I was unclear. I was referring to “all possible physical theories (there are only countably many, if they are to be expressed in finitely many symbols)”, which seems to be the same object Nelson uses. He takes that finiteness to imply that scientific naturalism can’t explain a possibly infinite amount of data. One can analyse this in several ways; my first reaction was that I remembered someone using Gödel to argue that physics is forever extendable if necessary.

I see that you mean now by not needing falsifiability in your analysis. I still feel that Occam tells us to junk completely the theories that have low probability. I’ll have to think about that.

In his most recent post, Cosmic Variance’s Mark Trodden talks about one of the presentations we both saw at last week’s meeting in Ishcia, where he explains one of the hot new techniques for analyzing cosmological data, the (so-called) Bayesian Evide…

I have no idea how Bayesian analysis of theories tells us anything about the truth or falsity of naturalism. I’ll have to think about the implications of Godel’s theorem.

Second, you use some sort of Bayesian metaanalysis to put off falsifiability. Even though a frequentist may say Bayesian probabilities are about beliefs, apparently these beliefs enable decisions for setting parameters (to 5 sigma, say). Isn’t it reasonable to do that for each theory individually (and therefore get falsifiability)? Anyway, shouldn’t the razor support the use of one specific theory at a time?

The point I was making is that (1) you can never completely falsify any theory, and (2) there is no need to falsify theories. If you need to know the magnetic moment of the electron then you can just perform the weighted average of the predictions made by all possible theories (weighted by the likelihood that each theory is true). If one theory (for instance, QED) is much better supported than any of the competing theories, then its posterior probability will be almost 1, and so its predicted value will dominate. You don’t ever need to consider which theory is best supported by the evidence, because the Bayesian probabilities automatically take all evidence into account.

Now, of course in practice we can’t keep track of all possible theories and all the evidence for and against each, so we simplify matters by throwing out all except for one or two likely candidates. But if the complete Bayesian analysis were possible to do, it’s hard to see how it would ever give us a worse prediction than our current approach (except by luck).

Daryl,
I have two questions you could help me with.

First and foremost, a creationist philosopher (Paul Nelson) use much the same construction as you do to arge that naturalism isn’t sufficient for science. I have earlier seen Gödels first incompleteness theorem used to argue that since even some simple formal systems can not be given a computably enumerable axiom list, physics will be forever extendable. Which is it? I would like to see if Nelson is wrong here.

Second, you use some sort of Bayesian metaanalysis to put off falsifiability. Even though a frequentist may say Bayesian probabilities are about beliefs, apparently these beliefs enable decisions for setting parameters (to 5 sigma, say). Isn’t it reasonable to do that for each theory individually (and therefore get falsifiability)? Anyway, shouldn’t the razor support the use of one specific theory at a time?

Physicists tend to think in terms of theories being completely overthrown by some observation, but there is a sense in which no single observation is ever really sufficient to reject a theory. It’s always possible that any single observation, or even any finite collection of observations is a fluke. So there really is a subjective judgement going on in the background. How improbable must a fluke be before you are convinced that the theory was wrong, and the contrary evidence wasn’t just a fluke?

It seems to me that Bayesian probabilities allow you in principle to put off forever making the decision as to whether a theory has been falsified or not. Instead, you can imagine starting with all possible physical theories (there are only countably many, if they are to be expressed in finitely many symbols), with some subjective a priori notion of likelihood. Then every experiment performed can be used to adjust the posterior probabilities.

With this approach, as time goes on, Newtonian physics would start off with a low probability, but then would gradually rise with the successful predictions about projectiles and planetary motion, but then would drop sharply with the discoveries of quantum and relativistic phenomena. But it would never go to zero.

If you are trying to make a prediction, you would weight the predictions made by every conceivable theory, according to the posterior likelihood of each theory.