DNA
– I’CHING
Dow
[Editor's
Note:] This paper was presented to the 9th International
Conference on the Studies of I-Ching, University of
Hawaii, Hilo, Hawaii, August 17-21, 1992. See the original
at www.thomehfang.com/suncrates4/2Dow.html
I. Introduction
An article entitled "The Unusual Origin of the Polymerase Chain Reaction"
by Kary B. Mullis disclosed a surprisingly simple method
for making unlimited copies of the DNA fragment.[1]
On February 16, 1992, the New York Times attributed
to Mullis's 1983 invention the revolutionizing of molecular
biology and genetics.
What is surprising is Mullis's graphic presentation of this simple binary mathematical
formulation obviously corresponds to, if it is not identical
with, the hexagram progression sequence in China
arranged by Shao, Yung. (See appendix). Is this
striking similarity between these two merely an accident?
In view of the fact that the I-Ching sequence
is based on binary mathematics and its identity with
the binary table of Leibniz has been well established,
the identicalness between the DNA replication sequence
and the hexagram progression sequence would seem
difficult to dismiss as pure coincidence.
1. The DNA replication sequence is restricted to genes, a causal event except
for human inducement or manipulation, and is relatively
precise, automatic and derived from scientific experimentation.
2. The hexagram sequence, derived from the observation of nature for the purpose
of showing the way for human action, is all inclusive
and comprehensive in its prediction of events in uncertainty.
An attempt can be made to see the DNA sequence as a
microscopic empirical discovery to complement the macroscopic
I-Ching sequence in searching for a unified description
and understanding of world phenomena?
Advancement in quantum mechanics, fractal geometry and the chaos theory,
etc., tend more to reaffirm the values of the basic
philosophical assumptions in I-Ching's formulation
of the hexagram progression sequence. The DNA replication
sequence serves as one example. However, science describes
what it is but does not answer the question of why.
Therefore, a philosophical approach is necessary to
search for an answer to why these two phenomena are
similar.
In view of the fact that both sequences are derived from the observation of nature
and both deal with the problem of human beings, the
significance of this search for an explanation of their
similarity may shed some light on the efforts to reach
a unified theory which is able to explain the world.
We shall proceed in three steps to examine the issues.
They are: (1) An evolutionary view of the world with
a unique position of man or the material unity of man
and the world;
(2) A twofold complementary-contradictory process of
being and becoming of the world phenomena;
and (3) A binary representation of the world process
in our understanding of and predictions about the world.
II. The Material Unity of the World and the Position of Man
1. The Concept that Quarks and Leptons are the Basic Constituents of Man and
the World.
Aside from the admission of an omnipotent, omnipresent and omniscient external
creator of this world and man, the prevailing scientific
view of the origin of the universe is the theory of
the "Big Bang" initiated by George Gamow in
1948 and subsequently confirmed in 1964. It described
that not only were human beings part of nature, but
also originated from the same source which is beyond
the scope of science. This is known as the singularity.
Nevertheless, it is believed to be composed of undifferentiated
material, a "primordial soup," or "quantum
chaos," etc. The advancement of quantum mechanics
asserted that the basic constituents of this material
unity of which man is a part, consist of the fundamental
particles of quarks and leptons. The originator of this
concept of quarks and leptons, Murray Gell-Mann, conceded
that they are mathematical conclusions rather than those
made from direct observations or laboratory experimentation.
Obviously the composer of the hexagrams in I-Ching relied on direct visual
observations of nature in order to arrive at their statement
that the world including mankind originated from the
same source indescribable but implied to be a "primordial
chaos or fusion" (hung tung) the "great void
(tai xu)" "oneness" (tai yi, tai su),
or "yuan", etc. Despite its antiquity, this
speculation of singularity by I-Ching surprisingly
anticipated modern science. In view of this convergence
regarding the origin of the universe and its human beings
in the concepts enunciated in the I-Ching and
the prevailing concepts of modern science, we may venture
to say that the similarity which exists between the
hexagram sequence and the DNA sequence may be based
upon the same assumption of a material world unity.
Assuming that man and nature have evolved from the same stem and have the same
physical constituents, then, the patterns of their evolution
and the processes of their manifestation in diverse
and complex entities on various levels would have to
bear some similar marks though it may be to a greater
or lesser degree. Recent discoveries of the phenomena
of both the simple and yet the complex in fractals reveal
such a process. In the words of Benoit B. Mandelbrot,
the originator of fractal geometry, "a major and
surprising impact of fractal geometry and the chaos
theory is that in the presence of a complex pattern
there is a good chance that a very simple process is
responsible for it."[2] This discovery is known
as the principle of self-similarity, which seems to
underlie the structures and evolvements of all natural
entities.
The best example of self-similarity as revealed by fractal geometry is in
cauliflowers. The head of it contains branches or
parts. Compared with the whole, the branches are very
much the same, only smaller. These clusters again can
be decomposed into smaller clusters which again look
very similar to the whole as well as to the first generation
of branches. This process can continue but with limitations.
That is what is known as iteration which is synonymous
with the concept of feedback and dynamic laws. In other
words it is the same operation carried out repeatedly,
the output of one iteration being the input for the
next one. Macroscopically we see a shape or form
which is best described as a cluster. It is composed
of smaller ones which look almost identical to the entire
cluster however scaled down by some factor. On a microscopic
level in dealing with subatomic particles one can determine
the location and velocity of a particle at one instant
from the preceding instant whether or not the process
is discrete or continuous.[3] The technique of video
feedback and multiple reduction copy operates on this
principle with great success.
In view of the discovery that the infinite and complex phenomena of nature can
be seen as a result of the interactive process of the
most simple elements or events, the arrangement of
the hexagrams known as "double-up (chong gua) of
the lines" in I-Ching would seem to be part
of the same process as the DNA sequence. The similarity
between the two arises naturally since both systems
base their statements upon objective observations of
nature with the exception that the latter rests upon
a more scientific ground than the former. Nevertheless,
the relationship between the structural arrangements
are modeled after the relationships that are recognizable
as those existing in nature. In this sense the statement
in I-Ching that every thing or event in this
world contains a Yin and Yang would be consistent with
the structures of being and becoming of the world phenomena.
This may be due to the iteration process displayed in
the I-Ching sequence that inspired Zhu Shi-jieh
(1303) during the Yuan time to portray it in an arithmetic
triangle (Ssu Yuan Yuan), which computes the
binomial coefficients up to the eighth power. Since
that time it became the Pascal Triangle (1623-1662).
Both of these triangles could be related to the mathematical
structure of the hexagrams sequence.
As a consequence of this mathematical enlightenment of the iteration process,
the introduction of the Cantor set (George Cantor, 1845-1918)
in 1883 in graphics not only magnified the nature of
this similarity to that of the hexagram sequence (Appendix)
but also reinforced the concept of the ultimate unity
of world phenomena as Buddha had long ago contended.
In 1913, Ramanujan equaled the square with the circle,
one with two. To Ramanujan, the mathematics and metaphysics
lay side by side, inextricably intertwined.[4] To equal
a square to a circle is what topology has revealed to
us since the Cantor set is considered to have contributed
to the development of topology, the concept that the
complex and diverse world phenomena are temporary aggregations
of simple elements would seem to underlie all of these,
whether a DNA hexagram sequence or a Pascal triangle,
Cantor set or topology for they all are equivalent.
In the demonstration of Peitgen (p.157) fractal is in
strict correspondence to the decimal system. Therefore,
the graphic presentation of the hexagrams sequence in
the "Tai Ji Tu" ("Diagram of the Great
Ultimacy") of Zhou Dunyi and the mathematical presentation
of it in Shao, Yung's sequence or any other arrangements
are all ultimately the same. Furthermore, Peitgen showed
that the Cantor set is a solution to the problem of
equations (p.192). Thus the Confucian concept of zhong
(harmonious balance?) which explains the rule of change
in hexagram lines in Xici (Appended Commentaries)
may have conveyed the same meaning and function as the
mathematical meta sign of "=" in mathematics.
(2) Relational Holism: the Seven Stages of Progression and Eight Catgories
Admitting that the Cantor set is truly complex yet also self-similar and simple,
then, can the iteration process as shown in fractals
be infinite? From a purely linear view of progression
on a one-dimensional plane, the iteration process could
be infinite. However it would be contrary to the phenomena
of recycling observed in nature such as the alternation
of day and night, the four seasons and the birth and
death of human events and especially, contrary to the
manner in which topology and fractal geometry deal with
nature. Further discoveries in the phenomena of self
limit in fractal geometry and the notion of relations
in topology have become reconciled with the problem
of recycling.
In the first place, the iteration process has its limit. This is know as fractal
dimension. The concept of self-similarity is intimately
related to the concept of limit. Self-similarity may
hold true for a few orders of magnitude. Below a certain
scale, matter decomposes into a collection of molecules,
atoms, and elementary particles. It would be too infinitesimal
to consider further dissection in scaling down copies
of the complete object. It has been proved notably by
Gestalt's contention that parts can never be exactly
equal to the whole. Some variations must be accounted
for. There are several variations of mathematical definitions
of self-similarity. This is what is known as the phenomena
of statistical self-similarity. The Shannon limit in
computer operation may serve as another example. In
addition, the resulting miniature copies may be distorted
in some ways. In fractal geometry this is called self-affinity,
a sort of approximation of a self-similarity which is
not strictly or precisely identical. This principle
of approximation in epistemology has been well explored
in quantum mechanics. Therefore, the concept of similarity
is intimately related to the concept of limit in terms
of fractal dimension. Every limit is a negation which
leads to new qualities. In this sense, negation can
be seen as creation or positiveness as in the Chinese
saying "bu po bu li" (no destruction,
no construction). For example, the weight of a tree
is proportional to its volume. If the stem is increased
beyond a certain limit the strength of the wood will
not be sufficient to bear the corresponding pressure
and the existence of such an entity as a tree will be
at stake. The tension between volume and area also explains
why mountains do not exceed a height of seven miles,
or why different creatures respond differently to a
fall. A mouse by his own weight may be unharmed by a
ten story fall but a man may be injured in just falling.
As volume increases the hazards of falling from the
same height increases. The phenomenon of nonproportional
growth is at the heart of fractals. In the I-Ching
it is said that the way of heaven is to diminish the
strong and augment the weak so that all can survive.
Secondly, in topology a straight line can be bent into a circle, pitched into
a triangle or pulled out into a square. However, not
everything is topologically changeable. The basic relationship
between things does not change. For example, an intersection
of lines remains an intersection which is invariant
mathematically. A sphere may be transformed into the
surface of a horseshoe but never into a doughnut. For
topology deals with forms and shapes from a qualitative
point of view. Qualitative features will not change
when the objects are transformed. If the quality of
an object is changed the object will be no longer the
same and it will become something else. While the basic
objective of I-Ching is not a mathematical excursion
such as topology, the main efforts of the arrangers
of the hexagram sequence centers on the relationship
of one's action within the circumstances including time
and place or position and one's ability and capabilities
to predict the anticipated results. Six lines (yao)
were used to depict the relationship of the place value
of change to the constancy of the situation in a hexagram.
With a consultation to the models provided in the hexagram,
one can determine an intelligent choice on which to
act. Regarding the hexagram, the second and the fifth
lines are the most significant because they occupy the
central positions. The second is in the center of the
lower trigram, and the fifth is in the center of the
upper, while the third and the fourth lines are variable
with the first and the last as constant. Thus the structure
and functions of the hexagram sequence seem to reflect
the same meaning as the limit in fractals and the unchangingness
in topology. While molecular biology is primarily experimental
and advances are based on observations of a specific
biological system, a natural way to organize the sequences
is by taxonomy or evolutionary relationships. DNA exists
as a complex, three dimensional structure. Thus, the
use of topology and geometry in the study of the DNA
sequences is well accepted. The famous formula that
is related to topological property, the unchanged property
under continuous deformation of the DNA with geometric
quantities by James H. White, serves as an example.[5]
If the existence of humankind depends on the human gene
pool, the similarity of the DNA sequence with the hexagrams
sequence may signify a profound meaning for all of us
to explore.
Since the DNA polyinerase chain reaction reveals a cyclic process as pointed
out by Mullis,[6] it can be seen to display seven cycles
or seven stages. For example, the first copy may be
identified as the first cycle or the first stage; with
the two copies, the second cycle or stage; the four
copies, the third cycle or stage; the eight copies,
the fourth cycle or the fourth stage; the sixteen copies,
the fifth cycle or stage; the thirty-two copies, the
sixth; and the sixty-four copies, the seventh. Assuming
there must be an initial stage and assign it as zero
(0) and count the initial or the original one from which
the first copy derives, the sequence can be categorized
as an eight cycles or an eight stage process. In a seven
stage process the sequence will naturally consist of
64 copies or hexagrams in simple mathematics because
the initial one is zero (0). However, Mullis did not
engage in this sort of analysis.
What is puzzling is that these seven or eight stages of this evolutionary process
in the DNA sequence and the hexagram sequence, correspond
to the seven stages of evolvement in the origin of the
universe as conveyed by the theory of the Big Bang.
Why is this true? The Big Bang theory does not attempt
to explain nor to infer what came before or after the
sudden explosion. Genesis attributed it to the
creation of the world in six days plus one for resting
while the 24th hexagram, Fu (Returning) in I-Ching
depicts the recycling process in nature as running seven
days (qi re yi lai fu). What justified the seven days
cycle? The I-Ching left that for speculation.
However, the hexagram Fu explicitly conveys the meaning
of it as a turning point. It contends that recycling
is the process of a continuum. It occurs as the result
of self-regeneration within the mutual or reciprocal
relationship of the twofold structure and functioning
of world phenomena. In the arrangers' mind all that
is good in nature and society will eventually decline
once its limit is reached; when decay has reached its
limit recovery will begin. This is what the hexagram,
Fu, meant.
Recent studies on subatomic particles seem to reveal a similar seven-stage structure
and change though it can not be precisely matched with
the DNA or the hexagram sequences. Electrons which orbit
the atomic nucleus have a seven-layer shell. If we count
the initial one, it can be counted as eight-layer. Each
layer can hold limited numbers of electrons. The inner
one holds two electrons. Both the second layer and the
thirdlayer hold 8; both the fourth and the fifth, 18;
the sixth and the seventh, 32. Atoms may join one or
more other atoms to fprm the smallest stable particle
of an element or compound, the molecule. This process
takes place between electrons in each atom's outer shell.
The number in the outer shell dictates whether an atom
will donate electrons to another, share or seize them,
and how tightly the two will bond; if the outer shell
is filled, the atom is inert. This simple process of
exchange of the positions of electrons in a definite
number within seven layers constitutes the base of the
periodical table that accounts for the structures and
functioning of all elements in the known world. Is it
coincidental that the framers of the I-Ching
sequence based on their macroscopic observation of nature,
derived eight trigrams as the basis for projecting the
whole world of change? In view of the fact that Murray
Gell-Mann proposed to classify the particles according
to Buddha's eightfold path,[7] can it be outlandish
if we see the structure of the eight trigrams as an
ambitious primitive attempt to interpret the world of
change within the general frame of reference in eight
categories? It wouldn't be a surprise if the ancient
Chinese, particularly during the Han time, had observed
through the hatching of eggs the embryo process to acquire
a vague idea of a seven stage development, in addition
to the original one and thus inferred the eight trigrams.
For embryology has revealed such a process.
(3) Material Unity as the Basis of Cognition and the Unique Position of Man
Albert Szent-Gyorgyi who became a Nobel laureate due to his isolation of vitamin
C, confessed that in his search for the secret of life,
he ended up with atoms and electrons which have no life
at all. In his opinion, life is the expression of the
harmonious collaboration of all.[8] Therefore, the search
for an answer to the question of the origin of life
continues. At the present stage, the standard proposal
that life originated from a "primordial soup"
is equally disturbing. No one has even been able to
precisely describe it. The is no clear or viable theory
of the chemical origin of life. Modern science may delve
into the recesses of the brain to explain the phenomenon
of consciousness, yet there is an undeniable distinction
between the brain and the mind. While the brain may
be measured on physical terms, the mind is difficult
to quantify. Assuming that the brain is constituted
of atoms and atoms have the so-called "parallel
psychic components of some sort,"[9] it is still
difficult to explain, as the Gestalt theory has contended,
why the brain's consciousness is unified and not merely
the sum total of all atomic consciousness. Francis Crick,
the co-discover of DNA, even dreamed of a genetic code-carried
to earth from another planet.[10]
So long as we cannot describe the phenomena of the mind in terms of nerve fibers,
as some of the artificial intelligence researchers would
like to do whether or not we can study the mind remains
a profound challenge to the world. However, we can learn
about the phenomena of the mind by what the mind does.
This is involved with the most fundamental question:
How do we know and how do we perceive the world and
how do we anticipate events? The fact that we have survived
and made progress in civilizations proves that we know
it, but not completely. The most popular prevailing
explanation of this universal success of mankind at
the present level of knowledge is, perhaps, the corresponding
theory. This is that the mind is structured in a way
that corresponds directly with the structure of the
universe. In a sense it meant that there is a unified
ground state of consciousness in which the observer,
the process of observation, and the observed are unified
in a structure of self-interacting consciousness. This
subjective experience of the unified origin of what
we call subjective and objective existence constitutes
the essence of the corresponding theory.
Aside from philosophical support of the corresponding theory, the DNA encoding
could have been the link between the world and ourselves
in knowing. For example, we do not learn grammar. We
could not make this up. The ability to organize and
arrange words into intelligent sentences and the capacity
to identify the patterns of the world and link them
to instructions from parents, teachers and others are
said to have been genetically set through DNA messages
which stay with us throughout our life span and are
passed on to future generations. According to Heisenberg,
apart from the process of observation, we are finally
led to believe that the laws of nature which formulate
mathematically in quantum mechanics no longer deal with
particles themselves but with our knowledge of elementary
particles.[11] Modern physics has postulated the existence
of various dimensions or levels of reality which correspond
to different levels of conscious perception. In this
sense human beings occupy a unique position in their
relationship to the world. It is upon this premise that
the framers of I-Ching said that the coordinated
unity of the way of heaven, the way of earth and the
way of man is the essential structure for perception
and the basis for cognition.
From a broader perspective the theory of correspondence may imply the correlated
positions pronounced in the I-Ching, and particularly
expounded by Tung, Chung-shu's (Dong Zhong-shu) of the
Han time. Dong's exposition of the correlation theory
has failed in its test by experimental philosophy but
the rise of quantum mechanics has resuscitated it. In
an attempt to integrate the contents of perceptual experience
we have learned that the function of particularity becomes
effective only through its being correlated with the
function of universality (identify in whiteness as the
definition of white)! The function of univereality becomes
meaningful only in correlation with the function of
particularity (perceiving the particular object intuitively).
It is the correlation within one and the same structural
complex of the two basic functions, that of particularity
and that of universality, that make cognition possible.
Echoing Leibniz that there is a preestablished harmony
between the world and man which effects knowing. Andre-Marie
Ampere, the pioneer of the science of electrodynamics
has contended that our mind is structured in a way that
directly corresponds to the structure of the universe.
Otherwise, we wouldn't be able to ascertain any degree
of certainty.[12] Science has now conceded to idealism
that its objective reality is not given but is constructed
by the creativity of the mind. Without assuming a coordinated
system existing between the world and man, objective
reality cannot be constructed. What we have claimed
as the laws of nature are not the elements of nature
but the products of the human mind derived from their
observation and understanding of nature. It is on this
ground that we can explain the pure process of logic
which consists of seven stages, (e.g., 1. perceiving
the world; 2. selecting the features perceived; 3. discriminating
between them; 4. judging them; 5. analyzing them; 6.
constructing a theory or hypothesis; and 7. testing
them) which is universal. Some researchers have extended
a similar seven-stage view to analyze the development
of our moral reasoning process from childhood to old
age, e.g., 1. simple obedience; 2. conforming to group
behavior; 3. peer pressure; 4. duty orientation; 5.
legal considerations; 6. conscience orientation and
7. liberty). Karl Jung thought that behind every picture
or image of the nature and social behavior as well,
there stands an archetype deeply ingrained upon the
collective unconscious of humanity and available for
the individual to decipher. That is what the hexagrams
meant. Mandel-Brot demonstrated that mathematical fractals
in fact have many features in common with the shapes
found in nature. Indeed some characters such as the
Cantor set were already there.
(4) The Twofold Complementary and Contradictory Process in Being and Becoming
Specifically both the DNA sequence and the hexagram sequence are oriented towards
practical application. If we examine the predictability
of these sequences, they are almost incompatible. The
DNA replication must proceed with an accuracy of one
error in every iota base pair.[13] The fidelity of the
DNA replication is relatively assured. The predictability
of the hexagram sequence depends on the capability of
the inquirer to decipher their meanings. Therefore,
the margin of error in the hexagram sequence is enormously
greater than that of the DNA sequence. However, a careful
study would show that the difference between the two
in predictability is a matter of degree rather than
of kind. The tact that the terms of error, wrong or
mismatched can be introduced in the DNA replication
tells us that all is not absolute. If our heredity is
a chance occasion as biologists have stated, then, chance
also plays a critical role in hexagram predictability.
In three aspects we can illustrate that both DNA and
hexagram sequences reveal the same fundamental principles.
Their differences in predictability arise not from their
basic operation, but from the scope and degree of their
applications. These principles are: (1) the twofold
structure in continuous creating or change; (2) the
twofold polar relationship of complementarity in contradiction;
and (3) the reciprocal generation of each other in a
spiral progression.
1. The basic twofold structure in continuous renewal.
In their efforts to search for a unified theory of elementary particles and forces
with which to explain the material world, physicists
have found that all matter is built from just two classes
of elementary particles: leptons and quarks. Frank Wilczek
notices that "all the fundamental constituents
of matter came in matched pairs: For every kind of particle
there is an antiparticle. Indeed, a particle and an
anti-particle have often been discovered simultaneously
when the two were created as a pair by a high energy
collision in a particle accelerator."[14] It has
become apparent that all particles are coupled more
or less strongly to each other. Scientists do know that
the twofold nature of a positive and negative charge
constitutes the basic nature of the structure and functioning
of matter but they do not know why. Except in different
linguistic expressions, it is indisputable that both
the DNA sequence and the hexagram sequence consist of
basic pairs. For example, Yin must pair with Yang in
order to exist and function.
Either Yin or Yang by itself cannot exist (gu yin bu sheng; yu yang bu zhang).
The I-Ching says that this is the way of the
universe. Anyone can infer from the obvious event of
the pair-up between male and female for procreating
and contemplate that this twofold process of being and
becoming is not just for the material world of heaven
and earth but also for human events as well, as the
arrangers of the hexagram sequence had contended. The
advancement of our knowledge at the present stage seems
not to disapprove this theory. Microscopically the photons
display a twofold phenomena of particle and wave; the
DNA structure is a double helix; the 46 chromosomes
of a person are in 23 pairs except x and y which specify
sex, etc. On a macroscopic level, it has become common
knowledge that our brain hap two hemispheres and our
thought has both digital and analog processes; linguistic
expression consists of semantics and syntax, i.e., yes
and no; in the structure of any society throughout the
history of the world there has always been the twofold
stratification between the ruled and ruler or between
command and obedience, enemy and friends, internal and
external, etc; in economic activities there is the twofold
structure between haves and have not, gains and losses,
management and labor, monopoly and small enterprises,
diminishing returns and increasing returns, etc.; in
politics, the most cherished democracy is predicated
on a two-party system; there are conservatives and liberals,
subjects or citizens and rulers or government for and
against, in and out, etc.; in ethics and aesthetics
there is the twofold distinction between the virtuous
and the disreputable, the beautiful and the ugly, etc.
In civilizational development as a whole, a twofold
process has evolved in all things. For example, there
are the small vehicle and great vehicle schools in Buddhism,
idealists as opposed to realists in early Confucianism
and schools of reasoning and the school of intuition
in Neo-Confucianism, there are Catholics and Protestants
in Christianity, and Sunnis and Shiites in Islam, etc.
Sufficient is this example to warrant a statement that
the twofold structure of being and becoming is universal
and permeates all phenomena of the world. We take this
for granted because it is so obvious.
Ultimately it is the primordial relationship of pair-up which constitutes the
source of individuality, change and transformation,
and finally, our cognition of world phenomena. For instance,
logically in order for A to change into B both A and
B must be identifiable and individualized entities;
however, if either A or B is an absolute entity or all
by itself, then, A can not change into B. Therefore,
A and B are separate and yet in unity. A and B may vary
in their structures and identities but the twofold primordial
relationship does not vary. In our cognition of the
world the first fundamental step is to classify observable
objects into similar categories so that if we know A
we will be able to know B through A and so on. If A
is similar to B, then, B is not different from A. Therefore,
it is the twofold primordial relationship of similarity
and difference which provides the bases for classification,
or taxonomy, and thus for cognition. Without the twofold
relationships of unity in separation and similarity
in difference classification and cognition would be
impossible. Quantum mechanics reveals that the union
of a positive and a negative force or energy generates
a new object or a new phenomenon. The difference between
the positive and negative aspects is only a sign. Even
scientists do not know why this is true. Particles have
a property called spin. We can measure for spin in any
direction but cannot meaaure in two or more direction.
on any one given particle. Any measurement will yield
exactly one "up" (+) or "down" (-).
The spins are correlated.[15] While the identity of
an individual structure or object may vary, the relationship
between one individual and another in terms of their
union (of positive and negative aspects) to generate
a new individual will not vary. The relationship of
wife and husband will remain regardless of who the husband
and wife may be. This is how Kung Yinda defined the
concept of "yi" as having two meanings: changing
and unchanging (yi, yi ming er han erh yi). Change and
non-change are contradictory [contradistinctive] yet
complementary.
Recent studies in life science have further enlightened us through its discovery
of simplicity in diversity for the continuous renewal
of the evolutionary process. Research in the morphogenetic
field reveals how an initial assembly of identical cells
can give rise to differentiated cells that are spacially
organized and stabilized. In embryology there exist
physiological gradients within eggs during their development.
The difference in gradients provides the basis for difference
in gene positional information which enables the DNA
double helix to assume the way in which those forms
can recognize and interact with their cognate binding
proteins and enzymes. This essential information for
differentiation implies the existence of a coordinated
system within which cell position is specified and induces
their differentiation in neighboring cells.[16] This
discovery that the gradients in position can effect
change and transformation, would seem to shed light
on the contention in the hexagrams of the place value
to interpret through the position of lines the anticipated
course of development of events. The line 3 and the
line 5 in a hexagram have similar functions but are
in different positions. Therefore, their results would
be different. The line 5 occupies the most influential
and advantageous place according to the values of being
in the center and, therefore, strong. It is the place
of honor. This place of value in the functional sense
reminds us of the Chinese saying: There is not so much
apprehension about who has the title to govern but who
is in the position to take charge (bu pa guan, zhi pa
guan). For the sake of safety the driver of a car in
which the president of a country is riding is more critically
important than the president.
Since science does not fully explain why gradients exist in the relationships
of the DNA helix, the primordial relationship between
and within the primordial pair which prompts change
and transformation of world phenomena seems to have
meaning. Scientists, have learned that a profound difference
can result from the half integer spin of particles between
Fermion and Boson. Several protons are supposedly arranged
in close proximity to one another, the strong binding
force among them (Boson.) is much greater than the repulsing
weak force (Fermion). If one replaces each proton with
a neutron, the force remains unchanged. [17]
2. The Twofold Polar Relationship of Complementarity in Contradiction
In his recounting of the race to decipher the structure and the functions of
DNA, 25 years old James D. Watson pointed out the reason
why the venerable Linus Pauling fell behind him in this
discovery because of Pauling's inability to perceive
the contradiction that "A" and "Not A"
can accommodate each other in the manner of complementarity
between two opposite fundamental nucleotides but Watson,
in light of quantum mechanics, reasoned that the basic
DNA is helical and consists of two intertwining helices;
these are not identical but complementary so that if
one helix contains a quanine base the other helix contains
a thymine base. This pairing-up of DNA in a double helical
structure meant that if the base sequence of the two
intertwined chains were complementary to each other
but yet must run in opposite directions.[18] We see
emerging now a general agreement that the functional
distinction between the continuous and discontinuous
DNA replication is accomplished by a dimeric DNA polymerase.
The relationship of donor and recipient, lagging and
leading, affects locality and mismatches.[19]
What is amazing is that the fundamental principle upon which the hexagram sequence
is formulated is the same as the DNA sequence, though
it is not in one-to-one correspondence. By virtue of
the fact that the signs which symbolize the primordial
pair, Yin and Yang, with the broken line and the unbroken
line reflect their contradictory nature. Yet it is the
complementarity of Yin and Yang in their union that
generates a myriad of things and events ad infinitum.
The apparent order displayed in the hexagram sequence
is that any odd numbered hexagram and its even numbered
successor form a pair of opposites. The inverse of a
hexagram is found by turning the hexagrams upside down,
resulting in a pair in inverse. For instance, hexagram
Heng (continuity or perseverance) is followed by Tung's
retreat. Whether or not the principle of complementarity
in contradiction constitutes the ultimate truth of reality,
no one can be certain. Niels Bohr, one of the foremost
founders of quantum mechanics, proclaimed this as the
fundamental principal of quantum phenomena. Bohr considered
this power of explanation to encompass not only physics
but also chemistry, biology, psychology, sociology,
and even economics and history, or for that matter,
the whole spectrum of human existence and its manifestation
in the development of civilization. In this sense, the
attempt of the arrangers of the hexagram sequence to
realize an all inclusive interpretation and prediction
of events and things may have, philosophically if not
scientifically, anticipated the ideas of quantum mechanics.
With the advancement of science we began to realize that the mechanism which
deals with the cognition of the world, our thought processes
consists of a twofold mode of operation: synthetic and
analytic. The former deals with the dynamic and the
latter with the static; the analytic thought refers
to formal logic while the synthetic deals with dialectical
logic. Both serve as complementary methodological tools
and are not mutually exclusive. Bohr contended that
our consciousness is a stream of mental phenomena (analog),
the subject being our experiences of life, which can
not be concretized and atomized yet there exists a discontinuous
aspect (digital) from one moment to another which is
tightly coupled with every given object. These two aspects
complement each other though they are contradictory.
He noticed that this principle of complementarity in
contradiction is strikingly similar to that which William
James has contended all along.[20]
Due to the fact that economic activities comprise the most critical and also
a very large portion of our daily activities, we may
rely on some economic aspects to illustrate how the
principle of complementarity in contradiction operates
in human events. The function of the exchange of goods
and labor through a market mechanism in sustaining our
life, involves not only the twofold structure of buyer
and seller, supply and demand, as well as both the use
value and exchange value, but also the twofold relationship
of productivity and production in addition to diminishing
and increasing returns. For example, in an agrarian
and natural resource oriented economy the negative feedback
of diminishing returns dominates. The high price of
a product may lead to a drop in the price in the long
run because it stimulates production and discourages
consumption. In a highly technological and knowledge
oriented economy, the positive feedback of an increasing
return may occur to corner the market. The general trend
is that a free market often leads to monopoly and monopoly
requires a free market for its remedy. Buyer and seller,
monopoly and free market are contradictory yet they
complement each other to bring about overall economic
development in terms of the exchange of goods and services
to sustain life. In the production process the twofold
relationship between those who are in command and those
who are in obedience or between management and labor
is indispensable for attaining the objectives of production
regardless of whether the relationship is within a family
or in a social unit, etc. The only difference which
may occur, is that the potential to achieve the production
goal is tar greater in relationships of the heart than
otherwise. The relationship between command, (generally
a minor elite) and obedience (usually the ordinary majority),
is contradictory yet they are complementary from the
functional or relational point of view. This principle
can be extended to explain all phenomena in politics,
history, etc., as Bohr has contended. For that matter
the similar principle of being contradictory yet complementary
(xiang fan er xiang cheng) underlies the formulation
of the hexagram sequence, particularly in its interpretation
in the Appendices during the Han time and would
seem to anticipate the ideas of Bohr if not quantum
mechanics. Thus, indirectly there may' be some grounds
for the thread of identicalness between the DNA sequence
and the hexagram sequence via quantum phenomena from
the relational holistic point of view.
3. The Reciprocal Self Generation (of "Each Other") in a Spiral Progression
While Pauling had difficulty in visualizing the complementarity in contradiction,
Watson went beyond the boundary of classical physics
to prove this in DNA replication and pointed out the
polar relationship as the process for self-generation
in an evolutionary world. For DNA replication in Watson's
words requires the formation of a comple-mentary (negative)
image whose shape was related to the original (positive)
surface like that of a lock to a key. The complementary
negative image would then function as the mold (template)
for the synthesis of a new positive image.[21] Subsequent
studies have shown that in a typical gene, a segment
of DNA encodes a protein that has a specific sequence
of the base pair that makes up that segment of DNA and
corresponds to the specific sequence of the amino acids
that make up the protein. Decoding a gene requires first
that it be transcribed into a molecule of the messenger
RNA (,RNA) and then that RNA molecule be translated
into protein. These proteins are activators (helpers)
or repressors, (hinderers) depending on whether they
increase or decrease transcription. Genes are turned
"on" or "off" by enzymes called
RNA polymerases which transcribe genes into DNA.[22]
The presence of two DNA bound repressors affects transcription
in two ways: one negative, and one positive. The advancement
of this technology has reinforced the claims of psychophysicists
and has even prompted the contention of socio-biologists
who see human behavior in part as an adaption encoded
in the genes. The opening of this new vista in our attempt
to study human events has shed new light on the approach
to the I-Ching with its holistic endeavor.
Whether or not the symbolic description of the primordial pair, Yin and Yang
can be seen as a "template" of each other,
as with DNA, is a perplexing question in view of its
antiquity and ontological nature. There is no doubt
that the predictability of the hexagram sequence is
incompatible with the high fidelity of DNA replication.
However, the phenomenon of the reciprocal generation
of each other between the pair as shown in the DNA replication
would seem also to support the principles which underlie
the formulation of the hexagram sequence, particularly
if we take its graphic presentation in the Tai Ji
Tu by Zhou Dunyi as the starting point.
First, both sequences reveal an orderly non-linear spiral progression, knowable
except in their predictability. However, the difference
in predictability, between the two is a matter of degree
rather than of kind. For the value of predictability
lies in the degree of approximation. The recent discovery
of the principle of uncertainty ascertained through
quantum mechanics has postulated that in the process
of change the phenomena of certainty, order, symmetry,
determinant, and predictability go hand in hand with
uncertainty, disorder or chaos, indeterminant, and chance
or unpredictability. The enduring influence of the hexagram
sequence is derived not so much from its function of
fortune telling, but rather from its philosophical implications
notably among thinking people that have shaped the Confucian-Taoist
world-outlook towards life. In its non-linear mode of
progression, regularity in human events is seen as the
way to hold a harmonious balance in steering the course
within and among emergent situations. The fifth line
in a hexagram indicates regularity as being advantageous
due to its central position within the twofold structure.
Yet its regularity is subject to the intelligent choice
of an actor (li jian da ren). For example, in the hexagram
3, Tong, it says that movement in the midst of peril
gives rise to great progress and success through firm
correctness; the feeling that rest and peace has been
secured, should not be indulged. This certainly indicated
neither a linear progression nor a mechanistic determinant.
But history bears witness to the fact that it was over-indulgence
in luxury and comfort together with the abuse and corruption
of power which decimated the conquering Mongols and
not the battlefields. Alexander the Great died in the
arms of Persian princess.
Secondly, both sequences display a process of generating each other in their
interactions, though they are not precisely in correspondence.
The DNA replication starts with the separation of its
two identical chains. One chain serves as the template
for the synthesis of the other chain. The two new strands
of the offsprings are made on the two parental templates.
The essential task of gene replication comes from the
requirement that each base in the newly synthesized
chain is always composed of hydrogen bonds: an adenine-thyinine
pair held together by two hydrogen bonds was identical
in shape to a guanine--cytosine pair also held together
by at least two hydrogen bonds.[23] The DNA replication
stands out as the consequence of a double-helical structure
of the base sequences complementary to each other. Given
the base sequence of one chain, that of its partner
was automatically determined.
Comparing this process of replication in DNA with that which "Tai Ji Tu"
presented in hexagram progression such as that: (1)
when Yang reaches its zenith Yin emerges and vice versa;
(2) within the sphere of Yang there is an element of
Yin and vice versa; (3) when Yang waxes Yin wanes and
vice versa, the resemblance between the DNA sequence
and the hexagram sequence in the principle of reciprocal
generation and identity of the base pairs throughout
the process of change would seem apparent. The discovery
of self-similarity in fractals tells us that the output
of one iteration of Yin and Yang is the input of the
next. The interation process carried out in the DNA
sequence follows the same pattern as the hexagram sequence.
However, the chart of the DNA structure and the chart
of Tai Ji Tu appear radically different. The
DNA hydrogen bond served as both an intermediary to
hold the nucleotides chains together and the catalyst
protein to set it in motion. The graphic presentation
of the Tai Ji Tu stressed the polarity of Yin
and Yang in the most simple form and nothing more.
Again the difference between the two seems to reflect a philosophical approach
in Tai Ji Tu against the scientific approach
to DNA in dealing with the same reality of the twofold
structure in the self-progression process. Since topology
and fractal geometry indicate that a square can equal
a circle and the decimal system can be converted to
graphics, the difference between the charts of DNA and
Tai Ji Tu would seem superficial. In this sense
the square arrangement of hexagrams ascribed to King
Wen and the circle arrangement ascribed to Fu Xi formed
the same sequence. Therefore, the identity of mathematical
sequences between DNA and hexagrams remains standing.
But the philosophical value of the Tai Ji Tu even provided
some explanations of ultimate reality which science
is, so far, unable to address. The most simple question
of how the first catalyst originated continues to challenge
science. In 1989 Thomas R. Cech and his associate wan
the Nobel prize for their discovery that certain types
of RNA could act as their own enzymes. But their success
was secured with a great deal of human manipulation
in the laboratory. Could RNA have initiated itself at
the beginning of the universe? No one is sure.[24] This
reminds us that Thomas Aquinas asked the world: "Who
set the world in motion first?" Since 1982 the
"immortality" of genes began to hold true
and gene profiles now have become a booming business.
Not only is the reforming of the Darwinian theory of
evolution necessary, but also can the justification
of I-Ching granting a unique position to human
beings in the unity of heaven, earth and man in a trigram
be dismissed as nonsense?
Thirdly, both sequences display the principle that one helps is to generate the
other and in turn, regenerates itself. In this process
of reciprocal generation one or the other participant
is involved in its own negation. Microscopically in
particle physics every particle has an anti-particle
with which it can annihilate itself in order to be synthesized
into a new self. The study on genome imprinting shows
that when individuals produce sperm and eggs, the old
imprints are erased and new ones specific to the sex
of the individual are imposed on all the gamete chromosomes.
Philosophically, the occurrence of erasure in gene imprinting
can be seen as the negation of self or the extinction
of ego. In genome imprinting, the parents may contribute
identical genes to their offspring, but if the genes
have been differently imprinted they will not have identical
effects. Sex specific gene imprinting may influence
normal development and trigger diseases. The mismatching
of genes may be abnormal occurrence. Otherwise, human
beings would have no health problems.[25]
Not only do Mendelian genetics and the Darwinian theory of Evolution need revision
due to these recent discoveries, but also these events
have reinforced the concept of the non-permanence of
ego which was advocated by Buddhists, echoed by Hermann
Weyl[26] and employed by Confucians for their justification
of martyrdom. This erasure of parental imprinting ensures
the gene profile which otherwise would have no meaning.
Obviously the offsprings have inherited their parental
traits in order to continue the genetic line and heritage
but in the meantime, have their own uniqueness because
genes define the unique characteristics of every living
being. This process in a sense, reflects the twofold
structure of development in continuity. Following the
same line of reasoning, the hexagram sequence would
parallel this phenomenon. For example, hexagrams Qian
and Kun are derived from their parents Yin and Yang
but the meanings of the two hexagrams are no longer
the same even though they retain their primordial lines,
etc.
Macroscopically every thing or event identifiable in this world must have its
boundary in order to be individualized. In this sense
boundary means limits. Every thing or event in this
world has its own limit. Once the development of a thing
or event reaches its limit it is turned around to become
its opposite. Thus birth is the limit of death and vice
versa. The events of death and birth make room for each
other to go on indefinitely. The food chain may serve
as another example of self-negation which generates
the other. A fattened pig signals a slaughtering. Recent
discoveries in science are increasingly noticed in which
order and disorder (chaos) go hand in hand without one
excluding the other in the process of change. The Xu
Gua of Appendix X in the Zhouyi Zhengyi
(Correct Interpretation of the I-Ching of the Zhou
Dynasty) clearly stated this principle of negation
in the formulation of the hexagram sequence. The advancements
in computer science would seem to reinforce this principle.
The computer seems especially vulnerable to chaos. But
it is through computer usage that the theory of chaos
gained strength due to the tact that chaos tends to
follow certain very stable patterns.[27] Fractal geometry
has shown that similarity is not restricted. The infinite
and finite in a polygonal spiral are not precise.
(5) A Mathematical Description of the Twofold Polar Relationship of the World:
O:1Ñ Unity of DNA and Hexagram Sequences
Assuming that the world phenomena of being and becoming in things and events
can be conceived of as unity in diversity in a twofold
relationship of complementarity in contradiction then,
mathematics, the conceptual crystallization of our cognition
of the world, would be able to describe it in the most
simple and easily understood mathematical language.
If we are part of this universe, then, from a relational
holistic point of view, the ultimate material base of
our life, the genes and their linkage with the world
would have a twofold mathematical formula. The binary
mathematical formula of 0:1 would be appropriate to
describe the relationships of the world and could compute
the possible course of world development. Martin Gardner
observed this. Therefore, the power of 2 turns up everywhere
in mathematical and physical structures and the arrangers
of hexagram sequences apply it to almost everything
from crystal structures to the solar system and the
cosmos. In a momentary flash of intuition Z. D. Sung
perceived the way to generate the eight trigrams at
the corner of a cube. With 0:1 indicating the distance
from the corner of origin for each coordinate direction,
he showed that the eight members correspond to the eight
trigrams with complementary trigrams at diametrically
opposite corner of the cube.[28] As a consequence Sung
divided the cube into 64 smaller cubes, linked the 64
hexagrams with six-digit binary numbers at the six-dimensional
hypercubes and identified them to the 64 "modes"
of the classical syllogism. The major premise, the minor
premise and the conclusion of a syllogism can each be
of four different forms giving 64 possible "modes."
This is what, according to Gardner (p.110), C. Howard
Hinton's work on the Fourth Dimension, reveals
as well as his discovery that cells correspond to traditionally
valid syllogisms that the anomalous syllogism is valid
after all if one applies a more liberal interpretation
to the syllogism.
Not only has the binary basis of computer operation inspired new thinking in
mathematics leading to a rekindled interest in I-Ching
but also it has resuscitated the mathematical value
of the simplicity in the Yin-Yang iteration process.
Since quantum mechanics has provided the means to see
the microscopic world beyond our ordinary sense of perception
and fractal geometry has discovered the process of iteration
in which the phenomenon of extreme simplicity can evolve
into the phe-nomenon of extreme complexity, no one now
can ignore the hexagram sequence as pure fortune telling
or magic nor can one dismiss it as too obvious. A major
and surprising impact of fractal geometry is that within
the presence of complex patterns there Is a good chance
that a very simple process may be responsible for it.[29]
The compatibility between the Cantor set and the evolving
hexagram sequence can be easily discerned. It is at
the microscopic level that the DNA sequence parallels
the hexagram sequence. Prigogine argues that the complexity
which involves the interacting cells in a myriad of
organisms can be appreciated if one realizes that even
the most advanced organism issues from a single cell,
the zygote.[30] Gregory J. Chaitin of IBM concedes that
the regulatory genes of a developing embryo are in effect
a computer program for constructing an organism.[31]
Therefore because both the DNA and the hexagram sequences
have the same twofold base of binary structure, their
similarity is natural.
After all, Claude E. Shannon's contribution to the world was his successful application
of the principle of simplicity in complexity by switching
on and of f the electric current in computer operations.
Basically the unit of information in bits contains an
extremely simple choice between two opposite yet complementary
numbers: yes or no, absence or presence. Billions of
calculations can be made if the object to be counted
is measurable or quantifiable. On a single microchip
millions of components can be stored and retrieved in
a billionth of a second. Computers analyze the most
essential molecular phenomena such as DNA which is mathematically
elementary and involves straightforward algorithms because
of the microscopic simplicity of nature.
The problem which the hexagram sequence has is its texts that, as noted by Gardner,
are so ambiguous that no matter what hexagram is selected
and matched, it is always possible to interpret it in
such a way that it seems to apply to the question. For
basically in I-Ching any relevant event which
may happen is not causally linked, but it is causally
related in the sense of being a part of the vast cosmic
pattern. The essence of I-Ching's approach is
to compute the future of an event in question and not
to rely on prayer (ji shu er zhi lai). Chinese
fortune-telling is described as calculating one's decree
from heaven (suan ming or suan gua). Though the fidelity
of DNA replication is assured to a large extent, the
reliable prediction of its protein structure from the
amino acid sequence is still beyond both current mathematical
methods and computer science.[32] Both the DNA and hexagram
sequences represent the attempt of human beings to understand
nature as well as human events so that we may act according
to the possibility that we may manipulate events or
things to suit our purposes whatever they may be.
Ultimately, mathematics is the creation of the human mind. Yet ultimate singularity
is beyond the ability of any mathematical language to
describe. Even the proposal for the term "quantum
fuzz" to explain the Big Bang as well as many others,
is merely a proposal. The intro-duction of the concept
of zero, with its relationship to concrete reality designed
as "1" (one) is perhaps a necessary conceptual
medium to convey the being and becoming of world phenomena
in the most simple terms. In our cognition of the world
it has turned out that the solution to certain mathematical
problems requires the use of assumptions essentially
transcending arithmetics.[33] With the assumption of
zero to reflect the wholeness of the universe not only
can the oneness of the universe be ascribed to each
and every identifiable object in the world of reality
without losing itself, but also the interation process
can be described for the continuous generation of things
or events in this changing world. For zero itself does
not change. It can not be created or destroyed. It does
not have contradiction in a twofold positiveness or
negativeness. Yet zero is not nothing. It can operate
as the "hidden correlation" in what Bernhard
Riemann referred to as the Zeta function which fits
any smooth function locally yet is globally chaotic.[34]
With this the statement of "wu ji er tai ji"
in Tai Ji Tu can be explained. Numbers begin
with zero and are open ended to add on more zeros. This
may be why the basic Cantor set corresponds to the hexagram
and DNA sequences, because it is an infinite set of
points in the unit interval O:1, which can be interpreted
as a set of certain numbers. A binary tree can be constructed
for the Cantor set which explains the meaning of the
equation in mathematical functions.[35] Therefore, with
zero everything can be done; without it nothing can
be done. Thus have declared the inventors of the theory
of zero both in India and in China.
(5) Conclusion
Since the scientific DNA replication sequence reveals a binary mathematical structure
and the non-scientific hexagram progression sequence
also reveals the same, the value of the latter can not
be easily ignored as mystical or magical as some have
recently attempted. Despite the differences in the contents
between the two sequences the similarity between them
would seem to strengthen the shared ontological ground
as follows: (1) the diverse and complex world phenomena
derive from a singular unity, be it named as quantum
fuzz, quantum chaos, primordial soup, hung tong, or
wuji etc.; (2) from this unity the world evolves in
a twofold complementary and contradictory process; (3)
in this twofold structure every identifiable entity
contains a twofold polar relationship of being and becoming,
e.g. stable-dynamic, symmetry-asymmetry, etc.; (4) this
process exhibits an hierarchical order in simple iteration;
(5) in the polar relationship one generates the other
which in turn, generates itself in a recycling spiral;
(6) the meta-sign zero best conveys this order; and
(7) for practical applications the reformulation of
the hexagram sequence may be necessary in order to gain
viability in the contemporary world.
In view of the recent claim of the genetic algorithms, modeled after the DNA
sequence in the binary mathematical structure of 64
bits (strings) or possibilities for computer programming
to solve problems both physical and social, a question
naturally arises: Would this new development in computer
science lend credibility to the ancient attempt of the
hexagram sequence?[36]
With the contention that the genetic algorithms for computer application rely
on the principle of implicit parallelism for simulating
the events in question to project their evolving course
of development one may wonder if the concept of "implicit
parallelism" could coincide with the concept of
correlation upon which the hexagram sequence rests.
Notes
[1] Scientific American, April 1990. pp. 56-65.
[2] Heinz-Otto Peitgen,Fractals (New York: 1992), Springer-Verlag, p.20.
[3] Peitgen, p.20; Michael F. Barnsley, Fractals Everywhere, Academic Press,
N.Y. 1988.
[4] Robert Kaniel, The Man Who Knew Infinity (New York: MacMillan, 1991), p.
65-66; G. H. Hardy, ed., Collected Papers of Srinivasa
Ramanujan (1887-lÕ920) (New York: Chelsea, 1962), p.22.
[5] J. H. White, "An Introduction to the Geometry and Topology of DNA Structure,"
Michael S. Waterman, ed., Mathematical Methods for DNA
Sequences (Boca Raton, FL , CRC Press, 1989), p. 255.
[6] Mullis, p.64.
[7] Michael J. Long, Fundamentals of Elementary Particle Physics (New York: McGrew
Mill, 1973), p.170.
[8] Biology Today, San Rrancisco, California: CRM Books,1972), "Preface."
[9] Bernhard Rensch, Evolution Above The Specie Level (New York: Columbia University,
1960), p. 355.
[10] Life Itself (New York: Simon-Schuster, 1981), p.88.
[11] Werner Heisenberg, "The Representation of Nature in Contemporary Physics,"
Daedalus, Vol. 87, No. 3, 1958, pp. 95-108.
[12] L. Pearce Williams, "Andre-Marie Ampere" Scientific American,
Jan. 1989, p.27.
[13] Charles C. Richardson, et al, Molecular Mechanism in DNA Replication and
Recombinatio (New York: Wiley, 1990), "Preface."
[14] F. Wilczek, "The Cosmic Asymmetry between Matter and Anti-matter,"
Scientific American, Dec. 1980, pp. 82-102.
[15] Charles H. Bennett, "Quantum Cryptography," Scientific American,
May, 1989, p. 28.
[16] G. Nicoles & I. Prigogine, Self-Organization in Non-Equilibrium Systems
(New York: Wiley, 1977), p. 411.
[17] Gerald Holton, "Gauge Theories of the Force between Elementary Particles,"
Scientific American, June 1980.
[18] James D. Watson, The Double Helix (New York:New American Library, 1969),
pp. 125.
[19] Charles C. Richardson, et al, Molecular Mechanism in DNA Replication and
Recombination (New York: Wiley, 1990), "Preface."
[20] Gerald Holton, Thematic Origins of Scientific Thought (Cambridge: Harvard
University Press, 1973), p. 139.
[21] The Double Helix, p. 84-141.
[22] Mark Plashrine, "How Gene Activators Work? Scientific American, Jan.
1989, p.41.
[23] Watson, p. 125.
[24] John Morgan, "In the Beginning", Scientific America, Feb.1991,
p.119.
[25] Carmen Sapienza, "Parental Imprinting of Genes," Scientific American,
Oct. 1990, pp. 52-53.
[26] Philosophy of Mathematics and Natural Science (Princeton, 1949), p.75.
[27] Fractals, p.62.
[28] Martin Gardner, "Mathematics of the I-Ching," Scientific American,
Jan. 1974, pp. 108-113; A. K. Dewdney, "Computer
Recreations" Scientific American, Nov. 1984, pp.
19-28.
[29] Fractals, p. 20.
[30] Prigogine, p. 409.
[31] "Randomness in Arithmetic" Scientific American, July, 1988, p.85.
[32] Michael S. Waterman, et al, Mathematical Methods for DNA Sequence, Boca
Raton, FL.: CRC Press, 1989), "Preface."
[33] Kurt Godels, in Gregory J. Chaitin, "Randomness in Arithmetic",
Scientific American, July, 1988, p. 83.
[34] Martin C. Gutzwiller, "Quantum Chaos," Scientific American, Jan.
1992, p.84.
[35] Fractals, p.79.
[36] John H. Holland, "Genetic Algorithms," Scientific American, July
1992, pp. 66-72; Rick L. Riolo, "Survival of the
Fittest Bits," Scientific American, July 1992,
pp. 114-116.
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