Sunday, March 23, 2014

what: Formalization, Mechanization and Automation of Gödel's Proof of God's Existence

Got all that?

Looking up more "mathematical proofs" for the existence of God, I found a Christian Post article.
Two scientists have declared they have proved the existence of God using nothing more than advanced mathematics and a computer.
Yes, let's cue the typical arrogance of the idea that one can, without actually appealing to empirical reality, without spending a single second actually examining reality... learn the secrets of the universe by sitting down and doing math, and then doing nothing to empirically confirm the "conclusions".

Let's dig in.

Godel, who died in 1978, put forward a theory regarding God based on Modal logic that extends propositional and predicate logic that covers operators expressing modality. Modality is a linguistic construction that allows a person to attach expression to a belief set.
Yes, this will be very jargon heavy. Bear with me.
Godel stated that a higher power must exist because, by definition, God can have no creator and which nothing greater can be conceived. 
This statement makes no sense to me.

God must exist because God can have no creator? And which nothing greater can be conceived?  That's just a flat-out non-sequitur. Maybe the CP article is just mucking it up.

God must also be the pinnacle of homosexuality, because there must be an ultimate homosexuality that's greater than can be conceived. Really?
Furthering along those lines if God exists then we could not, by extension, conceive him as greater than if God physically existed, so, therefore God must exist.
I'm still a little fuzzy on the idea that things must exist because we can't conceive them... or can. It's not making any sense either way.

Can they show an example? You know... examples of this mechanism that happen in reality? Well, wait... they're proposing some kind of extra-reality solution... something else they can't demonstrate exists.

Let's let the article writer try to clarify.
... it is roughly the same as stating that if a person is required to believe in the existence of God for proof of a corporeal being than God has to exist or that person would not exist.
... what? Okay, let me see if I got this.

You guys made up a non-corporeal thing that you assert needs you to believe it first before you get the "proof" of its corporeal  version ... SO THEREFORE GOD EXISTS, otherwise the corporeal version wouldn't exist.

I'm not even sure that's a fair interpretation. I've re-read the above statement a dozen times. The language is a little too vague, I think. When they say "proof of a corporeal being", do they mean any old guy walking down the street, or specifically the corporeal version of the god?

The only reason the requirement for "believing in the existence of god as a requirement for proof" is because theists made it up out of whole cloth... not because it's something demonstrated in reality, but because they said so.

I may as well say that one must believe in Zeus, Thor and Odin simultaneously to get proof that the universe exists, and if they didn't exist, the universe wouldn't exist. Since the universe exist, the trio must exist... and it's all hinging on numerous flat assertions I've made.

So at this point, I give up hoping the article writer can correctly convey what the "science" says here. I dug around for the publication they're talking about, and here it is.

It's not long; 2 pages. Let's take a look.
Godel defines God as a being who possesses all positive properties.
The hell does that mean?

By "positive", do we mean numeric, or subjective assessment?

I don't consider creating an eternal torture chamber that everyone automatically goes to, unless we profess belief in the god, to be a particularly "positive" trait. So God doesn't qualify as God then?

Surely, they specify what's meant here.
He does not extensively discuss what positive properties are,
Of course not... and neither do the "scientists" in this publication.
... but instead he states a few reasonable (but debatable) axioms that they should satisfy.
And this is where these sorts of "proofs" break down. The proof is only as good as the axioms, assumptions and premises, which - like here in this article - are tailored to be best conducive to their goals.

This "God is a being who possesses all positive properties" definition reeks of that tailoring.

They probably mean "positive" as in numeric, since this is a mathematical "proof", after all. But lets look at the actual proofiness.

I had to take a screenshot, to preserve the formatting (Sorry, accessible users, though you aren't missing much).

Outside of bad assumptions, it's basically a bait-and-switch... sort of. If we replace all instances of "God" and replace it with "infinity", you'd probably get a better understanding of the structure of the argument:
Infinity is defined as the greatest number. Infinity will always be one more than the biggest number we can conceive. We can conceive of a the greatest number we're capable of conceiving, therefore, infinity exists. Infinity necessarily exists.
... and all the extra baggage that "God" typically comes with - such as the ability to create universes, life; being an intelligent entity, the capacity to love, and the OCD obsession with foreskins - is all cast aside for the duration of this "proof", where "infinity" is merely swapped out with "God". Then, when the proof is done, all that baggage is re-attached as "proven by association".

It's like the typical rendition of the Kalam Cosmological Argument. The best they can achieve is establishing a probability that - given what we currently know about how the universe works - the universe probably had some kind of starting mechanism. Everything "Godlike" beyond that, such as intelligence, creating life, etc, isn't even broached by the argument.

Godel’s proof is challenging to formalize and verify because it requires an expressive logical
language with modal operators (possibly and necessarily) and with quantifiers for individuals and properties.
I'm curious to know what these two knuckleheads (or chuckleheads, either/or), have done to verify the "proof" against reality.
Our computer-assisted formalizations rely on an embedding of the modal logic into
classical higher-order logic with Henkin semantics [4,3]. The formalization is thus essentially done in classical higher-order logic where quantified modal logic is emulated.
In our ongoing computer-assisted study of Godel’s proof we have obtained the following results:
– The basic modal logic K is sufficient for proving T1, C and T2.
– Modal logic S5 is not needed for proving T3; the logic KB is sufficient.
– Without the first conjunct (x) in D2 the set of axioms and definitions would be inconsistent.
– For proving theorem T1, only the left to right direction of axiom A1 is needed. However, the
backward direction of A1 is required for proving T2.

Okay. I'd like to know how their program worked, what assumptions it made, etc. After all, you know, peer review. Please, guys, show your work.
This work attests the maturity of contemporary interactive and automated deduction tools
for classical higher-order logic and demonstrates the elegance and practical relevance of the
embeddings-based approach. Most importantly, our work opens new perspectives for a computer assisted theoretical philosophy. The critical discussion of the underlying concepts, definitions and axioms remains a human responsibility, but the computer can assist in building and checking rigorously correct logical arguments. In case of logico-philosophical disputes, the computer can check the disputing arguments and partially fulfill Leibniz’ dictum: Calculemus — Let us calculate!
And nope, we're done. Those were the last two sections of the publication.

Yep, this certainly "opens new perspectives for a computer assisted theoretical philosophy", such as that it's as bullshit as the analog versions of the arguments. Garbage in, garbage out. If only they put as much effort into fixing the core logical fallacies (lots of begging the question, bait-switching, etc), as they did in programming something to try to confirm it.

It'd be more impressive to me if their whole project was trying to disprove/falsify it. The fact that me, as a layman, can pick out the egregious errors, isn't painting a flattering picture of this whole argument.

These people aren't dumb. Their intelligence is misapplied... channeling that intellectual energy into trying to justify fantasy, rather than put it to some constructive, useful effort. If they took the same level of effort, and put it into writing novels/books, they'd probably be best-sellers.

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