And I have often written that the longer one spends
in science, that science has a nasty habit of falsifying
your earlier work. I think it was Frege or Russell or
someone spent years on writing a book when it was
obsolete just before the publishing date from some
new results.

Anyway, the history of science shows us that even some of the most
"indubitable knowledge" some of the
most closely held beliefs turn out to be falsified by future science
discoveries. There was a time when it
was believed that Newton's physics would be everlasting, just as
absolute space and absolute time.

When I wrote the AP-adics book, I was attacking two
math ideas. One was a better formulation of P-adics from Hensel's p-
adics and so I came up with the AP-adics. But a second attack was to
further the idea that
Geometry was this:

Euclidean Geometry = Elliptic unioned to Hyperbolic Geometries

Written in short: Eucl = Elliptic + Hyperbolic

And where I use the three and only three number-systems as native to
one of those geometries such as this:

Doubly Infinites = +AP-adics unioned -AP-adics

But with this book of Correcting Math, I have run into
a serious problem. In the AP-adics book, those numbers are defined as
having a infinite-component
and in Doubly Infinites there is no finite-component.

In this book, I define Finite as 10^500 or less (inverse
included). And I throw out as meaningless the infinity
or infinite-numbers.

So this book of Correcting Math ruins my previous book of AP-adics.

But I can salvage the AP-adics book.

I simply retitle it as Eucl Geom. = Ellipt unioned Hyperbolic. So I
switch the emphasis to the geometry
aspect of that previous book.

Then I go ahead and define all the integers to 10^500
as the AP-adics

I define the Hyperbolic geometry numbers as the negative integers to
(-)10^500

For the numbers native to Euclidean Geometry I define
them as the Doubly Finites rather than the Doubly Infinites which has
numbers such as this:

333..33d999..99 where the symbol .. signifies the
upper bound of 10^500 (and inverse).

Simply put, I erase all infinite numbers and provide
only finite numbers defined as 10^500 upper bound.

The beauty of all numbers as finite, is a relief to Calculus, which
works as old Calculus but there is
never any problems of discontinuities, since there
never was any continuity to begin with. All of geometry
is discontinuous with holes in between all numbers.

So math becomes what Feynman became used to
in his old age with Quantum Electrodynamics of getting
rid of the infinites with renormalization procedures.

So in effect, I, Archimedes Plutonium, is not improving
Quantum Electrodynamics by renormalizing, but rather,
I am improving all of mathematics by renormalizing all of mathematics
and throwing out infinities.

So by defining Finite in mathematics as 10^500 or below, what I have
thus done is similar to what Feynman did for Quantum Electrodynamics--
renormalized.

But, also, now, I have to correct my previous book of its Geometry
with the concept of finite.

Since there are no infinities in mathematics because there are none in
physics, then we cannot have an
infinite line, afterall. That means all lines in Geometry
are finite and can go to the extreme of (-)10^500
endpoint to 10^500 endpoint. I spoke of a Euclidean
Geometry as an all finite geometry already and see no
problems in replacing out the old with the new.

But I do see some problems in defining a finite line in
Elliptic and Hyperbolic geometry. So it may be years
before I get back to that new edition and in the meantime can
anticipate those changes.

| = ) + (

The idea is that the Elliptic geometry is the inverse of Hyperbolic
geometry
and where the application of a curve ) on another curve ( ends up with
a
straight line segment.

Now to translate that into number algebra we simply go to inverses
where +2 is the inverse of -2. So if we have a curve ) that is all
positive
finite numbers applied to a curve ( of all negative finite numbers,
they
cancel leaving behind a straight line segment.

Or, we can do the transform via multiplication inverse where curve )
is positive integers and curve ( is multiplicative inverses cancelling
to form a straight line segment.

And I set up Euclidean geometry as a finite geometry going from one
extreme end of
(-)10^500 to the other extreme end of 10^500

Both the Elliptic and Hyperbolic geometry can be represented by a
sphere and where
one is concave outwards, the other is concave inwards and putting the
two together
they cancel one another yielding a Euclidean straight line segment.

(1) Euclidean geometry is the Elliptic geometry unioned with
Hyperbolic geometry
(2)  | = )(

In (2) we see how the union occurs in that we simply reverse a
triangle's
sides of concavity into a reverse concavity yielding a straight line.

And I should mention a third equation in this process of connecting
geometry
to algebra:

(3) Given a Doubly Finite Number which is native to Euclidean Geometry
such as
1000..00d3333..33 where the symbol ".." indicates these are restricted
by a boundary
of finite as 10^500 decimal place value. We have:
1000..00d333..33 = 1000..00r + r333..333
So we have numbers in Elliptic geometry that are the integers and
numbers in
Hyperbolic geometry which we considered to be the decimal fractions.

In the 3rd edition I should detail how the Calculus is rendered more
clear
and easier with the definition of Finite as the boundary at 10^500
(with its
inverse).

Now there is one last item I want to convey before leaving. It is an
item I
brought up in the AP-adics book of considering these three different
Infinite-Integers.
(a) 1000....0000 = South Pole and 100% of the distance from North Pole
to South Pole
in Elliptic geometry.
(b) 1000....0000 = 10% of the distance from North Pole to South Pole
and is one decimal
place missing of (a) since it is 10%
(c) 0000....00100...0000 that number which is peculiar or strange. It
definitely exists
if you demand that Infinite Numbers exist. (Although I no longer
demand any infinity
concept and that Physics is all finitary.)

I bring this topic up, here at the end, because I suspect this
peculiar number offers a proof
that the definition of Finite versus Infinite requires one to pick out
a "known finite number"
such as 10^500 and declare all other numbers beyond as the realms of
incognitum or
infinite.

I believe this strange number 0000....00100....00000 is what Old Math
believed as Finite
in the best definition of Old Math had for "finite." If the reader
recalls, the best that the
Old Math ever did for finite definition was to say that a number is
finite if its string of digits
leftward ends in zeroes. For example: 747 was finite in Old Math
because it was
0000....0000747. Noone ever bothered the old-timers of math because
the discover that
a number has a FrontView with BackView blinded them of ever asking
whether
07777.....77777 would be a finite-number according to the best
definition of finite by the
old math and its old timers.

So a number like 0000....00100....00000 seemed to fit the bill for the
Old Math,
old timers definition of finite as ending in zeroes leftwards. And
noone can say
exactly where that "1" digit was. It was far smaller than 10% of the
distance
to the South Pole from North Pole. In fact, one could say that the
distance
was an infinitesimal distance from the North Pole in the direction
heading for
the South Pole.

So I leave this book with this strange number, and I suspect this
number can
prove that the only way of defining Finite versus Infinite is to
actually pick out
a large finite number such as any of the Planck Units in physics and
to say--
that is the end of Finite and that all numbers beyond are either
meaningless
since there is no more physics going on, or that all numbers beyond
are in
the realms of incognitum to infinite.

So this special number is a tool to use in a proof that to define
Finite in mathematics
requires one to select a large finite number and call it the boundary
between
finite and infinite, or finite and meaningless. And since Physics has
no
infinities, well, it means that infinity is meaningless.

Bon voyage.

I need to wrap my head around Plutonium day which is rapidly coming
up-- Nov 7,
auf wedersehn

thus:
what the Hey?...
EEE's result is just the old "trivial" one,
a triple of (1,0,1), a degenerate pythagorean trigon
(if using "two" for the exponent, n).

he used to couch it in the "nonequality
of 1.0000... and 0.9999...," which notion he seems
to have dropped.

lest ye fugeddabowdit,
http://wlym.com/~animations/fermat/Observations20on%20Diophantus.pdf

thus:
well, that was a _Peter_ Michelson.  He  and
Smolin are some kind of freaked, that they'd worry
about the idea of the index of refraction, varying
for different kinds of Newtonian "photons;" but,
how can a zero-mass point-particle have a frequency, or
a wavelength?

lest ye fugeddabowdit,
wlym.com/~animations/fermat/Observations20on%20Diophantus[1].pdf

thus:
I am lying about numbertheory, and
the number, 1.0000...;
who gives a floating fart?

thus:
original sources (and "sourcebooks") are really good,
such as the below-linked Ouvre de Fermat for number-
theory, and Bernoulli/L'Hopital's calculus textbook.
(Euclid, not so much, as an encyclopedia, although
he did supply new stuff, they say -- and
Langlands says that Book 7 needs a lot of work; I do have
a nice latter-day textbook on synthetic trigon geometry, but
it's in French, so it's hard work.)

thus:
of course, and the electrons can't go faster
than light *even if*
they might already be orbitting the nucleus
at such a velocity.

thus:
I could see that he got rid of the gamma function, but
it'll be a while before that is clear to me; so,
I asked about a problem he wrote about, before.

m'brain:
L'Ouvre: http://wlym.com/~animations/fermat/index.html