|
|
Re: FLT Discussion: Simplifying
Posted:
Jan 20, 2001 4:36 PM
|
|
In article <3A6842EB.B7633609@is.SPAMBLOCK.dal.ca>,
gus gassmann <gassmann@is.SPAMBLOCK.dal.ca> wrote:
> jstevh@my-deja.com wrote:
>
> > Sigh. Let me remind you of the facts. You are given that x and y
are
> > nonzero integers, and that x^2 + y^2 = 0. You notice that
> >
> > x^2 + y^2 = (x+sqrt(-1)y)(x-sqrt(-1)y), and let me explain that
> > further,
>
> Around here somewhere is your Assumption number 1: There exist nonzero
> integers x and y for which x^2 + y^2 = 0.
It's a proof by contradiction.
>
> >
> > What we're doing is noting that if there were this thing that
> > multiplied times itself to give -1, then we'd have that
factorization.
> >
> > We assume that such a thing exists.
>
> This is your Assumption number 2.
Your statement is somewhat redundant, since I just used the
word "assume".
>
> >
> > So, knowing that I have this odd object that is not an integer, but
is
> > *interacting with integers* I follow standard rules for integers.
>
> This is Assumption number 3. You don't know if you can work with
> this object (call it sqrt(-1), call it i, call it the Harris
constant, I
> don't care)
> in the same way you can work with the integers.
Which is again, just an argument that the proof is too long.
I think some of you wish to have your cake and eat it too.
(x+sqrt(-1)y)(x-sqrt(-1)y) = x^2 + y^2 is true for a wide variety of
x's and y's.
I agree that it probably wouldn't be true if x is an apple and y is an
orange, or even if x is an apple and so is y.
However, there has been an insistence on saying something like, x is a
complex number and y is a complex number, when they could just as
easily be integers with a finite ring.
So, yes, you *could* say that once you reach the conclusion that sqrt(-
1) does not produce a result in integers that the proof is complete as
the contradiction is reached.
>
> > ***That leads to a contradiction.***
>
> Maybe so, but which of your three assumptions is untenable? You'd
> like it to be assumption 1, but you (YOU, James Steven Harris!)
> have to rule out the other two possibilities. Otherwise your proof
> is incomplete, as in: unfinished, not done, containing a gap.
What do you call it when people keep making the *same* mistake over and
over and over again?
>
> > What some of you appear to be arguing is that when I realize that
this
> > thing, sqrt(-1) is not an integer, I must stop, and pull out a book
on
> > complex number theory.
>
> No. What people are saying is that once you realize that this thing,
> sqrt(-1) is not an integer, you must stop, and define (or state) what
> you mean by the operations x + sqrt(-1)*y, x - sqrt(-1)*y and
> (x + sqrt(-1)*y) * (x - sqrt(-1)*y). And you have to _prove_ for
> these objects (because they are NOT integers) that you can infer
> from (x + sqrt(-1)*y) * (x - sqrt(-1)*y) = 0 that either
> x + sqrt(-1)*y = 0 or x - sqrt(-1)*y = 0. And when you do
> _that_ proof, you have to make damn sure that it does not
> somewhere rely on another statement, namely that
> x^2 + y^2 = 0 has no solutions in the positive integers.
> Otherwise, your argument is circular.
>
> > ***But I'm doing a proof by contradiction.***
> >
> > I'm looking for something that pushes me outside of integers because
> > that's what I want to prove must happen!
> >
>
> But why don't you then simply write x^2 + y^2 = 2*[(x^2)/2 +( y^2)/2]
> and point out that this pushes you out of the integers already?
Remember,
> no one forces you to introduce sqrt(-1) into the proof, so the only
one
> doing the pushing is yourself.
There's a matter of practicality. After all, my *goal* is to
***solve*** a problem. While your goal may be to make academic debate.
>
> > You say the textbooks do it a different way, and I assume that
assaults
> > your sense of order for me to show it this way because if everybody
has
> > done it one way, you seem to assume that any other way is wrong.
> >
> > Then prove it's wrong!!!
> >
>
> OK. The example has been given several times before. Use the 2x2
matrices
> and embed the integers in them as follows:
>
> If x is an integer, associate with it the matrix | x 0 |
> | 0 x |
>
> Addition is done componentwise, like ordinary matrix addition:
>
> | x 0 | + | y 0 | = | x+y 0 |
> | 0 x | | 0 y | | 0 x+y |,
>
> multiplication is also ordinary matrix multiplication.
> (I can spell it out for you, but I assume that you can do
> matrix multiplication on your own.)
>
> Verify for yourself that these objects behave exactly like the
> integers with respect to addition and multiplication. The two
> rings (of integers and matrices of this form) are isomorphic,
> and it does not matter which representation you work with.
>
> Now define the symbol i = | 0 -1 |
> | 1 0 |.
>
> Verify that i*i = -1. (Again this can be spelled out if you wish.)
>
> Verify further that (x+iy) * (x-iy) = x^2 + y^2.
Well, *define* dog crap to be perfume, and try giving it to your
girlfriend.
Come on!
Here's where we're back to arguing semantics, and it's what makes me
think that many of you secretly wish to be lawyers.
Basically people this fellow is arguing that my proof has some gap
because I haven't taken a paragraph or two at the top to inform all of
you that x and y aren't matrices.
Well folks, you're going to need more than a paragraph, and I here
state that by this principle all mathematical proofs in existence have
gaps because none of you have handled all the possibilities that have
to covered because there are an infinite number of objects *besides*
matrices that can behave oddly and you'd better cover them too.
Cool, so there doesn't exist a valid math proof on the planet by these
arguments. *All* of your proofs and all proofs created by human beings
past, present, and future are gapped.
Remember, the start of the actual proof *does* begin by saying that x
and y are nonzero integers.
Sent via Deja.com
http://www.deja.com/
|
|
