The preliminary remarks.
First of all, we must distinguish the potential and
actual infinity. In his book “The pillar and ground of the Truth”
Florensky says: “The chief errors which are constantly made in
discussion of the infinite appear owing to the neglect of the fundamental
and wholly elementary distinction between actual
and potential infinity”.
1. Potential infinity.
Sometimes this kind of infinity is called as a bad
infinity. It is because the potential infinity is an infinite
incompleteness. The elementary example of this kind of infinity is a
A B C D
As we see, nothing prevents us to continue
indefinitely this line. We would continue this line indefinitely but we
never shall receive its whole. But
the incompleteness is a sign of imperfection. Therefore the potential
infinity is called as a bad infinity.
The other example of potential infinity is numbers’ succession:
1, 2, 3, 4, 5, 6, 7, 8…
2. Actual infinity.
The concept of actual infinity was known to
ancient Greeks and medieval thinkers. One can say that the Zeno’s
paradoxes (“Achill and a turtle”, “An arrow”) are based on actual
infinity. In XV century the famous theologian Nicolas of Cusa had
(actual infinity) and indeterminatum
(potential infinity). Still earlier, Thomas Aquinas distinguished positive
(actual) infinity from private
(potential) infinity. In XIX
century, the german mathematician George Kantor had given a mathematical
theory of actual infinity.
First of all, the actual infinity is infinity
of the whole. In this sense it is a perfect
infinity. Let’s take any segment.
We shall suppose that it is equal to one meter.
Evidently, we can divide it into two
parts and then divide into two parts each part. How
long we can do it? - Infinitely.
It means the finite segment contains the infinity. We can see that
it is a finished infinity.
We are faced following question: can we say that the segment (=1
meter) is a part of infinity? It is obvious that this segment is less than
infinity. But we receive here the paradoxical property of actual infinity:
the part of actual infinity is equal to the whole infinity. It is
expressed by the theorem of Kantor: finite
great number cannot be equivalent to its parts; infinite great number has
a parts equivalent to it.
II. The philosophical and
theological interpretation of actual infinity by Florensky.
The scientific knowledge is based on two
pillars: intuition and reason. The intuition is a kind of immediate
knowledge. For example, I see
the book on this table. One can say that the existence of this book is
given immediately by looking at it. I don’t need to prove the existence
of this object by reasoning, inasmuch it is given immediately. The
knowledge which we receive by means of our senses (seeing, hearing etc) is
a human experience.
But the scientific knowledge is not limited by
our experience. For example, nobody can see the elementary particles about
which the modern physical science speaks. This book is an object of my
vision, but the elementary particles are objects of our reason. Using
scholastic terminology one can say that elementary particles are objects
of intellectual vision.
It seems that the intuition and the reasoning,
the experience and the reason, have nothing in common. The intuition is an
immediate knowledge, whereas the reasoning is a mediate knowledge.
Nevertheless, the intuition and the reason are two kinds of identity.
Let’s consider the elementary perception of this book. One can say that
the book and its perception is the same. My vision of this book coincides
with the book itself. On the other hand, the perfect form of any reasoning
is an equation.
So, the theoretical physics is a system of equations. We can
conclude that the intuition and the reasoning are subordinated to a
metaphysical law of identity: A=A. Leibnitz,
a great philosopher and mathematician of the 17th and 18th
centuries believed that the metaphysical law of identity is the highest
law of human knowledge.
Classical rationalism of the 17th
and 18th centuries has been inspirited by an ideal of full
knowledge. What is a full knowledge? It is a synthesis of two sorts of
identity – intuition and reasoning – in the highest identity, i.e.
“the identity of thinking and being”. I would like to concern here the
distinction between the truths of
fact and truths of reason which has been introduced by Leibnitz. According to
him the experience give us only the truth of fact. Returning to our
example we say the book which I see on the table is the fact of our world.
As distinct from that the truths of reason which examples give us a
mathematics and logic are open only to reason. The project of general
mathematics (“mathesis universalis”) which we find in Leibnitz’
works had for an object the inclusion of the truths of fact into the
truths of reason. In this case a science could explain any fact of the
world. In other words it should be an actual knowledge of the world as the
whole, or knowledge of the world in
This claim of classical rationalism is
considered by Florensky as his Achilles’ heel.
Florensky affirms that the synthesis of experience and reason has a
In other words, it is impossible to include the truths of fact into
the truths of reason without rest. For example, the classical physics give
the formula of universal gravitation. But nobody can explain the nature of
universal gravitation’s force. In this sense the gravitation is an
inexplicable fact of our world. Another example: modern cosmology speaks
about the origin of our Universe from the Big Bang. Thus,
cosmology cannot explain the Big Bang itself. It is only a fact. In result,
a science cannot explain the world as a whole.
Let’s draw an important conclusion: if
the full synthesis of experience and reason is impossible, it means that
the scientific knowledge of our world is a potentially infinite process.
The progress of scientific knowledge is regressus
in indefinitum. Florensky says: “But this essence of reason is also
its Achilles’ heel. Regressus in
indefinitum is given in potentia,
as a possibility but not in actu,
not as a finished reality, a reality that is realized at a given time and
in a given place”.
But if the full synthesis of experience and reason is impossible it
means that there is a gap between them. It
is impossible for us to grasp the world in its actual infinity, it is
impossible to have a complete experience of the world. About this
Florensky says: “An impenetrable wall and uncrossable see; the
deadliness of stagnation and the vanity of unceasing motion; the
obtuseness of the golden calf and the eternal incompletion of the Tower of
Babel, i.e., a stone idol and “ye shall be as gods” (Gen. 3:5);
present reality and never finished possibility; formless content and
contentless form; finite intuition and boundless discursion – those are
Scylla and the Charybdis on the way to certitude. A very sad dilemma! The
first way out is to embrace obstinately the self-evidence of intuition,
which in the last analysis is reduced to the givenness of a certain
organisation of reason. The second way out is to plunge hopelessly into
reasonable discursion, which is empty possibility, to descend lower and
lower into the depths of motivation.”
From here follows that the Identity (A=A) is
not the metaphysical law of Being. Those
philosophers of the 17th and 18th centuries as
Descartes, Spinoza and Leibnitz saw the gap between the experience and
reason. However they saw in it the imperfection of our reason that will be
to overcome in a future. From this purpose the philosophers of the 17th
and 18th centuries developed the doctrine about method.
They supposed that the reason armed by the method will be capable
of full knowledge of the world. As
we already know full knowledge will be a unity of Thinking and Being. The
example of this unity has been given by Descartes in his famous statement
“Cogito ergo sum” (I think, consequently I am).
The metaphysical sense of that statement is as follows: thinking and being is the same. Descartes’ statement was an ideal
of scientific knowledge; it means the full dissolution of being in
According to Florensky it is impossible to
overcome “an impenetrable wall and uncrossable sea” dividing Being and
thinking. No method can help us here. The law of Identity is a partial but
not universal law.
are faced with a difficult question. I have just been told that the
identity is a partial law. But in this case what
limits the identity? We
can answer: the
identity is limited by Being as the Other.
I would like to use an example which I constantly give in my
lectures. We can say “Socrates is a man”, but we cannot say “A man
is Socrates”. About what does this example speak?
We can see that “is” is not the identity of “Socrates” and
a “man”. Otherwise the statement “a man is Socrates” would be true.
In other words, the “is” is neither “Socrates”, nor a “man”.
Consequently a “being” is a pure
difference, i.e. the Other.
We can also understand why a Being escapes from reason in potential
infinity. First, a rational knowledge is a process of identification. In
this case, a Being
as the Other is something else than rationality. Secondly, we must
strongly distinguish “being” from “existence”. The question
addressed to existence is the question “what is it?” But the question
addressed to being is the question “why”. Let me take an example that
is more concrete. I hear the noise behind the window and I ask “What is
it?” One may answer: “it is a rain”. I have here a perfectly
complete answer. I see that it is raining. The rain exists
as a fact of our world. But if
I ask “why is it raining?” the answer may be:
because there were rainclouds.
But if I ask again “Why were there rainclouds?” one may answer:
“it is because there is the atmospheric pressure”.
Then I can ask: “But for what reason is there an atmospheric
pressure?” We can see that every answer is incomplete. A Being escapes
from our questions in potential infinity. We have the following paradox
here: explaining a drop of rain requires the explanation of the Universe.
Here we can see that the rationalism of Modern
Time is based on substitution of Being by Identity. This substitution has
been declared by the famous thesis of Leibnitz: we
can recognize anything as existent if we can think of it without
absence of contradiction is an identity. In other words, that we cannot
think without contradiction does not exist.
“Only A that is equal to itself and unequal
to what is not A is considered by rationality as genuinely existent, as
το ον, το οντως
ον, as “truth”. On the other hand, to everything that is
unequal to itself or equal not tο itself, rationality refuses to
attribute genuine being, it ignores it as “non-existent”, or as not
truly existent, as το μή
ον. Only the first,
i.e. “existent”, is recognized by rationality, which rejects the
second, i.e. the “non-existent”. Rationality pins on this
“non-existent” the label το μή ον, does
not notice it, making us believe that it does not exist at all. For
rationality only an affirmation about the “existent” is truth. By
contrast, a declaration about the “non-existent” is, strictly speaking,
not even a declaration. It is only δοζα, an
“opinion”, only the appearance of a declaration, devoid of the power
Since the 17th century a scientific
reason became the legislator of Being. The rationalism of Modern Time is
“usurpation” of Being as the Other by reason as identity. This
usurpation is declared as the identity of thinking and being.
However a Being is a border of identity: not
a difference that is a special case of identity, but on the contrary is
every identity a special case of
Being as the Other. It
means that the law of identity needs a justification itself. What is this
justification? It is the Truth of Being or Being as the Truth. We already
know that a reason is a process of identification. But if the Being is the
Other it means that the Truth of Being is above reason. One can simply say:
the Truth of Being is a Revelation.
“Such is absolute Truth, if it exists. In it,
the law of identity must find its justification and ground. Abiding above
all ground that is external to it, above the law of identity, the Truth
grounds and proves this law. The Truth contains the explanation of why
being is not subject to this law”
Here we encounter a new consideration. We saw
that a process of scientific knowledge is potentially infinite. So, the
Truth of Being is above the
potential infinity. What
does it mean? Look at this number’s succession;
1, 2, 3, 4, 5, 6, 7, 8….
One can see that each member is less than the posterior
and more than the previous. Hence, the number which is outside of
potential infinity cannot be more or less: it
is actually more than any number. In other words, it is actual
infinity or absolute Unity.
“Thus, if the Truth exists, it is real
reasonableness and reasonable reality. It is finite infinity and infinite
finitude or – to use a mathematical expression – actual infinity, the Infinite conceived as integral Unity, as one
Subject complete in itself. But complete in itself, the Truth carries in
itself the whole fullness of the infinite series of its grounds, the depth
of its perspective”
Summing up, we can say that the
Being as the Other is the Unity as actual infinity. Besides, it is an absolute
Truth or the Truth as Revelation.
But how can the absolute Truth reveal itself in our world?
We shall give the following answer: the absolute Truth is revealed
as hypostasis, as a person.
What is hypostasis?
Discussing the properties of actual infinity we
saw that an actual infinite’s part is equal to the whole infinity. We
can repeat the theorem of Kantor: a finite
great number cannot be equivalent to its parts; an infinite great number
has parts that are equivalent
to it. Here we have a mathematical
analogy of hypostasis. We have to remember that the concept of
hypostasis is directly connected with
ομοουσιος which has been
introduced by Athanasius the Great. The Greek’s word ουσια
(essence) have the synonym υποστασις.
Unlike the first which designated a concept or a thing, the last
designated a person. For example: “Socrates is a man”. The “man”
is Socrates’ essence, whereas Socrates himself is hypostasis. In the 4th
century Christian theology discussed the problem of mutual relation
between the persons of the Trinity. In short, this problem is as follows:
God’s essence is infinity but his persons (Father, Son and the
Holy Spirit) are something definite. Therefore each person is a part of
divine infinity. It was the position of Arius: three divine persons have
different degree of divine essence: only God-Father possesses the greatest
degree of being; God-Son is less than God-Father but more than the
Holy-Spirit. Using the word
affirmed the similarity between the divine persons. We can here see that the
relation between the persons of the Trinity is disposed by Arius in order
of potential (bad) infinity. But we saw that potential infinity is the
image of imperfection. Hence, the divine being is imperfect. It has been
understood by Athanasius the Great. The
concept ομοουσιος means “one
essence with three hypostases”. Hence, according to Athanasius every divine person contains all completeness of the divine’s essence.
Hypostasis is a part which is equal to the whole or the divine
person who contains all infinity of the divine’s being.
I would like to finish my report by the words
of Florensky: “It is the victory over the law of identity that raises a
person above a lifeless thing and makes him a living center of activity”
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