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From: John Jones on 26 Mar 2010 19:04
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.
From: Frederick Williams on 26 Mar 2010 19:26
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 can't go on, I'll go on.
From: Niklasro(.appspot) on 26 Mar 2010 19:47
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
or
2. Assume it's false and deduce a contradiction.
Say method 1 or 2 works. Must then both?
From: Jesse F. Hughes on 26 Mar 2010 22:03
John Jones <jonescardiff(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)
From: Immortalist on 26 Mar 2010 23:11
On Mar 26, 7:03 pm, "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)

> > Goedel proved that any set of axioms at least as rich as the
> > axioms of arithmetic has statements which are true in that set of
> > axioms, but cannot be proved by using that set of axioms.
>
> What does "true in that set of axioms" mean?

Not those axioms needed to figure this set of axioms out?

Actually it doesn't necessarily mean that. If "the set of axioms" is
that set of axioms containing all possible axioms base upon the
numerals 1,2,3,4,5,6,7,8,9, or 0, each used as many times as
necesarry, to produce all possible axioms, does in fact contain all
axioms, including the axioms explaining those axioms, but is logically
impossible at this time since we cannot yet count to infinity.

Russell's paradox represents either of two interrelated logical
antinomies. The most commonly discussed form is a contradiction
arising in the logic of sets or classes. Some classes (or sets) seem
to be members of themselves, while some do not. The class of all
classes is itself a class, and so it seems to be in itself. The null
or empty class, however, must not be a member of itself. However,
suppose that we can form a class of all classes (or sets) that, like
the null class, are not included in themselves. The paradox arises
from asking the question of whether this class is in itself. It is if
and only if it is not. The other form is a contradiction involving
properties. Some properties seem to apply to themselves, while others
do not. The property of being a property is itself a property, while
the property of being a cat is not itself a cat. Consider the property
that something has just in case it is a property (like that of being a
cat) that does not apply to itself. Does this property apply to
itself? Once again, from either assumption, the opposite follows. The
paradox was named after Bertrand Russell, who discovered it in 1901.

There is a solution to the Epimenides Paradox (a.k.a. the Liar's
Paradox).

The paradox goes like this:

1. Epimenides is a Cretan.
2. Epimenides states, "All Cretans are liars."

On the face of it, this appears to be a paradox. Epimenides, being a
Cretan, must either be a liar or a truth-teller. Thus his statement
must be either true or false. But if it's true, then he (being a
Cretan) must be a liar, so the statement can't be true. On the other
hand, his statement is false, then he can't be a liar, so the
statement must be true. This is a paradox.

Or so it would seem. Actually, the trouble lies in the interpretation
of the statement "The statement 'All Cretans are liars' is false".

The solution goes like this:

p1. Epimenides is a Cretan.
p2. Epimenides is either a liar or a truth-teller.
p3. His statement is either true or false.

Assume that there is more than one Cretan:

p4. There is more than one Cretan.

Also assume that Epimenides is indeed a liar:

p5. Epimenides is a liar.
p6. Thus Epimenides's statement is false.
p7. Thus "All Cretans are liars" is false.
p8. Thus not all Cretans are liars.
p9. Thus some (one or more but not all) Cretans are not liars.
p10. Thus at least one (but not all) of them is a liar.
p11. Thus Epimenides, a Cretan, could be a liar.

We assumed that Epimenides was a Cretan (p1) and a liar (p5).
Therefore, there is no paradox.

Another way of looking at it is to realize that, unless there is only
one member of a set, then the negation of "all members of the set",
i.e., "not all members of the set", is not "no members" but "some
members".

If Epimenides is the only Cretan (so the set of Cretans has only one
member), then there would be a paradox, since "not all Cretans" would
mean "no Cretans".

................

(1) This sentence is false.

If (1) is true, then (1) is false. On the other hand, assume (1) is
false. Because the Liar Sentence is saying precisely that (namely that
it is false), the Liar Sentence is true, so (1) is true. We've now
shown that (1) is true if and only if it is false. Since (1) is one or
the other, it is both.

http://david.tribble.com/text/liar.htm
http://www.google.com/search?hl=en&ie=UTF-8&oe=UTF-8&q=solution+to+the+liars+paradox


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