't bother to critique it enough since I'm already at the "infinity everything exists is plausible" conclusion and didn't need to be convinced of it with further arguments.
A crisper answer for why R^4 can only have aleph_0 people, I think, is that people must take up space (have nonempty topological interior, which they don't share). Any open set in R^4 will contain a rational point. There are aleph_0 many rational points in R^4. (You could have continuum-many people if you they could have literally 0 volume, but that's not allowed in our world afaict.)
>Fourth, even if one isn’t initially inclined to accept moral non-natural realism, they should think it’s likely because of its wonderful explanatory power.
I both reject non-naturalist realism and reject that it has literally any explanatory power at all, given that I don't even think it's intelligible (FWIW, that also means I can't directly assign a probability to bit being true, since I don't think it the claim that non-naturalist moral realism is true is a proposition).
Even if others don't accept that it's unintelligible, they at least aren't obliged to agree that it's an especially good explanation for anything. Whether or not it serves as a "good explanation" will itself turn on contentious claims they may be free to reject.
This is a great post title. :)
I appreciate the rebuttal and I'm linking it at the end of my OP now, ty! I don't find it persuasive, and it seems we're just talking past each other at this point.
I think one major disagreement is that I accept SIA but I believe it provides 0 evidence rather than infinite evidence. If you accept infinite universes across infinite multiverses, then it follows that if ANY discrete thing exists (whether it be a particle or an insanely complex non-repeating pattern or a thing in between or things OOMs more small/fundamental or more big/complex than these examples) then yes, infinites of that discrete thing exist out to as many Beths as you care for. And this is true of all the things, there's no reason to limit these discrete things to just humans. This doesn't actually give evidence for anything. It's just another way of saying "yes I think there are infinite multiverses" or "no I don't think that" and neither statement is useful for anything at all.
All the actual work is being done by assuming goodness is fundamental and a god exists. If someone doesn't already share those assumptions the argument has no persuasive power. If someone does already share those assumptions, the argument is unnecessary.
I don't think we have any way to resolve this disagreement with words.
I say "your god" rather than "God" to clarify that we're talking about your beliefs rather than a thing that exists in reality.
I did use quotation marks for a summary rather than a direct quote, which I apologize for! Old, bad habit. I'm trying to revert to 'single-quote' when it's not an actual quote, but I slip up sometimes. I hope from context it's clear it's not a direct quote ^^;
> You should, therefore, be no more than 90% confident that they’re all wrong.
Pshhh. :) I have a lot of practice being confident that mystics are wrong, I'm not about to throw in the towel now!
The self-indication assumption says from your existence you get evidence that there are more people. By that principle, you'll get infinitely strong evidence there are infinite people. I agree other conscious beings should also reason about the self-indication assumption in the same way.
A Bayesian should reason in terms of probabilities. If you're, say, 5% confident that God exists, because this is so much more strongly predicted, you should now be a theist.
From the existence of a rock with a chip you gain infinitely strong evidence there are infinite rocks with a chip. You don't need to limit this to minds.
Why would anyone be 5% confident that your god exists? Are you 5% confident the gravity goblin exists? Such a creature is already more likely than your god, since we have evidence of gravity. Also there's infinite rocks that are attracted to each other, thus providing infinite evidence for the goblin.
No, it only applies to minds. The core intuition behind SIA--that if there are more people the odds are higher that I'd be one of them--doesn't apply to the other objects, because if there are more objects, while the odds any particular object would exist are higher, on account of there being more of them, the odds that I'd witness that one is lower. Thus, taking into account the specific evidence--my table exists and I witness it--we don't get any update in favor of more chairs existing, except insofar as it means more people will have my precise experience of seeing a chair. In contrast, your existence is different--if I exist, I'd necessarily know about it, while there being more chairs makes it less likely I'd see any particular chair.
Why have a 5% credence prior to seeing this argument? Well, because God is simple, lacks arbitrary limits, has a higher prior than naturalism, explains why anything exists, explains the physical world, laws, complex structures forming, fine-tuning, the laws applying to the stuff that exists, consciousness, psychophysical harmony, structures (brains) that produce consciousness arising, and about a dozen other things. In short, you believe in God for the same reason you believe in dark matter--it explains a lot https://benthams.substack.com/p/god-best-explains-the-world
There is no reason to privilege minds aside from really liking minds and thus being sure they MUST have cosmic importance of some sort. That is not a valid argument. The odds that you personally witness something don't matter.
>God is simple
nope
>lacks arbitrary limits
haha, lacks all sorts of things XD
>has a higher prior than naturalism
oof, no
>explains why anything exists
no
>explains the physical world
me: --quickly checks all explanations about the physical world-- Hm, another no
>laws
...? no
>complex structures forming
definitely not
>fine-tuning
no
>the laws applying to the stuff that exists
Ah! oh wait, you said this one already, bonus-no.
>consciousness
lol no
>psychophysical harmony
I don't know what this, so hey, maybe this one! But given the track record so far, I'm not going to take your word for it
>structures (brains) that produce consciousness arising
are we just double-counting No's from earlier in this list for the fun of it?
>and about a dozen other things.
hmmmmmm....
OK, I'm just being cheeky there, which is a sign I should bow out. I've had this debate at least a hundred times (maybe hundreds? definitely in the triple digits tho) back in the late 90s through late 00s. (I started out on the theist side but was won over fairly quickly, so the majority of the time was just ever greater refinement into epsilon with each new argument). I'm not the person to go through all of this with you with, but hopefully someone will take up the mantle. It can be a lot of fun! Remember that no serious person is going to start with anything close to 5%. They're either already a theist, or already disillusioned, if they've spent any serious amount of time on this.
Some people are undecided. But again, arguments are for the purpose of convincing the undecided. You don't use arguments to convince people who are, based on philosophical confusion, almost certain you're wrong.
I'd be happy to have you on the YouTube channel to discuss any of these points.
Hm. I'll think on it. I'd want to really start at something basic, like "how can you define goodness in a way that it can even conceivably be thought of fundamental to reality and existing without a valuer." If your god's first attribute is that he's the incarnation of this fundamental aspect of the universe, you first gotta give people a reason to think that's a sane sentence to even utter.
For quick clarification - eg: you know god either made 1 thing or 100 thing. In a sample of the universe you observe at least 1 thing. Odds are higher he made 100 thing, as that was more likely to result in finding at least one thing in a random sample. This thing doesn't need to be a mind. Also note that "god made X thing" is an assumption of the hypothetical, and thus cannot be used to provide evidence for what was already assumed.
Yes that's true. But the only thing that matters is how many things are observed by observers that you might currently be.
Imagine that we don't live in a universe anything like ours. It's one of those universes that isn't "finely tuned," so there's no opportunity for anything like complex life. There is no multiverse or anything like it, no parallel universes - it's all that exists. However, it's sufficiently vast and lasts sufficiently long, that for just a fraction of a second, there's a Boltzmann brain, and it thinks, "Hm, I exist, ergo infinitely more people must exist," even though all its sense data is "there is literally nothing in the entire universe except you," before it dies and all consciousness ceases to exist.
Apparently the Boltzmann brain was completely, 100% rational, given the SIA. There's a reason, however, that people are expected to not merely guess populations, but instead perform censuses.
> The claim is that if there are two theories, and one of them says that N times more people exist than another, your existence confirms that theory by a factor of N. This is the self-indication assumption—it’s overwhelmingly plausible—and Eneasz accepts it! But if this is true, your existence gives you infinitely strong evidence that infinite people exist. <
Isn't this a reductio ad absurdum of the SIA? There never have been and never will have been an infinite number of people. If SIA tells you that there are (or will be) an infinite number of people, then it must be wrong.
Alternatively, doesn't the combination of SIA and the fact that there have only (and forever will have only) been finitely many people prove that an all-good God does not exist? After all, if an all-good God existed then an infinite number of people would exist, and they don't.
Of course part of the standard beliefs about God is omnipotence, so God could (presumably) create an infinite number of people, thus validating the SIA. But we have no evidence that God has done such a thing -- none of the Holy Books claims such a thing has happened!
> Finally, because God is good, and would want to make either all possible people or some huge portion of them <
This looks like another weak link in your argument. What is your evidence that God would want to make so many people? AFAIK he only made two to four people (Adam and Eve definitely, maybe Lilith, and maybe Jesus). God specifically told A&E to be fruitful and multiply. He didn't say "But don't sweat it. I'll be making infinitely many more just like you in any case." Kinda seems like our (finite!) reproduction is part of His plan. Your claim here might actually be heretical, and believing heretics is definitely not what God wants us to do. :-)
OTOH, if that claim is false then it saves the SIA and God from the devastating arguments you (presumably unintentionally) provided.
How do you know there aren't infinite people? They might exist in the multiverse or in our infinitely big universe.
I'm not a Christian, I don't think the Bible ever says the number of people God makes. But if God exists, he'd have a moral reason to make all possible people if it's good to do so (which it is), so all possible people would exist.
> Isn't this a reductio ad absurdum of the SIA? There never have been and never will have been an infinite number of people. If SIA tells you that there are (or will be) an infinite number of people, then it must be wrong.
How do you save SIA from this reducto ad absurdum, though? Whatever evidence you get of infinite people not being able to exist it can't possibly outweight the infinitely strong anthropical update in favor of infinite people that SIA provides. You need to be absolutely sure that infinite people can't possibly exist, to rescue SIA, and you can't arrive to this view just by mere updating on available finite evidence.
So a world with more people in it is more likely to contain me than a world with fewer people. Doesn't equating the ratio of population between those worlds with the ratio of the likelihood of being in each world (given that I exist) rely on the probability of each person being independent? That is, a world with 100 times more people is presented as being 100 times more likely, but I am much less likely to exist in a world that never contained my parents than a world that did. Since my probability and my parents' probabilities are not independent, I would think that a world with 100 times more people would be more likely but something less than 100 times more likely.
Yes, this is correct. What matters isn't the raw number of people but the share of people that you might currently be where you discount people if you're less likely to be them. So, if theory 1 says there are 10 people and theory 2 says there are 2, theory 1 is 5x better, because it makes it 5x likelier that there'd be someone like me. But if theory 1 says there are 10 people named John, and I'm Matthew, and theory 2 says there are 2 Matthews, theory 2 is better.
I don't think your argument that a physical universe cannot contain more than aleph0 observers works, as it seems to be top strong. It would rule out that physical reality could be continuous. Suppose that we have a rock which is continuos. It would contain beth1 non-overlapping parts. This certainly seems possible. But for each non-overlapping part, it would require beth1 sentences to describe it's spacial relation to each other non-overlapping parts, which cannot be done with only aleph0 possible sentences. So your argument would rule out space being continuous, it seems.
It also rules out universal composition for a universe which is infinitely large. If the universe is infinitely large, and there is not only a finite bounded region which contains stuff, there will be at least aleph0 things (for example rocks) in that universe. But on universal composition, the number of composite objects consisting of those rocks would then be the power set of the set of those rocks, which would at least have cardinality beth1. Now, the spacial relations of all these composites cannot be described using aleph0 sentences, meaning you should rule out the possibility of infinite space and universal composition.
But this seems like far too strong conclusions from an argument based on symbol manipulation, if you will. Though you may of course just bite the bullet and say that the positions I outlined can be dismissed out of hand.
You can't have more than aleph null observers of more than arbitrarily small size or less than some arbitrarily large size I think. If you have people made of every subset of some number of points, then a finite sentence can't pick out one particular one (over the other arbitrarily large number of subsets).
But why does a sentence have to be able to pick out one over the other for them to actually exist? We can perfectly well decribe a scenario where beth1 (or even beth2) observers, all of which are distinct and all of which are both larger than point-size and smaller than some finite size. So while we can't list the whole set of those observers using a sentence, we can still describe a scenario where they exist. That surely seems good enough?
Yes, so if any observer can be described in finite characters there can only be aleph null. That will be true in typical infinite worlds, but by having weird things like infinitely small people, it won't apply.
So, take a brain and one of its neurons. The brain is conscious both with and without the neuron, and the neuron contains aleph0 non-overlapping parts (though we should surely also think that a continuous neuron containing beth1 non-overlapping parts is possible). Now, on univeralism, for each part of the neuron, there will be a mereological fusion of that part and the rest of the brain, and each of these will be conscious (and so observers). And since the neuron has aleph0 non-overlapping parts, and since the set of parts is the powersets of the non-overlapping parts, it will have beth1 parts, and there will thus be beth1 observers there. Now each of these will be roughly the size of a brain, so there is no shenanigans with infinitely small people. But according to your argument we can be sure that such a scenario is impossible, because we don't have enough sentences to list out all the observers. But that just doesn't seem to be a reasonable criterion for possibility.
After all, I believe you think that the real numbers exist (or are in some sense "real"). We also cannot list out all of these. And what about the powerset of the real numbers, and the powerset of the powerset etc. I take it you think all these exist in some sense. But if those things can exist without us being able to describe them, why the exception with physical stuff/observers?
But most of those things can't be picked out by a sentence! If there are infinite members, the idea that there's 1, 4, 5, 9...and so on, picking out numbers forever, can't be picked out by any sentence of finite length.
Also, even if you can get Beth 1, that's not enough (I don't think Beth 2 is enough, but def not Beth 1).
Sure, I am also not objecting to the argument as a whole - especially if you think that there are more than beth2 possible people, then it really begins to be difficult to see how to get that in a physical world. I just have a gripe with the specific argument you gave, as I don't think it is sufficient.
But, I mean, most real numbers can't be picked out by a sentence either, right? You can't write down pi in a single sentence. You can of course use the sign π, but that just denotes pi. If we allow that, then we can do the following for the observers:
Denote the brain minus the neuron with "B". We can then say one of the observers is {B}. Another of the observers is {B, a}, where a is a non-overlapping part of the neuron, and another is {B, a, b} etc.
Doing this, we can pick out some of the observers, just like we can pick out some of the real numbers.
On eternalism plus mereological universalism plus the continuity of time, it's easy to get Beth-2 objects. If you choose a particle and an arbitrary collection of moments at which it exists, take the mereological sum of the time-slices of the particle at those moments. Voila, Beth-2 choices are possible. Although this observation is more nitpicking than anything and is probably not the kind of thing BB had in mind.
Why stop at Beth 2? No master why you think you should stop there, you have infinite evidence that Beth 3 is more likely! And again with Beth 4…
This leads to a contradiction.
SIA: N times more people in a hypothesis means that that hypothesis is preferred by a factor of N. (I buy it for finite N)
SIA to infinity: if hypothesis A has a larger infinity of people than hypothesis B, A is infinitely more likely than B. (This is the bad axiom).
Probability axiom: probabilities are real numbers in [0,1] (Claude says that infinitesimals can’t save us either)
Probability axiom: the sum of all mutually exclusive and exhaustive possibilities = 1. If the number of people is card1, then the number of people is not card2, and the number of people must have some value.
Fact about infinity: there’s no largest infinity
Therefore we need a function f that maps infinite cardinals to probabilities such that f(card1)/f(card2) = card1/card2 that satisfies the requirements above. The math here is difficult set theory, but both ChatGPT4o and Claude 3.5 Sonnet were confident that such a function cannot exist.
Therefore SIA to Infinity is false.
There's no biggest infinite set! But maybe you should think this shows it's not a set--it's absolute infinite, the size of the collection of truths. That's what wins the arms race.
Alternatively, you can think what matters isn't absolute numbers but the share of possible people that exist.
But then there’s certainly a size even bigger than the collection of truths. Your reasoning for why the collection of truths is the biggest is certain to be false.
It is the maximum size--it is at a point that truly transcends size, where everything capable of multiplication will be smaller.
You’re just stringing together words at this point. Your view of anthropic lead to a direct logical contradiction by saying that the number of people in existence has no size, (or is bigger than itself).
Not sure why you give this contradiction so little credence, and heavily unintuitive unrestricted SIA so much.
Can you explain this more simply?
Oh yeah, I was describing what you were talking about in terms of 3.
There are technically only continuum many hyperreals on the most common construction of "the" hyperreals. Although I think you can instead construct them in less typical ways with larger ultraproduct to get an arbitrarily large number of them.