
Re: A Point on Understanding
Posted:
Dec 28, 2012 10:30 AM


On Thu, Dec 27, 2012 at 2:16 PM, kirby urner <kirby.urner@gmail.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 coopted by any subsect or religious body for purely > its own purposes, although of course they're welcome to piggyback, 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 epsilondelta 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 factdenier  and if he or she persists in such factdenial, 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.

