The essential part in our world isconstituted by the Universe
raised at the rank of consciousness. The depth of the world, that of
informatter and of energymatter,
cannot have reason, cannot create models, cannot be conscious,
cannot have knowledge. It is true that we had to give informatter
certain properties providing ourselves with a deep landmark, thus
explaining the qualities unexplainable through today's forms of
material existence. However this raises the question of a Physics
and a Mathematics of the intro-openness, to determine the relationship
between structured and unstructured, to study their interdependence,
towards a future power of acting in the unstructured zone.
4. The importance of the notion of
intro-open system is that it allows for the rational tackling
of certain phenomena that until now have not made the object
of science. One has to see to what extend and between what limits
the intro-open systems can be fitted within todays systems theory
which in principle is a theory of open systems. Reducing the
intro-openness to an openness could be, in theory, a first approximation
that could explain certain behaviors of the man-society-universe
systems; however, in the long run, one should reach higher approximations
dealing with the foundation of the problem, i.e. the relationship
between unstructured and structured.
The open system implies having some information
about what is to be found outside the system, about the structures
outside it. The intro-openness of the system leads us towards
an unstructured zone, an exchange zone in which certain informational
events can take place, without excluding the possibility of
certain psychological processes too. The informatter is seen as
a form of material existence: when structured, it is attached
to the Universe; and when unstructured, it determines the intro-openness.
Then one should constitute a science of information for
all the levels of the material world. The information, though
widely used as a concept, does not come into the description of
a large number of the processes we currently consider. The problem
of real structures and of the system cannot be separated from
the notion of information. Without information no structuring
is possible, not the birth of quantum world, nor of macroscopic
world, of man, of society, of Universe. It is no exaggeration
to look for the informational aspect in all the manifestations
that surround us, including the nature of the forces in the
Universe18
and in society19.
The set, as an abstract notion, is a mathematical object, but
it needs a subject to bear this notion; for the subject the set
is a piece of information. Can we detach the notion of "set"
from the nature of our mental process of abstractization ? Obviously
not, due to the very paradoxes to which set theory arrives and
which are reflecting the particularities of human knowledge. Any
set supposes an observer, a man. While the paradoxes of the set
theory (and other similar ones) can become the object of a true
new philosophical experiment implying the whole human reality.
Since the model of the existence is imagined
as a ring, or a multitude of rings, does it then mean that the
existence, without being systematical, has a cybernetic characteristic?
The notion of cybernetics refers to the process
of leadership,with its decisions and feed-back loops. But in no
case can we put up the problem of leadership in the existence.
It takes place by itself. But it takes place in cycles, an universe
being only a link in a cycle. There is no consciousness, no leadership,
to supervise these cycles. The universe is not a control loop,
but it can determine an act of will and creation thus structuring
the informatter in a way to give birth to a new universe, to a
new essential experiment, even by its own disappearance. The consciousness
of a universe like ours represents in itself the consciousness
of the existence. It takes place in the time of a universe. While
us, the vanishing elements, are at the same time everlasting
since we are part of the existence. And if the individual consciousness
is to disappear, and the social one too, then they can contribute
via the construction and the overcoming of nature, via creation,
to the most powerful form of manifestation of the existence itself.
The ring of the existence is full of poetry and of prose, of absurd
and of logic.
5. The model proposed in this book
is in close connection with the relationship between man and automaton.
Then the very sharp question arises of following the automaton
within the man, and we should prove that in all the brain
activities, in behavior and in spiritual life he remains an automaton.
Thus we have to try to embed as much as possible of the man in
the automaton, to see were we have difficulties following this
embedment and how could we overcome them.
The most advanced artificial automata known
today are the electronic computers. The
automaton20 is a discrete
dynamic system, meaning a system with discrete states, changing
from one state to the other at a certain pace, t1,
t2,... given from outside or depending on the functioning of the
system itself. There are non-autonomous automata, where
the state of the system changes only at certain commands (signals)
arriving from outside. A communication line can be interpreted
as such an automaton. The automata with autonomy, or internal
freedoms, have their internal states changing by themselves.
All the automata that we meet in practice or that we deal with
at practical level, including the nervous system considered as
a neuronic machine, are systems with a finite number of states.The systems with
finite states21,
currently known as finite automata, are not only some physical
structures like the electronic computer and the nervous systems,
but also computer programs, algorithms, and even formal
logics22.
Hao Wang proved23 using the electronic computer the nearly 400 theorems
of the restricted predicated logics of Principia
mathematica24.
Wang's experiment proved that it is possible to automatically
establish all the theorems of a logics manual. One should make
a difference between numerical calculations that use approximations
and proofs that use propositions. The formalization of logics,
and mathematical logics in itself, suggest according to Wang that
"the time has come for a new branch of applied logics, called
<inferential analysis> to appear and to treat proofs in
the same way numerical analysis treats
calculus"25.
Mechanized mathematics, in fact automated, raises the computer-automaton
to a level similar to the human mind, or it brings the latter
to the level of the automaton. In certain circumstances a proof
may require such a large number of steps that it becomes impossible
for the man to do it. This is what Wang calls a "deep"
theorem: a short formula that requires long demonstrations is
a deep theorem. A short expression that is equivalent to some
very long other expressions is a "rich"
concept26. The
newspapers told us that by using 12,000 computer hours one proved
the four-color theorem. Practical experience taught us that four
colors are sufficient to do a map, but mathematics could only
prove down to five colors due to the large amount of logical procedures
to be performed. K. Appel, W. Haken et. al. of Illinois University
managed to make a program to prove the theorem using an electronic
computer. Checking this proof, assuming that it is valid, needs
doing it again on a different computer, with a new version of
the program, since it is not possible to check it using human
means. We see that the social checking of the truth requires new
methods that imply an automaton (the first program) checking an
other automaton (the alternative program).
In the end one should look for an automated
variant of the whole mathematics (and with all that can be potentially
abstracted from this) which does not appear as an extremely difficult
task for the existent mathematical background of the human intellect.
A mathematical branch can nowadays be derived from another, the
geometry is itself reduced to algebra and logics. With certain
starting points, giving certain logics of the machine, one
can demonstrate an extensive mathematics:
"The natural and
the real numbers theories can be derived from set theory. If the
theorems are mechanically generated from set theory, then any
set of isomorphic theorems with the axioms of the real (or natural)
numbers, will determine expressions that can be taken as definitions
for the basic concepts of the real (or natural) numbers theory.
In such a case one can state that the machine can discover
definitions"27
Intro-Open Systems
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