Were you (or others here) not introduced to multiplication as repeated addition and exponentiation as repeated multiplication? How was it introduced to you? I don’t remember if I was taught this in school, but I viewed the commutativity of addition/multiplication geometrically: addition through the lens of stacking “sticks” of different lengths together and multiplication as area.
When I was in middle school I was also obsessed with higher operations and begun to accelerate my own math journey intending to conduct research in that field. I was also surprised to see so little work done there. Turns out it’s just an ugly area of math (compared to others) and I stopped really thinking about. But I don’t regret the time I spent discovering “theorems” and whatever and encourage you to do the same. I’ll bet in time you’ll reverse your opinions here, but who knows.
For your last paragraph: consider looking into how one might even define tetration at fractional hyper-powers. That’s the “easiest” case but it’s already non-trivial!
Were you (or others here) not introduced to multiplication as repeated addition and exponentiation as repeated multiplication? How was it introduced to you? I don’t remember if I was taught this in school, but I viewed the commutativity of addition/multiplication geometrically: addition through the lens of stacking “sticks” of different lengths together and multiplication as area.
When I was in middle school I was also obsessed with higher operations and begun to accelerate my own math journey intending to conduct research in that field. I was also surprised to see so little work done there. Turns out it’s just an ugly area of math (compared to others) and I stopped really thinking about. But I don’t regret the time I spent discovering “theorems” and whatever and encourage you to do the same. I’ll bet in time you’ll reverse your opinions here, but who knows.
For your last paragraph: consider looking into how one might even define tetration at fractional hyper-powers. That’s the “easiest” case but it’s already non-trivial!
Clearly OP was introduced to addition and multiplication as the coproduct and product in the category set.