On Dec 3, 2012, at 4:54 PM, Joe Niederberger <firstname.lastname@example.org> wrote:
> (What I was questioning was: "repeated subtraction twice, which is back to repeated addition". > Lost me there. Strikes me as a sleight-of-speech.)
I was thinking...
(3) * (-2) = (-2) + (-2) + (-2) = -2 - 2 - 2
Or the other way around ...
(-2) * (3) = - 3 - 3 (repeated subtraction) (I just realized so much for crabtree's subtracted once from itself theory, that was an unexpected bonus)
And with two negatives...
(-2) * (-3) = - (-3) - (-3) (thus, when you are subtracting a minus you are actually adding)
> > Going on, I've just been pointing out that this particular issue has a long and colorful history, and in some sense remains a conundrum. Look, people *do* learn the rule. That's not even my point, as if it stops 'em cold. > > But just look around the web today - this remains a stumbling block, a puzzler, an enigma to many people. Chalk one of for "formal reasoning" if you like -- people did not establish this rule here by common sense (alone) for sure, but the desire to understand via common sense endures, and is apparently frustrated, at least on an case-by-case basis. Time travel and un-losing poker hands are nice, but they just don't have the force of two apples plus three apples.
Isn't the context just getting more complicated? I mean, by the time you figure out the cars or un-losing poker hands, you have probably developed the elements (habits of mind) behind formal thinking.