On Thu, Dec 27, 2012 at 2:16 PM, kirby urner <email@example.com> wrote: >> ... >>> Contradiction? >>> >> >> In all these posts in this thread you have yet to actually state what >> the contradiction is supposed to be, since you have not actually put >> forth a contradiction, which is by definition a statement that is >> false in all its substitution instances - that is, for instance when >> doing truth tables, in the main column there would be nothing but but >> F's. (A tautology is by definition a statement that is true in all its >> substitution instances - when doing truth tables, in the main column >> of its truth table there would be nothing but but T's.) > > "Contradiction" is an English word that has survived the centuries > without being co-opted by any sub-sect or religious body for purely > its own purposes, although of course they're welcome to piggy-back, as > is their wont. > > I hope you're not so dismissive of student difficulties when they're > trying to get their minds around concepts with inherent difficulties.
Students need to be taught that in mathematics, they ought not throw the term "contradiction" around casually - it's just plain sloppy. To prove a contradiction one does not merely negate a given statement with a sloppy argument.
> > Picture it as a debate between two opposing sides if you like. You > are the judge and need to score each debater and declare a winner.
> > Resolved: the limit of |360 - v| as the number n of vertexes v on a > geodesic sphere increases to infinity is 0. > > Debater A (affirmative): simple epsilon-delta proof will do the job. > As we all learned in calculus, if I can give you a small epsilon e > such that |360 - v| < epsilon, when n (number of vertexes) > delta, > and if I can show that for any epsilon, no matter how small, a > corresponding delta might be found, then | 360 - v | < epsilon indeed > has 0 as its limit as n -> infinity. QED. > > Debater B (negation): we know conclusively and without doubt from > Descartes' Deficit, that |360 - v|, no matter how small, is never 0, > because the difference, however vanishing, contributes to a total of > 720 and this number holds constant regardless of your delta or > epsilon, so I piss on your "proof". > > So there's your p & ~p where p = proposition (the resolution).
Yes, it is true: Merely negating a mathematical result p does result in the form p & ~p.
But so what?
If p is actually a proved statement, a theorem, then person B is simply showing himself or herself to be a fact-denier - and if he or she persists in such fact-denial, to be a crackpot.
The problem is, if a person does not first do the careful work needed to prove that a mathematical result is actually incorrect, then that person's claim that that result is incorrect - along with his or her sloppy argument - shows himself or herself to be mathematically inept and ignorant and a sloppy thinker.
Again: Students need to be taught that in mathematics, they ought not throw the term "contradiction" around casually - it's just plain sloppy. To prove a contradiction one does not merely negate a given statement with a sloppy argument.