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From: Niklasro(.appspot) on 28 Mar 2010 17:36
On Mar 28, 11:22 am, Frederick Williams
<frederick.willia...(a)tesco.net> wrote:
> 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?
Modus tollens outline: assume it's solved. Derive a contradiction.
Done.
From: John Jones on 28 Mar 2010 18:39
Frederick Williams wrote:
> 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 don't know. That sort of thing wouldn't interest me in any case.

In order for a mathematical problem to be solved we have to assume that
the problem is presented in mathematical syntax.

Therefore, a proof ought to be able to distinguish syntax from mere
marks on paper. But a proof doesn't do that.
From: Jesse F. Hughes on 28 Mar 2010 19:09
"Niklasro(.appspot)" <niklasro(a)gmail.com> writes:

> On Mar 28, 11:22 am, Frederick Williams
> <frederick.willia...(a)tesco.net> wrote:
>> 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?
> Modus tollens outline: assume it's solved. Derive a contradiction.
> Done.

You obviously don't know what modus tollens is. Aside from that, your
suggestion is, of course, brilliant. How hard could it be, after all?

--
"Just because you're ... in a Ph.d program it does not mean that
you're up to the challenge of being a real mathematician. Only those
who have a purity of mind and dedication to the truth as the highest
ideal have a chance." --James Harris, as Sir Galahad the Pure.
From: Frederick Williams on 29 Mar 2010 10:32
John Jones wrote:

> In order for a mathematical problem to be solved we have to assume that
> the problem is presented in mathematical syntax.

By "mathematical syntax" do you mean a formal language in the logician's
sense? If so then you are clearly wrong since such things were only
invented in the nineteen century (though they were presaged by Leibniz
before then).

> Therefore, a proof ought to be able to distinguish syntax from mere
> marks on paper. But a proof doesn't do that.


--
I can't go on, I'll go on.
From: John Jones on 30 Mar 2010 08:35
Frederick Williams wrote:
> John Jones wrote:
>
>> In order for a mathematical problem to be solved we have to assume that
>> the problem is presented in mathematical syntax.
>
> By "mathematical syntax" do you mean a formal language in the logician's
> sense? If so then you are clearly wrong since such things were only
> invented in the nineteen century (though they were presaged by Leibniz
> before then).
>
>> Therefore, a proof ought to be able to distinguish syntax from mere
>> marks on paper. But a proof doesn't do that.
>
>

erg
If you ask "is P solvable" then we can both agree what P is. It doesn't
matter what P is as long as we are both agreed as to what it is.

My point is that ONCE we are agreed as to what P is, then if P is not
solvable, then it isn't what we thought it was.
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