> But for aliens counting, which I think is itself arguable
I honestly can't imagine how you can reach this conclusion with any rigour. Do you agree that aliens will need to consume some energy source to stay alive, which we will call "food"? Do you agree that an understanding of "there's no food in my environment", "there's some food in my environment", and "there's lots of food in my environment" would be selected for? I certainly hope so, so at the very least they will understand the differences between zero, non-zero and "many".
The only way this wouldn't happen is if the environment is so rich in abundance that there is never any absence of food. But this is impossible, because even single-celled life by necessity will reproduce to consume all available resources until it reaches an equilibrium matching the rate of food production. So any intelligent species will necessarily evolve in an environment of scarcity where zero and non-zero will be implicitly understood.
Since intelligent life will necessarily evolve in scarcity, quantifying the amount of food is a useful trait that would be selected for. This is why we've now proven that numerous "non-intelligent" animals can count, including salamanders, chicks, mosquitofish, honeybees and more. Intelligent life needs to understand where they are, what they have and what they will need in the future. This involves quantifying, aka counting, no way to escape it.
> There's a vast gap between counting and what mathematics is and encapsulates.
Yes, but you posited intelligent aliens that have their own math. The conclusion that they would not understand zero and repeated application of a construction over zero to build non-zero quantities is impossible. It is the very root of building a theoretical structure of any kind, so if they have math of any kind, they have some kind of counting system that will have an isomorphism to ours.
I honestly can't imagine how you can reach this conclusion with any rigour. Do you agree that aliens will need to consume some energy source to stay alive, which we will call "food"? Do you agree that an understanding of "there's no food in my environment", "there's some food in my environment", and "there's lots of food in my environment" would be selected for? I certainly hope so, so at the very least they will understand the differences between zero, non-zero and "many".
The only way this wouldn't happen is if the environment is so rich in abundance that there is never any absence of food. But this is impossible, because even single-celled life by necessity will reproduce to consume all available resources until it reaches an equilibrium matching the rate of food production. So any intelligent species will necessarily evolve in an environment of scarcity where zero and non-zero will be implicitly understood.
Since intelligent life will necessarily evolve in scarcity, quantifying the amount of food is a useful trait that would be selected for. This is why we've now proven that numerous "non-intelligent" animals can count, including salamanders, chicks, mosquitofish, honeybees and more. Intelligent life needs to understand where they are, what they have and what they will need in the future. This involves quantifying, aka counting, no way to escape it.
> There's a vast gap between counting and what mathematics is and encapsulates.
Yes, but you posited intelligent aliens that have their own math. The conclusion that they would not understand zero and repeated application of a construction over zero to build non-zero quantities is impossible. It is the very root of building a theoretical structure of any kind, so if they have math of any kind, they have some kind of counting system that will have an isomorphism to ours.