55

Reality and Imagination

The Multidimensional Geometry of Time/Space

Turing Machines

No Time for Time

Imagination is not reality. Or is it? Did one just

imagine something or other? If perception alone is the

totality of reality then it certainly does not matter what

another perceives or if science makes any sense trying to

find objective reality and non-subjective truths.

Imagine that a semi-trailer traveling at 80 mph is

coming down on your car stalled on a single lane highway.

Then imagine that the giant truck is actually a kumquat as it

crashes into your auto. Would the truck be smashed to

pieces? Will you wipe kumquat juice off your windshield?

The rigorous demands of mathematics often show that

great truths have simple expressions. Getting to those truths

may be very complex. In simple algebra the age-old

problems of solving simple equations may be highly

complex. For example, consider x2 - 1 = 0.

We know: if x2 - 1 = 0, then x2 = 1 and x2 = 1.

So, x = 1 because both 1 are solutions to the original

equation. This is fully defined by the process of taking

square roots. This is true regardless of one's perception.

But now suppose we look at an equally simple

algebraic equation such as x2 + 1 = 0.

56

We know: if x2 + 1 = 0, then x2 = -1 and x2 = -1.

But, for real numbers, all those on the real number line,

R1, that is, the numbers used in our "real" world, there is no

such number. The square root of a negative number does

not exist in reality. Or does it?

Almost four centuries ago mathematicians wondered,

what if there is a solution to x2 + 1 = 0? It could not be a

real number and not part of our "real world", but just for

the sake of argument they called this number i, a truly

"imaginary" number. The metaphysics of numbers

reappeared. The more "reasonable" in mathematics and

science denied it. We only "imagine" this number i to solve

the equation x2 + 1 = 0 because we define i = -1

so, x = i is the non-real real solution for x2 + 1 = 0.

But this unreal imaginary metaphysical number made

mathematics and physical science explode into a Universe

of infinite possibilities that united thought, imagination,

matter, energy and time. In a general sense it was a new

quanta. Every field of science began to use the imaginary

numbers to achieve a plethora of real tangible results.

In short order, mathematicians were able to find ways

to graph these imaginary numbers in a very real way. The

foundation for all this imagining was found in the solutions

of the univariate quadratic polynomial equations,

ax2 + bx + c = 0, a 0. In those cases where the solution

was not a real number, we see varied solutions of the form

i, where and are Real numbers and i = -1.

57

The complex number is often found in using

the quadratic formula to solve equations of the form

ax2 + bx + c = 0, a 0.

This formula is taught to high school and college

students alike, over and over until it becomes a

mathematical mantra. Rarely, except for those that may be

philosophically or metaphysically inclined, is the full

meaning of this simple process of defining imaginary

numbers explained.

Just like religion, Quantum Physics may be said to be

in "awe" of the mathematics of its own Annihilation and

Creation Theory. It uses a myriad of calculations using "i"

that yield results in the "search and find" of subatomic

particles. That is, let's look for some particle "" and in

short order, there it is. This boggles the "imagination", but

the particle conjured via imagination is "really" there.

Though the mathematics is very complex in all of this,

wave equations, particle/wave duality, etc., with arguments

and debates on the parameters, nonetheless, it becomes an

ironic analog to "seek and ye shall find". The use of

i = -1 is a sine qua non to these inquiries and

correlations.

58

If we graph the complex numbers and show their

mathematical relationship to the real numbers, we gain

insight into a manifold array of scientific applications that

also include biology, (biomathematics and biophysics),

engineering, chemistry, and on and on. We can gain

another glimpse into the correlation of metaphysics to our

limited view of "reality".

Let x and y be any real numbers. Our complex number

z as with our solution of quadratic equations, may be

simply expressed as x + yi, where x is a real number or a

point on the real number line, and y is a real number

multiplying our imaginary number i = -1 given by yi. (y

0 because the imaginary part would "disappear". This is

written as a complex number z = x + yi, x & y R1, y 0.)

If we graph this complex number "z" we have:

The point y on the imaginary axis above may be

written as yi because it is on the imaginary axis or viewed

59

as being the imaginary part of z = x + yi. Some solutions to

simple quadratic equations have these part real and part

imaginary components, as well as purely imaginary ( x = 0)

or completely real (y = 0). The interplay of this is yet

another view of our imaginary "ordered" real world.

To further our equalizing of the imaginary world to

the material reality of our world, consider a real

dimensional space, say a two dimensional plane with real

coordinates (x, y). Its coordinates are real numbers. We

make a one-to-one correspondence from our imaginary

numbers z = x + yi that exhausts the range of the real

coordinates (x, y) in its plane, (a bijection). This is a simple

calculation and the correspondence is an objective truth.

We used the complex number z, z = x + yi. Look at the

graph of z above. Note the imaginary axis y and the real

axis x. This correspondence of the x and y parts of z is a

map to the real ordered pair (x, y). This ordered pair is an

arbitrary point corresponding to an arbitrary complex

number z. C 1 (The Complex Plane) is the set of all complex

numbers z of the form x + yi. The imaginary z remains in

its domain yet corresponds to a purely real coordinate,

(x, y) which lives in the Real plane.

The point (x, y) also defines every point in the Real

two-dimensional plane, R2. We have a perfect one-to-one

correspondence between Real Numbers expressed as

ordered pairs in R 2, (the "Cartesian Product" for two

number lines, simplistically written as R 1 R 1 = R 2) and

the Complex Numbers are C 1. Thus, we not only have a

60

natural one-to-one correspondence between C 1 and R 2, call

this map f, but also an inverse one-to-one correspondence

from R 2 back to C 1 call this f -1, and both of these

"exhaust" or use up all of their prospective ranges.

Thus, f(z) = (x,y) and f -1 (x,y) = z. In addition we can

show that these maps are continuous. This satisfies the

mathematical requirement for spaces to be homeomorphic

and although they are not identical in some measures and

calculations, they are of the same form, which is the

definition of homeomorphic.

yi

z = x + yi

Imaginary

Axis (yi)

x

Real

Axis (x)

ordered

pair (x, y)

Real

y-axis

x

Real

x-axis

y

f (z)

f -1 (x, y)

61

In addition, the unit circle can be considered as the

unit of complex numbers, the set of complex numbers z

C 1 whose measure or absolute value is 1, z = 1. Each z is

also of the form:

z = e it = cos (t) + isin (t)for all t.

This relation is called Euler's formula. It has far

reaching implications especially in mathematics and

physical science.

(cos t, sin t)

1

y

x t

x

y z

The point on the circle z has

"length" 1 and can be

z = x + yi or z = e it or

z = cos(t) + isin(t)for all

angles t,

equivalently.

62

Even in medical research the idea of imagination and

especially the circulation of energy, (Pneuma, Prana, Chi

Ki, etc.), is now a real phenomenon because it affects

positive outcomes in patient's health and longevity. It is a

medical intervention in more and more areas of traditional

medicine. The mathematics of energy and quanta and the

use of imaginary dimensions are being imbued into bio-

logical sciences as an exegesis of mind/body phenomenon.

A caveat to all this is how do we measure imagination

beyond our unit circle and homeomorphisms? Can we link

the processes of inverses and projective geometry that we

explored in Part III, especially with our foray into the

metaphysics and mathematics of the I Ching? How do we

explain our complex number z as a single number solution

to polynomials, (the Fundamental Theorem of Algebra is at

stake), that are isomorphic to R 2 and thus must be

expressed in a two-dimensional plane C 1? How do these

varied dimensions interplay with imagination, matter,

energy, space, time, and even thought itself?

Some time ago I was critical of Einstein's mathematics

and physics to which I was criticized vehemently. Not that

my math is perfect, nor have I met such an animal with

perfect unflawed calculations. Some time later a disciple of

Einstein's, Hans Obanion, wrote a book on "Einstein's

Mistakes". In it he writes:

"in desperation he (Einstein) turned to his friend

Grossmann, exclaiming, 'Grossmann, you must help me,

or I'll go crazy!'

63

Curved Three-Dimensional space - or, even worse,

curved four-dimensional spacetime- is impossible to

visualize. If our three dimensional space is curved, it must

be curved into some dimension beyond three dimensions.

Our mind is attuned to three dimensions, and it does not

permit us to visualize anything with more than three

dimensions. Some mathematicians claim they can

visualize a curved three-dimensional space, but if so, they

are crazy, that is, crazy in the sense of abnormal. The best

a normal person can do is to visualize a curved surface

such as the surface of an apple or the surface of the

Earth. Such a surface is a two-dimensional curved space,

which curves into the visualizable third dimension.

The curved four-dimensional spacetime of general

relativity curves into a fifth, sixth,…or even a tenth

dimension. But since we can't step out of our four-

dimensional spacetime to contemplate its curvature from

"outside," we will have to focus on those features of the

curved geometry that we can measure within the four-

dimensional space, without stepping out into any extra

dimensions."

The idea of "desperation" and going "crazy" is a

danger to anyone stepping outside the "philosopher's cave".

It is the classic you're damned if you do and damned if you

don't. This is why the intrepid scientist, researcher, and

psychoanalyst Carl Jung insisted that one must have a firm

foundation in his or her cultural background before treading

in the metaphysical arenas.

64

So, Obanion, disciple of Wheeler, disciple of Einstein

makes a poignant point. We may take it a step further. How

does one visualize a point that is defined to have no

dimensions at all? Yet, points are pointed to in mathematics

and physical science routinely. How does one visualize a

line that has only length and no width? We nonetheless

measure their slopes define their intercepts on the Cartesian

and Complex planes as well as in higher dimensional

spaces.

We can only see "readily" what is finite is true, yet we

are receptive to the infinite. A line or a point is merely a

representation but we "visualize" with a vision that is

beyond our two eyes. The other side of the coin plays out

as well.

There are many mathematicians and physicists that

look at an infinite line as a representation of one-

dimensional space. To this infinite extension of every line

they sometimes postulate a point at infinity. This can be

done if we set up predicates akin to proving the moon is

made out of blue cheese. Does a line turns back on itself at

infinity and create a kind of circle? What is imagination

and what is poppycock?

If we examine the paradoxes of Zeno, Hilbert, Russell

et al., we may have learned that reason is not enough and

that there can be no assumed static condition placed on

infinity. The first row of our infinite matrix does not end

with the first term. Nor does the first element in our infinite

vector representations, (s1, s2, s3, t1, t2, t3) have s1

automatically become s2 or t2 by postulating something we

65

do not know. It is akin to assuming what we are trying to

prove.

So, let us consider how mathematics and science

views direction and how direction takes us out infinitely

far. Most reasonable mathematicians and scientists use

reason to be reasonable. We can show the Cartesian Plane,

R2 to have four directions, the positive direction on the x-

axis, the negative direction on the x-axis, the positive

direction on the y-axis, and the negative direction on the y-

axis.

( , + ) (+ , + )

( , ) (+ , )

Here, God is in His Infinite Heaven and all is right

with the world. There are indeed four directions and they

do indeed exhaust the Real or Cartesian plane, R2. All four

quadrants are included and the possibility of all directions

- x

-

- y

-

+ x

+

+

+y R 2

66

even to the "point" of considering the signs of all ordered

pairs is also included.

Now Obanion along with other scientifically minded

says there are then six directions in three-dimensional

space, the +x-axis, the x-axis, the +y-axis, the y-axis, the

+z-axis and the z-axis. (Here we use z as simply the third

axis for three-dimensional space not the complex number

z.) This is the usual view of science and mathematics but

all too often this assumption may get us into difficulties.

Although it is true that there are six directions relative

to an enumeration of the axes that describe R3 our own

three-dimensional space, this does not cover the idea of

- x

-

- y

-

+ x

+

+

+y +

+ z

z

67

direction in three space very well. Using the same

formulation that completely defined direction in R2 we now

have, ( +, +, + ), (+, +, - ), ( +, -, +), (-, +, +), (+, -, -),

(-, +, -), (-, -, +), and (-, -, -,). Applying this we see that we

have once again found the totality of direction in R3, that is,

considering the signs of the "ordered-triples" (x, y, z). But

there are eight octants that formulate direction fully and

not the limited six directions of axes often glossed over.

In four-dimensions as is minimally required to

calculate space/time physics, we will now have sixteen

directions of R4 or 16 hexadecants. The same is true for R5

R6, and so on.

So for the number of directions in a multi-dimensional

real space Rn, we have: R2 = 22 = 4, R3 = 23 = 8,

R4 = 42 = 16, R5 = 2 5= 32, R6 = 26 = 64. This is again two

things (+, -) taken n at a time exactly as our I Ching solid

and broken lines and found in the binary expression of our

General Continuum Hypothesis. All of this is founded on

the simple combinatorics once again given by, 2C n = 2n.

It should also come into our worldview that this binary

way of expressing an interplay of order/disorder, yin/yang,

plus/minus, solid/broken, zero/one, etc. is inherent in the

Real/Imaginary homeomorphisms of mathematics noting:

C 1 R2, C 2 R4, C 3 R6, …C n R2n.

It may be further noticed that the odd powers of Rn are

not expressed in the above correlation of the Real and

68

Complex Planes. This involves a wider view that is often

misplaced in much of modern mathematics. I will leave that

for a later date.

We can now turn to the father of computer science,

Alan Turing. Much is written about Alan Turing's life and

what stands out is that he "broke" the law of Great Britain

by being homosexual. He was tried, though not imprisoned

long, forced to be castrated via estrogen injections, and

finally was said to have committed suicide to which his

friends exclaimed "murder it was"!

We note that Turing was an ingenious cryptographer

who was instrumental in saving his country during World

War II, and that his innovations in computer engineering

led to our modern day computers, and a philosopher and

logician as well as a mathematical biologist and much

much more. Like his many predecessors studying the

infinite he met an unkindly demise.

Alan Turing

As Turing developed his mathematics, science, and

philosophical talents, his atheistic and material view of the

Yet the mathematics of Turing

being without peer in various arenas

of mathematics, when he faced the

death of a friend as often is the case,

he gave up any religious inclination

and became a devout atheist in spite of

his general inability to define "God"

required by his study of own logic.

69

world underwent some amelioration. He began a quest for

"artificial intelligence" that hopefully did not interfere with

his experience that the all is material. To this day the

"Turing Test" and his biological theories are a gold

standard for many measures of artificial intelligence and

the mathematical structures that underlie life itself.

What is not known about Turing and may be simply

implicit was how he thought about his Turing machines and

forms of intelligence. Leibniz and others maybe as far back

as the Pythagoreans had sought after mechanisms that

could calculate. Turing wanted his "machines" to think for

themselves or at least in concert with his thought. Turing

thought thought to be material as well.

The intrepid and often vilified analytic psychologist,

philosopher and M.D., Carl Jung jumped into the

Carl Jung

` Yet in his experiment with the I Ching a remarkable

experience was documented and preserved. It was Jung's

metaphysical waters from his early

childhood. In any case, he was a

disciplined scientist that influenced many

areas of both psychoanalytic research as

well as metaphysics. His views were

often seen as disconnected. Indeed, in his

treatment of yoga and energy, his

attempts to coincide eastern metaphysics

with modern medicine and psychology

raised many questions on if such things

could even be done.

70

attempt to clarify discrepancies with higher intelligence of

an almost religious experience, agencies, the collective

unconscious and the realization of "self".

His experiment was documented thus:

"If the meaning of the Book of Changes were easy to

grasp, the work would need no foreword. But this is far

from being the case, for there is so much that is obscure

about it that Western scholars have tended to dispose of it

as a collection of "magic spells,"…

"Our science, however, is based upon the principle of

causality, and causality is considered to be an axiomatic

truth…"

"…we know now that what we term natural laws are

merely statistical truths and thus must necessarily allow

for exceptions…"

"…The ancient Chinese mind contemplates the cosmos in

a way comparable to that of the modern physicist, who

cannot deny that his model of the world is a decidedly

psychophysical structure…"

"…For this purpose I made an experiment strictly in

accordance with the Chinese conception: I personified the

book in a sense, asking its judgment about its present

situation, i.e., my intention to present it to the Western

mind…"

71

"…not even the strangeness of insane delusions or of

primitive superstition has ever shocked me. I have always

tried to remain unbiased and curious - rerumnovarum

cupidus (eager to learn anew). Why not venture a dialogue

with an ancient book that purports to be animated? There

can be no harm in it…"

"…In accordance with the way my question was phrased,

the text of the hexagram must be regarded as though the I

Ching itself were the speaking person. Thus it describes

itself as a caldron, that is , as a ritual vessel containing

cooked food. Here the food is to be understood as spiritual

nourishment…"

"…The ting, as a utensil pertaining to a refined

civilization, suggests the fostering and nourishing of able

men, which redounded to the benefit of the state…Here

we see civilization as it reaches its culmination in

religion. The ting serves in offering sacrifice to God…The

supreme revelation of God appears in prophets and holy

men. To venerate them is true veneration of God. The will

of God, as revealed through them, should be accepted in

humility…"

Jung's answer from a book was accurate and timely.

He spoke to "other" and received an answer he could not

reconcile with "self". Turing it is said thought the ideal

"computer to have its own inherent intelligence". If one

wrote a question on a paper the computer would answer.

Jung felt this experience was his. What is the essence of

intelligence and thought becomes a bigger question.

72

73

If one takes a straight edge and moves it from

Hexagram 1 down to Hexagram 16 and next from the

second row (or column if the orientation is changed) from

Hexagram 17 to Hexagram 32, and so on with rows 3 and

4, then a pattern of motion is seen. This motion in this

binary representation becomes visibly "computer-like".

The view of all this regarding the metaphysics of

agencies, psychoanalysis, mathematics, or application to

the sciences is a step that is not only difficult but highly

controversial even though the metaphysics of today MAY

be the physics of tomorrow.

But what about perception, reality, subjective and

objective truths? Let's travel back in time with our thoughts

and try to make ourselves present circa 500 BC. Choose

your coordinates to visit the Academy of Pythagoras,

keeping in mind how we began this paper on the

mathematics, physical science, and metaphysics of .

Like manifolds in our four-dimensional space/time

discussions, we too have come a full circle in our

discussion to which we can now extrapolate instead of

hyperbolate. But before we consider our perception versus

an objective reality, as women and men of reason we

should examine what reason and logic entail and hopefully

not in-tale.

As a species many of us prize logical thought and shun

what is illogical. As the paradoxes of mathematics have

74

shown us, there are some things, logical or not, that

transcend our logical acumen. This is the alogical. We may

add to this that there may be hidden variables such as in our

look at Relativity and Quantum Mechanics. In other words

there are always latent variables that should be sought out.

The patent is right in front of our noses, but it too may be

something to apprehend in terms of logic, illogic, and

alogic. After as much of this as we can stomach, where are

brains often descend to, we may begin to comprehend.

ALOGICAL

LOGICAL

ILLOGICAL

Patent

Latent

We recall the circumference of

a circle as C = d = 2r. In this

circle the diameter is 2 as

shown on the graph. The

circumference of this circle "C"

is then simply 2. The radius of

the circle is d/2 = 2/2 = 1.

Any half of the circle's

circumference is then ½ C.

In this circle C = 2 so ½ C is

then ½ of 2 which is .

Let us "unfurl" or "straighten the

top half of our circle to the right

until it is perpendicular to the

fixed point at x = 1, and the

x-axis as shown below.

x

y

1/2 1 - 1 - 1/2

1

1/2

- 1/2

- 1

½ C =

75

1/2 1

The vertical line is the ½ C

or ½ circumference straightened

and unfurled from the semi-circle

above. This length is then also ,

which is measured now as a line

from the x-axis at the point x = 1

to the top of the line at the point

(x, y) which is (1, ). This is the

change in y value from the x-axis

(y = 0) which is the point (1, 0) to

the top of the line at point (1, ).

This length or rise or change in y

is often written as y, where

y = in the example on the left.

The change in the x-value is simply

the distance from zero to x, the

"run" and written as x or x - 0 = x

in the example on the left.

Next we draw a line from the (x, y)

point at the top of our straightened

½ circumference arc point, (1, ),

through the origin point (0, 0) of

the graph below.

Lines in the plane are of the form

y = mx + b, where y and x describe

all the points (x, y) on the line and

so m (slope) and b (y-intercept) can

be any Real number.

- 1 - 1/2

- 1/2

- 1

x

y

1

1/2

y =

x = 1

(1, )

76

- 1/2

- 1

x

y

1

1/2

- 1 - 1/2

y

=

x = 1

1 1/2

We now have a line through the

origin with b = 0 and a slope

m = , that is, y = x.

Since y = mx + b is the slope-

intercept form of every line ( m =

slope and b is the y-intercept) in the

plane, by choosing our original

circle to be of different sizes and in

a different position in the plane and

then "unfurling" our top or bottom

semi-circle circumference, we can

represent EVERY line with a factor

of slope , an uncountable number

of lines. These are of the form:

y = ax + b, a,b R and a0.

The slope of any line,

a roof, hill, ramp or

anything can be seen

as the ratio of the rise

to the run.

= slope = m

The point where the

graph intersects the y-

axis is called the y-

intercept usually

denoted by "b" and

here is the point (0, 0)

That is, y = 0.

Rise

Run

the

line

y =

x y

x =

77

It must be noted that this line is not limited to the

plane R2 but can extend out in "space" R3 or R4 or be

viewed in some other "plane" in R3 or R4 that is rotated in

that space. With modification we can extend the same

concept of these -slope lines into non-Euclidean and other

curved spaces.

More important is what I shall term the "Columbus

Effect". We all know the story of Columbus watching the

masts of ships as they came in from and went out to the

horizon. The gnostic ancients were well aware that the

earth was not a flat surface that one could fall off of and

subsequently be eaten by a giant dragon. Religion and

government insisted, sometimes on penalty of death or

imprisonment, that one must accept the earth as being flat.

With our representation of a circle becoming a line we

have a natural and logical view of the illogical. The line is

straight but it came from a semi-circle and once again may

be transformed into a semi-circle with a myriad and

uncountable types of curved lines in between the straight

line and the circle. Because of the -factor inherent in the

slope of the family of lines we constructed, the curve is

always latent in spite of our perceived "straightness".

When we walk on a "straight" road our perception is

that we are ambulating in the straight-line patent to our

senses. As with the ships at the horizon, we are indeed on a

curved surface that is latent to our perception. Our -slope

line above is patently straight but the curve creating it

makes it latently a curve, especially in terms of potentials

and possibilities. (cf. Harmonic Analysis and Brownian

Motion). In higher mathematics as found in Differential

Geometry, the patent curves on curved surfaces such as our

78

own earth are latently straight. More of that at another

place in time.

This "flat-view" is not very different than the Steven

Hawking event-horizon near a black hole where once one

falls into it, the gravity dragon rips them to shreds. But the

reality of patent and latent is readily observable in the

construction of our line from a circle. We perceive what is

patent, the line, we fail to see the curve, the latent. This also

holds vice versa. At any "horizon" there should not be a

limited view of the line without considering the latent circle

and the putative "edge of the earth". The event horizon is

simply better stated as a horizon event. New data in

astrophysics contradicts the Hawking view.

As we present these basic realities of mathematics,

physical science, and metaphysics, Swiss researchers have

begun to photograph the latent and patent interplay of light,

as a wave AND a particle. This duality is at the heart of

Quantum Physics, Taoism, and much of higher mathematics

permeating into the arenas of all branches of science.

"The lower layer of the image

shows “packets” of energy

exchange between electrons and

photons, which results in a

visualization of particles of light,

while the top layer shows the

wave-nature of the light, in

“blips” of standing waves: "

(Swiss Federal Institute of

Technology in Lausanne in

Switzerland…Fabrizio Carbone,

who led the research team that

designed the technique …)"

79

This goes to our sometimes ludicrous non-geometric

view of "time travel' as well. If we limit ourselves to a

simplistic directional view of time, traveling "forward" in

time, say along the x-axis that may now represent the time

dimension in space/time, time travel then is not a problem.

We do it everyday, or more correctly, it does us.

Moving back in time now presents an "impossibility"

to those that are more "reasonable". The problem becomes

one of causality. That is, if we move backwards in time

there may be observable events such as hearing a door slam

as it opens instead of closing. Maybe everything becomes a

reversed motion as an apple "falling" from the ground and

onto a tree. Such visions can be seen in video but none of

us can make sense of it patently or in the mundane.

On the other hand, if time was more than the singular

linear direction "visualized" in much of physical science,

we might see varied routes of "back in time travel" via the

so-called 5th dimension of science fiction. Recall above

our directional perspective of Real Space as not simply the

enumeration of axes. Work in a fifth dimension is common

not only to science fiction but to many metaphysicians,

physical scientists and sane mathematicians, to say nothing

of those that have been pursued by men in white coats with

butterfly nets.

How we work in R5 is not completely patent either,

for we have a five-tuple and a vector and a vector space no

less, that can be represented by (s1, s2, s3, t1, t2) where

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the s terms represent space and the t terms time. Is there a

t3 also latent? More of this question at some other "time".

"Breaking the Time Barrier" is of course non-trivial.

There we might see a "finality" to mortality. Biological

issues come to the forefront. In this light I might attempt to

present another way of looking at being and nothingness,

entropy, the horizon-event, and what it means to be in the

form of a semi-articulate monkey held hostage by time. Let

us take nihilism one step further and in a quasi-articulate

form attempt to explain its workings as an objective reality.

The Nothingness Theorem

"There is not a point of nothingness in a

neighborhood of somethingness."

We once again see our binary representation of 0

and 1. The famous Italian mathematician Giusseppi Peano

developed much of the logic and rigor of our modern

mathematics. Some of this was a result of the issues raised

by Cantor. The proof of this may be a bit intricate but

suffice it to say that we need to define a point.

A point has no dimension. Somehow we manage to

"point" to it and use it in manifold ways. If we address the

predicate of a nullity and conclude because a point does not

exist, it is there because we can point to it. Is this alogical

or illogical, being or not being, or parsing words?

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The symbolism I introduce anew is exactly the genre

of mathematics that created not only vehement and ugly

arguments, injected god into a mathematical question, but

as we have seen, drove many mathematicians, scientists,

and philosophers into the de facto loony bin.

How can one not cringe on apprehending and trying to

comprehend the implications of our "Nothingness

Theorem" or the other side of the coin of the measurement

of nothingness as = {} = {1 element}? Are these a

contradiction or simply an alogical construction?

We might construct a highly mathematical look at our

existing/non-existing point in the midst of something. If we

use the standard increments of getting closer and closer to a

limit, indeed a definition of continuity itself, and use the

smallest of smallest positive measures, traditionally given

by the symbols and , where > 0 and > 0, we still get

into the same quagmire of contradictions in picturing

dimensionality and nothingness.

In more abstract Space as defined in Topology, the

same problem remains. Even in the infamous Non-Metric

Spaces where there is no measure whatsoever, the

contradictions remain.

Thus, without trying to find God patently or latently,

we can turn to the macrocosmic view of the same

Point, pun, as usual, intended.

No Points at Infinity Theorem

"There can be no point at infinity."

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Although it is popular to assert points at infinity, they

are simply predicates. Mathematics, physical science, and

metaphysics fall into a quandary when predicates are taken

for granted. We should never assume what we are trying to

prove save for the cause of contradiction. Unlike our

Nothingness Theorem, here the issue is not only the

existential point, but also the nature of infinity itself.

Much ado has been made of points at infinity

sometimes postulating a line curving back in space as some

sort of geometrical and . If a point at infinity or the

definition of infinity is not well defined, making an

assumption that is a covert conclusion may bite the

mathematician on his or her own "posterior point". Rings

and circles also may have similar caveats in that some

extraneous motion or point of origin may be latent or

undefined as well.

Yet the ideas of pictorial representations of Multi-

dimensional Space beyond our limited view of Space/Time

and higher geometries, may sometimes be like the masts on

ships at the horizon. In some sense, there is that nothing,

not a singular point to obscure the dynamics of views

beyond the physical science, id est, the metaphysics.

Whether vis á vis to our "Nothingness at a Point Theorem"

or our "No Infinity Point Theorem", simply the process of

approach to any pseudo-limit, may help us get a glimpse

into a wider worldview or higher dimension. Both the

masts and the curve become apparent.

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We need not be fooled by a crystal ball that pretends

to disclose the future or be buffaloed by a Hog-Bison, a

rather ostentatious beast that exudes matter as some kind of

cosmic excrement. If we are mindful that what is

observably patent may not exhibit the whole picture, we

might ride that point towards infinity, undeluded and

without the specter of ignorance, especially to what is

latent.

There is so much more we could examine but for now

suffice it to say as the Bard on Avon put it, "'Tis mad

idolatry to make the service greater than the god".

John Kotsias

A-B Technical College

Asheville, NC

April 2, 2015

Georg Riemann Henri Lebesgue Max Planck

William Shakespeare