
Re: FLT Discussion: Simplifying
Posted:
Jan 16, 2001 8:57 PM


In article <942k1a$i6f$1@nnrp1.deja.com>, jstevh@mydeja.com wrote: > In article <G78GGv.7ID@cwi.nl>, > "Dik T. Winter" <Dik.Winter@cwi.nl> wrote: > > In article <9400ps$b5g$1@nnrp1.deja.com> jstevh@mydeja.com writes: > > > Given x^2 + y^2 = 0, x and y nonzero integers, show that no > solution > > > exists. > > > > > > Proof by contradiction: > > > > > > (x+sqrt(1)y)(xsqrt(1)y) = x^2 + y^2 = 0, so > > > > > > x = sqrt(1)y *or* x = sqrt(1)y. > > > > James, you have to *prove* that last step > > As you can see, some people will force you to outline simple steps > that others would find unnecessary.
The others would not force you to outline the steps because they can see how it can be done. Some people who cannot follow the justification of a statement within a proof can ask you to outline the steps for that statement's justification. When you are asked by such people, then you should provide the missing steps. If the steps are simple, then you should not have any problem.
This is similar to statements in your FLT proof. Some people who cannot follow the justification of a statement within your FLT proof have asked you to outline the steps to justify it. You have brushed aside their request as you are doing above for the proof of x^2+y^2=0 having no integer solutions. But the missing steps can be filled in for justifying the statements in the proof of the latter. Since they can be filled in, we are asking you to do the same for the missing steps (or gaps) in your FLT proof.
> That bit of arbitrariness is used by some people to claim that > a proof isn't.
It is not arbitrariness. A person can request the missing steps in any proof, especially if he or she does not see what they are.
We do come to a problem when we are down to first principles. For example, I might be asked to justify the following step (I will assume that I am working in the field of complex numbers, which you have refused to do for some reason):
Claim: (x+sqrt(1)y)(xsqrt(1)y) = x^2 + y^2
I could reply that I am using the distributive law of multilication to get:
(x+sqrt(1)y)(xsqrt(1)y) = (x+sqrt(1)y)*x + (x+sqrt(1)y)*(sqrt(1)y)
You might now ask me to justify this step as being an application of the distributive law of multiplication in the field of complex numbers (not objecting to the distributive law itself). Now we are getting into first principles (substitution of expressions for universal variables). Thus, the discussion of the proof can bog down at this point, with no progress to be made.
However, the missing steps that you are being asked to supply do not even remotely fall in the realm of first principles.
 Bill Hale
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