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Topic: A Point on Understanding
Replies: 66   Last Post: Jan 6, 2013 11:07 PM

 Messages: [ Previous | Next ]
 Paul A. Tanner III Posts: 5,920 Registered: 12/6/04
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:
>> ...
>>>

>>
>> 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.

Date Subject Author
12/15/12 Robert Hansen
12/17/12 kirby urner
12/18/12 Robert Hansen
12/17/12 Haim
12/18/12 Robert Hansen
12/26/12 kirby urner
12/26/12 Robert Hansen
12/26/12 kirby urner
12/27/12 Paul A. Tanner III
12/27/12 kirby urner
12/27/12 Robert Hansen
12/27/12 kirby urner
12/27/12 Robert Hansen
12/27/12 kirby urner
12/27/12 Robert Hansen
12/27/12 kirby urner
12/27/12 Robert Hansen
12/27/12 kirby urner
12/28/12 Paul A. Tanner III
12/28/12 kirby urner
12/28/12 Paul A. Tanner III
12/30/12 kirby urner
12/30/12 Louis Talman
12/30/12 kirby urner
12/30/12 kirby urner
12/30/12 kirby urner
12/30/12 Paul A. Tanner III
12/30/12 kirby urner
12/30/12 Paul A. Tanner III
12/27/12 Domenico Rosa
12/27/12 Robert Hansen
12/27/12 Richard Strausz
12/27/12 Domenico Rosa
12/30/12 Joe Niederberger
12/30/12 Paul A. Tanner III
12/30/12 Robert Hansen
12/30/12 Joe Niederberger
12/31/12 Robert Hansen
1/2/13 kirby urner
12/30/12 Joe Niederberger
1/1/13 Paul A. Tanner III
12/31/12 GS Chandy
12/31/12 Robert Hansen
1/1/13 GS Chandy
12/31/12 GS Chandy
12/31/12 Robert Hansen
12/31/12 Joe Niederberger
1/1/13 Robert Hansen
1/1/13 GS Chandy
1/1/13 GS Chandy
1/1/13 GS Chandy
1/1/13 Robert Hansen
1/1/13 Haim
1/1/13 Joe Niederberger
1/1/13 Joe Niederberger
1/1/13 Paul A. Tanner III
1/1/13 Louis Talman
1/2/13 Paul A. Tanner III
1/1/13 Joe Niederberger
1/2/13 Paul A. Tanner III
1/2/13 GS Chandy
1/2/13 Joe Niederberger
1/4/13 Joe Niederberger
1/5/13 GS Chandy
1/5/13 GS Chandy
1/6/13 Robert Hansen
1/6/13 GS Chandy