Great post. I think substack is great because it allows everyone to write high quality effort post reflecting their own experience and expertise that I otherwise wouldn’t have access to! I’ll admit I only understood about 40% of this, but I still greatly appreciated it
Good post. Typing on phone. Inb4 he just gerrymanders the "illuminated region" and also he claims priors dont matter even though there is some number so low a posterior wont overcome it. Finally, his specification of simplicity corresponding to prior assignements is completely cooked up - he frames it as if there is such a thing as an ideal language where the complexity of a causal model corresponds to the linguistic complexity and he knows God is simple in that language (wtf). Finally, there are deep questions about the use of Bayes thm for theories of everything and whether it even makes sense in non-decision-theoretic contexts. Theres more that can be said. As I say great post, but he has little dialogue trees for lots of the things you say that bottom out in nothing and then he will go off and like stop engaging with you and post about eugenics or something instead before circling back and making the same claims you objected to.
Thank you for offering a thoughtful, good-faith criticism of Bentham Bulldog use of Bayes rule—it shares one of my concerns about his probabilistic arguments: that they may not be well-defined. Discussing mathematics in natural language carries real risks of ambiguity and misinterpretation.
It looks very relevant — something to read on the plane today, perhaps.
Edit: after a quick skim, yes it is indeed relevant, I need to think about this more but my core objection would not be to any of the examples, they’re fine and I think demonstrates the power of the premise, but with extrapolating that intuition built on “you are given precisely two scenarios” to more complex arguments where you are NOT given some scenarios by fiat. Bentham’s Bulldog raised much the same point about my lack of thought concerning SIA, so I’ll need to think about this a bit more.
Excellent piece! I’ve also tried to take a more mathematical approach to responding to Bentham (shameless plug below [1]), but I was talking more about measures of complexity and tried to be less technical than you. And of course I also glossed over certain measure problems in my not-fully rigorous argument.
Anyway, your point about the monster group was brilliant. I hadn’t heard that one before.
As for SIA, the problems run a lot deeper than just an ill-defined measure. If you’re applying SIA specifically to the question “does God exist?” then of course you hit a measure problem because as you pointed out it’s impossible to ask that question probabilistically. But even if you make toy examples where the probability spaces are entirely well-defined, the SIA is simply incorrect, and you can see this pretty easily if you actually try to write out the relevant equations. I have a longer piece I’m writing about it which should be ready in a few weeks if you’re interested. (I’m a slow writer and I have another piece I’m planning to put out first).
Also, it’s a minor point, but you mention that God could not select from a non-measurable set, but this isn’t true. God would, presumably, act non-randomly, and one can definitely non-randomly select from a non-measurable set (for instance, selecting 0 from the Cantor set). Since God is presumably defined as omnipotent, omniscient, and wanting to create life, P(fine-tuning | God) would be 1. Obviously P(fine-tuning | not God) is still completely undefined, so it’s not like this changes anything.
Oh wait I’ve read that piece! It’s really excellent and I fully agree, except for the flaw that I think someone in the comments pointed out, that some of the computability theorems you relied upon are classical, right? While true, I’m certain an equivalent argument exists for a fundamentally quantum universe that demonstrates effectively the same thing (how can God be anything but extremely K-complex?), so I don’t think it detracted from the point you made.
As for the SIA I would be really interested to read your piece, because I’d only ever heard of it from Bentham’s Bulldog and (according to him) it implied a bunch of bizarre conclusions, at least one of which seems verifiably false.
I think the argument about God selecting from a set was trying to say that even if God selects non-randomly from a tiny but continuous interval, the a priori probability we should expect of Him choosing any particular value is still zero.
But yes, if pressed I will concede that P(something|specific characteristics of a creator) is obviously theology, could easily be 1, and definitely not math, so I think the argument still works.
Thanks! I can message you the piece when it’s done.
The issue with the approach wasn’t quite that computational theories are classical, because quantum states can be simulated with classical computers given exponential overhead (up to some precision). There are a couple issues, such as that classical computers just can’t deal with infinities - no Turing machine could simulate an infinite universe. Also, if you wanted to include multiverses in your simulation (for instance, to handle fine-tuning) then you would run into exactly the same measure problem you spent this whole post describing.
I also found out afterwards that the famous (and somewhat eccentric) computer science Schmidhuber had explored similar ideas years ago, and proposed some solutions that at least solves the precision issues that some commenters mentioned: https://arxiv.org/abs/quant-ph/0011122
Yes, I do tend to think this. Extrapolating from the natural to the supernatural feels like a category error to me too, but I also really wanted to write a measure theory article, so that’s what I went with.
That's the point though, BB's arguments are explicitly framed in the language of natural science. You have priors, observations to explain, and possible entities producing causal effects. The kind of God he's arguing for is framed as an entity within a network of entities. It's entirely modeled on the natural world.
It needs to be said that this is far from the only possible conception of God. Check out Scott Lipscomb's substack for a completely different alternative (and IMO much better).
I think the issue is that the argument wants to have it both ways: do fine-tuning arguments gain purchase on God? Is the hypothesis of a God who wants to create the universe for life made less likely by the existence of suffering and evil? Often the answer is, "no, there's something special about God that we're not meant to be trying to understand God in this reductive way"--but we don't apply that reasoning when we try to think about fine-tuning of the universe. If God is a hypothesis and we can understand his motivations anthropomorphically, then what else does the hypothesis predict? Do we see observe those predictions fulfilled? Or if not, then why do we make one exception for God's desire to create the universe?
The domain error comes from the the fact that mathematical terminology gives a veneer of respectability to otherwise incorrect arguments (see also: Trump’s tariff equation). BB knows this, and he knows just enough math to put together an argument that sounds convincing to people without any formal math training, but unfortunately he doesn’t know enough math to see why his argument is wrong.
I think the argument was made in good faith, and use of Bayesian probability is an attempt to reason with some discipline. But yes, even the best mathematics cannot bear the burden of "proving" an unlimited mind and goodness exists.
I’ve raised these same concerns in comments before as well, and I think there are even larger concerns.
1) Bertrand’s paradox. Even if you did have a well defined sigma algebra, its not clear how to pick a uniform prior (in the language of your post, you can have the sigma algebra, and the event space, but picking P is not necessarily easy)
2) Universes with different laws of physics, or uncomputable laws of physics, or individual universes which are too large to be measured make assigning a probability space impossible.
3) The space of all *logically possible* universes is way way way way way bigger than universes permissible with ZFC. It is not even clear if it makes sense to apply probability theory, which is built upon ZFC, to the larger universe of *all logically possible universes*. Logical possibility is unfathomably big. Im not even sure if “all logically possible universes” is coherent (modal logic notwithstanding)
I appreciate you writing this. I am vaguely an agnostic atheist, because I don't think that there's a good way for us to know whether God exists with an amount of confidence that would sway me to take one side. Accordingly I've vaguely agreed with "we don't know how to assign probabilities to fundamental physics", but I haven't had the math to back it up (which is funny because I will graduate from UChicago with a physics degree in like a week).
> I agree evil is a problem but I think theists can explain it, and God is just much likelier than a limited God that we should take the probabilistic hit and still believe in it.
and I think this is not a very epistemically honest way of approaching criticism of your argument. "I think someone else can explain it" is not a sufficient answer, and any attempt to deal with the problem of evil is going to result in a million burdensome details (https://www.lesswrong.com/w/burdensome-details) (I hope I'm using that correctly).
Just from looking at the universe, it appears that if god does exist, it is a clockwork universe kind of god. So even if I concede the existence of some noncontingent entity that goes around creating stuff it thinks is interesting, I'm still unconvinced by the "and the most interesting thing is perfectly good" part.
This is an interesting article--basically raising the measure problem--but I find it weirdly triumphalist. In that original article I did not address the measure problem but I have elsewhere (here for example https://benthams.substack.com/p/fine-tuning-made-by-god?utm_source=publication-search). You shouldn't be totally confident that the measure problem wrecks my view if you haven't read what I--or people on my side--say about the measure problem!
I give a lot of more specific objections in the linked post. But crucially we can *obviously* say that things are improbable even from an infinite set. So, for instance, there's an infinite set of things that the nearest aliens might be thinking about presently. An infinite number of them involve tacos. But still, I would guess with high confidence that the nearest alien to me is not thinking about tacos!
First off, thanks so much for reading it! I'm a fan.
I'll definitely need to read your thoughts on the measure problem before thoroughly responding, but "crucially we can *obviously* say that things are improbable even from an infinite set" is not true in a rigorous sense, and that's what I spill so much ink and Latex arguing about.
But to your second point, from a strict physicalist perspective aliens thinking about tacos IS well defined! You just aren’t going to enumerate the space or the sigma-algebra, but huge and uncountable isnt the same as ill-defined.
Formally, to try to define the algebra you would take something like the Borel set over the universe’s phase space, in which some, presumably small subset of those configurations is aliens thinking about tacos. That’s monstrously unlikely AND well defined. The space of all physical theories is NOT well defined, which is the key distinction.
I think you’d have to have e.g. some theory of how multiverses spawn random universes to give that a proper formal treatment.
Bayes rule, for me is an intellectual super power. You clearly know that and so does another excellent writer on Substack Nate Silver. It lets you think probabilisticly about everything from poker to existence of god 😎
Also on the anthropic stuff, I don't really get what your objection is. Like, do you accept SIA? Do you accept that if you accept SIA you should think there's a giant infinity worth of people? Where do you get off the boat? It's all very well to note that I have not done fancy shmancy sigma algebra or whatever, but much less clear where you think the argument goes wrong? I do slightly feel like I'm getting Eulered.
(also just want to say, absolutely love the title!)
It's actually not true that "Even the most generous naturalistic scenarios (infinite multiverse, infinite universe) can only produce aleph-null people - the smallest infinity."
It's actually pretty easy to construct scenarios which produce numbers of people in larger cardinalities. One of the other comments mentions Cosmological Natural Selection. You might think this isn't the case, given that each universe only produces a finite amount of black hole singularities. But actually, after quantum mechanics is evolved for, assuming the many-worlds interpretation, each node in the CNS multiverse tree actually represents a QM multiverse consisting of an infinite number of parallel universes. The result is an infinite fractal tree of universes, where zooming into any node shows an infinite number of outgoing branches leading to nodes which also produce an infinite number of branches.
Notably, this scenario also solves the objection you have with Tegmark about Boltzmann Brains, because those are extremely complex entities which nevertheless do not assist in the production of singularities. They are therefore selected against.
Oh man, I'm so happy you liked the title, and frankly flattered you bothered to read this at all.
I'll have to read your arguments on the measure problem before responding to those, I just want to let you know that I have definitely not thought deeply enough about the SIA and whether it makes sense to me -- its inclusion here was because I came across it, a bunch of arguments about aleph-null numbers of people and infinite probability and it looked like the exact same undefined space to me. I'll get back to you!
Thanks! I did appreciate the article. Way more thoughtful than a lot of what people write in reply to my stuff, so thanks for that!
Out of curiosity: have you read any things that philosophers or physicists who promote fine-tuning have said about the measure problem? Because the article doesn't engage with any of their work. Sorry this is going to sound a bit harsh and I don't quite know a way of saying it in a way that isn't harsh, but if you're going to write a big article about the measure problem, it seems worth reading what those who think the measure problem objection isn't fatal say.
I confess I have not, I have read YOUR article(s) and then the entry on the Stanford Encyclopedia of Philosophy, which I thought was sufficient for a blog post.
I probably ought to read some of the actual technical work, especially after being so politely called out. Alas, your articles are much more engaging than academic literature, hence my lack of desire to go beyond, though, still wrong, I argue ;)
Small quibble: I don't understand your mention of AC. If we take on not-AC we can prove that all sets are measurable.
Bigger quibbles: What do you mean P(God|measurable sets)? The set of measurable sets is not an element of the algebra, it is itself the algebra. You claim there's no natural choice of algebra. The natural choice of algebra is the Borel algebra over a Euclidean representation of the parameter space. Then you say that the measure theorist says the probability of picking any particular value from a continuum is zero. This is wrong. Singletons are Borel in R^n. Zero-inflated priors, for example, are bread and butter Bayesian statistics
It was inevitable that an actual mathematician show up at some point and pick out my errors. I will try my best to respond.
For your axiom of choice quibble, are you referring to this section?
"It’s odd, don’t you think, that if you pick a random subset of the real line, it is guaranteed to be unmeasurable with probability 1? Yet all the sets we use for, well, anything, are always measurable? Is P(God|measurable sets) therefore high? Well, no. This is simply the natural consequence of the axiom of choice."
What I was trying to do was draw an analogy between unintuitive consequences and simpler principles that imply them, and use the axiom of choice, which is fairly simple but has really unintuitive consequences, as an example. I may have chosen a poor one here, is the proof that you're referring to this one? https://en.wikipedia.org/wiki/Solovay_model. You're right that I am implicitly assuming AC here, though choosing or not choosing AC for your universe model is of course a choice. An axiom of axiom of choice?
"What do you mean P(God|measurable sets)?"
I think this is just sloppy notation on my part, the set of measurable sets is absolutely just the algebra -- this should say something more like P(God | the surprising fact that measurable sets exist and physics occurs on them), which is of course completely undefined, and was a rhetorical flourish. This wasn't meant as a formal probabilistic statement itself, and I acknowledge the irony in a post about making imprecise math statements.
"You claim there's no natural choice of algebra. The natural choice of algebra is the Borel algebra over a Euclidean representation of the parameter space."
I'm not 100% sure which part of the article this one refers to. I don't think I say anywhere that conditional on choosing a particular reasonable representation of the parameter space you can't define an algebra... Do you mean the part where I assert "Same problem" in choosing the sigma algebra if your sample space is undefined? Here I think my intent is clear, of course you can't choose a sigma algebra over an undefined space. If your space is defined, sure, but my point there is that it isn't defined, so I think the argument that the algebra is undefined holds, if trivially.
"Then you say that the measure theorist says the probability of picking any particular value from a continuum is zero. This is wrong. Singletons are Borel in R^n. Zero-inflated priors, for example, are bread and butter Bayesian statistics"
Yes, I agree you could construct something like this, a "our universe's constants" inflated prior or something like that. Would you be happier with also imposing an absolute continuity condition on the measure?
Generally, I think a lot (all?) of your points show the peril of trying to essentially write a measure theory textbook as a blog post, I inevitably have some elisions and errors between the concept I'm pointing to and a fully correct definition, which is entirely my fault.
I don't think any of the points you make are fatal to the overall argument, as this is mostly just imprecision on my part, as well as me forgetting about point masses being allowed in Borel sets. There's no good way to justify putting specific point masses in your theory, of course, but you can and it's well defined. This is my bad.
I watched a YouTube video about the Vitali set. If I understand correctly, it can occupy an arbitrarily small interval on the real line. Is that correct? When using the axiom of choice, simply choose a member of the set which happens to fall in that narrow interval.
Yes, this is correct as I understand it. In essence, when you choose a member of each equivalence class over the rationals you can always choose it from some arbitrarily narrow interval by "shifting" by a rational number to be within that interval.
Well said. I wish more creationists understood the limitations of the fine-tuning “argument.” It can be very useful AFTER adopting the premise that a creator exists because it can then suggest things about the nature of that creator. In this case, the problem II authentically becomes problem I as you technically have a set of possibilities. However, you set yourself up for failure the moment you try to use fine-tuning against atheist beliefs.
Finally, there is the none mathematical application of fine-tuning. And that simply is just saying “I see a kind of harmony in the relationship of physical laws. Maybe that shows intention. Cool!”
Great post. I think substack is great because it allows everyone to write high quality effort post reflecting their own experience and expertise that I otherwise wouldn’t have access to! I’ll admit I only understood about 40% of this, but I still greatly appreciated it
Yeah I don't know what the hell any of this means but I'm glad Substack exists as a place for it to exist and actually get noticed.
Good post. Typing on phone. Inb4 he just gerrymanders the "illuminated region" and also he claims priors dont matter even though there is some number so low a posterior wont overcome it. Finally, his specification of simplicity corresponding to prior assignements is completely cooked up - he frames it as if there is such a thing as an ideal language where the complexity of a causal model corresponds to the linguistic complexity and he knows God is simple in that language (wtf). Finally, there are deep questions about the use of Bayes thm for theories of everything and whether it even makes sense in non-decision-theoretic contexts. Theres more that can be said. As I say great post, but he has little dialogue trees for lots of the things you say that bottom out in nothing and then he will go off and like stop engaging with you and post about eugenics or something instead before circling back and making the same claims you objected to.
Thank you for offering a thoughtful, good-faith criticism of Bentham Bulldog use of Bayes rule—it shares one of my concerns about his probabilistic arguments: that they may not be well-defined. Discussing mathematics in natural language carries real risks of ambiguity and misinterpretation.
I believe this post could also be relevant to the discussion: https://mon0.substack.com/p/benthams-bulldog-ruins-my-weekend
It looks very relevant — something to read on the plane today, perhaps.
Edit: after a quick skim, yes it is indeed relevant, I need to think about this more but my core objection would not be to any of the examples, they’re fine and I think demonstrates the power of the premise, but with extrapolating that intuition built on “you are given precisely two scenarios” to more complex arguments where you are NOT given some scenarios by fiat. Bentham’s Bulldog raised much the same point about my lack of thought concerning SIA, so I’ll need to think about this a bit more.
If you’re interested, I just made a comment on that post explaining where the math goes wrong in the first pro-SIA example on that article.
On BB’s SIA article?
On mon0’s article.
Huh, that’s weird I checked and didnt see it. The comments icon claims 12 comments, but I count only 11. Odd. I’ll check again later.
Very weird. Anyway I think the comment is here: https://mon0.substack.com/p/benthams-bulldog-ruins-my-weekend/comment/121824242
Excellent piece! I’ve also tried to take a more mathematical approach to responding to Bentham (shameless plug below [1]), but I was talking more about measures of complexity and tried to be less technical than you. And of course I also glossed over certain measure problems in my not-fully rigorous argument.
Anyway, your point about the monster group was brilliant. I hadn’t heard that one before.
As for SIA, the problems run a lot deeper than just an ill-defined measure. If you’re applying SIA specifically to the question “does God exist?” then of course you hit a measure problem because as you pointed out it’s impossible to ask that question probabilistically. But even if you make toy examples where the probability spaces are entirely well-defined, the SIA is simply incorrect, and you can see this pretty easily if you actually try to write out the relevant equations. I have a longer piece I’m writing about it which should be ready in a few weeks if you’re interested. (I’m a slow writer and I have another piece I’m planning to put out first).
Also, it’s a minor point, but you mention that God could not select from a non-measurable set, but this isn’t true. God would, presumably, act non-randomly, and one can definitely non-randomly select from a non-measurable set (for instance, selecting 0 from the Cantor set). Since God is presumably defined as omnipotent, omniscient, and wanting to create life, P(fine-tuning | God) would be 1. Obviously P(fine-tuning | not God) is still completely undefined, so it’s not like this changes anything.
[1] https://open.substack.com/pub/onemanynone/p/benthams-bulldog-doesnt-understand?r=7wyej&utm_campaign=post&utm_medium=web&showWelcomeOnShare=false
Oh wait I’ve read that piece! It’s really excellent and I fully agree, except for the flaw that I think someone in the comments pointed out, that some of the computability theorems you relied upon are classical, right? While true, I’m certain an equivalent argument exists for a fundamentally quantum universe that demonstrates effectively the same thing (how can God be anything but extremely K-complex?), so I don’t think it detracted from the point you made.
As for the SIA I would be really interested to read your piece, because I’d only ever heard of it from Bentham’s Bulldog and (according to him) it implied a bunch of bizarre conclusions, at least one of which seems verifiably false.
I think the argument about God selecting from a set was trying to say that even if God selects non-randomly from a tiny but continuous interval, the a priori probability we should expect of Him choosing any particular value is still zero.
But yes, if pressed I will concede that P(something|specific characteristics of a creator) is obviously theology, could easily be 1, and definitely not math, so I think the argument still works.
Thanks! I can message you the piece when it’s done.
The issue with the approach wasn’t quite that computational theories are classical, because quantum states can be simulated with classical computers given exponential overhead (up to some precision). There are a couple issues, such as that classical computers just can’t deal with infinities - no Turing machine could simulate an infinite universe. Also, if you wanted to include multiverses in your simulation (for instance, to handle fine-tuning) then you would run into exactly the same measure problem you spent this whole post describing.
I also found out afterwards that the famous (and somewhat eccentric) computer science Schmidhuber had explored similar ideas years ago, and proposed some solutions that at least solves the precision issues that some commenters mentioned: https://arxiv.org/abs/quant-ph/0011122
Oh really? That is actually even more interesting, I’ll have to look at the paper. Please do message me the piece!
Isn't there also something more fundamentally off? Like a domain error?
It's like trying to use mathematics to prove or disprove that a parent loves their child.
Even Gödel's rigorous system is not sound.
Yes, I do tend to think this. Extrapolating from the natural to the supernatural feels like a category error to me too, but I also really wanted to write a measure theory article, so that’s what I went with.
That's the point though, BB's arguments are explicitly framed in the language of natural science. You have priors, observations to explain, and possible entities producing causal effects. The kind of God he's arguing for is framed as an entity within a network of entities. It's entirely modeled on the natural world.
It needs to be said that this is far from the only possible conception of God. Check out Scott Lipscomb's substack for a completely different alternative (and IMO much better).
I think the issue is that the argument wants to have it both ways: do fine-tuning arguments gain purchase on God? Is the hypothesis of a God who wants to create the universe for life made less likely by the existence of suffering and evil? Often the answer is, "no, there's something special about God that we're not meant to be trying to understand God in this reductive way"--but we don't apply that reasoning when we try to think about fine-tuning of the universe. If God is a hypothesis and we can understand his motivations anthropomorphically, then what else does the hypothesis predict? Do we see observe those predictions fulfilled? Or if not, then why do we make one exception for God's desire to create the universe?
The domain error comes from the the fact that mathematical terminology gives a veneer of respectability to otherwise incorrect arguments (see also: Trump’s tariff equation). BB knows this, and he knows just enough math to put together an argument that sounds convincing to people without any formal math training, but unfortunately he doesn’t know enough math to see why his argument is wrong.
I think the argument was made in good faith, and use of Bayesian probability is an attempt to reason with some discipline. But yes, even the best mathematics cannot bear the burden of "proving" an unlimited mind and goodness exists.
I’ve raised these same concerns in comments before as well, and I think there are even larger concerns.
1) Bertrand’s paradox. Even if you did have a well defined sigma algebra, its not clear how to pick a uniform prior (in the language of your post, you can have the sigma algebra, and the event space, but picking P is not necessarily easy)
2) Universes with different laws of physics, or uncomputable laws of physics, or individual universes which are too large to be measured make assigning a probability space impossible.
3) The space of all *logically possible* universes is way way way way way bigger than universes permissible with ZFC. It is not even clear if it makes sense to apply probability theory, which is built upon ZFC, to the larger universe of *all logically possible universes*. Logical possibility is unfathomably big. Im not even sure if “all logically possible universes” is coherent (modal logic notwithstanding)
I appreciate you writing this. I am vaguely an agnostic atheist, because I don't think that there's a good way for us to know whether God exists with an amount of confidence that would sway me to take one side. Accordingly I've vaguely agreed with "we don't know how to assign probabilities to fundamental physics", but I haven't had the math to back it up (which is funny because I will graduate from UChicago with a physics degree in like a week).
But even taking Bentham's argument for the existence of god as given, I still don't buy that his god is perfect or infinitely good (i.e. his god is not God). One of Bentham's old comments against someone who brings up the problem of evil (https://benthams.substack.com/p/the-fine-tuning-argument-simply-works/comment/56147619?utm_campaign=comment-list-share-cta&utm_medium=web&comments=true&commentId=56147619&utm_source=substack) says
> I agree evil is a problem but I think theists can explain it, and God is just much likelier than a limited God that we should take the probabilistic hit and still believe in it.
and I think this is not a very epistemically honest way of approaching criticism of your argument. "I think someone else can explain it" is not a sufficient answer, and any attempt to deal with the problem of evil is going to result in a million burdensome details (https://www.lesswrong.com/w/burdensome-details) (I hope I'm using that correctly).
Just from looking at the universe, it appears that if god does exist, it is a clockwork universe kind of god. So even if I concede the existence of some noncontingent entity that goes around creating stuff it thinks is interesting, I'm still unconvinced by the "and the most interesting thing is perfectly good" part.
This is an interesting article--basically raising the measure problem--but I find it weirdly triumphalist. In that original article I did not address the measure problem but I have elsewhere (here for example https://benthams.substack.com/p/fine-tuning-made-by-god?utm_source=publication-search). You shouldn't be totally confident that the measure problem wrecks my view if you haven't read what I--or people on my side--say about the measure problem!
I give a lot of more specific objections in the linked post. But crucially we can *obviously* say that things are improbable even from an infinite set. So, for instance, there's an infinite set of things that the nearest aliens might be thinking about presently. An infinite number of them involve tacos. But still, I would guess with high confidence that the nearest alien to me is not thinking about tacos!
(Cross-posting from reddit discussion here)
First off, thanks so much for reading it! I'm a fan.
I'll definitely need to read your thoughts on the measure problem before thoroughly responding, but "crucially we can *obviously* say that things are improbable even from an infinite set" is not true in a rigorous sense, and that's what I spill so much ink and Latex arguing about.
But to your second point, from a strict physicalist perspective aliens thinking about tacos IS well defined! You just aren’t going to enumerate the space or the sigma-algebra, but huge and uncountable isnt the same as ill-defined.
Formally, to try to define the algebra you would take something like the Borel set over the universe’s phase space, in which some, presumably small subset of those configurations is aliens thinking about tacos. That’s monstrously unlikely AND well defined. The space of all physical theories is NOT well defined, which is the key distinction.
I think you’d have to have e.g. some theory of how multiverses spawn random universes to give that a proper formal treatment.
This is one of the best posts I’ve read on this site. Thorough and informative, thank you!
Wow thank you so much
Bayes rule, for me is an intellectual super power. You clearly know that and so does another excellent writer on Substack Nate Silver. It lets you think probabilisticly about everything from poker to existence of god 😎
Also on the anthropic stuff, I don't really get what your objection is. Like, do you accept SIA? Do you accept that if you accept SIA you should think there's a giant infinity worth of people? Where do you get off the boat? It's all very well to note that I have not done fancy shmancy sigma algebra or whatever, but much less clear where you think the argument goes wrong? I do slightly feel like I'm getting Eulered.
(also just want to say, absolutely love the title!)
It's actually not true that "Even the most generous naturalistic scenarios (infinite multiverse, infinite universe) can only produce aleph-null people - the smallest infinity."
It's actually pretty easy to construct scenarios which produce numbers of people in larger cardinalities. One of the other comments mentions Cosmological Natural Selection. You might think this isn't the case, given that each universe only produces a finite amount of black hole singularities. But actually, after quantum mechanics is evolved for, assuming the many-worlds interpretation, each node in the CNS multiverse tree actually represents a QM multiverse consisting of an infinite number of parallel universes. The result is an infinite fractal tree of universes, where zooming into any node shows an infinite number of outgoing branches leading to nodes which also produce an infinite number of branches.
Notably, this scenario also solves the objection you have with Tegmark about Boltzmann Brains, because those are extremely complex entities which nevertheless do not assist in the production of singularities. They are therefore selected against.
Oh man, I'm so happy you liked the title, and frankly flattered you bothered to read this at all.
I'll have to read your arguments on the measure problem before responding to those, I just want to let you know that I have definitely not thought deeply enough about the SIA and whether it makes sense to me -- its inclusion here was because I came across it, a bunch of arguments about aleph-null numbers of people and infinite probability and it looked like the exact same undefined space to me. I'll get back to you!
Thanks! I did appreciate the article. Way more thoughtful than a lot of what people write in reply to my stuff, so thanks for that!
Out of curiosity: have you read any things that philosophers or physicists who promote fine-tuning have said about the measure problem? Because the article doesn't engage with any of their work. Sorry this is going to sound a bit harsh and I don't quite know a way of saying it in a way that isn't harsh, but if you're going to write a big article about the measure problem, it seems worth reading what those who think the measure problem objection isn't fatal say.
I confess I have not, I have read YOUR article(s) and then the entry on the Stanford Encyclopedia of Philosophy, which I thought was sufficient for a blog post.
I probably ought to read some of the actual technical work, especially after being so politely called out. Alas, your articles are much more engaging than academic literature, hence my lack of desire to go beyond, though, still wrong, I argue ;)
This is just what I said about basically ignoring the objection and saying "lets just pretend and carry on anyway with states I select"
I argued here that even accepting SIA for the sake of the argument, it doesn't follow that there's a giant infinity of people:
https://mon0.substack.com/p/benthams-bulldog-ruins-my-weekend/comment/86091008
Small quibble: I don't understand your mention of AC. If we take on not-AC we can prove that all sets are measurable.
Bigger quibbles: What do you mean P(God|measurable sets)? The set of measurable sets is not an element of the algebra, it is itself the algebra. You claim there's no natural choice of algebra. The natural choice of algebra is the Borel algebra over a Euclidean representation of the parameter space. Then you say that the measure theorist says the probability of picking any particular value from a continuum is zero. This is wrong. Singletons are Borel in R^n. Zero-inflated priors, for example, are bread and butter Bayesian statistics
It was inevitable that an actual mathematician show up at some point and pick out my errors. I will try my best to respond.
For your axiom of choice quibble, are you referring to this section?
"It’s odd, don’t you think, that if you pick a random subset of the real line, it is guaranteed to be unmeasurable with probability 1? Yet all the sets we use for, well, anything, are always measurable? Is P(God|measurable sets) therefore high? Well, no. This is simply the natural consequence of the axiom of choice."
What I was trying to do was draw an analogy between unintuitive consequences and simpler principles that imply them, and use the axiom of choice, which is fairly simple but has really unintuitive consequences, as an example. I may have chosen a poor one here, is the proof that you're referring to this one? https://en.wikipedia.org/wiki/Solovay_model. You're right that I am implicitly assuming AC here, though choosing or not choosing AC for your universe model is of course a choice. An axiom of axiom of choice?
"What do you mean P(God|measurable sets)?"
I think this is just sloppy notation on my part, the set of measurable sets is absolutely just the algebra -- this should say something more like P(God | the surprising fact that measurable sets exist and physics occurs on them), which is of course completely undefined, and was a rhetorical flourish. This wasn't meant as a formal probabilistic statement itself, and I acknowledge the irony in a post about making imprecise math statements.
"You claim there's no natural choice of algebra. The natural choice of algebra is the Borel algebra over a Euclidean representation of the parameter space."
I'm not 100% sure which part of the article this one refers to. I don't think I say anywhere that conditional on choosing a particular reasonable representation of the parameter space you can't define an algebra... Do you mean the part where I assert "Same problem" in choosing the sigma algebra if your sample space is undefined? Here I think my intent is clear, of course you can't choose a sigma algebra over an undefined space. If your space is defined, sure, but my point there is that it isn't defined, so I think the argument that the algebra is undefined holds, if trivially.
"Then you say that the measure theorist says the probability of picking any particular value from a continuum is zero. This is wrong. Singletons are Borel in R^n. Zero-inflated priors, for example, are bread and butter Bayesian statistics"
Yes, I agree you could construct something like this, a "our universe's constants" inflated prior or something like that. Would you be happier with also imposing an absolute continuity condition on the measure?
Generally, I think a lot (all?) of your points show the peril of trying to essentially write a measure theory textbook as a blog post, I inevitably have some elisions and errors between the concept I'm pointing to and a fully correct definition, which is entirely my fault.
I don't think any of the points you make are fatal to the overall argument, as this is mostly just imprecision on my part, as well as me forgetting about point masses being allowed in Borel sets. There's no good way to justify putting specific point masses in your theory, of course, but you can and it's well defined. This is my bad.
I watched a YouTube video about the Vitali set. If I understand correctly, it can occupy an arbitrarily small interval on the real line. Is that correct? When using the axiom of choice, simply choose a member of the set which happens to fall in that narrow interval.
Yes, this is correct as I understand it. In essence, when you choose a member of each equivalence class over the rationals you can always choose it from some arbitrarily narrow interval by "shifting" by a rational number to be within that interval.
"He does math with English" is a great line!
Well said. I wish more creationists understood the limitations of the fine-tuning “argument.” It can be very useful AFTER adopting the premise that a creator exists because it can then suggest things about the nature of that creator. In this case, the problem II authentically becomes problem I as you technically have a set of possibilities. However, you set yourself up for failure the moment you try to use fine-tuning against atheist beliefs.
Finally, there is the none mathematical application of fine-tuning. And that simply is just saying “I see a kind of harmony in the relationship of physical laws. Maybe that shows intention. Cool!”
Great essay.
I strongly resonated with footnote three, although I’m a believer and in my case it’s mostly responding to audits.