The way you think of it is precisely the mathematical way of thinking about n-dimensions.
Mathematically, an n-dimensional space is R^n, that is, the cross product of the real numbers (R) with itself n times. Each copy of R lets you specify a number independently of all the others.
Mathematics is just careful abstraction. If you are abstracting away certain aspects of your problem and carefully making deductions about what is left, you are doing mathematics. As you point out, this can be taken too far but any tool can be used poorly in the hands of a novice.
Mathematically, an n-dimensional space is R^n, that is, the cross product of the real numbers (R) with itself n times. Each copy of R lets you specify a number independently of all the others.
Mathematics is just careful abstraction. If you are abstracting away certain aspects of your problem and carefully making deductions about what is left, you are doing mathematics. As you point out, this can be taken too far but any tool can be used poorly in the hands of a novice.