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Alexandros comments on Open Thread June 2010, Part 4 - Less Wrong
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I started writing something but it came up short for an article, so I'm posting it here:
Title: On the unprovability of the omni*
Our hero is walking down the street, thinking about proofs and disproofs of the existence of a god. This is no big coincidence as our hero does this often. Suddenly, between one step and the next, the world around her fades out, and she finds herself standing on thin air, surrounded by empty space. Then she hears a voice. "I am Omega. The all-powerful, all-knowing, all-good, ever-present being. I see you have been debating my existence with true purity of heart, so I have decided to provide you with any evidence you request". Once the shock wears off, our hero runs through the list of possible requests she could make. Healing the sick? Perhaps the reanimation of a dead person? Some time-travel? Maybe this could still be doubted. How about creation of a solar system? Or a universe? Maybe a proof of P vs. NP? Alas, our hero realises that any evidence she could request would only be proof of the power of Omega to produce just that thing, not an inclusive proof.
What's more, our hero knows that her thinking is subject to the operation of her mind and the readings of her senses, something she cannot trust in the presence of a vastly overpowering entity. The lower bound of power required of Omega to produce any experience for our hero is much lower than the power to create universes. It is the ability to control only the senses of our hero, become a kind of hypervisor, and simulate all requests. While this is great power indeed, the distance from there to omnipotence is great indeed. Similarly for omniscience, omnipresence, and omnibenevolence.
Our hero does not ask anything of Omega, and their meeting ends uneventfully, at least in terms of new universes being created, or problems thought unsolvable being solved. She does realise though, that omnipotence, omniscience, omnipresence, and omnibenevolence are not properties that can be verified by a human. If this is the definition of a god that theists are working with, then it is not only undisprovable, it is also unprovable. Taking knowledge to be 'justified true belief', a belief in an omni* god can never be justified, putting if firmly in the territory of the unknowable. The strongest claims that can be reasonably made are that of a being that is very powerful, very knowledgeable, etc. But that is not nearly as interesting.
Now, I have posted a question along those lines in this thread before, with little response. What I would like your feedback on is whether this is a reasonable argument, whether I've gotten something completely wrong in my epistemology, and whether there have been similar arguments made by others. All help appreciated, cheers.
Wait... a being which, while possibly not omni-anything, is likely very powerful, offers to provide her any evidence she likes, and she considers and rejects the "healing the sick" and "resurrecting the dead" plans?
Not to mention a solution to the P=NP problem (or the Riemann Hypothesis)?
A super-powerful agent who is desperate to prove itself to her! That's the perfect opportunity! Unless she messes up the requested 'proof' she has can become a demi-god, just below the Omega (until Omega cracks it with her).
That should result in an exponentially growing multiverse of universes, with each universe self-replicating on a sub-nano-second time frame while simultaniously expanding in size and neg-entropy, all arranged for maximum Fun. Still not proof of Omnipotence but hey, it'll do.
That's a good point. Any ideas on how to mend the hole?
Have Omega offer to provide the proof but then will ask for an answer to the question of whether he is actually omni*. If the answer is incorrect he will destroy the world, if correct he will let the world continue with whatever changes were made by the "wish". There is also the choice not to play.
You would have to make him non omni benevolent though.
Can we not get around this by using randomly chosen questions? And then we have IP=PSPACE, so anything that's in PSPACE, he can relatively quickly convince us he can solve. Obligatory Scott Aaronson link.
Thanks for that link, it was quite good. Any chance you could elaborate a bit on the IP=PSPACE identity?
No, I don't really know complexity theory at all, so I couldn't really tell you any more than Wikipedia could.
What if your hero asks to be made omniscient, including the capacity to still be able to think well in the face of all that knowledge?
Throw in omnibenevolence if you like, but I think you get some contradictions if you ask omnipotence. Either that, or you and Omega coalesce.
How could you test your omniscience to be sure it's the real thing?
Asking to modify yourself may be a useful strategy, (or maybe not, as you note) but it's not something that's available to philosophers trying to prove the existence of a god. As far as we know that is :)
It's possible that looking at how you'd test something which claims to be omniscience would give some pointers to finding unknown unknowns and unknown knowns.
Or also show you if there are unknowable unknowns?
An unknowable unknown: I shot a rocket across the cosmic horizon. On the rocket was a qGrenade set to detonate on a timer. Did my Schrödinger's rocket explode when the timer went off in my Everett branch?
I don't see that decoherence would occur in that case.
This once again explains why "reality" is a largely meaningless concept.
Wow. I maybe understand where you are alluding to, but I'm not sure I'm reverse engineering the thoughts right. Explain for me?
Whether or not it's meaningful, it's certainly useful, especially by Phillip K. Dick's definition: "Reality is that which, when you stop believing in it, doesn't go away."
I'm pretty sure unknowability would have to be proven rather than shown.
Nothing is provable to the level you demand (well, pretty much nothing, cogito ergo sum and all that). Given that none of the omni* are well defined, the question doesn't mean much either.
Are you saying that it's an inference problem and after enough pieces of evidence we should just accept omnipotence (for instance) as the best hypothesis with a high degree of confidence, as we trust gravity now? How about the mind control problem?
Also, what you say about the omni* being not well defined sounds interesting. can you elaborate?
That's exactly what I'm saying, and you're right to point out that mind control will always be a more probable explanation than omnipotence (as will mental illness). If I knew that something would continue to apeear omnipotent, I would just treat it as omnipotent (which equates to "accepting the simulation" if the actual explanation is mind control).
Omnipotence is badly defined because it leads to questions like "Can Omega create a rock so heavy that Omega cannot lift it?", can omnipotent beings create logical contradictions? Can they make 2+2=3? Omniscience leads to similar problems, can Omega answer the halting problem for programs that can call Omega as an oracle? Omnibenevolence is the least paradox ridden, but the hardest to define. Whose version of good is Omega working toward?
Your logic is ok. By the way Thomas Aquinas thought along this lines, but in different direction. However discussing scholastic here doesn't make much sense (if it can make sense at all).