Hal Finney wrote
> In other words, just as the falsity of the GC would seem to imply
> that there must be a specific number which ISN'T the sum of two
> primes, so the falsity of the PP would seem to imply that there
> must be a specific distance to a point which is the farthest
> possible point of intersection. For either proposition to be
> unprovable would seem to imply that there is no such number or
> point, which would mean that they are true. In the case of the
> Parallel Postulate, we can measure the distance to the
> intersection point as the lines approach parallelism.
I think that I see the flaw here, but I had to write a little
story before I could, and so, even if it is redundant, I might
as well post the story since someone besides me may need it too:
Okay, so we are standing in the lobby in the Restaurant at the
End of the World, and overhearing an interesting conversation
between a well-known great 20th century mathematician and Euclid.
I mean to say that with the help of some Tiplerian technology,
they've both been resurrected.
In the middle of their discussion, a student rushes up to
the mathematician and says, "I've discovered that Goldbach's
Conjecture is unprovable!"
"Nonsense," says the mathematician. "That is impossible.
If you had, then you would have proved it true, because
you would have prevented any possibility of me exhibiting
an even number with the (easily verifiable) property that
it is not the sum of two primes. Go away, charlatan!"
Euclid takes this all in, and is very impressed. But
just then, another student rushes up to Euclid, thinking
to have some fun. "By the gods, Mr. Euclid, I can show
that your parallel postulate can be neither proved nor
disproved from your first four axioms!"
Euclid starts talking almost without thinking: "That
is impossible. Suppose that you had shown such a
thing. Then that would mean that for any point on
the (lower) line, the upper line couldn't possibly
cross there, and so you would have shown that the PP
is true---not undecidable! Go away, charlatan!"
"But sir," retorts the student, "I've shown that you
can take it (the PP) and not reach a contradiction,
or that you can leave it and not reach a contradiction.
Yes, in some models it's quite true, but in others,
Being a genius, Euclid somehow sees the point. But the
first student was still listening in on all this, and
noting Euclid's ashen countenance, decides he can pull
the same thing on the 20th century mathematician:
"Listen," he says, "you can have a model where every
even number is the sum of two primes", or, "you can
have a model where some even number IS NOT the sum
of two primes".
"And what becomes of that number in the first model?",
angrily roars the 20th century mathematician. "You see,
we are talking about THE SAME NUMBER, idiot! A particular
integer is a REAL THING by the grace of GOD! (screaming)
SEVENTEEN IS PRIME NOT BECAUSE OF ANY STUPID AXIOMS BUT
BECAUSE IT **IS** PRIME!!!". At this point, the 20th
century platonist mathematician becomes apoplectic and
has to be taken outside for some fresh air.
Says Euclid to the Riemannian geometry student, "So if
you are right, then a HalFinney point, i.e. a point that
is at a maximum "distance" from where a transversal cuts
two parallel lines, in your model---is that the right
word?---of course wouldn't have that property in a model
where parallel lines never meet. Hmm. I guess that's
another indication that geometry is richer than
arithmetic," he says with a smile.
So I'll guess from my little story that all models
of arithmetic that have enough axioms to support the
concept of "even number" either all have a the same
even number that isn't the sum of two primes, or all
have no such even number.
We reify "seventeen" and give it a particular identity,
and rightfully so, because you can prove many things
about seventeen, whereas "a point" is an undefined
concept in geometry.
> In other words, just as the falsity of the GC would
> seem to imply that there must be a specific number
> which ISN'T the sum of two primes,
evidently in any model,
> so the falsity of the PP would seem to imply that
> there must be a specific distance to a point which
> is the farthest possible point of intersection.
but only in some particular model, that "distance"
depending on which.
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