reality and imagination the multidimensional geometry of ... b tech... · 55 reality and...
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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.
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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.
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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.
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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
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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
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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)
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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.
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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!'
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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.
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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
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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
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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
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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
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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.
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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.
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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…"
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"…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.
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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
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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, )
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- 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 =
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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
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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 …)"
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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