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Archimedes Plutonium wrote:
> Squaring circle and cubing the sphere were not allowed in Old Math
> because they
> never well-defined what it means to be infinite. In New Math it is
> allowed and it is what
> gives the B-number uniqueness.

Alright, I have a new method of attack on both (a) where the B-number
is located and
(b) why it is unique of all numbers.

It occurred to me, actually in my sleep, that the Ancient quest for
squaring the circle or
cubing the sphere was a quest that involved the "phony, imprecise
definition of infinity."
That mathematics as a subject is all about precision, and the reason
that the circle is not
able to be squared nor the sphere to be cubed is not something
intrinsic to mathematics,
but what is at fault is the inability to precision, well define
infinity. Suppose we define infinity
as that of a B-number of 100 which then causes us to realize that
1/100 is the end of
the microworld. That means that pi is 3.14 and 3.141 is in infinity
territory, not trustworthy
mathematics. With that definition of infinity as a B-number of 100,
means we obviously can square a circle or cube a sphere, however, our
square and cube are partly imaginary-infinity.

The Limit concept that is central to Calculus is another concept that
does not really exist, except to sweep and clean up after the ill-
defined infinity. Once we know that math has a
pure intrinsic B-number such as 10^618, then we would make statements
about the unit pseudosphere that it is finite in volume because its y-
axis component has become smaller
than 10^-618 before its x-axis component reaches 10^618, (which the
author Snowshoe was
grappling with in Old Math).

In these long months that I was searching for the B-number and the
uniqueness of the B-number, well, I am getting mightly closer to both
answers. It involves the Ancient quest of
squaring the circle and cubing the sphere.

So let us use just circle, sphere, square and cube at the start here.
And later switch to pseudosphere and sphere.

The Volume of sphere is 4/3(pi)R^3 and volume of cube is LWH (which
are all the same).
The Area of sphere is 4(pi)R^2 and area of cube is 2WW + 4WL (where
W=H, just in case
I need to refer to rectangular solid.)

Now the idea that B-number is in the vicinity of 10^618 involves the
fact that only at
10^618 do I need the value of pi to be 206 digits long in decimal
because the volume
is 10^206 cubed, and that pi will be exactly that value with zeroes
thereafter and used
in the surface area formula where the volume and surface area of
circle, sphere, square
cube equal each other exactly. And only this large number of 10^618
serves that purpose.

That is, only this large number is able to eke out a exactitude of
where volume, surface area,
pi with so many digits, square, cube can come together and be
equilibrated.

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

   We were meant for each other.

or

   This car is meant to turn on a sixpence.

I’m having difficulty working out what "meant" means in such cases.  Has
any study been made of a modal operator, say #, which appears in such
formulations as

  #P(a)    – a is (was, etc) meant to have property P.

(Compare: P(a): a has property P.)

  #R(a,b)  – a is (etc) meant to stand in relation R to b.

(Compare: R(a,b): a stands in relation R to b.)

I just about understand

  I meant to buy a box of soap powder yesterday.

But "This car is meant…" I’m struggling with. Also I don’t know if the
"is", etc, is significant.

[Sorry, it's a vague question which I suspect is meant (:-) for some
philosophy group or other, but which one?]
—
Needle, nardle, noo.

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What things about this structure (model) should I investigate?

I have been working on an mathematical essay which at this point
culminates in the presentation of a structure and I would like to go
further and find out the nature of this structure.

Upon request I can provide a link to my essay as it is now which would
give a clear explanation of the structure I have in mind.

Thank you for your time.

- — -

Archimedes Plutonium wrote:
> Well the B-number is going to define the realm of mathematics that is
> trustworthy, and beyond
> lies Quantum duality logic, no longer the trustworthy Aristotelian
> logic that is essential for
> mathematics to survive.

> But there is a nagging question on my mind about angles and angle
> measurement. We all grew
> up with 90 degrees and 360 degrees and triangles of 180 degrees. We
> all grew up under that system, but was it a natural mathematical
> system? Are angles more like a operator of math, say like addition? Or
> are angles more like numbers of math, where they exist and we can play
> around with them by using operators on them?

> So angles in mathematics is a oddfellow compared to numbers and
> operators on numbers.

> Could we say that angles are just the same as saying circle or ellipse
> or parabola or hyperbola? So that when we say 0 degrees we are saying
> circle or when we say 60 degrees
> we are saying parabola and when we say 180 degrees we are saying
> triangle. But that really
> does not work, does it, and that angle seems to take an independent
> place in mathematics, much like that of area or volume. So that an
> angle seems to be more akin to a characteristic
> such as volume or area and we would have an angle.

> The definition of angle that I would endorse, involves the idea of a
> full complete rotation as
> 360 degrees and that anything less than 360 is marked by two line
> segments starting at
> a starting line segment of 0.

> Angles seem to be basic to mathematics, just as volume and area.

> So the nagging question is, since I discovered the B-number and that
> if it is 10^618 would mean that between 0 and 10^-618 there exists no
> more numbers of mathematics that are
> trustworthy, nor beyond 10^618. So the question is, since we have a B-
> number, can mathematics have a more refined definition of angles? Can
> mathematics have a **natural
> and pure numbers to denote angles rather than 90 degrees or 180
> degrees**.

> Far earlier in this book, I spoke of the idea that the Golden Ratio
> Log Spiral must start not
> at 0 for that would spoil its equiangularity trait, but must start at
> one of these 10^-618 points
> to retain equiangularity of its log spiral curve.

> So here I reopen that curiosity by asking, since we have a B-number,
> does that radically
> change and alter our angle measure system in mathematics? Does the B-
> number setup
> a entirely different angle system?

> Would the number 10^-618 be angle of 1 degree? and then 2×10^-618 be
> angle of 2 degree
> in the log-spiral?

> So that the log-spiral, whether golden ratio or otherwise in
> conjunction with the B-number
> imposes a Natural Angle System upon mathematics? So that the Golden
> Ratio Log spiral
> starts at 0 point and the next point that it intercepts is 17×10^-618
> where the 17 is for the
> 17 degrees of the Golden Ratio Pitch of the log spiral.

> So let me start a Conjecture: mathematics, having a B-number, forces
> math to have a
> Natural Angle System, rather than its current arbitrary and artificial
> angle system.

Some are going to say, why change the Angle System we have where 90
degrees is a right triangle and 180 degrees the interior angle sum of
triangles and 360 degrees a complete revolution on a circle. So why
change it.

And the answer is that it was all arbitrarily picked as the angle-
system when math actually has a natural angle system.

This new and natural angle system involves the existence of the B-
number the border between Finite versus Infinity and where
Aristotelian Logic is no longer upheld so that physics duality logic
is dominant. Math can only survive in linear logic of Aristotelian
where the truth value is all or none of only true or false.

So when you have a B-number, you have its inverse also
so that 10^618 has 10^-618 and that is the smallest number with the
next smallest as 2×10^-618. With the B-number 0 becomes something of
added meaning, whereas the Old Math made people think that 0 was just
as any other type of number. In New Math there are no numbers between
0 and 1×10^-618, and if there are numbers there, they are not
trustworthy of mathematics. So in New Math there is no continuum of
numbers for there are always holes and gaps between numbers.

So now, did Old Math rely on a continuum for its definition of angle?
Not necessarily for what it had was degrees then divided by 60 minutes
of angles then divided by 60 seconds of angles, and noone ever needed
angles of 10^-60 precision let alone 10^-618 precision.

But what I want to start in this post is a Natural Angle System, not
arbitrarily picked by the ancient Babylonians
with their 60, 90 and 180 and 360 system. I want a system of angles
that exists naturally in mathematics.

So given the B-number as 10^618 and its inverse of
10^-618 and given the fact that we have golden ratio
log spirals with their pitch of 17 degrees, and given the fact that
log spirals must, mandatorily, forced to be
equiangular all along the spiral curve. That forcing demand
of equiangular has a problem of the log spiral as it is just
created at the 0 point. And given that we must have a B-number, those
demands give rise to a Naturally Defined Angle System.

So what is that very first turn or complete revolution of the
Golden Ratio Log Spiral if its first point in being borne is the 0
point, and since the first hundred points are from
1×10^-618 to 2×10^-618 to 1×10^-616. So what points along the golden
ratio log spiral were taken up by the log spiral for its first
revolution? Let us say, for brevity and clarity that the first hundred
points were those listed from
1×10^-618 to 1×10^-616. By taken up, I mean that if we were to make a
plot of Cartesian coordinates that these numbers are on the log spiral
curve. And since those hundred numbers made one complete revolution.
Thence we can say that 360 degrees in Old Math is really 10^-616
in New Math for the golden ratio log spiral and that 90 degrees of Old
Math becomes 25×10^-618 in New Math.

Now we are going to use the Golden Ratio Log Spiral since it is unique
to define this New Math Angle System.
And for brevity and clarity let us assume the hundred numbers above.
So that in New Math, when one asks for a right angle, one is asking
for 25×10^-618 out of 100×10^-618

Now whether this leads to anything new and important, if the system is
valid, will remain to be seen. We do know what the arbitrarily chosen
angle system of Old Math of 0 to 360 degrees lead to– nothing of
importance other than the fact it works for recognition of what is
wanted.

The fact that you call an angle 90 degrees had no relationship to
intrinsic math than if you called a right angle as 25 degrees where
360 was 100.

Now there was an improvement in the history of mathematics with radian
measure so that pi meant 180 degrees. But here I am actually basing
the entire Angle System on the numbers of mathematics itself. So this
new system is natural to mathematics and not arbitrary in any sense of
that word.

Anyway, it is a first beginning for Angle Systems. And there is one
concept that forces me to do things like this.
the concept or principle that if something that is around for a long
time that has never been improved and which was started by arbitrarily
picking, then that must be improved with a new system.

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