The probability of a god existing

[Darth Wong]

Your calculus based approach is quite elegant. Using this approach, would you say that it is fair to say that the probability of a monotheistic god existing is P(god)=0 and that the probability that there is no monotheistic god is P(god^C), where god^C is the compliment of god existing (i.e. god does not exist)? Then, solving for P(god^C) is:

P(god^C)=1-P(god)

P(god^C)=1-0

P(god^C)=1

Its really quite amazing how once you learn some introductory/intermediate math, how much you can think for yourself. At the same time though, it really makes you look at the world with sheer amazement at how people can believe in such highly, highly, statistically improbable things and it makes you feel quite alone, being one of few people who know the reality; it really is a delusional world.

There's a "eureka" moment in the average science student's university career when he truly groks the idea of conceptualizing the world around him in terms of mathematics.

[R. U. Serious]

Maybe I've lost my sense of wonder, but realizations like this:

would you say that it is fair to say that the probability of a monotheistic god existing is P(god)=0 and that the probability that there is no monotheistic god is P(god^C), where god^C is the compliment of god existing (i.e. god does not exist)? Then, solving for P(god^C) is:

P(god^C)=1-P(god)

P(god^C)=1-0

P(god^C)=1

Its really quite amazing how once you learn some introductory/intermediate math, how much you can think for yourself.

I find about as amazing, as the realization that if you have 2 apples and you take one away, ou have exactly one left. The complement of zero probability being 100% probability is equally obvious.

At the same time though, it really makes you look at the world with sheer amazement at how people can believe in such highly, highly, statistically improbable things and it makes you feel quite alone, being one of few people who know the reality; it really is a delusional world.

If such basics amaze and excite you so much, you might want to get some paper towels when learning about Bayes and how it fools most people's intuitive reasoning. Try reading Innumeracy, or really any book by John Alan Paulos. To wet your appetite:
http://www.edge.org/3rd_culture/gigeren ... print.html
http://yudkowsky.net/bayes/bayes.html

[Fire Fly]

Maybe I've lost my sense of wonder, but realizations like this:

would you say that it is fair to say that the probability of a monotheistic god existing is P(god)=0 and that the probability that there is no monotheistic god is P(god^C), where god^C is the compliment of god existing (i.e. god does not exist)? Then, solving for P(god^C) is:

P(god^C)=1-P(god)

P(god^C)=1-0

P(god^C)=1

Its really quite amazing how once you learn some introductory/intermediate math, how much you can think for yourself.

I find about as amazing, as the realization that if you have 2 apples and you take one away, ou have exactly one left. The complement of zero probability being 100% probability is equally obvious.

And what is so wrong about stating and writing the obvious? It might be obvious to you but believe it or not, there are people out there who do not realize the obvious when it is staring them right in their face with bold letters, written in flashing neon purple. There's a reason why people who write research papers state the obvious: it eliminates any potential confusion.

At the same time though, it really makes you look at the world with sheer amazement at how people can believe in such highly, highly, statistically improbable things and it makes you feel quite alone, being one of few people who know the reality; it really is a delusional world.

If such basics amaze and excite you so much, you might want to get some paper towels when learning about Bayes and how it fools most people's intuitive reasoning. Try reading Innumeracy, or really any book by John Alan Paulos. To wet your appetite:
http://www.edge.org/3rd_culture/gigeren ... print.html
http://yudkowsky.net/bayes/bayes.html

I've already acknowledge the error in my original proposition and I've already stated that what I was doing was quite vague and arbitrary. But that is the nature of religion; if you even want to try to remotely discuss it in a logical manner, you have to set out some basic, wide, overarching assumptions or definitions, so long as you provide some sort of a case for why you chose what you chose.

[Hugh]

But that is the nature of religion; if you even want to try to remotely discuss it in a logical manner, you have to set out some basic, wide, overarching assumptions or definitions, so long as you provide some sort of a case for why you chose what you chose.

Discuss religion in a logical manner? In the real world?

That said, I think I can rationalize the apparently contradicting claims of the various religions. You see, most people have no sense of scale. When they say "God created the human race and all the world", they actually mean their own tribe and the piece of land they occupy. I've seen a similar attitude during the 2005 floods. You know, people wailing "we're all gonna die! everyone will perish under the waves!". But if you looked from a helicopter, only their village was flooded - it just happened to be in a depression of the terrain. But to them, that village was the whole world.

[Starglider]

The whole point of the exercise is to create a definition specific enough that you can apply a computational prior to it.

So you alter the definition in order to suit the computation, rather than altering the computation in order to suit the definition?

No, you just need a definition of the universe in question built up from logical primitives (i.e. a mathematical one). 'Intelligence' is far from a primitive. To do that you have to model 'god' as a computational entity. There are some straightforward ridiculously powerful intelligence models (e.g. AIXI-TL) that can serve as a baseline. Minimum limits are unfortunately highly speculative, but statements like 'gods mind encompasses the entire complexity of the universe' ratchet up the extant complexity to ridiculous orders of magnitude. The problem with that is that physics seems to conserve initial definitional complexity, not amount of raw computation (physics is horribly compute intensive) or description length of the evolved system. I've seen attempts to argue that a creator is likely because the minimal loogical substrate for intelligence is rather simpler than our own universe's physics, and such systems must occupy a much larger fraction of possibility space, and thus from something similar to Bostrom's simulation argument our own universe is likely to be embedded in such a (definitionally) simpler universe. IMHO this is reasonable enough given the basic assumptions, but pointless because such assumptions are hopelessly speculative while we still don't have a good appreciation of the possibility space being selected from, i.e. a strong answer to 'why does anything exist at all'.

As long as you are not talking about an incredibly broad class (eg- "anything supernatural"), the logic still holds and you have failed to identify any flaw with it

No such flaw; it should be obvious that the probability of any one of a large set of mutually exclusive items must be very small if you have a reasonably indifferent prior, which an objective view of human religion must be (religion is essentially a delusion about the value of subjective intuitive conclusions that allows people to ignore this). For normal debate you can take the fact that various religious people condemn each other to hell as evidence of mutual exclusivity even down to something as close as protestantism and catholicism. But if you want to feed philosophical systems into a formal reasoning system, which various people I know do, the mutual exclusivity has to be defined by some logical incompatibility in formal models that renders the sets disjoint. I don't know anyone who's gone as far as trying to fully model a real human religion, as that would involve spending years trying to formalise various brands of nonsense for no real change in the output. So usually it's fairly broad things like 'a deliberate intelligent creator of the universe exists'.

I never made any such assumption. x is infinite as long as there is an infinite number of possible ideas for supernatural deities. You don't need infinite universes in order to say that there are an infinite number of possible ideas for supernatural deities.

The number of possible ideas is bounded by the size of the intelligences that imagine them, and ultimately the substrate available to encode them. If the universe is finite in extent (likely) and the holographic information bound holds (very likely) then the total information content of our own universe is finite. Any cosmological definition must be encoded into our universe somehow (right now it has to fit in a single human mind), so the total description length is finite, thus the total number of ideas that can actually be imagined is finite. The total number of ideas would be infinite if the universe itself was infinite, or if you're recursing and imagining the set of ideas that could be present in all universes, but that's circular logic. The numbers here are so large that for all practical purposes they might as well be infinity, but the difference is relevant for a rigorous formal analysis.

Are you suggesting that if you have, say, a billion ideas for supernatural deities, it will become impossible to think of any more variations?

Yes, though the number is much bigger than a billion. The number of possible holy books the size of the bible in gramatically correct English is somewhere north of a ten to the power of one billion, but it's still finite.

Math is only as good as the reasoning it's based on, and you are adding a completely unnecessary factor by assuming that the number of possible ideas is tied to the complexity of the universe

That seems pretty straightforward to me. At the limit, it's impossible to have an idea with a minimum description length longer than the universe that contains it. In practice ideas have to have a substantially smaller MDL than the mind that contains them, which for humans is probably in the petabytes for a lossless mind MDL and gigabytes for a really complicated idea that you can hold in your head.

and/or the number of universes in existence

The number of universes in existence isn't directly relevant; it may be that only our universe exists, but to generate a probability you have to look at a selection from the set of possible universes that could exist. The underlying assumptions for this are that all possible universes are consistent and computable (causality is an optional extra). These axioms aren't really founded on anything other than matching observation and hopeful optimism, but if they don't hold we might as well pack up and go home. Specifically for a rational intelligence those assumptions are made because it's necessary to apply the available tools to the problem, and there are no other candidate mechanisms for tackling the problem.

[Darth Wong]

I find about as amazing, as the realization that if you have 2 apples and you take one away, ou have exactly one left. The complement of zero probability being 100% probability is equally obvious.

Yeah, I didn't really comment on that part of his post because it seemed a bit redundant to point out that if P(God) = 0, then P(not God) must be 1

[Starglider]

Yes, though the number is much bigger than a billion. The number of possible holy books the size of the bible in gramatically correct English is somewhere north of a ten to the power of one billion, but it's still finite.

Though in practice the fact that you could theoretically write circa 10^1,000,000,000 mutually exclusive religious books isn't a central tennent, because the prior while fairly naive isn't likely to be uniform. Basic Kolmogorov alone will strongly favour (allocate most of the probability mass to) short simple religions where the scope for variation is much more limited - unlike human intuition formal reasoning doesn't suffer from the 'adding more detail actually increases subjective probability' fallacy. Secondly though strictly exclusive, the vast majority of religions have significant shared complexity, and when you do a recursed Bayesian probability analysis the probability of theory components being correct is a vital part of the outcome. For the purposes of making decisions the point probability of a complete theory (in this case religions) is important, but the group probability of sets of theories sharing a common element is often at least as important if not more important for action selection. Fortunately as I've noted pretty much anything religious, even at the conceptual level, comes out with a trivially small probability, even when generalising, for any reasonably sane set of axioms.

[Darth Wong]

Basic Kolmogorov alone will strongly favour (allocate most of the probability mass to) short simple religions where the scope for variation is much more limited - unlike human intuition formal reasoning doesn't suffer from the 'adding more detail actually increases subjective probability' fallacy. Secondly though strictly exclusive, the vast majority of religions have significant shared complexity, and when you do a recursed Bayesian probability analysis the probability of theory components being correct is a vital part of the outcome.

You are still making the mistake of confusing the world's actual religious ideas with all of the possible religious ideas. Any probability analysis which is bound to the known characteristics of real religions is completely irrelevant to the analysis I performed earlier, which was based on the number of possible religious ideas, no matter how complex or unlikely to actually gain a following. Your analysis relies on the completely unfounded assumption that a probability analysis for an exclusive member of the "cannot be proven false" class of ideas should be limited by reducing that class to ideas that are found or are likely to be found in actual religions.

[Starglider]

You are still making the mistake of confusing the world's actual religious ideas with all of the possible religious ideas.

I'm not confusing them, but perhaps I'm not being clear. Three general classes of entity have been covered so far. One is the various actual religions that humans have come up with and the general nature of dogma we tend to come up with. I don't have much to say on this other than that it's often so broken as to have an instant zero probability (due to internal contradictions) and that it isn't worth the time for anyone to do objective logically positivist modelling. Second is the space of all possible religious cosmologies, which is a small subset of the space of all possible ideas. This space is mind-bogglingly vast, but it is finite, because there is a finite limit to the description length of the corresponding models. You could argue that the complexity of the cosmology within which we are embedded is not bounded by the complexity of the part of it that we can observe or utilise for modelling, but that strikes me as circular logic and somewhat pointless. Even an infinite set will not give infinitesimal probabilities for finite MDL models if you're using a tapering (i.e. at least somewhat Kolmogorov) prior, which most sensible people do (because it's essentially formalised Occam's razor). This actually strengthens the atheist position, because a completely indifferent prior over an infinite set implies infinitesimal probability for any given religion (and is thus strongly agnostic) but not an obviously low probability for the category of creator-less universes, whereas a complexity-tapering prior dismisses the entire category of creator-based religions as incredibly unlikely. Which brings me to the third category, general concepts, classes and sub-elements of cosmological/theological models. We're not debating those here, I just thought I'd mention that some people are working on detailed computational models of them, I've looked at some of them, they generally support the strong atheist (probably of a creator = very small, probability of elaborate dogma = effectively zero, probability of typical human religion = literally zero) position and that it is somewhat relevant to my field (AGI).

Your analysis relies on the completely unfounded assumption that a probability analysis for an exclusive member of the "cannot be proven false" class of ideas should be limited by reducing that class to ideas that are found or are likely to be found in actual religions.

No, I'm certainly not claiming that; my nitpicks of your argument are just that the set is very large but finite (the physical capacity to have ideas is separate from what particular ideas happened to turn up in history), and that sensible priors over it are nonuniform, neither of which really changes the outcome (except to clearly favour atheism over agnosticism for the latter).

I was just adding some flavour on what it looks like when people try to tackle these issues at a truly objective level; for normal human debate you can use simple pieces of maths that rely on human interpretation to have meaning, whereas if you seriously want to ask an AI system 'what is the probability of their being a God' (which we can't really do yet, but some people are working on it in advance) you have to define everything all the way from the abstract theological notions right down to mathematical/logical primitives, which is a hell of a job and creates hugely complicated models.

[Kuroneko]

Why does there need to be a finite upper bound on the complexity of possible monotheistic religions, even if it true that there is one for human beings (extremely likely) or perhaps the universe (far from clear)? It seems very reasonable at first glance--after all, what good are religions that no one is capable of holding--but you seem to be forgetting that the issue is not really about religions as human practices but about possible deities. By making this stipulation, you implicitly assume that any Creator-entity is almost surely accessible (i.e., with probability 1) to human beings or similar at least in principle. Why?

But let's back up a bit.

That's a solid axiom, but it means that when you're comparing the cardinality of different classes of universe that we might be in you're invoking transfinite mathematics.

Speaking of transfinite mathematics, if they're classes, they might not have any cardinality whatsoever. But that the entire class of "metaphysical possibilities" (whatever that might mean) forms a measure space at all is already a rather hefty assumption.

For a finite total complexity, some (usually very small) fraction of possible universes correspond to a particular metaphysical belief ... . Remove the complexity bound ... and you have to look at how the cardinality ratio between 'metaphysical belief is true' and 'metaphysical belief is false' changes as the complexity increases.

Why would one be looking at cardinalities rather than the measures at this point? (Or even that much.)

But if you want to feed philosophical systems into a formal reasoning system, which various people I know do, the mutual exclusivity has to be defined by some logical incompatibility in formal models that renders the sets disjoint.

It doesn't seem to be at all necessary to go that far. All we need to suppose is that for some collection of monotheistic religious beliefs, every pairwise intersection forms a null set (i.e., one having zero probability of occuring), but not necessarily the empty set.

The number of possible ideas is bounded by the size of the intelligences that imagine them, and ultimately the substrate available to encode them.

However, there is no reason to assume that there is any being in our universe capable of holding the ultimate "correct" monotheistic religious belief. That would imply that we're all out of luck, but in the end it's also quite possible.

If the universe is finite in extent (likely) ...

(Emphasis mine.) Why? (Or even more pertinently, why is it relevant?)

Are you suggesting that if you have, say, a billion ideas for supernatural deities, it will become impossible to think of any more variations?

Yes, though the number is much bigger than a billion. ...

It is possible that no monotheistic religion is "correct" (in some theologically appropriate fashion, e.g., for salvation) unless it also identifies the correct number of toes possessed by the deity in question. Since this can be any natural number, there is at least a countable infinity of possible distinct monotheistic religions. That holding some of those possible religions would be impossible for any actual human being (for the exact reasons you state) is correct but is of very dubious relevance. The same goes for the possibility that the entire universe might be incapable of holding a being of sufficient complexity.

[Starglider]

By making this stipulation, you implicitly assume that any Creator-entity is almost surely accessible (i.e., with probability 1) to human beings or similar at least in principle. Why?

I don't assume it implicitly; I explicitly stated that I was making that assumption (in the sense of being conceivable by intelligences in this universe). AFAIK there are no objective grounds for this assumption, because even if we knew that the observable universe was finite, we'd still have no evidence against the entire universe being embedded in something inaccessible and indefinitely more complex besides an appeal to Occam's razor. The motive for making it is simply that no useful analysis of cosmologies beyond the scope of what can be tackled directly by the reasoning system is possible, nor is it likely to be useful under a general Kolmogorov prior. Recursed expected utility will simply discard generalising beyond modellable completely-theoretical cosmologies as a waste of time (probably with a residual null hypothesis in the probability assignment). If and when we ever have an answer to the question 'why does anything exist at all' this situation may change.

But that the entire class of "metaphysical possibilities" (whatever that might mean) forms a measure space at all is already a rather hefty assumption.

Ignoring low-level grounding issues (the AI equivalent of various problems in the foundations of mathematics), a 'metaphysical possibility' is a formal model of a set of universes that can select that set from a formal model of the totality of possible universes. A finite description length (i.e. conceivable models) doesn't make the selected sets finite (for some models it will, for others it won't) but it does make the set of possible models finite. Clearly an unconstrained description length permits an infinite number of models - even if the set of possible universes was finite, which it presumably isn't, there would be an infinite number of computationally equivalent models. As I understood it, the original argument here applied to actual religions, that humans can conceive of, and less directly the set of metaphysical propositions that any given finite formal reasoning system can evaluate for probability against a prior. The total set of models with no description length constraint is infinite by definition, but it isn't possible to reason about that except indirectly. For a tapering prior, all that will happen when you remove the constraint is that a small residual amount of probability mass will end up in a 'non-evaluable model' category, which will decline as you increase the capacity of the reasoning system (size of the mind) but can never be eliminated.

Why would one be looking at cardinalities rather than the measures at this point? (Or even that much.)

I've only done this with huge-but-finite sets, more than can be evaluated, but small enough to model directly in abstract. This generalises fairly well to the infinite case when the base prior tapers enough that what you can't model has a trivial probability of being relevant (under reflection). There may well be a more abstract method for reasoning about relative probabilities that uses some other measure over transfinite discrete sets, but if there is I don't know about it, nor have I seen any other measures used in this application. I've used arbitrary measures in probabilistic reasoning about continuous models based on real numbers (dealing with infinitesimals rather than infinities - the main reasoning system I'm working on sometimes uses local measures rather than simple integration to reason about these kind of sets), but I only have a minimal knowledge of transfinite maths (enough to know how much I don't know).

It doesn't seem to be at all necessary to go that far. All we need to suppose is that for some collection of monotheistic religious beliefs, every pairwise intersection forms a null set (i.e., one having zero probability of occuring), but not necessarily the empty set.

I am not clear if by 'belief' you mean a complete religion (cosmology model) or a single constraint on such a model ('Thor likes thunderbolts'). For the general case, some models will be equivalent, some will be subsets of other models, and others will contradict, which if you've fully formalised the model can be traced back to axioms and the set of maximally inclusive weaker parent models that include both generated. Simply making the supposition doesn't tell you anything that helps assign probabilities to specific models (and constraints on models, which are what theory components essentially are, ignoring context issues in shared complexity, which is really the same thing recursed a bit anyway).

However, there is no reason to assume that there is any being in our universe capable of holding the ultimate "correct" monotheistic religious belief. That would imply that we're all out of luck, but in the end it's also quite possible.

Something I've both glossed over; there's a big difference between a constraint, a partial model and a complete model. All of them select a subset of universes from the set of possible universes, but only the later is predictive rather than definitional and only the later is fully predictive in the sense of being able to generate a complete history from a state. The final step in the specificity threshold is a complete description of a universe, which is presumably identical to an 'actual' instantiation of that universe from the viewpoint of reasoning systems embedded in it (where 'actual' refers to a universe that isn't a simulation or natural phenomenon embedded inside a prerequisite larger context). A complete description of physics would be fully predictive (though probably nondeterministic for the real universe; it's fully predictive of a future history tree) of the observable universe, but a theological model would characterise it as incomplete, describing an additional realm (presumably occupied by supernatural entities of some sort).

Clearly it's not possible to completely describe a universe within itself. It may or may not be possible to specify a fully predictive model, a partially predictive model or even an objective constraint selecting what actually exists from what could potentially exist (assuming those are actually different rather than merely a question of location, but AFAIK that distinction doesn't impact on this argument other than making it easier to talk about in English). As I noted, the main reason for assuming that we can is that it's pointless to worry about what we can't analyse, other than by assigning a residual probability to it via some sort of Kolmogorov. If you don't do that (and use an indifferent prior), you'll get a strongly agnostic result; no expected utility to direct reasoning about anything metaphysical because the probability is always effectively zero (even indirect measure-based comparison of infinite classes of belief won't work because you've abandoned any sort of discrimination in the base prior). That would probably be fine for practical purposes; the end result is similar, formal reasoning systems (i.e. AGIs) aren't going to start acting religious, but since Kolmogorov is applicable (and extremely useful) for everything else it would be a needless and (AFAIK) pointless bifurcation of the base prior mechanism.

It is possible that no monotheistic religion is "correct" (in some theologically appropriate fashion, e.g., for salvation) unless it also identifies the correct number of toes possessed by the deity in question. Since this can be any natural number, there is at least a countable infinity of possible distinct monotheistic religions.

Hypothetically yes. It would be impossible to actually hold this belief if the number is sufficiently high that humans can't encode it. In which case no salvation would be possible. Two problems with this; firstly we can enumerate the set with a trivial algorithm and analyse it perfectly well (in terms of comparing it to other models) on those terms. For this argument to work it has to be irreducable complexity that prevents any sort of direct analysis (i.e. comparison with other models/constraints). Secondly for practical purposes the infinite number of possible toe-counts where no salvation is possible reduce to a single consequence for the observable universe, so only the finite subset that we can encode (plus one aggregate possibility for all the numbers we can't) matter. Again you can almost certainly get around this, it'll just take a more complicated example.

That holding some of those possible religions would be impossible for any actual human being (for the exact reasons you state) is correct but is of very dubious relevance.

It's certainly true that the universe has no obligation to be comprehensible, and my earlier arguments about what's worth reasoning about while critical to the issues I mentioned (in AI) aren't directly relevant to the original question. However the application of a Kolmogorov (equivalent) prior, i.e. some sort of Occam's razor, is definitely relevant, essentially because you take a huge probability hit for multiplying toes unnecessarily.

[Darth Wong]

So you dismiss possibilities that will not yield a "useful analysis", while discussing the probability of an idea that is itself immune to useful analysis? Do you understand that if one introduces the logic of discounting possibilities that are a waste of time to consider, then this entire exercise is completely pointless?

[Starglider]

So you dismiss possibilities that will not yield a "useful analysis",

A little meta first; I've semi-hijacked the thread at this point to talk about an interesting theoretical area (computational philosophy) related to my own field (though I'm not specifically an expert in it). The logic used for this is painstakingly built up from formal axioms, but it's often hard to translate into the kind of semi-colloquial arguments that take huge complex and fuzzy concepts (like 'intelligent creator') as mutually understood primitives which humans usually debate with. AI in general has this problem but philosophy-from-first-principles is worse than usual. I don't have a well-worked out personal position to defend on most of the specific open issues in computational philosophy - I'm just familiar with the issues and arguments and thought I'd detail them a bit - but I do have a strong position on elements that are directly shared with artificial general intelligence, as well as the really simple stuff (conclusions like 'any human religion has an effectively zero probability of being true') which most people here agree on anyway.

Dismissing possibilities that can't be analysed is a tricky issue. A simple dismissal is a clear logical fallacy; one of the many possible ways of conflating expected utility (in this case cognitive utility of attempting an analysis) with actual probability of being true in the real world. It only really becomes an issue at the absolute extremes of cognition, particularly in bootstrapping a usable, consistent base prior from simple indifference on every disjoint set of models considered (the simplest possible baseline, but only minimally consistent). I mentioned a clear case where you can and should do such a dismissal above; when the practical result of all the possibilities you can't model is exactly the same. More complex equivalences where entire transfinite classes of model map can be deemed equivalent to single finite exemplar crop up pretty often too, but (AFAIK) this isn't true for the general case of metaphysical propositions, though it seems to cover the vast majority of the examples people bring up. There is a strong bound on observable consequences of any extra-universal complexity, the complexity of the observable universe itself, but that doesn't necessarily help, since we're dealing with propositions such as 'souls go to heaven' which will eventually render that currently-extrauniversal complexity relevant to the observer (plus there are propositions that directly shatter any local complexity bounds by restructuring the universe anyway). The single strongest reason for dismissal is still essentially an elaboration of Occam's razor; any reasonable complexity-tapered base prior will make the prior for the entire set of models you can't analyse so low that even the hypothetical expected utility of further analysis (i.e. the chance that a detailed analysis would find a reason to assign those super-complex theories a nontrivial probability if you could actually do such an analysis, given that this has not occured for all the theories you could analyse) is also effectively zero.

while discussing the probability of an idea that is itself immune to useful analysis?

Which idea do you mean? Metaphysical propositions in general aren't immune to useful analysis, it's just very difficult, not assured to generate much discrimination and unlikely to have a bearing on any real world decisions. I support the work of a few people looking at doing this from first principles, because I'd like to know what happens if we ask an AGI 'does god exist' before we actually build one.

Do you understand that if one introduces the logic of discounting possibilities that are a waste of time to consider, then this entire exercise is completely pointless?

Not so, and a topic change in any case (for a strict interpretation of 'waste of time', which you may not have intended). The problem with infinite sets of propositions is that they cannot be analysed or applied as a workable belief by any finite intelligence. There's no waste of time about it; it's impossible unless you can use tricks to reduce the infinite set to a finite one, which doesn't work in the general case of unbounded minimum description length. Generally if there's no way to even complete the analysis of a candidate model, we have to discard it and invoke our prior for 'residual probability to assign to stuff we can't model even in principle' to allocate probability mass to it. The question of what's worth considering when the capability to analyse the model exists, but when resources are limited (i.e. picking the most worthwhile of the still utterly vast set of human-conceivable metaphysical notions to subject to a rigorous formal analysis) is a separate issue and a rather more common one. The answer to that is concrete cognitive expected utility, which ultimately comes from recursed composite Bayes (in a rational system). Allocation of cognitive effort over various time scales is a central issue in AGI, but unfortunately the logic isn't really adaptable to human debates, since no one wants to write or read a 100 page probabilistic logic trace just to find out why further debate of a particular idea is probably a waste of time. Again though, it's important to working out how fully rational reasoners deal with the concept of religion.

[Wyrm]

Dearheart, you need to empty your cup of stupid, for it runneth over.

One of the first things you learn in applying mathematics is to make assumptions that simplify your problem, not to complicate it. Your discussion of complexity is a distractor as it leads us with a discussion of what kinds of gods are believable and finding the probability of those and those gods only. Mike's argument doesn't need to restrict us to believable gods — indeed, the fact that there are equally probable gods that we can never believe is kind of the point.

[wolveraptor]

I never agreed with the statistical method of determining the probability that God exists. You're making the assumption that each religion is equally likely to be true. Since almost every religion makes at least 1 claim that can be tested, you discount many religions altogether, without giving them the benefit of a probability, however small it may be. For example, the Norse religion predicts a land of giants to the south. This is obviously false. Christianity says Christians can drink poison and their Lord will save them. This is also false. There is no need to give these two religions, and many others, even a chance of being veracious.

[OmegaGuy]

Why would there not be an infinite number of possible gods?

Say I could imagine a god with 1 head, or a god with 2 heads, a god with 3 heads, *skipping forward a bit* a god with 2.456e89763 heads, etc.

Infinite possibilities.

[Darth Wong]

You know, for someone who says that his ideas are too complex to describe here, you're sure posting an awful lot of verbiage.

So you dismiss possibilities that will not yield a "useful analysis",

A little meta first; I've semi-hijacked the thread at this point to talk about an interesting theoretical area (computational philosophy) related to my own field (though I'm not specifically an expert in it). The logic used for this is painstakingly built up from formal axioms, but it's often hard to translate into the kind of semi-colloquial arguments that take huge complex and fuzzy concepts (like 'intelligent creator') as mutually understood primitives which humans usually debate with. AI in general has this problem but philosophy-from-first-principles is worse than usual. I don't have a well-worked out personal position to defend on most of the specific open issues in computational philosophy - I'm just familiar with the issues and arguments and thought I'd detail them a bit - but I do have a strong position on elements that are directly shared with artificial general intelligence, as well as the really simple stuff (conclusions like 'any human religion has an effectively zero probability of being true') which most people here agree on anyway.

Plain-english translation: "I have studied AI for a long time and I'm trying to shoehorn this argument into the framework of the AI problems that I'm used to". In other words, you are altering the parameters of the problem to suit your preferred method of analysis, rather than altering the method of analysis to suit the parameters of the problem. Do you honestly not understand what's wrong with that?

Dismissing possibilities that can't be analysed is a tricky issue. A simple dismissal is a clear logical fallacy; one of the many possible ways of conflating expected utility (in this case cognitive utility of attempting an analysis) with actual probability of being true in the real world. It only really becomes an issue at the absolute extremes of cognition, particularly in bootstrapping a usable, consistent base prior from simple indifference on every disjoint set of models considered (the simplest possible baseline, but only minimally consistent). I mentioned a clear case where you can and should do such a dismissal above; when the practical result of all the possibilities you can't model is exactly the same. More complex equivalences where entire transfinite classes of model map can be deemed equivalent to single finite exemplar crop up pretty often too, but (AFAIK) this isn't true for the general case of metaphysical propositions, though it seems to cover the vast majority of the examples people bring up. There is a strong bound on observable consequences of any extra-universal complexity, the complexity of the observable universe itself, but that doesn't necessarily help, since we're dealing with propositions such as 'souls go to heaven' which will eventually render that currently-extrauniversal complexity relevant to the observer (plus there are propositions that directly shatter any local complexity bounds by restructuring the universe anyway). The single strongest reason for dismissal is still essentially an elaboration of Occam's razor; any reasonable complexity-tapered base prior will make the prior for the entire set of models you can't analyse so low that even the hypothetical expected utility of further analysis (i.e. the chance that a detailed analysis would find a reason to assign those super-complex theories a nontrivial probability if you could actually do such an analysis, given that this has not occured for all the theories you could analyse) is also effectively zero.

Plain English translation: "I still think that the number of possible metaphysical ideas is limited by the complexity of the observable universe: a totally groundless assumption which I will keep using because I hope no one will notice if I express it in a sufficiently overcomplicated way. If backed into a corner I will invoke Occam's Razor, even though Occam's Razor dismisses all metaphysical ideas anyway, thus making the whole exercise moot".

while discussing the probability of an idea that is itself immune to useful analysis?

Which idea do you mean? Metaphysical propositions in general aren't immune to useful analysis, it's just very difficult, not assured to generate much discrimination and unlikely to have a bearing on any real world decisions. I support the work of a few people looking at doing this from first principles, because I'd like to know what happens if we ask an AGI 'does god exist' before we actually build one.

You're missing the whole fucking point. You artificially limit the bounds of "possible" metaphysical ideas with a variety of assumptions that you stubbornly cling to even though you know they cannot be defended. You assume that the complexity of metaphysical ideas is limited by their manifestation in the observable universe, hence limited by the laws of the observable universe. You assume that classes of metaphysical ideas can be condensed into a single meta-idea, even though that is irrelevant to the question of picking a specific exclusive religious belief. You assume that it's OK to use Occam's Razor to reduce the size of the "no evidence but can't be disproven" data set, even though the introduction of Occam's Razor eliminates the entire set. And as far as I can tell, you cling to these assumptions, dressed up in thoroughly unnecessary but carefully long-winded verbiage, because that allows you to use your preferred methods of analysis that you spent a lot of time learning. Well too fucking bad, but every problem does not become a nail just because you worked hard to become a hammer.

Not so, and a topic change in any case (for a strict interpretation of 'waste of time', which you may not have intended). The problem with infinite sets of propositions is that they cannot be analysed or applied as a workable belief by any finite intelligence.

Oh for fuck's sake .. news flash: they don't need to be analyzed! Nowhere in my probability argument is it necessary to analyze the entire set of possible metaphysical ideas. One need only point out that it is unlimited by its very definition. Indeed, many conceptions of God explicitly state that he is beyond comprehension, remember? Or did you block that out of your mind too, in your zeal to alter the problem rather than altering your solution?

And I don't know where the fuck you learned calculus, but calculus does allow us to employ the concept of infinity to arrive at useful conclusions. The entire field is based upon the use of infinity in limits. The idea of infinity being somehow off-limits for rational analysis is bullshit.

There's no waste of time about it; it's impossible unless you can use tricks to reduce the infinite set to a finite one, which doesn't work in the general case of unbounded minimum description length. Generally if there's no way to even complete the analysis of a candidate model, we have to discard it and invoke our prior for 'residual probability to assign to stuff we can't model even in principle' to allocate probability mass to it. The question of what's worth considering when the capability to analyse the model exists, but when resources are limited (i.e. picking the most worthwhile of the still utterly vast set of human-conceivable metaphysical notions to subject to a rigorous formal analysis) is a separate issue and a rather more common one. The answer to that is concrete cognitive expected utility, which ultimately comes from recursed composite Bayes (in a rational system). Allocation of cognitive effort over various time scales is a central issue in AGI, but unfortunately the logic isn't really adaptable to human debates, since no one wants to write or read a 100 page probabilistic logic trace just to find out why further debate of a particular idea is probably a waste of time. Again though, it's important to working out how fully rational reasoners deal with the concept of religion.

Plain-English translation: "I don't see how you can work with infinity, so I will arbitrarily reduce the set to a finite one so that I can use my preferred tools on the problem". See above.

[Kuroneko]

I'm very suspicious of the entire complexity-based approach. Yes, in some sense it is an analogue of Ockham's razor, but any such complexity measure with be critically language-dependent. Instead of having a complexity-based prior, we should be maximizing information entropy instead of the distribution because we have no justification of one thing over any other--which is actually precisely what the original analysis has done (just not in those terms).

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I don't assume it implicitly; I explicitly stated that I was making that assumption (in the sense of being conceivable by intelligences in this universe). AFAIK there are no objective grounds for this assumption, ...

Alright. So why make it?

... because even if we knew that the observable universe was finite, ...

The observable universe is finite, and every astronomer knows this. As for the universe itself, this we don't know, nor is there enough evidence to believe it's even likely.

As I understood it, the original argument here applied to actual religions, that humans can conceive of, and less directly the set of metaphysical propositions that any given finite formal reasoning system can evaluate for probability against a prior.

Not quite. The possibility that every in-universe being is "out of luck" even in principle as far as the "correct" religion is concerned is something that has to be acknowledged in any such analysis, and that means not limiting complexity. It's not even a very outlandish notion--the expectation that universal Creator-entity notices, much less cares about, entities so many orders of magnitude smaller than the universe (which might not even be finite) seems to be a more epistemically special position.

This generalises fairly well to the infinite case when the base prior tapers enough that what you can't model has a trivial probability of being relevant (under reflection).

Why the principle of reflection? Evidence is not a factor for this, so this sort of pseudo-Bayesian approach seems to be inappropriate. (At least I think that's what you're referring to.) Mr. Wong's analysis supposes that there is no evidence at all to favor any one over any other, in which case one should be maximizing the information entropy of the distribution--which is exactly what a uniform distribution does.

There may well be a more abstract method for reasoning about relative probabilities that uses some other measure over transfinite discrete sets, but if there is I don't know about it, nor have I seen any other measures used in this application.

The previous approach maximally embraced indifference and gave them all equal probability, since it only considers competing monotheistic belief systems (i.e., they contain the claim that every other one is false). A uniform distribution is always possible if the requirement for the probability measure (mapping events to their probabilities) to be continuous is optional (i.e., the probability measure may be only finitely additive, or equivalently the algebra of events not necessarily a σ-algebra), in which case it would not be a Kolmogorov probability. This is only a concern for a countable infinity of possibilities.

[Patrick Degan]

I would simply point out this to Starglider: even with my own piss-poor mathematical skills, I still remember one lesson from my classes that the object of the exercise is always to cancel out redundant terms, not add to them.