Thursday, December 25, 2008

God exists (with probability one)

Bonjour Gab. Joyeux Noël. C'est un jour tout particulier pour toi, alors profites-en.

Tu n'aimes pas quand, à la messe de minuit, les imposteurs envahissent ton église pour prier maladroitement à tes côtés, alors je ne prétendrai pas partager ton enthousiasme. Par contre, je voulais te rappeler que moi aussi je crois en dieu — juste pas en Dieu, avec une majuscule. Tu dois te dire que je dis ça pour te faire plaisir. Qu'à mes amis athéistes, je dis le contraire avec autant de conviction. Eh bien en fait... ouain, ça ressemble pas mal à ça.

Et c'est pourquoi ce post est, en quelque sorte, un cadeau de Noël. Pour toi, je fais l'effort de choisir un camp officiel et de l'afficher clairement. Bon, peut-être pas aussi clairement que si je l'avais mis sur Facebook, mais en tout cas c'est là où on arrive quand on tape mon nom sur Google ces temps-ci.

Le reste du post est mon argument "anthropique" que tu connais déjà. Alors, si tu préfères profiter du reste de la journée pour penser au petit Jésus et profiter de Noël, c'est un moment idéal pour fermer cette fenêtre.


The above French comment isn't necessary to understand the following argument. Feel free to disagree violently with my hypotheses or with my reasoning.

The first of these wild hypotheses is that mathematical truths actually exist, in some parallel Platonic world of ideas. My actual belief is stronger: in a twisted interpretation of modal logic, I believe that necessary truths, such as mathematical tautologies, necessarily exist; whereas the facts about our universe are merely ordinarily true, and therefore physical things only happen to exist. I have been led to these unusual beliefs by a convincing argument using the anthropic principle, which I will not repeat here.

Now that we accept the hypothesis, we have to accept a very, very large number of existing things. True statements about numbers. True statements about sets. True statements about true statements. True statements about programs. True statements about programs running games. True statements about a program running an extremely realistic simulation of our physical universe. True statements, one for each instant, describing the entire contents of the simulated universe at that instant. Oh, did I mention that I also assume that our universe is both computable and indistinguishable from a simulation of itself?

At this point I usually picture the statements in the sequence of universe contents to be shown true one after the other, unfolding the story of the universe at the rate at which we feel it happening. But I have come to realize that time is a universe-specific concept, making it implausible to imagine time flowing by in the world of ideas. Because of this, I must admit that I'm not sure how consciousness should enter the picture, but at this point I am at least reasonably sure that (a possibly consciousness-less version of) our universe exists (what a relief!), and even necessarily exists.

An atheist would stop here and point out that since the universe necessarily exists, there is no need to postulate a god powerful enough to bring it into existence. And I totally agree. Nothing so far has argued for or against the existence of a god. Against the lack of evidence, some people (like me!) prefer to assume a god-less model by default, while others prefer to stick to their own favorite meme. But some of us reason one step further.

My next guest in this series of implausible hypotheses is that some humble universe inhabitants are so powerful that they count as gods in their own right. In my book a god is an entity which creates, maintains, and possibly interacts with one or more universe(s). And since I've already assumed that our universe could be nothing more than a simulation, a god could be little more than a programmer with infinite time on his hands. One can certainly imagine a universe similar to ours in which this was true, a universe which would also necessarily exist. In fact, some of us claim that our universe is itself suitable for setting up the kind of infinite computation which running sub-universes require.

With infinite time on his hands, it's not at all a waste for him to run all finite universes, one after the other, until the non-existing end of time. Surprisingly, it's also possible to run all infinite universes, if he interleaves them cleverly. And while you can't assume that god's unsearchable ways will involve enumerating all programs, we've got so many gods in our collection of necessarily existing universes that I'm sure we can find one who likes to do just that.

Running every possibility, god will eventually create a sub-universe looking just like ours and one just like his, in which a sub-god will be running sub-sub-universes including one like ours and one like his, and so on. We end up with infinitely many identical copies of god, and most importantly, infinitely many identical copies of universes indistinguishable from ours.

Let's recapitulate. We have now established the existence of many, many worlds, some of them like ours, some of them different. At the top level are infinitely many of them, all distinct. One of them is indistinguishable from ours. Infinitely many are extremely close to ours, like the many-worlds of quantum mechanics, but not exactly the same. Those top-level worlds exist by virtue of the first hypothesis, the existence of the world of ideas. Again, nothing so far encourages or precludes a pre-top-level god who created that world, so I default to a god-less model. In addition to the single top-level world which is like ours, infinitely many others exist as sub-universes, created by something in a higher-level universe. Those universes do have some sort of god, namely, that thing which created them. It could very well be a machine without a programmer, or a programmer without a machine.

Many-world models are suitable for probabilistic arguments. For example, if a many-world theory postulates a particular distribution of worlds, it should be possible to calculate the conditional probability of an universe having such and such property given that intelligent beings inhabit it. This could be used to make observable predictions about our universe, and test the theory. Today, however, I'm going to be lame and make a non-observable prediction: the conditional probability of an universe being god-less, given that it is indistinguishable from ours, is zero. After all, there is only one such god-less universe, the one at the top level, divided by infinitely many equally probable indistinguishable universes. Hence, god exists with probability one.