(Moscow)

I. The acquaintance.

The topic of infinity is one of the central themes, which tie science and religion today. Infinity is one of the main attributes of God in the world religions. But such an understanding was formed not at once. Greek Antiquity understood infinity by its own way. Mainly, for the thought of Antiquity the infinity is something formless, becomeless and, hence, imperfect. In Pythagorean list of the opposites the infinity stands on the side of the bad(evil). The being in Greek thought is closely tied with the cathegory of measure and limit. Infinity appears here as a limit - less, a bound - less, something almost not - existing - m h o n . The infinite is something closed to chaos, and sometimes it is identified with the last. The infinite is brought by Plato and Aristotle together with the category of matter as a formless sub-stance of things, understood only by an "illegitimate thinking". At the same time, there are some thinkers in Greek philosophy, who use more positive understanding of the infinite. This is, at first, Anaximander. For him the first principle of cosmology is apeiron, an unlimited and boundless essence (from Greek a p e i r o n - the boundless), which all the things arise from and return to. One can name here, also, the atomists Leucippes and Democrites: their infinite empty space contains the infinite number of atoms, making the infinite number of worlds. But nevertheless, the prevailing point of view on infinity is other, the one expressed by Aristotle. For the last infinity exists only potentially, as an opportunity of the limitless changing: "Infinity is the matter for the perfectness of magnitudes and the whole only as an possibility, and not as a reality". So, there is no actual infinite body; Cosmos is finite as well; the infinite sequence of reasons doesn�t exist too. The actual infinity is given neither to the senses nor to the reason. The potential infinity exists, according to Aristotle, for numbers - as the growing series of integers, and for magnitudes - in the direction of diminishing, as the potentially infinite division of a segment. It is interesting that Greek mathematics, depended directly on this circle of ideas, imagines always its "straight lines" and planes as finite (even if arbitrary large).

The theme of infinity comes positively in the European science together with Christianity: God is infinite in the genuine sense, as omnipotent, omniscient and infinitely good. Thus, the human thought finds here the infinite essence for reflection, in contrast to the Greek tradition of Antiquity. But nevertheless almost a thousand years the European thought takes no new steps in the cognition of infinity. God reveals Himself, at first, as a way of salvation, of adoration and only then, to a degree, as the one of cognition. The infinity of God is revealed only "apophatically" (from Greek a p o j a t i k o V - denialable). It means: God is good, strong, generous, loving, knowing infinitely more than all in the world, but this infinity is not done as a subject of contemplation. Of course, there is the immediate way of the cognition of God: the ladder of mystical experience. But as mystics themselves teach, on the high levels of this ladder all the contemplations have to go out: the high degree of man�s cognition of God is an ecstasy, not a contemplation; it is an entering in the absoluteness of God�s mystery, in God�s darkness...

Nevertheless some scholastic authors begin to speculate about infinity. It is to them that we are obliged with the distinction of the potential and actual infinity, or in more medieval terms, the syncategorimatical and categorimatical infinity (Petrus Hispanus, William of Heytesbury, Grigorius de Arimino and some others). Duns Scotus began the revision of the Aristotelian view on infinity. He stressed that man, created as an "image of God", was given by Creator the spesific infinite ability, which is the base of all our speculations about infinity. Nicolas of Cusa was the thinker, whose teaching about the identity of the Absolute Maximum and the Absolute Minimum have contributed, may be, most to the legalization of the actual infinity in the European thought. But we are interested here more in the scientific approaches to the infinite, not pure philosophical ones, and therefore, we pass to the XVII century.

II. The seeming mastering of infinity.

The pure symbolic notion of infinity, designating just the experience of the spiritual intercourse with God, didn�t meet the cognitive ambitions of XVII century to see "clearly and distinctly". The modern time wanted to understand infinity and began to speculate about it. But how to justify this notion for human reason ? The majority of XVII century�s philosophers, including Descartes, retreats in the face of this problem�s difficulties. But one of the inventor of the differential calculus, G.-W. Leibniz takes a new step. Leibniz understood very good that it was impossible to "master" infinity, so to say, naively, without some new postulates. He formulates some new principles, to wich all his philosophical and scientific constructions are subordinated. Thus, the principle of continuity and that of the constancy of law allow Leibniz to carry the characteristics of the finite to the infinite and vice versa. "The properties of things are always and everywhere the same as now and here", - sounds Leibniz� principle of constancy of law. It helps to justify the differential calculus: one can realize the constructions in the infinitesimal ( the �infinitesimal triangles", the "infinitesimal straight lines" etc.) as usual finite ones.

The arbitrariness of these new postulates irritated many people. G.Berkeley opposed strongly to the idea to carry the features of the finite to the infinite. B.Pascal, - who himself made a definite contribution to the creation of the calculus of infinitesimals and, more else, to that of the projective geometry, which also uses positively the notion of the infinitely distant point, - understood deeply the role of infinity in science. But his attitude to the infinite was quite different from Leibniz�. Two infinities, one of the infinitely grand and the other of the infinitely small, between which man stands both in the material and intellectual sense, plunged Pascal into deep religious awe. The actual infinity, realized by man, reveals to us, according to Pascal, the infinity of God�s power and wisdom, inconceivable by human reason. On this high level, the cognition is impossible as a strategy of the audacious "unmasking" of mysteries. Here one can reach cognition only as a gift, seeking it in humility, as a revelation...

But nevertheless, the differential and integral calculus were invented and they "worked". Without any sufficient justification of their foundations they had been continuing their active development till the second half of the XIX century. It is interesting to remind the case of French famous mathematician A.Cauchy. Being the strong Catholic, and orientated on the Aristotelian tradition, he rejected the existence of the actual infinite set both in the world and in the mind. Having excluded the use of the infinitesimals in his version of the differential calculus, he nonetheless, couldn�t banish the actual infinity from his definition of the limit.

III. The "yawnings" of Cantor�s set theory.

Cantor tried at first to construct the set theory using the "naive" intuition of set. But one understood very soon that it was very uncertain way. To use the actual infinity we have to have some its fixed properties and so, we have to postulate them. Beginning from Zermelo we use the axiomatic form of the set theory. But the adoption of these axioms is very controvercial question. E.g. this is the case of the famous axiom of choice. Its formulation seems to be very simple: For any set M, there is at least one mapping g such that for each non-empty subset M' of M, g(M')Î M' (Zermelo's form). This axiom is widely used in mathematical analysis. Its role in the set theory is not less important. The proof of the comparability of cardinals, and, so, the construction of the scale of powers, are based on this axiom. But from the very beginning by no means all the mathematicians approved this axiom. It was not very surprisingly, because from it followed, e.g., such an unexpected result, as Banach-Tarski's paradox: It is possible to divide the sphere on the finite number of parts and make of them two spheres of the same size as the first one. Thanks to K.Goedel�s (1939) and P.Cohen's (1963) works, it was established that axiom of choice was independent from the other axioms of Zermelo - Fraenkel set theory. Instead of axiom of choice some alternative ones were proposed, e.g. the axiom of determinateness (1962). Using these new axioms we can construct new non-Cantorean set theory, new and very unusual mathematics.

The other famous example is Cantor�s continuum hypothesis. The founder of the set theory hoped to prove that the power of the continuum is just next after a₀ (the power of the natural series: 1,2,3, ... ) cardinal a₁ :

2^a₀ = a₁

In more general sense this proof would mean that the continuum can be "constructed out of points". But neither Cantor himself nor others succeded to prove this. By the efforts of Goedel and Cohen it was proven that the continuum hypothesis was independent of the axioms of Zermelo - Fraenkel set theory. And what is more, Cohen was inclined to consider the continuum hypothesis more likely to be false: the continuum is an "incredibly great" set, which it is not possible to approach by any gradual process of construction. All happens so, as if continuum contains a specific yawning metaphysical openness, a possible "gap for God".

The opportunity of this metaphysical openness in the "entrails" of the continuum one can realize, also, in the theory of the so called chaotic dynamic systems. The behavior of these non-linear systems, describing some processes in hydrodynamics, plasma physics, chemistry, biology, meteorology, cosmology, becomes immensely complicated under certain conditions. Although the equations themselves, which correspond to these processes, seem express the triumph of determinism, but in fact, because of the instability of these systems and the immense variety of possible attractors, we can't predict their behavior. Small perturbations of both the coefficients of the equations and the initial conditions lead to irreversible reconstructions of the macroscopic effects. Our inability to establish infinite precision in the definition of data and calculations does not allow us to govern the behavior of these processes, in spite of the seeming determinism. All this poses the question that there could be other kinds of governing of chaotic systems, which also can be interpreted theologically. God can either use the holistic "top-down causality" (J.Polkinghorn), or just intervene in the process as an infinitely precise Super-Calculator. The chaos turns out to be so complicated system that it makes to remember the Biblical: "The wisdom of this world is foolishness before God" (1 Cor. 3,19).

IV. Infinity as a "ladder to Heaven".

Some constructions of Cantor pose more immediately the question of the transition from mathematics to religion, from the pure intellectual contemplation to the religious communion to the Absolute Reality. This is tied with Cantor�s scale of all the ordinals W . It was realized that it was impossible to consider the set of all the ordinals W, as the whole. Otherwise, W would have, as every well-odered set, its own ordinal b, greater than all the ordinals of W. But because W contains all the ordinals, then we would have b> b, what is impossible (Burali-Forti�s paradox). Cantor tried to go out of this dead-end by the new notion of the inconsistent collection. Not every collection(Vielheit). is a set (Menge). The set means the collection, which it is possible to think as a whole without contradiction. E.g. the "set of the all which can be thought" (R.Rucker named it "a Mindscape") is not such a collection. Thus, theory of sets deals only with the consistent collections. And W is , also, an inconsistent one. But why Cantor didn't worry W to be as if outside the theory of sets ? J.W.Dauben is convinced that it is immediately tied with Cantor's religious interpretation of the set theory. Cantor was sure the scale of the transfinite ordinals to climb up to the Absolute, the infinity of God, which can be neither increased or decreased. And so, it is not surprising that the notion of this scale is associated with contradictions. This high unity, which is greater than both all the finite and transfinite, this Genus supremum has, already, the ontological nature, according to Cantor. Thus, the scale of ordinals turns out to be Cantor's specific Ontological Argument , side by side with Anselm's and Descartes' ones. At the same time, the scale W (and the corresponding scale t [tau] of all the cardinal numbers, alephs) is a specific way to God, a "ladder to Heaven", going from the pure mental costructions to the Being itself.

Here, the question of principle arises: how this pure intellectual ascent to God is possible? The traditional religious and mystical practice always uses the image of the ladder, as a metaphor of the ascent to God. One of the main Christian textbooks of the spiritual life, written by John Climacus (VI - VII c.), the Abbot of Sinai, was just named "Ladder to Paradise". This ladder is the succession of the spiritual purifications and initiations, neccessary to near God: the penitance, the fasting, the memory of death, the fighting with passions, the winnig of love to man and God. But is it possible the pure scientific intellectual ladder, going to God ? It is interesting that already from the time of Pythagoreans the theme of cognition (gnwsij) and that of the religious purification were considered together. Pythagoreans used an asceticism (vegetarianism, abstemiousness); they cultivated the probation of silence. Mathematics and music were used here as a technique for the purification of soul, too.

The theme of link of the spiritual growth and the scientific creative work goes through all the history of the European science. And here in these Cantorean reflections we meet the new revival of this old Pithagoreism in all its scientific and religious depth.

V. Conclusion.

The constructions of the non-Cantorean set theories, non-Cantorean models of continuum testify that our axioms of the infinite are rather artificial, accidental and, so to say, "importunate" with respect to infinity. The theories of set turn out to be usually the incomplete theories, what means that they contain some right propositions, which can be neither proved nor refuted (in a definite sense). More interesting is that it is not possible to complete them anyhow: one can add any new axioms, but the theories will never become complete. In other words, in our idea of the actual infinity there is always the great "gap", which can be a reason for the very unpredictable features of the systems using this idea. Notwithstanding that we could discern some structures in the infinite, it remains a mystery for us. Wherever one uses infinity - the foundations of the set theory, the foundations of the theory of probability, the chaos theory, the structure of continuum, free will, the problems of the unconscious etc. - one finds inevitably out the theological "umbilical cord" of the infinite: the yawning metaphysical openness to the Absolute.