There were many, many amazing discussions at the just-finished Applied Category Theory workshop in Leiden, but my favorites were two that discussed the need for an “approximate” category theory, especially in applications to the sciences.
One of the first things we realized (even if it took about an hour to figure out) was that we were really talking about three different problems:
- A kind of “
-category theory” inspired by Ross Duncan’s problems in modeling real quantum circuits: every quantum process represented in a string diagram has some sort of known noise property, and the experimenter may want to search through or optimize these noise parameters. So in some sense, as the experimenter executes through the string diagram, she needs to keep track of a probability distribution through all of it.
- A “categorical numerical analysis” inspired by Jules Hedges’s experience with economic models: the idea is that a given string diagram may output a particular numerical program, but equivalent string diagrams (modulo rewriting) may give rise to different numerical programs which, due to problems with how numerical errors propagate across different computations, may actually give different results! A kind of “non-faithful computation”; the errors are structured by the diagrams, but they don’t respect the structure of the diagrams!
- The larger theme of “scientific category theory”, which has several origins:
- the desire for a partial or weak composition that allows one to more easily build and present working scientific models within CT without the baggage of satisfying all the categorical properties on the nose. (Update: Prakash Panangaden recently informed me that he, Radu Mardare, and Gordon Plotkin had developed along these lines called quantitative algebraic reasoning. Update 2: Walter Tholen just gave a talk at ACT 2019 on approximate composition using enriched category theory.)
- The desire to infer categorical structures directly from data.
- Hierarchical / multi-scale / coarse-graining models that allow one to create and overlay models at different levels of observation, over different parts of systems.
- The overall desire for more numerical predictions.
In the second session, we focused more closely on (3) above, and thought about how / why category theory has failed to address the needs of scientists:
- artificiality of
and
“inherited from pure math”, though this is to some degree solved by diagrammatic syntax
- we have no benchmarks against which we can compare ourselves with other, more traditional approaches. But perhaps we can come up with some, using CT?
- we talk up the ability of CT to provide “scale”, but how do we measure scale, and where are the examples / benchmarks that really test “scale”? People have a hard time coming up with these!
- we can’t interact with data very well / very directly! Typically, the data informs a particular model, which then has to be described as an algebra over some category
- the lack of computational implementations generally, or at least methods that output numerical predictions. Aleks Kissinger pointed out that one can see this in the very language of applied CT: our models don’t predict, they recover some existing piece of math
- we don’t have a good calculus for dealing with uncertainty, which is a bigger deal in science than in engineering. Intuitively, engineering benefits more from compositionality than science—think of zooming out versus zooming into a system: zooming out amounts to building with known objects, while zooming in is fundamentally underdetermined, since many things may be possible given the context
- we don’t have a good way of building (and testing) models iteratively, “taking the real world as our model”
- category theorists think that category theory is super cool and want to communicate results directly to scientists… but that’s a catastrophic mistake
One thing that came out of this discussion: we agreed that we would like a way of talking about stuff that is morally, kind-of true, and that having such a theory—something like a weak or partial composition—would make it easier to use category theory as a medium for applied math. (One description we came up for this: “moral category theory”.) My suspicion is that this will require more developments in pure math, rather than more hacking around. For example, what if I could construct a “friendlier” topos in which the models I construct always work out, even when, considered more formally, the compositions are completely mis-typed?
In any case, I’m sure there are many more things that could go into the list of failures above. This is work I’m really interested in, and I would love to work with people to sketch out the problems and the parameters of a more scientific category theory!
Just a comment.
I am an outsider to CT and a ‘generalist’ but following Baez’s course –i was into theoretical bio (applied math)— where CT is rarely seen (old speculative work by Rashevsy and Rosen did and was seen as sort of ‘off the wall’).
I like the title of this blog. (Category theory certainly failed me, or maybe i failed–i definateley failed alot of courses. ) But I’m interested in the general issue—sort of like ‘how you go from lab to the marketplace’ or ‘does any of this math (or say cosmology) have any relevance to ‘practical’ aspects of life (like modeling) or is it more like art, theology, religion, philosophy –just something done for itself?
From my perusal it does look applicable but for me its like being 5 years old again and learning the alphabet—except i’m not 5. Is there any use for me to learn this language—either some ‘user friendly form’ like current computers (as opposed to ancient ones only for experts) which i could actually apply to solving ‘real world problems’ (econ, ecology, society, etc. as done in nonlinear dynamics/complexity theory) , or if not (since those applications require skills similar to ones for climbing Mt Everest—i just hike small mountains and leave those to experts) maybe i could write some pop sci book like CT for dummies (pop math/sci books are like videos people who can’t climb watch of mountain climbers).
Anyway, i’d be interested in ideas on how you might induce people who don’t speak category theory language to learn some–should they, or at least more people? And how much (you can get a GED up to a PhD in other fields)?
Also sometimes technical experts in some fields invite non-experts like philosophers and sociologists to comment on what they do (eg medical doctors and researchers invite bioethicists to comment.)
I wonder if there is any space for that. (Philosophers and historians of science and scientists often are pretty seperate groups.)
Anyway any thoughts you (or other CT people) have on that that is my interest.
—————-
( My main applied interests were in things like stochastic differential equations (and their path integral representations, so you can turn them into a ‘quasi-Hamiltonian/Lagrangian system that looks like classical mechanics), and also statistical mechanics of networks (neither of which i am expert in.)
To an extent I’m sort of like people who can’t code (dont know python, R,blockchain, etc.) but can use google search, or dont know much math but may be able to use a calculator or mathematica.
I once tried to learn CT on the my own (as i’m doing now) —it seemed either it or set theory was the ‘most fundamental’ (and i’m learning people are sort of putting the two together). (My view was they were doing the same thing in different ways—sort of like doing ‘matrix mechanics’ versus ‘wave mechanics’ in quantum theory (or path integrals, Dirac notation, etc.) or dividing a cake by cutting it from the top versus the side.
But doing both set and category theory is like learning russian and chinese–too much work for me. Also in my world nonlinear dynamics was like ‘english’–the local dialect. Others are luxuries only a few can afford to learn. (my website is sort of a joke so i usually think its best not to list it but i did anyway.)
here are some thoughts on some of those weaknesses that became apparent and that you listed.
Many years ago (1989) I wrote a book with Jean-Marc Cordier, called: Shape Theory: Categorical Methods of Approximation, Mathematics and its Applications. I this we explored the idea of using approximations to knowledge of (in particular) spatial forms by sytems of more precise models (i.e. already identifiable). This was intended as an approach to exactly your third point namely:
>Hierarchical / multi-scale / coarse-graining models that allow one to create and overlay models at different levels of observation, over different parts of systems.
BTW the book is still in print with Dover if any one is interested. (I do not get royalties any longer!)
The basic principle is that category theory can handle the precise nature of the models and then the various forms of model give approximative information about the objects being studied.
This was also the theme of a preprint: What ‘shape’ is space-time? Preprint arXiv:gr-qc/0210075 in which the relationships between observations at a given scale can be assembled into a model. Refinements of models are discussed in papers by Jon Gratus and myself (A Spatial View of Information, Theoretical Computer Science, 365, (2006), pp. 206 – 215. ) and extended versions in Dagstuhl Seminar Proceedings:
A geometry of information, I : Nerves, posets and differential forms, in Spatial Representation: Discrete vs. Continuous Computational Models, Dagstuhl Seminar Proceedings (04351), (http://drops.dagstuhl.de/opus/volltexte/2005/126).
A geometry of information, II : Sorkin models, and biextensional collapses, in Dagstuhl Seminar Proceedings (04351), (…/opus/volltexte/2005/127). Recent spinoff from this is in the Journal of Experimental and Theoretical Artificial Intelligence, December, 2018 in two papers by Fields and Glazebrook.
I could go on, mentioning homotopy coherence in which composition is not precisely defined only up to deformation, and then also probablistic nerves as directed spaces, and so on, but the point is that category theory does not really fail in these situations, rather the limitations it has in its present form need addressing as my attempts that I have mentioned are all inadequate for what I had hoped to do. The machinery is there and in various combinations it should be able to push things forward. Many of the categorical lego pieces have been around for 30 or more years but not all the combinations of fitting them together in sensible ways have been tried. Of course, new tools of a categorical flavour will also need developing as you say.
One final point is that as many of the categorical ideas have a geometric or diagrammatic form looking at ideas from just outside category theory in directed homotopy, topological data analysis, rewriting theory etc, is an excellent way to proceed, mixing the notions from them into the pool of ideas being used. (Good luck! and keep in touch as I am interested in peoples musings on these areas.)
Josh,
(I tried to post this yesterday but it got lost somewhere. If the other did `get through’ you can decide if you post both as I make slightly different points here.)
Your 3.3 says`Hierarchical / multi-scale / coarse-graining models that allow one to create and overlay models at different levels of observation, over different parts of systems.’
Exactly this was a motivation for some work back in the 1980s which was published in book form as `Shape Theory: Categorical Methods of Approximation, Mathematics and its Applications’ which is still available (Dover republication).
Other work looked at the use of Chu spaces to handle uncertainty and there was a preprint by me on the arXiv:What ‘shape’ is space-time? Preprint arXiv:gr-qc/0210075. Some of this may be of relevance.
Of course, indeterminacy of composition is inherent in homotopy coherence but putting probability distributions on hom sets has not been combined with that homotopy approach.
I believe the so called failures are not failures just holes in our understanding. Many of the themes of Applied Category Theory as John Baez has identified are, as he says, ones that were opened up years ago but in the climate of the times they did not attract `Brownie points’ and so could not be developed.
For instance, I discussed with my ex-colleague Lucy Kuncheva on aspects of pattern recognition, mixing some categorical ideas into the ideas from the mainstream of that area. it is hard but well worth doing as it reveals new categorically flavoured mathematics. (That research with Lucy came to nothing because our department was shut down by the University and I was out of a job. Someone had decided we were not getting enough Brownie points! People have to keep publishing in the mainstream journals so as to attract grant money, peer recognition, (Research assessment exercises), etc. but there is limited time.) Another problem is that few category theorists have had to teach subjects like Operational Research (OR) and so have not encountered the AI aspects of such subjects. As our department shrunk, I was teaching OR and including Petri-nets, discrete event systems, etc. in undergrad courses, using a light amount of CT, (in fact the students seemed to like that material.)
I think that many of the building blocks of applied category theory are `out there’, some need much more development of course. It is not a failure of category theory, however, that they have not yet been used in applications. Most of the points you mentioned have been studied to some extent, but until there is a large enough body of people interested in pushing them forward, there will not be recognition of the subject area nor impetus to identify the lacunae.
Hi Tim, thanks for your comment and the references! Re: your main point, whether some of these problems should be regarded as “failures” or as lacunae in current understanding—I think it’s both. In the back of my head what I mean by “failure” is “we should let someone else take care of this”. I believe in the power of the language, and that with sufficient effort and #s of PhDs we can build all these potential applications in CT style. The question is whether the kind of abstraction that takes place in CT (or applied CT) is the right approach, given a bunch of scientific and psychological constraints, for solving certain problems. And, if not, then what sorts of problems is CT the “right” approach for? My admittedly-slim understanding of the history is that certain problems received CT-style solutions, but the solutions just did not get much traction (for a variety of reasons, some of which you name above).
Hi Josh,
Perhaps the point is that CT does surprisingly often shed conceptual light on problems. It is perhaps less good at actually solving them as that requires input from other sources so ends up not being thought of as being CT. ‘Solutions’ are very hard things to say what they are in the areas of interest, but the intuitions from category theory can and do help.
I was involved in the planning of a project to look at a large multiscale ecological system. As long as I did not claim to have the `solutions’ the input and insights that a categorical / compositional viewpoint made seemed to go down very well with systems theorists and practioners and also with some of the more mathematically minded ecologists. (Unfortunately the project though highly rated at the initial stages was not funded due to some sourpuss of a referee!)
Again, my comments should be taken as ‘a view from the outside’, but I was also looking at possible applications of CT to social and ecological systems.
(There is another CT—catastrophe theory—named by Rene Thom in a famous book that was also a sort of attemted TOE (theory of everything). (I once almost did and independent study class on that book under a well known history of math professor,, partly because I was sort of too advanced for anything in that college at the time. But I read some of the book and reviews, which were all negative, so I didn’t do it, though I thought the book was good in its own way , though pretty complex for me (lots of differential geometry) , and it wasn’t really a TOE since it only applies to smooth manifolds if i recall). It was discussed also in economics (eg the book by J B Rosser ‘from catastrophe to chaos: a general theory of economic discontinuities’. Rosser is the son of the Rosser who developed lambda calcuklus in math logic).
For example, the paper ‘the behavioral approach to systems theory’ is a title which I think could be found in a psychology, systems science (a sort of partly outdated term but precursor to nonlinear dynamics), or even ecological and ethological (animal behavior) journals. The IEEE 2007 review paper from my reading seems to say in different terms what is said in other ways in papers I’ve read by people in other fields (though many do not do much technical computations or modeling , or if they do, its in their own dialects, which range from relatively simple graph theory type models (eg Heider balance, ecological networks) to ones using statistical mechanics.
I remember from elementary school through undergraduate college one would learn rudiments of set theory–Peano arithmatic , ZFC, issues like Cantor’s diagonal argument, CH ,etc —-though you might not use nor see any of it ever again. (I was told to learn differential equations, and the current generation learns Python.) I wonder if a very simple CT book for people from grade school through undergrad would be useful—if only to learn the basic terminology.
Most of the biologists and psychologists I’ve come across do not have much use for mathematical biology or psychology . (Azimuth blog discusses some of the ‘state of the art’ research in those fields—which relies heavily on nonequilbrium statistical physics formalism). Some psychologists I’ve come across are ‘clinical’ or ‘social work’ academics, who deal with ‘real world problems’ (inner city and rural problems of addiction, violence, depression, autism, ADD and ADHD, school dropouts and bad test scroes, and so on). Those are the problems i was sort of trying to model.
Its a sort of ‘compositional’ problem. How do you add apples , oranges, land, air, money, biochemicals, etc? Maybe you Kant.
The problem I’ve seen with CT is that it seems to be ultimately a prop-up for philosophical flights of fancy. Dig deep enough and I find something about “the Trinity”, “the formula of myth”, “Lacan”, or some such nonsense.
Unless you can convince the public that CT is real math, it’ll never fly.