Sunday, September 20, 2009

Reflection on John Norton's argument

Here and referred to below.

Suppose we have a physical quantity Q that must take its value between 0 and 1, inclusive. Let us stipulate that our current knowledge has absolutely nothing to say about the possible value of Q. It is very tempting to say that a priori, Q is uniformly distributed on [0,1]. This is nothing but a theoretical prejudice. We would in effect be saying that our utter lack of evidence means that it is nine times more likely for Q to lie between 0 and 0.9 than for Q to lie between 0.9 and 1. In reality the support for Q to take on some set of values is exactly equal to that for Q not to take on that set of values (as long as the set is not the entire set or equivalently we do not try to assert that Q takes on no value whatsoever.)

This may be easier to see if we map [0,1] to [0,infinity). There is no uniform distribution on [0, infinity); and our theoretical prejudice explicitly manifests itself when we try to assign a probability measure for Q on [0,infinity).

In a physical theory, we need a physical reason to believe that Q has a probability distribution. In Norton's language, we need a randomizer in order to induce probabilities. E.g., in many situations we have a good reason to believe that a quantity follows a normal distribution, because it arises as an aggregate from a large number of underlying processes and the law of large numbers holds. In statistical mechanics, we have good reason from Hamiltonian mechanics to assume that a system in thermal equilibrium is well represented by assuming a uniform distribution over its microstates. And so on.

2 comments:

CapitalistImperialistPig said...

We try to make sense of the world in terms of our current world picture - it seems to be part of our internal programming. Where we lack guidance from theory, assuming any particular kind of Bayesian prior is merely a guess. Which is one more reason I consider any kind of anthropic reasoning pretty worthless.

CapitalistImperialistPig said...

I think that it is very difficult to go beyond the ancient prescription that science should make testable quantitative predictions.

More sutleties are involved when we consider puzzles like the distribution of intelligent life in the universe. There we actually have a number of well-tested theories that bear on the question, even though we have only one lonely example.