Prof., PhD., BTh. Vladimir Katasonov

 

Interplay of science and religion:

the case of G.Cantor's set theory

 

аааааааааааааааааааааа For a philosopher of science Georg Cantor's theory of sets represents one of the most amazing example of interrelation of science and religion. Everyone who becameа even a little absorbed in the history of the invention of the theory of sets, inevitably feels the significance of religious problems for the development of this fundamental mathematical theory. At the same time, in his correspondence with theologians Cantor tried to propose some theological innovations, which were naturally connected with basic premises of his theory. This interplay of Cantor's scientific and theological views was grounded, of course, in the profound religious needs of Cantor's personality and in the peculiarities of his upbringing[1]. Besides that, the central notion of theory of sets - actual infinity -а was traditionally connected with theological problems.

ааааааааааааааа Cantor himself stressed very persistently the ties between science and religion. To his mind, these ties were mediated by metaphysics. This understanding we can find in the letter to priest Thomas Esser, 1 Febrary 1896. At first, Cantor indicates "...an indissoluble bond, which ties together metaphysics and theology; because, on the one hand, the latter is the guiding star, according to which the former directs its ways, and from which it receives the light, when the natural and ordinary lamps fail; and on the other hand, for its scientific development and perfomence, theology requires the service of a general philosophy"[2]. At the same time, Cantor stresses that "the every broadening of our view of the domain of that which is possible to create, must, therefore, lead to a broadened cognition of God"[3].

аааааааааааааааааааааа In its turn metaphysics was considered by Cantor as the base of science (and mathematics, as well) : "The foundation of the principles of mathematics and natural sciences is the lot of metaphysics; this considers them as its children, servants and assistants, from whom it must not remove its oversight, whom it must guard and control all the time as a Queen-bee, which is sitting in a beehive and sending thousands of diligent bees into a garden in order that they gather the nectar of flowers everywhere, and then, under her control, work the nectar together into an excellent honey; and which have to bring her the building stones for the finishing of her palace from the remote kingdoms of physical and spiritual natureФ[4].

аааааааааааааааааа But mathematics is the special type of science. According to Cantor, theory of sets itself is a kind of metaphysics, as well: "The general theory of sets...belongs totally to metaphysics. You would prove it easily yourself, if you evaluated the degree of generality of the notions of the cardinal number and the order type, these main concepts of set theory, and besides, if you noticed that the thought in these notations is absolutely abstract and there is no place for phantasyФ[5].

ааааааа аааааааааWe can show Cantor's understanding of the interdependence of levels of cognition by the scheme:

 

ааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааа Theology

 

ааааааааааааааааааааааааааааа Mathematicsаааааааааааа ааааааааааааMetaphysics

 

ааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааааа Science

 

аааааааааааааааааа It is more interesting to discuss the concrete examples of mutual influence of mathematics and religion in Cantor's work.

 

Theology influences mathematics

 

ааааааааааа In his polemics with the opponents of actual infinity Cantor very often showed their "refutations" of existence of infinity to be just petitio principii. From the very beginning they demanded from an actual infinite set to have either the properties of a finite one or satisfy the definitions of potential infinity. Cantor rightly insisted one had to investigate without any prejudices the new mathematical object itself, and not to attribute to it beforehand some habitual properties.

аааааааааааааааааа This distinguishing of actual and potential infinity was very important for Cantor. He repeated tirelessly that potential infinity is not genuine infinity[6]а and that the theory of sets does not deal with this kind of infinity. According to Cantor, the true infinity, i.e. actual infinity takes place between the finite and the Absolute. The latter means the infinite in God. The Absolute can neither be increased or decreased. It can't be studied mathematically. The true infinite can exist either in nature (Transfinitum ) or in the mindаа (transfinite numbers ). Every infinite set has his transfinite power. A transfinite number corresponds to every specially ordered infinite set (a well-ordered set ). In contrast to the Absolute, one can infinitely increase these latter numbers.

аааааааааааа The opposition to the legitimating of actual infinity in mathematics had a long ago established philosophical and theological ground. From the Middle Ages it became natural to use the predicate infiniteа only in respect to God. In the new philosophy the overwhelming majority of philosophers - Descartes, Locke, Kant, Hegel, - with minor exceptions ( Leibniz, at first ) considered the actual infinity as an essence without existence in the created world. In this perspective, to "handle" the actual infinity meant to "handle" God Himself, to "domesticate" God, "to bring down" God from Heaven to earth. In theology this titanic enterprise was traditionally seen as a very dangerous breaking of boundaries between the Creator and the creation. It was symbolized usually by the myth about the rebellion of the Titans, or the building of the Bible's Tower of Babel. In theology it was usually considered as some sort of pantheism.

ааааааааааааааааа Cantor knew all this very well. For him the reality of actual infinity and of transfinite numbers - both in nature and the mind - was guaranteed, at first, by their existence in God'sа reason. So, in his letter of 1895 to French mathematician Ch.Hermit Cantor doesn't agree with Hermit's opinion the number to have the reality like that of the natural things : "Let me notice that the reality and the absolute conformity of the natural numbers to laws seems to be stronger than that of the sensory world. This is only so for a very simple reason, namely, that the natural numbers, both separately and in their actual infinite totality, exist in GodТs mind as the eternal ideas in the highest degree of realityФ[7]. And what about the existence of actual infinite sets in the mind of man ? Here, Cantor was inclinedа rather to some Platonic understanding of the existence ofа mathematical essences. He himself named his position the "moderate Aristotelian realism", but some scholars are more inclined to call it "strong Platonism"[8] . At the same time, Cantor agreed man couldn't manage actual infinity like some usual thing. For man, the notion of actual infinity is not the conceptus rei proprius ex propriis, but only the conceptus proprius ex communibus, Cantor stressed[9] . So, man can manage it only symbolically, with the help of general predicates, exceptions and examples. Thus, partly against his own will, Cantor led mathematics to its modern formalistic mood.

аааааааааааааа The main difficulties of the theory of setsа began with the discovery of so called paradoxes. Burali-Forti published the first of them in 1897. If we consider the set of all ordinal numbers W, then, according to Cantor's theory, it has its own ordinal number b , which is greater than every member of W. But W contains all the ordinal numbers, so, b belongs W, as well. And we have b>b, what is impossible.

аааааааааа Cantor tried to escape this through the notion of inconsistency[10]. Not all collectionsа <Vielheit >а are setsа <Mengen>. There are collections which can't be thought as complete wholes (e.g. the collection of all ordinal numbers, or the collection of "all, that can be thought"). Cantor calls such a collection inconsistent. The system תּ(taw)а of all the cardinals (א -"alephs") turned out to be such an inconsistent collection, as well. So, one could always obtain transfinite cardinals greater than every given one and it was impossible to consider all the cardinals as a whole. Thus, as if that potentiality, which Cantor tried to escape transcending the sequence of natural numbers (the first transfinite powerа א₀ ), returned again in a new form...

ааааааааааааааааа Here, the natural question arose: how far does this scale of alephs go ? There was the natural "ceiling" for the growth of powers, the Absolute itself. Could one say that alephs rise "just up to" the Absolute, to the Infinite of God ? Dauben is sure of this. According to him, just this consideration was the basis of Cantor's calm attitude to paradoxes : "Essentially, he <Cantor> had recognized the impossibility of subjecting of entire succession of transfinite numbers to exact mathematical analysis. The nature of their existence as a unity in the mind of God constituted a different sort of perfection, and Cantor was not disturbed that it was beyond his means to comprehend it precisely"[11] . Thus, this scale of transfinite numbers ( as well, as the cardinals ) turned to be a "ladder on Heaven", leading up to the Absolute...

аааааааааа аааааWe see here the striking example how theology directs the scientific constructions. The impossibility to consider the whole series of alephs was paradoxical for almost all the mathematicians. It testified to the logical inconsistency of the main concepts of theory of sets. This was so for almost everybody, but not for Cantor himself. According to his thought, the ladder of alephs ascended up the Absolute, and just because of God's incomprehensibility in principle, there was not any paradox in the fact of inconsistency of the set of all the alephs. Where, for others, it was a problem to resolve, for Cantor himself, it was not a problem. In the letter to the English mathematician Grace C. Young in 1908 Cantor wrote about this: "I have never proceeded from any "Genus supremum" of the actual infinite. Quite the contrary, I have rigorously proven that there is absolutely no "Genus supremum" of the actual infinite. What surpasses all that is finite and transfinite is no "Genus"; it is the single, completely individual unity in which everything is included, which includes the "Absolute", incomprehensible to the human understanding. This is the "Actus Purissimus" which by many is called "God"[12] .

аааааааааааа But Dedekind asked: on what grounds can one state that the "first" infinite set , the set of all natural numbersа N ( having power א₀а )а is a consistent collection ?а Cantor answered[13] unsatisfactorily: one had to accept it as an axiom, just like the axiom of the consistency of large finite numbers. So mathematics turned to be a very conventional science. Exploring the domain of the infinite Cantor wanted to investigate reality itself and his slogan here was Newton's famous: Hypothesis non fingo[14] . But it turned out that in this exploration one had to accept the premises, whose sense was very far from any understanding... They were only hypotheses.

аааааааааааааа The other hypothesis was the continuum-hypothesis, which Cantor tried to prove all his life. K.Godelа and P.Cohen showed the continuum-hypothesis neither could be proved nor refutated in the axiomatic system of Zermelo-Fraenkel set theory. But, moreover, Cohen considers the continuum-hypothesis to be rather wrong and the power of the continuum to be higher than any alephs[15] ... Cantor hoped that the power of continuum to be the second aleph (א₁ ). This would mean that in some sense, the continuum "is made from points" by some gradual process of building[16] . Cohen thinks this is impossible. Following Cohen's reflections, it seems that we return to the point of view of Antiquity, where the continuum was considered as an irresolvable essence...

аааааааааааааа Cantor had other opinion. He believed in science, in the power of human reason to understandа the "constructions" of all, or almost all the things, including the natural ones, in the terms of abstract set theory.

 

Science influences theology

 

аааааааааааааааааа Personally, Cantor was sure that set theory had been given to him by Higher Power. He wrote to his friend the Swedish mathematician G. Mittag-Leffler in 1883: "I am enough far from attributing my discoveries to personal merit, because I am only an instrument of a higher power, which will continue to work long after me, in the same way as it manifested itselfа thousands of years ago in Euclid and Archimedes..."[17] .According to Cantor, the theory of sets, because of the high abstractness of its principles, belongs to metaphysics. Speaking about the attempts at theoretical considerations of actual infinity, especially in XVII century, Cantor drew the conclusion almost all of them to be unsatisfactory and unfruitful. The theory of sets only was the first break to new horizons:╗ It is by me that for the first time, the correct study of the infinite is presented to Christian philosophy. I know certainly and definitely that it will accept this study, the only question being whether that will happen now or later, after my deathФ[18].

аааааааааа Thus because of this metaphysical significance of set theory, Cantor considers it may find an "application" in theology. So, particularly, the famous Biblical statement in Sap.XI,21 "Omnia in pondere, numero et mensura disposuisti" must be understood, according to Cantor, as a testimony of the existence of actual infinite sets in creation." There is no here "in numero finito", - Cantor says[19] . Traditionally this passage was used as an argument against the possibility of actual infinity in creation. So, this "application" of set theory to theology practically meant its rebuilding. In mathematics, according to modern mathematician P.Vopenka, the mathematics based on Cantor's set theory turned into the mathematics ofа Cantor's set theory. In theology as well, the application of theory of transfinite numbers meant the reconstruction of theology, to make it more compatible with the theory of sets...

ааааааааа The problem of real existence of infinite sets (in "physical" world) was the subject of Cantor's constant concern and anxiety. The inventor ofа the set theory considered the essence of mathematics to be in its freedom[20] . So, pure logical consistency was enough for the legal existence of new mathematicalа objects in science. Cantor believed this logical consistency to be tied by some fundamental, but not very clear way with the transsubjective or, as he called it himself, transient reality, i.e. the realа "physical" existence of scientific objects and truths of scientific theories. Cantor claimed, at least in the beginning of development of his theory, the transfinite numbers are absolutely consistent. But the belief in their transient reality demanded a justification...

аааааааааа So, in the letter to Cardinal Franzelin Cantor tried to prove theologically the existence of infinite sets (Transfinitum). According to this proof, God, because ofа His highest Perfection, has the possibility to create all that is logically consistent, hence, the actual infinite sets as well. Then, because of His Benevolence and Magnificence, He necessarily creates the Transfinitum. Cardinal Franzelin rightly objected: "He who infers the necessity of a creation from the infiniteness of the Benevolence and Magnificence of God, must maintain, that everything creatable is indeed created from eternity; and that before the eyes of God there is nothing possible, that His Omnipotence could call into existence"[21] . Such a conclusion contradicted to Cantor's own definition of Transfinitum, which had to be increasable, and, generally speaking, this statement led to pantheism... But Cantor didn't consider Franzelin's argumentation as final[22].

аааааааааааааааа One would like to remark here. Notwithstanding that our main concern is the cognitive dimensions of the interplay science and religion, we can't completely ignore the significance of social positions of science and religion in different periods of history. There is a big difference in the dialogue of science and religion in XIX century or, e.g. in the XIII one. In the latter case controversies about Aristotelian science led to famous Paris' condemnation of 1277. Researchers had to obeyа this condemnation, and the reappraisal of Aristotelian science began, which lead, generally speaking, to the emergence of modern science. But in the end of XIX century Cardinal Franzelin's reproaches of pantheism to Cantor were not so effective. Cantor reserved his own opinion and he continued his efforts to change the understanding of some points of traditional theology, to make it more fitting to his mathematical theory. The same we can see today, when just because of the high social position of science, theological thought experiences a strong pressure from scientific knowledge, which leads, sometimes, to the serious theological innovations[23].

 

Knowledge and symbols

 

аааааааааа The last act of Cantor's "theological drama" took place in 1905. As it is well known, from 1884 he suffered the serious nervous breakdowns. In the following almost every year Cantor had to spend a part of time in various mental hospitals. But between the periods of illness and depression he could work and even published in 1895 his best exposition ofа set theory "Beitraege zur Begruendung der transfiniten Mengenlehre". Having left Halle's Nervenklinik in the spring of 1905 Cantor wrote to Mrs. Young in London the curious letter, where he told, in particular: "As you know, I had been hermetically secluded 5,5 month (from 17 Sept. to 1. March) from the world, except few visits from my family...The Muse afforded to me I employed to a renewed study of our Bible with opened eyes and postponing all prejudices. The result has been highly remarkable, as you will see by a little pamphlet (anonymous) of half a sheet, that I will send you perhaps in a week"[24] . The highly remarkable result is a 12-pages pamphlet named: "Light from the East. A conversation of a master with his pupil about the essential points of the original Christentum. Told by pupil himself Georg Jacob Aaron, cand. sacr. Theol.Ф[25]. In this conversation the master explains to his pupil that Christ was the natural son of Joseph of Arimathea. The last turns to be the embodiment ofа God the Father Himself...

аааааааааа It would not be so important to dwell on this "strange" interpretation, born in a mental hospital, if one did not see a striking closeness of such a "theology" to the spiritual impulses of Cantor's mathematical work. All his scientific life Cantor fought for the idea that human reason can manage, can "domesticate" actual infinity, which traditionallyа was the most characteristic symbol ofа God Himself. Moreover, Cantor believed the scale of his transfinite powers to ascend up to the Absolute, the Infinity of God. This passionate will to take down the actual infinity from its "celestial throne" was stopped by neither the apories nor the difficulties ofа justification of the backgrounds of set theory. And here, in Cantor's monstrous interpretation of Gospel's historyа in "Ex Oriente Lux", as if the circle have been closed. In this "taking down" of God from Heaven to earth, one can see as if the symbol of Cantor's scientific efforts...

аааааааааааааааа The historical research makes me more inclined to accept J.H.Brooke's third position in his interpretation of the relationship of science and religion, i.e. the historical lessons of this relationships "are far from simple"[26]. Behind the interplay of religion and sciences we reveal the unity of a person, who tries to express his experience of Truth, values orientations and Cosmos, both in science and theology. Metaphysics, likely, plays here the most significant role.

 

The special role of metaphysics

 

ааааааааааа In the history of genesis of Cantor's set theory and the interplay of his theological and scientific ideas we can find some features, which have the general significance. Let's imagine once more the situation. Cantor said: because of existence of the set theory, we have to understand some theological issues by new way. At the same time, notwithstanding his initial conception ofа mathematics ("the essence of mathematics is in its freedom") Cantor needed some support from other domains of culture, and especially in the fundamental claim of the theory: the existence of an actual infinite set. We see that Cantor, in order to guarantee this support, begins a great effort of cultural rebuilding, which can be better designated by that famous Russian word perestroyka. Theology has to be in such order that the set theory would have ontological sense, which would support, in its turn, theology ! This is the logic of such perestroyka ! At first glance we see here the classical hermeneutical circle. As though some general principle seeks to find its embodiment both in mathematics and theology.

аааааааааааааа аThis situation is not something extraordinarily new. One sees it in the history of culture rather often. For example, Leibniz, when he tries to find a justification to his Calculus of infinitesimals, again, seeks to explain the basis of the Calculus by some architectonical principles, which his theology must be submitted to, as well. These principles have metaphysical sense. Leibniz knew explicitly how formulate his principles: the principle of continuity, the principle of constancy of law[27] etc. In Cantor's case we are in more difficult situation. And here, both Cantor's theology and set theory becomes symbols of a hidden metaphysics.

ааааааааааа How could we describe the structure of this metaphysics ? The word metaphysics we understand here as a definite set of general representations about the being and the kinds of its cognition. What are the most significant points of Cantor's metaphysics ? We can express them in the following way:

I. Ontologicalа moments : Especially, the concrete ideas about the characteristics ofа the High Cause ( Supreme Power, Creator ), about what is more suitable for this High Cause. Thus,

A)а Cantor considered it was appropriate that the Supreme Power necessarily created all which is creatable, i.e. logically consistent (the passing from God's Perfection to His Benevolence, which we have discussed above). One should note here the opposite position of Cardinal Franzelin: there is the mystery of the limits of creation; we don't know, why God created only that, which He has created, and not all that is possible to create.

B) The existence of actual infinity in the world is for Cantor a necessary quality of the world, reflecting the infinity of God Himself. At the same time, there is the usual Christian symbol of the intensive infinity in the world: the embodiment of God's Logos in Jesus Christ, as the symbol ofа the infinite God's love to man. But this is not enough for Cantor's understanding of the relationships between God and creation. Cantor's views demand the presence of extensive infinity in the world, as the appropriate reflection of the infinite Creator.

II. аEpistemological elements: man, as an image of God, can to an extent "understand infinity" (primarily, existing in God's reason). Cantor's position here had its own peculiarities:

A)а Man can "understand" the infinity not directly, but only with the help of symbols, general predicates and examples; not as conceptus rei proprius ex propriis, but as conceptus proprius ex communibus. It was the great idea, a heritage of the Renaissance: to pretend to express God's perfections symbolically. Not by chance Cantor felt a closeness of his theory to Leibniz' logical and mathematical constructions. Leibniz seriously tried to invent a sort of logical calculus, mathesis universalis, by means of which "all the problems", concerning both nature and moral sphere, could be resolved. Characterizing this tradition, O.Becker wrote: "Being led by his thought, audacious without precedent, that the power of human symbolism extends to God as well, Leibniz, possibly stronger than all other philosophers, felt this mysterious motive, this inspiring, assaulting the Heaven, breaking the boundaries of eidetic thinking, flight of Western mathematics, Northen German in its kernel"[28] . Cantorа also worked within this tradition: the actual infinity, primarily existing in God's mind, must be accessible to human reasoning, also.

B)а There was, also, a very strong claim in Cantor's views, a sort of his own "ontological proof". The scale of powers, alephs, ascends, according to Cantor, up to the Absolute itself, to the single individual unity in God, which has an ontological nature. Thus, the constructions of our mind become, by some mysterious way, the reality of the ontological sphere, "Actus Purissimus", as Cantor himself named it. The scale of alephs was, as we said above, a sort of "ladder to Heaven". Traditional Christianity knew many spiritual "ladders to Heaven". One of them, which may be the best known, was the book "Ladder to Paradise" ofа St.John Climacus, the Abbot of Sinai's monastery (VI-VII century). It is a sort of textbook of spiritual life, the guide of ascent to spiritual perfection. The book was very popular both in Eastern and Western (especially medieval ) Christianity. But, it was a spiritual ladder to Heaven, whereas Cantor proposed a purely intellectual one ! Spiritual life, spiritual ascent are unthinkable without mystery, without mysteries of spiritual initiations, accompanying the way upwards (mysticism !). But, willy-nilly, Cantor has built just a scientific ladder to Heaven, to God, by the definition free of any shadow of mysticism... This ladder upwards was, at the same time, "the ladder of descent" of actual infinity. And, paradoxically, it was in a sense, the ladder of descent ofа God Himself, which was symbolically reflected in the interpretation of "Ex Oriente Lux"...

ааааааааааааааааааааааааааааааааааа *аааааааа *аааааааа *

ааааааааааааа Metaphysics is an intermediate sphere between the domains of positive religion (as revelation) and positive science. In this sphere human freedom playsа a decisive role. It is human freedom that decides here what must be the norms of being and cognition. Metaphysics carries weight science and religion. For the former metaphysics is the methodological guide and teacher; for the latter, it is the interpreter and sometimes a ... competitor. All the history of collisions between the different denominations and religions, struggles against heterodoxies, is that of the struggle for authentic understanding of God's revelation. This understanding is one always existing in the light of some metaphysics. At the same time, all the history of modern philosophy is also that of metaphysics, initially. Regarding the interdependence of science and religion, metaphysics plays here quite often a very important role. Intellectual principles, value orientations, accompanied by human passions, are embodied in metaphysical principles, organize science, and try also to deform theological structures, in order to give scientific theories religious sanctions, as well. Such the case of G.Cantor's set theory.