[nine_k]

If one cannot apply mathematics to concrete things, what use does the mathematics have, besides mental gymnastics?

If mathematics is applied poorly to reality, it's usually because an inadequate model is used, not because attempting this is a wrong thing to do.

[sadar_]

First of all, when I said concrete things I did not mean for example theoretical physics concepts. I mean things like economics[0] or philosophy[1].

I am saying that mathematical abstraction is not the right tool to get a better understanding for a lot of subjects. My questions was asking if this is an instance of someone with a categorical hammer and seeing categorical nails everywhere, or that it is a super natural fit.

Most mathematicians (I know) do mathematics for the mathematics. It is in itself a goal that does not need an external use. So yes, mathematics for a lot of mathematicians is just mental gymnastics that turned out to be very useful. Probably the reasons for some people's love for math and the usefulness are related.

[0] https://www.quora.com/Are-there-applications-of-group-theory...

[1] https://en.wikipedia.org/wiki/Alain_Badiou#Mathematics_as_on...

[pensativo]

Category theory is a formal semantics for (or alternative to, depending on one's perspective) type theory and thus functional programming. It's been widely used (for example, in Haskell, ML, Agda, Coq, Idris, etc) not only for formal foundations but to derive many "smaller" practical applications. Many of the creations from that domain have been useful in many other languages. Are you unaware of this and asking something else?

It's an extremely good fit and highly productive, to answer your question directly.

[sadar_]

I am aware of these applications. I am not aware of any "smaller" practical applications that are derived thanks to the categorical interpretation, can you give any examples?

[danharaj]

https://arxiv.org/abs/1502.05947

In this paper we describe a functorial data migration scenario about the manufacturing service capability of a distributed supply chain. The scenario is a category-theoretic analog of an OWL ontology-based semantic enrichment scenario developed at the National Institute of Standards and Technology (NIST). The scenario is presented using, and is included with, the open-source FQL tool, available for download at categoricaldata.net/fql.html.

This is part of a series of work on applying category theory to databases. The initial work was to cast database concepts into categorical concepts, this led to a clarification of various concepts such as many kinds of SQL query being instances of limits and colimits. The theory was then used to extrapolate via category theoretic concepts to develop new database manipulation concepts.

This is a general recipe for applying category theory, though there are other approaches.

[pensativo]

I don't have any papers on this hard drive but databases, grammar formalisms and generally many data structures (you might know about monads, comonads, arrows, zippers, etc.) If you start digging on hackage, you'll find many interesting data structures, many of which were derived categorically or at least algebraically (they often cite the papers that inspired their implementation.)

Obviously all the work can be done without category theory but since the mid-2000's I gather that many insights have been gained by exporing various categories and their relations.

Edit: I think one of the main benefits is efficiency. Even if one doesn't start from categorical formalisms, you can later use them to pare things down to what's absolutely necessary.

[voidhorse]

I think you make a good point--there is probably a natural limit or reasonable degree of extension when it comes to using mathematics as an explanatory framework or domain language. The tendency to over-extend its application probably stems from its abstract nature--the more abstract your system or calculus, the more readily it's made to fit as a potential explanatory mechanism or language for expressing problems in a particular domain. Because mathematics is so darn abstract to begin with, it's difficult, if not impossible, to refute, a priori, someone attempting something like a Marxist topology or category theory or algebra or whatever. The application of mathematical concepts is really only refutable a posteriori, after we discover it wasn't actually a useful or sufficient explanatory mechanism for the domain. Because higher level mathematics often operates at a level in which it doesn't restrict the set of entities its variable terms could resolve to or reference, it's not possible to refute its application to anything as nonsense until we discover it as such in the process of applying it.

You also get into a bit of a tangle if you believe mathematics is not some separable set of properties or things 'in the world' but an expression of some fundamental laws of human reasoning--i.e. it captures the way in which, whether we like it or not we simply must reason, because that just happens to be how the brain is constructed. If this truly is the case, and mathematics captures some fundmanetal kernel of truth about 'proper' reasoning, it should, in theory, be widely applicable in many human endeavors. For example, someone might argue, from this perspective, that something like function application doesn't only express some property or rule of reasoning with numbers, but rather indicates a form of processing or mental operation that underlies many if not all forms of conceptual reasoning. (Frege and Leibniz are two historical figures who play with some of these ideas in their pursuit of a universal language for concepts).

[nine_k]

To me, neither philosophy nor economics are very concrete, because a lot of important information in their domains is either flimsy or unknowable.

Programming is very much unlike that; machines strictly follow known abstract principles.

[tupshin]

https://arxiv.org/abs/0903.0340

[mcguire]

"I am saying that mathematical abstraction is not the right tool to get a better understanding for a lot of subjects."

As a possible example, I found Categories for Software Engineering to be...unspectacular, both in terms of category theory and software engineering.

[andrepd]

Mathematics is not all about models, or real world applications. Pure mathematics is an end, in an of itself, for knowledge if not anything else.