Multiplication over the reals is commutative. Matrix multiplication of non-square matrices isn't. Multiplication in a Ring isn't necessarily commutative. Other algebraic structures also have non-commutative multiplication.
One could argue that these things aren't "multiplication" even if they are "products" since they don't satisfy all the properties of multiplication over the reals. But it is common to call the use of the product operation "multiplication", at least in cases where there's only one product operation to use. EG Geometric Algebra has Inner, Outer, and Geometric products, so calling them "multiplication" seems less common IME.
I really don't get the point you're making. If we're going to pull out random examples, monoids aren't guaranteed to be abelian; strings and concatenation form a monoid that's not abelian.
If you have enough mathematical sophistication to conceptualize a non-commutative ring, you're well past the point where naming conventions are even remotely an issue.
The original article was contrasting addition and multiplication on the basis that addends are called the same while factors are supposed to be called differently, which not only makes no sense (it's just a naming convention), but it also breaks down when you have more than two factors: what is the "c" in a x b x c called? Or we're talking about non-associative operations now?
My point is only that the commutative property is not inherent to all multiplication operations, so there can be a distinction between the operands. It's not necessarily a useful distinction, and in the usual use of multiplication it's utterly useless and only adds confusion.
But matrix multiplication is taught in high school (and usually promptly forgotten), it's not particularly advanced math.
Personally I'm of the opinion that the terminology is muddled. There's no need to distinguish the operands of a multiplication over any of the usual domains (reals, rationals, integers, etc). And when you reach the point where it does become important there's generally more than one product operation and we should stop calling it multiplication. "Matrix multiplication" is a bad term. You also typically wouldn't name the operands, since as you note there can be more than two!
Sure, my counterpoint was just that the same reasoning technically applies to addition as well, but as you note "matrix multiplication" is taught in high school, while my example wouldn't come up.
I agree not calling it multiplication would probably help, perhaps something like "linear transformation composition" might encourage students to keep it separated from real multiplication, I jsut found the argument in the original article kind of ridiculous to be honest.
Yea, math tends to reuse terms and notation across disciplines in ways that adds confusion rather than clarity. It’s much better to think of infinity for example as multiple independent concepts than assume it’s all the same idea.
One could argue that these things aren't "multiplication" even if they are "products" since they don't satisfy all the properties of multiplication over the reals. But it is common to call the use of the product operation "multiplication", at least in cases where there's only one product operation to use. EG Geometric Algebra has Inner, Outer, and Geometric products, so calling them "multiplication" seems less common IME.