On Tue, Dec 4, 2012 at 12:08 AM, Joe Niederberger <email@example.com> wrote: >>And so to say that that the theorem itself (and not merely such a particular proof of it) is based on such is not right. > > I said the sign rule for multiplying negative integers is not based on the fact that integers have a magnitude (together with a sign.) You think it is? > > Joe N
"In fact it emphasizes the fact the rule is based on the signs alone."
This is not correct.
I said :
"One of the standard proofs in abstract algebra textbooks of the theorem (-a)(-b) = ab for all ring elements a,b (including for rings that are not ordered) contains the equality (-a)(-b) = -(-(ab)).
Is this what you meant?
Regardless, there are ways to prove this theorem that does not use the prior theorem of additive groups x = -(-x).
And so to say that that the theorem itself (and not merely such a particular proof of it) is based on such is not right."
That is, the theorem (-a)(-b) = ab is about the elements of the sets and the unary function of taking additive inverses of elements, not about "the signs", which is about how to write "the additive inverse of".