Seeking a "Seeking Whence 'Seek Whence'" Sequence

Baez' series on network theory and information geometry answer a couple of your questions in a very accessible way.

Here Baez and Fong prove a version of Noether's theorem for Markov processes.

http://math.ucr.edu/home/baez/networks/networks_11.html

Earlier Baez talked about conservation of total probability.

http://johncarlosbaez.wordpress.com/2011/09/09/network-theory-part-8/

Here he relates the distribution of existing species to a prior and how Bayes' rule says how the number of each species changes over time.

http://johncarlosbaez.wordpress.com/2011/05/26/information-geometry-part-8/

It looks as if you (Will) are trying to make some sort of equivalence between the idea of "going meta" and the "principle of sufficient reason", but surely these are completely different things.

Going meta (in the sense used here): saying "OK, so is there a reason why this is the way it is, and if so what's the reason?"

Principle of sufficient reason: "Everything that is true is true because of some prior reason sufficient to make it true."

PSR is not at all the same thing as "going meta"; it's not even the same kind of thing as "going meta"; it is one particular opinion about what sort of answers one will get when one goes meta.

It doesn't seem to me that the PSR is credible enough to warrant the sort of amount of attention you're trying to give it here, and in particular tying it to LWers' fondness for meta-ness seems entirely out of order.

(It feels to me as if the nearest thing to the PSR in LW tradition is not the idea of "going meta" but the idea of Solomonoff induction. But I haven't thought this through very hard.)

First, recursive statements (aka "going meta") are powerful. As the history of mathematics teaches, seeking to restrict the ability to make recursive statements has enormous difficulty.

Second, recursive analysis of society (aka sociology) can provide very interesting insights, despite the fact that there is substantial social resistance to engaging in meta-level analysis. Hansonian signalling theory is a valuable perspective that could not be created without meta-level analysis of social behavior. Thus, lowing the resistance of the general population to meta-level analysis is relatively low-hanging fruit in raising the sanity line.

Most importantly, going meta often a mistake when trying to solve real-world problems. Going meta again is almost always a mistake.
To be concrete, when I work as a lawyer, the actual meaning of the rules is usually the only important level of analysis. Sometimes, thinking about the policies that justify the rule is helpful. But thinking about the policies that justify having policies is pointless and unhelpful. In short, it is appropriate to be skeptical of assertions of the value of "going meta," particularly assertions about the value of going "meta-meta."

A closely related idea, or maybe a special case, is at the heart of what we do here:

• When one makes a decision, one is also choosing to use a particular decision process.

• When one believes something, one is also choosing to trust a particular truth-seeking process.

The history of the idea of conservation laws; whether the discovery of conservation laws was fueled by PSR-like philosophical-like concerns (e.g. Leibniz?), by lower level intuitive concerns, or other means.

If I'm not mistaken, Maslama al-Majriti beat Antoine-Laurent de Lavoisier to the punch on the principle of the conservation of mass by about 8 centuries. As far as I can tell, he was closer to the practical side (like refining metals and creating medicines, not like proving theorems or arguing from first principles) of the theoretical-practical continuum (or, if you prefer, the philosopher-craftsmen continuum) than Leibniz.

I can't think of any interesting connection between conservation of probability and Liouville's theorem. Can you elaborate on what you're thinking about? Conservation of probability just tells us that the total probability must always add up to 1, so if an update increases the probability of some hypothesis, there must be a corresponding decrease in probability of some other hypotheses. Since phase space volume is a finite measure, it must of course satisfy this sort of conservation. If the volume of some region of phase space increases, there must be a corresponding decrease in the volume of some other region.

But Liouville's theorem tells us something different. It tells us that if we start with a region of phase space A and follow it along as it evolves under a Hamiltonian flow into another region B, then the volumes of A and B will be identical. This feature isn't forced on us by the mere fact that volume is a measure. The only constraint placed by conservation of probability (in this case, conservation of measure) is that the total volume of phase space must remain constant under the flow. But this is compatible with the volume of some subset of phase space changing as it evolves under the flow, as long as this change is couteracted by another change elsewhere.

The conservation of total phase space volume (or what you might call the conservation of probability) simply follows from the fact that phase space volume is a finite measure. This places a constraint on possible phase space flows. There is no flow that can increase the total measure of phase space. Liouville's theorem doesn't just follow from the nature of the volume measure; it follows from the geometric structure of phase space. In a Hamiltonian system, phase space is a symplectic manifold. The flow must preserve the symplectic structure, and a consequence is that it must preserve the natural volume measure. So here it's not just the general mathematics of measures that's relevant, it's the particular symmetries of Hamiltonian phase space. The symplectic structure of Hamiltonian mechanics places a further constraint on the flow, and this time it's a continuity constraint.

Additionally, if you look at the probability distributions that are actually used in statistical mechanics, these distributions don't obey the Liouville kind of conservation of probability, but they obviously obey the Bayesian kind of conservation of probability (since they are probability distributions). If the distributions obeyed Liouville's theorem, there would not be an increase in entropy. Increasing entropy requires the probability distribution to spread out, but Liouville's theorem forbids this. We get increase of entropy by working with coarse-grained probability distributions that aren't governed by Liouville's theorem

Zuckerberg just went meta. Reminder that in the novel, the architect of the metaverse was a billionaire using a Burroughsian information virus for mind control... 

I'm also looking for a discussion of the symmetry related to conservation of probability through Noether's theorem. A quick Google search only finds quantum mechanics discussions, which relate it to spatial invariances, etc.

If there's no symmetry, it's not a conservation law. Surely someone has derived it carefully. Does anyone know where?

Insufficiently meta.

Psychologically I think there are some good straws at which to grasp. Note how meta and recursion appeal to nerds a lot. Possible explanations include its extensive use in computer science and even GEB itself (now that's meta). Philosophers in general seem less interested in it, in general, in general, and common folk seem uninterested in it. Now recall that I am grasping at straws. Here is one of them: perhaps the more precise your thinking, the more you must seek whence.

This post should be moved to Meta.