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From: Archimedes Plutonium on 19 Jul 2010 03:26

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.

Until recently I would have thought that the Riemann Hypothesis RH was
unprovable because
it was another victim of this non definition of what it means to be a
finite-number. But
as it turns out, RH only needed a clever mind, the same said for Twin
Primes, Perfect Numbers conjectures and Goldbach.

So the most severe case of having to define finite-number was FLT in
Algebra and the Poincare Conjecture. And one can sympathize with why
geometry seems to glide along
without knowing the titantic iceberg of defining the boundary of
infinity. Not too many people
really think about infinity of the small as you approach the number 0
from 1 or the number
1 approached from 2. Not too many people in mathematics ever question
"absolute continuity".
Only perhaps a tiny few of all people that ever lived asked whether
the axioms of geometry
are compatible, consistent with infinity on the small scale. So it is
rather easy and understandable that no problems would ever come up
that directly question absolute-continuity, except of course the
Poincare Conjecture. And obviously there would be no proof
because internally it is false. Now there has been some brouhaha
recently of claims of an
alledged proof, but those are fakery-proofs.

And it is rather surprizing that the Calculus which is centered on the
small with its concept of
limit, surprizing that it seems to have glided past the awful omission
of a definition of finite
versus infinite boundary. But the Calculus has not really escaped
unscathed by the omission.
For it is not long when someone earnestly studies Calculus that you
get into so much thorny
questions of continuity. So I think the Calculus was never at ease
with continuity, and was fighting every day since the birth of
Calculus. Does the 10^-500 lower bound help Calculus?
I happen to think it strengthens the Calculus, in that when you reach
10^-500 and want to
talk about smaller, you simply say the same as dividing by zero, that
it is "undefined".

And that leaves only FLT and Poincare Conjectures as the most severe
victims of never a
precision definition of finite-number versus infinite-number. It seems
to me, that math
ducked the issues rather well without a precision definition. One
would think math of today,
having never defined the boundary should be in a whole lot worse
shape. Perhaps it is in
bad shape but that we are so climatized in our own era, that only
future generations can
look back and claim "how the 19th, 20th centuries were such Dark Ages
of mathematics".

We have two new geometries of NonEuclidean geometries of which we have
only begun to scratch the surface of this new subject, and perhaps
that is largely due to how much time
we waste on nonmath subjects of NP, 4 Color Mapping, Poincare
Conjecture, FLT, the
Cantor issue. But a whole bunch of other time wasters such as the
transcendental number
theory. So this precision definition of finite number is a full
sweeping cleaning up and cleaning
out of mathematics, and it sorely needs it.

It is worth noting and repeating, that the infinity defined without a
boundary and due to
quantum duality logic, that mathematics is destroyed by such an
infinity, and it makes
Cantor's infinities look like some harmless silly toy infinity.

So it is not a matter of "if" math defines the boundary, but when, and
that I am only the one
that started the process moving forward. Math is, afterall, the
science of precision and it is
its duty, its job to precision define.

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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