Be kind. I’m a physicist who has more interest in abstract math than most colleagues.
Obviously, when basic algebra was invented, it makes perfect intuitive sense. We can all see that if two numbers are equal and you add the same amount those numbers are equal. Same for commutative. The distributive property is more working the algebra.
So it’s surprising to me it had anywhere “to go”. For example, why should matrix algebra work? And other algebraic structures.
We think of math starting with concrete numbers, then go to matrices and tensors. And loads of other examples.
But why should the idea be able to expand?