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From: John Jones on 27 Mar 2010 23:26
Frederick Williams wrote:
> John Jones wrote:
>> That old mathematical chestnut - "x cubed + y cubed = z cubed"
>> seemed to be a historical problem for mathematicians. I'm not even sure
>> if it has been proved yet.
>
> What precisely? Is the problem to find real numbers x, y, z such that x
> cubed + y cubed = z cubed, or is it to find positive integers x, y, z
> such that x cubed + y cubed = z cubed, or something else?
>

I don't think real numbers are a possibility. The problem is portrayed
in the standard way.
From: John Jones on 27 Mar 2010 23:29
Niklasro(.appspot) wrote:
> On Mar 26, 11:04 pm, John Jones <jonescard...(a)btinternet.com> wrote:
>> That old mathematical chestnut - "x cubed + y cubed = z cubed"
>> seemed to be a historical problem for mathematicians. I'm not even sure
>> if it has been proved yet.
>> Of course, the proof of "x cubed + y cubed = z cubed" isn't difficult.
>> It's just that in a universe of an infinite number of proofs, we don't
>> know where to find the proof we want.
>> Why is this? There are no rules for finding proofs. Proofs aren't
>> sequenced like numbers. Proofs are all jumbled together. Unlike numbers,
>> there are no convenient Peano-type axioms for ordering and finding proofs.
>>
>> THE PROOF of x cubed + y cubed = z cubed-------
>>
>> "x cubed + y cubed = z cubed" is the answer, or solution to the proof
>> that we need to find.
>> "x cubed + y cubed = z cubed" is the starting condition.
>>
>> IF
>> "x cubed + y cubed = z cubed" is both theorem (starting condition or
>> question) and solution (answer),
>> THEN
>> proof is assured.
>>
>> BUT IF
>> "x cubed + y cubed = z cubed" is theorem (question) and not a solution THEN
>> proof is meaningless because "x cubed + y cubed = z cubed" is
>> necessarily neither theorem nor solution. That is, a theorem is a
>> vacuous notion in that it can be both present and absent.
>>
>> THE JOHN JONES CONCLUSION
>> When asking for a proof, the first thing we need to do is ask whether we
>> have a theorem or equation to prove, and not just a selection of empty
>> marks or signs that only seem to look mathematical.
>>
>> But here's the pinch. A proof, in principle, must illustrate the
>> difference between a vacuous mark and a semiotic indice (mathematical
>> sign). How does a proof do that?
>>
>> Come on then you mathematicians, let's have the answer.
> Obvious 2 methods are
> 1. Induction

You can't INDUCE that a sign is or isn't a mathematical or logical sign!


> or
> 2. Assume it's false and deduce a contradiction.

Falsehood isn't expressed by mere marks or signs, nor can such signs be
inductively or deductively proved.
> Say method 1 or 2 works. Must then both?
From: Niklasro(.appspot) on 28 Mar 2010 05:57
On Mar 28, 3:29 am, John Jones <jonescard...(a)btinternet.com> wrote:
> Niklasro(.appspot) wrote:
> > On Mar 26, 11:04 pm, John Jones <jonescard...(a)btinternet.com> wrote:
> >> That old mathematical chestnut - "x cubed + y cubed = z cubed"
> >> seemed to be a historical problem for mathematicians. I'm not even sure
> >> if it has been proved yet.
> >> Of course, the proof of "x cubed + y cubed = z cubed" isn't difficult.
> >> It's just that in a universe of an infinite number of proofs, we don't
> >> know where to find the proof we want.
> >> Why is this? There are no rules for finding proofs. Proofs aren't
> >> sequenced like numbers. Proofs are all jumbled together. Unlike numbers,
> >> there are no convenient Peano-type axioms for ordering and finding proofs.
>
> >> THE PROOF of x cubed + y cubed = z cubed-------
>
> >> "x cubed + y cubed = z cubed" is the answer, or solution to the proof
> >> that we need to find.
> >> "x cubed + y cubed = z cubed" is the starting condition.
>
> >> IF
> >> "x cubed + y cubed = z cubed" is both theorem (starting condition or
> >> question) and solution (answer),
> >> THEN
> >> proof is assured.
>
> >> BUT IF
> >> "x cubed + y cubed = z cubed" is theorem (question) and not a solution THEN
> >> proof is meaningless because "x cubed + y cubed = z cubed" is
> >> necessarily neither theorem nor solution. That is, a theorem is a
> >> vacuous notion in that it can be both present and absent.
>
> >> THE JOHN JONES CONCLUSION
> >> When asking for a proof, the first thing we need to do is ask whether we
> >> have a theorem or equation to prove, and not just a selection of empty
> >> marks or signs that only seem to look mathematical.
>
> >> But here's the pinch. A proof, in principle, must illustrate the
> >> difference between a vacuous mark and a semiotic indice (mathematical
> >> sign). How does a proof do that?
>
> >> Come on then you mathematicians, let's have the answer.
> > Obvious 2 methods are
> > 1. Induction
>
> You can't INDUCE that a sign is or isn't a mathematical or logical sign!
>
> > or
> > 2. Assume it's false and deduce a contradiction.
>
> Falsehood isn't expressed by mere marks or signs, nor can such signs be
> inductively or deductively proved.
>
> > Say method 1 or 2 works. Must then both?

OK: You assume there is no language. Say you have a lagnuage. Then
both methods are perfectly valid. Modus tollens and unduction.
From: Niklasro(.appspot) on 28 Mar 2010 05:58
On Mar 27, 2:03 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
> John Jones <jonescard...(a)btinternet.com> writes:
> > It's just that in a universe of an infinite number of proofs, we don't
> > know where to find the proof we want.
> > Why is this? There are no rules for finding proofs. Proofs aren't
> > sequenced like numbers. Proofs are all jumbled together.
>
> Yes!  Yes!  Now I understand it.
>
> This must indeed be why finding proofs is tough.  There's simply no
> way to order them.
>
> I see now that you *are* the resident Goedel expert after all.
>
> --
> "But what if I'm right [and have solved the factoring problem]?  And
> what if I can't contain the problem and, oh, in six months from now,
> you are desperately trying to find food as you run from humans who
> have turned to cannibalism to survive?"  -- James S. Harris (2/21/09)

There are rules for proving. Modus ponens, modus tollens, inductions...
From: Frederick Williams on 28 Mar 2010 07:22
John Jones wrote:
>
> Frederick Williams wrote:
> > John Jones wrote:
> >> That old mathematical chestnut - "x cubed + y cubed = z cubed"
> >> seemed to be a historical problem for mathematicians. I'm not even sure
> >> if it has been proved yet.
> >
> > What precisely? Is the problem to find real numbers x, y, z such that x
> > cubed + y cubed = z cubed, or is it to find positive integers x, y, z
> > such that x cubed + y cubed = z cubed, or something else?
> >
>
> I don't think real numbers are a possibility. The problem is portrayed
> in the standard way.

So you think that this problem:

Are there positive integers x, y, z
such that x cubed + y cubed = z cubed?

has not been solved?

--
I can't go on, I'll go on.
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