The language allows for greater efficiency which carried the ML to the DL era. In the same sense that you need calculus for physics. You could write everything down as additions and multiplications. You just will have a very very bad time...

what efficiency are we discussing here? efficiency of communication, or efficiency of computation?

Yes

so i would argue that efficiency of computation is nice but also somewhat accidental and not that important. but lets focus on efficiency of communication: my point is kinda exactly around this. in a field that is actually built around algebra, the communication efficiency would be MUCH higher.

because when we use a term such as "a norm" we get the definition, which is nice, but also a bunch of properties that hold for items of this kind. and if we dont actually rely on these properties later on, then its a waste in terms of what we communicated.

Oh I thought you were talking about communication within the electronics which are built specifically to perform the (for instance) norm optimally

btw, re the hardware example: the hatdware is not built to perform "norm". it is built to approximate norm over a floating point representation of real numbers. this is highly specific and doesnt enjoy the generality of algebra at all. i think it may not even be commutative and associative.

Yes yes. And everything can be broken down to mov and add. And yet if you try to describe and design things using those commands you'll not get very far.