I responded to the question "What is a number?" on X with the answer: What you need it to be. That was one of those intuitive ideas that just sprang into my mind.
For what it is worth, if I were allowed to redefine what a number is from scratch, I would suggest defining numbers to be elements of localizations of global fields. Then the term would actually have utility: numbers would be precisely the sort of thing that number theorists work with, which just makes sense. (It would make the term "number field" awkward, because while it would be a field of numbers, it would be a field of a special kind of numbers. But I'm sure a workaround can be found.)
Unfortunately, it would be a definition that is impossible to explain to almost all undergraduate math students, let alone 5-year-olds. It is perhaps best to keep "number" as a loose, non-mathematical term.
As a counterexample to your list of properties shared by all number systems, I would argue that the strictly positive integers are sometimes treated as a number system in their own right, so not all number systems have an additive identity.
An interesting edge case of numbers is the hyperreals. Despite the terminology of "the" hyperreals, this does not refer to a specific algebraic structure, but just imposes some conditions that the algebraic structure you're talking about is assumed to satisfy.
While it is true that not all number systems have an additive identity, I don't know of an example where they can't be extended to one that does. So I think it is fine to include it as a requirement: we don't lose any examples of objects that are called numbers.
Speaking of "the" hyperreals is indeed a little odd, although if you assume the Continuum Hypothesis, there is a unique hyperreal model that you get from ultrafilters. But that is probably jumping through way too many hoops just to be able to say "the" with a straight face.
Historically, the word "field" did also include objects with noncommutative multiplication (like the quaternions). Some time around 1970, if I remember correctly, there was a switch and fields were always understood to be commutative, with the more general noncommutative option known as a division algebra. Why? It turns out that commutative division algebras show up much, much more often. They are central to linear algebra, algebraic geometry, and so on. Noncommutative division algebras have their uses but they are more niche.
As for groups, we care about noncommutative examples at least as much as commutative ones.
I responded to the question "What is a number?" on X with the answer: What you need it to be. That was one of those intuitive ideas that just sprang into my mind.
Honestly, that's pretty consistent with observed usage.
The question shouldn't be if we can define a suitably comprehensive notion of number... but rather, what do we want to do with such a definition.
For what it is worth, if I were allowed to redefine what a number is from scratch, I would suggest defining numbers to be elements of localizations of global fields. Then the term would actually have utility: numbers would be precisely the sort of thing that number theorists work with, which just makes sense. (It would make the term "number field" awkward, because while it would be a field of numbers, it would be a field of a special kind of numbers. But I'm sure a workaround can be found.)
Unfortunately, it would be a definition that is impossible to explain to almost all undergraduate math students, let alone 5-year-olds. It is perhaps best to keep "number" as a loose, non-mathematical term.
As a counterexample to your list of properties shared by all number systems, I would argue that the strictly positive integers are sometimes treated as a number system in their own right, so not all number systems have an additive identity.
An interesting edge case of numbers is the hyperreals. Despite the terminology of "the" hyperreals, this does not refer to a specific algebraic structure, but just imposes some conditions that the algebraic structure you're talking about is assumed to satisfy.
While it is true that not all number systems have an additive identity, I don't know of an example where they can't be extended to one that does. So I think it is fine to include it as a requirement: we don't lose any examples of objects that are called numbers.
Speaking of "the" hyperreals is indeed a little odd, although if you assume the Continuum Hypothesis, there is a unique hyperreal model that you get from ultrafilters. But that is probably jumping through way too many hoops just to be able to say "the" with a straight face.
Please do make a list of things-that-look-like-numbers-but-they-are-not.
Why is you vanilla field commutative, but your vanilla group is not?
Historically, the word "field" did also include objects with noncommutative multiplication (like the quaternions). Some time around 1970, if I remember correctly, there was a switch and fields were always understood to be commutative, with the more general noncommutative option known as a division algebra. Why? It turns out that commutative division algebras show up much, much more often. They are central to linear algebra, algebraic geometry, and so on. Noncommutative division algebras have their uses but they are more niche.
As for groups, we care about noncommutative examples at least as much as commutative ones.