“Undecidability, Uncomputability and the Unity of Physics. Part 1.”

That's the title of a post by Tim Palmer at Backreaction. Here's the opening:

Our three great theories of 20th Century physics – general relativity theory, quantum theory and chaos theory – seem incompatible with each other.

The difficulty combining general relativity and quantum theory to a common theory of “quantum gravity” is legendary; some of our greatest minds have despaired – and still despair – over it.

Superficially, the links between quantum theory and chaos appear to be a little stronger, since both are characterised by unpredictability (in measurement and prediction outcomes respectively). However, the Schrödinger equation is linear and the dynamical equations of chaos are nonlinear. Moreover, in the common interpretation of Bell’s inequality, a chaotic model of quantum physics, since it is deterministic, would be incompatible with Einstein’s notion of relativistic causality.

Finally, although the dynamics of general relativity and chaos theory are both nonlinear and deterministic, it is difficult to even make sense of chaos in the space-time of general relativity. This is because the usual definition of chaos is based on the notion that nearby initial states can diverge exponentially in time. However, speaking of an exponential divergence in time depends on a choice of time-coordinate. If we logarithmically rescale the time coordinate, the defining feature of chaos disappears. Trouble is, in general relativity, the underlying physics must not depend on the space-time coordinates.

So, do we simply have to accept that, “What God hath put asunder, let no man join together”? I don’t think so. A few weeks ago, the Foundational Questions Institute put out a call for essays on the topic of “Undecidability, Uncomputability and Unpredictability”. I have submitted an essay in which I argue that undecidability and uncomputability may provide a new framework for unifying these theories of 20th Century physics. I want to summarize my argument in this and a follow-on guest post.

What interests me is simply the conjunction mentioned in the opening line,  general relativity theory, quantum theory and chaos theory, which featured in a post in January, The third 20th-century revolution in physics [non-linear dynamics]. Here are two (of three) observations I made at the end of that post:

• It seems to me that quantum mechanics and relativity are focused on explanatory principles whereas non-linear dynamics tends more toward description, description of a wide variety of phenomena. Moreover quantum mechanics and relativity are most strongly operative in different domains, the microscopic and macroscopic respectively.
• Computation: many cases there are various computational paths from the initial state to the completion of the computation. As a simple example, when adding a group of numbers, the order of the numbers doesn't matter; the sum will be the same in each case. In the case of non-linear systems successive states in the computation 'mirror' successive states in the system being modeled so the temporal evolution of the computation is intrinsic to the model rather than extrinsic.

Does that second observation imply that (something like) computation is inherent in the physical nature of the universe and is not merely an intellectual operation carried out by various artificial means?