God Doesn't Cook the Books

Sketching a Potential Path from Conservation of Evidence to Conservation of Energy

The notion that physics and computation are basically the same thing has become almost commonplace by now.  The notion that logic and computation are the same is even more solidly established, via the Curry-Howard correspondence that lets one map each logical axiom system into a corresponding computational model.

At a simple level: If we know the values characterizing the state of a physical system at a certain point in time, and know the laws of physics via which the system evolves, then figuring out what values characterize the state of the system at a future point in time is a matter of logical derivation from the axioms of the initial values and the physics laws.   It becomes an almost pointless question whether to say that the universe is carrying out this logical derivation, or carrying out a different sort of process that is mathematically equivalent to this logical derivation.

(The same perspective can be taken if one’s initial data is the future state of a system, or a combination of past and future state, or a set of values given on the the boundary of some region etc.)

The quantum mechanical version of this story is particularly natural, with “the universe as a quantum computer” emerging as a perfectly practical and useful way to think about the universe as a quantum dynamical system.

This parallel leads me to wonder which key aspects of physics can be explained in conceptually straightforward ways as manifestations of computational or logical phenomena.

For instance, a key role is played in physics theories by conservation laws, in particular conservation of energy.   One then wonders whether conservation of energy can be viewed as a result of some computational or logical phenomenon.  

In particular: In designing computational probabilistic reasoning systems, a key issue is to avoid “double counting of evidence” — I.e. indirectly doing statistics in a way that counts the same piece of evidence twice.   Is this sort of “conservation of evidence” somehow the root of, or a variant way of looking at, “conservation of energy?

Linear logic — in which each axiom in a proof begins with a certain specific number of copies, and can only be used in the proof this given number of times — is a logical rather than statistical take on conservation of evidence.   

After reading and thinking about related matters a bit, I’ve come to the tentative conclusion that, yes, conservation of energy probably can usefully be viewed as a spacetime-continuum specialized version of conservation of evidence.   I haven’t had time to try to construct a fully rigorous argument in this regard, so what you’ll get for now is this suggestive blog post instead.   If this post inspires someone else to take time and effort to work out more details and see if this holds up — and if so under what conditions — that will be awesome.

A Couple Notes on Conservation Laws

I’m going to focus on conservation of energy here.  Other conservation laws are also important, but it may be they can all be reduced to conservation of energy given other reasonable assumptions.  For instance in classical physics elementary arguments suffice to derive conservation of momentum from conservation of energy plus assumptions of invariance among reference frames.

I also note that, while energy conservation doesn’t hold globally in General Relativity, it does hold locally.

I have wondered if one may be able to reverse that argument and derive the Einstein equation from an assumption of "mass-energy curves space in a way that ensures local energy conservation.”

“Locally” in a relativistic sense is the sort of frame in which one can assemble a coherent body of observational evidence from the perspective of a particular observer.  So the locality of energy conservation in GR fits in fine with an evidential-conservation perspective.

Gorard’s discrete version of GR on spatial hypergraphs seems like one framework in which one could attempt an “Einstein equation as a manifestation of energy conservation" argument.

Physics Dynamics as Linear Logic Computations (?)

So how might we map conservation of energy into some sort of conservation of evidence?

There’s some recent, tasty computer science theory that seems to point in this direction.   While it’s a cutting-edge research field and some of the basics are yet to be worked out over the next few years, there is significant partial progress on Curry-Howard mapping from linear, reversible computing (which comes close to capturing dynamics of quantum systems without measurement) to a variant of linear logic with least and greatest fixed points.

Loosely, it seems that in this case, the conservation of operations in linear logic (which relates closely to "no double counting") corresponds on the computing side to conservation of the number of computing operations done, which (assuming a fixed energy cost for each computing operation) comes down to conservation of energy.

I recall here a morphism I observed some time ago btw the algebra of linear logic and the algebra of uncertain reasoning, which suggests that linearity assumption among computational operations corresponds to assumption of not double-counting evidence,

It seems we can look at the actual execution of physics dynamics as a (reversible) computation of the form

Initial or boundary conditions AND laws of physics <==> conditions on the rest of the domain AND laws of physics

If we assume a linear logic where each piece of evidence needs to be accounted for, and the evidence in question (apart from the laws of physics part) are statements about physical systems during some interval, then the equivalence of conservation-of-energy to conservation-of-evidence pretty much follows if the individual logical inference rules obey energy conservation, i.e. the amount of energy needed to represent the conclusion is the same as the amount of energy needed to represent the premises...   

So if we can do reversible linear logic using  elementary operations that are conservative in, say the sense that a Fredkin gate is (whose output has the same Hamming weight as its input, a conservation property that seems to translate into conservation of energy in simple implementations) then the parallel seems to work...

I suspect there may also be connections btw reversible linear logic and Jordan-Banach algebras that have not been fleshed out, which would then connect this line of thinking to the correspondence John Baez summarizes between Noether’s Theorem (which underlies physics conversation laws) and these JB algebras.

So, fudging a bit here and there, we have arguments something like

Algebra of uncertain inference w/o double counting <==> additive/multiplicative linear logic 

Linear logic w/ least+greatest fix-points <==> linear reversible computing

Linear reversible computing ==> computing that conserves energy

Conservation of energy ==> conservation of momentum

General relativity ==> (? <==) Local conservation of energy 

Lots of gaps but the conceptual picture gets clearer...

It seems that with some work one could plausibly frame an argument that the conservation laws in physics are “just” a different way of looking at “conservation of evidence” in logic.

I.e. in the vein of Einstein’s (apparently incorrect) maxim “God does not play dice,” we could say “Conservation laws exist because God doesn’t double-count evidence within any local reference frame.”

Or, in a wizzy phrasing suggested by my dad after reading an earlier version of this article: “God doesn’t cook the books.” ;)