"Entropy" (in the "log of number of possibilities" sense) and "probability" are "in the mind" in another, stronger sense.

Aha! So it would seem the original sense that "Energy" is "realer" (more like Apple) than Entropy is because Entropy is associated with Probability, and Bayesian Probability, the local favorite, is more in the mind than other things because its accurate estimation requires information about the state of knowledge of the person estimating it.

So it is proposed there is a spectrum "in the mind" (or dependent on other things in the mind as well as things in the real world) to "real" (or in the mind only to the extent that it depends on definitions all minds would tend to share).

We have Sexiness is in the mind, and thinking it is in reality is a projection fallacy. At the other end of the spectrum, we have things like Energy and Apple which are barely in the mind, which depend in straightforward ways on straightforward observations of reality, and would be agreed upon by all minds that agreed on the definitions.

And then we have probability. Frequentist definitions of probability are intended to be like Energy and Apple, relatively straightforward to calculate from easy to define observations.

But then we have Bayesian probability, which is a statement which links our current knowledge of various details with our estimate of probability. So considering that different minds can have different bits of other knowledge in them than other minds, different minds can "correctly" estimate different probabilities for the same occurrences, just as different minds can estimate different amounts of sexiness for the same creatures, depending on the species and genders of the different minds.

And then we have Entropy. And somebody defines Entropy as the "log of number of possibilities" and possibilities are like probabilities, and we prefer Bayesian "in the mind" probability to Frequentist "in reality" definitions of probability. And so some people think Entropy might be in the mind like Bayesian probability and sexiness, rather than in reality like Energy and Apple.

Good summary? I know! It is!

So here is the thing. Entropy in physics is defined as That is, the entropy is very deterministically added to a system by heating the system with an unambiguously determined amount of energy dQrev, and dividing that amount of energy by an unambiguously determined temperature of the system. That sure doesn't look like it has any probabilities in it. So THIS definition of Entropy is as real as Energy and Apple. And this is where I have been coming from. You me and an alien from Alpha Centauri can all learn the thermodynamics required to build steam engines, internal combustion engines, and refrigerators, and we will all find the same definitions for Energy and Entropy (however we might name them), and we will all determine the same trajectories in time and space for Energies and Entropies for any given thermodynamic system we analyze. Entropy defined this way is as real as Energy and Apples.

But what about that "log of number of possibilities" thing? Well a more pedantic answer would be, that the number of possibilities has nothing to do with probabilities. I have a multiparticle state with known physics of interactions. Its state when first specified, the possibility it initially occupies, has a certain amount of energy associated with it. The energy (we consider only closed systems for now) will stay constant, and EVERY possible point in parameter space which has the SAME energy as our initial state shows up on our list of possibilities for the system, and every point in parameter space with a DIFFERENT energy than our initial state is NOT a possible state of this system.

So counting the possibilities does NOT seem to involve any Bayesian probabilities at all. You, me, and an alien from Alpha Centauri who all look at the same system all come up with the same Entropy curves, just as we all come up with the same energy curves.

But perhaps I can do better than this. Tie this in to the intuition that entropy has something to do with probabilities. And I can.

The probabilities that entropy has to do with are FREQUENTIST probabilities. Enumerations of the physically possible states of the system. We could estimate them mathematically by hypothesizing a map of the system called parameter space, or we could take 10^30 snapshots of the physical system spread out over many millenia and just observe all the states the system gets into. Of course this second is impractical, but when has impractical ever stopped a lesswrong discussion?

So the real reason Entropy, Energy and Apple are "real" even though Bayesian Probability like Sexiness is "in the mind" is because Entropy is unambiguously defined for physical systems in terms of other unambiguous physical quantities "Energy" and "Temperature." (BTW, Temperature is Average Kinetic Energy of the particles, not some ooky "in the mind" mind thing. Or for simplicity, define temperature as what the thermometer tells you.)

And to the extent you love Bayesian probability so much that you want somehow to interpret a list of states in parameter space that all have the same energy as somehow "in the mind," you just need to realize that a frequentist interpretation of probability is more appropriate for any discussion of entropy than is a bayesian one: we use entropy to calculate what systems we know "enough" about will do, not to estimate how different people in different states of ignorance will bet on what they will do. If we enumerate the states wrong we get the wrong entropy and our engine doesn't work the way we said it would, we don't get to be right, in the subjective sense that our estimate was as good as it could be given what we knew.

I hope this is clear enough to be meaningful to anybody following this topic. It sure explains to me what has been going on.