Date: Dec 28, 2012 10:30 AM
Author: Paul A. Tanner III
Subject: Re: A Point on Understanding
On Thu, Dec 27, 2012 at 2:16 PM, kirby urner <email@example.com> wrote:
>> 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.