According to the best-systems account of laws (BSA), the fundamental
laws of nature are the axioms of the system that are true and optimize a
balance of informativeness and brevity in a perfectly natural language
(i.e., the language cuts reality perfectly at the joints). There are
some complications in probabilistic cases, but those will only make my
argument below more compelling.

Here is the issue I want to think about: There are many reasonable
ways of defining the “balance of informativeness and brevity”.

First, in the case of theories that rule out all but a finite number
of worlds, we can say that a theory is more informative if it is
compatible with fewer worlds. In such a case, there may be some natural
information-theoretic way of measuring informativeness. But in fact, we
do not expect the laws of nature to rule out all but a finite number of
worlds. We expect them to be compatible with an infinite number of
worlds.

Perhaps, though, we get lucky and the laws place restrictions on the
determinables in such a way that provides for a natural state space.
Then we can try to measure what proportion of that state space is
compatible with the laws. This is going to be technically quite
difficult. The state space may well turn out to be unbounded and/or
infinite dimensional, without a natural volume measure. But even if
there is a natural volume measure, it is quite likely that the
restrictions placed by the laws make the permitted subset of the state
space have zero volume (e.g., if the state space includes inertial and
gravitational mass, then laws that say that inertial mass equals
gravitational mass will reduce the number of dimensions of the state
space, and the reduced space is apt to have zero volume relative to the
full space). So we need some way of comparing subsets with zero volume.
And mathematically there are many, many tools for this.

Second, brevity is always measured relative to a language. And while
the requirement that the language be perfectly natural, i.e., that it
cut nature at the joints, rules out some languages, there will be a lot
of options remaining. Minimally, we will have a choice point about
grouping, Polish notation, dot notation, parentheses, and a slew of
other options we haven’t thought of yet, and we will have choice points
about the primitive logical operators.

Finally, we have a lot of freedom in how we combine the
informativeness and brevity measures. This is especially true since it
is unlikely that the informativeness measure is a simple numerical
measure, given the zero-volume issue.

We could suppose that there is some objective fact, unknowable to
humans, as to what is the right way to define the informativeness and
brevity balance, a fact that yields the truth about the laws of nature.
This seems implausible. Absent such a fact, what the laws are will be
relative to the choice of informativeness and brevity

measure *ρ*. We might have
gotten lucky, and in our world all the measures yield the same laws, but
we have little reason to hope for that, and even if this is correct,
that’s just our world.

Thus, the story implies that for any reasonable informativeness and
brevity measure *ρ*, we have a
concept of a law_{ρ}.
This in itself sounds a bit wrong. It makes the concept of a law not
sound objective enough. Moreover, corresponding to each reasonable
choice of *ρ*, it seems we will
have a potentially different way to give a scientific explanation, and
so the objectivity of scientific explanations is also endangered.

But perhaps worst of all, what BSA had going for it was simplicity:
we don’t need any fundamental law or causal concepts, just a Humean
mosaic of the distribution of powerless properties. However, the above
shows that there is enormous complexity in the account of laws. This is
not ideological complexity, but it is great complexity nonetheless. If I
am right in my preceding post that at least on probabilistic BSA the
fact that something is a law actually enters into explanation, and if I
am right in this post that the BSA concept of law has great complexity,
then this will end up greatly complicating not just philosophy of
science, but scientific explanations.