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 x 2 - 1 = 0. We know: if x 2 - 1 = 0, then x 2 = 1 and x 2 = 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 x 2 + 1 = 0.
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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.
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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
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
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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