I'm not a chemist and my math is not amazing. Could someone explain how this would be used for, say, population dynamics, as mentioned in the article?
I think it's not unreasonable to consider Petri nets as a generalization of automata (finite or infinite) with concurrency. Just like automata, they find applications in many places depending on how you further extend or restrict them.
We are working to capture this categorically as well. We are perfectly conscious of how a discrete system won't help you if you have an Avogadro number of things going around. Just give us time. :)
It puts the different models on the same footing (and hence allowing you to unify tools and do more work)
Also, I'd even argue that biology/chemistry needs to embrace categorical methods and we would see some deep discoveries.
Soon these methods will be usable by many people for all kinds of purposes. You can think of cryptographic contracts, business process execution, game theory, functional reactive programming, quantum protocols and digital or analogue electronics.
The only difference is what category the diagrams live in, or put another way, which semantics are you assigning to thee boxes and wires.
You can stay tuned on statebox.org and process.io
Chemists and biologists not caring can have a lot to do with hyperspecialization and fear of math. Category theory is still relatively esoteric.
The fact that workers in the field with the presumed "applications" don't seem to care is pretty telling. My guess is that working at a level of generality where subject-specific insights are suppressed automatically limits the usefulness of the work.
[0]: http://philsci-archive.pitt.edu/8890/1/critique_sep10.pdf
The fact that mathematical foundations don't have applications now does not mean they won't have applications in the future. Gregorio Ricci-Curbastro's work was also considered pretty useless until Einstein decided to build General Relativity over it. If it were for opinions like this one, progress in many field would be incremental at best.
I don't know anything about ZX calculus, but if people are using it to solve real-world problems, that sounds good to me. What I object to is pure mathematicians giving "applications" of their work that aren't useful or real. And when the authors allude to applications in chemistry and biology, and no chemists or biologists are doing anything with category-theoretical analyses of Petri nets, I think it's reasonable to point this out.
Human cultures are more complex than that. Capitalistic pressures to produce results have had a noted impact on research and development. There will likely need to be a practitioner willing to do the field work to bring about the big "Aha!" for that field. Based on the silo'd history of so many fields, it's a safe bet to say that may be necessary in every field where CT can be applied.
Again, it's pretty telling specific applications aren't mentioned, and instead there's just vague gesturing. How can we expect scientists to "do the field work" necessary to bring the applications to fruition if we can't even tell them what those applications are?
Also, regarding your first sentence, the vast majority of influential mathematics was indeed "immediately acknowledged," because it made progress on some important problem. Perfectoid spaces are a recent example.
I feel you are one of those people that just hate category theory because they consider it too abstract and useless for any purpose. We get that a lot, from a lot of people. Still, since Grothendieck, categorical methods are absolutely central in modern algebraic geometry and topology, and this centrality is only destined to grow, because CT is the best theory we have to manage emerging complexity in describing systems. In my opinion, the approach of "if it's not immediately useful then it's useless" really will lead you farther and farther away to understand modern developments in applied mathematics.
Practitioner apathy and willingness to dismiss things as not useful can reflect a lack of intellectual humility in their cultures and lives.
The generality is useful if one practices seeking subject-specific insights from the generality. It sometimes leads to new options I haven't even thought of or leads me to reconsider decisions.
"Useful" is a judgment and most judgments like this are made on too short of a timeline. What's actually happening is someone hasn't found a use for the thing labeled as "useless" and is blaming it on the thing, themselves, or where the thing came from. Instead, a growth mindset focused on learning, accountability, and responsibility might be "I did this thing and the outcome I wanted didn't occur. I haven't yet learned how to use it for said outcome."