Does Julia have this quality?
I'm not suggesting it does, I'm genuinely curious for anyone that knows this area well.
I'm not suggesting it does, I'm genuinely curious for anyone that knows this area well.
Multiple dispatch, as in Julia, Mathematica, CommonLisp, etc, does get you much closer to this vision than single dispatch. To an extent, but then you're on your own again.
Maybe consider compilation as a collaboration of a computer and a wetware compiler. You have knowledge and do analysis which you are unable to share with the computer compiler. For silly tiny illustration, you might know some size variable has to be an integer power of 2, but most language's type systems don't let you tell the compiler that. Or think of writing a datatype in old C - a struct and a bunch of functions. You have a mental model of how they should all hang together, and then you mentally compile that down and hand emit C code and tests. Or you could hand emit assembly code and tests instead. But the C compiler is the more helpful collaborator. You can talk about more (with some sacrifices), and more easily. Now given say two structs and a bunch of associated functions, multiple dispatch allows weaving them together without pains which in single dispatch so discourage weaving. Which is better, but still, you've a couple of structs, a bunch of functions, and a still limited ability to express your intent so the computer compiler can help you with it.
So you can have multiple dispatch of + and - , and more easily add new numeric types, but any relationship between + and - is still all in your mind - the compiler has no idea that you think them connected.
Would "first-class explicit algebras" involve the language allowing redefining of operator precedences? Julia doesn't allow that. Would it require support for customizable associativity? You can painstakingly do that by defining custom types for every intermediate operation, but it isn't part of base Julia.
The original problem in the thread's start, "builtin_object <operator> your_object", is neatly solved by multiple dispatch. But the ask in your grand-parent comment is much bigger, and Julia only has slightly more support for that compared to other languages (i.e. it's a bit less of a pain in the ass to create such algebras, but you'll still have to create it yourself).
> redefining of operator precedences?
That can be confusing, but it's a fun thought. Maybe type-sensitive parsing? Or maybe "^--- the precedence of this operator is ambiguous between the theories visible at this point - please disambiguate which theory's operator you intended, either Int32:+ or ..."?