I was surprised to find Julia uses the multiplication method for string concatenation. For me it feels as wrong as using bitshift operators for I/O but maybe I'm missing something.
Multiplication for concatenation makes perfect sense: it is commutative, associative, and consistent with mathematical notation. A space would be even better. And the empty string would be great, but maybe difficult to implement if you want to allow multiple-letter variable names, as julia seems to do.
Even if using multiplication to join strings may be consistent with some properties of mathematical notation, it's less consistent with the usual meaning of the words (I checked the cosine similarity using GloVe 6B words embeddings, the similarity between "add" and "join" is 0.4 while between "multiply" and "join" is only 0.03), so it's probably the balance between being the general purpose and math oriented language.
> For example, the language does not let you overload + for a non-commutative operation
It does allow it? I'm not even sure how one would disallow non-commutative operations. How would you know if a definition is commutative?
> how one would disallow non-commutative operations. How would you know if a definition is commutative?
I don't think that this is possible without solving the halting problem. But in practice, you can do that by documenting this enforcement and making it unfeasible to overload a commutative operator with non-commutative code. For example, the callers of the overloaded commutative operator can and do assume commutativity to optimize compilation; as in, you fill a matrix with "f(i+j)" and it may only evaluate the upper-half of the matrix (this is even more interesting for the associative case). I think that Mathematica does a similar thing, I recall several symbols for operators to be overloaded assuming certain symmetries. As another example, in C++ you must overload "<" with something that is an order relation, otherwise the sort function fails. Being more fancy, a test option of the interpreter may run each call of the operator in a random order, etc.
Also, related to Mathematica, using juxtaposition for product is very natural to mathematicians. It is indeed hard point of friction when moving from Mathematica to Maple or Matlab.
Or the sum operator for string concatenation, which is the epitome of non-commutative operation.
I like your point of view, but still I'm skeptical of function and especially operator overloading. Shouldn't these semantic constraints that you mention be enforced by the language? For example, the language does not let you overload + for a non-commutative operation, and so on.