Can God appear in a puff of logic?

Saint AnselmLogic is a tricky thing. Any sound argument must rely on it, but it is easy to build seemingly sound and logical arguments that are still wrong or fail to apply to the real world. Fuzzy or wrong premises, shortcuts in reasoning, as well as plain fallacies such as circular reasoning, are easy to obfuscate, and apologists are kings at this game. It's what they do: take the conclusion they want to reach, and then build the rationalization for it. A prime example of this is the age-old ontological argument for the existence of God, that I will be looking at in details in this post.

The argument is that because we can conceive of a perfect being (defined by the impossibility to improve it), then it must exist for surely existing is better than not existing. Really? We'll see.

But first, let me quote Douglas Adams...

Now it is such a bizarrely improbable coincidence that anything so mindbogglingly useful could have evolved purely by chance that some thinkers have chosen to see it as a final and clinching proof of the non existence of God.
The argument goes something like this: "I refuse to prove that I exist," says God, "for proof denies faith, and without faith I am nothing."
"But," says Man, "the Babel fish is a dead giveaway isn't it? It could not have evolved by chance. It proves that you exist, and so therefore, by your own arguments, you don't. QED."
"Oh dear," says God, "I hadn't thought of that," and promptly disappears in a puff of logic.
"Oh, that was easy," says Man, and for an encore goes on to prove that black is white and gets himself killed on the next zebra crossing.

What Douglas Adams articulates so brilliantly here is that with badly defined premises and "pure logic", you can prove anything and its opposite, and that therefore you can prove nothing. There is no such thing as a puff of logic of course, as puffs are physical, and logic is mathematical, independent of the physical world, and therefore utterly unable to puff. Of course, I could have quoted Hume and Kant to pretty much the same effect, but this is a lot more fun, isn't it?

To drive the point home, let me paraphrase a reverse formulation of the argument I found in the comments of Ambrose's recent post on the subject:

We can conceive of maximal evil, for which one cannot possibly imagine anything more evil. Surely, it must exist, as something maximally evil would be quite benign if it didn't exist, and would assuredly be more evil if it existed. Therefore, it exists.

Oops. Putting empirical credibility aside, it doesn't look any more or less logically sound than the original argument. So where's the flaw?

What most people call "pure logic" is actually much trickier to define than they may think. I learned that in France a little more than 20 years ago when I was preparing the entry contest for college. One of the students in my class was an orthodox Jew, convinced that the world was 6000 years old, but also a genius, who had already explored Mathematics way farther than any of us. What he taught me was that words are not appropriate to do mathematics. One must be absolutely formal in order to avoid talking nonsense. Here is the example he used, also known as Russel's paradox:

A mathematical set is a pretty simple entity, right? It is defined by its elements. OK, so now consider the set of  non-auto-inclusive sets, defined as the set of all sets that do not contain themselves. Well, that set cannot include itself, by definition, because all its elements are non-auto-inclusive. Therefore, it must include itself since it doesn't.

Uh? Yeah, exactly. Mathematics don't have paradoxes, they only have reductio ad absurdum. This so-called paradox only proves that the naïve concept of set we used here is inconsistent. In particular, the notion of a set of all sets can't be rigorously defined, although an English formulation of it seems to present no challenge. This is known as the naïve set theory, and it had to be replaced by something much more rigorous, which eventually led to a re-foundation of all of Mathematics by the Bourbaki group. This is an eminently modern idea that  Anselm of Canterbury, Kant, Leibniz, Descartes or Plantinga could not possibly have known. We need to apply formal logic in order to determine what in the ontological argument is valid formal logic and what constitutes its premises and hidden assumptions.

Several people have done exactly that with varying success, but the attempt that I find the most interesting consisted in feeding the argument into a computer algorithm that automatically proves mathematical theorems. If that wasn't awesome enough, the good news is that the algorithm not only showed the logical soundness of the argument, it was actually able to simplify it and reduce the assumptions to a single one. The bad news is that this remaining assumption is not trivial. Here it is:

If the conceivable thing than which nothing greater
is conceivable fails to exist, then something greater than it is conceivable.

Makes sense? Suffice it to say that this still needs independent justification that cannot be reduced to formal logic. Back to square one are we? You can still argue one way or the other, but you are outside of the realm of logic doing so, which pretty much means that the argument, while quite subtle and logically sound, is not a complete proof of the existence of God.

Before I conclude this post, I'd like to point out that such attempts to make God appear in a puff of logic are not only doomed logically, they also constitute poor theology (assuming for a second there is such a thing as good theology). For really, doesn't it degrade the idea of God to reduce it to something that can be described and constrained by mathematical expressions? Doesn't that bring him down to the realm of the natural?

Tags: Religion, Math, Science

Sunday, July 17, 2011 6:30:00 AM

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  • bleroy said

    That's quite simple: we are all puzzled by how bad and inconsistent it all is and still, people take it seriously. We just want to provoke you into giving us all you have, just to see if there is going to be some intellectually challenging stuff coming out of it, at least. No such luck though.

  • Ambrose Little said

    I would say that it's not bad theology but vastly incomplete, as I in fact did say in my post.

    Needless to say, I'm not convinced by your post that there's a problem with the argument. It seems to be more about asserting that it is inappropriate than actually dealing with the argument itself, and I also am not convinced of that. But then, I am not trained in mathematics. ;)

  • bleroy said

    Ambrose, you might have misunderstood the post. The argument is actually perfectly fine and sound, which was a surprise to me and is quite neat seeing how old it is. The point is to understand which parts are logic and which parts are not. It is a surprise to me that the parts that are not logic are that small. But small is not nothing. That it exists does not invalidate the argument, but it invalidates that it is sufficient to assert anything about the real world unless you first assert the premise (hardly a surprise as far as I'm concerned). You are still free to assume that this last premise is true or false but it is a matter of opinion, not logic. I don't think there is a way around that one.
    To be clear, this does not prove the non-existence of God, it shows what it takes to be able to apply this specific argument. It takes nothing out of your faith, so you can and should feel free to accept the maths here, I think.
    As for the theology thing, it's kind of a joke and a provocation as I actually don't take theology very seriously in general, so I really don't have much to say about it.

  • bleroy said

    Let me quote the conclusion of the last paper I pointed to (http://mally.stanford.edu/Papers/ontological-computational.pdf), hoping that can clarify the conclusion, which as far as I can tell is as sound as it gets, coming as it does from formal logic:
    "In the case in which the description [...] denotes and the object it denotes exists, the antecedent of Premise 2 is false, making Premise 2 true. But the defender of the ontological argument can take no comfort from such an observation, since it defends Premise 2 by using the conclusion of the ontological argument. That is, if she uses the existence of the conceivable thing than which no greater thing is conceivable to prove Premise 2, she is guilty of circular reasoning. She needs an independent argument to support the premise. Thus, arguments (1) and (2) above show that the defender of the ontological argument needs independent support for two claims: that the definite description denotes and that Premise 2 is true. [...] Though the simplified ontological argument is valid, Premise 2 is questionable and to the extent that it lacks independent justification, the simplified argument fails to demonstrate that God exists."

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