"For centuries, scientists have attempted to identify and document analytical laws that underlie physical phenomena in nature..." and so goes the rather bombastic opening salvo from a recent Science article titled Distilling Free-Form Natural Laws from Experimental Data by Schmidt and Lipson. It's the kind of work that follows the well-trodden path of the logical positivists who tried to subvert science into a branch of logic. Although the logical positivist took a near fatal beating from Gödel's theorem, there are some who want to keep the dream alive. Or in this case, it is the attempt to reduce the scientific process to a computable process.

According to my machine-learning life-line (Dr. Mark Reid), this article represents a huge advance. The article describes an algorithm that deduces analytical equations from the analysis of observations made on several mechanical systems. These guys were able to identify the subtle tweaks needed to let the system find invariants in a reasonable amount of time, a major breakthrough in machine-learning.

Yet the paper suggests that these methods can be applied to "all physical laws", which rhetorically suggests the method can be widened to many different branches of science. This, I think is a massive overstatement.

Let me explain. Many physics undergraduates cut their teeth on Goldstein's Classical Mechanics, an exhaustive encylopedia of mechanical systems that has served as the standard text of classical mechanics, that slowly builds the formal machinery of mechanics from Newton's equation to the abstract formalism of Lagrangians and Hamiltonians. Goldstein is not a pleasant read. Later on, if sufficiently motivated, they might crack open Feynman Lectures on Physics (a rarity in the science literature in that the book is genuinely fun) where much of the messy guts of mechanics is exposed. But only a few physics undergrads will ever venture onto Lev Landau's slim volume Course of Theoretical Physics: Mechanics where the formal properties of mechanics are properly explained in some 80 terse pages, so terse that I've had to read the book several times. It's the kind of mechanics book where Newton's three laws of motion are not even mentioned.

I bring up Landau's "Mechanics" because not many people have studied it and it is there that Landau points out that if you have any system that is deterministic with respect to coordinates and velocity, you will end up with a conserved Langragian, from which you can derive a conserved Hamiltonian. If you look at the systems studied in the Science paper, they are all mechanical systems, and the observed data are coordinates and velocities. If you assume the system is deterministic, then you can be sure that there must be a conserved Lagrangian or Hamiltonian based on the coordinates and velocities. The algorithms identified by Schmidt and Lipson will only work on deterministic mechanical system, which would be obvious to anyone who's studied Landau.

Unfortunately, there are not many systems that have such beautiful analytical properties, so it is hard to see how this system can be applied to other systems. For instance, could it work for the Schrödinger equation, the workhorse equation in everything from chemistry to semiconductor physics? The Schrödinger equation is deterministic in a very loose sense, and it is the wave function that is conserved, not the observable probabilities! In biology, we have an incredible amount of data on genomes, on genes, and interaction maps. Unfortuantely, we do not have any equivalent Lagrangians for them.

In some ways, this article illustrates one of the points made by the great Canadian philosopher of science, Ian Hacking, that physics was the first science to be developed was no accident. It was because the data for theorizing about planets are the easiest to measure in the natural world. This data came in the form of careful measurements of the motion of the stars and planets, made not originally for science, but for commerical purposes in the need for accurate navigational charts. These precious measurements of planetary motions allowed Kepler and Brahe and Newton to theorize about planetary orbitals and interplanetary forces. Fortunately for them, the forces that dictate planetary orbitals, at least from a non-relativistic approximation, are beautiful determistic systems that, as Landau could well appreciate, could be derived from the coordinates and velocities only.