9. CONCLUSIONS.
COGNITIVE VISUALIZATION
OF LEIBNIZ'S MONADOLOGY
1. The natural mirror-like 1-1-correspondence between the set D of standard
hyper-real proper fractions and non-standard hyper-natural (integers)
of the transfinite w-order
is established.
2. The connection between Cantor's transfinite integers and hyper-natural
numbers of Non-Standard Analysis is established.
3. The new type of Cantor's w-like
transfinite integers is discovered.
4. The informal incompleteness of Cantor's transfinite integers Theory
is proved.
5. The intrinsic natural extension of classical Cantor's w
-Theory is constructed and the natural solution of Continuum Problem in
its week formulation 3 is offered in the framework of this extension.
6. The natural cognitive visual representation of the hyper-real numbers
system is proposed.
7. The natural cognitive visualization of Leibniz's Monadology is realized:
every vertex xÎ TR
(a "real" Monad) is a root of the same TR (a
"real" Universe outgoing to the right), and every vertex x-Î
TL (an "integer" Mirror-Monad) is a root of the same
TL (an "integer" Mirror-Universe outgoing to the left).
- See Fig. 2.
LIEBNIZ [14] (FOR VISUAL THINKING SOULS):
9. Indeed, each Monad must be different from every other…
These Monads are the real atoms of nature and, in a word, the elements
of things.
11. It follows from what has just been said, that the
natural changes of the Monads come from an internal principle, since an
external cause can have no influence upon their inner being. (Theod. 396,
400.)
12. But, besides the principle of the change, there must
be a particular series of changes [un detail de ce qui change], which constitutes,
so to speak, the specific nature and variety of the simple substances.
13. This particular series of changes should involve a
multiplicity in the unit [unite] or in that which is simple. For, as
every natural change takes place gradually, something changes and something
remains unchanged; and consequently a simple substance must be affected
and related in many ways, although it has no parts.
16. We have in ourselves experience of a multiplicity
in simple substance, when we find that the least thought of which we
are conscious involves variety in its object. Thus all those who admit
that the soul is a simple substance should admit this multiplicity in
the Monad;
51. But in simple substances the influence of one Monad
upon another is only ideal, and it can have its effect only through the
mediation of God, in so far as in the ideas of God any Monad rightly claims
that God, in regulating the others from the beginning of things, should
have regard to it. For since one created Monad cannot have any physical
influence upon the inner being of another, it is only by this means that
the one can be dependent upon the other. (Theod. 9, 54, 65, 66, 201. Abrege,
Object. 3.)
53. Now, as in the Ideas of God there is an infinite
number of possible universes, and as only one of them can be actual,
there must be a sufficient reason for the choice of God, which leads Him
to decide upon one rather than another. (Theod. 8, 10, 44, 173, 196 sqq.,
225, 414-416.)
56. Now this connexion or adaptation of all created things
to each and of each to all, means that each simple substance has relations
which express all the others, and, consequently, that it is a perpetual
living mirror of the universe. (Theod. 130, 360.)
57. And as the same town, looked at from various sides,
appears quite different and becomes as it were numerous in aspects [perspectivement];
even so, as a result of the infinite number of simple substances, it is
as if there were so many different universes, which, nevertheless are
nothing but aspects [perspectives] of a single universe, according
to the special point of view of each Monad. (Theod. 147.)
58. And by this means there is obtained as great variety
as possible, along with the greatest possible order; that is to
say, it is the way to get as much perfection as possible. (Theod.
120, 124, 241 sqq., 214, 243, 275.)
62. Thus, although each created Monad represents the
whole universe, it represents more distinctly the body which specially
pertains to it, and of which it is the entelechy; and as this body expresses
the whole universe through the connexion of all matter in the plenum,
the soul also represents the whole universe in representing this body,
which belongs to it in a special way. (Theod. 400.)
63. The body belonging to a Monad (which is its entelechy
or its soul) constitutes along with the entelechy what may be called a
living being, and along with the soul what is called an animal. Now this
body of living being or of an animal is always organic; for, as every
Monad is, in its own way, a mirror of the universe, and as the universe
is ruled according to a perfect order, there must also be order in
that which represents it, i.e. in the perceptions of the soul, and consequently
there must be order in the body, through which the universe is represented
in the soul. (Theod. 403.)
ACKNOWLEDGEMENTS. - The author wish to thank the International
Science Foundation of G.Soros (Grant No. ZZ5000/114, 1995), the Russian
Humanitarian Scientific Foundation (Grant No. 98-03-04348), the Russian
Basic Researches Foundation (Grant No. 98-01-00339) and Ministry of Science
and Technologies of Russian (Grants No. 05.04.1179, No. 05.04.1221, 1996-1997)
for the financial support of this work.
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