"Number" is an informal concept in mathematics. The real numbers are a specific concept with multiple formalizations.
A model of a number system is an assembly of behaviors, rules, or objects which behave like the numbers you want. I don't see the circularity here, and given that they are tackling an informal concept of number, I don't see any loss of clarity either.
Now, it is true that there are different kinds of numbers in math, but each different kind of number is formally defined. All of those definitions are interrelated, and collectively they define what it means to be a number. There is nothing informal about it.
> All of those definitions are interrelated, and collectively they define what it means to be a number. There is nothing informal about it.
Then you could simply point to the formalism which unites various mathematical activity under some universal model of number. I will reiterate the claim again — "number" is an informal concept among mathematicians and they are fine with that.
A model of a number system is an assembly of behaviors, rules, or objects which behave like the numbers you want. I don't see the circularity here, and given that they are tackling an informal concept of number, I don't see any loss of clarity either.