#46 Listing of the major mistakes of the old math and their Reals extensions; new textbook; "Mathematical-Physics (p-adic primer) for students of age 6 onwards"
Archimedes Plutonium
Algebraic Fields and Rings when they should have focused on Geometry
as to what Numbers are native to what geometry.
(1) By old math I mean all math before 1993 when I started posting that
the Counting Numbers were really the P-adics.
(2) The mistake that old math had the notion they were tops over
physics, when in fact, mathematics is a tiny subset compartment of physics.
(3) The mistake that mathematics had only one type of Number and all
others were side-extensions of Reals
(4) The Peano axioms and the development of the Reals from Naturals then
Rationals then Irrationals and finally a Dedekind Cut and then the
Complex Number extension, that entire program is deeply flawed because
it is never linked by any solid anchor of geometry itself. There are
two separate, independent and different geometries as Euclidean and then
the bundled Elliptic plus Hyperbolic.
(5) Physics tells us Particle Wave duality of all the important things
in the Universe. Thus, Numbers have duality and come in two types. This
is something the old math never had the tiniest of idea, that Numbers
must come in two types. Just as geometry comes in two types.
(6) Symmetry plays a huge role in physics and so it should in
mathematics. Since Reals are infinite rightward strings, then any
intelligent person would easily come to realize the other major type of
number is infinite leftward strings. Symmetry would also say that Doubly
Infinites is nonsense.
(7) So we have two great systems of Numbers which are independent of one
another just as there are two great systems of Geometry and are
independent of one another. These are Euclidean compared to
Elliptic/Hyperbolic. And the Numbers are Reals compared to P-adics.
(8) Knowing these facts then we go back and look at Natural Numbers and
Complex Numbers and ask ourselves, are these part of Reals or are they a
part of P-adics. As for the Counting Numbers or Natural numbers the
answer is quite simple. The Natural Numbers or Counting Numbers are a
subset of the P-adics and are not members of the Reals. One of the very
first proof theorems of mathematics is that of the Infinitude of Primes.
How can you have infinitude of primes if the Reals are only finite
portion string leftwards? You can not. The P-adics are infinite leftward
string and so you can have an infinity of primes in P-adics.
(9) Although the Complex number of (i) which is also called imaginary
number since in Reals it is the square root of (-1) had evolved in the
history of mathematics to make the Reals complete to the operations of
roots. However, the P-adics have a native numbers of square root of (-1)
such as the P-adic of (-)....000001 itself. When we take the root of the
P-adic (-)....000001 we end up with the same number since all operations
on negative P-adics ends up with a negative signed answer.
(10) Instead of wasting much of the 20th century on Algebras, they
should have spent that time on finding what MODEL best models Riemannian
and Lobachevskian geometry. I ended up finding this "best model" for it
is the sphere itself where one hemisphere is Elliptic geometry
(Riemannian) and the antipodal hemisphere becomes Hyperbolic geometry
(Lobachevskian). Having thus found the world's best Model for
NonEuclidean geometry, I can use that as the anchor for revealing much
of the structure of the P-adics. Such things as ...99999 although it
behaves like the Real (-1) it is altogether different and it is the
world's largest integer. I called it the Infinity Integer.
(11) And probably the most annoying mistake of the 20th century which
annoyed and frustrated me to no end in the 1990s and 2000s decade was
the base dependency of the P-adics of the 20th century. This base
dependency of 3-adics versus 5-adics without students ever seeing all
the P-adics in their mind's-eye all at once is largely responsible for
why math was retarded in the 20th century. It would be like all students
before they reached University were taught how to turn one number in
base 2 into some other base such as 10, when in fact, math is more than
simply changing bases. So this textbook shows any student who reads and
learns from it, what the P-adics are in full sight. Just as we learn
the structure of Reals as the Decimal Reals, this textbook teaches the
structure of P-adics as Decimal P-adics. And let the ivory towered hair
twirling professor of mathematics worry about his/her nitpicking and
nattering Algebra on P-adics, for they are so lost in fog that they
cannot explain or teach the truth.
Archimedes Plutonium
www.iw.net/~a_plutonium
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
a_plutonium
>
> (11) And probably the most annoying mistake of the 20th century which
> annoyed and frustrated me to no end in the 1990s and 2000s decade was
> the base dependency of the P-adics of the 20th century. This base
> dependency of 3-adics versus 5-adics without students ever seeing all
> the P-adics in their mind's-eye all at once is largely responsible for
> why math was retarded in the 20th century. It would be like all students
> before they reached University were taught how to turn one number in
> base 2 into some other base such as 10, when in fact, math is more than
> simply changing bases. So this textbook shows any student who reads and
> learns from it, what the P-adics are in full sight. Just as we learn
> the structure of Reals as the Decimal Reals, this textbook teaches the
> structure of P-adics as Decimal P-adics. And let the ivory towered hair
> twirling professor of mathematics worry about his/her nitpicking and
> nattering Algebra on P-adics, for they are so lost in fog that they
> cannot explain or teach the truth.
>
Let me talk alot more about the last point I made above in (11). Once
I or anyone
else opens up the P-adics to easy operations of add, multiply,
subtract and divide, where
it is as easy as in Reals, then the vast amount of progress in
understanding the P-adics
comes about. When school children ask their teacher what is 1/2 of ....
999999
and what angle does it represent and when they ask is this number of
5....000000
larger than 49....9999999, then we have opened up the P-adics. Taken
the P-adics
away from the ivory towered nattering nutter who is lost in some silly
algebra.
Once we get schoolchildren talking about prime numbers that come in
strings of three
instead of the mere twin-primes such as 29 and 31. Where we have
Triplet primes because
the P-adics can construct such triplet primes, do we make vast
progress in mathematics.
Once we have schoolchildren able to Add, to Multiply to Subtract and
Divide P-adics almost
as easily as they do Reals, then the world of mathematics is better
off.
But that will not happen unless we give them P-adics without base.
Where we give them
the Decimal P-adics just as mathematics education is now entirely
devoted to Decimal
Reals.
People and students alike have to be able to see in their mind's eye
what something is. They
can never do that if they are never taught the Decimal Reals but
taught silly base 2 and base
whatever without ever a anchor. The theorems of mathematics are base
independent, meaning
that all you need is Decimal Reals. Likewise for P-adics in that the
20th century had little to no
progress with P-adics and were seen as some remote and arcane rather
useless tool. They were seen
as useless because no-one could picture them. No-one could visualize
them. All could visualize
the Reals because they are the points in a line-segment of Euclidean
Geometry.
So this textbook clears up P-adics and thrusts them forward as the
great Number system that rivals
the Reals. This textbook allows anyone who reads it, to have a full
picture in their minds what the P-adic
numbers are. The best the old math did was a frontis page in a book by
Koblitz showing a crude
artist sketch of 3-adics. That is the pitiful best that the old math
could do for P-adics.
How do I arrest control of the P-adics from those nattering nutter
algebraists who spend most of their
time twirling their hair? I do it by giving young students the way to
ADD, to MULTIPLY to Subtract
and Divide and to Exponentiate and to take Roots and to perform almost
all the operations that exist
in Reals. So if a student can do a operation with Reals, they can do
it with Decimal P-adics.
This sort of reminds me of political powers such as dictators who
steal away the freedom of speech
and expression of its people and when some technology comes along that
the dictators cannot control
and then there is a outburst of freedom of speech and expression. So
long as the P-adics were in
control of ivory towered math professors who could only see them as
based on primes and algebras
and would not allow students to develop the P-adics to where they
could "see them all" and operate
on them, well, P-adics were going nowhere under the control of math
professors.
Other Me
to spell "The Delicious Rump Man".
"Archimedes Plutonium" <a_plu...@hotmail.com> wrote in message
news:4703C941...@hotmail.com...
Proginoskes
wrote:
> These are the major mistakes of the old math [...]
At least they never proved the contradiction 0 * 0 = -1 and called it
a "victory".
Once again, AP can't create a consistent numerical system, so he has
to try to tear down the old one.
--- Christopher Heckman
a_plutonium
finding the roots of positive
P-adics and then attaching a negative sign to the final answer. So
that in the case of
the square root of (-)....000001 in Negative P-adics is the square
root of ....00001
and then attaching a negative sign.
I also reported that addition and multiplication and division in +P-
adics has its final
answer as "another positive P-adic", so that the final answer never
leaves the +P-adics
or in terms of geometry, never leaves the hemisphere that is Elliptic
geometry as per
addition, multiplication and division.
But it gets sticky here with subtraction. And maybe I need some time
to work out the
bugs in subtraction. Since the +P-adics never leave the +P-adics under
add, multiply,
and divide would it make sense that subtract leaves the +P-adics. Well
look at
the +Reals under add, multiply, divide and they do not leave the
+Reals under those
operations. But they could leave the +Reals for -Reals under
subtraction. So I think
I am alright.
For example
.......000000002 - .......9999999 = (-) ......9999997
Another example
5....00000000 - 49.....999999 = .....000001
How did we get ....000001 ?
As by all operations in P-adics are exactly similar to Reals and where
the final answer
is what digits remain the same after successive place value.
So we have
50 - 49 = 01
then
500 - 499 = 001
then
5000 - 4999 = 0001
and so we know that .....000001 is the final answer.
But the subtraction in the first example
.......000000002 - .......9999999 = (-) ......9999997
goes like this:
So we have
2 - 9 = -7
then
02 - 99 = -97
then
002 - 999 = -997
and thus we conclude our final answer.
But here the problem is that subtraction takes us out of the positive
hemisphere
of the P-adic sphere. The positive hemisphere is Elliptic geometry and
the negative
hemisphere is Hyperbolic Geometry. With Reals, subtraction may put the
final
answer into negative Reals yet the geometry remains the same
Euclidean. But here
in P-adics, the negative P-adics are a different geometry.
So I have to evaluate this circumstance. Does it make commonsense? I
am
too tired tonight and will pick up on this probably tomorrow.
Proginoskes
> A few articles back, I said that the roots of -P-adics was like
> finding the roots of positive
> P-adics and then attaching a negative sign to the final answer. So
> that in the case of
> the square root of (-)....000001 in Negative P-adics is the square
> root of ....00001
> and then attaching a negative sign.
Looks like AP is saying that if A*B = C, then (-A)*(-B) = -C. This is
almost as bad of his mistake as 0*0 = -1, since, in order for it to be
true, we lose the distributive property.
> For example
>
> .......000000002 - .......9999999 = (-) ......9999997
>
> Another example
>
> 5....00000000 - 49.....999999 = .....000001
This is not a well-defined operation, since you could also have
50....0000000 500....00000
- 4....9999999 OR - 4....99999
_______________ ____________
46....0000001 496....00001
> As by all operations in P-adics are exactly similar to Reals and where
> the final answer
> is what digits remain the same after successive place value.
> [...]
> So we have
> 50 - 49 = 01
> then
> 500 - 499 = 001
> then
> 5000 - 4999 = 0001
This is true only in base 10, so this operation really is base-
dependent. If we work in base 11, we'd find that
50 - 49 = 02
500 - 499 = 012
5000 - 4999 = 0112
and so on, which would give us an answer of ....1112, if we follow
AP's lead. My other objection also applies in base 11.
The issue of what ....9999 + ....0001 is will also show that AP is
really thinking of base 10.
--- Christopher Heckman
a_plutonium
Archimedes Plutonium wrote:
>
> So I have to evaluate this circumstance. Does it make commonsense? I
> am
> too tired tonight and will pick up on this probably tomorrow.
>
I love my curiousity, for whether tired or not tired it runs at full
pace.
In Euclidean geometry we have no sense of a physical meaning for
negative
Reals compared to positive Reals other than the negative Reals are on
the other
side of zero. So a direction from zero is the meaning of negative
Reals.
But in Elliptic joined with Hyperbolic on a sphere we get this
incredibly rich view
of what negative sign compared to positive, in that the negative P-
adic is concave
inwards whereas positive P-adic is concave outwards. The quantity
maybe the
same for both a positive p-adics as a negative p-adic such as ....
00002
and (-)....000002 but the only difference is one is concave inwards
and the other
concave outwards.
So, by subtraction we are really doing what when we do
.......0000002 - ......9999999
we are doing this
.....000002 + (-) .....99999
So, everywhere we have subtraction we simply replace the subtraction
with
addition and we replace the positive adic with a negative adic.
So now it makes sense that the ....000002 cancelled out (-) .....00002
portion
of (-) .....99999 leaving (-) .......999997.
I guess the important thing to learn in this is that the negative sign
is opposite concaveness
of positive sign. So concavity is equivalent to sign in Elliptic and
Hyperbolic geometry.
In Euclidean geometry which is the Reals, the negative versus positive
sign
is merely direction from a fixed point zero.
This is going to get real interesting when I discuss square root of
(-1) for it was
long thought that this imaginary number in Reals is a rotation. And in
P-adics
the square root of (-1) involves concavity. So here is an interesting
clash of
ideas. So if the square root of (-).....000001 is the same number,
then can I say
that such a operation was a rotation?
Perhaps one can say that, since in the above example with ....00002
the concavity
cancelled one another.
But in taking the square root of (-) .....000001 we multiply the same
number by
itself and end up with the same number. So can one visualize that if
we start with
-1 in -P-adics and do this operation -1 X -1 = -1. It is obvious that
no concavity
was cancelled but that one had to rotate about the point -1 and end up
with -1.
Not that one rotated completely around the sphere from -1 to come back
to -1, but
that one rotated around a single point and ended up at the same point.
And the same can be said for square root of (-) .....000004 which is
(-)....00002
that we have -2 X -2 does not equal +4 but equals -4 and so we can say
that square root of not only -1 is a rotation but the square root of
all negative P-adics
is a rotation around a point.
Proginoskes
> [...]
> But in taking the square root of (-) .....000001 we multiply the same
> number by
> itself and end up with the same number. So can one visualize that if
> we start with
> -1 in -P-adics and do this operation -1 X -1 = -1.
What is -1 * 1, then?
--- Christopher Heckman
a_plutonium
I should have gone to bed about an hour ago for I have a long day
ahead of me
tomorrow.
But I cannot put any thoughts aside.
> And the same can be said for square root of (-) .....000004 which is
> (-)....00002
> that we have -2 X -2 does not equal +4 but equals -4 and so we can say
> that square root of not only -1 is a rotation but the square root of
> all negative P-adics
> is a rotation around a point.
Having reread the above, it occurred to me that square root of -1 is
special
over square roots of other negative numbers such as -4, because -1 is
the
same as -1 whereas -2 is not the same as -4.
So this brings up a physics question as well as mathematics. Is the
concept of
rotation of square roots of negative numbers special with that of -1.
Apparently
it is since we cannot say that a rotation around -2 ends up at -4.
Probably none of this is new to mathematics since they have been using
only square
root of -1 and never bothering with any other square roots of negative
numbers.
So that rotation is uniquely reserved for -1 only.
So it is not new but it is new for education purpose in that I have
perhaps explained
rotation for square root of -1 that noone else has ever explained.
In physics, when Quantum Mechanics was discovered in its early years
and the quantum
numbers of N, L, m_L and m_s that the fourth one was not imputed until
some years
afterwards when it was obvious that particles in physics had an
intrinsic spin.
So what that tells me is that particles in physics have some of their
properties over
in Elliptic geometry of a hemisphere and some of their properties
reside over in Hyperbolic
geometry of a rotation which Quantum Mechanics calls m_s for spin.
a_plutonium
> > -1 in -P-adics and do this operation -1 X -1 = -1.
>
> What is -1 * 1, then?
>
The only break from Reals operations is that multiply and divide in
negative P-adics
attaches a negative sign to all final answers.
So that when we have a negative P-adic of (-).....000001 X ....000001
the final answer is -1.
The idea is that *sign* in P-adics is concavity and so when you
multiply or divide in
concave inward the answer will be the same concavity, likewise concave
outward.
The difference between multiply and divide in Reals versus P-adics is
that in P-adics
a negative multiply negative ends up as negative.
In add and subtract of unlike signs, the concavity of one cancels part
of the concavity of the
other. In multiply and divide, the concavity at the end is the same as
the concavity
before the operation.
a_plutonium
>
> Looks like AP is saying that if A*B = C, then (-A)*(-B) = -C. This is
> almost as bad of his mistake as 0*0 = -1, since, in order for it to be
> true, we lose the distributive property.
>
You are still in Reals and in love with Galois Algebra group theory.
This is a sphere surface with its intrinsic coordinate points of P-
adics,
so I better not have a distributive property for the sphere does not
obey distributive rules.
And, someone should have proven, or perhaps I should be the first one
to prove that in Galois Algebras, that Euclidean geometry is the only
geometry
and the Reals are the only numbers that obey Galois theory. And that
the P-adics
cannot obey Galois group or ring or field theories. The easy proof of
this would
probably be to show that No Galois theory can cope with two zeroes of
a -0 and
+0 as North and South Poles of a sphere.
>
>
> > For example
> >
> > .......000000002 - .......9999999 = (-) ......9999997
> >
> > Another example
> >
> > 5....00000000 - 49.....999999 = .....000001
>
> This is not a well-defined operation, since you could also have
>
> 50....0000000 500....00000
> - 4....9999999 OR - 4....99999
> _______________ ____________
> 46....0000001 496....00001
>
You do not understand Front loaded P-adics representation versus Rear
loaded
P-adics
The numbers 5.....0000000 and 4....999999 have to occupy the same
point at infinity
of the "5" and "4". You cannot scoot around the front digit.
> > As by all operations in P-adics are exactly similar to Reals and where
> > the final answer
> > is what digits remain the same after successive place value.
> > [...]
> > So we have
> > 50 - 49 = 01
> > then
> > 500 - 499 = 001
> > then
> > 5000 - 4999 = 0001
>
> This is true only in base 10, so this operation really is base-
> dependent. If we work in base 11, we'd find that
>
> 50 - 49 = 02
> 500 - 499 = 012
> 5000 - 4999 = 0112
>
> and so on, which would give us an answer of ....1112, if we follow
> AP's lead. My other objection also applies in base 11.
This textbook gets rid of bases for P-adics. Just as all theorems of
mathematics
gets by with Decimal Reals. I convert all P-adics into Decimal P-
adics.
What is true for Decimal Reals is true for Reals in base 2. What is
true for Decimal
P-adics is true for 2-adics or 3-adics.
Adics in different bases is a waste of time that only professors of
mathematics indulge
in along with their misguided notions of Galois Algebra. No wonder the
20th century
of mathematics was mostly a waste of time.
a_plutonium
(snipped)
>
>
> >
> >
> > > For example
> > >
> > > .......000000002 - .......9999999 = (-) ......9999997
> > >
> > > Another example
> > >
> > > 5....00000000 - 49.....999999 = .....000001
> >
> > This is not a well-defined operation, since you could also have
> >
> > 50....0000000 500....00000
> > - 4....9999999 OR - 4....99999
> > _______________ ____________
> > 46....0000001 496....00001
> >
>
> You do not understand Front loaded P-adics representation versus Rear
> loaded
> P-adics
>
> The numbers 5.....0000000 and 4....999999 have to occupy the same
> point at infinity
> of the "5" and "4". You cannot scoot around the front digit.
>
I wanted to go to bed early tonight. Early for me is 1am because I
want to pour some
concrete tomorrow. But an active mind and curious mind never seems to
tire. (Shame
that an automobile tire is spelled the same as sleep tire.)
Anyway after making the above post some hours earlier tonight I went
away
thinking about the number 04.....9999999 and the number 004....999999.
They are
P-adics since they are a possible digit arrangement as per the
definition of P-adic
as all possible digit arrangements of infinite leftward strings.
And this post is another historic mathematics post, and historic for
all human
thought, because it changes our understanding of mathematics and
physics.
Because I am happy to announce I can count from 1 to .....9999999
It took me 14 years to arrive at this post, because I started the idea
that Counting
Numbers are the P-adics in 1993. In the decade of the 1990s I tried to
prove the
Counting Numbers were the P-adics and I did it by showing the Peano
Axioms were
flawed. But the trouble with that is that noone will believe me or
accept the assertion that
P-adics equals the Counting Numbers
or
P-adics are one and the same as the Natural Numbers
No matter how I showed the Peano Axioms are flawed and contradictory,
noone
is going to believe the P-adics are the Counting Numbers until the day
I show how
to count from 1 to .....9999999. Well, it is 2007, some 14 years from
1993, and I am
happy and proud to announce that I can count from 1 to ....999999 and
thus proving
beyond doubt that the P-adics are the Counting Numbers.
What allows me to show this is the Front View Representation of P-
adics and the
idea that there is an angle equal to every P-adic number. That the
Euler Identity
of (e)^ 2(pi)(i) = 1 gives us the fact that ......99999999 is approx
180 degrees
So, while taking a bathe before going to bed, I had the realization
that I could now
count all the P-adics in a sequence or progression from 1 to .....
999999
The idea came from those strange numbers of 004...9999999. So what
angle is
that number if 9....99999 is approx 180degrees and shy of 180 degrees
by one unit
distance away from the South Pole as -0.
So what angle is the number 004...999999? And the answer lies in this:
+0 is 0 degrees
1.....00000000 is approx 10 percent of 180 degrees
2.....0000000 is approx 20 percent of 180 degrees
3....00000000 is approx 30 percent of 180 degrees
4.....00000000 is approx 40 percent of 180 degrees
5.....0000000 is approx 50 percent of 180 degrees
6....00000000 is approx 60 percent of 180 degrees
7.....00000000 is approx 70 percent of 180 degrees
8.....0000000 is approx 80 percent of 180 degrees
9....00000000 is approx 90 percent of 180 degrees
So a P-adic such as any ending in 0s such as .....0000000231
which we know of as the Counting Number 231 , all of those with zeroes
are in the 0 degrees to 10 percent of 180 degrees
The P-adic of .....77777 is front view loaded as 7...7777 and thus is
about
77 percent of 180 degrees.
What is the angle of 004....9999999 ?
Well it is in the range of 0 to that of 1....000000 and thus the
number
004...9999999 is less than 10 percent of 180 degrees.
By the ability of giving an angle for every P-adic and the ability to
tell which P-adic is
larger than another P-adic allows me to count all the P-adics in a
sequence from
1 to 9...99999999
In the 1990s when I showed how flawed and contradictory the Peano
Axioms were, there
was no way that anyone was going to believe me, as long as I was
unable to Count the
P-adics from 1 to ....999999. Today I can do that.
I can tell you which of two P-adics is larger and then assign a degree
angle.
This textbook has many new important discoveries, but this one
probably tops them
all.
Gottfried Helms
conclusion, that finally there is no use in it for me.
One may do a bit of arithmetic within this concept, and may base
all considerations primarily on the sum of a periodic part of in-
finite length) and an aperidoc part, let's first think of finite length.
Considering the periodic type only one can easily see, that an
arithmetic of this is a map of the same arithmetic of rational
numbers between -1..0, where ...9999 =: -1 , and for instance
....3333 := -1/3 and so on.
One may apply addition, subtraction, multiplication and division
based on this mapping and the mapping will always be consistent.
Strings, which are sums of periodic strings and finite aperiodic
substrings can still be handled in the same way as the rational
numbers between 0..-1 summed with integers (the aperiodic summand
is just the mapping of a natural number).
Products of periodic strings get aperiodic, since they cannot
be mapped to a rational number from(0..-1)
Let x element( 0..-1) y element(0..-1) then
x*y not element(0..-1)
and must be expressed by an aperiodic expression.
That was nearly all what I was distilling out from this and was
sort of interesting. One could proceed and discuss roots, exponentials
and complex functions - but, well... it may be useful as a
compact notation for modular valuation, why not.
Gottfried Helms
--
---
Gottfried Helms, Kassel
a_plutonium
Gottfried Helms wrote:
> Some years ago I played around with these ideas and came to the
> conclusion, that finally there is no use in it for me.
> One may do a bit of arithmetic within this concept, and may base
> all considerations primarily on the sum of a periodic part of in-
> finite length) and an aperidoc part, let's first think of finite length.
>
Diving in without an anchor. Diving in with Reals as the only Number
system
and where all other numbers are extensions of Reals. That you suspend
or dangle
Complex numbers from one limb of the Reals and you suspend and dangle
P-adics from another branch or limb of the Reals.
Diving into P-adics without a single thought that they represent some
geometry.
A geometry different from what Reals represent. And never using the
geometry that
is native to Reals or native to P-adics.
Diving in only with Galois Group theory. As if Group theory was more
important
and taking precedence over Geometry. In fact, I suspect I can prove
that Galois
Group theory is confined to only Reals and confined to only Euclidean
Geometry.
That the book by Koblitz and others that base their P-adic development
on Galois
Group theory are all wrong.
> Considering the periodic type only one can easily see, that an
> arithmetic of this is a map of the same arithmetic of rational
> numbers between -1..0, where ...9999 =: -1 , and for instance
> ....3333 := -1/3 and so on.
> One may apply addition, subtraction, multiplication and division
> based on this mapping and the mapping will always be consistent.
>
That is a wrong development, because you never tell us what the
Geometry is.
P-adics are the numbers of a sphere. So unless you develop the Reals
alongside
Euclidean Geometry and develop the P-adics alongside Elliptic/
Hyperbolic Geometry,
you have no development at all.
Reals are infinite rightward.
That means something equally important and equally as great as the
Reals and independent
of the Reals exists which is infinite leftwards.
That something else is P-adics and their geometry of Elliptic/
Hyperbolic.
Prior to 1993 Mathematics had one house where Reals occupied the
entire house and where
NonEuclidean geometries were expected to fit into this house.
After 1993 Mathematics has two houses, one where Reals and Euclidean
and Galois Group theory
reside. A second house that is totally different and far away where P-
adics and NonEuclidean geometry
reside. Physics confirms the two houses as Particle and Wave. The
world and reality is dualistic,
even mathematics is dualistic.
> Strings, which are sums of periodic strings and finite aperiodic
> substrings can still be handled in the same way as the rational
> numbers between 0..-1 summed with integers (the aperiodic summand
> is just the mapping of a natural number).
>
This is still the old hanging on to P-adics as if they were a
differently dressed or labeled
Real. Where you cannot get over the idea that .....999999 is not -1
but is the largest
Decimal P-adic Integer. In the old Monty Python TV shows where the
comedians dress up
as old women and talk in high pitched voices; they may appear and act
like old women but
they are in reality men.
> Products of periodic strings get aperiodic, since they cannot
> be mapped to a rational number from(0..-1)
> Let x element( 0..-1) y element(0..-1) then
> x*y not element(0..-1)
> and must be expressed by an aperiodic expression.
>
> That was nearly all what I was distilling out from this and was
> sort of interesting. One could proceed and discuss roots, exponentials
> and complex functions - but, well... it may be useful as a
> compact notation for modular valuation, why not.
So you end it with the thought that by dressing up P-adics to be like
the Reals is their
only significance. That the Monty Python actors are only as good as
how they can fake
being old women.
The mistake of 20th century mathematics is that they never realized
Galois Group theory
is narrowly confined to only Reals and Euclidean Geometry and that 1/2
of the world of
mathematics is missing in action.
That the 1/2 of the world of mathematics is not Reals but P-adics and
totally separate and independent of
Reals. And these P-adics are the points of NonEuclidean geometries.
When historians write the history of mathematics and when they get to
the 20th century, people of
the 30th century will see the 20th as a period of "Dark Ages". Where
so much of mathematics had
been corrupted and convoluted. Where there was little anchor to guide
20th century math.
>
> Gottfried Helms
>
>
> --
> ---
>
> Gottfried Helms, Kassel
In the 1990s I claimed P-adics equalled, or were the same thing as
Natural Numbers. I claimed
.....999999 was a Natural Number as well as the idempotents in 10-
adics. I claimed Irrational P-adics
were Natural Numbers. I showed in the 1990s where the Peano Axioms
were in error and mistaken.
I showed how to correct the Peano Axioms. Noone listened to me, and
the reason they did not listen
is because I could not count from 1 to .....999999. Well, I have some
good news. By 2007, I can count
from 1 to .....9999999.
We count this way:
.....0000000
then
....0000001
then
.....0000002
then
.....0000003
then
until we get to one of the largest numbers known to physics which is 2
x 10^180
..........000000000000000000000000000000000000000000000000000002
166386284733232094762487827434976872857384758472847682784754
762857834738766540083478362786131476532828748671673467465743
620639486576747127346574638888889128746378768129187190273588
then we get to some more big numbers in physics such as the number of
Coulomb Interactions inside a plutonium atom which is about 188!
But all these numbers are tiny compared to the distance we have to go
since
we have not even come close to this number
1......0000000000000000000000000000000000000000000000000000000
And this number is only about 10 percent of the Counting Numbers for
we have an infinity of Counting Numbers to traverse before we get to
20 percent of the Counting Numbers represented as
2......000000000000000000000000000000000000000000000000
But back to the number 1....00000000000
What preceded that number and what will succeed that number?
The preceding number for 1.....000000000 is the number
09....9999999999
The digit at the *point of infinity* for these two numbers are "1" and
"0"
respectively. We can visualize the Counting Numbers around 100 that
the
preceding number is 099 and the succeeding number is 101
So the succeeding number to 1....00000000 is the number 1....0000001
and we fill in all the Counting Numbers between 1....0000000 to
2....000000000 much as we fill in all the counting numbers from 100 to
200.
When we reach the number
9......0000000000 we are 90 percent finished counting and this number
is
approx 90 percent of 180 degrees on the surface of a sphere.
When we finally reach the counting number 9....99999999 we are only
one unit
distance shy of -0 the South Pole on a sphere. And this number
9....99999
is the world's largest integer just as 0 is the world's smallest
positive integer in
Reals. Reals do not have a largest integer because Reals are finite
strings
leftward.
Proginoskes
> > Looks like AP is saying that if A*B = C, then (-A)*(-B) = -C. This is
> > almost as bad of his mistake as 0*0 = -1, since, in order for it to be
> > true, we lose the distributive property.
>
No, just the distributive property. Pay close attention:
(-1) * (-1) + (-1) * 1 = (-1) * ((-1) + 1) = 0
(-1) * (-1) + (-1) = 0 (from elsewhere in this thread)
(-1) * (-1) = 1 (adding 1 to both sides).
This is a real proof.
> > > For example
>
> > > .......000000002 - .......9999999 = (-) ......9999997
>
> > > Another example
>
> > > 5....00000000 - 49.....999999 = .....000001
>
> > This is not a well-defined operation, since you could also have
>
> > 50....0000000 500....00000
> > - 4....9999999 OR - 4....99999
> > _______________ ____________
> > 46....0000001 496....00001
>
> You do not understand Front loaded P-adics representation versus Rear
> loaded
> P-adics
Maybe because you've never posted a clear explanation of them.
> The numbers 5.....0000000 and 4....999999 have to occupy the same
> point at infinity
> of the "5" and "4". You cannot scoot around the front digit.
Why not? After all, 50.....00000 has an infinite number of 0's after a
5, and 5.....0000 has an infinite number of zeros after the 5. Since
"all infinities are equal", this means
50....0000 = 5....0000.
> > > As by all operations in P-adics are exactly similar to Reals and where
> > > the final answer
> > > is what digits remain the same after successive place value.
> > > [...]
> > > So we have
> > > 50 - 49 = 01
> > > then
> > > 500 - 499 = 001
> > > then
> > > 5000 - 4999 = 0001
>
> > This is true only in base 10, so this operation really is base-
> > dependent. If we work in base 11, we'd find that
>
> > 50 - 49 = 02
> > 500 - 499 = 012
> > 5000 - 4999 = 0112
>
> > and so on, which would give us an answer of ....1112, if we follow
> > AP's lead. My other objection also applies in base 11.
>
> This textbook gets rid of bases for P-adics. [...]
No, it doesn't, and that's the point. "P-adics" are strictly base 10,
which means they ARE base-dependent, regardless of what you say.
If you say that 8 * 3 + 1 = 25, then you are using base 10 arithmetic;
it is not "baseless".
--- Christopher Heckman
Proginoskes
> Gottfried Helms wrote:
> [...]
> This is still the old hanging on to P-adics as if they were a
> differently dressed or labeled
> Real. Where you cannot get over the idea that .....999999 is not -1
> but is the largest
> Decimal P-adic Integer.
You still haven't answered my question: What is .....999999 + 1 ?
--- Christopher Heckman
porky_...@my-deja.com
well, it's a Little Bigger (if you know that joke).
Proginoskes
deja.com> wrote:
> On Oct 4, 6:23 pm, Proginoskes <CCHeck...@gmail.com> wrote:
>
> > On Oct 4, 10:30 am, a_plutonium <a_pluton...@hotmail.com> wrote:
>
> > Where you cannot get over the idea that .....999999 is not -1
> > > but is the largest
> > > Decimal P-adic Integer.
>
> > You still haven't answered my question: What is .....999999 + 1 ?
>
Yes, but AP might not. And I doubt he'd find it funny, but ...
"Who is bigger, Mrs. Bigger, or her baby?" "Her baby's a little
Bigger."
--- Christopher Heckman
a_plutonium
a_plutonium
>
> You still haven't answered my question: What is .....999999 + 1 ?
The +P-adics start at +0 as the North Pole and they form a semisphere
as lines of longitude only 1/2
of the line of longitude for they all end as one unit away from the
South Pole -0. Or, depending on
where you want to start, the +P-adics can start at -0 and where .....
9999999 is one unit distance
away from the North Pole of +0. So in addition, as in multiplication
there often are two answers.
Adding 1 to an integer that is one unit away from the South Pole is
-0. Adding 1 to an integer
that is one unit away from the North Pole is +0.
Both +0 and -0 in P-adics are imaginary numbers, just as the square
root of -1 in Reals
is imaginary. So when you add one more unit to the largest possible
Infinite Integer the
answer is imaginary -0.
....9999999 + ....0000002 = (-) .....000001
and
....9999999 + .....00000002 = (+).....0000001
a_plutonium
a_plutonium wrote:
> >
> > You still haven't answered my question: What is .....999999 + 1 ?
>
> The +P-adics start at +0 as the North Pole and they form a semisphere
> as lines of longitude only 1/2
> of the line of longitude for they all end as one unit away from the
> South Pole -0. Or, depending on
> where you want to start, the +P-adics can start at -0 and where .....
> 9999999 is one unit distance
> away from the North Pole of +0. So in addition, as in multiplication
> there often are two answers.
>
> Adding 1 to an integer that is one unit away from the South Pole is
> -0. Adding 1 to an integer
> that is one unit away from the North Pole is +0.
>
> Both +0 and -0 in P-adics are imaginary numbers, just as the square
> root of -1 in Reals
> is imaginary. So when you add one more unit to the largest possible
> Infinite Integer the
> answer is imaginary -0.
>
> ....9999999 + ....0000002 = (-) .....000001
>
> and
>
> ....9999999 + .....00000002 = (+).....0000001
>
The number ....999999 or written front view as 9...99999 is the
world's largest integer and
has an angle measure of approx 180 degrees. It is not quite 180
degrees because it is one
unit away from the South Pole if the North Pole is +0.
So, depending we would expect that if you add approx 180 degrees with
180 degrees you
would expect to end up with approx 360 degrees.
So we have
.....9999999 + .....9999999 = (-) ......9999998
.....9999999 + .....9999999 = (-) ......0000002
depending on whether ....99999 ends up one unit away from North Pole
+0 or South Pole -0.
The above gives an answer of 180 degrees + 180 degrees = 360 degrees
a_plutonium
Archimedes Plutonium wrote:
>
> ....9999999 + ....0000002 = (-) .....000001
>
> and
>
> ....9999999 + .....00000002 = (+).....0000001
That is a mistake of (+) .....00001 for that should have been (-) ....
99999
depending on whether the Negative P-adics are reverse ordered or
whether they are ordered as Negative Reals.
The number 9...99999 is approx 180 degrees and ....00002 is so
miniscule
that it is almost 0 degrees so adding 180 to 0 degrees we expect the
answer to be another 180 degrees.
So (-)....99999 is approx 180 degrees
and (-) ....00001 is approx 180 degrees from +0
David R Tribble
> These are the major mistakes of the old math where they hyped on
> Algebraic Fields and Rings when they should have focused on Geometry
>
> (7) So we have two great systems of Numbers which are independent of one
> another just as there are two great systems of Geometry and are
> independent of one another. These are Euclidean compared to
> Elliptic/Hyperbolic. And the Numbers are Reals compared to P-adics.
>
> (8) Knowing these facts then we go back and look at Natural Numbers and
> Complex Numbers and ask ourselves, are these part of Reals or are they a
> part of P-adics. As for the Counting Numbers or Natural numbers the
> answer is quite simple. The Natural Numbers or Counting Numbers are a
> subset of the P-adics and are not members of the Reals. One of the very
> first proof theorems of mathematics is that of the Infinitude of Primes.
> How can you have infinitude of primes if the Reals are only finite
> portion string leftwards? You can not. The P-adics are infinite leftward
> string and so you can have an infinity of primes in P-adics.
1. Since p-adics form the true basis for numbers, how do we
represent something simple like sqrt(2) in p-adic form?
2. Since p-adics have an infinite portion string leftwards, do
they also have digit holes like the uncompleted transcendental
decimal fractions?
a_plutonium
>
> 1. Since p-adics form the true basis for numbers, how do we
> represent something simple like sqrt(2) in p-adic form?
>
> 2. Since p-adics have an infinite portion string leftwards, do
> they also have digit holes like the uncompleted transcendental
> decimal fractions?
Remember in P-adics as this textbook defines them as infinite strings
leftward of all possible digit arrangement
and that these are points of the surface of a sphere.
And that the operations on these points is as close as possible to the
operations in Reals where
the final answer is unchanging digits in place value.
So the square root of 2 in P-adics would be the square root of 2 in
Reals except for the fact that
P-adics have a finite string rightwards so that square root of 2 in P-
adics is .....000001r414
Yes there are holes everywhere in P-adics because one of the main
functions or purpose of
Euclidean geometry and the Reals that represent Euclidean geometry is
continuity. That between
any two points, any two numbers, exists another point, another number.
In P-adics, there is no
concern about continuity but the concern in these geometries of
Elliptic and Hyperbolic is
curvature and rotation and disjoint myriad numbers. Picture a world
where all things are composed
of spheres, tiny spheres of atoms and larger spheres of planets and
stars and larger still of galaxies
and the entire cosmos one big atom itself. So in this Cosmic view of
spheres nested in other spheres
is the notion of Elliptic and Hyperbolic geometry. So the P-adics
concerns itself with curvature
and rotation and size of spheres, whereas Euclidean and Reals concerns
itself with continuity and
zero curvature.
So this teases us with an interesting Conjecture. Is it impossible to
have continuity and to have
curvature? My intuition tells me it is impossible to be continuous and
NonEuclidean. Proof of this
conjecture would probably easily show that infinite leftward strings
have a natural curvature
where ....999999 ends up being near -0 or +0 and where infinite
rightwards never gets close
to 0.
Major Quaternion Dirt Quantum
ct 4, 9:50 pm, David R Tribble <da...@tribble.com> wrote:
> Archimedes Plutonium wrote:
> > These are the major mistakes of the old math where they hyped on
> > Algebraic Fields and Rings when they should have focused on Geometry
> > as to what Numbers are native to what geometry. [...]
>
> > (7) So we have two great systems of Numbers which are independent of one
> > another just as there are two great systems of Geometry and are
> > independent of one another. These are Euclidean compared to
> > Elliptic/Hyperbolic. And the Numbers are Reals compared to P-adics.
>
> > (8) Knowing these facts then we go back and look at Natural Numbers and
> > Complex Numbers and ask ourselves, are these part of Reals or are they a
> > part of P-adics. As for the Counting Numbers or Natural numbers the
> > answer is quite simple. The Natural Numbers or Counting Numbers ar a
> > subset of the P-adics and are not members of the Reals. One of the very
> > first proof theorems of mathematics is that of the Infinitude of Prim
> 2. Since p-adics have an infinite portion string leftwards, do
Proginoskes
> a_plutonium wrote:
>
> > > You still haven't answered my question: What is .....999999 + 1 ?
> > Adding 1 to an integer that is one unit away from the South Pole is
> > -0. Adding 1 to an integer
> > that is one unit away from the North Pole is +0.
>
> > Both +0 and -0 in P-adics are imaginary numbers, just as the square
> > root of -1 in Reals
> > is imaginary. So when you add one more unit to the largest possible
> > Infinite Integer the
> > answer is imaginary -0.
>
> > ....9999999 + ....0000002 = (-) .....000001
>
> > and
>
> > ....9999999 + .....00000002 = (+).....0000001
>
> The number ....999999 or written front view as 9...99999 is the
> world's largest integer and
> has an angle measure of approx 180 degrees. It is not quite 180
> degrees because it is one
> unit away from the South Pole if the North Pole is +0.
So the P-adics are arranged in a circle, not along a line, and as you
go around this circle, in the direction from ...0000 to ...0001, you
get bigger and bigger numbers, by repeatedly adding 1, until you get
back to ...0000 itself. This means that ...0000 is larger than itself,
which throws the whole idea of an ordering out the window.
If, of course, you want to be consistent.
If you want to stick with a linear ordering, you would have to deal
with questions like: Which is bigger: ....2222 or ....1313?
--- Christopher Heckman
G. Frege
<a_plu...@hotmail.com> wrote:
>
> adding 180 to 0 degrees we expect the answer to be another
> 180 degrees.
>
Sure?
--
E-mail: info<at>simple-line<dot>de
a_plutonium
addition of ....999999 + ....000002
The quandary is whether the Antipodal Semisphere of negative P-adics
of the South Pole as -0
whether the next unit point is (-)....000001 or whether it is (-) .....
9999999
When someone finds something that is the "truth", it often jumps off
the page and tells the person
which is the correct "next move". Truth guides its ownself.
So, I have three possibilities that this situation implies and I will
take the English cities of London, Brighton
and Cambridge as examples of +0, ....000001 and see what Cambridge is.
Assigning London as +0 is straightforward and no problems with that.
Since London is +0 and
I am moving in the direction of Brighton on a line-of-longitude then
Brighton is obviously
....0000001 and there is no problem with that. Nor is there any
problem of seeing that the antipodal
point of London is the point somewhere in Antarctica which is -0 and
one unit distance before
I reach -0 is the point ....9999999. So far everything is alright with
points +0 and ...00001
and ....999999 and -0. The difficulty arises as to what is the point
once I reach -0, what is the
next unit point in the Negative P-adics? Is it (-).....00001 or is it
(-) .....99999. The same question
is asked of London and Brighton and what is Cambridge? Is Cambridge
(-) .....000001 or is
Cambridge (-) .....999999
So I have three possible options which come to my mind now:
(a) Cambridge is (-).....9999999
(b) Cambridge is (-) .....000001
(c) Cambridge *has to* be both (-) .....999999 and (-).....000001
As I said earlier, the truth almost speaks for itself when you find
the true Model of Hyperbolic
geometry which this sphere surface is a true model.
I am inclined to think the answer is (c) that Cambridge, in Hyperbolic
geometry, when London
is +0 and Brighton is +....00001 then Cambridge is simultaneosly both
(-) ....00001
and (-) .....99999
I am inclined to think both because of something in physics in Quantum
Mechanics that Reality
becomes all the possibilities and in this case Cambridge has maximum
possibilities in being
both (-)....99999 and (-).....000001 simultaneously.
So that when I do the addition of .....9999999 + .....00002 , I end up
getting both (-)....00001
and (-).....99999
Another favor in that interpretation is the idea that in
multiplication we often end up with at least
two answers, in recognition that points come always in pairs on a
sphere of point and antipodal
point. So if Multiplication always yields at least two points, then to
maintain symmetry, addition
on a sphere has to deliver, always, at least two points. And thus the
so called swing shift ability
of the Antipodal 0 degree meridian once we reach -0 then the next
jumping off point is both
(-)....000001 and also (-).....9999999 and likewise, once we fix
London as +0, then Cambridge
is simultaneously boh (-).....000001 and (-) .....999999. It is to
maintain the symmetry that Quantum
Mechanics demands.
Although the truth has not leapt off this page and screamed at me that
both are correct answers,
it is interesting how when one finds the true Model, that it guides
the conversation.
lwa...@lausd.net
> So the P-adics are arranged in a circle, not along a line, and as you
> go around this circle, in the direction from ...0000 to ...0001, you
> get bigger and bigger numbers, by repeatedly adding 1, until you get
> back to ...0000 itself. This means that ...0000 is larger than itself,
> which throws the whole idea of an ordering out the window.
>
> If, of course, you want to be consistent.
>
> If you want to stick with a linear ordering, you would have to deal
> with questions like: Which is bigger: ....2222 or ....1313?
This is the first time I've participated in any of
Archimedes Plutonium's p-adic threads. But just as
with Tony Orlow, Ross Finlayson, and others among
the so-called "cranks," we can make AP's arguments
more rigorous by using hyperreals.
We already know that one can construct the real
numbers as equivalence classes of Cauchy sequences,
and one can think of the p-adics as sequences of
standard natural numbers that converge wrt a new
norm (the p-adic norm). Thus we identify the
10-adic number ....2222 by the sequence:
a = {2, 22, 222, 2222, ...}
and ...1313 by the sequence:
b = {3, 13, 313, 1313, ...}
both of which converge 10-adically.
Now AP wishes that all of his 10-adics be
comparable, but Mr. Heckman argues that these
two are incomparable.
But what if considered the sequences above
to be hyperreals? After all, _every_ sequence
of reals corresponds to a hyperreal. Then
since the hyperreals are linearly ordered, we
can define < on the adics to be the
restriction of < on the hyperreals to the
set of equivalence classes of sequences
corresponding to the adics.
So which is greater, ....2222 or ....1313? It
depends on which ultrafilter was used to
construct the hyperreals. Assuming that our
sequences are indexed starting with one
(rather than zero), so b_1 = 3, b_2 = 13,
b_3 = 313, b_4 = 1313, and so on, we have
that a < b if the set of odds is in the
ultrafilter, and a > b if the set of evens
is in the ultrafilter.
Now AP tells us that his p-adics are "front
loaded," so that they have leftmost as well
as rightmost digits. Obviously ....2222 has
leftmost digit 2, but ....1313 would have
leftmost digit 1 if the set of odds is in
the ultrafilter, but 3 if the set of evens
is in the ultrafilter.
Notice that AP also has numbers such as
50000....0000, for example. This would
correspond to the sequence:
c = {5, 50, 500, 5000, ...}
Notice that although the sequence converges
10-adically to zero, in the hyperreals this
is distinct from zero. Then one can
determine the leftmost digit of any
sequence by using the ultrafilter to
compare the sequence to:
{1, 10, 100, 1000, ...}
{2, 20, 200, 2000, ...}
{3, 30, 300, 3000, ...}
and so on.
If we consider the identity sequence:
I = {1, 2, 3, 4, ...}
then the largest number in AP notation:
d = {9, 99, 999, 9999, ...}
is just 10^I - d. In general, the set of
AP-adics is merely the set of nonnegative
hyperintegers less than p^I.
If we take the ring of hyperintegers and
mod out the ideal generated by p^I, then
we obtain formulae such as
....9999 = -1
just as in the usual p-adics. And as was
already mentioned in the AP threads, we
have ring homomorphisms:
f: Z[sqrt(2)] -> Z*/(7^I)Z*
g: Z[i] -> Z*/(5^I)Z*
(Z* = set of hyperintegers).
Notice that Tony Orlow has also come up with
some ideas similar to AP's. But TO defines
....9999+1 to be a new number 1:0000...0000
(which is similar to Z* without modding out
any ideals).
Also, notice that the set of hyperintegers
that correspond to the 2-adics are a proper
subset of the set corresponding to the
10-adics (since 2^I < 10^I). But TO wants
the set of T-riffics to be the same without
concern with whether the base is two, ten,
or any other base. So he comes up with
numbers of different "lengths," and uses
logarithms (in essence, couting up to e
rather than p^I, so that the number of
digits is log_p(I) rather than I), and
identifies 1:000...000 as "Big'Un." But this
would contradict the Fundamental Principle
of Arithmetic, extended to infinite values
via the Transfer Principle (for how can
Big'Un be a power of two, three, and every
prime, in order to be able to find
log_2(Big'Un), log_3(Big'Un), etc.). I
am not sure of AP's opinion on this matter.
Archimedes Plutonium
lwa...@lausd.net wrote:
> On Oct 5, 1:23 am, Proginoskes <CCHeck...@gmail.com> wrote:
>
>>So the P-adics are arranged in a circle, not along a line, and as you
>>go around this circle, in the direction from ...0000 to ...0001, you
>>get bigger and bigger numbers, by repeatedly adding 1, until you get
>>back to ...0000 itself. This means that ...0000 is larger than itself,
>>which throws the whole idea of an ordering out the window.
>>
>>If, of course, you want to be consistent.
>>
>>If you want to stick with a linear ordering, you would have to deal
>>with questions like: Which is bigger: ....2222 or ....1313?
>
>
> This is the first time I've participated in any of
> Archimedes Plutonium's p-adic threads. But just as
> with Tony Orlow, Ross Finlayson, and others among
> the so-called "cranks," we can make AP's arguments
> more rigorous by using hyperreals.
>
Why such a low opinion of me, which will likely only come back and haunt
you later on. If you cannot avoid calling someone a name, then do not
bother to even engage in a conversation.
> We already know that one can construct the real
> numbers as equivalence classes of Cauchy sequences,
> and one can think of the p-adics as sequences of
> standard natural numbers that converge wrt a new
> norm (the p-adic norm). Thus we identify the
> 10-adic number ....2222 by the sequence:
>
The trouble with your scheme and your post and your
ideas of HyperReal and Real and P-adics is that you,
as well as almost every math professor starts off with
Galois Group theory. And forces every number to obey some
sort of Galois structure.
So why can you not start off with the highest Logical Starting
Point.
How do we construct the Reals so that they are the points
of Euclidean Geometry.
Not how to start the Reals so they obey Galois Group or Ring or
Field theory.
But how do we start Reals so they are the numbers that are NATIVE
to Euclidean Geometry.
So the second question is that we have not only Euclidean Geometry,
but we have found there is Elliptic and Hyperbolic Geometry.
So, the wise and intelligent and logical person does not ask next
what happens to Reals if some Galois theory is changed or altered.
Does not ask what happens to P-adics or HyperReals as Cauchy
sequences.
The intelligent and logical and wise person asks for the second question:
Since REALS are the numbers that are native and intrinsic to Euclidean
Geometry, then what numbers are native and intrinsic to NonEuclidean
Geometries.
The mistake that WALKE makes throughout this post is that his foundation
for all that he says is rooted in Algebra of Galois manipulations and
definitions of operations. This is a very poor and lousy foundation to
be holding any conversation on how to construct REALS or any other
Number System.
The only acceptable Foundation where anyone can discuss Reals or any
other number system is via GEOMETRY.
The REALS make sense and come alive only because they are the intrinsic
native numbers that forms Euclidean Geometry. Galois theory happens to
describe the algebra for Reals. But when it comes to NonEuclidean
Geometry, one must throw away Galois theory, throw away Cauchy
Sequences. One must focus only on NonEuclidean Geometry and as to what
Numbers can occupy each point in Elliptic and Hyperbolic Geometry.
The HyperReals of Walke are nothing more than the Reals with a costume
on them. The P-adics of Walke, of Koblitz of Gouvea, of Amice, of
Bachman, of Iwasawa, of Mahler, of Robert and many others was nothing
more than costumed Reals.
Reals are only infinite strings rightward and that leaves a huge
gigantic hole in Mathematics for there are no numbers that fill the
other half of the picture, except for P-adics that are different than
the Reals.
Not until WALKE and others enter P-adics with the idea that they arise
not by Galois algebra manipulations, but come into existence because
they are the unique numbers that fill the surface of NonEuclidean
Geometries.
I shall answer many of the mistakes WALKE makes below.
> a = {2, 22, 222, 2222, ...}
>
> and ...1313 by the sequence:
>
> b = {3, 13, 313, 1313, ...}
>
> both of which converge 10-adically.
The P-adic .....2222 and ......131313 are very easy to describe and
place on a sphere surface, since ....999999 is approx 180 degrees and
front-viewed as 9....9999 where .....22222 is front viewed as 2...2222
would place 2...2222 as an angle arc measure of 22 percent of 180
degrees and where the number.....1313 is front viewed as 13....1313
would place it as about 13 percent of 180 degrees
>
> Now AP wishes that all of his 10-adics be
> comparable, but Mr. Heckman argues that these
> two are incomparable.
>
> But what if considered the sequences above
> to be hyperreals? After all, _every_ sequence
This is just another game of algebra, but what is
wanted is understanding of the numbers that compose
Elliptic and Hyperbolic geometry. Not some fake game,
and not some men dressed up with costumes pretending to
be old moms in a Monty Python sketch.
> of reals corresponds to a hyperreal. Then
> since the hyperreals are linearly ordered, we
> can define < on the adics to be the
> restriction of < on the hyperreals to the
> set of equivalence classes of sequences
> corresponding to the adics.
>
> So which is greater, ....2222 or ....1313? It
> depends on which ultrafilter was used to
No ultrafilter is required in the *True P-adics*
since the angle measure of ...222 is 22 percent
of 180 degrees whereas ....1313 is only 13 percent
of 180 degrees
> construct the hyperreals. Assuming that our
> sequences are indexed starting with one
Karl Heuer mentioned that I study the HyperReals
in 1993 when I started this program of P-adics
Resolution. I never to this day have looked up
what the HyperReals are. And I probably never will,
just as I do not care to stop at some tourist junk
shop where they craft some tourist trinkets and
babbles and pottery.
> (rather than zero), so b_1 = 3, b_2 = 13,
> b_3 = 313, b_4 = 1313, and so on, we have
> that a < b if the set of odds is in the
> ultrafilter, and a > b if the set of evens
> is in the ultrafilter.
>
> Now AP tells us that his p-adics are "front
> loaded," so that they have leftmost as well
> as rightmost digits. Obviously ....2222 has
> leftmost digit 2, but ....1313 would have
> leftmost digit 1 if the set of odds is in
> the ultrafilter, but 3 if the set of evens
> is in the ultrafilter.
Well in P-adics there are two different numbers
where one is 13....1313 and there is another
P-adic which is 31....1313 and where one is
13 percent of 180 degrees and the other is
31 percent of 180 degrees. We have no problem
with this as long as the person concerned wants
13...1313 or wants 31...1313.
The problem here is not the math but the wants and
desires of the person. Transposed pi as a P-adic
as .......951413 cannot be front-viewed since we have
no idea as to what digit would be at the "point of infinity".
Yet there is a transposed pi that is front loaded viewed as
314159.......r
And this transposed pi has either an angle of 31 degrees or 314
degrees when 9....99999 is approx 180 degrees.
>
> Notice that AP also has numbers such as
> 50000....0000, for example. This would
> correspond to the sequence:
>
> c = {5, 50, 500, 5000, ...}
>
Don't get me wrong, there is some use of Cauchy sequence as per
my definitions of P-adics since I use Cauchy sequences in operations
on P-adics. So I am not rebuking or repulsing all your suggestions. I
am just rebuking your starting foundation for talking about Numbers
where you have no reference to GEOMETRY and that is your largest mistake.
My P-adic number of 50...000000 is approx 50 percent of 180 degrees
and so would be a 90 degree angle and is midway between +0 and
9...999999
But the important feature of 50....000000 is that it is one unit
distance of arc further away from 49....999999
And when you divide 9....99999 by ....000002 (what we call 2) then
the answer is 49....9999999 So that 49....00000 is truly the midpoint
of 9....999999 and 50....0000000 is one unit arc greater than
49....99999
> Notice that although the sequence converges
> 10-adically to zero, in the hyperreals this
> is distinct from zero. Then one can
> determine the leftmost digit of any
> sequence by using the ultrafilter to
> compare the sequence to:
>
> {1, 10, 100, 1000, ...}
> {2, 20, 200, 2000, ...}
> {3, 30, 300, 3000, ...}
>
> and so on.
Sorry, I see this as just playing a game of algebra and
not of any mathematical content. As if one plays crossword
puzzles as distinct from writing a journal paper on a subject.
Take away Galois Group theory from mathematicians
with a college degree in mathematics and they fall apart
just as taking away a stuffed teddy bear from a 5 year old.
The P-adics start not from a conversation about Algebra and
Galois Group theory. The P-adics and Reals start by GEOMETRY
and the asking of questions what are the *native numbers
of those geometries* Only after we know what numbers are native,
then we can figure out what algebras they possess.
So we start with Geometry and find what Numbers fit the geometry,
and much later do we ever worry about algebras and operations.
a_plutonium
Archimedes Plutonium wrote:
>
> 314159.......r
>
> And this transposed pi has either an angle of 31 degrees or 314
> degrees when 9....99999 is approx 180 degrees.
>
Sorry that should be 31 percent of 180 degrees which is about 60
degrees
and that the 314 is also 60 degrees and not 314 degrees.
Now I do not think there is any significance to transposed pi , except
perhaps if
on compares transposed (e) of 271828.......r since it is 27 percent of
180 degrees
And knowing of the fact that pi on the surface of a sphere varies in
value from 2 to
that of 3.14159...... and that Reals-pi is the upper limit of value of
pi on a sphere
surface of tiny circles on a sphere surface
So that if one found the variance of (e) on a sphere surface and
compared its
variance to the (pi) variance would the angle of 60 degrees come to
significance.
Major Quaternion Dirt Quantum
guy's thought on hyperreals, but
I seriously doubt their applicability, although
p-adics are "just" the application
of decimal properties to the integers,
"just" like infinitessimals relate to infinities.
as for your adumbration of Galois,
you seem to have a bizarre catagory
of what is mathematics,
which is actually the study that you promote,
of geometry, astronomy, music and
the higher arithmetic (numbertheory). so,
your "programme" of "outlining the p-adics"
as a necessary front-end to the quadrivium,
amounts to "p-adics are what ever I mean,
when I say the word, p-adics, but
only then," as you refuse to "do the math"
of Hensel et al at any level
of comprehension -- as far as any one
of your fans, hereinat, can comprehend
in your vast typing prowess (lots
of scanning for larger key words e.g.).
good luck with that;
any thing is possible for an amateur,
like you or me, but science has always used,
what ever math was available:
the p-adics are excellent for some things,
two ... BUT,
you should call what you are making,
the AP-adics, to avoid *any* possibility that
any one could be confuzed (there is none, but
the suggestion is aimed at your own mental hygiene,
Dood .-)
thus:
hm, modular and/or automated building foundations
(501c3). I just found an advert in *The Nation*
for a book, _Broken Buildings, Busted Bridges;
How to Fix America's Trillion-dollar Construction Industry_,
reviews hereat invited. "...it operates
with an efficiency..of the old SU." (now, of course,
we in the US can order all of the components
from every where else in the world,
with no other recourse, if
we can afford the freightings.)
thus:
yeah, like, you're writing the textbook
on that ****, like AP-adics?
anyway, we allready have quaternions etc.;
how is your polysignatory any improvement?
(not that quaternions are written in stone,
except for on that bridge in Eire .-)
personally,
I don't think, reading should be taught
til about age eleven, and this is not just because
of my own experience with the sitcom-duration span
of attention; also,
it would save them from the double-whammy
of Potterism in film & literary formats (NB:
foremost, Harry Potter is a literary referant,
as well as simply an adumbration
of the imperialist "public school" a l'Oxford,
where Dame Maggie and Sir Tony got their policy,
Hey, George, let's you and Saddam fight!)
of course, by exposure to geometry and
the higher arithmetic, they'll have
had some exposure to letteracy (sik),
as well as linguistic expression
of the other elements of *mathematics*
(Latin: quadrivium;
trivium, the "3 Rs" of a literate slave).
> I do believe that people in this age range would understand the
> polysign context nicely:
> http://bandtechnology.com/PolySigned
> Especially since you mention graph theory this new form which is
> arguably number theory directly implies the graphics that result and
> so extending their knowledge of the real line upward in sign will aid
> them in processing the real values and help prevent sign errors.
> Simply teaching the three-signed numbers is teaching complex numbers
> so limiting the instruction to three-signed is sensible and the
> extension follows naturally very easily. This then also teaches the
> distributive law etc. since multiplication of three-signed values
> generally requires such usage. The math could be kept to integer
> values for ease of graphing and processing.
thus: NB: could have been in *American Scientist*.
> so, did anyone find a workable integer value
> for Avagadro's # ??... I seem to recall an article, or
> the letters back from the article in *The Sciences*,
> about that, somewhat inconculusive (if not fully multivalued).
thus:
what I'm saying: study the p-adics;
tell us what happenned to you!
--14 Italian Senators Call for Cheney Impeachment
Aug. 1, 2007 (EIRNS)-
The Lyndon LaRouche Political Action Committee
(LPAC) issued the following release today.
Fourteen members of the Italian Senate have signed a call "to the
Members of Congress to support Rep. Kucinich's House Resolution 333
for the Impeachment of Dick Cheney."
http://larouchepub.com/pr/2007/070801italian_senators_call.html
David R Tribble
>> (8) Knowing these facts then we go back and look at Natural Numbers and
>> Complex Numbers and ask ourselves, are these part of Reals or are they a
>> part of P-adics. As for the Counting Numbers or Natural numbers the
>> answer is quite simple. The Natural Numbers or Counting Numbers ar a
>> subset of the P-adics and are not members of the Reals. One of the very
>> first proof theorems of mathematics is that of the Infinitude of Prim
>
David R Tribble wrote:
>> 2. Since p-adics have an infinite portion string leftwards, do
>> they also have digit holes like the uncompleted transcendental
>> decimal fractions?
Major Quaternion Dirt Quantum wrote:
> pardonnez-moi, mais, quelle est la "digital hos?"
> [forgive me, but, what are "digital holes"?]
It's from an earlier post by AP, where he theorizes that
transcendental decimal fractions are "incomplete" because
they have "holes in the digits" at some far-away digit
position (probably somewhere around the 10^-38 digit).
I can't say I can make any sense of his "theories".
David R Tribble
>> 1. Since p-adics form the true basis for numbers, how do we
>> represent something simple like sqrt(2) in p-adic form?
>
a_plutonium wrote:
> Remember in P-adics as this textbook defines them as infinite strings
> leftward of all possible digit arrangement
> and that these are points of the surface of a sphere.
>
> And that the operations on these points is as close as possible to the
> operations in Reals where
> the final answer is unchanging digits in place value.
>
> So the square root of 2 in P-adics would be the square root of 2 in
> Reals except for the fact that
> P-adics have a finite string rightwards so that square root of 2 in P-
> adics is .....000001r414
Wow, that's not even close to a right answer.
Are you sure you know what the p-adics are?
And why are you using base 10 for your p-adics?
Here's a slightly easier question that might help:
1b. How do we represent a rational that is arbitrarily close
to sqrt(2) in p-adic form? (It's okay to show your answer
as a 10-adic, even though it's not a p-adic.)
David R Tribble
> This is the first time I've participated in any of
> Archimedes Plutonium's p-adic threads. But just as
> with Tony Orlow, Ross Finlayson, and others among
> the so-called "cranks," we can make AP's arguments
> more rigorous by using hyperreals.
Doubtful. I wouldn't go down that particular rabbit hole
if I were you. (You have been warned.)
You might want to do a little background research by
Googling for AP and his works before proceeding.
Most of his published "books" are threads in these
newgroups.
And you're going to find it hard to get any answers
to direct questions to AP.
Jesse F. Hughes
> You might want to do a little background research by
> Googling for AP and his works before proceeding.
> Most of his published "books" are threads in these
> newgroups.
"Most"? Have I missed something.
--
"Sorry, wakeup to the real world. You're on your own dependent on me
as your guide. Luckily for you, I'm self-correcting to a large extent,
so if the proof were wrong, I'd tell you. It's not wrong."
--- James Harris confirms that his proof is correct.
Proginoskes
> David R Tribble <da...@tribble.com> writes:
>
> > You might want to do a little background research by
> > Googling for AP and his works before proceeding.
> > Most of his published "books" are threads in these
> > newgroups.
>
> "Most"? Have I missed something.
AP has redefined (like he re-defines a lot of things but uses the old
word) "book" to mean "series of Usenet posts". That way he doesn't
have to bother with things like referees, who would point out that his
works lead to contradictions like 0 = -1.
--- Christopher Heckman
a_plutonium
between P-adics
and the Sphere Model which models both Elliptic and Hyperbolic
geometry
simultaneously.
The sphere with North Pole as +0 and South Pole as -0 comprise the
Positive
P-adics where +0, ....00001, .....00002, .... stretchs to .....99999
which is one unit
short of -0. So this Semisphere is the Positive P-adics and is
Elliptic geometry.
The Antipodal Semisphere is Negative P-adics and is Hyperbolic
Geometry
because when joining a triangle in Positive P-adics with its antipodal
triangle
in the other Semisphere, the concave outward cancels the concave
inward
leaving us a Euclidean flat plane geometry triangle.
But let us check on other features of these Semispheres such as
parallel lines.
In Elliptic geometry there are no parallel lines and in the Semisphere
there are
no parallel lines since they all meet at both +0 and -0. Now with the
Negative P-adics
in the Antipodal Semisphere, it is a little more difficult to see that
all lines are
parallel. And lines here are the same as in Positive P-adics of lines
of longitude
that are cut in half. So how does Hyperbolic geometry as the Antipodal
Semisphere
obey all lines are parallel? That is rather easy, in that the +0 and
-0 are imaginary
in P-adics and so the +0 and -0 are included in Positive P-adics, but
not in
Negative P-adics. So we end up with a Poincare disc that models
Hyperbolic geometry
and since there is no intersection at the points +0 and -0 for the
Antipodal Semisphere
then all lines are parallel in the Antipodal Semisphere. I may find a
even better
answer in the future for there is some moments of weakness in that
explanation.
Since +0 and -0 are imaginary to P-adics, why would they be included
in the Positive
P-adics but not the Negative P-adics. So I need a stronger argument
here.
Another geometrical feature is that the Equator line is never
involved, but only the
lines of longitude. Other than being a line that runs across the
midpoint of ....99999
and (-) .....999999. So the Equator line is half 4...999999 and half
(-)4....999999
I would have thought the Equator line had a more significant role in
both the geometry
and P-adics.
Now I want to comment on visualizing Hyperbolic geometry on the sphere
surface. It
is easy to visualize Elliptic geometry as that of us walking around on
the surface of a
Hemisphere. But to visualize Hyperbolic geometry we must picture
ourselves walking
around on the inside of a huge sphere that is hollow. So walking
around on the
outside is concave outward and walking around on the inside of a
hollow sphere is
concave inward (or I may have that turned around, depending on my
memory of
what it means to be concave inward versus outward.) So this is the
general picture
of how a sphere is both Elliptic and Hyperbolic all in one. That we
walk on the outside
surface of one hemisphere and then walk on the inside hollow antipode
hemisphere
for Hyperbolic geometry.
And the parallel line issue was taken care of by +0 and -0
intersection of all Positive
P-adics but +0 and -0 are missing in Negative P-adics forcing all the
lines in
Negative P-adics to be parallel.
The issue of sum of angles of triangles is taken care of since the
concave outward
of one hemisphere cancels the concavity of the antipodal triangle
leaving a Euclidean
triangle.
An interesting feature that comes out of Positive and Negative P-adics
reminds me
of the physics of magnetism in that you always have two poles for
magnet (not
discussing Dirac's monopoles). And I own the book "Electricity and
Magnetism"
Berkeley Physics Course volume 2, 1965 where the cover has a sort of
picture that
resembles the North and South Pole and lines of force.
So apparently the laws of electricity and magnetism are all written in
P-adics
and not Reals.
Now I wonder if in physics, the Maxwell Equations and laws of
electricity and magnetism
also have a "no importance value" attached to an Equator line as does
the Sphere Model?
And perhaps the sphere model can help us to picture the meaning of
negative
charge versus positive charge. Perhaps the difference between the
electron
charge versus the proton charge is that the -0 in protons is surround
by +...9999
and (-)....99999 whereas the -0 in electrons is surrounded by +....
99999
and (-) .....00001. So that charge in physics, is how the Negative P-
adic structure
is laid out against the Positive P-adic hemisphere.
Proginoskes
wrote:
> [...]
> Well in P-adics there are two different numbers
> where one is 13....1313 and there is another
> P-adic which is 31....1313 and where one is
> 13 percent of 180 degrees and the other is
> 31 percent of 180 degrees. We have no problem
> with this as long as the person concerned wants
> 13...1313 or wants 31...1313.
That wasn't the question, though; the question concerned ....1313.
Although it is relevant if AP can tell is whether it is 13....1313 or
31....1313.
and now, da Capo:
> lwal...@lausd.net wrote:
> [...]
> > This is the first time I've participated in any of
> > Archimedes Plutonium's p-adic threads. But just as
> > with Tony Orlow, Ross Finlayson, and others among
> > the so-called "cranks," we can make AP's arguments
> > more rigorous by using hyperreals.
>
> Why such a low opinion of me, which will likely only come back and haunt
> you later on. If you cannot avoid calling someone a name, then do not
> bother to even engage in a conversation.
AP's next book will be titled "How to Win Friends and Influence
People". He's still researching it.
And I don't see him calling AP a crank (not in anything that AP has
quoted). I see him referring to AP as a "so-called crank", but he
never says "AP is a crank." He also offers a way to approach the
problem without ending up in a contradiction.
> [...]
> The P-adic .....2222 and ......131313 are very easy to describe and
> place on a sphere surface, since ....999999 is approx 180 degrees and
> front-viewed as 9....9999 where .....22222 is front viewed as 2...2222
> would place 2...2222 as an angle arc measure of 22 percent of 180
> degrees and where the number.....1313 is front viewed as 13....1313
> would place it as about 13 percent of 180 degrees
> [...]
> Well in P-adics there are two different numbers
> where one is 13....1313 and there is another
> P-adic which is 31....1313 and where one is
> 13 percent of 180 degrees and the other is
> 31 percent of 180 degrees. We have no problem
> with this as long as the person concerned wants
> 13...1313 or wants 31...1313.
That wasn't the question, though; the question concerned ....1313.
Although it is relevant if AP can tell is whether it is 13....1313 or
31....1313.
> [...]
> Take away Galois Group theory from mathematicians
> with a college degree in mathematics and they fall apart
> just as taking away a stuffed teddy bear from a 5 year old.
That was an interesting enough sentence to comment on. AP is unaware
that not all mathematicians are Algebraists, and in fact may go
through their entire careers without using Galois Group Theory. My
research results wouldn't be invalidated if Galois Group theory was
proven false, for instance.
The humor in this situation (which I use since AP has tried to post
humor in his responses) is that if AP did this for all aspects of his
life, then when he stubbed his big toe, instead of going to a
podiatrist, he'd go to a gynacologist, reasoning that both are doctors
and should be able to treat his condition equally well. Or when a pipe
bursts, he calls an electrician because they both "fix things".
--- Christopher Heckman
a_plutonium
>
> That wasn't the question, though; the question concerned ....1313.
> Although it is relevant if AP can tell is whether it is 13....1313 or
> 31....1313.
>
I see where I need to spend a whole chapter devoted to simply front-
viewing
of P-adics.
When presented with any Rational P-adic it can be front-view loaded.
Whenever
presented with an Irrational P-adic, it can never be front-viewed.
P-adics have no transcendentals because a transcendental is a number
that has
place values that require one of these digits of 0,1,2,3,4,5,6,7,8,9
but is empty of
a digit. So a transcendental number is a number with a hole in it and
missing digits
and is growing and dependent on the Cosmic Clock.
So all Rational P-adics can be front-view loaded.
So what is the front-view of .....1313 ?
The answer follows from taking the square root of .....999999. One
square root
answer was ....999999r9 but a second answer was 316....r
So the rule for front-view is to take the most leftward digit of a
repeating block as
the front-view digit.
The most leftward digit of the repeating block of 13 is "1"
So ....1313 front-viewed is
13........1313
Which is an angle of about 13 percent of 180 degrees and the point
lies 13 percent
arc distance away from +0 and 87 percent arc distance away from ....
999999
a_plutonium
Now I may have made a mistake in the above in that the square root of
.....99999 with its 31622.... answer maybe irrational, and probably is
so.
And if it is irrational then I contradicted myself in the claim that
Irrational
P-adics cannot be front-view-loaded. Apparently some irrationals can
be front view loaded such as 31622.....r that is provided it is
irrational.
And that other P-adics such as the idempotents which are irrational
cannot
be front-view loaded.
If that is true, I have no answer at this moment as to why some
irrationals can
be front-viewed while others cannot. Although something tells me that
some
irrationals are already front view loaded.
Proginoskes
> Someone wrote this question:
I'm beginning to think AP isn't constipated.
> > That wasn't the question, though; the question concerned ....1313.
> > Although it is relevant if AP can tell is whether it is 13....1313 or
> > 31....1313.
>
> I see where I need to spend a whole chapter devoted to simply front-
> viewing
> of P-adics.
> [...]
> When presented with any Rational P-adic it can be front-view loaded.
>
> So the rule for front-view is to take the most leftward digit of a
> repeating block as
> the front-view digit.
>
> The most leftward digit of the repeating block of 13 is "1"
>
> So ....1313 front-viewed is
>
> 13........1313
(1) But I can also look at ...1313 as the block 31 repeated forever to
the left with a 3 tacked on the right, which means the most leftward
digit is a 3.
(2) ...131313 ought to be bigger than ...131312, but AP says that ...
131313 is 13...131313, and ...131312 is 31...131312, making ...131312
bigger than ...131313.
--- Christopher Heckman,
President of the Association for the Preservation
of Contradiction-Free Mathematics
a_plutonium
If I remember the Poincare disc that models Hyperbolic geometry that
the lines were diameters around the disc but that the center was
removed
so that all of them were parallel.
So is that a sufficient reason to say that the Sphere Model of both
Elliptic
and Hyperbolic is that the poles of +0 and -0 belong to the Elliptic
hemisphere
and the antipodal hemisphere is negative P-adics with +0 and -0
removed because
the positive P-adics own +0 and -0.
In the saddle shape model of Hyperbolic geometry we again so no trace
of any
point that acts as a "center" or intersection point for lines.
Perhaps one aid to visualization is that the huge hollowed out sphere
that an architect
makes where we write on the surface the points +0 and -0 as the poles
and we
write ....00001 as one unit arc distance from +0 and ....99999 as one
unit arc
distance away from -0 and as for the Antipodal hemisphere we mark the
inside hollow
hemisphere starting at (-).....1 and at the opposite end writing
(-).....999999.
So the two numbers of +0 and -0 can be owned by only one of the
geometries and
not by both.
David Bernier
> I seemed to have fallen into a bit of a quandary over the recent
> addition of ....999999 + ....000002
Concerning your #64 and the Poincaré disk representation of the
hyperbolic plane:
(a) yes, I think you're right that diameters are (some of) the straight
lines.
(b) no, I don't think the circle's center is removed; there's another
way of
getting infinitely many lines that don't cross some diameter.
(c) my understanding is that arcs of a circle (any circle) that lie
completely within the disk are also representations of lines in the
hyperbolic plane *if* they intersect the circular boundary of the
disk at right angles.
---
In the representation, say drawn on paper, of the Poincaré disk, the
arcs mentioned in (c) aren't straight. But given points A and B on
the arc, the shortest route to go from A to B in the hyperbolic plane
is by following the part of the circular arc "between" A and B.
I don't think points on the boundary of the Poincaré disk represent points
in the hyperbolic plane. If a snail started at the the center of a big
Poincaré disk
and advanced along a radius at a steady speed in the hyperbolic plane,
what one would observe on the "map" of HP is a point moving slower and
slower
towards the boundary, but never managing to reach it.
Something that might help to explain what the "map" of HP can do /
cannot do is to consider
a Mercator projection map of the globe. In the Mercator projection,
polar regions
get blown up compared to regions nearer the equator, and the North and South
Poles would never appear on the map, however big the map was made.
David Bernier
Jesse F. Hughes
Right, I understand that. What I was asking is why David said "most"
rather than "all".
--
"Quincy, would you rather do epistemology or conceptual analysis?"
"You know what? I'd rather fight on an aircraft carrier.... And Mama
and Baba (Papa) would fight on an aircraft carrier, too."
-- Quincy P. Hughes, age 3 1/2
a_plutonium
David Bernier wrote:
> a_plutonium wrote:
> > I seemed to have fallen into a bit of a quandary over the recent
> > addition of ....999999 + ....000002
> [...]
>
> Concerning your #64 and the Poincaré disk representation of the
> hyperbolic plane:
>
> (a) yes, I think you're right that diameters are (some of) the straight
> lines.
>
> (b) no, I don't think the circle's center is removed; there's another
> way of
> getting infinitely many lines that don't cross some diameter.
>
I think Poincare disc means 2nd dimension and so the center has to be
removed since you cannot have any other lines in 2nd dimension
> (c) my understanding is that arcs of a circle (any circle) that lie
> completely within the disk are also representations of lines in the
> hyperbolic plane *if* they intersect the circular boundary of the
> disk at right angles.
>
Well here we are in 3rd dimension and exactly the same Model as my
Semisphere
where the lines are arcs of lines of longitude.
It is interesting that you say intersect the circular boundary of the
hemisphere at right
angles which is precisely what the lines of longitude do in the
Semisphere only they
all intersect at right angles with the two poles.
So the Poincare disk in 3rd dimension is a exact copy of the
Semisphere as Negative
P-adics.
Now I need to get the bugs out of -0 as a South Pole.
> ---
>
> In the representation, say drawn on paper, of the Poincaré disk, the
> arcs mentioned in (c) aren't straight. But given points A and B on
> the arc, the shortest route to go from A to B in the hyperbolic plane
> is by following the part of the circular arc "between" A and B.
>
> I don't think points on the boundary of the Poincaré disk represent points
> in the hyperbolic plane. If a snail started at the the center of a big
> Poincaré disk
> and advanced along a radius at a steady speed in the hyperbolic plane,
> what one would observe on the "map" of HP is a point moving slower and
> slower
> towards the boundary, but never managing to reach it.
>
> Something that might help to explain what the "map" of HP can do /
> cannot do is to consider
> a Mercator projection map of the globe. In the Mercator projection,
> polar regions
> get blown up compared to regions nearer the equator, and the North and South
> Poles would never appear on the map, however big the map was made.
>
> David Bernier
Archimedes Plutonium
a_plutonium
Well it was because of symmetry
that called me to say the South Pole was -0. Symmetry because the
Negative P-adics would say (-)....00001
is preceded by -0.
But does Physics have two zeroes? Does physics have a positive and
negative zero? In this case
the Poles are imaginary points affixed to Elliptic Geometry as the
point of intersection so that all
lines intersect.
Trouble with -0 even though imaginary has a negative sign and Elliptic
geometry is all positive signed
individual numbers. So that is a big bug
Another bug is that of operations. What is +0 + -0 or what is +0/-0
So alot of bugs emerges.
There is a very simple solution here. -0 as the South Pole behaves and
acts as it was a 180 degree
angle. So just eliminate the name -0 and call it 180 degrees.
Yes, I do believe that gets me out of trouble with the South Pole.
So all the points on a sphere are either Positive P-adics which
includes the imaginary number +0
and includes the imaginary number called 180 degrees as the South
Pole, and the Negative P-adics.
So, now, that relieves me of the pain that -0 is a negative number
while Positive P-adics have only
positive numbers including +0 and that 180 degrees is a positive
number.
And it relieves me of the fact that -0 would have to be included in
Negative P-adics since it has a negative
sign but it cannot be included because it would be a point of
intersection of all lines in the Hyperbolic
geometry.
And the 180 degree name eliminates the troubles with operations using
the South Pole.
Perhaps someday, someone in physics may need a name for a phenomenon
that resembles a
-0 in contrast to a +0, but for the Model of Elliptic and Hyperbolic
geometry a name such as
-0 causes bugs in the system and the better name is 180 degrees.
Note: that I have been calling ....999999 as approx 180 degrees and
that is a good description
because ....999999 is one unit arc distance short of 180 degrees.
P.S. Also I never liked -0 because of my proof that Infinity comes in
only one type and destroying
Cantor's transfinites. The proof went on the lines that if Infinity
comes in different types then the number
0 comes in different types and by opening Pandora's box of a -0 would
devaluate that proof.
So by following symmetry, I made a mistake, for there is no -0 in
mathematics. There is only one
unique zero in mathematics and we can call it either 0 or +0.
a_plutonium
Since 180 degrees is often known as (pi) degrees we can also call the
South Pole as (pi).
We can do this because both 0 and (pi) are imaginary numbers tacked
onto Elliptic geometry.
And we can reconcile that fact with the idea that in Elliptic Geometry
there is no 180 degree
triangle and there is no (pi) since (pi) is exclusive to Euclidean
Geometry. In the Elliptic geometry
the value of pi is a variable ranging from 2 to that of an upper bound
of 3.14159......
So this is all coming together very nicely. So the Positive P-adics
has two numbers that are
imaginary and tacked onto the Positive P-adics. Those two imaginary
numbers are 0 and 180 degrees
= (pi) Those two imaginary numbers are the North Pole and South Pole
respectively.
This even concords with the Euler Identity that (e)^ 2(pi) (i) = 1
where the pi is 180 degrees and
2(pi) is then 360 degrees or 0 again and where (i) is the imaginary
number 0. So you have
(e) ^ (0 x 0) which is 1.
So, everything is falling into place with this Model of a Sphere as
both Elliptic and Hyperbolic geometries
simultaneously. And where you have Positive P-adics forming a
Semisphere that is Elliptic geometry
and you have Negative P-adics as the Antipodal Semisphere forming
Hyperbolic geometry.
About the only bugs remaining to fix in this Model is the operations
of add, subtract, multiply and divide
so that they satisfy the GEOMETRY, not someones wish of satisfying a
Galois Algebra. Once those
bugs are worked out, then in retrospect do we look to see what Galois
Algebra of Group or Ring or
Field theory that these numbers satisfy. I speculate the P-adics
satisfies no Galois Algebra and I speculated
earlier that Galois Algebra is confined to only the Reals and
Euclidean Geometry.
So the old P-adics of Hensel to Koblitz were not P-adics at all but
dressed up costumed Reals, just
as the actors in Monty Python are not old moms but men comedians
pretending to be old moms.
David R Tribble
>> You might want to do a little background research by
>> Googling for AP and his works before proceeding.
>> Most of his published "books" are threads in these newgroups.
>
Jesse F. Hughes wrote:
>> "Most"? Have I missed something.
>
Proginoskes writes:
>> AP has redefined (like he re-defines a lot of things but uses the old
>> word) "book" to mean "series of Usenet posts". That way he doesn't
>> have to bother with things like referees, who would point out that his
>> works lead to contradictions like 0 = -1.
>
Jesse F. Hughes wrote:
> Right, I understand that. What I was asking is why David said "most"
> rather than "all".
For some reason (probably based on the discussion of him
in Wikipedia) I was under the impression that AP had posted
some "papers" on the net in the past, perhaps on some
personal web page. I guess that's not true, and no, it does
not surprise me in the least that all of his "publications" are
in newsgroups.
David R Tribble
> Why did I ever say the South Pole was -0, a zero distinct from +0?
> negative zero?
Most people would answer that question by asking what
the difference between them is, i.e., what is (+0) - (-0).
David R Tribble
> Note: that I have been calling ....999999 as approx 180 degrees and
> that is a good description
> because ....999999 is one unit arc distance short of 180 degrees.
What is the magnitude of that "unit arc distance"?
Is it smaller than 1 degree?
lwa...@lausd.net
> > > That wasn't the question, though; the question concerned ....1313.
> > > Although it is relevant if AP can tell is whether it is 13....1313 or
> > > 31....1313.
>
> > I see where I need to spend a whole chapter devoted to simply front-
> > viewing
> > of P-adics.
> > [...]
> > When presented with any Rational P-adic it can be front-view loaded.
>
> > So the rule for front-view is to take the most leftward digit of a
> > repeating block as
> > the front-view digit.
>
> > The most leftward digit of the repeating block of 13 is "1"
>
> > So ....1313 front-viewed is
>
> > 13........1313
Although I know that AP has already rejected my use of the
hyperreals to describe his p-adics, I can still formalize what
he's saying in the hyperreals.
Suppose our ultrafilter contains the set of factorials:
{1, 2, 6, 24, 120, 720, 5040, 40320, 362880, ...}
Then one can prove that, for every natural number n, the set
of all multiples of n is in the ultrafilter (since each such set
is a superset of a cofinite subset of the set of factorials.) We
see that this agrees with AP's notion that if a string of digits
is purely periodic, then the front-loaded version begins at the
start of the period:
13...1313131313
31...3131313131
456...456456456
564...564564564
645...645645645
7890...78907890
8907...89078907
and so on.
> (1) But I can also look at ...1313 as the block 31 repeated forever to
> the left with a 3 tacked on the right, which means the most leftward
> digit is a 3.
Using this particular ultrafilter, we see that it doesn't
apply to strings with digits "tacked on." Thus:
13...1313131311
13...1313131312
13...1313131313
13...1313131314
13...1313131315
31...3131313131
31...3131313132
31...3131313133
In each case the leftmost digit must agree with the digits in
all but finitely many even positions (starting with the rightmost
digit as the first position).
> (2) ...131313 ought to be bigger than ...131312, but AP says that ...
> 131313 is 13...131313, and ...131312 is 31...131312, making ...131312
> bigger than ...131313.
And therefore ...131313 is greater than ...131312 in this ultrafilter.
Indeed,
...131313 is greater than ...131312 in _any_ ultrafilter, since every
term
of the sequence used to form the hyperreal:
{3, 13, 313, 1313, 31313, 131313, ...}
is greater than every corresponding term of:
{2, 12, 312, 1312, 31312, 131312, ...}
In fact, ...131313 - ...131312 is exactly 1, since it corresponds to
the
constant sequence:
{1, 1, 1, 1, 1, 1, ...}
which is the hyperreal 1.
Proginoskes
I guess it got too expensive for him. And/Or he found out that Usenet
archives are free.
--- Christopher Heckman
Proginoskes
> [...]
> Although I know that AP has already rejected my use of the
> hyperreals to describe his p-adics, I can still formalize what
Is there a good primer on the hyperreals online? What you're posting
looks interesting, but I have the feeling I'm missing out on some of
the details.
BTW, my favorite extension of the real numbers are the "surreal
numbers" discovered by John Horton Conway. Numbers are given in terms
of one definition, and I seem to recall reading that that definition
creates a superset of the hyperreals (which is an ordered field).
--- Christopher Heckman
lwa...@lausd.net
> On Oct 6, 9:32 pm, lwal...@lausd.net wrote:
>
> > [...]
> > Although I know that AP has already rejected my use of the
> > hyperreals to describe his p-adics, I can still formalize what
> > he's saying in the hyperreals. [...]
>
> Is there a good primer on the hyperreals online? What you're posting
> looks interesting, but I have the feeling I'm missing out on some of
> the details.
Here is the Wikipedia page:
http://en.wikipedia.org/wiki/Hyperreal_number
and the Wikipedia page has two links near the bottom:
http://www.math.wisc.edu/~keisler/chapter_1a.pdf (1.6 MB)
http://mathforum.org/dr.math/faq/analysis_hyperreals.html
(Actually, there are three links, but the third is hardly
for beginners at all -- it involves the existence of a
countably saturated extension of the reals.)
Notice that these links focus more on the infinitesimals
and their usefulness in calculus, but AP's p-adics are
more like infinite numbers than infinitesimals.
> BTW, my favorite extension of the real numbers are the "surreal
> numbers" discovered by John Horton Conway. Numbers are given in terms
> of one definition, and I seem to recall reading that that definition
> creates a superset of the hyperreals (which is an ordered field).
The surreals have been mentioned in these set theory
threads as well, though not as often. In particular,
WM's (Wolfgang Mueckenheim's) theory about the binary
tree reminds me of the surreals, where Conway draws a
tree beginning with zero, then branching out to the
dyadic rationals before reaching the infinite(simal)
surreal numbers.
Proginoskes
any of the other newsgroups.)
On Oct 8, 3:15 pm, lwal...@lausd.net wrote:
> On Oct 7, 12:58 am, Proginoskes <CCHeck...@gmail.com> wrote:
>
> > On Oct 6, 9:32 pm, lwal...@lausd.net wrote:
>
> > > [...]
> > > Although I know that AP has already rejected my use of the
> > > hyperreals to describe his p-adics, I can still formalize what
> > > he's saying in the hyperreals. [...]
>
> > Is there a good primer on the hyperreals online? What you're posting
> > looks interesting, but I have the feeling I'm missing out on some of
> > the details.
>
> Here is the Wikipedia page:
>
> http://en.wikipedia.org/wiki/Hyperreal_number
I glanced at this page but couldn't grok it right off. Then I moved to
MathWorld's hyperreals page but was *very* disappointed.
> and the Wikipedia page has two links near the bottom:
>
> http://www.math.wisc.edu/~keisler/chapter_1a.pdf (1.6 MB)
> http://mathforum.org/dr.math/faq/analysis_hyperreals.html
>
> (Actually, there are three links, but the third is hardly
> for beginners at all -- it involves the existence of a
> countably saturated extension of the reals.)
>
> Notice that these links focus more on the infinitesimals
> and their usefulness in calculus, but AP's p-adics are
> more like infinite numbers than infinitesimals.
Don't tell him that. 8-) His argument that "there is only one
infinity" is based on the assumption that 1/infinity is zero, when it
should actually be an infinitesimal.
I read the "Chapter 1a" link, and have a few comments.
(1) The Transfer Principle looks like voodoo mathematics to me. In
fact, it seems to lead to a contradiction (based on the idea whether
hyper-hyperreals exist). I'm sure it's not, and I can even see where
I'm going wrong, but not why.
Here goes: Consider the statement
(Ee)(Ax in R)(0 < x ==> e < x)
which is a basic axiom of hyperreals; namely, that there is at least
an infinitesimal other than 0. Now let's use the Transfer Principle on
it. (I'm sure this is where the error is being made, but I don't know
why, based on what I've read.)
(Ed)(Ax in R*)(0 < x ==> d < x)
which guarantees that this statement is true. So let d be as given (a
positive number of some type that is less than any positive
infinitesimal). Now d is less than any positive real number, so d is
an infinitesimal. Hence the statement above is true when
x = d, which means d < d. Contradiction.
(2) The "Standard Part Principle" states that "Every finite hyperreal
number is infinitely close to exactly one real number." Isn't this
provable? Certainly, there is at most one real number which can
satisfy the above condition, as no real number is infinitely close to
any other real number.
To show that there is a real number that satisfies the condition, let
S be the set of all real numbers less than a given hyperreal b; then
take the supremum of S (which is a real number, since the set S is
bounded above by any upper bound of b).
(3) Is there a way to generate ALL of the infinitesimals? The paper I
read will define new infinitesimals in terms of the one that is given
to exist, but I can't get a grasp on when you know that you have them
all. (From what you've posted, along with a brief look at the Dr. Math
page, it seems that you can generate infinitesimals from sequences of
real numbers, which suggests that there might be more of them than
real numbers.)
(4) The definition of the derivative which sticks to the real numbers
looks intuitive to me, but the definition which uses hyperreals looks
like it needs some justification. Why should you choose the standard
part of the difference quotient? (I'm looking for an answer other than
"because it works in all the simple cases.")
(5) I was reminded of the following quote: "In this book it is spoken
of the sephiroth and the paths, of spirits and conjurations, of gods,
spheres, planes and many other things which may or may not exist. It
is immaterial whether they exist or not. By doing certain things
certain results follow; students are most earnestly warned against
attributing objective reality or philosophical validity to any of
them."
--- Christopher Heckman
Dave Seaman
> (Only posted to sci.math, as this part of the thread is off-topic for
> any of the other newsgroups.)
> (1) The Transfer Principle looks like voodoo mathematics to me. In
> fact, it seems to lead to a contradiction (based on the idea whether
> hyper-hyperreals exist). I'm sure it's not, and I can even see where
> I'm going wrong, but not why.
> Here goes: Consider the statement
> (Ee)(Ax in R)(0 < x ==> e < x)
I think you mean
(Ee in R*)(Ax in R)(0 < x ==> 0 < e < x).
But this statement is neither fish nor foul. It's not a statement about
R, because it contains a reference to R*. It's not an internal statement
about R*, because it contains a reference to the external set R.
> which is a basic axiom of hyperreals; namely, that there is at least
> an infinitesimal other than 0. Now let's use the Transfer Principle on
> it. (I'm sure this is where the error is being made, but I don't know
> why, based on what I've read.)
> (Ed)(Ax in R*)(0 < x ==> d < x)
The transfer principal takes a statement about R and turns it into a
statement about R*. But the statement above is not about R, and
therefore the TP does not apply.
--
Dave Seaman
Oral Arguments in Mumia Abu-Jamal Case heard May 17
U.S. Court of Appeals, Third Circuit
<http://www.abu-jamal-news.com/>
Michael Press
<1191912149.2...@o80g2000hse.googlegroups.com>
,
Proginoskes <CCHe...@gmail.com> wrote:
> (5) I was reminded of the following quote: "In this book it is spoken
> of the sephiroth and the paths, of spirits and conjurations, of gods,
> spheres, planes and many other things which may or may not exist. It
> is immaterial whether they exist or not. By doing certain things
> certain results follow; students are most earnestly warned against
> attributing objective reality or philosophical validity to any of
> them."
Actually, this makes a great deal of sense.
Not that I would undertake the author's course of study.
The author says that certain terms are undefined,
and that focusing on the undefined terms is
counter-productive. The goal is to observe
the effects upon oneself while following
the author's course of study.
--
Michael Press
Proginoskes
> On Tue, 09 Oct 2007 06:42:29 -0000, Proginoskes wrote:
> > (Only posted to sci.math, as this part of the thread is off-topic for
> > any of the other newsgroups.)
> > (1) The Transfer Principle looks like voodoo mathematics to me. In
> > fact, it seems to lead to a contradiction (based on the idea whether
> > hyper-hyperreals exist). I'm sure it's not, and I can even see where
> > I'm going wrong, but not why.
> > Here goes: Consider the statement
> > (Ee)(Ax in R)(0 < x ==> e < x)
>
> I think you mean
>
> (Ee in R*)(Ax in R)(0 < x ==> 0 < e < x).
That's more restrictive, and true. ((Later: Yes, I forgot to mention
that e should be positive. Good catch.)) It also suggests that, in
order to show that one particular object exists, that you have to find
a set of related objects, which seems excessive and unnecessary.
After all, we say that i^2 = -1, not that some complex number, when
squared, equals -1.
> But this statement is neither fish nor foul. It's not a statement about
> R, because it contains a reference to R*. It's not an internal statement
> about R*, because it contains a reference to the external set R.
>
> > which is a basic axiom of hyperreals; namely, that there is at least
> > an infinitesimal other than 0. Now let's use the Transfer Principle on
> > it. (I'm sure this is where the error is being made, but I don't know
> > why, based on what I've read.)
> > (Ed)(Ax in R*)(0 < x ==> d < x)
>
> The transfer principal takes a statement about R and turns it into a
> statement about R*. But the statement above is not about R, and
> therefore the TP does not apply.
Okay. But now, back to the question that motivated this: Are there
"hyper-hyperreals"? That is, given the theory of hyperreals, can you
suggest that there is yet another type of number which is positive and
less than all positive hyperreals?
--- Christopher Heckman
Dave Seaman
> On Oct 8, 11:54 pm, Dave Seaman <dsea...@no.such.host> wrote:
>> On Tue, 09 Oct 2007 06:42:29 -0000, Proginoskes wrote:
>> > (Only posted to sci.math, as this part of the thread is off-topic for
>> > any of the other newsgroups.)
>> > (1) The Transfer Principle looks like voodoo mathematics to me. In
>> > fact, it seems to lead to a contradiction (based on the idea whether
>> > hyper-hyperreals exist). I'm sure it's not, and I can even see where
>> > I'm going wrong, but not why.
>> > Here goes: Consider the statement
>> > (Ee)(Ax in R)(0 < x ==> e < x)
>>
>> I think you mean
>>
>> (Ee in R*)(Ax in R)(0 < x ==> 0 < e < x).
> That's more restrictive, and true. ((Later: Yes, I forgot to mention
> that e should be positive. Good catch.)) It also suggests that, in
> order to show that one particular object exists, that you have to find
> a set of related objects, which seems excessive and unnecessary.
> After all, we say that i^2 = -1, not that some complex number, when
> squared, equals -1.
Ok, fair enough. But if you don't say what e is or what kind of order
relation is being used, I don't see how you can apply the transfer
principal.
>> But this statement is neither fish nor foul. It's not a statement about
>> R, because it contains a reference to R*. It's not an internal statement
>> about R*, because it contains a reference to the external set R.
>> > which is a basic axiom of hyperreals; namely, that there is at least
>> > an infinitesimal other than 0. Now let's use the Transfer Principle on
>> > it. (I'm sure this is where the error is being made, but I don't know
>> > why, based on what I've read.)
>> > (Ed)(Ax in R*)(0 < x ==> d < x)
>> The transfer principal takes a statement about R and turns it into a
>> statement about R*. But the statement above is not about R, and
>> therefore the TP does not apply.
> Okay. But now, back to the question that motivated this: Are there
> "hyper-hyperreals"? That is, given the theory of hyperreals, can you
> suggest that there is yet another type of number which is positive and
> less than all positive hyperreals?
Yes. In fact, you can find such numbers in the surreals. Every ordered
field can be realized as a subfield of the surreals.
Proginoskes
> On Tue, 09 Oct 2007 22:32:19 -0000, Proginoskes wrote:
> > On Oct 8, 11:54 pm, Dave Seaman <dsea...@no.such.host> wrote:
> >> On Tue, 09 Oct 2007 06:42:29 -0000, Proginoskes wrote:
> >> > (Only posted to sci.math, as this part of the thread is off-topic for
> >> > any of the other newsgroups.)
> >> > (1) The Transfer Principle looks like voodoo mathematics to me. In
> >> > fact, it seems to lead to a contradiction (based on the idea whether
> >> > hyper-hyperreals exist). I'm sure it's not, and I can even see where
> >> > I'm going wrong, but not why.
> >> > Here goes: Consider the statement
> >> > (Ee)(Ax in R)(0 < x ==> e < x)
>
> >> I think you mean
>
> >> (Ee in R*)(Ax in R)(0 < x ==> 0 < e < x).
> > That's more restrictive, and true. ((Later: Yes, I forgot to mention
> > that e should be positive. Good catch.)) It also suggests that, in
> > order to show that one particular object exists, that you have to find
> > a set of related objects, which seems excessive and unnecessary.
> > After all, we say that i^2 = -1, not that some complex number, when
> > squared, equals -1.
>
> Ok, fair enough. But if you don't say what e is or what kind of order
> relation is being used, I don't see how you can apply the transfer
> principal.
Then I guess my mistake was not being able to recognize when exactly
the Transfer Principal can be used; it's only a guess based on on-line
sources (which themselves admit it's tricky).
> >> But this statement is neither fish nor foul. It's not a statement about
> >> R, because it contains a reference to R*. It's not an internal statement
> >> about R*, because it contains a reference to the external set R.
> >> > which is a basic axiom of hyperreals; namely, that there is at least
> >> > an infinitesimal other than 0. Now let's use the Transfer Principle on
> >> > it. (I'm sure this is where the error is being made, but I don't know
> >> > why, based on what I've read.)
> >> > (Ed)(Ax in R*)(0 < x ==> d < x)
> >> The transfer principal takes a statement about R and turns it into a
> >> statement about R*. But the statement above is not about R, and
> >> therefore the TP does not apply.
> > Okay. But now, back to the question that motivated this: Are there
> > "hyper-hyperreals"? That is, given the theory of hyperreals, can you
> > suggest that there is yet another type of number which is positive and
> > less than all positive hyperreals?
>
> Yes. In fact, you can find such numbers in the surreals. Every ordered
> field can be realized as a subfield of the surreals.
I meant: in the hyperreal theory (Nonstandard Analysis)? Of course,
there are surreal numbers there:
( 0 | {all positive hyperreals} ) is such a number. (Which raises an
interesting question, similar to one I asked before: Is there a (say)
countable set S of hyperreals so that every hyperreal is greater than
at least one element of S? Or an easy description of how to get all of
the hyperreals? For instance, for the reals, you can use decimal
expansions to accomplish the same thing.)
It seems that if a theory points out that there are gaps in the real
number line, that it ought to be able to fill them, or say that there
is no way to ever fill them completely. I was wondering which (if
either) NSA claimed.
--- Christopher Heckman
Dave Seaman
>> >> I think you mean
I take it you mean, can one repeat the construction that produces R* and
thereby come up with a larger field R** of hyper-hyperreals? I'm sure
you can, and in fact, I think my observation about the surreals leads to
exactly that conclusion. There is a subfield of the surreals that is
isomorphic to R*, and there is an extension field S (still within the
surreals) that is isomorphic to R**.
> ( 0 | {all positive hyperreals} ) is such a number. (Which raises an
> interesting question, similar to one I asked before: Is there a (say)
> countable set S of hyperreals so that every hyperreal is greater than
> at least one element of S? Or an easy description of how to get all of
> the hyperreals? For instance, for the reals, you can use decimal
> expansions to accomplish the same thing.)
> It seems that if a theory points out that there are gaps in the real
> number line, that it ought to be able to fill them, or say that there
> is no way to ever fill them completely. I was wondering which (if
> either) NSA claimed.
Fill them completely in what sense? Only one ordered field is complete,
and it has no infinitesimals. But again, I think the surreals serve as
the ultimate target that you are searching for here. Every ordered field
is isomorphic to some subfield of the surreals. You can't get any more
"filled up" than that.
Proginoskes
Complete in the sense that (R*)* = R* (which is evidently false).
> Only one ordered field is complete,
> and it has no infinitesimals. But again, I think the surreals serve as
> the ultimate target that you are searching for here. Every ordered field
> is isomorphic to some subfield of the surreals. You can't get any more
> "filled up" than that.
True, but I'm asking about the path to "filling up" via the hyperreal
idea.
The hyperreals have a stated purpose: Simplifying proofs involving
Calculus of functions of one real variable. Hyper-hyperreals ought to
have a similar purpose, even if it's not used that often.
--- Christopher Heckman
Proginoskes
> (Only posted to sci.math, as this part of the thread is off-topic for
> any of the other newsgroups.)
>
> On Oct 8, 3:15 pm, lwal...@lausd.net wrote:
>
> > On Oct 7, 12:58 am, Proginoskes <CCHeck...@gmail.com> wrote:
>
> > > On Oct 6, 9:32 pm, lwal...@lausd.net wrote:
>
> > > > [...]
> > > > Although I know that AP has already rejected my use of the
> > > > hyperreals to describe his p-adics, I can still formalize what
> > > > he's saying in the hyperreals. [...]
>
> > > Is there a good primer on the hyperreals online? What you're posting
> > > looks interesting, but I have the feeling I'm missing out on some of
> > > the details.
>
> > Here is the Wikipedia page:
>
> >http://en.wikipedia.org/wiki/Hyperreal_number
>
> I glanced at this page but couldn't grok it right off. Then I moved to
> MathWorld's hyperreals page but was *very* disappointed.
>
> > and the Wikipedia page has two links near the bottom:
>
> >http://www.math.wisc.edu/~keisler/chapter_1a.pdf(1.6 MB)
> >http://mathforum.org/dr.math/faq/analysis_hyperreals.html
This Dr. Math page is the best of the three links.
> (1) The Transfer Principle looks like voodoo mathematics to me. In
> fact, it seems to lead to a contradiction (based on the idea whether
> hyper-hyperreals exist). I'm sure it's not, and I can even see where
> I'm going wrong, but not why.
>
> Here goes: Consider the statement
>
> (Ee)(Ax in R)(0 < x ==> e < x)
>
> which is a basic axiom of hyperreals; namely, that there is at least
> an infinitesimal other than 0. Now let's use the Transfer Principle on
> it. (I'm sure this is where the error is being made, but I don't know
> why, based on what I've read.) [...]
Having made my way through the Dr. Math page, I finally know what
exactly the error is: It's because the (Ee) is not bound; there is no
set involved, which contains e. As such, it cannot be "transferred" to
the hyperreals.
--- Christopher Heckman
G. A. Edgar
Proginoskes <CCHe...@gmail.com> wrote:
> Okay. But now, back to the question that motivated this: Are there
> "hyper-hyperreals"? That is, given the theory of hyperreals, can you
> suggest that there is yet another type of number which is positive and
> less than all positive hyperreals?
Of course you can. But of course that is another nonstandard model of
the reals. Your first nonstandard model is not uniquely determined
anyway, so maybe your second model is my first model!
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
Major Quaternion Dirt Quantum
publishable format, he'd have to be able
to comprehend his critics;
do you really think that he does, now?
sure, it's related to the comprehensability
of his own sub-book-length MSs --
a word is what I think it means,
whn I type it!... just like for every one, else.
> > > Most of his published "books" are threads in these
> > > newgroups.
>
> > "Most"? Have I missed something.
>
> AP has redefined (like he re-defines a lot of things but uses the
thus:
although a bicycle is dynamically stable,
the tetrapod is stable when one wheel lifts
-- still "tripodal" on grounding. of course,
nothing is dynamically "stable"
without a lot of testing; apparently,
they tested that tricar on one kind of test, so....
>Neither is a quad pod.
thus:
so, this seems to show that
there are 48 permutations of three objects (or,
What?), regarding Yaw, Pitch and Roll?...
anyway, I can only use tripolar coordinates,
as a matter of preference & ability!
> > I have a tracking system that gives me the rotation of an object in
> > the 3 euler angles in the order YPR. I have to do something with these
> > values and hand them to a system that visualizes refering objects by
> > applying their 3 stored angles in rotations ordered RPY.
> Quite a few years ago I built a system named ARTIBODIES for
> visualisation of articulated bodies and dynamical simulation of rotating
> rigid bodies. Rotations and orientations are represented internally by
> quaternions system-wide, and only when viewing and manipulating
> rotations and orientations from a user interface Euler angles come
> into play. So far the software design considerations.
> I put the relevant representation conversion programs on my website at
http://www.xs4all.nl/~jemebius/Eulerang.zip, reachable from
http://www.xs4all.nl/~jemebius/Eea.htm<http://www.xs4all.nl/~jemebius/index.html.
> Conversions run among representations by quaternions, by angle and axis,
> by Euler angles following the avionics convention (YPR or ZYX), and by
> Euler angles following the theoretical physics convention (ZYZ).
> In my opinion the best you can do is to convert from YPR to quaternion and
> subsequently from quaternion to your RPY Euler angles. I bet this yields
> also the fastest code in the framework of need4speed. (RPY: another of
> the 48 possible Euler angle conventions)
> My software does not cover quaternion to RPY, but
> I guess that the RPY and YPR conventions
> differ no more than a reversal of 3D rotation matrix c.q. quaternion multip.
> Perhaps my procedures provide some examples and some guidance in writing the
> quaternion-to-RPY conversion yourself.
thus:
foremost, Harry Potter is a literary referant,
as well as simply an adumbration
of the imperialist "public school" a l'Oxford,
where Dame Maggie and Sir Tony got their policy,
Hey, George, let's you and Saddam fight!)...
of course, by exposure to geometry and
the higher arithmetic, they'll have
had some exposure to letteracy (sik),
as well as linguistic expression
of the other elements of *mathematics*
(Latin: quadrivium;
trivium, the "3 Rs" of a literate slave).
--14 Italian Senators Call for Cheney Impeachment
Aug. 1, 2007 (EIRNS)-
The Lyndon LaRouche Political Action Committee
(LPAC) issued the following release today.
Fourteen members of the Italian Senate have signed a call "to the
Members of Congress to support Rep. Kucinich's House Resolution 333
for the Impeachment of Dick Cheney."
http://larouchepub.com/pr/2007/070801italian_senators_call.html
Proginoskes
wrote:
> In article <1191969139.537099.148...@d55g2000hsg.googlegroups.com>,
>
> Proginoskes <CCHeck...@gmail.com> wrote:
> > Okay. But now, back to the question that motivated this: Are there
> > "hyper-hyperreals"? That is, given the theory of hyperreals, can you
> > suggest that there is yet another type of number which is positive and
> > less than all positive hyperreals?
>
> Of course you can. But of course that is another nonstandard model of
> the reals. Your first nonstandard model is not uniquely determined
> anyway, so maybe your second model is my first model!
Well, that would make the "first model" (the hyperreals) not well-
defined ("ill-defined"?), which is a bad thing for theory. If your
hyperreals aren't isomorphic to my hyperreals, then what I say about
"mine" might not agree with what you say about "yours".
OTOH, when considering the complex numbers, "my" value of "i" might be
the negative of "yours". However, there is an isomorphism (a
homomorphism, in fact) which will translate statements about my "i"
into statements about "your" "i".
--- Christopher Heckman
lwa...@lausd.net
> On Oct 10, 4:38 am, "G. A. Edgar" <ed...@math.ohio-state.edu.invalid>
> wrote:
> > Of course you can. But of course that is another nonstandard model of
> > the reals. Your first nonstandard model is not uniquely determined
> > anyway, so maybe your second model is my first model!
>
> Well, that would make the "first model" (the hyperreals) not well-
> defined ("ill-defined"?), which is a bad thing for theory. If your
> hyperreals aren't isomorphic to my hyperreals, then what I say about
> "mine" might not agree with what you say about "yours".
According to both the Wikipedia and Dr. Math links above, it is
possible
to prove that any two hyperreal fields are isomorphic, but there's a
catch --
the proof requires, of all things, the Continuum Hypothesis. Indeed,
it is
equivalent to CH, so thus in ZFC+~CH one can have two non-isomorphic
hyperreal fields.