[contravariant]

If you were working with the standard field then why would you need to bother defining the general concept of subtraction?

If you want to define the concept of subtraction then you probably don't want to assume you can multiply elements with an integer. Not that it can't be done but in general it will be a lot easier to define the additive inverse directly (if one exists).

[godelski]

I mean but under the example that the author is doing they are working with standard integers (integrals). So this appears to me to still be the standard addition and subtraction. And with standard R1 subtraction it is the same as inverse of addition.

I mean if we move into different coordinates and different fields, then yeah things change, but I don't see what the issue is with the example given here.