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From: Archimedes Plutonium on 19 Jul 2010 03:57


Archimedes Plutonium wrote:
> It turns out that a precision definition of finite-number versus
> infinite-number had two
> severe cases in all of mathematics, one in Algebra as the Fermat's
> Last Theorem
> and the other in geometry as the Poincare Conjecture.
>

Well, wait a minute here, perhaps Kepler Packing suffered much more
than did
the Poincare Conjecture. I guess why I picked Poincare is because it
deals so directly
with the ill-defining of the continuity as we go smaller than 10^-500.
But the Kepler Packing
was never proveable for it never defined the infinity boundary of
10^500 for the large and
10^-500 for the end of the small.

The Kepler Packing is roughly over 300 years old yet the Poincare
Conjecture is over 100
years old.

Now if in Kepler's time, if it were announced to Mr. Kepler that
infinity is the boundary at
10^500 and you cannot go smaller than 10^-500, would Mr. Kepler then
have proven his
own conjecture? I think he would have by imagining such a container
and then approaching
the container walls with his unit spheres. Kepler would have realized
that the maximum packing is going to have to do some adjustments as
the spheres approach the walls, adjustments so that the hexagonal
closed packing is not the only pattern but that some
opportunistic changes of the pattern near the walls, delivers a more
dense packing.

And if Mr. Poincare had been advised that mathematics needs a
precision definition of the
tiny and that there is no absolute continuity, without much doubt, Mr.
Poincare would have
retracted his conjecture, and possibly entered a new conjecture with
that feature of 10^-500
gaps in mind.

And it goes to show that mathematics can hobble along quite well, even
though it had
major flaws of definition for the majority of its history. It goes to
show that a science can still
be productive even though it has a major flaw at its core. I suppose
the analogy is the
leaching or blood letting in Medieval times by the science of
medicine. It held back medicine, but it still plodded along. That is a
nice analogy to mathematics, that without a precision
boundary of finite-number versus infinite-number is the leaching and
bloodletting of students
in mathematics.

Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
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