Tuesday, August 29, 2006

Gödel's Proof of the Existence of God

This summer I gave a workshop on "Mathematics and Religion", and among the items I covered was a rather arcane and strange proof of the existence of God by Kurt Gödel, the mathematician who proved that a mathematical theory which includes number theory can never be complete, and it will never prove by itself that it does not contain contradictions. His argument is this: God is that entity that has all positive properties. Existence is a positive property. Therefore, God has existence; he exists. Another version was presented by Anselm centuries ago: if there is a God, it is a necessary truth that he exists; in other words, "God exists" is like 2+2 = 4 instead of like "blue jays have feathers". It is conceivable that a world could exist that has a God in it. Therefore, God must exist in all worlds.

I will make an analogy to demonstrate that Gödel's argument is fallacious. Let's take instead a young man seeking a female partner. He has preferences; he prefers women who like Vangelis over those that prefer Van Halen; he prefers Unitarian Universalists over Methodists, and so forth. So he defines properties of women. He says that a property P of women is Athena-positive if he prefers women who have property P over those that don't. This relation has some properties of its own: If a property P is Athena-positive, then not-P is not Athena-positive and vice versa. The conjunction P1 and P2 and … and Pn of a bunch of Athena-positive properties is itself an Athena-positive property, because he clearly wants his woman to have all these properties. In other words, Athena-positive defines on the set of properties something mathematicians call an ultrafilter.

He now defines Athena (he picks this goddess because he admires her properties a lot) to be that woman who has all Athena-positive properties. He now thinks about whether such a woman exists. He notes that existence is a property. For example, Brooke Shields satisfies the property but Scarlett O'Hara does not. Further, he notes it's an Athena-positive property, because he would rather have a real woman to date and be able to relate with than a figment of his imagination. Since Athena is defined as that woman who has all Athena-positive properties, and since existence is an Athena-positive property, he concludes that Athena exists. "Well dagnabit", he says. "If indeed she does exist, why hasn’t she dropped me a line?"

He is also looking for a job. He has preferences; for example, he would rather be an actor than a janitor. So he defines properties of jobs. He says that a property P of jobs is Calling-positive if he prefers jobs which have property P over those that don't. Like with Athena-positive properties, Calling-positive properties form an ultrafilter. He defines his Calling as being that job that has all Calling-positive properties. Since he would rather have a job that exists instead of one that doesn't exist (and therefore provides him no money), existence is a Calling-positive property. Since the Calling is defined as that job that has all Calling-positive properties, and since existence is Calling-positive, he concludes that his Calling, his ideal job, really does exist. "If so, I would like the company with that job opening to contact me immediately.", he thinks.

Do you notice the problem with both of these ideals? The problem is that you can define Athena to be a woman with all Athena-positive properties. That does not necessarily mean she exists. Even if existence is an Athena-positive property, all that we have shown is that all Athenas that exist and have all Athena-positive properties exist. That does not necessarily mean that one exists. Saying that existence is a certain type of property does not mean that one exists. The same holds true with his Calling.

And the same holds true with Gödel's argument. All that Gödel proved was that all Gods with all positive properties that exist, exist. He has not shown that one necessarily exists. For if he did, all of us would be in perfect marriages and in ideal jobs. A look into the homes and offices of people in our society will reveal that that is emphatically not the case.


Blogger Sbiis Saibian said...

This reminds me of a very strange thought experiment someone came up with once.

The thought experiment was as follows :

Imagine there is a disease that can kill you even if your not infected. It follows that even if it doesn't exist you would die. So why is everyone in the world not dead ?

Even stranger is that the person also hinted at a possible solution : because there is a cure which can save you even if you don't take it.

Yet there seems something very flawed about this kind of reasoning. It's almost as if logic can compel things to suddenly pop into our reality upon thinking something up, and yet, they fail to do so ... so something is off with this line of thinking.

The flaw seems to be a hidden assumption that the object already exists. It boils down to saying something like "there is an object which exists whether or not it actually does".

Many arguments for the existence of God seem to be of this variety. You start with the assumption that their may or may not be a God, but it doesn't matter because logic says there must be one either way.

In any case if there is an object which exists whether or not it actually does, God would certainly seem to be the only likely candidate; Not that that proves anything though. :)

-- Sbiis Saibian

8:50 PM  

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