Are the laws of nature first order?

I think it's a pretty common to think that the laws of nature should be formulated in a first-order language. But I think there is some reason to think this might not be true. We want to formulate the laws of nature briefly and elegantly. In a previous post, I suggested that this might require a sequence of stipulations. For instance, we might define momentum as the product of mass and acceleration, and then use the concept of momentum over and over in our laws. If each time we referred to the momentum of an object a we had to put something like "m(a)⋅dx(a)/dt", our formulation of the laws wouldn't have the brevity and elegance we want. It is much better to stipulate the momentum p(a) of a as "m(a)⋅dx(a)/dt" once, and then just use p(x) each time.

But our best-developed logical formalism for capturing such stipulations is the λ-calculus. So our fundamental laws might be something like:

• ∀p(p=λa(m(a)⋅dx(a)/dt)→(L₁(p)&...&L_n(p)))

instead of being a rather longer expression which contains a conjunction of n things in each of which "m(a)⋅dx(a)/dt" occurs at least once. But the λ-calculus is a second-order language. In fact, it seems very plausible that encoding stipulation is always going to use a second-order tool, since stipulation basically specifies a rewrite rule for a subsequent sentence.

So what if the language of science is second order? Well, two things happen. First, Leon Porter's argument against naturalism fails, since it assumes the language of science to be first-order. Second, I have the intuition that this line of thought supports theism to some degree, though I can't quite justify it. I think the idea is that second-order stuff is akin to metalinguistic stuff, and we would expect the origins of this sort of stuff to be an agent.