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Pavel Florensky: 
An actual infinity 
as the ontological basis for Truth

Ruslan Loshakov



I. 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 direct line.


                  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 distinguished   infinitum (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 actu.

This claim of classical rationalism is considered by Florensky as his Achilles’ heel.  Florensky affirms that the synthesis of experience and reason has a partial character.  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”.[1]  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.”[2]   

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 thinking. 

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.

  We 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 contradiction.   An 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 of declaration”.[3]

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”[4] 

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”[5]

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 ομοιουσιος, Arius 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”[6]

[1] Pavel Florensky. The Pillar and Ground of the Truth. Princeton University Press, Prinston , New Jersey , 1997, P. 26

[2] Ibid

[3] Ibid, P. 24

[4] Ibid, P.34

[5] Ibid, P.33

[6] Ibid, P. 59

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