Retrospective on Mathematical Boundaries Workshop

Primarily written by Evan Miyazono (with help from Manuel Baltieri and others) - mistakes my own

Minimum viable introduction

• We ran a workshop on Mathematical Boundaries from April 10-14. This was the successor of the Conceptual Boundaries workshop Feb 10-12

  • The overlap in participants and approach was fairly low (notably lower than intended, due to availability restricting participation, which in turn led to a natural difference in approach)

  • Intent:

    • The first workshop was intended to develop a sense of what one might want to do with boundaries, and explore possible avenues

    • This event was more focused on making mathematical design decisions that would lead to a more concrete model that was opinionated enough to be useful (the natural question becomes “useful for what”)

• You’re probably here because you want to see the outputs, so let’s get to them:

Outputs from the workshop

Here are write-ups started during writing sessions during the workshop

• Manuel Baltieri 1: Crossing boundaries 

• Manuel Baltieri 2: Fighting for boundaries

• Kevin Carlson: Nondeterministic dynamical systems and crossing boundaries  

• Martin Biehl 1: Gliders and similar phenomena in (categorical) systems theory 

• Martin Biehl 2: Towards a more general law of requisite variety 

• Owen Lynch: Grothendieck lenses for functors into 2Cat 

• Sophie Libkind: Ontological commitments for boundaries

• Nathaniel Virgo: Boundaries and Good Regulators

Noting that I’m getting these to you before I’ve read them, so don’t expect me to be able to answer questions about them.

Also worth noting, Nathaniel Virgo and Martin Biehl participated in this panel discussion at a later workshop in Kyoto, where we discussed some of the issues that came up at the boundaries workshop

General structure from the workshop

The general daily structure was scheduled to be “a talk and a breakout session before lunch, then a breakout session and a longer-form discussion after lunch,” though we weren’t particularly strict adherents. 

We found on Thursday (day 1 of 4) that the group wanted to continue discussing after Martin’s interesting talk and ending up doing more like “A talk and a discussion, followed by breakouts after lunch.”  Thursday breakouts were (1) a session on trying to work out a cocategorical formalism for specifying things via wholes in which they participate, rather than by composing together their parts and a session on, and (2) an idea to formalize / keep track of gliders as non-deterministic or possibilistic closed dynamical systems.

Friday morning Nathaniel gave a talk on control theory that was so engaging we reached a consensus on pointing the rest of the workshop towards fleshing out adjacent ideas.

• One breakout the rest of the day Friday was focused on choosing formalisms for various words in Nathaniel’s talk, and resulted essentially in Sophie’s blog post.

• The other one ended up focusing on an idea of generalizing the law of requisite variety resulting in Martin’s second write-up.

Saturday was primarily time for writing down outputs (learnings from last time: have a big block of time to support people in generating written artifacts), and also included a small breakout group on nondeterminism (that one led to Kevin’s blog post).

Sunday morning some individuals started departing and we had some visitors, most activities involved chatting about a wide range of topics after an intense few days.

Next steps

• We're still genuinely interested in boundaries and would like to see additional work happen. We're exploring funding options for work on these open problems, so email me (evan@atlascomputing.org) you would like to work on them.

• One possible next step is setting up a workshop adjacent to a conference which most of the Mathematical Boundaries Workshop participants are likely to attend

  • Interestingly, it seems like the attendees were split somewhat [40]/[40]/[20]% between researchers who seem most likely to attend conferences exclusively in [applied category theory], [artificial life], and [cross-domain and para-academic conferences like this one], which I think makes this goal hard, but also makes the conversations at such an event particularly interesting.

Evan’s personal takes

Here’s some notes that are very specific to me.

• How it differed from the first one:

  • Chris, Manuel, and I set out with the intent of bringing people together to build mathematical models of boundaries.  As a result, we ended up inviting more people with stronger math background, and people who we expected, based on prior interactions and training, to be inclined toward formalizations and reach for math as a tool.

• Where I could have done better:

  • It wasn’t ex ante clear that much moderation would fall to me; there was some hope that davidad would be able to attend, but through no fault of his, he was unable.

  • Believing that I knew enough math to even moderate this workshop was probably my greatest act of hubris since at least founding Atlas Computing.  I knew I didn’t have enough background knowledge to contribute, but I thought at least I would be able to make proposals that could be iterated on to reach a local equilibrium, but others were far better than I at identifying what the participants agreed was a better starting point.

    • Huge thanks to Manuel Baltieri and Brendan Fong for taking the reins.

• What’s next from here:

  • I’m not sure how involved in logistics, curation, or moderation of future boundaries workshops I’ll be.  I’ll likely advocate for their utility, and potentially support aspects like fundraising and translation, but I think I’d be happy if others took up the mantle. (To be fair, that’s what I said before the first and second workshops as well, though 😅)

    • This could be particularly compelling if it invited participation from a broader conference – if someone would like to propose this event as a side event to a relevant existing conference, please reach out!

  • To the extent that davidad’s ARIA program is focused on building a github for science, but not a monorepo of science, I think it could be really valuable to have the following:

    • If you have two “repositories” of interoperable / composable scientific theories, we should be able to identify boundaries and define boundary violations in each “repository” in a way that we’re confident that specifying a boundary violation in one scientific model (combination of scientific theories) is sufficient to confidently identify the same boundary violation in another scientific model.

    • At this point, Manuel, Brendan, and I are discussing what it would look like to organize a continuation on this theme.  On the bright side, this starts highlighting and framing concrete problems that could be solved.  On the other hand, pursuing solutions to this specific problem could also significantly diverge from the original VAPE formulation from Critch’s «boundaries» formulation. 

Lastly, here are some random assorted brief insights that I liked:

• Some boundaries are (sets of) physical boundaries. Others are parameter regimes, and might be better called “margins” or “viability regimes”. These seem sufficiently distinct that they’re worth calling by different names. “Membranes” may work well for singling out the “physical” boundaries, which don’t have to actually be literally made out of matter but should demarcate an agent’s “body” from its environment, rather than the space of happy states for an agent from its space of sad states.

• Models could be defined as low-loss compressions of the environment and agents could be defined as models that scale in complexity with the scope of the universe unless you ascribe them some telos or desires.

Feel free to comment here, or reach out to via email (first name at domain.org).