> In modern math, _algebra_ or _modern_ algebra or _abstract_algebra, is the study of structures over sets with defined operations on the elements of that set.
> Math is not the study of numbers, but the relationships between them.
I'd say more modern math has replaced the word "numbers" with the more abstract concept of "objects". This makes math truly the study of abstraction.
I actually wish we taught more of this math early on[0]. Children seem to be quick to grasp many of the fundamentals of important structures like groups, fields, and algebras. I find that many of these concepts have fundamentally shifted how I think and can be used on a daily basis, without the need of writing formulas or using formal semantics.
It's odd that it takes getting up upper division undergraduate education in math (or sometimes from neighboring fields) to learn what the field is even fundamentally about. It's akin to teaching people that programming by teaching people how to use a word editor. It's such a narrow aspect and no surprise so many are so fundamentally confused.
[0] I'm unconvinced the "new maths" programs were a complete failure. Just because we didn't get it right on the first attempt doesn't mean we should have thrown the baby out with the bath water.
I actually wish we taught more of this math early on[0]. Children seem to be quick to grasp many of the fundamentals of important structures like groups, fields, and algebras. I find that many of these concepts have fundamentally shifted how I think and can be used on a daily basis, without the need of writing formulas or using formal semantics.
It's odd that it takes getting up upper division undergraduate education in math (or sometimes from neighboring fields) to learn what the field is even fundamentally about. It's akin to teaching people that programming by teaching people how to use a word editor. It's such a narrow aspect and no surprise so many are so fundamentally confused.
[0] I'm unconvinced the "new maths" programs were a complete failure. Just because we didn't get it right on the first attempt doesn't mean we should have thrown the baby out with the bath water.