[derbOac]

Interesting essay. Not sure it really provides insights into the problem spaces I'm interested in personally, but it's interesting to see a formal connection to mereology.

One of the problems in many of these domains is that the potential higher-order interaction space is so large that it's impossible to make inferences about them (in an exploratory way at least). So in genetics for example, there's a lot of genes, and the number of potential combinations of causal factors is huge. Unless you have an a priori reason to think a particular P-way combination of factors is important, it's impossible to search for them because the resources required to make inferences about the P-way interactions exceeds any computational resources available to study them.

This is kind of the idea of emergence, that at some point the information involved in representing a set of higher-order interactions becomes too great to actually represent, so we measure some property of the system that summarizes these interactions instead.

I think the problem scientifically always is knowing whether that information required actually is too large, or whether we just don't understand what is exactly involved. Unknown unknowns or something like that, but where some of the unknowns might in fact be fundamentally unknowable.

I've always found it might be useful to have an estimate of the predictability of some system from some level of analysis of predictors, so we at least know how much we can ever expect to explain from them. In some cases I think this might be doable and others impossible.

[abelaer]

(OP here) That's a good point---these mereologies tend to grow very fast with system size. Exponential is not even so bad. There is the 'redundancy' mereology for example, which scales with the Dedekind numbers. This appears quite often in information theory and neuroscience, but quickly becomes intractable.

As I see it, emergence comes in two flavours: a higher-order interaction among microscopic parts is already emergent in the sense that it is a non-atomic thing that determines the behaviour of atoms (I use atoms to refer to the 'singletons' or smallest elements of the theory, not necessarily physical atoms). But you're completely right in saying that there is another sense of emergence which only really happens for a 'thermodynamic' number of atoms. The difference seems somehow captured by the contrast between:

-- the whole is more than the sum of the parts -- the whole is less than the sum of the parts.

Both are commonly called emergence! If it turns out that you don't need to keep track of all birds in a flock to describe its behaviour, then we call that emergent because the whole is somehow less than the sum of the parts.

Your example of genetics is interesting, because it is actually what got me interested in this problem in the first place. I spent most of my PhD struggling with calculating up to 7-point interactions among genes, and you indeed need some clever tricks to make this tractable. I used causal discovery methods to rule out most potential interactions based on conditional dependencies. This is now a piece of open-source software: https://www.embopress.org/doi/full/10.1038/s44320-024-00074-...

[throwfar]

And yet the finding of the work is that the formal connection is common to many different fields bravely making progess against the unknown. More generally, our operative epistemic premise is that we will continue to learn and understand, not that impossibility will overwhelm.