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\title{A CYBERNETIC METAPHYSICS}

\author{Valentin Turchin}
{The City College, the City University of New York}

\node{Metaphysics}
% part-of: Philosophy

Metaphysics is supposed to answer the question ``What is the nature of
things?'', or something like that. Before trying to answer this question we
should ask a counterquestion: ``What is meant by the {\sl nature} in this
context?'' Indeed, we want to understand clearly the meaning of the
question before getting involved in answering. In this paper we interpret
the metaphysical quest from the vantage point of what we denote as {\sl the
cybernetic epistemology} ([Tur90]), its main principle being that the
meaning of language constructions is defined by the extent to which it can
be used as a model of a part of the world, or as a means of constructing
such models. Then we present the beginnings of a metaphysics inspired by
the cybernetical approach to the world, which is seen in terms of
functioning, organization and control, in contrast with the view of the
physicist, who features space, time and matter. Starting from these
premises we come to a philosophy where ontological priority belongs to {\sl
action}.

A metalanguage is still a language, and a metatheory a theory.
Metamathematics is a branch of mathematics. Is metaphysics a branch of
physics?

`Meta' in Greek means over, and -- since when you jump over something you
find yourself behind or after it -- it is also understood as behind and
after. The word `metaphysics' is said to originate from the mere fact that
the corresponding part of Aristotle's work was positioned right after the
part called ``physics''. But it is not unlikely that the term won a ready
acceptance as denoting this part of philosophy because it conveyed the
purpose of metaphysics, which is to reach beyond the nature ({\sl`physis'})
as we perceive it, and to discover the ``true nature'' of things, their
ultimate essence and the reason for being. Such a theory would obviously be
priceless for judging and constructing more specific physical theories.

%This use of the prefix `meta' is somewhat different %from the way we
understand it in the 20-th century. A metatheory is a %theory about another
theory, which considered as an object of %knowledge: how true it is, how it
comes into being, how it is %used, how it can be improved, etc. A
metaphysician, in contrast, %would understand his knowledge as a knowledge
about the world, %like that of a physicist (scientist, generally), and not
as a %knowledge about the scientific theories (which is the realm of
%epistemology).

If so, metaphysics should take as honorable a place in physics as
metamathematics in mathematics. But this is very far from being the case.
It would be more accurate to describe the situation as exactly opposite.
Popularly (and primarily by the ``working masses'' of scientists and
engineers), metaphysics is considered as almost opposite to physics, and
utterly useless for it (if not for any reasonable purpose). We shall argue
below that this attitude is a hangover from the long outdated forms of
empiricism and positivism.

A detractor of metaphysics would say that its propositions are mostly
unverifiable, if intelligible at all, so it is hardly possible to assign
any meaning to them. Thales taught that everything is water. The
Pythagoreans taught that everything is number. Hegel taught that everything
is a manifestation of the Absolute Spirit. And for Schopenhauer the world
is will and representation. All this has nothing to do with science, which
allows only precise and verifiable statements.

But what does it mean for a statement to be verifiable? According to the
naive view on the nature of our language, known as the
reflection-correspondence theory, language, like a mirror, creates certain
images, reflections of the things around us. But some images can also be
arbitrarily fabricated. A statement is true if it corresponds to reality,
otherwise it is false. To verify a statement we simply compare it with
reality. Any expression of our language which cannot be immediately
interpreted in terms of observable facts, is meaningless and misleading.

This viewpoint
in its extreme form, according to which all unobservables must be banned
from science, was developed by the early nineteenth-century positivism
(August Comte). Such a view, however, is unacceptable for science. Even
force in Newton's mechanics becomes suspect in this philosophy, because we
can neither see nor touch it; we only conclude that it exists by observing
the movements of material bodies. Electromagnetic field has still less of
the naively understood reality. And the situation with the wave function in
quantum mechanics is simply disastrous.

The history of the Western philosophy is, to a considerable extent, the
history of a struggle against the reflection-correspondence theory.
We now consider language as a material to create models of reality.
Language works as a whole, and should be evaluated as a whole. It is not
necessary that every specific part of it should be put in a direct and
simple correspondence with the observable reality. A language and a theory
expressed in it are good if they help make correct predictions about the
world around us.

Human language is a multilevel system.
On the lower levels, which are close to
our sensual perception, our notions are almost in one-to-one correspondence
with some conspicuous elements of perception. In our theories we construct
higher levels of language. The concepts of the higher levels do not replace
those of the lower levels, as they should if the elements of the language
reflected things ``as they really are'', but constitute a new linguistic
level, a superstructure over the lower levels.

Top-level theories of science are not simply deduced from observable facts;
they are constructed by creative acts, and their usefulness can be
demonstrated only afterwards. Einstein wrote that foundations of physics
are not directly extr acted from experience, but come only by ``free
fantasy''.

This ``free fantasy'' is the metaphysician's. Democritus, and then Epicurus
and Lucretius taught that the world is an empty space with atoms moving
around in it. In due time this concept gave birth to classical mechanics
and physics, which is, unquestionably, science. At the time of its origin,
however, it was as pure a metaphysics as it could be. The existence of
atoms was no more verifiable than that of the Absolute Spirit. Physics
started as metaphysics.

When we understand language as a hierarchical model of reality, i.e. a
device which produces predictions, and not as a true static picture of the
world, the claim made by metaphysics is read differently. To say that the
real nature of the world is such and such means to propose the construction
of a model of the world along such and such lines. Metaphysics creates a
linguistic structure -- call it a logical structure, or a conceptual
framework -- to serve as a basis for further refinements. Metaphysics is
the beginning of physics; it provides fetuses for future theories. Even
though a mature physical theory fastidiously distinguishes itself from
metaphysics by formalizing its basic notions and introducing verifiable
criteria, metaphysics, in a very important sense, {\sl is physics}.

When Thales said
that all is water, he did not mean that quite literally; he surely was not
that stupid. His `water' should rather be translated as `fluid', some
abstract substance which can change its form and is infinitely divisible.
The exact meaning of his teaching is then: it is possible to create a
reasonable model of the world where such a fluid is the building material.
Is not the theory of electromagnetism a refinement of this idea? As for the
Pythagoreans, the translation of the statement ``everything is number'' is
that it is possible to have a numerical model of the Universe and
everything in it. Is not the modern physics such a model?

Building intelligent machines is similar to buiding physical theories. Such
machines must be able to create and use models of the world around them.
Inescapably, we turn to our own intelligence as a model for artificial
intelligence. But our brain is a huge and very complicated system. We do
not yet know it works and cannot mimic its operation. What we usually do is
take the description of the world in terms of our natural language as the
primary reality of which our intelligent machines should construct models.
The problem is that this reality is not, actually, primary. It is a
hodge-podge of the products of brainwork which have been shaped on various
occasions and at different times in the process of biological evolution.
It is impossible to achieve good results on this basis as long as we cannot
imitate the brain processes in their totality.

What we should do is imitate the development of the brain, the way it has
evolved, and not the complex system it has become now. We must copy only
the principles, but apply them to some new fresh-made concepts implemented
in the material which we know how to work with -- unlike the actual nerve
nets of the human brain. We must take some most primary, basic elements of
the world and conctruct artificial intelligence on this basis. It is
metaphysics that can direct us toward these most primary elements, as well
as the right ways to operate with them.

The meaning of metaphysics is in its potential. I can say that Hegel's
Absolute Spirit is meaningless for me, because at the moment I do not see
any way how an exact theory can be constructed on this basis. But I cannot
say that it is meaningless, period. To say that, I would have to prove that
nobody will ever be able to translate this concept into a valid scientific
theory, and I, obviously, cannot do that.

It may take quite a time to translate metaphysics into an exact theory with
verifiable predictions, or a new series of intelligent machines. Before
this is done, metaphysics is, like any fetus, highly vulnerable. The task
of the metaphysician is hard indeed: he creates his theory in advance of
its confirmation. He works in the dark. He has to guess, to select, without
having a criterion for selection. Successes on this path are veritable
feats of human creativity.

\node{From Kant to Schopenhauer}
% part-of: Metaphysics

That sensation is the main, and maybe the only, source of our knowledge was
noticed already by the ancient Greeks. In the new time, Berkley and Hume
stressed this in a very strong manner: things {\sl are} our sensations.
But rationalists still believed that some crucial ideas are inborn and have
nothing to do with the imperfection of our sense organs.

Kant synthesized empiricism and rationalism by seeing knowledge as
organization of sensations by our mind. Space, time, and other categories
are not given us in sensation. They are our forms of perception, the way we
organize sensations. This is how {\sl synthetic judgments a priory} become
possible. They reveal the working methods of our mind which are inborn and
do not depend on sensations.

In the light of cybernetics, Kant's ideas are surprisingly modern. We say
that sensations are at the
input of our cognitive apparatus, the nervous system. This input is then
processed by a huge hierarchical system. As the signals move up in the
hierarchy, sensations become perceptions (there are no sharp boundaries, of
course).

Mach and Einstein would be, probably, impossible without Kant. They used
the Kantian principle of separating elementary facts of sensations and
organizing these facts into a conceptual scheme. Einstein's analysis went
further from the intuitive space-time picture given by classical mechanics
down to the level of separate measurements, and resulted in reorganization
of measurements into a different space, the four-dimensional space-time of
the relativity theory. This space-time is now as counterintuitive as it was
in 1905, even though we have accustomed to it. Hence what we call the
paradoxes of the relativity theory. But they do not bother us. We use a bit
less of neuronal models, and a bit more of symbolic models, that is all.

In quantum mechanics, the physicists went even further. They rejected the
idea of a material body located in the space and time as the ultimate
reality. The space-time continuum is left as a mathematical construct, and
this construct serves the purposes of relating micro-phenomena with
macro-phenomena, where it has the familiar classical interpretation. But
elementary particles lost their tangible character. In the relativity
theory, observations (measurements) at least belong to the same universe as
the basic conceptual scheme: the space-time continuum.
In quantum mechanics, on the contrary, there is a gap between micro-world
and macro-world, between what we believe {\sl to really exist}, i.e.
quantum particles and fields, and what we take as the basic observable
phenomena, which are all expressed in macroscopical concepts: space, time
and causality.

Here we face the most intriguing part of metaphysics: the concept of `real
existence'.
Cybernetic epistemology, according to which all meaningful statements are
hierarchical models of reality,
has a double effect on the concept of existence. On the one hand,
theoretical concepts, such as mechanical forces, electromagnetic and other
fields, and wave functions, acquire the same existential status as the
material things we see around us. On the other hand, quite simple and
trustworthy concepts like a heavy mass moving along a trajectory, and even
the material things themselves, the egg we eat at breakfast, become as
unstable and hazy as theoretical concepts.

One can argue that since every theory, in the last analysis explains and
organizes observable facts, which all are, and will always be, macroscopic
facts, there is simply no need in the concept of real, or ultimate,
existence.
This is formally true. But we still feel a need for our theory to give an
answer to the question of ultimate existence. What is the ultimate reality
of physics? This question is not meaningless. Its meaning is in the quest
for a theory which would start with concepts bel ieved to correspond to
that ultimate reality, and then step by step construct observables from
these ``really existing'' things. Somehow it seems that such a theory has
better chances for success. If we have a theory of that kind, and it
constructs the world from some things --- call them `ex-why-zeds' --- and
the theory is born out by experiment, then we can say that the ex-why-zeds
{\sl do really exist}, and that the world really consists of ex-why-zeds.
Ontologically, this will be as certain as when we say that the apple is in
a bowl on the basis of seeing it and touching it.

The contemporary quantum mechanics does not meet this requirement. It
starts with space-time continuum, which in no sense exists. It is only a
form of our perception. Suppose we are determined to construct a theory
which is built as required above. How should we go about the construction
of such a theory? We must go further down in the hierarchy of neuronal
concepts. Space and time must
not be put in the basis of the theory. They must be constructed and
explained in terms of really existing things. An attempt must be made to
identify the most essential, pervasive, primordial elements of experience.
Kant's metaphysics had served as the philosophical basis for the modern
theories of physics. We see now that a further movement down is required.
Thus let us turn to the development of metaphysics after Kant.

Kant introduced the concept of the thing-in-itself for that which will be
left of a thing if we take away everything that we can learn about it
through our sensations. Thus the thing-in- itself has only one property: to
exist independently of the cognizant subject. This concept is essentially
negative; Kant did not relate it to any kind or any part of human
experience. This was done by Schopenhauer. To the question `what is the
thing-in- itself?' he gave a clear and precise answer: it is {\sl will}.
The more one thinks about this answer, the more it looks as a revelation.
My will is something I know from within. It is part of my experience.
Yet it is absolutely inaccessible to anybody except myself. Any external
observer will know about myself whatever he can know through his sense
organs. Even if he can read my thoughts and intentions -- literally, by
deciphering brain signals -- he will
not perceive my will. He can conclude about the existence of my will by
analogy with his own. He can bend and crush my will through my body, he can
kill it by killing me, but he cannot in any way perceive my will. And still
my will exists. It is a thing-in-itself.

What is then therest of the world as we know it? Schopenhauer answers: a
`Vorstellung'. This word was first translated into English as an `idea',
and then a `representation'. Both translations are not very precise. In the
Russian language we have a word for it which is a literal translation of
the German `Vorstellung': `predstavleniye'. `Vorstellung' is something that
is put in front of you. It is a world picture we create ourselves -- and
put in front of us, so that to some extent it screens the real world. This
aspect of Vorstellung is not properly reflected either in `idea' or in
`representation'.

Schopenhauer's formula for all that exists is:

[Exists][Exists] the\ world = will + representation [Exists][Exists]

\eon

\node{Will}
% part-of: Metaphysics

In our thought and language we distinguish two different classes of
elements about which we say that they exist: those expressing what we know,
or think we know, and those expressing what we are striving for and intend
to do. We unite the elements of the first class to referred as {\sl
knowledge}, and the elements of the second class as {\sl will}. They are
not isolated from each other. Our goals and even our wishes depend on what
we know about our environment. Yet they are not determined by it in a
unique way. We clearly distinguish between the range of options we have and
the actual act of choosing between them. As an American philosopher
noticed, no matter how carefully you examine the schedule of trains, you
will not find there an indication as to where you want to go.

Another way to describe the relation between will and knowledge is as a
dichotomy between `I' and `not-I', or between subject and object. The
border between them is defined by the phrase `I can'. Indeed, the content
of my knowledge is independednt of my will in the sense that I cannot
change it by simply having some intentions or preferences. On the contrary,
I can change my intentions without any externally observable actions. I
call it my will. It is the essense of my `I'.

It is only {\sl my} will, i.e. the will of the subject of knowledge, that
exists as will. {\sl Its} will, and {\sl their} wills, if they exist (of
course, they do), exist only as my representations.

If Kant's view of knowledge has a clear cybernetic interpretation, then
even more so has Schopenhauer's view of the world. His formula is borne out
by the practice of cyberneticians during the last decades. We try to
understand ourselves by building cybernetic creatures and computer programs
which model intelligent behaviour. Our artificial models of intellect
consist of two parts: a device that collects, stores and processes
information; and a decision maker -- another device that keeps certain
goals and makes choices in order to reach these goals, using the
information from the first device. Thinking about ourselves in those terms
we speak about knowledge and will. It is there, and there is nothing beyond
it. \eon

\node{Freedom}
% part-of Metaphysics

The concept of will assumes the existence of {\sl freedom} to exercise the
will. Thus recognizing will as a cornerstone of being, we do the same for
freedom. For the mechanistic worldview of the nineteenth century freedom
was a misconcept, a nuisence which escaped satisfactory definition within
the scientific context. For us freedom is the very essence of the things,
and, first of all, of the human person.

However, in many minds, science is still associated with the deterministic
picture of the world, as it was in the nineteenth century. This picture,
was as follows.

Very small particles of matter
move about in virtually empty three-dimensional space. These particles act
on one another with forces which are uniquely determined by their
positioning and, possibly, velocities.The forces of interaction, in their
turn, uniquely determine, in accordance with Newton's laws, the subsequent
movement of particles. Thus each subsequent state of the world is
determined, in a unique way, by its preceding state. Determinism was an
intrinsic feature of the scientific worldview of that time. In such a world
there was no room for freedom: it was illusory. Humans, themselves merely
aggregates of particles, had as much freedom as wound-up watch mechanisms.

In the twentieth
century the scientific worldview has undergone a radical change. It has
turned out that subatomic physics cannot be understood within the framework
of the naive realism of the nineteenth century scientists. The theory of
relativity and, especially, quantum mechanics require that our worldview be
based on critical philosophy, according to which all our theories and
mental pictures of the world are only devices to organize and foresee our
experience, and not the images of the world as it ``really'' is. Thus along
with the twentieth-century's specific discoveries in the physics of the
microworld, we must regard the inevitability of critical philosophy as a
scientific discovery -- one of the greatest of the twentieth century.

We now know that the notion that the world is ``really'' space in which
small particles move along definite trajectories, is illusory: it is
contradicted by experimental facts. We also know that determinism, i.e. the
notion that in the last analysis all the events in the world must have
specific causes, is illusory too. On the contrary, freedom, which was
banned from the science of the nineteenth century as an illusion, became a
part, if not the essence, of reality.

There is genuine freedom in the world. When we observe it from the outside,
it takes the form of quantum-mechanical unpredictability; when we observe
it from w ithin, we call it our free will. We know that the reason why our
behaviour is unpredictable from the outside is that we have ultimate
freedom of choice. This freedom is the very essence of our personalities,
the treasure of our lives. It is given us as the first element of the world
we come into.

Logically, the concept of free will is
primary, impossible to derive or to explain from anything else. The concept
of necessity, including the concept of a natural law, is a derivative: we
call necessary, or predetermined, those things which cannot be changed at
will.
\eon

\node{Action}
% part-of: Metaphysics

\line{\sf \hfill Am Anfang war die Tat.} \line{ \hfill {\it Goethe}.}
\smallskip

Will is manifested in {\sl action}. Will and action are inseparable. Our
understanding of will is this: will is not a description of options the
subject has, nor is it a list of preferences of the subject -- these are
all representations; will is the quality that allows to choose between the
(possible) options and act. Action and will are two faces of essentially
the same phenomenon, and in the philosophy we are developing, action is its
perceivable part. A human subject that performs an action usually sees it
from within as an action of his will. But not every action, of course, is
an action of a human subject; we regard them as manifestations of some
other, non-human, wills. We rewrite Schopenhauer's formula as follows:

[Exists][Exists] the\ world = action + representation [Exists][Exists]

If we are looking for the ultimate undoubted reality, we must turn to
action, and not to the space-time picture of the world. For a picture is
only a picture, a representation which changes from one subject to another,
from one theory to another; while action is an irrefutable reality.

Our knowledge of anything in the world starts with sensations. Sensations
are not things. They do not have reality as things. Their reality is that
of an event, an action. Sensation is an {\sl interaction} between the
subject and the object, a physical phenomenon.

Consider the concept of action in the context of physics. According to our
present understanding of the world, all the variety of events we observe
result from elementary acts of interaction between elementary particles.
These acts constitute unquestionable reality, while both our theory, and
our intuitive picture of the world, are only representations of reality.

Furthermore, it is the physical quantity of {\sl action} that is quantized
by Plank's constant [Exists]h[Exists].
This can be seen as an indication that action should have a higher
existential status than space, time, or matter. Of course, it is not
immediately clear whether the concept of action as we understand it
intuitively and the physical quantity that has the dimension of energy by
time and called `action' are one and the same, or related at all. That the
physicists use the word `action' to denote this quantity could be a
misleading coincidence. Yet the intuitive notion of an action as
proportional to the intensity (intuitive understanding of energy) and the
time does not seem unreasonable. Furthermore, it is operators, i.e., {\sl
actions} in the space of states, that represent observable (real!) physical
quantities in quantum mechanics, and not the space-time states themselves!

Even if we reject these parallels and intuition as unsafe, it still remains
true that neither space, nor time, nor matter are characterized by a single
constant indestructible quantum, but a combination of these. Is it not
natural to take this combination as a basis for the picture of the world
--- if not for a unifying physical theory?

The purpose of metaphysics is to find in our experience the most
fundamental elements or aspects of the world. We take actions as such,
which means that in the model of the world we are constructing the lowest
level of representations consists of representations of actions.

\node{Action ontology}
% part-of: Action

Action and event are prominent features of reality, so philosophers
explored them since very long. The idea that events may be, in some sense,
more primary than space and time has been appearing, now and then. Russell,
in his treatment of time and space, takes ``as raw material'' events,

\quote{
which are to be imagined as each
occupying a finite continuous portion of space-time. It is assumed that two
events can overlap, and that no event recurs.}

The motive is to explain and justify the continuing use by the physicists
of instants in time and points in space while rejecting at the same time
Newton's conception of absolute time and space, which conferred to instants
and points a great deal of ontological primacy. Using events, Russel
defines instants as follows:

\quote{
An `instant' as I propose to define them, is a class of events having the
following two properties: (1) all the events in the class overlap; (2) no
event outside the class overlaps with every member of the class.}

Russel's event is, essentially, a set-theoretic concept. It is a set of
instants of time. It remains so even when instant (element) is defined
through set (event) as the intersection of a certain family (class) of
sets. This class is nothing but the set of all events which include a given
instant.
Typically for a set-thereticall approach, all these concepts are static, do
not really involve action.

Action proper is treated by several contemporary philosophers, including
Aune, Davidson, Quine, Sellars and others. To quote Bruce Aune [Aun88],

\quote{
Perhaps the most controversial aspect of so called action theory is its
subject matter. This subject matter is generally said to be (or to concern)
actions, but different philosophers conceive of actions in radically
different ways. For some philosophers, actions are abstract entities --
states of affairs, propositions, sets, or even ordered pairs of some kind.
For others, actions are distinctively concrete entities located in space
and time. Another group of philosophers, among whome I include myself, have
even denied that actions are required for a reasonable action theory,
insisting that agents or actors will suffice as the theory's sole objects.}

For us, actions are concrete, in the sense that they constitute the primary
reality and not man-made representation, such as propositions, sets etc.,
which are referred in the quote as `abstract entities'. At the same time we
cannot say that actions are located in space and time; it is space and time
that are constructed from actions. The high ontological status of action
which we maintain is usually rejected on the grounds that action is not
`fundamental' enough. Thus Aune writes:

\quote{
According to an agent theory, although agents clearly exist and may truly
be described as acting in this or that way, it is philosophically
misleading to say that their actions or deeds also exist.}

Aune justifies his view-point by using the concept of fundamental
realities, which is traced back to Aristotle. A pile of stones is less
fundamental than the stones which make it up. If a pile exists at all, then
it does so in some weaker form than the stones. We read further in Aune's
paper:

\quote{
Philosophers holding what are known as substance ontologies contend that
the fundamental objects of our world are continuants -- and nothing but
continuants. A continuant is a thing like a man,a marble, or a tree:
something that, as Aristotle said, persists in time and can undergo change.
Philosophers holding an agent theory often accept such an ontology. If
fundamental objects are continuants, then changes, events, and therefore
actions are not fundamental objects; their existence is derivative at
best.}

Our view is exactly the opposite: actions constitute the reality which we
perceive through our sense organs; agents are representations, elements of
a language, which we use to construct models explaining the observed
actions. This view is {\sl action ontolgy}.

The reason why we espouse this ontolgy is inseparable from our cybernetic
epistemology. We do not take the concept of existence as given intuitively,
nor do we think about it as a reflection of `real' existence. For us,
fundamental things, i.e. things that have the greatest degree of existence,
are those we start with when we construct our models of the world: the
cornerstones of construction. To pick up those cornerstones we look for the
most certain, unquestionable facts of life, and we find that they are
actions (the word {\sl fact} itself comes from the Latin {\sl facere},
which means to do or to make). Moreover, our knowledge itself, i.e. a
collection of the world's models, is nothing but action; the only meaning
of a model is in its operation. Representations are arbitrary: a matter of
convenience; the changes in representations must correspond to the changes
in reality if we want to have a true model. The translational relativity
(symmetry) of space is the best illustration of this. When we think of an
isolated point, we do not yet think about space.
Space is created by certain actions: moves, or shifts. These shifts are a
measurable reality. The points in space have meaning only with respect to
some reference system. The same shift on a line can be seen as a shift from
point 3 to point 7 or from point 12 to 16. The choice of a reference --
i.e. representation -- system is arbitrary. \eon

\node{Agent}
% part-of: Action

When we speak of an action, we speak also of an {\sl agent} that performs
the action. An agent is the carrier of will, the entity that chooses
between possible actions. We do not see agents, we see only what they are
doing. But we use the concept of agent to create models of the world.

When we speak of actions
of human beings we know very well what the agent is: just the person whose
action it is. We reconstruct this notion, of course, starting from our own
`I'. When we speak of such animals as dogs, we again have no doubt in the
validity of the concept {\sl agent}. This reasoning can be continued down
to frogs, worms, amoebas, trees, and inanimate objects, without any
convincing arguments for stopping. When we say: ``the bomb exploded and the
ship sank'', are there any reasons to object against understanding this in
the same way as if we were speaking abut people and dogs? After all, if the
bomb was not very big, the ship might or might not sink, depending on the
{\sl ship} itself, the ship as a whole. Notice that even given a definite
ship and a definite time, the result might not be uniquely predetermined.

And what about an {\sl act} (sic!) of radioactive decay? It is definitely
an action, but whose action is it? The physicist could say that the agents
here are electrodynamic and chromodynamic fields. This make sense because
of the theory the physicist has. If we do not have such a theory, we simply
say that there is a special agent for each possible act of radioctive
decay. At each moment in time this agent makes a choice: to decay or not to
decay. This immediately explains the exponential law of radioactivity.

Introduction of agents is, speaking informally, our first theory of the
world. The primary instance of an agent for a human being is itself. So, it
is not surprising that in primitive societies the concept of agent is
understood anthropomorphically: as something which is very similar, if not
identical, to ourselves. Hence the animism of primitive thinking:
understanding of all actions as initiated by various kinds of spirits or
other imaginary creatures.

The development of modern science banned spirits from the picture of the
world. But agents, cleared from anthropomorphism, still remain, even though
the physicists do not call them so. What is Newtonian force if not an agent
that changes, every moment, the momentum of a body? Physics concentrates on
the description of the world in space and time; it leaves -- at least at
present -- the concept of angent implicit. We need it explicitly because of
our metaphysics based on the concept of action, not to mention the simple
fact that cybernetics describes, among other things, the behavior of human
agents. (This last field of application of cybernetics is, of course, one
of the reasons for our metaphysics).
\eon

\node{Emergence}
% part-of: Action

Agents come into, and go out of, existence. For centuries philosophers
grappled with a problem: how to distinguish simple (``quantitative'')
changes from the cases where something really ``new'' emerges. What does it
mean to be ``new'', to emerge? In our theory this intuitive notion is
formalized as the coming of a new agent into existence. An action can lead
to an emergence of new agents.

Take radioactive decay. A neutron
suddenly {\sl chooses} to break down into a proton, electron and neutrino.
Whatever agents could have been involved into the events around the neutron
do not exist anymore. New agents, such as the interaction between the
newborn proton and electron emerge.

In the case of complex actions, such as the birth of a baby, we can argue
about the exact time of the event, because we have more than one reference
system in which to describe actions. As a member of society, the baby
emerges at birth. As an object of embryology it emerges at the moment of
egg fertilization.
\eon

\node{Event}
% part-of: Action

When we ignore the agent, we speak of actions as events. Event is an action
abstracted from the agent. \eon
\eon

\node{State}
% part-of: Metaphysics

The idea of a {\sl state} of the world, or some part of the world, is
familiar to everybody who took elementary courses in science. It is usually
considered so basic that there is no need and, probably, possibility of
definition. But we need a definition, because the only undefinable element
in our metaphysics is action.

Our definition is: a state of a part of the world is the set of actions
that are possible in this state, with their probabilities, if this concept
is applicable.
Two states in which all the same actions are possible and equally probable
are the same state, because there is no way to distinguish between them.

For example, if the state of affairs is such that there is an apple on the
table in front of me, I can reach it and pick it up. If there is no apple
this is impossible. If the moon is on the night sky, I can execute the
action of observing it. For this purpose I rotate my head in a certain way
and keep my eyes open. An atom is in an excited state when it can emit a
photon.

There are states of a type different from what we have considered above,
the states for which our definition is not suitable. If I feel pain, or am
frustrated, or elated, angry, or complacent, this has no effect on the
actions I can take. It affects only the choices I am going to make
selecting from the same set of possible actions. Indeed, if my hand is over
a gas heater and hurts (say, gently, for plausibility), I still have the
choice between keeping the hand where it is, or withdrawing. But,
obviously, the more it hurts, the more likley I am to withdraw it.

Thus we come to distinguish between:
\list a physical state, which is a set of possible actions for the subject
of knowledge and other agents; and \list a mental state, which influences
the choices to be made by the subject, but does not alter the set of
possible actions.

\ni This dichotomy clearly reflects the fundamental feature of the world as
consisting of the will and representation. A physical state is that of
representation. A mental state is a state of the subject's will. When we
use the concept of an agent in out model of the world we may endow agents
with some `mental' states. However, such a state remains physical for all
other agents: part of the representation of the world. It is only the
subject's mental states which belong to the category of will. Other agent's
will is for me only a representation.

When speaking of ``states'' without any of the two adjectives, we shall
mean physical states.

We have already had a number of reasons for considering action the most
fundamental observable reality. Seeing action in the context of state
provides more of it. The concept of state is a strong contender for the
most basic role. Indeed, the standard beginning of a theory is to introduce
states of the objects of the theory, whatever they are, and then define
actions, which are understood as certain changes of the state. But when we
define an action as a change of the state, we introduce something new,
which is
not present in the idea of a state; change is an event i.e. an action
abstracted from the agent that performs it. Thus we cannot avoid
introducing action as an undefinable element. At the same time the state of
the world can be defined through action, as we have just demonstrated.
\eon

\node{The algebra of actions}
% part-of: Metaphysics

Algebra is the part of mathematics that deals with operations. Operations
are actions. Since action is the basic reality of the world, algebra is the
beginning of all beginnings.

\node{Domain}
% part-of: The algebra of actions

A set of actions is referred to as a {\sl domain}. It is a formalization of
the idea of `a part of the world'. The models we are given by nature and
construct artificially are never universal. They are always applicable only
to some part of reality. This part is the domain of the model. Since states
of that part of reality are defined by sets of actions, the domain of a
model also defines the set of all states which can, in principle, exist: it
the powerset (the set of all subsets) of the domain. The actul set of
possible states may be a subset of this powerset.

When we apply a model (in particular, a theory), we assume that only those
actions take place that are within the domain. Make an action which is not
included in the domain, and the whole theory may become out of place. The
states of the world are defined as subsets of the domain of the model.
Other actions are ignored; they may be either irrelevant, when they have no
impact on the legitimacy of the model, or prohibited, when they make the
model unapplicable.
\eon

\node{Null action}
% part-of: The algebra of actions

We call a {\sl null action} the absence of any action. An action which is
not null is a {\sl non-null} action.
\eon

\node{Composition}
% part-of: The algebra of actions

There are two ways to unite actions into a composite action.

If [Exists]a_1[Exists] and [Exists]a_2[Exists] are actions then their {\sl
sequential composition}, denoted as [Exists](a_1;a_2)[Exists], or just
[Exists]a_1a_2[Exists], is the action which is performed by first
performing [Exists]a_1[Exists] and then, immediately, [Exists]a_2[Exists].

We say that action [Exists]a_1[Exists] is {\sl sequential part of} action
[Exists]a[Exists], if there exists an action [Exists]a_2[Exists] such that
[Exists]a_1a_2=a[Exists].

If [Exists]a_1[Exists] and [Exists]a_2[Exists] are actions then their {\sl
parallel composition}, denoted as or [Exists](a_1\parallel a_2)[Exists],
is the action which is performed by performing [Exists]a_1[Exists] and
[Exists]a_2[Exists] in parallel.

We say that action [Exists]a_1[Exists] is {\sl parallel part of} action
[Exists]a[Exists], if there exists an action [Exists]a_2[Exists] such that
[Exists]a_1 \parallel a_2=a[Exists].

\node{Process}
% part-of: Composition

A composite action is referred to as a {\sl process}. \eon
\eon

\node{Inverse action}
% part-of: The algebra of actions

Given an action [Exists]a[Exists], we denote as [Exists]\bar{a}[Exists] the
action such that [Exists]a\bar{a}[Exists] and [Exists]\bar{a}a[Exists] is
an action which returns the world to the same state as before this compsite
action. For a given [Exists]a[Exists] the inverse action
[Exists]\bar{a}[Exists] may or may not exist. If it does exist,
[Exists]a[Exists] is {\sl reversible}, otherwise {\sl irreversible}. \eon

\node{Continuous vs. discreet}
% part-of: The algebra of actions

A {\sl domain} is {\sl continuous} if for every action [Exists]a[Exists]
there exist two actions [Exists]a_1[Exists] and [Exists]a_2[Exists] such
that neither is null and the composite action [Exists]a_1a_2[Exists] is the
same as [Exists]a[Exists]. Note, that the requirement for
[Exists]a_1a_2[Exists] and [Exists]a[Exists] is {\sl to be the same}, and
not just lead to the same state.

An action [Exists]a[Exists] is {\sl elementary} if [Exists]a \not= a_1
a_2[Exists] for any two non-null actions [Exists]a_1[Exists] and
[Exists]a_2[Exists].

A domain is {\sl discrete} if every action from it can be represented by a
finite sequence of elementary actions.

A domain may have discrete and continuous subdomains. \eon
\eon

\node{Cognitive action}
% part-of: Metaphysics

Among the possible actions which constitute a state of the world there are
some which are performed by the subject of knowledge with the explicit
purpose of increasing the subject's knowledge. We shall call them {\sl
cognitive actions}. Actually, there is no sharp boundary between purely
cognitive actions and other actions which serve different purposes. Each
action can be considered cognitive to the extent it takes part in the
formation of a model of the world.

When I rotate my head to see
the moon, this is, obviously, a good cognitive action. To take an apple is
to a great extent a cognitive action for a baby who is just now forming the
concept of an external object. It is also cognitive for a blind man who has
no other way to know that an apple is there. But if you take an apple after
you have seen it, this hardly adds much to your mental model of the world.
The act is alimentary rather than cognitive. \eon

\node{Modeling scheme}
%MODEL, part-of: KNOW, Knowledge

The basic principle of the cybernetic epistemology is that knowledge is a
{\sl model} of (a part of) the world. Below we define the concept of a
model as it is most commonly used (see Fig.1).

{\midinsert \smallskip
\begtt{
\setbox1 = \hbox{[Exists]R[Exists]}
\wd1 = 0in
\setbox2 = \hbox{[Exists]r_1[Exists]}
\wd2 = 0in
\setbox3 = \hbox{[Exists]r_2[Exists]}
\wd3 = 0in
\setbox4 = \hbox{[Exists]M_a[Exists]}
\wd4=0in
\setbox5 = \hbox{[Exists]w_1[Exists]}
\wd5=0in
\setbox6 = \hbox{[Exists]w_2[Exists]}
\wd6=0in
\setbox7=\hbox{[Exists]a[Exists]}
\wd7=0in
~
~ \box4
~ \box2 \box3
~ -{-}-{-}-{-}-{-}-{-}-{-}-{-}-{-}-{-}-{-}-{-} {\bf model}
~ | |
~ | |
~ |\copy1 |\box1
~ | \box7 |
~ \box5 | |\box6
~ -{-}-{-}-{-}-{-}-{-}-{-}-{-}-{-}-{-}-{-}-{-} {\bf world}
~
\endtt}
\medskip
\centerline{\rm Fig.1. The modeling scheme} \endinsert}
\smallskip

A model is a system which includes: (1) a certain subsystem, whose states
will be referred to as {\sl representations}; (2) a {\sl representation
procedure} [Exists]R[Exists], such that if performed when the state of the
world is [Exists]w[Exists] it causes a certain representation [Exists]r =
R(w)[Exists] in the system;
(3) a family [Exists]R_a[Exists] of {\sl modeling procedures} which depend
on a possible action [Exists]a[Exists] in the world and convert one
representation into another. For the system to be a (correct) model, the
modeling procedures must have the following property. Let the world be in a
state [Exists]w_1[Exists], and suppose that system [Exists]S[Exists] makes
an action [Exists]a[Exists], as a result of which the state of the world
becomes [Exists]w_2[Exists]. Then the modeling procedure
[Exists]M_a[Exists] applied to the representation [Exists]r_1 =
R(w_1)[Exists] produces the representation [Exists]r_2[Exists] of the
resulting state [Exists]w_2[Exists]:

[Exists][Exists] M_a(R(w_1)) = r_2 = R(w_2) [Exists][Exists].

Thus by applying [Exists]M_a[Exists] to [Exists]r_1[Exists] the system can
predict, to some extent, the development of events in the world resulting
from its action [Exists]a[Exists]. This definition describes equally well
the case of a dog catching in flight a piece of sugar, and an astronomer
who computes the position of a planet in the sky. In the first case, the
model is built in the material of nervous cells of the dog's brain, in the
second case from the signs that the astronomer writes on paper when he
makes computations.

Mathematically, a model is defined by the representation function
[Exists]R(w)[Exists] and the family of modeling functions
[Exists]M_a(r)[Exists], for possible actions [Exists]a[Exists] of the
system. When we fix an action [Exists]a[Exists], we have what is known as a
{\sl homomorphism}. Thus a model is a family of homomorphisms. \eon

\node{Observation}
% part-of: Metaphysics

Observation is an abstraction of knowledge from the impact of our cognitive
actions on the world.
The state of the world [Exists]w_i[Exists] in Fig.1 and the action of using
the procedure [Exists]R[Exists] are not really separable: it is an action
of cognition, interaction of the subject and the object. When we speak of
observation, we assume that there exist cognitive actions which only serve
us to acquire knowledge, but have no influence on the phenomena we observe.

For instance, when we watch a party of billiards, the positions of the
balls are registered by means of the light thrown on the balls and
reflected into our eyes. We rightly believe that the effect of the lighting
on the movements of the balls is negligible, so we speak about the play in
a complete abstraction from the way we know about it.

This separation is not always possible.
Quantum mechanics deals with actions so elementary that the means we use to
know of them cannot be abstracted from. Our usual ``classic'' notions of
space and time include the a bstraction of observation. Indeed, the mental
construction of a reference frame uses actions of shifting and waiting
(doing nothing) which are assumend to have no effect on the studied
processes; for quantum-mechanical processes this assumption is not valid,
and the classic space-time frame of reference looses its legitimacy,
becomes meaningless. One should not use in mental constructions and
experiments the things known not to exist. \eon

\node{Objective description of the world} % part-of: Metaphysics

By objective description of the world we mean, first, a description in
terms of some {\sl objects}, and second, a description which is, as much as
possible, ``objective'' in the usual sense, i.e. impersonal, not depending
on the cognitive actions, or other features, of the subject of knowledge.
As we shall see, the use of the same word in these two meanings is not
accidental: a description can be ``objective'' because it is a description
in terms of objects.

\node{Object}
% part-of: Objective description of the world

Suppose I am aware of a tea-pot on the table in front of me. This is a
result of my having
the mechanism of abstraction in the brain. I recognize the image on my
retina as belonging to a certain set of images, the abstraction `tea-pot'.

But there is more to it. I perceive the tea-pot as an {\it object}. The
object `tea-pot' is certainly not a definite image on the retina of my
eyes; not even a definite part of it. For when I turn my head, or walk
around the table, this image changes all the time, but I still perceive the
tea-pot as the same object. The tea-pot as an object must, rather, be
associated with the transformation of the image on my retina which results
from the changing position of my eyes. This is, of course, a purely visual
concept. We can add to it a transformation which produces my tactile
sensations given the position and movements of my fingers.

The general definition of an object suggested by this example consists of
three parts.
\smallskip

\ni (1) First we define a set [Exists]R_{ob}[Exists] of representations
which are said to represent the same object;
in our example this set consists of all images of the tea-pot when I look
at it from different view-points, and possibly, my sensations of touching
and holding it.

\ni (2) Then from the set of all possible actions we separate a subset
[Exists]A_{cogn}[Exists] of actions
which will be referred to as {\sl cognitive}; in our case
[Exists]A_{cogn}[Exists] includes such actions as looking at the tea-pot,
turning my head, going around the table, touching the tea-pot etc. -- all
those actions which are associated with the registration of the fact that a
tea-pot is there.

\ni (3) Finally, we define a family of functions [Exists]f_a(r)[Exists],
where for every cognitive action [Exists]a \in A_{cogn}[Exists], the
function

[Exists][Exists] f_a: R_{ob} \to R_{ob} [Exists][Exists]

\ni transforms a representaiton [Exists]r \in R_{ob}[Exists] into
[Exists]f_a(r) = r'[Exists] which is expected as a result of action
[Exists]a[Exists]. \smallskip

The most important part here is the third; the first two can be subsumed by
it. We define an object [Exists]b[Exists] as a family of functions
[Exists]f_a[Exists]:

[Exists][Exists] b = \{f_a: a \to A_{cogn}\} [Exists][Exists]

The set [Exists]A_{cogn}[Exists] is the domain of the index
[Exists]a[Exists]; the set [Exists]R_{ob}[Exists] is the domain and
co-domain of the functions of the family.

When I perceive an object [Exists]b[Exists], I have a representation
[Exists]r[Exists] which belongs to the set [Exists]R_{ob}[Exists]; I then
execute some cognitive actions, and for each such action [Exists]a[Exists]
I run my mental model, i.e. perform the transformation [Exists]f_a[Exists]
on [Exists]r[Exists]. If this anticipated representation
[Exists]f_a(r)[Exists] matches the actual representation [Exists]r'[Exists]
after the action [Exists]a[Exists]:

[Exists][Exists] f_a(r) = r' [Exists][Exists]

\ni then my perception of the object [Exists]b[Exists] is confirmed;
otherwise I may not be sure about what is going on. Observing a tea-pot I
check my actual experience against what I anticipate as the result of the
movements of my head and eyeballs. If the two match, I perceive the tea-pot
as an object. If I travel in a desert and see on the horizon castles and
minarets which disappear or turn topsy-turvy as I get closer, I say that
this is a mirage, an illusion, and not a real object.

The concept of an object is naturally (one is tempted to say, inevitably)
arises in the process of evoluiton. It is simply the first stage in the
construction of the world's models. Indeed, since the sense organs of
cybernetic animals are constantly moving in the environment, these actions
are the first to be modeled. In the huge flow of sensations a line must be
drawn between what is the result of the animal's own movements, and other
changes which do not depend on the movements, are {\sl objective}. Looking
for objectivity is nothing else but {\sl factoring out certain cognitive
actions}. Function [Exists]f_a[Exists] factors out the action
[Exists]a[Exists] by predicting what should be observed when the only
change in the world is the subject's taking of the action
[Exists]a[Exists]. If the prediction comes true, we interpret this as the
same kind of stability as when nothing changes at all. The concept of
object fixates a certain invariance, or stability, in the perception of a
cybernetic system that actively explores its environment.

The metasystem transition from representations to their transofrmations is
a step towards objectivity of knowledge. Actions and, in particular,
sensations are intimately tied to the agent, the subject of knowledge. An
object is a transformation and prediction of actions. The very fact that
prediction is possible indicates that the transformation depends less on
the subject of knowledge, the `I', and more on the `not-I'.
This does not ensure a complete objectivity; alas, there is no such thing.
But a jump from a representations to a transformation of representations
verified by the practice of correct predictions, is the only way we know to
increase the informally understood objectivity.

When we perceive a tea-pot as an object, we have a lot of cognitive actions
to factor out: we can walk around it, grasp it, rotate it in from of our
eyes etc. But often we observe things from afar and that is about all we
can do, as, fro instance, when we observe a star and still call it an
object. Well, from the viewpoint of our theory, we always associate with an
object some kind of stability, and stability exist obly with respect to
action. In the case of a star, this is the stability with respect to
varying conditions of observation. We can observe `the same' star at
different times and factor out the differences in time by taking into
account the rotation of the sky around the Earth's axis. The same is true
with respect to the movement of the observer around the Earth's surface.
The more we know of astronomy and physics, the greater number of properties
of the object will we discover, such as the constancy of the star's
spectrum etc.

We also must include into the concept of cognitive actions the more
sophisticated and esoteric actions which were not among those actions for
which evolution created human brain, but emerge as a result of the
development of science. We get involved in this kind of actions when we
construct huge accelerators of elementary particles and set up experiments
to explore how the particles interact. As an apple and other physical
bodies are invariants in the processing of input information by the brain,
so an electron and other elementary particles are invariants of the
scientific symbolic models of the world. We can measure the charge of the
electron in many different ways -- which all are various cognitive actions
-- but after making all the computations required by the theory, we still
come to the same number (within the error). The same with mass, spin, etc.
So an electron is, for us, an object, as real as an apple. One could
qualify this statement by noticing that the existence of electrons depends
on the legitimacy of our physical theory, which is not absolute. True
enough. But who are we to claim that the legitimacy of our brain as a
collection of models is absolute? \eon

\node{Hierarchical modeling}
%part-of: Objective description of the world

We defined an object as a family of transformations [Exists]f_a(r)[Exists]
on the set of representations which predicts the representation resulting
from a given cognitive action [Exists]a[Exists]. This is the same
definition as the general definition of a model, where [Exists]f_a[Exists]
is the modeling (prediction) function [Exists]M_a[Exists]. The specificity
of an object is, first, in the domain of the index [Exists]a[Exists], which
is a set of cognitive actions [Exists]A_{cogn}[Exists]; and second, in the
way the
functions [Exists]f_a[Exists] of the family are further used.

The domain [Exists]A_{cogn}[Exists] includes only actions we deem external
to the intuitively understood {\sl essence} of the object. We call them
cognitive because they allow us to separate the object from other
phenomena, to see it from different sides --
often literally -- an at the same time not to change the object itself
beyond recognition.

As for the use of [Exists]f_a[Exists], it serves not to provide a needed
prediction, but only to confirm, by checking the prediction against
reality, that we do deal with a given object.
Then on the basis of this information we, probably, will make a prediction
which is needed as such. We see here a hierarchy of two models: a model
that, having primary sensory data as input, recognizes an object, and a
model which uses this object as input. In order to use a model as an
object, it must be {\sl objectified}, represented by a material object.
This act of representation is of the same nature, and plays the same role
as representation in models. In this way a hierarchy of models is
constructed, where each next level is a representation of the
transformation of the representations of the preceding level.

A hierarchical model of this kind is shown in Fig.2.

{\midinsert \smallskip
\begtt{
\setbox1=\hbox{[Exists]r^1_{11}[Exists]}
\setbox2=\hbox{[Exists]r^1_{12}[Exists]}
\setbox3=\hbox{[Exists]r^1_{21}[Exists]}
\setbox4=\hbox{[Exists]r^1_{22}[Exists]}
\setbox5=\hbox{[Exists]r^2_{11}[Exists]}
\setbox6=\hbox{[Exists]r^3_{1}[Exists]}
\setbox7=\hbox{[Exists]r^3_{2}[Exists]}
\setbox8=\hbox{[Exists]w'_1[Exists]}
\setbox9=\hbox{[Exists]w_1[Exists]}
\setbox0=\hbox{[Exists]w_2[Exists]}
\wd1=0in
\wd2=0in
\wd3=0in
\wd4=0in
\wd5=0in
\wd6=0in
\wd7=0in
\wd8=0in
\wd9=0in
\wd0=0in
~ [Exists]M_a[Exists]
~
~ - - - - - - - - - - - - - - - - - {\bf language/conceptions}
~ \box6 \box7
~
~
~ - - - - - - - - - - - - - - - - - {\bf brain/perceptions}
~ \box5
~
~
~ - - - - - - - - - - - - - - - - - {\bf sense organs/senasations}
~ \box1 \box2 \box3 \box4
~ [Exists]a'[Exists] [Exists]a[Exists]
~
~ - - - - - - - - - - - - - - - - - {\bf world/action}
~ \box9 \box8 \box0
~

\endtt}
\medskip
\centerline{\rm Fig.2 Hierarchical modeling scheme} \smallskip
\endinsert}

\ni It consists
of the ground level to be modelled (the world), and three levels of
representations. This scheme is constructed by combining several simple
modeling schemes shown in Fig.1. On the ground level we see, as in Fig.1,
the states [Exists]w_1[Exists] and [Exists]w_2[Exists]; and the ultimate
goal of the hierarchy is still to predict some features of
[Exists]w_2[Exists] on the basis of the initial state [Exists]w_1[Exists]
and the action [Exists]a[Exists]. In Fig.1 these features were expressed in
terms of representation of the first level [Exists]r_2[Exists]; in the
three-level scheme they are expressed in terms of representation of the
third level [Exists]r^3_2[Exists].

Now let us see how the goal of the model is achieved. The representation of
[Exists]w_1[Exists] is [Exists]r^1_11[Exists]; this might be, e.g., the
image of a tea-pot on the retina of my eye. But I perceive this image as a
part of {\sl the object} tea-pot. The entire primary modeling scheme is
reproduced here: it consists of the states of the world [Exists]w_1[Exists]
and [Exists]w_1'[Exists], the cognitive action [Exists]a'[Exists], and the
representations [Exists]r^1_{11}[Exists] and [Exists]r^1_{12}[Exists]. Here
[Exists]w_1'[Exists] is an intermediate state of the world resulting from
the cognitive action [Exists]a'[Exists] (e.g. that of my coming closer to
the tea-pot). The transformation of [Exists]r^1_{11}[Exists] into
[Exists]r^1_{12}[Exists] is part of my perception of the tea-pot when I
make allowance for [Exists]a'[Exists]. The whole scheme of the perception
of a tea-pot may be executed any number of times, depending on the time
scale of the processes involved; in particular, it is possible that there
is
no time to check the model property for [Exists]r^1_{12}[Exists], so that
the tea-pot will be perceived on the basis of [Exists]r^1_{11}[Exists],
which is just a glimpse of it. In any case the transofrmation of
[Exists]r^1_{11}[Exists] into [Exists]r^1_{12}[Exists] is objectified as a
representation [Exists]r^2_{11}[Exists] on the second level of
representations.

In a similar manner the other primary models work, which are shown
partially in Fig.2. A transformation of [Exists]r^2_{11}[Exists] becomes
represented by [Exists]r^3_1[Exists], and the edifice of transformations
and representations on the predicted side of the model is constructed
analogously. The height of the hierarchy in Fig.2 is three, so on the third
level we see the desired prediction [Exists]M_a[Exists]: if
[Exists]r^3_1[Exists] and the action [Exists]a[Exists] is taken, then
[Exists]r^3_2[Exists].

Iy is very important that representations of the [Exists]i[Exists]-th level
are abstractions from {\sl transformations} of representations of the
[Exists]i-1[Exists]-st level, not from {\sl representations themselves}.
Indeed, if the representations of higher levels were abstractions of
representations of lower levels, then all lower (intermediary) levels of
representation would have been unnecessary. Let the representation function
of the [Exists]i[Exists]-th level be [Exists]R_i(r_{i-1})[Exists]. Then we
could combine the functions of each two neighboring levels into one:
[Exists][Exists] R_i(r_{i-2}) = R_i(R_{i-1}(r_{i-2})) [Exists][Exists] Thus
a hierarchy of representation levels in this case could have been only a
question of expediency; it would not really give any additional power.
Repeated abstraction leads to loss of contents, and ultimately to the
concept `something', about which we can say nothing. With the hierachical
modeling as we describe it, the creation of each new level is a {\sl
metasystem transition} (see [Tur77]); it provides new possibilities and can
be repeated indefinitely. \eon

\node{Sensations, perceptions, conceptions} % part-of: Objective
description of the world

We hypothesize that the hierarchical modeling scheme, as described above,
is actually implemented in human brain and produces our many percepitons of
objects in the world, where objects may be understood in the widest sense
as islands of stability in the see of sensations. We do not know how many
levels of this hierarchy is there, and we speak of a hierarchy in a very
loose sense. Some models may use models of various lower levels and be used
by more than one model of higher levels.

We intend the three-level modeling scheme in Fig.2 as a rough picture of
human knowledge. The first representation level is constituted by
sensations. All levels of the brain hierarchy we compress in one level:
that of perceptions. The third level is that of human language and the
models we construct in language on the basis of perceptions. Here again we
merge many actual levels into one.

Sensations are produced by our sense organs. Perceptions are formed and
used within the brain. Conceptions are created by ourselves while we create
new, linguistic, models of the world. The triad {\sl sensation, perception,
conception} seems close in meaning to Kant's usage of these terms. We leave
it to the reader, though, to judge on it.
\eon
\eon

\node{Space}
% part-of: Metaphysics

Among the most elementary actions known to us are small displacements ``in
space''. We have put it in the quotes, because people have accustomed to
imagine that some entity, called ``space'' exists as a primary reality,
which creates the possibility of moving from one point of this space to
another. Our analysis turns this notion topsy-turvy. Only actions
constitute observable reality; space is nothing but a product of our
imagination which we construct from small displacements, or shifts, of even
smaller objects called points. If [Exists]x[Exists] is such a shift, then
[Exists]xx[Exists] -- the action [Exists]x[Exists] repeated twice -- is a
double shift, which we would call in our conventional wisdom a shift at the
double distance in the same direction. On the other hand, we may want to
represent a shift [Exists]x[Exists] as the result of another shift
[Exists]x'[Exists] repeated twice: [Exists]x = x'x'[Exists]. It so happens
that we can make three different kinds of shifts, call them [Exists]x, y,
z[Exists], none of which can be reduced to a combination of the other two.
At the same time any shift [Exists]w[Exists] can be reduced to a properly
chosen combination of shifts [Exists]x, y, z[Exists]. So we say that our
space has three dimensions. \eon

\node{Time}
% part-of: Metaphysics

When we do nothing for a while we still feel that something has happened:
we say that some ``time'' has pas sed.
In terms of actions, doing nothing is a special type of action. If we
denote it by [Exists]t[Exists], then
[Exists]tt[Exists] is an action of waiting for two times longer than with
[Exists]t[Exists].

We often say that all real processes take place in space and time. The
meaning of such statements is that in addition to what really goes on, we
imagine some reference actions of consecutive shifts (``in space'') and
waits (``in time'') and esatblish relationships between these actions and
actual objects and processes. Thus, in accordance with Kant's view, space
and time are not observable realities, but our ways to organize experience.

When we think about space, we treat those shifts which create the space as
instantaneous. In fact, however, all actions have a time component. Space,
as we understand it intuitively, is a complete abstraction from time.
Respectively, our intuitive time is abstracted form space and from
everything that happens in space. This intuition, when formalized into a
theory, gave rise to classical mechanics. It treats time as sort of a flow
which goes on uniformly in every point of space.

If the ultimate reality is that of actions, then there must be as many
times as there are agents. When an agent acts, the time that elapses is
just one of the characteristics of the action. Thus whatever is happening,
there must be its own time measure for it. This truth was dramatically
demonstrated by Einstein's relativity theory.

When we measure time, we take some repetitious process, like the swinging
of a pendulum, for a model of other processes. We may say, for instance,
that John needes 80 `pendulums' of time to smoke up a cigarette. In terms
of the modelling scheme (Fig.1), the state when John is lighting his
cigarette is [Exists]w_1[Exists]; the state when he extinguishes it is
[Exists]w_2[Exists]; the representation function [Exists]R[Exists] is
registration of the current value of the counter; [Exists]r_1[Exists] and
[Exists]r_2[Exists] are the states of the counter at the beginning and end
of smoking.

\node{Objectification}
% part-of: Time

We often want to think and speak about an action or process as something
definite, constant -- in other words, as an object. Then we {\sl objectify}
it, i.e. replace the process, in reality or in our imagination, by an
object. Objectification is a kind of metasystem transition: a process
becomes an object to be manipulated by a metasystem. Strictly speaking, we
should say `becomes represented by an object', but `becomes an object' is
also admissible, because as a rule the representation would be such as to
allow to reproduce the process, maybe even with variations. Hence a partial
equivalence between a process and its objectification.

One common case of objectification is replacement of a process by its
definition. For instance, we define algorithms as computational processes
which we expect to be executed in a certain fixed manner. The definition of
an algorithmic process is an object, usually a text in some formal
language.
The semantics, i.e. the meaning, of the language is provided by a machine
which executes the process in accordance with its definition. The famous
Turing machine is an example.

The mapping of an action onto its representation, which is made in the use
of a model, is also a form of objectification: representations are objects.
\eon

\node{Historical record}
% part-of: Time

Consdier a model [Exists]M[Exists] in which certain states of the world
[Exists]w_1[Exists] and [Exists]w_2[Exists] are represented, respectively,
by [Exists]r_1[Exists] and [Exists]r_2[Exists]. Recall that states of (a
part of) the world are, in fact, {\sl cognitive actions} used by
[Exists]M[Exists] to create representations of the world. Since
[Exists]w_1[Exists] and [Exists]w_2[Exists] are actions, we can ask
ourselves: how will [Exists]M[Exists] represent the composite action
[Exists]w_1 w_2[Exists]? Typically, each model would have a certain set, a
tool-bag of ``atomic'' actions to register the states of the world.
Therefore, to represent the sequential composition of those, a method must
either exist in [Exists]M[Exists],
or to be built as an addition to [Exists]M[Exists]. Depending on the
physical nature of the representations [Exists]r_i[Exists], various methods
may be used; for example, if [Exists]r_i[Exists] are symbols we can simply
concatenate them: [Exists]r_1 r_2[Exists]; then a sequence of
state-registering actions [Exists]w_1, w_2, \ldots, w_n[Exists] will be
represented as [Exists]r_1, r_2, \ldots, r_n[Exists]. No matter what the
actual method of combining representations is, we can think of the result
as a sequence of representations. We shall refer to it as a {\sl historical
record}.

\node{Memory}
% part-of: Historical record

The subsystem of the subject of knowledge which keeps historical records is
its {\sl memory}.
\eon

Historical records add nothing to our formal concepts of model and
knowledge. States are always actions, whether they are split into sequences
or not. It is a technical detail, after all. I can consider every state of
the world in my knowledge as a sequence of actions [Exists]w_1, w_2,
\ldots, w_n[Exists] which covers all my life from the moment I was born.
Then my predictions will always be functions of the representation
[Exists]r_1, r_2, \ldots, r_n[Exists] . This, of course, does not exclude
predictions which take into account only the latest member
[Exists]r_n[Exists] of the sequence. If the system has memory, then memory
becomes a component of each current representation; its role in predicting,
though, may vary in wide limits.

\node{Present, Past, Future}
% part-of: Historical record

The last atomic representation at the moment of making a prediction is
referred to as {\sl the present}. The whole sequence of memorized
representation, the total historic record, is {\sl the past}. The
representation resulting from prediction is {\sl the future}.

It is clear from this definition that the duration of time which is
included in the present depends on the cognitive actions which determine
the state of the world. In the context of writing a diary, the current day
is the present. In the work of a computer, one millionth of a second may
mean a long past state of affairs. \eon

\node{Real time vs. model time}
% part-of: Time

Henri Bergson was first to notice and emphasize the difference between {\sl
real time}, in which we live and act, and the objectified time of history
and physics. Imagine a pendulum which at each swing puts a mark on a moving
tape. We have a historical record of `how the time is moving'. This
historic record is an object at every moment we look at it. We use it as a
model of reality. We shall refer to the marks on the tape as representing a
{\sl model time}. It is very much different from the real time.

Real time is such that two moments of it never coexist. In model time the
moments coexist as different objects in some space. Thus Bergson calls
model time a projection of real time on space. Bergson's real time is
irreversible. Model time is reversible: we read historical records equally
well from left to right and right to left. The seemingly inconceivable
feature of Feynman's diagrams, the movement of a particle in the direction
opposite to time, is explained simply by the fact that the time of physical
theories is model time, i.e. a spacial phenomenon.

The solution of the fundamental metaphysical problem is the key to the
understanding of time. As long as you see the world as abiding, at each
moment of time, in a certain state, your time will be reversible, because
there always remains the possibility of returning to one of the passed
states. You may invent some tricks to prevent such things from happening in
your theory, but a theory which requires patches at the very base is
seriously deficient.

If, on the contrary, the only reality of the world is action, then time is
irreversible from the beginning, because a new action does not revoke a
previous action, as a new state does, but builds on it. If in the sequence
of two alternating states [Exists]w_1w_2w_1w_2w_1w_2\ldots [Exists] etc.
[Exists]w_1[Exists] and [Exists]w_2[Exists] are understood, literally, as
states of the world, then each time a state returns, the world is the same,
as if the time were running on and back. If the states are cognitive
actions, the reality is continuously changing:

[Exists][Exists] w_1,\ w_1w_2, \ w_1w_2w_1, \ w_1w_2w_1w_2, \
w_1w_2w_1w_2w_1 \ldots [Exists][Exists]

\ni There is no return. Action is cumulative.

In the contemporary science, real time shows up in probability t heory and
its applications, such as statistical physics. Probability is a
characterization of certain actions, not states. In the theory of
probability we deal with acts of choosing of a number of possibilities. If
there are ten possible actions then the probability of actually doing one
of them is [Exists]10_{-1}[Exists]. If in the second step there is again
the same number of options, the probability of the composition is
[Exists]10_{-2}[Exists]. Suppose that the first step was from state
[Exists]w_1[Exists] to [Exists]w_2[Exists], and the second step takes the
system back to [Exists]w_1[Exists]. The probability of each choice in one
more step will be [Exists]10_{-3}[Exists], not [Exists]10_{-1}[Exists]. In
the next
step it will be [Exists]10_{-4}[Exists] etc. It is cumulative; there is no
way back. This is why the time of statistical physics is irreversible.

In the world where the ultimate reality is action, time differs very much
from the time of classical mechanics. The former is irreversible, the
latter reversible: we can designate this difference as a difference in
macro-structure. There is even a greater difference in micro-structure. Our
real time is closer to Bergson's {\sl duration} than to mechanical
infinitely divisible time. Unlike the mechanical (model) time, real time is
quantized: a quantum of time is the time involved in one action. An
elementary action has a duration within which it makes no sense to speak
about distinct moments of time. Real time does not consist of a continuum
of infinitely small time coordinates. To make small times real,
corresponding actions must exist. As it happens, this is not quite simple.
We know from physics that there is no universal quantum of time; time
quanta depend on actions. To perform an action with the duration
[Exists]t[Exists] (by the order of magnitude), we must involve an amount of
energy [Exists]h/t[Exists], so the expression `a point in time', which
presumes a duration of [Exists]t = 0[Exists], presumes an infinite energy.
If it has any meaning, then, possibly, only for the state of the world just
before the big bang. \eon
\eon

\node{Individual physical body}
%part-of: Metaphysics

We saw that an object is more than just a complex of sensations: it
includes a certain measure of stability with regards of our cognitive
actions. When we speak of a physical body we have in mind more than just an
object. We include the object's {\sl historical record}. Hence `a cat' is
an abstraction applicable to any cat. The pet cat I may have at home is an
on-going historic record of my having it.

The historic record of a physical body may not be actually known in full,
not even partially. The important thing is that to speak of a physical body
we must associate some, may be abstract, historical record with our direct
perception of it. This record makes physical bodies, in principle, {\sl
identifiable}. Thus we consdier the terms `identifiable' or `individual'
body or object as synonimous with `physical' body.

One of the most beautiful features of quantum mechanics is the way a
philosophical analysis is translated into theory and experimentally
confirmed. We can speak of individual physical bodies only if we can
actually identify them by some means. We cannot do that to elementary
particles when there are no impenetrable boundaries between them. Therefore
they must be considered indistinguishable. This requires symmetrization of
the wave function, which leads to various physical consequences brilliantly
confirmed by experiment. Electrons are not physical bodies in the usual
sense, because they are not individually identifiable bodies.
\eon

\node{Causality}
% part-of: Metaphysics

Let us analyze the meaning of the statements of the form:
`[Exists]C[Exists] is the cause of the effect [Exists]E[Exists]', or simply
`[Exists]C[Exists] causes [Exists]E[Exists]'. Here [Exists]C[Exists] is an
action, and [Exists]E[Exists] a state of the world. Sometimes we speak of
the cause [Exists]C[Exists] as a state too, but what we mean then is that
this state leads inescapably to a certain action which causes
[Exists]E[Exists]. On other occasions, we speak of the effect
[Exists]E[Exists] as an action, but this, again, for the same reason that
this action is an immediate result of the situation caused by
[Exists]C[Exists]. So, in the mainstream, the cuase is an action, and the
effect a state of affairs.

One could try to explain the meaning of causality by saying that whenever
[Exists]C[Exists] takes place, [Exists]E[Exists] follows. But this is not a
satisfactory explanation. We see that after a day a night always comes, but
the day is not the cause of the night; the cause is rather the rotation of
the Earth. A simple observation of the sequence [Exists](C,E)[Exists] is
not sufficient. [Exists]C[Exists] and [Exists]E[Exists] might be two
effects of a common cause, or this could be just a coincidence. It was
noted by many thinkers that causal relation is more than just sequence in
time.

In search of the true meaning of causality consider the case when the cause
[Exists]C[Exists] is {\sl my own} action. By `my own' we mean, as usually,
a reference to the subject of knowledge. So, `if I perform
[Exists]C[Exists], there will be [Exists]E[Exists]'. Our interpretation of
this statement is the pair of predictions [Exists]A[Exists] and
[Exists]B[Exists] in Fig.3.

{\midinsert \smallskip
\begtt{
~
~ True True True False
~ -{-}-{-}-{-}-{-}-{-}-{-}-{-}-{-}-{-}-{-}- -{-}-{-}-{-}-{-}-{-}-
~ | | | |
~ | | | |
~ S| E| S| E|
~ | | | |
~ | C | | do nothing |
~ -{-}-{-}-{-}-{-}-{-}-{-}-{-}-{-}-{-}-{-} -{-}-{-}-{-}-{-}-{-}-
~ (A) (B)
\endtt}
\medskip
\centerline{\rm Fig.3 Two predictions involved in causal relationship}
\centerline{\rm when the agent is the subject of knowledge } \endinsert}
\smallskip

\ni In words: if a situation recognized by abstraction [Exists]S[Exists]
holds, and I do action [Exists]C[Exists], then a situation characterized by
abstraction [Exists]E[Exists] takes place. But if in the same situation I
do nothing, then [Exists]E[Exists] does not occur. This precisely matches
our intuitive understanding of causality.

Now suppose that [Exists]C[Exists] is not my action, but that of some other
agent. It may come as a surprize that this, seemingly innocuous,
substitution radically changes the picture. We must remember that I can
perceive an action directly
only when it is my own action; all other actions I perceive only as
representations created by my sense organs. Therefore, the action
[Exists]C[Exists] must now be a part of the initial situation S. Let this
complex situation be represented as [Exists]S+obs(C)[Exists], where
[Exists]obs(C)[Exists] stands for the fact that I observe
[Exists]C[Exists].

Suppose we try to express the meaning of causality in this case as
consisting of the pair of predictions in Fig.4.

{\midinsert \smallskip
\begtt{
\setbox4=\hbox{[Exists]S+obs(C)[Exists]}
\wd4 = 0in
~ True True True False
~ -{-}-{-}-{-}-{-}-{-}-{-}-{-}-{-}-{-}-{-}- -{-}-{-}-{-}-{-}-{-}-{
~ | | | |
~ | | | |
~\box4 | E| S| E|
~ | | | |
~ | do nothing | | do nothing |
~ -{-}-{-}-{-}-{-}-{-}-{-}-{-}-{-}-{-}-{-}- -{-}-{-}-{-}-{-}-{-}-{
~ (A) (B)
~
\medskip
\centerline{\rm Fig.4 A wrong interpretation of causal relationship} }
\endinsert}
\smallskip

One can see that this is not a correct expression of causality. This is
only an observation of time sequences, which we have rejected above. It
could be that there exists a real cause [Exists]C'[Exists] which first
causes action [Exists]C[Exists], and then situation [Exists]E[Exists]. For
example, I can, on some days, have scrambled eggs for breakfast. On such
days I break two eggs and fry the content in a frying-pan. An external
observer will see that when I break eggs, I always fry them; when I do not
break eggs, I do not fry them. This is an if-then relationship, and it can
be a useful model of my uninterrupted breakfast. But it would be an error
to conclude that breaking eggs is the {\sl cause} of frying them.

Then is there any meaning in speaking about a cause which is not an action
of the subject of knowledge? We believe, there is, but it is a result of a
certain {\sl reduction} of this case to the basic case where the cause
[Exists]C[Exists] is the subject's action. There is an additional element
introduced by the subject of knowledge in understanding causality in the
case of sombody's else action. It is a tacit assumption that I could,
somehow, allow or prevent the action of which I think as a cause. The
interpretation of the statement that an action [Exists]C[Exists] of some
agent causes [Exists]E[Exists] as the statement that if I, the subject of
knowledge, {\sl allow} [Exists]C[Exists] to take place, then
[Exists]E[Exists] ensues; however, if I somehow {\sl prevent} it, there
will be no [Exists]E[Exists]. This view of causality is pres ented in
Fig.5.

{\midinsert \smallskip
\begtt{
~
~ True True True False
~ -{-}-{-}-{-}-{-}-{-}-{-}-{-}-{-}-{-}-{-}- -{-}-{-}-{-}-{-}-{-}-
~ | | | |
~ | | | |
~ S| E| S| E|
~ | | | |
~ | allow C | | prevent C |
~ -{-}-{-}-{-}-{-}-{-}-{-}-{-}-{-}-{-}-{-}- -{-}-{-}-{-}-{-}-{-}-
~ (A) (B)
~
\medskip
\centerline{\rm Fig.5 Two predictions involved in causal relationship}
\centerline{\rm when the agent is not the subject of knowledge } }
\endinsert}
\smallskip

The actual feasibility of my preventing an action is no issue. The meaning
of my statements is that they provide certain models of the world. It is
not necessary that every model be directly verifiable. Some models
(metamodels) help create other models, which produce directly verifiable
predictions.
When I say that rotation of the Earth is the cause of alternation of day
and night, the meaning of it is, according to our scheme, that if I stop
the rotation of the Earth, the effect of alternation will disappear.
Clearly, I cannot stop the Earth from rotationg, but this does not make the
statement less true or useful. When I think of the rotation of the Earth, I
actually have in mind a model of the Solar system, where celestial bodies
are represented by balls of appropriate (and convenient for my imagination)
sizes. In this model I can actually stop the Earth's rotation and see what
happens. This is the true meaning of our causal relationship. Since a model
of a model is a model, we often do not care to make distinction between our
models and our models of models.

As required by our epistemology, we explained causal statements as certain
models of the world. We might now ask: what about the concept of model
itself? Does it not include the idea of causality?

The answer is: yes, it does. In the use of models causality is exploited
twice. First, whenever we run the machinery of the model, we expect that
the switching on of the model produce a definite effect. Second, when we
compare our predictions with the reality, we expect them to match. The fact
that there are such phenomena as models of the world is, essentially, the
same fact as the existence of causality.

\node{Laws of nature}
% part-of: Causality

The laws of nature are models of the world with a wide range of
applicability.

Predictions we make using models, in particular, laws of nature, are
expressed in terms of representation functions, or abstractions.
Abstractions are many-to-one mappings. There are always more than one state
of the world that are compatible with prediction. The prediction of a
representation [Exists]r[Exists] is the restriction of the expected states
of the world to those states [Exists]w[Exists] which are mapped on
[Exists]r[Exists] by the representation function [Exists]R[Exists]:
[Exists]R(w) = r[Exists].

The mechanistic worldview saw the laws
of nature as something that uniquely prescribes how events should develop,
with indeterminacy resulting only from our lack of knowledge; for
contemporary science the laws of nature are only restrictions imposed on a
basically non-deterministic world. It is not an accident that the most
general laws of nature are conservation laws, which do not prescribe how
things must be, but only put certain restrictions on them. \eon
\eon

\node{Number}
% part-of: Metaphysics

The {\sl number} is a conceptual scheme, an abstraction of the second level
from {\sl specific numbers}: 1, 2, 3, ... etc. The abstraction procedure to
recognize specific numbers is {\sl counting}.

Counting is based on the ability to divide the surrounding world into
distinct objects. This ability emerged quite far back in the course of
evolution; the vertebrates appear to have it in the same degree as humans
do. The use of specific numbers is a natural integrated description
complementary to the differential description by recognizing distinct
objects. This ability would certainly be advantageous for higher animals in
the struggle for existence. And cybernetic apparatus for counting could be
very simple -- incomparably simpler than for recognition of separate
objects in pictures.

Yet nature, for some reason, did not
give our brain this ability. The numbers we can directly recognize are
ridiculously small, up to five or six at best (though it can be somewhat
extended by training). Thus the number 2 is a neuronal concept, but 20 and
200 are not. We can use them only through counting, creating artificial
representqations in the material external to the brain. The material may
be, and was historically, fingers and toes, then pebbles, notches etc., and
finally sophisticated signes on paper and electronic states of computer
circuitry. For theoretical purposes the best is still the ancient-style
representation where a chosen symbol, say {\bf 1} stands for one object.
Thus 2 is {\bf 11}, and 5 is {\bf 11111}. This is referred to as the {\sl
unary} representation of numbers. \eon

\node{Infinity}
% part-of: Metaphysics

In mathematics and philosophy we find two concepts of infinity: {\sl
potential infinity}, which is the infinity of a process which never stops,
and {\sl actual infinity} which is supposed to be static and completed, so
that it can be thought of as an object.

\node{Potential infinity}
% part-of: Infinity

The cybernetic philosopy readily embraces the concept of potential
infinity; in fact, it is hard to see how we can avoid it. We say that a
process {\sl ends} when it reaches a certain stage. In a particular case,
we can define the end stage so that it never takes place. As every general
notion, this is an abstraction: we have abstracted from the practical
impossibilty to run any real process infinitely, and in fact, for very
long. In this abstraction, no matter how long we have run the process, we
always can do, or observe, the next step. This is why this infinity is
called potential. At every specific stage the process involves no more than
quite a finite reality; it is infinite only potentially. \eon

\node{Actual infinity}
% part-of: Infinity

For {\sl actual infinity}, which is thought of as static and completed, we
have no place in our system of concepts. On the intuitive level, we cannot
imagine anything that would qualify as actual infinity, because neither we,
nor our evolutionary predecessors ever had anything like that in
experience. When we try to imagine something infinite, e.g., infinite
space, we actually imagine a process of moving from point to point without
any end in sight. This is potential, not actual, infinity.

On a more formal level we can demonstrate the incompatibility of the
concept of actual infinity with our cybernetic understanding of meaning.
Indeed, suppose that some abstraction [Exists]r[Exists] represents the
concept of an infinite object, and we use it while constructing a model.
According to our semantics, there must exist a representation function
[Exists]R[Exists] which recognizes whether a given state of the world
belongs to this concept, an if so, results in [Exists]r[Exists]. Moreover,
the function [Exists]R[Exists] must, by definition, always require a finite
time for coming up with a definite result. If the ``infinite'' object can
be always recognized as such in a finite number of steps, it is not
actually infinite, because it can be adequately replaced by a finite
object. If, in accordance with the intuitive concept of an infinite object,
its recognition
may require infinite time, then the representation function will have to
work, at least in some cases, infinitely, but then it is not a valid
representation function. Thus we cannot use the concept of actual infinity
at all.
\eon

As an example, consider the process of counting. We can imagine it going on
infinitely if we do not count real sheep or apples, but simply produce
consecutive numbers. Let us represent numbers in the unary system; then the
process is:

\centerline{ {\bf 1}, {\bf 11}, {\bf 111}, ... etc. {\sl infinitely} }

When we say that this process is infinite, we mean that whatever is the
current number, we can add one more {\bf 1} to it. Thus we deal with
potential infinity.

To convert it into an actual infinity, we must imagine an object that
includes in itself all whole numbers.
We call it the set of all positive whole numbers. Suppose that such a thing
exists. How would it be possible for an abstr action function
[Exists]R[Exists] to distinguish it from other objects, e.g. from the set
of all whole numbers with the exception of the number
[Exists]10^{50}[Exists]? Intuitively, [Exists]R[Exists] must examine the
infinite number of the elements of the set. Since this is impossible to
achieve in any finite time, the needed function [Exists]R[Exists] does not
exist.

What we can do, however, is to create an {\sl objectification} of the
process which generates all whole numbers. A machine which initiates this
process (and, of course, never stops) is such an objectification. This
machine is a finite object. It can be made of metal, with an electric motor
as the agent causing the process of work. Or we can describe it in some
language addressing a human agent, but requirung only simple ``mechanical''
actions uniquely defined at each stage of the process. Such descriptions
are known as {\sl algorithms}. If we use the English language for writing
algorithms, this machine, to be referred to as [Exists]N[Exists], could be
as follows:

\medskip
{\sl At the initial stage of the process produce the number {\bf 1}. At
each next stage take the number produced at the preceding stage and produce
the number obtained by putting {\bf 1} on the right side of it.'' }
\medskip

Now we can say that the set of whole numbers {\sl is} [Exists]N[Exists]. We
have no objections against sets defined in this way. Their meaning is
crystal clear. Their infinity is still potential.

The problem with contemporary mathematics is that many sets it operates
with {\sl cannot} be represented by finite mechanical generators. They are
{\sl uncountable}.
The question of the nature and meaning of these sets is, in the eyes of
contemporary mathemticians and philosophers, wide open. Yet their
usefulness is abundantly demonstrated. If there were no way to interpret
uncountable sets in our philosophy, it would undermine our claim that the
basic principles on which we build our philosophy are universally
applicable and sufficient.

Fortunately, we {\sl can} interpret set theory, as well as classical and
intuitionist logic, in our terms (see [Tur87]). Our interpretation assigns
quite definite {\sl meanings}, in our sense of the word, to the basic
concepts of logic and set theory, and does it without any recourse to the
concept of actual infinity. \eon

\node{God}
% part-of: Metaphysics

In a philosophy where the basis of existence is actions and agents, the
concept of God is naturally introduced. We observe a lot of hierarchical
relations between agents. The scope of one ``rank-and-file'' act of
electron scattering is very small. In comparison, my decisions {\sl
control}, i.e. change conditions for, a great deal of processes in the
tissues of my body, down to elementary subatomic interactions. God may be
understood as the agent on the highest level of the control hierarchy in
the world. What my mind is for my body, God is for the whole of the
Universe.

This definition, of course, is not a proof of existence. We can develop
both theistic and atheistic versions of cybernetic metaphysics. In the
atheistic version the accent is made on the concept of self-organization of
matter. According to this view, the control hierarchies of the world grow
up from the bottom, so that nothing is over them, no agent constructs the
hierarchies from above. According to the theistic conception the creation
of the Universe was the starting act of God. The laws of nature are
manifestations of God's will, which also directs the universal evolution of
the world.

The fate of a metaphysical theory is decided when it leads to verifiable
predictions, or at least a widely accepted scientific theory. Nobody has
yet derived a scientific theory from the existence of God. But to prove
that there is no God, it is not sufficient to develop theories which work
without God; it is necessary to derive predictions from the {\sl
non-existence} of God. This has not yet been done either. So we shall do
our best if we watch developments along both theistic and atheistic lines
of thought.
\eon
\eon

\bigskip

\ni {\bf References}
\medskip

\bib{Aun88} {Aune, Bruce, `Action and Ontology', {\sl Philosophical
Studies}, vol. 54, pp. 195-213.}

\bib{Rus48} {Russell, Bertrand, {\sl Human Knowledge: Its Scope and
Limits}, Simon and Schuster, New York, p.269 etc.}

\bib{Tur77} {Turchin, V.F. {\sl The Phenomenon of Science}, Columbia
University Press, New York, 1977.}

\bib{Tur87} {Turchin, V.F. A constructive interpretation of the full set
theory, {\sl The Journal of Symbolic Logic}, vol.52, pp.172-201, 1987.}

\bib{Tur91} {Turchin, V.F. Cybernetics and Philosophy, {\sl Proceedings of
the 8-th International Congress of Cybernetics and Systems}, 1991 (to
appear).}



