[johncolanduoni]

ZFC does not suffer from Russel’s paradox, since it doesn’t allow a “set of all sets”. If it did, the search for new foundations would be much more widespread. New foundations are usually only considered seriously once they can be shown to be relatively consistent with ZFC or the slightly stronger but still uncontreversial TG set theory.

There are people working on categorical foundations, but the main reason for lack of broader popularity or drive is that most mathematicians don’t do work that is “foundational” in that sense. For example, if you’re an analyst, you generally don’t care exactly how your real numbers are built (dedekind cuts, Cauchy sequences, etc.). You only care that they satisfy a certain set of properties, you can define functions between them, and that’s about it. Most mathematical reasoning at this level is insensitive to differences in proposed foundations, except “constructive” foundations that don’t have the law of excluded middle (that are unpopular since they make many proofs harder and some impossible).

What is very popular in mathematics is using category theory at a high level; basically every field uses its concepts and notation to some degree at this point. New foundations are only really relevant to this program in that they may make automated proof checking easier, which is also not a widespread practice in mathematics.

[spekcular]

It is not true that "basically every field uses its concepts and notation to some degree at this point." In particular this is false for mainstream combinatorics, PDE, and probability theory, to give a few examples.

In fact, I would suggest that most mathematicians don't care about category theory at all.

[bitdizzy]

This all hinges on "mainstream". For example, in combinatorics, combinatorial species are a vast organization of the all-important concept of generating function. They were developed by category theorists and are most tidily organized along categorical lines. If you don't think this is close enough to mainstream, I can't dispute that. It's a value judgment.

There is often an undercurrent of category theory within a subject that maybe most people are not privy to. Anything to do with sheaves or cohomology (which I know factors into some approaches to PDEs) are using categorical ideas.

Every generation, it seems, has some contingent of serious mathematicians who consider category theory marginal in their field of interest. But every generation, that contingent grows smaller as more mathematics as practiced is brought into the fold. Maybe they're coming for you next :)

[spekcular]

Respectfully, I disagree. The question of what's mainstream and valued by the community is empirical and can be answered by looking at what's published in the leading combinatorics journals. And anyone can check those out and see that categories are basically absent. So as a sociological fact, I maintain it's far from the mainstream.

Whether combinatorialists ought to elevate certain work is of course of a question of value, but it's also a different question.

Also, in no way are sheaves or (co)homology essentially category-theoretic ideas. It's possible to develop and use these ideas without mentioning categories at all (and e.g. Hatcher's introductory textbook does just this, although he mentions in an appendix the categorical perspective later). In general I think it's good to remember that homological algebra and category theory are not the same subject. Sure, I can develop a theory of chain complexes over an arbitrary abelian category, but most of the time you just need Hom and Tor over a ring. (Again, see Hatcher.)

Finally, I'm not sure there has been a serious uptake in category theory in the mainstream of some field of mathematics since, I don't know, at least 50 years ago? We've understood for a while now what it's good and not good for. This hasn't stopped people from trying to inject it in fields where it doesn't do any good (e.g. probability), but for that reason those attempts are mostly ignored.

[bitdizzy]

> Respectfully, I disagree. The question of what's mainstream and valued by the community is empirical and can be answered by looking at what's published in the leading combinatorics journals. And anyone can check those out and see that categories are basically absent. So as a sociological fact, I maintain it's far from the mainstream.

I think it is also a reasonable interpretation to take "mainstream" as "pertaining to the main subject matter of the field". Anyway, I think it is the case that the mainstream of combinatorics or probability is yet so big that a particular researcher or even group of researchers can be comfortably in the mainstream and yet have never cared for or even heard of some other line of research that is also mainstream.

The founding paper of combinatorial species [1] has hundreds of citations including many in what I gather are top journals in combinatorics, and even some in the Annals of Probability. So, what are we to make of that? Some people who are serious enough about combinatorics or probability to get published in serious journals have read, perhaps understood, and maybe even taken seriously some of these categorical ideas?

In any case, I respect your viewpoint. In my youth I was a bit category-crazy, trying to use it to organize all of my mathematical knowledge. I'm much more prudent about it these days but I'm still an optimist that we will find more unifying ideas in mathematics through it.

[1] https://www.sciencedirect.com/science/article/pii/0001870881...

[spekcular]

It may the case that combinatorics is large, with mathematicians focused on their own particular sub-specialities, but I still don't believe category theory can properly be construed as "mainstream combinatorics" in any real sense.

Regarding your claim about that paper being cited by papers in top journals, I checked the first five pages of citations in Google Scholar for combinatorics and probability journal papers.

It's cited once in an offhand way in the concluding discussion of "The Cycle Structure of Random Permutations." No categorical concepts are used there. Ditto for the "Independent process approximations" paper (except now cited in the introduction). Another one-sentence mention in the context of background literature appears in "Tree-valued Markov chains derived from Galton-Watson processes." Same for "A Combinatorial Proof of the Multivariable Lagrange Inversion Formula" and "Bijections for Cayley Trees, Spanning Trees, and Their g-Analogues."

There's a one-sentence mention with actual (slight) mathematical content in "Limit Distributions and Random Trees Derived from the Birthday Problem with Unequal Probabilities." But you don't need categories to prove the bijection they're referring to, from what I understand.

In none of these instances is the work used in a substantive fashion, and unless I missed something no paper features the word "category." It's getting cited because authors have a duty to survey any potentially related background literature.

(On page 6 I found "Commutative combinatorial Hopf algebras," which does use a functor form that paper. It was published in a journal that is decent, but very far from the top.)

So, I think we can reject the notion that Joyal's paper has seriously influenced the fields of combinatorics or probability.

I'm sorry to harp on this, but I see claims like yours about the importance of category theory thrown around a lot on here, and often I feel that they're clearly wrong. So I thought it would be good to provide some details this time around.

[bitdizzy]

I should have demonstrated my picks. I didn't just look for citation but also looked for at least some nontrivial review of the results of the paper^.

I pick these two papers. The first is literally about adjunctions pertaining to combinatorial species whereas the second devotes an entire section to reviewing the theory and stating results that it uses in a way that I don't really understand. I'm going to read the first paper though because it's relevant to my interests :)

Rajan. The adjoints to the derivative functor on species

https://www.sciencedirect.com/science/article/pii/0097316593...

Panagiotou, Konstantinos; Stufler, Benedikt; Weller, Kerstin. Scaling limits of random graphs from subcritical classes.

https://projecteuclid.org/euclid.aop/1474462098

The fact that I have to go rummaging around for these examples kind of proves your point, doesn't it? I don't think category theory will lead to any fantastic new results in the fields we're discussing, but the bar is quite low for it to be useful as an organizational tool.

^ I searched for the word functor of course! :)

[Koshkin]

> most mathematicians

Agree, but only on the level on which they do not care about the abstract math (algebra, topology, etc.) in general. As soon as you step into the territory of the abstract math, especially where different disciplines blend, such as homology and cohomology, category theory (and its diagram language) helps a lot to clarify things. (Incidentally, a lot of this stuff is now part of the "applied math" as well, having found its way into theoretical physics, for example.)

[spekcular]

I'm not sure I'm comfortable characterizing the fields where category theory is useful as "abstract math." Modern PDE is plenty abstract, for example. Probably it's better to say that the usefulness of category theory is proportional to the problem's distance from algebraic topology and algebraic geometry.

I also am reluctant to characterize theoretical physics as "applied math." I haven't seen anyone who calls themselves an applied mathematician use category theory in a substantive way (where here I am thinking about numerical computing, mathematical biology, and so on).

[joe_the_user]

I have only an MA in math - but I have the impression mathematicians can be arranged on spectrum between ad-hoc theorem provers and giant machinery builders, between those like Paul Erdős and those Alexander Grothendieck.

The thing to keep in mind is that a mathematical structure can be expressed as instance of any number of more general mathematical structures ("mathematical machinery"). The structures that useful, however, are those that are "illuminating", a somewhat vague criteria but one which generally include a structure facilitates and unifies proofs of important theorems. Wiles' proof of Fermat's Last Theorem showed the value of many forms of this mathematical machinery [1], including category theory.

At the same time, I think things like Godel theory of cutting down proofs and Chaitin's Omega constant give a suggestion that the some number of "important" theories within a given axiomatic system will have long proofs that don't necessarily benefit from the application of a given piece of mathematical machinery, whatever that machinery.

And think that's related to efforts to apply category theory to programming via functional programming. I feel like it's an effective method for certain kinds of problems but that you get a certain problematic "it's the best for everything" evangelism that doesn't ultimately do the approach favors.

[1] https://en.wikipedia.org/wiki/Wiles%27s_proof_of_Fermat%27s_...