# How can we formally define the laws of a physical universe?

Have any philosophers come up with a workable formal, mathematical definition for what the laws of an arbitrary physical universe might be? Such a definition would need to allow specification of different possible laws of physics, as well as the actual current universe state or history. It would need to be as general as possible, not tied down to any particular theory of physics. And it would need to allow us to reason about it mathematically, e.g. defining morphisms between different possible laws.

As an example of the kind of thing I'm looking for, consider this definition:

The laws of a physical universe are a set of propositions P that may hold within possible universes that obey the laws, and a set of subsets of P, interpreted as saying that for each subset, there is a universe obeying the laws in which all elements of that subset hold.

Or this one:

The laws of a physical universe are given by a set X of instantaneous states, and a function δ: X, R -> X where δ(x, t) gives the new state of the universe from initial state x after time t has passed, subject to the constraint δ(δ(x, t1), t2) = δ(x, t1+t2).

Or this one:

The laws of a physical universe are a set of propositions P, and a set of directed acyclic graphs with nodes labeled in P, with the edges indicating causation relationships.

• If it as general as possible, not tied down to any particular theory of physics, and mathematically reasoned, how is it different from just mathematics? Take any mathematical theory and call its models universes, it doesn't get more general than that. Your examples are definitely "tied down" to conventional physics, classical time and causality, even the multitude of string theories has more variety. May 17 at 23:31
• @Conifold the first example isn't tied to classical time or causality. But I'm not sure it has enough structure to do what I want with it (e.g. define when one system simulates another or is a subsystem of another). Models of an arbitrary mathematical theory seem perhaps more complex a structure than necessary, with all the complication of predicates with different arities, quantifiers, and so on. Also I don't see how to define the subsystem relationship for them. May 17 at 23:36
• @Conifold the first example I gave is general enough to include any mathematical theory; the propositions of the theory, together with the sets of mutually consistent propositions, are sufficient to define the mathematical theory. Actually the third one is also general enough to include any mathematical theory, if you interpret the causation relationship A->B as "B can be deduced from A (and from its other parents)" May 17 at 23:39
• 'an arbitrary physical universe'. Meaning what? Positions in the string state-space? phys.org/news/2014-12-universe-dimensions.amp The state-space of varying the dimensionless universal fundamental physical constants that seem to parameterise our universe? How universey are you talking? Perhaps you will find Tegmatk & Wu's AI Physicist arxiv.org/abs/1810.10525 Hawking considered Godel Incompleteness to imply a complete set of physical laws can never be recursively enumerable, so I'm not sure your quest is meaningful. May 18 at 1:56
• @CriglCragl I'm looking for a formalism that would be at least general enough to describe any of the following universes/systems: Conway's game of life, any other cellular automaton (including continuous-time and continuous-space ones), general relativity, Newtonian physics, quantum mechanics, Turing machines, pushdown automata, string theory, finite state machines. I don't care about whether the universe or its laws are recursively enumerable or not. But I am interested in describing the A-is-a-subsystem-of-B relationship. May 18 at 2:02