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Gödel's ontological proof (wikipedia.org)
12 points by hhm on Jan 3, 2009 | hide | past | favorite | 9 comments

"The conventional view of the task of the philosopher is to solve seemingly intractable problems of philosophy using logical analysis (for example, the problem of free will, the relationship between mind and matter, what the good or the beautiful or the true consist of, and so on). However, Wittgenstein argues that these problems are, in fact, "bewitchments" that arise from philosophers' misuse of language.

In Wittgenstein's view, language is inextricably woven into the fabric of life, and as part of that fabric it works relatively unproblematically. Philosophical problems arise when language is forced from its proper home and into a metaphysical environment, where all the familiar and necessary landmarks and contextual clues are absent - removed, perhaps, for what appear to be sound philosophical reasons, but which lead, for Wittgenstein, to the source of the problem. Wittgenstein describes this metaphysical environment as like being on frictionless ice: where the conditions are apparently perfect for a philosophically and logically perfect language (the language of the Tractatus), where all philosophical problems can be solved without the confusing and muddying effects of everyday contexts; but where, just because of the lack of friction, language can in fact do no actual work at all. There is much talk in the Investigations, then, of "idle wheels" and language being "on holiday" or a mere "ornament", all of which are used to express the idea of what is lacking in philosophical contexts. To resolve the problems encountered there, Wittgenstein argues that philosophers must leave the frictionless ice and return to the "rough ground" of ordinary language in use; that is, philosophers must "bring words back from their metaphysical to their everyday use."


It's a highly technical and (I think -- though of course I'm no Goedel) bullshit-rich version of St Anselm's ancient "ontological argument" for the existence of God. ("Ontological argument" is a pretty stupid name. Blame Kant.)


0. Use modal logic. (Goedel's "proof" requires that you be able to say things like "for some x, necessarily P(x)", which -- if you think about modal logic in terms of possible worlds, which many people do -- means that you need to be able to think of particular objects existing in multiple worlds. It is far from clear that this really makes sense.)

1. Assume that all properties of things can be classified into "positive" and "non-positive" properties, and that a few boring technical axioms about "positivity" hold. Intuitively, the "positive" properties are supposed to be the ones it's good for something to have. (I see no reason to believe that there's any notion of positivity that's close enough to the intuitive one but that has the technical properties Goedel wants.)

2. Say that something is "godlike" if it has all positive properties. (We're aiming to prove that something godlike exists. Note that even if everything else works, this will only be a proof of the existence of God if "positive" really does have something like its intuitive meaning. That's why the tension between that meaning and all the technical requirements for "positive" properties is important.)

3. Theorems: (a) for every positive property, it's possible that something exists that has that property, and (b) in particular it's possible that something godlike exists. (The fact that these really are theorems, at least if you use a suitably chosen modal logic, is one reason why I think it unlikely that any notion of "positivity" exists that both satisfies Goedel's conditions and matches up with intuition.)

4. Say that x "essentially has property P" if x has P, and any other property Q that x has is a necessary consequence of P. (Kinda weird, but never mind.)

5. Say that x "necessarily exists" if all its essential properties are necessarily instantiated; i.e., for every essential property P of x it's necessary that there's something with P. (It seems to me to be stretching it to call this "necessary existence", but never mind.)

6. Claim that "necessary existence" is a positive property. (Seems pretty arbitrary. Anyway, this is the key point at which Goedel's "proof" makes contact with Anselm's.)

7. Now it turns out that we can put the pieces together and deduce that there necessarily exists something godlike.

The "proof" depends on taking a very fuzzy intuitive notion, that of something being "positive" or "good", assuming that it can be treated with the utmost formality, assuming that a bunch of highly technical assumptions apply to it (e.g., "if a property is positive, then necessarily it is positive" -- which only even makes sense if you go beyond first-order modal logic, and I'm buggered if I can see why it should be true), and then seeing what follows.

This is essentially the same procedure that yields the following (absurd) proof that God doesn't exist: If God existed, he could make something too big for even God to move; but if God existed, nothing could be too big for him to move; contradiction. This is, I repeat, absurd, but its absurdity is of just the same sort as Goedel's argument depends on.

It is fascinating that someone as sharp as Goedel could argue this. But then again, this might be exactly why he never published the "proof" - he could see there were holes in it big enough to drive a truck through.

What is the meaning of this?

"although he did not go to church, was religious and read the Bible in church every Sunday morning"

Yeah, I was baffled by that too. Maybe "in church" should be "in bed"? Maybe he didn't go to the church services, but found some time every Sunday morning when the church wasn't in use to sit there and read the Bible?

... Ahaha, bingo. Hao Wang's "Reflections on Kurt Goedel":


"In January 1978, G's wife told me that G read his Bible in bed on Sundays."

Time to go fix the Wikipedia article.

(The WP article actually references another of Wang's books for this. That's googleable too, and also says "bed" not "church".)

cf. also Alvin Plantinga's modal-logic-using Ontological argument:


God, by definition, is that than which a greater cannot be thought.

Doesn't this assume some kind of total ordering of thoughts and wouldn't that lead to a Cantor-style paradox?

I don't think so. (Congratulations, you've found something that isn't wrong with the ontological argument!)

1. You can define something to be "godlike" if nothing greater than it can be conceived, in which case the conclusion of the ontological argument (if it worked) would be that at least one godlike thing exists. That's probably enough for anyone who actually wants to use the argument; they'd probably say that "obviously" uniqueness is a kind of perfection, or something.

2. The nearest thing I can see to a Cantor-style paradox would be if somehow the totality of thinkable things were necessarily "greater" than any particular thing. But "greater", whatever it's meant to mean (the vagueness of the terms is one of the problems with the usual ontological argument) isn't the same as "bigger", and you could probably get away with arguing that the totality of all thinkable thoughts isn't so "great" because it involves inconsistencies, or something.

(Given some of the other arguments Anselm makes, which involve saying e.g. that something that causes greatness must itself be great, I think he would have had trouble making that last argument. But the ontological argument is so weak that I feel one ought to make all possible excuses for it :-). )

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