A model of knowledge of the existance of god

Say I have two urns:

One is filled with numbered green balls, all of which lie in the range of 1 to 100.
The other is filled with numbered red balls all of which lie in the range of 1 to 500.

I draw a sequence of balls from a single urn, announce the numbers and I then ask you what color you think the urn is and to what degree of probability. Notice though, that the distribution of numbers is not specified as uniform within that range, it’s possible that there are twice as many odd numbers as even numbers for example.

Obviously, if there is a single ball >= 101, then you can assert with 100% probability that the urn is green. However, there’s no possible sequence of balls that could definitively prove a red urn.

It seems, without knowledge of the distribution, it’s impossible to be able to give precise probabilities for any other outcome. However, I could argue for certain properties of the probability. For example, it seems intuitive that the drawing of an additional ball under 100 can never decrease my posterior probability of a red urn, it can only leave it unchanged or increase it.

Also, it seems intuitive that drawing of “1, 3, 3, 2, 4, 3, 6, 8” provides less evidence of a red urn than “99, 93, 100, 96, 99, 98, 96, 100” because the first sequence can be easily explainable under both the green urn and red urn scenario but the second distribution can only be easily explainable under the red urn scenario. In fact, you might even be able to generalize it and say “the closer a ball is to 100, the greater it increases the posterior probability”

I view this as analogous to the problem of the existence of god. Under a certain formulation of god*, the space of possible universes in which god does not exist is a strict subset of the space of possible universes in which god does exist. It’s therefore strictly impossible to prove that god does not exist. However, you can view each observation as drawing a ball out of an urn and each observation can be consistent with an atheistic or supernatural interpretation of the world.

I think this analogy has helped me clarify both how to correctly think and more importantly, how to explain to others how to think about the existence of god. Yes, it’s strictly true that you can’t disprove the existence of god but by putting it into concrete terms of the urns, they can understand how evidence can be gathered for either side.

However, the urn model is still not perfect. It seems intuitively obvious to me that watching a rock fall over and over again should only have an insignificant effect on my posterior probability. Additionally, the space of supernatural universes is not 5x bigger than atheistic universes, it’s many googleplexes bigger, perhaps possibly infinitely bigger. However, it also seems intuitive to me that the size of the range of the urn should not have an effect on how to evaluate the posterior probabilities.

Any thoughts?

  • In this case, a being who possesses powers outside of the realms of scientific law but not tied to any particular religious interpretation of God.

I’d say that if you can prove it, it ain’t god.

This seems to be an epistemological question/discussion much more general than confinement to the simple question of God’s existence.

I’m glad you realize that much depends on the probability distribution of balls in the urns, which can be all kinds of things. But you also clearly have certain properties in mind for what a reasonable distribution to postulate would be (and why not? Without this, we’d have no such thing as inductive logic). Along these lines, let me ask: why do you feel drawing a ball like 99 should give more reason to believe in the red urn than a ball like 1?

It’s late, so maybe I’m missing something, but did you reverse red and green here?

This makes no sense whatsoever unless you’re assuming a basically random distribution (and even then it’s pretty sketchy). Whoever filled your urns could have a sense of whimsy. Or just like numbers in the high 90’s.

Also confused by this usage of “subset”. The space of possible universes in which god does not exist is a proper subset of possible universes. So is the space of possible universes where god exists. But one can’t be a subset of the other. Further, you don’t know anything * a priori * about the size of “supernatural” universes, so you can’t state anything about its relation to “atheistic” universes.

he did flip the red/green urn there a bit, but you know what he meant.

That said, drawing 1, 2, 3, 4 etc doesn’t prove a leaning towards either urn just as drawing a 90, 91, 92, etc doesn’t, not if the balls are randomly distributed and you have no knowledge of said distrobution before. Now, if each urned contained EVERY number between 1-100 and 1-500, respectively, than you could make this arguement. But say if each urn had maybe 10 balls that were randomly distributed, one could contain ‘90-99’ while the other contained ‘90-98’ and ‘500’

That said, proving the existance of god is equally as impossible of proving the lack of existance of god, it’s just not something with factual, conclusive evidence supporting or dismissing it.

It’s an intuition, not a solid statistical

This could be a bit confusing because yes, the space of universes are disjoint sets, you can’t have something which is an atheistic universe and a supernatural universe. But the set of observations is a strict subset. There exists a set of observations you could make which is only consistent with a supernatural universe but every set of possible observations you could make about an atheistic universe is also possible in a supernatural universe (If you assume god is present but makes no detectable action upon the world). In the example of the urns, red is a disjoint set from green but 1…100 is a strict subset of 1…500.

I’ve very carefully delineated out god from Gods. Yes, you can add a bunch of other parameters onto the definition of god such that it does become possible, in principle to disprove it and that’s another road to take but in this case, I’m using god in the most general sense possible.

After talking to my friends, I realised the OP was slightly confusing because I was making two seperate points:

  1. It seems possible to prove theoretical points about statistical systems even if you explicitly remain ignorant about distributions

  2. Using urns and balls seems like a simpler and more intuitive way of explaining the logical error in the claim “you can’t prove that god does not exist” better than occam’s razor or other attempts to people not already familiar with the arguments.

With regard to point 1, it’s very much an intuitive argument rather than a mathematical one. I guess, thinking about it even further, what I’ve implicitly assumed is that distributions will be weighted towards simplicity (or kolmogorov complexity if you want to get technical).

I don’t get at all why proving the existence of god would be challenging at all. It seems it could be posited as a scientific hypothesis just like any other. Here’s what would convince me god existed:

A black box appeared in the world which was a universal compressor. Any stream of data fed into it, it could compress to 50% the size and then feeding that corresponding stream back into it would produce the original text with 0% loss.

Assuming we do all the due dilligence to make sure it’s not doing any clever tricks or using any side channels, that would convince me. Of course, you run into the problem of maybe you didn’t take something into account but it’s the same problem with any scientific hypothesis. We feel comfortably saying that gravity exists even though the same criticisms could apply.

Other miracles that would convince me:

The sun standing still for 24 hours
The red sea turning from water into blood
A rod turning into a snake and then back into a rod again
Repeatable and measurable examples of ESP
Repeatable and measurable examples of being able to tell the future
Magic like how it works in any fantasy novel

A universal compressor is mathematically impossible; there are 2^N bitstrings of length N, and only 2^N -1 bitstrings of length less than N, so there can be no injective function from the former to the latter. It’s like saying “Here’s what would convince me God existed: A black box appeared in the world with an odd number divisible by 6 written on it.”

The rest I have no objection to.

None of those would be proof, since they could be faked by beings more powerful than us, but less so than God. Especially since, unlike a natural law, you need to take in account the possibility of deliberate deception - gravity isn’t trying to trick you, a claimant to godhood might. And once you get up into the power range where they can completely fool your senses and alter your memories, there’s no way for you to prove anything at all. Even if you think you succeeded, or saw something you’d consider proof, you couldn’t tell if the memory was real or not. Or, for that matter if your opinion of what counted as proof was your own idea in the first place.

And of course there’s the possibility that you are hallucinating without help from some pseudo-god, which is more likely than either a pseudo god or a violation of physics created by an actual God ( whatever that may mean ). If I see a rod turn from a rod to a snake and back again, someone slipping drugs into dinner is more likely than anything so unlikely being real.

I think we’d need at least some definition of god to be able to separate the possible universes into those that contain one and those that don’t, or to argue that there are more of the former kind than there are of the latter. (Indeed, it isn’t even clear whether universes containing something we would generally identify with a deity can exist at all, i.e. if the assumption of the existence of ‘powers outside of the realms of scientific law’ is consistent with the assumption of the existence of a universe.)

Also, as for your urn example, it seems perfectly consistent with your premises that both urns only contain balls in the 90-100 range, and thus only drawing balls with those numbers isn’t evidence either way.