You are assuming that additive inverse is unique. Also you are assuming that -1(-1) is some value at all; you could write an extremely similar proof that leaves you with 0/0 = 1 and it would be faulty to conclude that 0/0 actually is 1.
> You are assuming that additive inverse is unique
Suppose that some number a has two additive inverses,x and y.
a + x = 0 = x + a
a + y = 0 = y + a
Consider
x + a + y = (x + a) + y = 0 + y = y
x + a + y = x + (a + y) = x + 0 = x
Thus x=y, and so the additive inverse is unique.
> you are assuming that -1(-1) is some value at all;
We can define it to have a value and then derive the properties of the value. We can show it to be consistent by creating an explicit model. Such things are relatively easy to do, but require a level of detail inappropriate for this context.
> you could write an extremely similar proof that
> leaves you with 0/0 = 1 and it would be faulty to
> conclude that 0/0 actually is 1.
Additive inverses are unique in any ring. The subtle things I skipped over because of the technicalities involved is how do I know the distributive property holds for negative integers. Indeed, what is a negative integer? How does one get them from the natural numbers? Suffice it to say that this can all be defined in a consistent, precise way and everything works out.
Actually, we only think that it can be described in a consistent way. We have proven it consistent using set theory , but Godwell's theorems tell us that we cannot prove a system to be consistent without using something outside of said system (unless the system is inconsistent). This means that there is some level in our chain of proofs that cannot be proven consistent.
