This meeting took place on the 14th of June 2021. We summarize and discuss parts of Behavioral Mereology by Brendan Fong, David Jaz Meyers, and David Spivak.
This text models the relationships between parts and wholes as being one of passing constraints on behavior. A behavior can be a set, or more generally, a type living in a suitable topos. The text uses trajectories in some phase space as a canonical example from physics. From the set of behaviors on a larger part, we can restrict to a smaller part. An example we use extensively in the meeting is: two people are doing the waltz, which is one dance among many. When we restrict to an individual partner in the dance, we are no longer "seeing" a waltz, but something like "leading" or "following". We can restrict further to just viewing the feet of one individual. It's clear that there are only certain behaviors of the feet which are compatible with the waltz - if your feet are tap dancing, you're likely not waltzing. It's also clear that certain behaviors of another part of the body, like a steady heartbeat for example, are compatible with waltzing but are not determined by it like your feet are. In this way, the paper formalizes compatibility and determination as relations of behaviors.
The authors also show how constraints can be viewed as selecting a set of behaviors (in a topos, this is modeled by a morphism to the subobject classifier). From a constraint on one part of a system, we can conceive a constraint on another part. For example, a constraint on a whole can require that it exhibits a behavior that, when restricted to the part, satisfies another constraint. Using adjoint functors, the authors derive two new "inter-modalities", allowance and ensurance. These generalize the compatibility and behavior relations to constraints on relations. From these we can derive a language for asserting how different parts of the system relate to one another. Finally, the text shows how allowance and ensurance also generalize possibility and necessity from Kripke semantics.
During the end of the meeting, our discussion revolved around the question of novelty and systems which are "open" in that their behaviors (or which behaviors are/become relevant) are not yet known. This seems to be a basic feature of a certain class of systems that arise "organically" within, say, a sociopolitical situation. We also notice parallels between the topos of behavior types offered in the text and Badiou's phenomenology.