Sure! First, here are references, in case you want to deep dive:
1. http://lucacardelli.name/Papers/Binary.pdf
2. https://www.researchgate.net/publication/221321423_Parasitic...
Second, asymmetric multimethods give something up: symmetry is a desirable property -- it's more faithful to mathematics, for instance. There's a priori no reason to preference the first argument over the second.
So why do I think they are promising?
1. You're not giving up that much. These are still real multimethods. The papers above show how these can still easily express things like multiplication of a band diagonal matrix with a sparse matrix. The first paper (which focuses purely on binary operators) points out it can handle set membership for arbitrary elements and sets.
2. Fidelity to mathematics is a fine thing, but it behooves us to remember we are designing a programming language. Programmers are already familiar with the notion that the receiver is special -- we even have a nice notation, UFCS, which makes this idea clear. (My language will certainly have UFCS.) So you're not asking the programmer to make a big conceptual leap to understand the mechanics of asymmetric multimethods.
3. The type checking of asymmetric multimethods is vastly simpler than symmetric multimethods. Your algorithm is essentially a sort among the various candidate multimethod instances. For symmetric multimethods, choosing which candidate multimethod "wins" requires PhD level techniques, and the algorithms can explode exponentially with the arity of the function. Not so with asymmetric multimethods: a "winner" can be determined argument by argument, from left to right. It's literally a lexicographical sort, with each step being totally trivial -- which multimethod has a more specific argument at that position (having eliminated all the candidates given the prior argument position). So type checking now has two desirable properties. First, it design principle espoused by Bjarne Stroustroup (my personal language designer "hero"): the compiler implementation should use well-known, straightforward techniques. (This is listed as a reason for choosing a nominal type system in Design And Evolution of C++ -- an excellent and depressing book to read. [Because anything you thought of, Bjarne already thought of in the 80s and 90s.]) Second, this algorithm has no polynomial or exponential explosion: it's fast as hell.
4. Aside from being faster and easier to implement, the asymmetry also "settles" ambiguities which would exist if you adopted symmetric multimethods. This is a real problem in languages, like Julia, with symmetric multimethods. The implementers of that language resort to heuristics, both to avoid undesired ambiguities, and explosions in compile times. I anticipate that library implementers will be able to leverage this facility for disambiguation, in a manner similar to (but not quite the same) as C++ distinguishes between forward and random access iterators using empty marker types as the last argument. So while technically being a disadvantage, I think it will actually be a useful device -- precisely because the type checking mechanism is so predictable.
5. This predictability also makes the job of the programmer easier: they can form an intuition of which candidate method will be selected much more readily in the case of asymmetric multimethods than symmetric ones. You already know the trick the compiler is using: it's just double-dispatch, the trick used for "hit tests" of shapes against each other. Only here, it can be extended to more than two arguments, and of course, the compiler writes the overloads for you. (And it won't actually write overloads, it will do what I said above: form a lexicographical sort over the set of multimethods, and lower this into a set of tables which can be traversed dynamically, or when the types are concrete, the compiler can leverage monomorphize -- the series of "if arg1 extends Tk" etc. is done in the compiler instead of at runtime. (But it's the same data structure.)
6. It's basically impossible to do separate compilation using symmetric multimethods. With asymmetric multimethods, it's trivial. To form an intuition, simply remember that double-dispatch can easily be done using separate compilation. Separate compilation is mentioned as a feature in both the cited papers. This is, in my view, a huge advantage. I admit, this I haven't quite figured out generics will fit into this -- at least if you follow C++'s approach, you'll have to give up some aspects of separate compilation. My bet is that this won't matter so much; the type checking ought to be so much faster that even when a template needs to be instantiated at a callsite, the faster and simpler algorithm will mean the user experience will still be very good -- certainly faster than C++ (which uses a symmetric algorithm for type checking of function overloads).
To go a bit more into my "vision" -- the papers were written during a time when object-orientation was the dominant paradigm. I'd like to relax this somewhat: instead of classes, there will only be structs. And there won't be instance methods, everything will be a multimethods. So instead of the multimethods being "encapsulated" in their classes, they'll be encapsulated in the module in which they're defined. I'll adopt the Python approach where everything is public, so you need to worry about accessibility. Together with UFCS, this means there is no "privileging" of the writer of a library. It's not like in C++ or Java, where only the writer of the library can leverage the succinct dot notation to access frequently used methods. An extension can import a library, write a multimethod providing new functionality, and that can be used -- using the exact same notation as the methods of the library itself. (I always sigh when I read languages, having made the mistake of distinguishing between free functions and instance methods, "fix" the problem that you can only extend a library from the outside using free functions -- which have a less convenient syntax -- by adding yet another type of function, an "extension function. In my language, there are only structs and functions -- it has the same simplicity as Zig and C in this sense, only my functions are multimethods.)
Together with my ideas for how the parser will work, I think this language will offer -- much like Julia -- attractive opportunities to extend libraries -- and compose libraries that weren't designed to work "together".
And yeah, Claude Code and Gemini are going to implement it. Probably in Python first, just for initial testing, and then they'll port it to C++ (or possibly self-host).