I'm imagining Bentham doing the Two Buttons / Daily Struggle meme, where one button is resisting the urge to destroy, and the other button is how utterly terrible and philosophically confused Carrier's response is.
God really gives his hardest battles to his strongest soldiers.
“Oh no! I forgot to check census statistics! I had thought earth had infinite people, rather than 8 billion. How could I have made so foolish an error.” I almost spilled out my iced tea laughing
FWIW, I was the one who shared your post with Richard Carrier, and I have now shared this response of yours with him as well. I'd like to see how exactly he will respond to this.
I guess that my issue with the SIA is that there remains the difficulty of proving it. I mean, I can easily think of many people who don't exist in this universe. Bentham Bulldog's sister, for instance. Where's the evidence that they exist in other universes? One can make conjectures in regards to this, but where specifically is the hard evidence? In this universe, many people exist whose existence is extraordinarily unlikely, but many people whose existence is extraordinarily unlikely don't exist as well.
The evidence would be in your existence. And it isn't necessary that they exist, it works just as well if you only require that there be a number of people of the same cardinality as the number of possible people.
I don't think it necessarily does. You can believe in an atheistic multiverse.
In fact, there are numerous physical theories already used that are a multiverse of sorts. E.g. many physicists prefer the many-worlds theory. Inflationary cosmology does that as well, I think. None of these are as large as the size of multiverse that the argument that Matthew is presenting would ask for.
(Context, about me: I'm a Christian, I find multiverses somewhat repugnant, but also think that SIA is probably a strong argument for them.)
I think it might. I'm not familiar enough with this. That said, there are different sizes of infinities, and so it might depend on what size of infinity we need. But that's a good point, I'd have to check, and I'm not sure how to do that most straightforwardly.
I think that there is an extraordinarily massive amount of different permutations of reality in many-worlds theory. For instance, US President James Abram Garfield lives vs. Garfield dies in 1881. Then beyond that: Garfield signs the Chinese Exclusion Act vs. vetoes it in 1882. Garfield wins vs. loses reelection in 1884. Et cetera.
Sure. Cardinality refers to how many things we are. Things have the same cardinality if we can put them in a one-to-one correspondence. We're used to thinking about finite numbers, where if you add one, it changes the cardinality. But when we're talking about infinite numbers, you can have different sets with the same cardinality. For example, the function f(x)=2x puts the set of integers into a one-to-one correspondence with the even numbers, even though the one is a subset of the other. So they have the same cardinality.
There are different size infinities—some infinites cannot be put in a one-to-one correspondence with each other. What I was saying was that I do not think his anthropic argument would necessarily imply that every possible person exist, but it would suggest that you could make a one-to-one correspondence between existing people and possible people.
Let me know if any part of that is unclear, that's still not quite layman's level, but it's closer.
Wow! I feel like Richard is pretty good on certain points, but that was really embarrassing for him! Especially how confident and dismissive he is while simultaneously giving incredibly misinformed and stupid points, lol
Just a stylistic point: I think it would help if you broke up your sentences more. Having multiple asides in a sentence or nested if clauses can be cognitively taxing for the reader
Forgive me if this is a point you have covered before, but you say "Because the number of people that could exist is a very large infinity—far more than the number of numbers—if there’s no God, it’s very unlikely that such a number of people would exist." Why exactly should we expect it to be unlikely for that large infinity of people to exist given atheism? I understand the point you're making that a God who wants to create would create a very large infinity of people, I just don't understand why that many people couldn't also turn up under atheism.
You say "To get at least Beth 2 people, one needs an extreme degree of gerrymandering and shenanigans—a natural multiverse doesn’t get Beth 2 people, nor does a universe that’s infinite in size. God can easily make Beth 2 people, but if he doesn’t do that, the odds of there being Beth 2 people are very low." How does God manage to create Beth 2 people within the multiverse without himself having to pull shenanigans, and what if I grant that shenanigans are required but simply assert the truth of atheism + shenanigans?
I'm fairly sympathetic to this argument. It would seem like modal realism, or a tegmark iv multiverse would get you there, without a God being necessary. Not sure if the typical universe in such situations would look like our world, ours is unexpectedly simple.
"if a coin gets flipped that creates one person if heads and ten if tails, then if you get created from this process, you should think the coin came up tails at 10:1 odds"
RAA: change the rules to heads: zero people, tails:10. Then when you get created what are the odds of tails? That's simple causality so it's 100%. If you're prepared to allow assymmetry in this case you can't deny it in the weaker case.
> Let N be the size of the collection representing the number of possible people:
> There are at least N people (deduced from the fact that you exist).
Am I silly? How is it possibly the case that one’s own existence allows one to deductively infer that there must be the maximal possible number of people. This seems to completely obviate any distinction between the actual and the possible.
And I will once again say, for posterity, that all the irreconcilable paradoxes on both sides of the anthropic debate are resolved by simply not using the mere fact one’s own existence to make inferences about the nature of your reference class. It’s entirely possible to make all the conclusions one would actually need to do anything based on other data.
Keep in mind that, "maximal possible number" does not necessarily mean that every possible human exists. But it would mean that the same cardinality of humans exists as of possible humans. Which would be the same number as the maximal possible number. (At least, I think that would be what the argument would say.) If, say, every other possible person existed, then I'd imagine the anthropic argument would only suggest a 2:1 amount of evidence in favor of the larger universe, not the infinite ratio that it argues for when compared to a finite number. 2:1 can easily be overcome by other reasons or sources of evidence.
That said, it's still murky to me how infinities interact with probability theory. Surreals and hyperreals can form nice fields that we can do addition and multiplication on, but those are only ordinal infinities, not cardinal infinities, and we need cardinal infinities here.
Your last paragraph amounts to asking us to ignore facts about the world. You can do so, of course, but dismissing evidence out of hand is not good for truth-seeking.
Want to say: I'm very appreciative of you actually understanding the argument and raising serious objections to it. I can't tell you the number of people (e.g. Carrier) who are confident it's wrong but then raise super lame objections.
Which bit of the math do you dispute? Like, the only thing needed for the anthropic argument to get off the ground is that if your existence is likelier if there are more people, you should think the number of people that exists is the most there could be. That seems pretty straightforward!
I wasn't trying to address objections to it (but thank you for the kind remark), though, I suppose it was an objection. I was mostly trying to address Vikram. The first paragraph is not an objection, but just clarifying what the result of the argument might be.
"Like, the only thing needed for the anthropic argument to get off the ground is that if your existence is likelier if there are more people, you should think the number of people that exists is the most there could be."
I don't think that's quite the right principle, though it might be true: the more straightforward one is that, if SIA is true, you should expect that the universe has per some objective measure, a non-infinitesimal fraction of all possible people. In practice, that's the same number, because of how cardinal infinities work, so what you said is technically true, but it doesn't necessitate that literally every possible person exist.
This is obviously true in that you shouldn't have an appreciable difference in probability between a multiverse where every possible person exists, versus one where all but one exists, at least, evidentially. (Perhaps the one where every person exists might have a greater likelihood on prior probability.)
My comment regarding cardinal utilities was just what I said, that it's kind of murky to me, and I agree that handling infinites is a problem that we all have to deal with to some extent. I think we're likely to end up with Pascal-style infinities in utility theory that we have to deal with as well. (Or maybe worse—might the utility representation diverge??) Ways to try to avoid such things tend to have other bad effects.
I don't agree that "more straightforward one is that, if SIA is true, you should expect that the universe has per some objective measure, a non-infinitesimal fraction of all possible people."
All you need to do is compare the relative probabilities between theories on which there are the number of possible people and smaller numbers. I agree it's murky how to do the comparisons regarding different numbers of people, each equal to the number of possible people, but I don't think the argument needs to assume anything about how to do that. I elaborate more on this in the last section here
For the purpose of the anthropic argument, this doesn't make a difference, it would still require the same large cardinalities.
But I still think that I'm correct to say that this does not mean that you should be sure that every possible person exists, in that, as I said, the probability that all but one exist should have roughly the same amount of evidence going for it, and you should be able to do a similar sort of thing for any number of the same cardinality, even if it omits many possible people. You would only have infinite evidence towards cardinalities that are smaller.
I don’t know if this was answered in some post or comment but one thing I have trouble understanding with infinites and probabilities is: how to compare probability of infinities with each other.
If something is infinitely likelier because its cardinality is infinitely bigger, then that means that the infinite has probability 1 and the other 0. That’s the only way the math would work.
But then you can again imagine an infinity of people whose cardinality is infinitely more than the first infinite (that list of infinities is never ending by the way). What is then the probability of this? It should be 0 since it’s an exclusive event with the first one but it would be absurd since this set has more people.
I put this comment on one of your earlier posts without response:
Mathematically, you can’t compare probabilities of items drawn from different infinite sets. The no-fair-lottery principle states that you can’t create a uniform probability distribution across an infinite set. Your argument states, more or less:
>> Theism implies that bigger infinite cardinalities of people (e.g. Beth 2) are realized, whereas any “naturalistic” multiverse is presumably only Aleph Null (or some lesser cardinal) in the number of people who exist.
But to turn that into a “factor of advantage” (e.g. “the theistic theory is infinitely more favored”) you need to treat those cardinalities as if “being one of Beth 2 many possible people” is strictly more probable than “being one of Aleph null many possible people,” in some sense. You end up requiring a well‐defined probability measure that says
Probability(Beth 2 cardinal of persons) > Probability(Aleph Null cardinal of persons)
in a way that “weights” your personal existence. But precisely because there is no uniform way to sample from these sets, the measure comparisons become murky or undefined. It runs into the no‐fair‐lottery principle: there is no consistent uniform distribution over “all possible persons” at cardinalities like Beth 2.
I learned about it in a discrete maths class a little while ago and realized that if it is true, it does break the argument. Take a look here https://sites.pitt.edu/~jdnorton/teaching/paradox/chapters/infinite_lottery_chances/infinite_lottery_chances.html#mozTocId16440 for a decent summary, the most important thing to recognize is the additivity paradox. Though he does mention various solutions, and there are many out there, as far as I can tell none of them solve Mr. Bulldog's very specific issue of creating a definition wherein picking something from one infinite set can be more likely than picking from another infinite set.
> Your last paragraph amounts to asking us to ignore facts about the world. You can do so, of course, but dismissing evidence out of hand is not good for truth-seeking.
I don't think I'm telling someone to "ignore" the fact about one's own existence. I am suggesting that one's own existence implies nothing about the size of one's reference class, since every method of using it to divine that fact produces paradoxes with no resolution. There are plenty of facts that don't bear on some questions. Does the fact 123+456 = 579 bear on the size of my reference class? No. Obviously the arguments for your own existence shedding light on the question are more complex, but I submit that their ultimate result is nil.
Asserting that I am "ignoring" a relevant fact begs the question of whether its possible for the fact to bear on the question you want to apply it to. So far, it seems like SIA/SSA is just arbitrarily deciding what paradoxes one can tolerate. I think my solution is superior, since it requires accepting no paradoixes, and wouldn't seem to impact any actual human activity in the usual ways that ignoring evidence harms "truth-seeking".
> Keep in mind that, "maximal possible number" does not necessarily mean that every possible human exists.
I am aware of what Matthew means by "maximum possible number". My question was about one's own existence necessarily meaning that the maximum number of possible people must exist.
> If, say, every other possible person existed, then I'd imagine the anthropic argument would only suggest a 2:1 amount of evidence
I get what you're saying, but this line of argument really just sounds like "2*0 is twice as large is 1*0" to me.
Question for you: How exactly would SIA be falsified? After all, even if we live in a world without God and thus have a small number of living humans, one would be able to use this small set of living humans--*regardless of who they are*--as evidence for the SIA.
Well, SIA says your existence is likelier if more people exist. If the number of people that existed was below N, 0% of possible people would exist, and thus your existence would be infinitely unlikely. At least, that's true if you buy SIA, which you can see the linked post for a defense of.
Even if the number is finite there's a non-zero chance of my existence! Or do you mean "infinitely less likely than if there were infinitely more people"? This wouldn't be unreasonable, though these probabilistic arguments about infinity are potentially dodgy. I'd consider it possible that the prior of it being a certain infinity scales downwards as fast as the infinity scales upwards, or something like this.
Are we sure that it's actually possible for an infinite amount of a real thing to exist? Maybe this is often accepted, I haven't looked into it much. The amount of (functionally-distinct) possible brains is finite, so even large finite numbers can achieve "there are loads of exact replicas of you", and even this might get a bit messy - do two consciousnesses overlap?
Going from aleph0 to aleph1 is also gonna require some weird stuff! I think even an infinite-radius universe couldn't contain an uncountable amount of humans, as "distance" feels intrinsically countable. So does time. Countably many countably big universes for countably long time... still results in countably many humans, or aleph0.
Regarding that last paragraph, I agree with you about the physical parameters of our universe only being able to hold countable things, but I think the relevant question is what's logically possible, instead of what fits within our current model of physics. We don't need to think that all of reality fits within our universe.
"You shouldn’t even trust that the sun will rise tomorrow, for there are just as many people with your exact experiences for whom the sun rises as for whom it doesn’t."
Does this hold? If I draw a real number uniformly from the interval (0, 1), what's the chance its first decimal place is a 3? This is well-defined to have chance 0.1, even though the cardinality of these options is the same as the cardinality of the whole set.
I'll probably need to spend a lot more time reading your posts tbh, bit busy rn to do super deep dives so apologies if I missed something!
Where specifically (and in which article)? I wasn't always sure which ones were reductios from certain positions, and which ones you hold. As it happens, I think I accept SIA (at least, I'm a thirder in the Sleeping Beauty problem, which I'm guessing is roughly equivalent?). I'm curious for your take on the solution to this problem: https://en.wikipedia.org/wiki/Induction_puzzles#Countably_Infinite-Hat_Variant_without_Hearing . We end up with Aleph0 "correct" people and a finite number of "incorrect" people, and yet each person has a 50% chance of being correct!
Anyway while reading ultimate-guide, there's another question/objection I'd like to ask/make. Unless I've misunderstood the argument, I don't see how it shows that the number of truths is more than the size of any set: doing this with conjunctions doesn't seem to work, because (A∧B∧...)∧(C∧D∧...) is the same proposition as (A∧B∧C∧D∧...). If we started from "Beth 10 (non-conjunctive) truths", then yes there'd exist Beth 11 conjunctive truths, each being the conjunction over a subset of the (original) truths. But there's no contradiction here, because if we try to do that again nothing happens (any conjunction of a subset of this set gives a result that's already in the set). And regardless, with any finite-sized brain, surely there's some large finite number n such that "for it to be possible to imagine a truth, there must be some way of conveying it in under n characters/words", in which case "distinct imaginable truths" remains finite.
And about the amount of people, I don't think it's unreasonable for a naturalist to argue that once you account for "tiny differences that almost certainly won't matter" then you can get this number finite. If the universe is quantised, you get this for free. Otherwise, you can get something like "for any c < 1, there exists a large finite number n such that if you created n distinct 'lives', I would have at least c confidence that one of them was identical (or completely-imperceptibly-different) to mine".
And if souls were limited to aleph null, why would it follow that "the number of souls just runs out when you get to aleph null"? Hilbert's hotel can always take more people! Aleph0 + 1 = Aleph0, so you'll never be "out of souls".
Btw I love your stuff, and I agree with lots of what you write, so don't worry about all the criticism lol
(I'm totally on board with "you exist, therefore massively update towards large universe/multiverse", but I'm skeptical that standard Bayesianism (and expected value, and so on) extend nicely to infinities)
If the intermediatory stage in the anthropic argument is that we should think there are Beth 2 'people' and instead of 'people' I said 'homo sapiens', would the argument still be true? If not, what does the word 'people' mean?
If you have already answered these questions, please feel free to simply link.
Hmm. It depends on your priors and how the universe works.
Say there's a 10% chance in one universe.
Double the number of universes and double the number of people: this apparently (almost) doubles your chances but there's also a 1% chance that "you" will come up twice. That makes no sense so the maths can't work like that.
Perhaps awareness arises outside the universe then there's the 50:50 split into which universe and after that the probability is unchanged at 10%.
But then we've added the problem that your awareness is picked from an ?infinite pool of your peers and the chances of YOU being incarnated into one of a finite number of bodies is zero. So that doesn't work either.
Only solution I can see is to recycle a single soul.
I think Carrier’s treatment of your earlier article was unfair on its face (and needlessly mean!).
I did want to nitpick one claim in your article, though - the claim that “if you assign a 0% prior to an infinite multiverse, you have to think that no matter how strong the scientific evidence was for an infinite multiverse, you should always reject it.”
As intuitive as this is, I don’t think probability maps to modality (and in particular, to our beliefs about what is actual) this way. It is entirely possible, even sensible, to accept the actuality of an event with a 0% prior. Zero-probability events are not merely possible, but occur all the time - any actuality taken from an infinite sample space has a hard 0.0% probability of being the case. If you pick a random real number between 0 and 1, that is a zero-probability event. The odds that you are the person you are (rather than any of infinitely many possible alternatives) is a hard 0.0%. The odds that you scratch your nose at the time you do, in the manner you do, rather than in any other possible way or at any other possible time, are strictly 0%. The odds that the gravitational constant is the amount it is, rather than some other amount, are 0%.
My suspicion is that a lot of our commonsense intuitions, when it comes to infinities, and in particular infinite probability spaces, are dangerously misguided. If I have a jar with balls of infinitely many colors, the odds I pull out any given ball (one in infinity - 0 percent) are *the exact same* as the odds I pull it out ten or a hundred times in a row (one in infinity^100 - also 0 percent).
My suspicion is further heightened when we try to do this sort of probabilistic argumentation across even more exotic infinite sample spaces, like the set of all possible worlds, or the set of all possible sets of natural laws. I don’t think Bayesian analysis gives us the sort of rigor we want it to, here - even when it yields intuitive-sounding premises for deductive argumentation.
Suppose I pick a real number between 0 and 1, and I tell you the number I picked is 0.36 + (467^-1000007000) (It is).
This is an infinitely improbable number for me to have picked, and the evidence you have that I picked it doesn’t seem to be “infinitely strong” (though I’m not sure what exactly you mean by that). And yet, couldn’t it be reasonable to conclude that that is in fact the number I picked?
To say that the odds of picking that number are nonzero, we would need to say, “for each world in which Gumphus selected ‘0.36 + (467^-1000007000),’ there are only finitely many possible worlds in which he did not.”
Intuitively, this is not the case, because there are infinitely many numbers between 0 and 1.
We might suspect that short, easy-to-say numbers like “0.12” are, empirically, more likely to get picked than long, eldritch strings. But there’s no way to draw a hard boundary around which numbers people are able to pick, which is what we’d need to do to say that any particular number has a *nonzero* chance of getting picked. To establish this we’d need to say, for instance, that it’s simply impossible for people to pick numbers with 4729264958 or more digits (which is fairly easy to do - just divide any irrational number by any number larger than it, and you get a qualifying answer with infinitely many digits). Or something like that. Until we can, there is no nonzero probability we can possibly assign.
>To say that the odds of picking that number are nonzero, we would need to say, “for each world in which Gumphus selected ‘0.36 + (467^-1000007000),’ there are only finitely many possible worlds in which he did not.”
That's only true if things are of equal probability.
But if you got that number, and are asserting that it was from a truly random distribution, I absolutely should not believe you. There's no way you get an algebraic number (indeed, a rational one!) by chance if they're equally probable.
But wait, now that I'm getting to that last paragraph, you're now talking about people choosing numbers by hand? Yes, we can absolutely draw bounds around it. You're definitely only going to be picking computable numbers, for one thing.
But there's clearly a finite number of numbers you can pick. You have one lifetime to describe a number. That's a finite number of syllables you can say, or characters you can write. There's therefore a finite number of descriptions you can make, and so a finite number of possibilities. There you go, nonzero.
> That's only true if things are of equal probability.
Sort of. If there is some property the number I picked possesses which (A) is necessary to be a possible answer, and (B) is shared by only a finite number of other real numbers between zero and 1, then we can say, yes, this isn't a zero-probability event after all, this number-picking case is a bad example of a zero-probability event, and I should choose another one (perhaps one of the other three I listed in my initial comment).
But it's by no means obvious that such a property exists (a finite number of numbers that can be described in all possible lifetimes? For all possible methods of description and forms of notation? Isn't it impossible to provide even one example of a number that could not be picked, let alone some comprehensive category of non-pickable numbers which provably includes all but a finite set?). And even if one does seem plausible to you, this is intended more as an illustration of my earlier point. Zero-probability events occur. Modal claims like "X is possibly the case" or "X is not possible" are not what probability models. Probability is a model which tells us the extent to which a prediction is justified based on samples from a representative population. If P(X) = 0, that does not mean X is impossible, or that X must not be the case - just that, based on whatever assumptions we used to calculate P(X), there is no reason to predict X. In many non-metaphysical contexts, this distinction is unimportant. (But not all! See, e.g., https://math.stackexchange.com/questions/236998/conditional-probability-and-division-by-zero).
The intuitions we have about probability break down when sampling from infinite sample spaces, including the space of all possible worlds, meaning that *all probabilistic arguments which rest on those intuitions contain a hidden premise - that the intuitions are applicable.* Facts comparable to (A) and (B) in the first paragraph must be demonstrated for any metaphysical argument which draws from probabilistic methods, including the anthropic argument, to establish that the sample spaces at issue are in fact finite and we aren't trying to apply Bayes' theorem to zero-probability events.
I'm imagining Bentham doing the Two Buttons / Daily Struggle meme, where one button is resisting the urge to destroy, and the other button is how utterly terrible and philosophically confused Carrier's response is.
God really gives his hardest battles to his strongest soldiers.
“Oh no! I forgot to check census statistics! I had thought earth had infinite people, rather than 8 billion. How could I have made so foolish an error.” I almost spilled out my iced tea laughing
FWIW, I was the one who shared your post with Richard Carrier, and I have now shared this response of yours with him as well. I'd like to see how exactly he will respond to this.
I guess that my issue with the SIA is that there remains the difficulty of proving it. I mean, I can easily think of many people who don't exist in this universe. Bentham Bulldog's sister, for instance. Where's the evidence that they exist in other universes? One can make conjectures in regards to this, but where specifically is the hard evidence? In this universe, many people exist whose existence is extraordinarily unlikely, but many people whose existence is extraordinarily unlikely don't exist as well.
The evidence would be in your existence. And it isn't necessary that they exist, it works just as well if you only require that there be a number of people of the same cardinality as the number of possible people.
BTW, does multiverse (many-worlds) theory imply God’s existence?
I don't think it necessarily does. You can believe in an atheistic multiverse.
In fact, there are numerous physical theories already used that are a multiverse of sorts. E.g. many physicists prefer the many-worlds theory. Inflationary cosmology does that as well, I think. None of these are as large as the size of multiverse that the argument that Matthew is presenting would ask for.
(Context, about me: I'm a Christian, I find multiverses somewhat repugnant, but also think that SIA is probably a strong argument for them.)
Please pardon my ignorance, but couldn't many-worlds produce an infinitely large multiverse?
I think it might. I'm not familiar enough with this. That said, there are different sizes of infinities, and so it might depend on what size of infinity we need. But that's a good point, I'd have to check, and I'm not sure how to do that most straightforwardly.
I think that there is an extraordinarily massive amount of different permutations of reality in many-worlds theory. For instance, US President James Abram Garfield lives vs. Garfield dies in 1881. Then beyond that: Garfield signs the Chinese Exclusion Act vs. vetoes it in 1882. Garfield wins vs. loses reelection in 1884. Et cetera.
> it works just as well if you only require that there be a number of people of the same cardinality as the number of possible people.
Can you please rephrase it in such a way that a layman such as myself can understand, please?
Sure. Cardinality refers to how many things we are. Things have the same cardinality if we can put them in a one-to-one correspondence. We're used to thinking about finite numbers, where if you add one, it changes the cardinality. But when we're talking about infinite numbers, you can have different sets with the same cardinality. For example, the function f(x)=2x puts the set of integers into a one-to-one correspondence with the even numbers, even though the one is a subset of the other. So they have the same cardinality.
There are different size infinities—some infinites cannot be put in a one-to-one correspondence with each other. What I was saying was that I do not think his anthropic argument would necessarily imply that every possible person exist, but it would suggest that you could make a one-to-one correspondence between existing people and possible people.
Let me know if any part of that is unclear, that's still not quite layman's level, but it's closer.
Wow! I feel like Richard is pretty good on certain points, but that was really embarrassing for him! Especially how confident and dismissive he is while simultaneously giving incredibly misinformed and stupid points, lol
Just a stylistic point: I think it would help if you broke up your sentences more. Having multiple asides in a sentence or nested if clauses can be cognitively taxing for the reader
Carrier is among the worst of the worst when it comes to arguing against theism.
Forgive me if this is a point you have covered before, but you say "Because the number of people that could exist is a very large infinity—far more than the number of numbers—if there’s no God, it’s very unlikely that such a number of people would exist." Why exactly should we expect it to be unlikely for that large infinity of people to exist given atheism? I understand the point you're making that a God who wants to create would create a very large infinity of people, I just don't understand why that many people couldn't also turn up under atheism.
https://benthams.substack.com/p/the-ultimate-guide-to-the-anthropic
You say "To get at least Beth 2 people, one needs an extreme degree of gerrymandering and shenanigans—a natural multiverse doesn’t get Beth 2 people, nor does a universe that’s infinite in size. God can easily make Beth 2 people, but if he doesn’t do that, the odds of there being Beth 2 people are very low." How does God manage to create Beth 2 people within the multiverse without himself having to pull shenanigans, and what if I grant that shenanigans are required but simply assert the truth of atheism + shenanigans?
God is omnipotent. He can do whatever shenanigans he wants. Atheism can invoke shenanigans, but they're low probability given atheism.
I'm fairly sympathetic to this argument. It would seem like modal realism, or a tegmark iv multiverse would get you there, without a God being necessary. Not sure if the typical universe in such situations would look like our world, ours is unexpectedly simple.
Indeed. As I’ve written I think those collapse induction https://benthams.substack.com/p/the-ultimate-guide-to-the-anthropic?utm_source=publication-search
"if a coin gets flipped that creates one person if heads and ten if tails, then if you get created from this process, you should think the coin came up tails at 10:1 odds"
RAA: change the rules to heads: zero people, tails:10. Then when you get created what are the odds of tails? That's simple causality so it's 100%. If you're prepared to allow assymmetry in this case you can't deny it in the weaker case.
> Let N be the size of the collection representing the number of possible people:
> There are at least N people (deduced from the fact that you exist).
Am I silly? How is it possibly the case that one’s own existence allows one to deductively infer that there must be the maximal possible number of people. This seems to completely obviate any distinction between the actual and the possible.
And I will once again say, for posterity, that all the irreconcilable paradoxes on both sides of the anthropic debate are resolved by simply not using the mere fact one’s own existence to make inferences about the nature of your reference class. It’s entirely possible to make all the conclusions one would actually need to do anything based on other data.
Keep in mind that, "maximal possible number" does not necessarily mean that every possible human exists. But it would mean that the same cardinality of humans exists as of possible humans. Which would be the same number as the maximal possible number. (At least, I think that would be what the argument would say.) If, say, every other possible person existed, then I'd imagine the anthropic argument would only suggest a 2:1 amount of evidence in favor of the larger universe, not the infinite ratio that it argues for when compared to a finite number. 2:1 can easily be overcome by other reasons or sources of evidence.
That said, it's still murky to me how infinities interact with probability theory. Surreals and hyperreals can form nice fields that we can do addition and multiplication on, but those are only ordinal infinities, not cardinal infinities, and we need cardinal infinities here.
Your last paragraph amounts to asking us to ignore facts about the world. You can do so, of course, but dismissing evidence out of hand is not good for truth-seeking.
Want to say: I'm very appreciative of you actually understanding the argument and raising serious objections to it. I can't tell you the number of people (e.g. Carrier) who are confident it's wrong but then raise super lame objections.
Which bit of the math do you dispute? Like, the only thing needed for the anthropic argument to get off the ground is that if your existence is likelier if there are more people, you should think the number of people that exists is the most there could be. That seems pretty straightforward!
Of course, there's a further problem of how exactly we do probabilistic reasoning with infinite numbers of people. But that's everyone's problem https://en.wikipedia.org/wiki/Measure_problem_(cosmology)#cite_note-6
I wasn't trying to address objections to it (but thank you for the kind remark), though, I suppose it was an objection. I was mostly trying to address Vikram. The first paragraph is not an objection, but just clarifying what the result of the argument might be.
"Like, the only thing needed for the anthropic argument to get off the ground is that if your existence is likelier if there are more people, you should think the number of people that exists is the most there could be."
I don't think that's quite the right principle, though it might be true: the more straightforward one is that, if SIA is true, you should expect that the universe has per some objective measure, a non-infinitesimal fraction of all possible people. In practice, that's the same number, because of how cardinal infinities work, so what you said is technically true, but it doesn't necessitate that literally every possible person exist.
This is obviously true in that you shouldn't have an appreciable difference in probability between a multiverse where every possible person exists, versus one where all but one exists, at least, evidentially. (Perhaps the one where every person exists might have a greater likelihood on prior probability.)
My comment regarding cardinal utilities was just what I said, that it's kind of murky to me, and I agree that handling infinites is a problem that we all have to deal with to some extent. I think we're likely to end up with Pascal-style infinities in utility theory that we have to deal with as well. (Or maybe worse—might the utility representation diverge??) Ways to try to avoid such things tend to have other bad effects.
Nice, makes sense.
I don't agree that "more straightforward one is that, if SIA is true, you should expect that the universe has per some objective measure, a non-infinitesimal fraction of all possible people."
All you need to do is compare the relative probabilities between theories on which there are the number of possible people and smaller numbers. I agree it's murky how to do the comparisons regarding different numbers of people, each equal to the number of possible people, but I don't think the argument needs to assume anything about how to do that. I elaborate more on this in the last section here
https://benthams.substack.com/p/i-talked-with-the-most-prolific-critic?utm_source=publication-search
see also section 4.4 here https://benthams.substack.com/p/the-ultimate-guide-to-the-anthropic
For the purpose of the anthropic argument, this doesn't make a difference, it would still require the same large cardinalities.
But I still think that I'm correct to say that this does not mean that you should be sure that every possible person exists, in that, as I said, the probability that all but one exist should have roughly the same amount of evidence going for it, and you should be able to do a similar sort of thing for any number of the same cardinality, even if it omits many possible people. You would only have infinite evidence towards cardinalities that are smaller.
Oh yes I agree with that.
I don’t know if this was answered in some post or comment but one thing I have trouble understanding with infinites and probabilities is: how to compare probability of infinities with each other.
If something is infinitely likelier because its cardinality is infinitely bigger, then that means that the infinite has probability 1 and the other 0. That’s the only way the math would work.
But then you can again imagine an infinity of people whose cardinality is infinitely more than the first infinite (that list of infinities is never ending by the way). What is then the probability of this? It should be 0 since it’s an exclusive event with the first one but it would be absurd since this set has more people.
This would be a problem, but there's a maximum infinity--too big to be a set--so that's what you should think is the number of people.
I put this comment on one of your earlier posts without response:
Mathematically, you can’t compare probabilities of items drawn from different infinite sets. The no-fair-lottery principle states that you can’t create a uniform probability distribution across an infinite set. Your argument states, more or less:
>> Theism implies that bigger infinite cardinalities of people (e.g. Beth 2) are realized, whereas any “naturalistic” multiverse is presumably only Aleph Null (or some lesser cardinal) in the number of people who exist.
But to turn that into a “factor of advantage” (e.g. “the theistic theory is infinitely more favored”) you need to treat those cardinalities as if “being one of Beth 2 many possible people” is strictly more probable than “being one of Aleph null many possible people,” in some sense. You end up requiring a well‐defined probability measure that says
Probability(Beth 2 cardinal of persons) > Probability(Aleph Null cardinal of persons)
in a way that “weights” your personal existence. But precisely because there is no uniform way to sample from these sets, the measure comparisons become murky or undefined. It runs into the no‐fair‐lottery principle: there is no consistent uniform distribution over “all possible persons” at cardinalities like Beth 2.
Where should I look to learn more about this?
If you're interested, a more succinct summary comes from this ChatGPT conversation, without all the extra probability theory stuff. https://chatgpt.com/share/67b5cd98-3d24-8012-be57-c98646099227
I learned about it in a discrete maths class a little while ago and realized that if it is true, it does break the argument. Take a look here https://sites.pitt.edu/~jdnorton/teaching/paradox/chapters/infinite_lottery_chances/infinite_lottery_chances.html#mozTocId16440 for a decent summary, the most important thing to recognize is the additivity paradox. Though he does mention various solutions, and there are many out there, as far as I can tell none of them solve Mr. Bulldog's very specific issue of creating a definition wherein picking something from one infinite set can be more likely than picking from another infinite set.
> Your last paragraph amounts to asking us to ignore facts about the world. You can do so, of course, but dismissing evidence out of hand is not good for truth-seeking.
I don't think I'm telling someone to "ignore" the fact about one's own existence. I am suggesting that one's own existence implies nothing about the size of one's reference class, since every method of using it to divine that fact produces paradoxes with no resolution. There are plenty of facts that don't bear on some questions. Does the fact 123+456 = 579 bear on the size of my reference class? No. Obviously the arguments for your own existence shedding light on the question are more complex, but I submit that their ultimate result is nil.
Asserting that I am "ignoring" a relevant fact begs the question of whether its possible for the fact to bear on the question you want to apply it to. So far, it seems like SIA/SSA is just arbitrarily deciding what paradoxes one can tolerate. I think my solution is superior, since it requires accepting no paradoixes, and wouldn't seem to impact any actual human activity in the usual ways that ignoring evidence harms "truth-seeking".
> Keep in mind that, "maximal possible number" does not necessarily mean that every possible human exists.
I am aware of what Matthew means by "maximum possible number". My question was about one's own existence necessarily meaning that the maximum number of possible people must exist.
> If, say, every other possible person existed, then I'd imagine the anthropic argument would only suggest a 2:1 amount of evidence
I get what you're saying, but this line of argument really just sounds like "2*0 is twice as large is 1*0" to me.
Question for you: How exactly would SIA be falsified? After all, even if we live in a world without God and thus have a small number of living humans, one would be able to use this small set of living humans--*regardless of who they are*--as evidence for the SIA.
Absolutely agreed, this kind of math on infinities doesn’t seem to work imo.
See my reply to Mastricht.
Well, SIA says your existence is likelier if more people exist. If the number of people that existed was below N, 0% of possible people would exist, and thus your existence would be infinitely unlikely. At least, that's true if you buy SIA, which you can see the linked post for a defense of.
Even if the number is finite there's a non-zero chance of my existence! Or do you mean "infinitely less likely than if there were infinitely more people"? This wouldn't be unreasonable, though these probabilistic arguments about infinity are potentially dodgy. I'd consider it possible that the prior of it being a certain infinity scales downwards as fast as the infinity scales upwards, or something like this.
Are we sure that it's actually possible for an infinite amount of a real thing to exist? Maybe this is often accepted, I haven't looked into it much. The amount of (functionally-distinct) possible brains is finite, so even large finite numbers can achieve "there are loads of exact replicas of you", and even this might get a bit messy - do two consciousnesses overlap?
Going from aleph0 to aleph1 is also gonna require some weird stuff! I think even an infinite-radius universe couldn't contain an uncountable amount of humans, as "distance" feels intrinsically countable. So does time. Countably many countably big universes for countably long time... still results in countably many humans, or aleph0.
Regarding that last paragraph, I agree with you about the physical parameters of our universe only being able to hold countable things, but I think the relevant question is what's logically possible, instead of what fits within our current model of physics. We don't need to think that all of reality fits within our universe.
Yes as I argue here https://benthams.substack.com/p/against-against-the-infinite
//Even if the number is finite there's a non-zero chance of my existence! //
This is what I deny and what SIAers deny. For the justification, see https://benthams.substack.com/p/the-ultimate-guide-to-the-anthropic
"You shouldn’t even trust that the sun will rise tomorrow, for there are just as many people with your exact experiences for whom the sun rises as for whom it doesn’t."
Does this hold? If I draw a real number uniformly from the interval (0, 1), what's the chance its first decimal place is a 3? This is well-defined to have chance 0.1, even though the cardinality of these options is the same as the cardinality of the whole set.
I'll probably need to spend a lot more time reading your posts tbh, bit busy rn to do super deep dives so apologies if I missed something!
I talk about why this is wrong later in the article.
Where specifically (and in which article)? I wasn't always sure which ones were reductios from certain positions, and which ones you hold. As it happens, I think I accept SIA (at least, I'm a thirder in the Sleeping Beauty problem, which I'm guessing is roughly equivalent?). I'm curious for your take on the solution to this problem: https://en.wikipedia.org/wiki/Induction_puzzles#Countably_Infinite-Hat_Variant_without_Hearing . We end up with Aleph0 "correct" people and a finite number of "incorrect" people, and yet each person has a 50% chance of being correct!
Anyway while reading ultimate-guide, there's another question/objection I'd like to ask/make. Unless I've misunderstood the argument, I don't see how it shows that the number of truths is more than the size of any set: doing this with conjunctions doesn't seem to work, because (A∧B∧...)∧(C∧D∧...) is the same proposition as (A∧B∧C∧D∧...). If we started from "Beth 10 (non-conjunctive) truths", then yes there'd exist Beth 11 conjunctive truths, each being the conjunction over a subset of the (original) truths. But there's no contradiction here, because if we try to do that again nothing happens (any conjunction of a subset of this set gives a result that's already in the set). And regardless, with any finite-sized brain, surely there's some large finite number n such that "for it to be possible to imagine a truth, there must be some way of conveying it in under n characters/words", in which case "distinct imaginable truths" remains finite.
And about the amount of people, I don't think it's unreasonable for a naturalist to argue that once you account for "tiny differences that almost certainly won't matter" then you can get this number finite. If the universe is quantised, you get this for free. Otherwise, you can get something like "for any c < 1, there exists a large finite number n such that if you created n distinct 'lives', I would have at least c confidence that one of them was identical (or completely-imperceptibly-different) to mine".
And if souls were limited to aleph null, why would it follow that "the number of souls just runs out when you get to aleph null"? Hilbert's hotel can always take more people! Aleph0 + 1 = Aleph0, so you'll never be "out of souls".
Btw I love your stuff, and I agree with lots of what you write, so don't worry about all the criticism lol
(I'm totally on board with "you exist, therefore massively update towards large universe/multiverse", but I'm skeptical that standard Bayesianism (and expected value, and so on) extend nicely to infinities)
Isn't he just assuming SIA? Then if a possible N people exist, and you exist, N people existing is most likely.
I’m confused. What’s the difference between this argument and the anthropic argument against fine tuning?
If the intermediatory stage in the anthropic argument is that we should think there are Beth 2 'people' and instead of 'people' I said 'homo sapiens', would the argument still be true? If not, what does the word 'people' mean?
If you have already answered these questions, please feel free to simply link.
See section 1 https://benthams.substack.com/p/the-ultimate-guide-to-the-anthropic. What SIA tells you to care about is the number o people that you might presently be.
Hmm. It depends on your priors and how the universe works.
Say there's a 10% chance in one universe.
Double the number of universes and double the number of people: this apparently (almost) doubles your chances but there's also a 1% chance that "you" will come up twice. That makes no sense so the maths can't work like that.
Perhaps awareness arises outside the universe then there's the 50:50 split into which universe and after that the probability is unchanged at 10%.
But then we've added the problem that your awareness is picked from an ?infinite pool of your peers and the chances of YOU being incarnated into one of a finite number of bodies is zero. So that doesn't work either.
Only solution I can see is to recycle a single soul.
I think Carrier’s treatment of your earlier article was unfair on its face (and needlessly mean!).
I did want to nitpick one claim in your article, though - the claim that “if you assign a 0% prior to an infinite multiverse, you have to think that no matter how strong the scientific evidence was for an infinite multiverse, you should always reject it.”
As intuitive as this is, I don’t think probability maps to modality (and in particular, to our beliefs about what is actual) this way. It is entirely possible, even sensible, to accept the actuality of an event with a 0% prior. Zero-probability events are not merely possible, but occur all the time - any actuality taken from an infinite sample space has a hard 0.0% probability of being the case. If you pick a random real number between 0 and 1, that is a zero-probability event. The odds that you are the person you are (rather than any of infinitely many possible alternatives) is a hard 0.0%. The odds that you scratch your nose at the time you do, in the manner you do, rather than in any other possible way or at any other possible time, are strictly 0%. The odds that the gravitational constant is the amount it is, rather than some other amount, are 0%.
My suspicion is that a lot of our commonsense intuitions, when it comes to infinities, and in particular infinite probability spaces, are dangerously misguided. If I have a jar with balls of infinitely many colors, the odds I pull out any given ball (one in infinity - 0 percent) are *the exact same* as the odds I pull it out ten or a hundred times in a row (one in infinity^100 - also 0 percent).
My suspicion is further heightened when we try to do this sort of probabilistic argumentation across even more exotic infinite sample spaces, like the set of all possible worlds, or the set of all possible sets of natural laws. I don’t think Bayesian analysis gives us the sort of rigor we want it to, here - even when it yields intuitive-sounding premises for deductive argumentation.
I wasn't assuming probability maps to modality. But if something has a prior of zero, to conclude it happened, you need infinitely strong evidence.
Why might that be the case?
Suppose I pick a real number between 0 and 1, and I tell you the number I picked is 0.36 + (467^-1000007000) (It is).
This is an infinitely improbable number for me to have picked, and the evidence you have that I picked it doesn’t seem to be “infinitely strong” (though I’m not sure what exactly you mean by that). And yet, couldn’t it be reasonable to conclude that that is in fact the number I picked?
Why would that be infinitely improbable? Are we assuming that you have a machine that can just give you random real numbers?
(If so, we should be absolutely shocked to have gotten something algebraic, out of a random selection. That should have a probability of 0.)
To say that the odds of picking that number are nonzero, we would need to say, “for each world in which Gumphus selected ‘0.36 + (467^-1000007000),’ there are only finitely many possible worlds in which he did not.”
Intuitively, this is not the case, because there are infinitely many numbers between 0 and 1.
We might suspect that short, easy-to-say numbers like “0.12” are, empirically, more likely to get picked than long, eldritch strings. But there’s no way to draw a hard boundary around which numbers people are able to pick, which is what we’d need to do to say that any particular number has a *nonzero* chance of getting picked. To establish this we’d need to say, for instance, that it’s simply impossible for people to pick numbers with 4729264958 or more digits (which is fairly easy to do - just divide any irrational number by any number larger than it, and you get a qualifying answer with infinitely many digits). Or something like that. Until we can, there is no nonzero probability we can possibly assign.
>To say that the odds of picking that number are nonzero, we would need to say, “for each world in which Gumphus selected ‘0.36 + (467^-1000007000),’ there are only finitely many possible worlds in which he did not.”
That's only true if things are of equal probability.
But if you got that number, and are asserting that it was from a truly random distribution, I absolutely should not believe you. There's no way you get an algebraic number (indeed, a rational one!) by chance if they're equally probable.
But wait, now that I'm getting to that last paragraph, you're now talking about people choosing numbers by hand? Yes, we can absolutely draw bounds around it. You're definitely only going to be picking computable numbers, for one thing.
But there's clearly a finite number of numbers you can pick. You have one lifetime to describe a number. That's a finite number of syllables you can say, or characters you can write. There's therefore a finite number of descriptions you can make, and so a finite number of possibilities. There you go, nonzero.
> That's only true if things are of equal probability.
Sort of. If there is some property the number I picked possesses which (A) is necessary to be a possible answer, and (B) is shared by only a finite number of other real numbers between zero and 1, then we can say, yes, this isn't a zero-probability event after all, this number-picking case is a bad example of a zero-probability event, and I should choose another one (perhaps one of the other three I listed in my initial comment).
But it's by no means obvious that such a property exists (a finite number of numbers that can be described in all possible lifetimes? For all possible methods of description and forms of notation? Isn't it impossible to provide even one example of a number that could not be picked, let alone some comprehensive category of non-pickable numbers which provably includes all but a finite set?). And even if one does seem plausible to you, this is intended more as an illustration of my earlier point. Zero-probability events occur. Modal claims like "X is possibly the case" or "X is not possible" are not what probability models. Probability is a model which tells us the extent to which a prediction is justified based on samples from a representative population. If P(X) = 0, that does not mean X is impossible, or that X must not be the case - just that, based on whatever assumptions we used to calculate P(X), there is no reason to predict X. In many non-metaphysical contexts, this distinction is unimportant. (But not all! See, e.g., https://math.stackexchange.com/questions/236998/conditional-probability-and-division-by-zero).
The intuitions we have about probability break down when sampling from infinite sample spaces, including the space of all possible worlds, meaning that *all probabilistic arguments which rest on those intuitions contain a hidden premise - that the intuitions are applicable.* Facts comparable to (A) and (B) in the first paragraph must be demonstrated for any metaphysical argument which draws from probabilistic methods, including the anthropic argument, to establish that the sample spaces at issue are in fact finite and we aren't trying to apply Bayes' theorem to zero-probability events.
The evidence *is* infinitely strong
an infinitely improbable number has an infinite no of digits (and can't be compresssed into any arithmetic shorthand), so you will never pick it.