Complex Systems and Quantitative Mereology(abeljansma.nl)
abeljansma.nl
Complex Systems and Quantitative Mereology
https://abeljansma.nl/2025/01/28/mereoPhysics.html
17 comments
(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-...
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-...
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.
This is a great philosophical motivation for https://en.wikipedia.org/wiki/Incidence_algebra
That's right! This is secretly about doing calculus on posets. You can actually generalise some notions from incidence algebras to other settings, like groupoids and categories, where you can play the same games. This is something I mostly haven't looked at yet, but I think it might be fun, and some people seem to have found it useful (for example: https://arxiv.org/abs/1809.00941 and https://arxiv.org/abs/2501.06662)
Studying probability right now and seeing how the powerset mobius function immediately implies the inclusion-exclusion formula is pretty cool. Thanks for this!
The Möbius function actually appears quite often in statistics/probability theory. I wrote more about this in the paper (https://arxiv.org/pdf/2404.14423), but in short: if you invert moments with the powerset Möbius function then you get central moments, but if you invert moments with the partition Möbius function, then you get cumulants. In fact, you can vastly generalise this by changing the mereology from partitions to ordered partitions etc.
Super incomplete thought: how does this point of view relate to Euler characteristic? Can I get to Euler characteristic by asking how to solve an equation for some quantities q in terms of some quantities Q?
Author of the post here: There is quite a deep connection actually. You can assign a simplicial complex to a partial order P with a max and a min element (0 and 1). Then the Möbius function on P calculates the (reduced) Euler characteristic of that simplicial complex as µ(0, 1)=\Chi. For example, if the partial order is a power set mereology (a Boolean algebra) on 3 elements, then the associated simplicial complex is a triangle, and µ(0, 1) = µ(\emptyset, {a, b, c})=(-1)^3, which is the correct answer as a triangle (without interior) is homeomorphic to a circle.
In a way, calculating quantities q through Möbius inversion is just calculating Euler characteristics, weighted by by Q. (with some caveats)
In a way, calculating quantities q through Möbius inversion is just calculating Euler characteristics, weighted by by Q. (with some caveats)
Hm, so maybe the following is one answer.
In this article, we fix a mereology and a kind of quantity Q that "decomposes" over it---in the sense that Q(p) = sum_{r <= p} q(r) for some function q(r)---and then see that Mobius inversion lets us solve for q in terms of Q. In terms of incidence algebras, we're saying: assume Q = zeta q, as a product of elements in an incidence algebra. Then zeta has an inverse mu, so q = mu Q.
In other situations, we might want to "solve for" a quantity Q that decomposes over some class of metrologies while respecting some properties. The "simpler" and more "homogeneous" the parts of your mereology, the less you can express, but the easier it becomes to reason about Q. A mereology that breaks me up into the empty set, singleton sets with each of my atoms, and the set of all my atoms admits no "decomposing quantities" besides a histogram of my atoms. An attempt to measure "how healthy I am" in terms of that mereology can't do much. On the other hand, if I choose the mereology that breaks me up into the empty set and my whole, all quantities decompose but I have no tools to reason about them.
I guess Euler characteristic could be an example of how the requirement of respecting a certain kind of mereology can "bend" a hard-to-decompose quantity into a weirder but "nicer" quantity. For example, say we're interested in defining a Q that attempts to "count the number of connected regions" of some object, and we insist on using a mereology that lets us divide regions up into "cells". Of course this is impossible, as we can see in the problem of counting connected components of a graph-like object: we can't get the answer just as a function of the number of vertices and edges. However, if we insist on assigning a value of 1 to "blobs" of any dimension, the "compositionality requirement" forces us to define the Euler characteristic. This doesn't help us much with graph algorithms in general, but gives us an unexpectedly easy way to, say, count the number of blob-shaped islands on a map.
I wonder if there are other examples of this?
In this article, we fix a mereology and a kind of quantity Q that "decomposes" over it---in the sense that Q(p) = sum_{r <= p} q(r) for some function q(r)---and then see that Mobius inversion lets us solve for q in terms of Q. In terms of incidence algebras, we're saying: assume Q = zeta q, as a product of elements in an incidence algebra. Then zeta has an inverse mu, so q = mu Q.
In other situations, we might want to "solve for" a quantity Q that decomposes over some class of metrologies while respecting some properties. The "simpler" and more "homogeneous" the parts of your mereology, the less you can express, but the easier it becomes to reason about Q. A mereology that breaks me up into the empty set, singleton sets with each of my atoms, and the set of all my atoms admits no "decomposing quantities" besides a histogram of my atoms. An attempt to measure "how healthy I am" in terms of that mereology can't do much. On the other hand, if I choose the mereology that breaks me up into the empty set and my whole, all quantities decompose but I have no tools to reason about them.
I guess Euler characteristic could be an example of how the requirement of respecting a certain kind of mereology can "bend" a hard-to-decompose quantity into a weirder but "nicer" quantity. For example, say we're interested in defining a Q that attempts to "count the number of connected regions" of some object, and we insist on using a mereology that lets us divide regions up into "cells". Of course this is impossible, as we can see in the problem of counting connected components of a graph-like object: we can't get the answer just as a function of the number of vertices and edges. However, if we insist on assigning a value of 1 to "blobs" of any dimension, the "compositionality requirement" forces us to define the Euler characteristic. This doesn't help us much with graph algorithms in general, but gives us an unexpectedly easy way to, say, count the number of blob-shaped islands on a map.
I wonder if there are other examples of this?
> any description purely in terms of pairwise relationships would miss the fact that the three rings are connected.
Here is one:
A is connected to B
B is connected to C
C is connected to A
This is a description in terms of pairs, and from it is trivial to deduce how the rings are connected.
Here is one:
A is connected to B
B is connected to C
C is connected to A
This is a description in terms of pairs, and from it is trivial to deduce how the rings are connected.
But they're not pairwise connected.
In the text:
In the text:
If you only consider two of the rings and ignore the third, then any pair can be smoothly separated.
The pairwise description you list correspond to a different link, namely, one where each pair is actually connected. For the shown rings, the 'correct' description would be:
A is not connected to B B is not connected to C C is not connected to A A, B, and C are connected
This seems paradoxical, but the paradox is resolved by the 'higher-order' linkage.
A is not connected to B B is not connected to C C is not connected to A A, B, and C are connected
This seems paradoxical, but the paradox is resolved by the 'higher-order' linkage.
I thought it was about Meteorology and got unreasonably excited for a second.
I thought it might be a typo and was thrilled to be introduced to a new subject!
I thought it was about metrology and got excited
Haha sorry to disappoint you. Now I want to do the mereology of meteorology though...
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.