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John Baez (May 08 2020 at 16:44):Here's a practical textbook on systems theory temporarily available for free from Springer:
It has some case studies at the end, like rice milling in Bangladesh, which gives this diagram:
Causal loop diagram of the supply chain for rice milling in Bangladesh
What's nice is that it carries out the analysis in detail. It would be nice to understand the whole story more category-theoretically.
sarahzrf (May 08 2020 at 17:01):damn look at the size of that diagram
sarahzrf (May 08 2020 at 17:01):to think it's probably still a huge simplification @_@
John Baez (May 08 2020 at 17:04):Yes! We mathematical types can contribute by clarifying and beautifying the concepts. Then real-world types can apply them at scale.
a13ph (May 08 2020 at 17:06):sarahzrf said:
damn look at the size of that diagram
I think it's pretty usual for systems dynamics.
Even basic materials which I've seen - they are using big diagrams like this one, when presenting examples and not just trivial cases for introduction of notation.
It's much worse if they are hand-drawn haphazardly...
John Baez (May 08 2020 at 17:07):Yeah, it's a bit funny that people complain about big diagrams in higher category theory when the subject of rice milling in Bangladesh is infinitely more complex.
sarahzrf (May 08 2020 at 17:29):almost added another line to that previous post saying something like "see this is why i like math, it's simpler than real life, dunno what ppl complain about"
sarahzrf (May 08 2020 at 17:30):plus we can actually understand it
sarahzrf (May 08 2020 at 17:30):unlike real life
sarahzrf (May 08 2020 at 17:30)::thinking:
Jules Hedges (May 08 2020 at 19:47):I'm almost certain that these diagrams can be read as string diagrams in some decorated/structured cospan category, but I'd really like to know which one
Jules Hedges (May 08 2020 at 19:48):A diagram like the one in the picture is missing too much information to do any mathematics with, you need to know the capacity of the nodes, the rate and delay of the edges, etc
Nathaniel Virgo (May 08 2020 at 20:45):Jules Hedges said:
A diagram like the one in the picture is missing too much information to do any mathematics with, you need to know the capacity of the nodes, the rate and delay of the edges, etc
There is at least a little you can say from just a diagram like that, at least if you interpret it as indicating the signs of the elements of the Jacobian matrix around a fixed point. For example, there's a technique called "loop analysis" in ecology that's about finding necessary and sufficient conditions for the fixed point to be stable when all you have is that kind of data. (I suspect the book will cover a little of that kind of thing.)
Nathaniel Virgo (May 08 2020 at 20:57):There's also something called "structural controllability" that I think is similar - it's about inferring from just the network structure which nodes you need to control in order to control the whole system. I don't know very much about that but I think this paper is quite well known (but the idea is a lot older) https://barabasi.com/f/329.pdf
John Baez (May 08 2020 at 20:58):Jules Hedges said:
A diagram like the one in the picture is missing too much information to do any mathematics with, you need to know the capacity of the nodes, the rate and delay of the edges, etc
In the book they go ahead and add that information, then use computers to simulate the system, then talk about "parameter estimation" where you compare the simulation with real data to guess what the numerical parameters in your model should be. That's what the book is about.
However, like Nathaniel I disagree about not being able to do any mathematics with a purely "qualitative" diagram like the one I showed. It'll be mathematics that doesn't involve real numbers. But there's still math you can do at this level.
One thing I'm interested in is functorially relating purely qualitative models to quantitative ones, or mixed quantitative-qualitative models where you have some numerical information of the sort you describe, but not all of it. That's a situation we often find ourselves in: having a mixture of quantitative and qualitative information about what's going on in a complicated system.
John Baez (May 08 2020 at 21:00):When I say "functorially", I mean for starters: there should be a functor from "quantitative models" of system dynamics to "qualitative models". (Both these terms need to be defined, and there's a choice of ways to do it.)
Jules Hedges (May 08 2020 at 21:00):Yes, I was just thinking something involving functors
John Baez (May 08 2020 at 21:01):But the really interesting part to me is dealing with mixed quantitative-qualitative models.
John Baez (May 08 2020 at 21:01):Those are a bit closer to the mess we call reality.
Morgan Rogers (he/him) (May 09 2020 at 11:09):John Baez said:
When I say "functorially", I mean for starters: there should be a functor from "quantitative models" of system dynamics to "qualitative models". (Both these terms need to be defined, and there's a choice of ways to do it.)
The first thing one might try is a forgetful functor that sends a real-number-decorated diagram to a diagram where the decoration is less structured, or where the decoration is forgotten completely. Expanding on this, for each fixed diagram shape the forgetful functor(s) factor through intermediate forgetful functors that forget parts of the information, indexed by subdiagrams.
A general information-optimization question is "which of these forgetful functors have adjoints"? That is: when can we recover an extreme, but valid, extension of a reduced data set to a larger data set on a diagram? More generally, when does there exist any extension of a given dataset, and is there anything we can infer with certainty about every possible solution from the information we have?
I anticipate that the recently developed Behavioral Mereology of @Brendan Fong, Myers and @David Spivak will find practical applications here, since it provides a framework for reasoning about exactly this kind of restriction/extension problem.
The "indexed by subdiagrams" aspect makes me suspect that @Andrew Pitts' tripos theory might deserve a place at the table, but I don't know enough about the scope of this theory to give a confident estimate of its potential contribution.
Daniel Geisler (May 09 2020 at 11:27):@John Baez said
When I say "functorially", I mean for starters: there should be a functor from "quantitative models" of system dynamics to "qualitative models". (Both these terms need to be defined, and there's a choice of ways to do it.)
Symmetry and topological conjugacy come to mind for qualitative models. I got dinged for trying to publish a paper on dynamics without discussing germs.
sarahzrf (May 09 2020 at 14:49):i don't think they wanted you to use germs
sarahzrf (May 09 2020 at 14:49):they were mentioning germs in order to describe someone else's theory that used them, so that they could use it to explain why they thought something they were saying was true
Jon Awbrey (Jul 21 2021 at 12:24):@John Baez said:
One thing I'm interested in is functorially relating purely qualitative models to quantitative ones, or mixed quantitative-qualitative models where you have some numerical information of the sort you describe, but not all of it. That's a situation we often find ourselves in: having a mixture of quantitative and qualitative information about what's going on in a complicated system.
This is something I've been working on. In a turn of phrase I once concocted, it's like passing from the qualitative theory of differential equations to the differential theory of qualitative equations.
I'm posting a few notes on the Chategory topic Differential Logic.