On Sun, Dec 2, 2012 at 8:42 PM, Joe Niederberger <email@example.com> wrote: >> People state the plus and minus sign when they mean the positive and negative sign. > > I'll simply note that Barry Manzur of Harvard (of the recently mentioned book "Imagining Numbers" -- thanks Dave Renfro) says "minus times minus" over and over. He's not hung up on these wordsmithing trivialities. > > What remains though, is that neither he nor anyone else here can convincingly make a case for the infamous rule to be either common sense or a mathematical certainty. >
Equality (-a)(-b) = ab for all ring elements a,b is a mathematical certainty in every ring, since it like every other theorem of rings is simply ultimately a result of the definition of a ring. (See your local abstract algebra textbook.)
Outside of a ring, with contrary definitions, we can derive contrary results.
Theorems derive from definitions (or axioms when used). Different definitions (or axioms when used) can yield different results. Why is this so hard to accept?
Note: Each mathematical theorem is an implication statement or is at least equivalent to one (if not in the form of one, it can be put into such a form). (That is, there is always some p and q somewhere or somehow such that p -> q is what is proved when theorems are proved, not p or q by itself.) That is, what is proved is that some "atomic" statements have provable relationships among themselves - the proved p -> q is what is the mathematical certainty, not p or q by itself.
Real world example of this basic idea if you need one: If it's raining outside, then it's wet outside. What is certain in a real world way here is not either clause, but their implication-based relationship.
That is, it's not correct to say that there is no certainty to be had in mathematics. It's just that it's a certainly that some conjunctions of statements can be negated with certainty. (Note: I speak of the definition of an implication, which [where "~" means logical negation] is p -> q if and only if ~(p & ~q).)
(Yes, I know. Some may not find any psychological satisfaction in such a type of certainty. But I for one think that's it's pretty good.)