>First of all, I’m optimistic that there will be some nice way to do the math surrounding probabilistic reasoning involving bigger infinites.
I think this optimism is misguided. Mathematicians have devoted an unfathomable amount of effort over the past century to rigorously formalizing probability theory, and none of it really comes close to working at the level of generality you seem to want and/or need it to work at. In fact, there's a certain sense in which the entire story of the development of probability theory since Kolmogorov is one of mathematicians collectively realizing "hey, we can't naively apply these intuitive probabilistic concepts to situations willy-nilly, but instead need to very tightly delimit the places where they have some hope of working out, and figure out which other places exhibit behavior too pathological for us to say anything meaningful about." For example, all the early 20th century work that went into distinguishing between measurable and non-measurable sets, all the precise and non-trivial conditions sufficient for us to get "disintegration theorems" that let us conditionalize on events of probability zero, all the non-standard analysis research into Loeb measure-based probabilities on which we can incorporate infinitesimals but only in *very* specific and limited ways (see internal vs. external sequences and so forth), etc. There are many others.
None of this is a disproof, of course. Maybe some probabilist will come along tomorrow and present a revolutionary new theory that's vastly more applicable somehow, even to all the exotic, low-structure probability thought experiments that philosophers want to run - and even though there's always been a ton of incentive to do this among thousands of the world's smartest researchers who would be motivated to do exactly this, and none of whom would have needed CERN-level funding to carry it out if were indeed possible. I just think the appropriate emotion here is pessimism rather than optimism.
> It’s already possible with some small infinites—for example, there’s a coherent sense in which there are more odd numbers than primes, in the sense that they have greater density.
There's a bunch of problems with this. First, density isn't really an intrinsic property of a set of objects by itself, even a merely countably infinite set. You also need to impose an ordering. In the case of natural numbers, there is a "natural" ordering by size. It wouldn't be difficult to rearrange these numbers so that the density of primes on this new order goes to 1, or even any other number between 0 and 1, or even be undefined. Now, you can (and many would) argue that the natural order on N should have special epistemic privilege or something over any more gerrymandered-seeming order, and that's fine, but the issue is that in thought experiments, there need not be any obvious/natural order at all. If God just tells you he's created a countably infinite number of epistemic duplicates of you such that X and a countably infinite number of other duplicates such that ~X, how are you going to come up with a density of X based on that? And, more importantly, *we are in fact always in this situation with respect to almost everything*, since there's always a small positive subjective probability that this situation I just named is actual!
The other issue is that densities arguably aren't really probabilities, even presupposing a privileged order. They violate countable additivity - which, OK, maybe you're willing to give up - but more importantly, they frequently fail to be defined for even basic questions. For example, on the natural order, what is the density of positive integers which (in base-10) begin with the digit "1?" The answer is that there isn't one. It keeps going up and down between different numbers (IIRC ~11.1% and ~55.5%) as you increase the maximum cutoff of positive integers you're looking at.
Maybe your answer to this is to just say we simply need to hold our tongues in instances like this where the density approach doesn't turn up a unique, well-defined answer, but my point is that this is almost always going to be the case in real life as opposed to highly artificial toy scenarios in which we stipulate away every source of mathematical inconvenience!
There were more things I wanted to object to, but this comment is probably already too long.
I think I'm most interested in what you described in the first section—how probability theory does or doesn't work with less typical structures and sets—but any of it, really.
Good comment. One small observation is that the construction of non-measurable subsets of the reals, requires the Axiom of Choice.
So if you want to keep the idea that it makes sense to uniformly and randomly select a real number between 0 and 1 (i.e. that you can flip a coin Aleph_0 times), then one option is to deny the Axiom of Choice. Personally I'd prefer to do this.
I think the proof of non-measurable sets also requires the measure to be countably additive---which our host also seems to be denying? In that regard, your observations that the standard (fnitely additive) measure over N is incomplete, and that no measure can be invariant under permutations, seem quite important! Especially since he is presumably implicitly appealing to some restricted notion of permutation invariance, in his argument that I shouldn't assign lesser probability to people in greater numbered rooms.
I don't know what the permutation invariance is, so I certainly wasn't implicitly appealing to it. I don't really follow the math behind this, I suppose my general thought is that "maybe some smart people will find some hard-to-predict way to solve them that clears up the problems."
I don't think the existence of non-Lebesgue-measurable sets is the problem here. All the philosophical difficulties in this domain are going to show up for fair lotteries over N rather than [0,1], and there's no measurability issue with this cardinal.
//Am I crazy, or is this just a blunt appeal to faith? If you're wrong about this argument, then you're not going to get into Heaven. Obviously if you die and wake up in heaven, then it doesn't matter whether hevaen is logically impossible, because you're already there. Yay! //
This was just a poetic way of saying there’s some unknown solution. Don’t reject infinites because a) math needs them b) physics suggests an infinite world c) space is likely infinitely divisible and d) anthropics points to infinite people.
> Don’t reject infinites because a) math needs them
Math doesn't *need* actual infinities. We can somewhat reason about infinities using math but it doesn't say anything about applicability of this reasoning to the real world.
> b) physics suggests an infinite world
> c) space is likely infinitely divisible
I don't think so. Can you show your sources for these two points?
> d) anthropics points to infinite people
Only a particular theory of anthropics. And you can't appeal to it to justify infinities as it requires infinities to be justified in the first place.
>Current observational evidence (WMAP, BOOMERanG, and Planck for example) imply that the observable universe is flat to within a 0.4% margin of error of the curvature density parameter with an unknown global topology. It is currently unknown if the universe is simply connected like euclidean space or multiply connected like a torus. To date, no compelling evidence has been found suggesting the universe has a non-trivial (i.e.; not simply connected) topology, though it has not been ruled out by astronomical observations.
Global topology is unknown, if the universe curves in on itself it's around 250× larger than the observable universe, no decisive evidence for its (in)finitude either way.
>> c) space is likely infinitely divisible
General Relativity posits this, but some alternates to GR don't, and contender successor theories like Loop Quantum Gravity posit discretized spacetime.
Problems with infinities strike me as the biggest sticking point for both the anthropic and fine-tuning arguments as well. Two quick technical points (this builds on Mark's comment above). First, in his 'Infinite Lotteries, Perfectly Thin Darts and Infinitesimals' (2012), Pruss shows that infinitesimal probability assignments in both the countably and uncountably infinite cases lead to pathological measures, which (e.g.) allow for setups where an antecedently improbable falsehood is given a posterior only infinitesimally less than 1 whatever the outcome of the experiment. (See also his 'Infinitesimals Are Too Small for Countably Infinite Fair Lotteries' (2013).) Second, as Mark says, there are orderings of the integers which yield a prime density which tends to 1 -- I just wanted to add to this point that Lewis makes a similar remark (and gives a nice illustration of these orderings) in response to Peter Forrest's charge that modal realism undermines induction (see Plurality of Worlds, section 2.5). (Modal realists might also be relieved to note that Builes's argument for the same claim in his 'Center Indifference and Skepticism' founders on the same problems with infinities. Centre indifference requires a modal realist to impose a uniform measure over a set of uncountably many epistemically possible centred worlds, but (infinitesimals aside) such a measure won't be normalisable.)
As you say, problems with infinities arise everywhere, so this can't be parlayed into an argument against SIA, per se. (It's also clear that this proves too much as an objection against the FTA -- there's surely at least some evidence that would count in favour of theism.) But one might worry that problems with infinities force a retreat from a probabilistic anthropic or fine-tuning argument towards e.g. arguments framed in explanation-theoretic terms, whose evidential force is less clear.
> Again, I don’t know if this is right, but I’m optimistic that when we get to heaven, God will have something convincing to say about how to do infinite anthropic reasoning (if we’re concerned about that at that point). It might be that we’re just thinking about infinites completely wrong and there’s some elegant solution that will make the puzzles evaporate, a bit like Newton’s new paradigm eliminated the puzzles for the theories that came before him, according to which things only exerted force by pushing on each other.
Am I crazy, or is this just a blunt appeal to faith? If you're wrong about this argument, then you're not going to get into Heaven. Obviously if you die and wake up in heaven, then it doesn't matter whether hevaen is logically impossible, because you're already there. Yay!
> It might be that we’re just thinking about infinites completely wrong and there’s some elegant solution that will make the puzzles evaporate, a bit like Newton’s new paradigm eliminated the puzzles for the theories that came before him, according to which things only exerted force by pushing on each other.
The "we might be thinking about everything wrong" can't be selectively applied only to theories you don't agree with. The chances are just as good you're totally wrong about anthropics.
> Given the weirdness surrounding infinity, I don’t think it’s a big problem when a very plausible view implies infinite weirdness. A lot of plausible theories imply weirdness when it comes to the infinite. Given this, I think the most likely possibility is either a) that infinity just sort of breaks the world and makes a bunch of paradoxes where there aren’t really precise facts or b) that we’re thinking about the infinite wrong and there will be some elegant solution that makes the puzzles evaporate or c) somewhere in between. If anthropics was the only thing that got wonky with infinites, that would be one thing, but when everything does, it’s time for a paradigm shift!
Why not just reject infinities, and all those things (which, frankly, all seem pretty odd to me already) on their face. I don't get why saying "well if infinity defeats a theory this takes out a lot of theories" is an argument against infinity defeating your theory. If no one has come up with a good resolution to the problem of infinity, then everyone who adopts any suchg theory has a big problem!
Maybe that would be different if literally every theory had problems just as big as those created by infinity? That's not something you've demonstrated, though.
> This—surprise surprise—favors theism. There’s something a bit suspicious about this if SIA is false—it tells you to adopt a very specific picture of ultimate reality that happens to be independently supported.
I admit that I am not your most intelligent commenter. Someone please explain to me how SIA's picture of reality is independently supported by... the SIA anthropic argument? This is the opposite of independent!
> it’s one thing to say that infinity is a problematic mess to be mostly ignored or figured out later, but it’s quite another thing to say that when your theory tells you that there are infinite people.
I was happy to see you acknowledge this, hoping that you would finally engage with the argument, I've tried to bring your attention to several times. And then got immediately dissapointed because you didn't. You've just repeated once again that other views also have problems with infinities, appealed to pure ungrounded hope that some new math will solve SIA's problem with infinities and then tried to deny the whole premise by saying:
> So while SIA poses some practical difficulties, given that infinite people existing is possible, the fact that SIA instructs you to think those infinite people are actual shouldn’t be an additional problem. If we know that there’s a possible world where X=3, then who cares if the theory tells you that X actually = 3 or just in a possible world—either way, there’d have to be some way to reason about it.
First of all, in this metaphor SIA isn't saying that X=3 in some possible world. SIA is saying that X=3 as a universal property among all worlds and then is unable to reason about it.
Secondly, maybe it's not, in fact, possible that infinite people exist! Then all the problems other theories have with infinities are resolved, but not SIA who is dead certain that infinities are real and can only hope for the salvation from some new math. That's a uniquely bad position for SIA.
> Maybe something similar is possible with God’s worlds—if we imagine that God made the world’s one by one, if for each finite number of world’s he’d make, they’d mostly be inductive, then induction is reliable.
If God makes worlds one by one then you have countable amount of worlds. And if for any finite number of world induction is reliable you will not get a situation where for half the worlds induction is not reliable
> For example, suppose we’re theists in an infinite world, trying to see if we should trust induction. Here’s an argument for why we should: we reason “I’m loved by God, and God wants those he love to generally not have false beliefs. Therefore, I should trust that I have mostly true beliefs, and that my mind is not totally fault—that induction works.” Even if the number of people with reliable reasoning equals the number with unreliable reasoning…so what?
If the implication "loved by God" -> "doesn't have false beliefs" is correct, which is itself unobvious, considering that God created our universe to resemble naturalistic one, then it would mean that God doesn not love the other half of the people. But then how do you know which half of the people you are in? Whether you are loved by God or not?
> Fourth, it might be that every being in the universe has generally reliable reasoning. This is what we should expect if there is a God, for he has no reason to make anyone generally deceived.
Isn't God supposed to create all possible people because existence is better than non-existence? That includes people who reason incorrectly. And as there are much more ways to reason incorrectly than reason correctly... it seems that most people are reasoning incorrectly at least to some degree. Which seems about right if you look at people in our world. For instance, you are very confident that SIA is correct, while I'm very confident that SIA is wrong. Someone of us is definetely mistaken!
> I do not know what to say about anthropic reasoning with the infinite. I admit that it’s incredibly weird.
Then... shouldn't you abstain from reasoning about infinities via anthropics?
In general, is your commitment to SIA even falsifiable? At this point, you know about a Dutch book against thirdism, several counterintuitive results that SIA produces, and the fact that we need some completely new math to make SIA work in the case it assumes to be most probable. What else should happen so that you changed your mind?
My take is that it proves that no uniform distribution over the natural numbers exist. So a priori to using SIA you should already have a decreasing probability with regards to the number of people, at least at the limit (as in, your probability could vary at small numbers, but should converge to zero). Then you use SIA and update those numbers, but multiplying a number that tends to zero to a number that tends to infinity is undecidable, so you don't get an answer of which numbers of people are more likely. So this is not a problem of SIA, it's a problem prior to SIA.
Well, you should! Because the only way I know of to resolve this paradox is to insist that all good probability distributions should be normalizable.
Your argument against my approach to anthropics was that: "If your betting advice causes 100% of people to lose their bets, then it’s bad advice!" But note that, in infinite cases, 100% of people losing their bets (according to your non-normalizable measure on N) is still compatible with a finite number of people winning their bets.
On your approach, something worse happens, which is that I can arrange a series of bets where ALL people lose money. Now ALL > 100%, as stated above.
Here is an example of such a bet. A devil comes to each person and offers them a bet, where the devil pays them $1, but if their room turns out to be #1 they have to pay back $10. Since the probability of being in room 1 is zero, they accept that bet. Then the devil offers them $.50, but if their room is #2 they have to pay back $10. Then $.25 that their room is not #3, etc., with the gains halving each time while the losses stay the same. The amount you gain converges to $2, but the amount you eventually pay is $10. So eventually, everyone loses $8.
Here is another example. Suppose the rooms are labelled by two natural numbers (n,m), with 1 person in each room. The devil comes to each person and offers to have them bet a large sum at 10:1 odds in their favor concerning which of their n and m is larger. (You can pick which one to bet on, and ties resolve in your favor). Since you have no information about which is bigger, you should always take this bet. (If necessary, you can flip a coin to decide which way to bet.) But then, the devil reveals to you the value of whichever number you bet was larger. At this point, you always say "OH NO, with probability 1, I am going to lose my bet!" Kindly, the devil now offers you the chance to cancel your bet, before learning the other number. For a very small fee, of course. You gladly take this offer. In the end, every person in all of the rooms pays the devil money.
(Note that in none of these cases is the devil relying on any kind of inside knowledge, or a Hilbert Hotel style transfer of money between people. Since he makes these offers to each person, no matter what.)
These are clever and I'll have to think more about them. Maybe I'd want to say that you should treat a sequence of probabilities that approach zero something like how you'd treat infintesimal probabilities--but it's a bit weird to think each individual bet is good. Also, maybe this scenario is impossible if you're a causal finitist.
In your previous post you shared his article about Quran copying from circulating legends , He didn’t give any reference in the article for his claim , but gave two reference when someone asked him
One is vague , he doesn’t specify which story he is talking about
Second one is “testament of Solomon “ which claims to be written pre Islam but even the link he provided say it is completed in the Middle Ages
Hope you verified his article before sharing it , so pls provide references
>First of all, I’m optimistic that there will be some nice way to do the math surrounding probabilistic reasoning involving bigger infinites.
I think this optimism is misguided. Mathematicians have devoted an unfathomable amount of effort over the past century to rigorously formalizing probability theory, and none of it really comes close to working at the level of generality you seem to want and/or need it to work at. In fact, there's a certain sense in which the entire story of the development of probability theory since Kolmogorov is one of mathematicians collectively realizing "hey, we can't naively apply these intuitive probabilistic concepts to situations willy-nilly, but instead need to very tightly delimit the places where they have some hope of working out, and figure out which other places exhibit behavior too pathological for us to say anything meaningful about." For example, all the early 20th century work that went into distinguishing between measurable and non-measurable sets, all the precise and non-trivial conditions sufficient for us to get "disintegration theorems" that let us conditionalize on events of probability zero, all the non-standard analysis research into Loeb measure-based probabilities on which we can incorporate infinitesimals but only in *very* specific and limited ways (see internal vs. external sequences and so forth), etc. There are many others.
None of this is a disproof, of course. Maybe some probabilist will come along tomorrow and present a revolutionary new theory that's vastly more applicable somehow, even to all the exotic, low-structure probability thought experiments that philosophers want to run - and even though there's always been a ton of incentive to do this among thousands of the world's smartest researchers who would be motivated to do exactly this, and none of whom would have needed CERN-level funding to carry it out if were indeed possible. I just think the appropriate emotion here is pessimism rather than optimism.
> It’s already possible with some small infinites—for example, there’s a coherent sense in which there are more odd numbers than primes, in the sense that they have greater density.
There's a bunch of problems with this. First, density isn't really an intrinsic property of a set of objects by itself, even a merely countably infinite set. You also need to impose an ordering. In the case of natural numbers, there is a "natural" ordering by size. It wouldn't be difficult to rearrange these numbers so that the density of primes on this new order goes to 1, or even any other number between 0 and 1, or even be undefined. Now, you can (and many would) argue that the natural order on N should have special epistemic privilege or something over any more gerrymandered-seeming order, and that's fine, but the issue is that in thought experiments, there need not be any obvious/natural order at all. If God just tells you he's created a countably infinite number of epistemic duplicates of you such that X and a countably infinite number of other duplicates such that ~X, how are you going to come up with a density of X based on that? And, more importantly, *we are in fact always in this situation with respect to almost everything*, since there's always a small positive subjective probability that this situation I just named is actual!
The other issue is that densities arguably aren't really probabilities, even presupposing a privileged order. They violate countable additivity - which, OK, maybe you're willing to give up - but more importantly, they frequently fail to be defined for even basic questions. For example, on the natural order, what is the density of positive integers which (in base-10) begin with the digit "1?" The answer is that there isn't one. It keeps going up and down between different numbers (IIRC ~11.1% and ~55.5%) as you increase the maximum cutoff of positive integers you're looking at.
Maybe your answer to this is to just say we simply need to hold our tongues in instances like this where the density approach doesn't turn up a unique, well-defined answer, but my point is that this is almost always going to be the case in real life as opposed to highly artificial toy scenarios in which we stipulate away every source of mathematical inconvenience!
There were more things I wanted to object to, but this comment is probably already too long.
Where might I learn more about these things?
Which things specifically?
I think I'm most interested in what you described in the first section—how probability theory does or doesn't work with less typical structures and sets—but any of it, really.
Good comment. One small observation is that the construction of non-measurable subsets of the reals, requires the Axiom of Choice.
So if you want to keep the idea that it makes sense to uniformly and randomly select a real number between 0 and 1 (i.e. that you can flip a coin Aleph_0 times), then one option is to deny the Axiom of Choice. Personally I'd prefer to do this.
I think the proof of non-measurable sets also requires the measure to be countably additive---which our host also seems to be denying? In that regard, your observations that the standard (fnitely additive) measure over N is incomplete, and that no measure can be invariant under permutations, seem quite important! Especially since he is presumably implicitly appealing to some restricted notion of permutation invariance, in his argument that I shouldn't assign lesser probability to people in greater numbered rooms.
I don't know what the permutation invariance is, so I certainly wasn't implicitly appealing to it. I don't really follow the math behind this, I suppose my general thought is that "maybe some smart people will find some hard-to-predict way to solve them that clears up the problems."
I don't think the existence of non-Lebesgue-measurable sets is the problem here. All the philosophical difficulties in this domain are going to show up for fair lotteries over N rather than [0,1], and there's no measurability issue with this cardinal.
//Am I crazy, or is this just a blunt appeal to faith? If you're wrong about this argument, then you're not going to get into Heaven. Obviously if you die and wake up in heaven, then it doesn't matter whether hevaen is logically impossible, because you're already there. Yay! //
This was just a poetic way of saying there’s some unknown solution. Don’t reject infinites because a) math needs them b) physics suggests an infinite world c) space is likely infinitely divisible and d) anthropics points to infinite people.
> Don’t reject infinites because a) math needs them
Math doesn't *need* actual infinities. We can somewhat reason about infinities using math but it doesn't say anything about applicability of this reasoning to the real world.
> b) physics suggests an infinite world
> c) space is likely infinitely divisible
I don't think so. Can you show your sources for these two points?
> d) anthropics points to infinite people
Only a particular theory of anthropics. And you can't appeal to it to justify infinities as it requires infinities to be justified in the first place.
>> b) physics suggests an infinite world
The summary from wikipedia is helpful:
>Current observational evidence (WMAP, BOOMERanG, and Planck for example) imply that the observable universe is flat to within a 0.4% margin of error of the curvature density parameter with an unknown global topology. It is currently unknown if the universe is simply connected like euclidean space or multiply connected like a torus. To date, no compelling evidence has been found suggesting the universe has a non-trivial (i.e.; not simply connected) topology, though it has not been ruled out by astronomical observations.
Global topology is unknown, if the universe curves in on itself it's around 250× larger than the observable universe, no decisive evidence for its (in)finitude either way.
>> c) space is likely infinitely divisible
General Relativity posits this, but some alternates to GR don't, and contender successor theories like Loop Quantum Gravity posit discretized spacetime.
You should write more poems!
Problems with infinities strike me as the biggest sticking point for both the anthropic and fine-tuning arguments as well. Two quick technical points (this builds on Mark's comment above). First, in his 'Infinite Lotteries, Perfectly Thin Darts and Infinitesimals' (2012), Pruss shows that infinitesimal probability assignments in both the countably and uncountably infinite cases lead to pathological measures, which (e.g.) allow for setups where an antecedently improbable falsehood is given a posterior only infinitesimally less than 1 whatever the outcome of the experiment. (See also his 'Infinitesimals Are Too Small for Countably Infinite Fair Lotteries' (2013).) Second, as Mark says, there are orderings of the integers which yield a prime density which tends to 1 -- I just wanted to add to this point that Lewis makes a similar remark (and gives a nice illustration of these orderings) in response to Peter Forrest's charge that modal realism undermines induction (see Plurality of Worlds, section 2.5). (Modal realists might also be relieved to note that Builes's argument for the same claim in his 'Center Indifference and Skepticism' founders on the same problems with infinities. Centre indifference requires a modal realist to impose a uniform measure over a set of uncountably many epistemically possible centred worlds, but (infinitesimals aside) such a measure won't be normalisable.)
As you say, problems with infinities arise everywhere, so this can't be parlayed into an argument against SIA, per se. (It's also clear that this proves too much as an objection against the FTA -- there's surely at least some evidence that would count in favour of theism.) But one might worry that problems with infinities force a retreat from a probabilistic anthropic or fine-tuning argument towards e.g. arguments framed in explanation-theoretic terms, whose evidential force is less clear.
> Again, I don’t know if this is right, but I’m optimistic that when we get to heaven, God will have something convincing to say about how to do infinite anthropic reasoning (if we’re concerned about that at that point). It might be that we’re just thinking about infinites completely wrong and there’s some elegant solution that will make the puzzles evaporate, a bit like Newton’s new paradigm eliminated the puzzles for the theories that came before him, according to which things only exerted force by pushing on each other.
Am I crazy, or is this just a blunt appeal to faith? If you're wrong about this argument, then you're not going to get into Heaven. Obviously if you die and wake up in heaven, then it doesn't matter whether hevaen is logically impossible, because you're already there. Yay!
> It might be that we’re just thinking about infinites completely wrong and there’s some elegant solution that will make the puzzles evaporate, a bit like Newton’s new paradigm eliminated the puzzles for the theories that came before him, according to which things only exerted force by pushing on each other.
The "we might be thinking about everything wrong" can't be selectively applied only to theories you don't agree with. The chances are just as good you're totally wrong about anthropics.
> Given the weirdness surrounding infinity, I don’t think it’s a big problem when a very plausible view implies infinite weirdness. A lot of plausible theories imply weirdness when it comes to the infinite. Given this, I think the most likely possibility is either a) that infinity just sort of breaks the world and makes a bunch of paradoxes where there aren’t really precise facts or b) that we’re thinking about the infinite wrong and there will be some elegant solution that makes the puzzles evaporate or c) somewhere in between. If anthropics was the only thing that got wonky with infinites, that would be one thing, but when everything does, it’s time for a paradigm shift!
Why not just reject infinities, and all those things (which, frankly, all seem pretty odd to me already) on their face. I don't get why saying "well if infinity defeats a theory this takes out a lot of theories" is an argument against infinity defeating your theory. If no one has come up with a good resolution to the problem of infinity, then everyone who adopts any suchg theory has a big problem!
Maybe that would be different if literally every theory had problems just as big as those created by infinity? That's not something you've demonstrated, though.
> This—surprise surprise—favors theism. There’s something a bit suspicious about this if SIA is false—it tells you to adopt a very specific picture of ultimate reality that happens to be independently supported.
I admit that I am not your most intelligent commenter. Someone please explain to me how SIA's picture of reality is independently supported by... the SIA anthropic argument? This is the opposite of independent!
> it’s one thing to say that infinity is a problematic mess to be mostly ignored or figured out later, but it’s quite another thing to say that when your theory tells you that there are infinite people.
I was happy to see you acknowledge this, hoping that you would finally engage with the argument, I've tried to bring your attention to several times. And then got immediately dissapointed because you didn't. You've just repeated once again that other views also have problems with infinities, appealed to pure ungrounded hope that some new math will solve SIA's problem with infinities and then tried to deny the whole premise by saying:
> So while SIA poses some practical difficulties, given that infinite people existing is possible, the fact that SIA instructs you to think those infinite people are actual shouldn’t be an additional problem. If we know that there’s a possible world where X=3, then who cares if the theory tells you that X actually = 3 or just in a possible world—either way, there’d have to be some way to reason about it.
First of all, in this metaphor SIA isn't saying that X=3 in some possible world. SIA is saying that X=3 as a universal property among all worlds and then is unable to reason about it.
Secondly, maybe it's not, in fact, possible that infinite people exist! Then all the problems other theories have with infinities are resolved, but not SIA who is dead certain that infinities are real and can only hope for the salvation from some new math. That's a uniquely bad position for SIA.
> Maybe something similar is possible with God’s worlds—if we imagine that God made the world’s one by one, if for each finite number of world’s he’d make, they’d mostly be inductive, then induction is reliable.
If God makes worlds one by one then you have countable amount of worlds. And if for any finite number of world induction is reliable you will not get a situation where for half the worlds induction is not reliable
> For example, suppose we’re theists in an infinite world, trying to see if we should trust induction. Here’s an argument for why we should: we reason “I’m loved by God, and God wants those he love to generally not have false beliefs. Therefore, I should trust that I have mostly true beliefs, and that my mind is not totally fault—that induction works.” Even if the number of people with reliable reasoning equals the number with unreliable reasoning…so what?
If the implication "loved by God" -> "doesn't have false beliefs" is correct, which is itself unobvious, considering that God created our universe to resemble naturalistic one, then it would mean that God doesn not love the other half of the people. But then how do you know which half of the people you are in? Whether you are loved by God or not?
> Fourth, it might be that every being in the universe has generally reliable reasoning. This is what we should expect if there is a God, for he has no reason to make anyone generally deceived.
Isn't God supposed to create all possible people because existence is better than non-existence? That includes people who reason incorrectly. And as there are much more ways to reason incorrectly than reason correctly... it seems that most people are reasoning incorrectly at least to some degree. Which seems about right if you look at people in our world. For instance, you are very confident that SIA is correct, while I'm very confident that SIA is wrong. Someone of us is definetely mistaken!
> I do not know what to say about anthropic reasoning with the infinite. I admit that it’s incredibly weird.
Then... shouldn't you abstain from reasoning about infinities via anthropics?
In general, is your commitment to SIA even falsifiable? At this point, you know about a Dutch book against thirdism, several counterintuitive results that SIA produces, and the fact that we need some completely new math to make SIA work in the case it assumes to be most probable. What else should happen so that you changed your mind?
This post is Christian, specificllay Byzantine
I think at this point we’ll get AGI before Matthew reads something about the optimality of meta-induction strategy for justifying induction :(
Haha, yeah, I need to get around to doing that! Ugh, there are so many things to read!
Possibly relevant: John Pittard has a paper in which he argues that God would have very good reason to exclude epistemically inhospitable worlds from creation. https://journals.publishing.umich.edu/ergo/article/id/2925/
What's your take on the 2 envelopes problem (https://en.m.wikipedia.org/wiki/Two_envelopes_problem)?
My take is that it proves that no uniform distribution over the natural numbers exist. So a priori to using SIA you should already have a decreasing probability with regards to the number of people, at least at the limit (as in, your probability could vary at small numbers, but should converge to zero). Then you use SIA and update those numbers, but multiplying a number that tends to zero to a number that tends to infinity is undecidable, so you don't get an answer of which numbers of people are more likely. So this is not a problem of SIA, it's a problem prior to SIA.
I haven't thought about it unfortunately.
Well, you should! Because the only way I know of to resolve this paradox is to insist that all good probability distributions should be normalizable.
Your argument against my approach to anthropics was that: "If your betting advice causes 100% of people to lose their bets, then it’s bad advice!" But note that, in infinite cases, 100% of people losing their bets (according to your non-normalizable measure on N) is still compatible with a finite number of people winning their bets.
On your approach, something worse happens, which is that I can arrange a series of bets where ALL people lose money. Now ALL > 100%, as stated above.
Here is an example of such a bet. A devil comes to each person and offers them a bet, where the devil pays them $1, but if their room turns out to be #1 they have to pay back $10. Since the probability of being in room 1 is zero, they accept that bet. Then the devil offers them $.50, but if their room is #2 they have to pay back $10. Then $.25 that their room is not #3, etc., with the gains halving each time while the losses stay the same. The amount you gain converges to $2, but the amount you eventually pay is $10. So eventually, everyone loses $8.
Here is another example. Suppose the rooms are labelled by two natural numbers (n,m), with 1 person in each room. The devil comes to each person and offers to have them bet a large sum at 10:1 odds in their favor concerning which of their n and m is larger. (You can pick which one to bet on, and ties resolve in your favor). Since you have no information about which is bigger, you should always take this bet. (If necessary, you can flip a coin to decide which way to bet.) But then, the devil reveals to you the value of whichever number you bet was larger. At this point, you always say "OH NO, with probability 1, I am going to lose my bet!" Kindly, the devil now offers you the chance to cancel your bet, before learning the other number. For a very small fee, of course. You gladly take this offer. In the end, every person in all of the rooms pays the devil money.
(Note that in none of these cases is the devil relying on any kind of inside knowledge, or a Hilbert Hotel style transfer of money between people. Since he makes these offers to each person, no matter what.)
These are clever and I'll have to think more about them. Maybe I'd want to say that you should treat a sequence of probabilities that approach zero something like how you'd treat infintesimal probabilities--but it's a bit weird to think each individual bet is good. Also, maybe this scenario is impossible if you're a causal finitist.
In your previous post you shared his article about Quran copying from circulating legends , He didn’t give any reference in the article for his claim , but gave two reference when someone asked him
One is vague , he doesn’t specify which story he is talking about
Second one is “testament of Solomon “ which claims to be written pre Islam but even the link he provided say it is completed in the Middle Ages
Hope you verified his article before sharing it , so pls provide references