## Friday, May 24, 2019

### A way forward on the normalizability problem for the Fine-Tuning Argument

The Fine-Tuning Argument claims that the life-permitting ranges of various parameters are so narrow that, absent theism, we should be surprised that the parameters fall into those ranges.

The normalizability objection is that if a parameter ξ can take any real value, then any finite life-permitting range of values of ξ counts as a “narrow range”, since every finite range is an infinitesimal portion of the full range from −∞ to ∞. Another way to put the problem is that there is no uniform probability distribution on the set of real numbers.

There is, however, a natural probability distribution on the set of real numbers that makes sense as a prior probability distribution. It is related to the Solomonoff priors, but rather different.

Start with a language L with a finite symbol set usable for describing mathematical objects. Proceed as follows. Randomly generate finite strings of symbols in L (say, by picking independently and uniformly randomly from the set of symbols in L plus an “end of string” symbol until you generate an end of string symbol). Conditionalize on the string constituting a unique description of a probability measure on the Lebesgue measurable subsets of the real numbers. If you do get a unique description of a probability measure, then choose a real number according to this distribution.

The result is a very natural probability measure PL (a countable weighted sum of probability measures on the same σ-algebra with weights adding to unity is a probability measure) on the Lebesgue measurable subsets of the real numbers.

We can now in principle evaluate the fine-tuning argument using this measure.

The problem is that this measure is hard to work with.

Note that using this measure, it is false that all narrow ranges have very small probability. For instance, consider the intuitively extremely narrow range from 101000 to 101000. Supposing that the language is a fairly standard mathematical language for describing probability distributions, we can specify a uniform distribution on the 0-length interval from 101000 to 101000 as U[101000, 101000], which is 23 characters of LaTeX, plus an end of string. Using 95 ASCII characters, plus the end of string character, PL of this interval will be at least 96−24 or something like 10−48. Yet the size of the range is zero. In other words, intuitively narrow ranges around easily describable numbers, like 101000, get disproportionately high probability.

But that is how it should be, as we learn from the fact that the exponent 2 in Newton’s law of gravitation had better have a non-zero prior, even though the interval from 2 to 2 has zero length.

Whether the Fine-Tuning Argument works with PL for a reasonable choice of L and for a particular life-permitting range of ξ is thus a hard question. But in any case, for a fixed language L where we can define a map between strings and distributions, we can now make perfectly rigorous sense of the probability of a particular range of possibilities for ξ. We have replaced a conceptual difficulty with a mathematical one. That’s progress.

Further, now that we see that there can be a reasonable fairly canonical probability on infinite sets, the intuitive answer to the normalizability problem—namely, “this range seems really narrow”—could constitute a reasonable judgment as to what answer would be returned by one’s own reasonable priors, even if these are not the same as the probabilities given above.

Oh, and this probability measure solves the tweaked problem of regularity, because it assigns non-zero probability to every describable event. I think this is even better than my modified Solomonoff distribution.

Bill Wood said...

Lunchtime conversations in Oriel reaping results!

Alexander R Pruss said...

Thanks, Bill, for the conversation!

Alexander R Pruss said...

The last sentence of the post is not justified. For while every L-describable *point* gets non-zero probability, there might be L-describable Lebesgue-null sets (e.g., countable ones) that have no describable points, and there is no guarantee that the procedure described in the post assigns non-zero probability to them. I am hoping, however, that one can dismiss these cases as too pathological to worry about in science.

Rob K said...

This is quite intriguing, but I can't quite follow the construction in your very succinct third paragraph. I take it that the strings are supposed to describe probability distributions that assign probabilities to every Lebesque-definable set of real numbers (including singletons). Are the distributions assumed to be countably additive? Could you give an example of how a string would define such a distribution? And, finally, where do we get the probability distribution over the strings themselves? There won't be a unique distribution.

Alexander R Pruss said...

Rob:

Yes, I assume the distributions are countably additive.

Here's an example of how a string defines a distribution: "The normal distribution with mean 0 and standard deviation 3.7" defines the normal distribution with mean 0 and standard deviation 3.7. In other words, descriptions in a mathematical language define mathematical objects.

A fairly canonical probability distribution over strings is easy. Let's say the alphabet has n symbols. Add an (n+1)st "end of expression" (EOE) symbol. Now independently uniformly randomly choose one of the n+1 symbols and repeat until you chose the EOE symbol.

But what still worries me most is my May 25, 2019 2:58 am remark.

Alexander R Pruss said...

A paper based roughly on this approach has just been accepted by Synthese. An earlier version of it is here: http://philsci-archive.pitt.edu/18550/1/RegularityNormalization.pdf