| >> The classification of several microstates into the same macrostate, is this not a distinctly observer-centred function? > It seems that way if we consider only our neat models, but it fails to explain why experimental measurements of the entropy of a given materials are consistent and independent of whatever model the people doing the experiment were operating on. Fundamentally, entropy depends on the probability distribution, not the observer. I am not sure that I agree with this -- it feels a little too "neat and tidy" to me. One could argue, for example, that these seemingly-emergent agglomerations of states into these cohesive "macro" units are an emergent property limitations of modelling based of the physical properties of the universe -- but there's no way to necessarily easily tell if this set of behaviors comes from an underlying limitation of _dynamics_ of the underlying state of the system(s) based on the rules or this universe or the limitations of our _capacity to model_ the underlying system based on constraints imposed by the rules of this universe. Entropy by definition involves a relationship (generally at least) between two quantities -- even if implicitly, and oftentimes this is some amount of data and a model used to describe this data. In some senses, being unable to model what we don't know (the unknown unknowns) about this particular kind of emergent state (agglomeration into apparent macrostates) is in some form a necessary and complete requirement for modelling the whole system of possible systems as a whole. As a general rule, I tend to consider all discretizations of things that can be described as apparently-continuous processes inherently "wrong", but still useful. This goes for any kind of definition -- the explicit definitions we use for determining the relationship of entropy between quantities, how we define integers, words we use when relating concepts with seemingly different characteristics (different kinds of uncertainty, for example). We induce a form of loss over the original quantity when doing so -- entropy w.r.t. the underlying model, but this loss is the very thing that also allows us to reason over seemingly previously-unreasonable-about things (for example -- mathematical axioms, etc). These forms of "informational straightjackets" offer tradeoffs in how much we can do with them, vs how much we comprehend them. So, even in this light, the very idea of modelling a thing will always induce some form of loss over what we are working with, meaning that said system can never be used to reason about the properties of itself in a larger form -- never verifiably, ever. Using this induction, we can extend it to attempt to reason then about this meta-level of analysis, showing that because it is indeed a form of model sub-selected from the larger possible space of models, that there is some form of inherent measurable loss, and it cannot be trusted to reason even about itself. And therein lies a contradiction! However, one could postulate that this form of loss results in any model necessarily has some form of "collision" or inherent contradiction in it -- theories like Borsuk-Ulam come to mind, and so we must eventually come to the naked depravity of picking some flawed model to analyze our understanding of the world, and hope to realize along the way that we find a sense of comfort and security in the knowledge that it is built on sand and strings, and its validity may unwind and slip away at any minute. A very curious ideal, indeed. |