Ontological Commitments for Boundaries

dynamical systems


When we do math about things we care about, committing to which mathematics represents which intuitions moves us forward.

In April, I attended the Mathematical Boundaries Workshop with many fellow Topos researchers. I had two big takeaways:

Here’s a little story from the workshop that captures a bit of those two takeaways.

On Friday morning Nathaniel Virgo gave a talk on the good regulator theorem. That afternoon, a small group of us wrote down all of the key words and mathematical symbols that he used. Our goal was to go through the talk slowly. We wanted to understand what were the key ideas in his talk. In other words, what is it that Nathaniel cares about? What are the bones of the house and what was Nathaniel’s choice of paint color?

Here is our list:

Everything written here — critically, even symbols like h: E\times A\to E \times A — is intuition. It is not formal. This is hopefully a little surprising. Certainly, when Nathaniel spoke, he was thinking about E and A as sets, \times as cartesian product, and h as a set map. But as a group, we hadn’t committed to such an ontology. In conversation it became obvious that each person was making a translation in their head. Kevin is thinking in terms of fibrations other than the subobject fibrations. David is thinking in terms of polynomial functors. In fact, we each have made different ontological assumptions. The shadow of these differences made our conversations fraught and halting because in every conversation was secretly a heated negotiation about ontology, aka what words mean. It’s not bad to negotiate about ontology. In fact, I think that is exactly the game we are playing when we do math about things that we care about. But at a workshop called Mathematical Boundaries, I want to negotiate about the ontology of words like “boundary” and “boundary violation”. I don’t want to be negotiating about the ontology of pre-requisite words like “system” and “dynamics”.

I believe that for a following workshop to make meaningful progress, the organizers must make ontological commitments. Exactly, what words and symbols need committing to and what those commitments are I think should be lovingly selected by the organizers. I think this really matters for the integrity of the group and for any artifact produced by the group.

With the list of intuitions in hand, we then set about proposing specific ontological commitments based in David Jaz’s double categorical systems theory:

Given a double operad algebra \mathsf{Sys}: \mathcal{O} \to \mathcal{O}(\mathbb{C}\textrm{at}) :

A commitment to such an ontology would mean that when I say “interface” I mean type in a double operad. When I say “system” I mean a choice of interface and an element of the algebra.

From this ontology there are obvious meanings for things like closed system and simulation but there are many intuitive ideas that still need to be pinned down in order to make sense to Nathaniel’s talk. These intuitions include:


Q: What on earth is an ontological commitment?

A: From Wikipedia, “an ontological commitment is an agreement to use the shared vocabulary in a coherent and consistent manner within a specific context”.

Q: Is the timescale of this commitment forever??!

A: Definitely not. But you should only change it, if you have a proof (or at least slightly more than just a vibe) that you absolutely have to. It may be the case that something further down the line in the workshop forces you to revisit your choice of ontological commitment. But in a well-managed group, I expect that the re-negotiation will then have buy-in from the participants.

Q: At a meta-level, is this secretly about boundaries?

A: Why yes it is! An ontological commitment is establishing a boundary between what is on-the-table and off-the-table for negotiation. Boundaries are really useful! The walls of my home, the borders of my nation, the temperature set-point of my thermostat all allow me to relax. If I trust the boundary, then I have a security that frees me up to do my best work. If I throw a party, then establishing, communicating, and maintaining boundaries (e.g. who is invited or not, what time it will end, is it substance free) is respectful to my guests. And, as in any healthy system, I have a boundary protocol for when I allow boundary violations. I have a lot more thoughts on this (and in fact I’d prefer to talk about all of this from the perspective of containers and integrity) but I’ll leave it here for now.

Q: You proposed one set of ontological commitments. What other ones might I consider?

A: Two obvious ones come to mind. (1) I proposed a heretically synthetic systems theory, but you might want to choose something more concrete. E.g. Fix the double operad of interfaces to be directed wiring diagrams. (2) Polynomial functors.