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.

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    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. – Conifold 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. – causative 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)" – causative 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. – CriglCragl 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. – causative May 18 at 2:02

Contemporary constructor theory first sketched by David Deutsch may fit your requirement according to reference here:

Constructor theory is a proposal for a new mode of explanation in fundamental physics, first sketched out by David Deutsch, a quantum physicist at the University of Oxford, in 2012. Constructor theory expresses physical laws exclusively in terms of what physical transformations, or tasks, are possible versus which are impossible, and why. By allowing such counterfactual statements into fundamental physics, it allows new physical laws to be expressed, for instance those of the constructor theory of information.

According to Deutsch, current theories of physics based on quantum mechanics do not adequately explain why some transformations between states of being are possible and some are not. For example, a drop of dye can dissolve in water but thermodynamics shows that the reverse transformation, of the dye clumping back together, is effectively impossible. We do not know at a quantum level why this should be so. Constructor theory provides an explanatory framework built on the transformations themselves, rather than the components... Information has the property that a given statement might have said something else, and one of these alternatives would not be true. The untrue alternative is said to be "counterfactual". Conventional physical theories do not model such counterfactuals. However, the link between information and such physical ideas as the entropy in a thermodynamic system is so strong that they are sometimes identified.

In short, the proposed constructor is metaphorically like a generalized physical version of Turing Machine to try to transform or reduce physics problems to constructible computational or informational problems...

  • Thanks, interesting stuff. I'm not sure it's exactly what I'm looking for; the language used seems informal (what specifically is an "input"? what specifically is a "constructor"? how do we measure approximate correctness?). Since a "constructor" must be built out of simpler parts such as atoms, it doesn't seem to me like this could be truly foundational. But perhaps I'm wrong. That's just my impression from reading arxiv.org/pdf/1210.7439.pdf – causative May 17 at 23:28
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    @causative your intuition makes sense since this is just a sketch and young theory within 10 years. Maybe math logic foundational research especially in quantified modal logic with counterfactual PWs is helpful for fundamental physical laws expression to identify contingent from impossible physical laws. Even in logic we have impossible worlds which may give some useful info. Also currently all physical laws are treated as contingent/possible, but information theory may "derive" some quantum physics theory such as why wave function has to employ complex number... – Double Knot May 17 at 23:46
  • @causative: Just to clarify, a 'universal constructor' is a conceptual generalisation by VonNeumann, of a 'universal Turing machine', but with the important difference that it can completely copy itself (this is more important a difference than it sounds). So, highly relevant to what can simulate what. – CriglCragl May 18 at 2:01

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