Non-Euclidean Geometries as a Source of Faith in God for F.M. Dostoevsky and His Characters (on the Example of Ivan Karamazov)

Ou Menglian PhD in Philosophy Graduate student, Department of Cultural Studies, St. Petersburg University 190121, Russia, Leningrad region, Saint Petersburg, ul. Khalturina, 15, sq. 211 omenglian@gmail.com Other publications by this author

Introduction

Euclidean geometry is considered as one of the basic axioms of space. After all, within the framework of an ordinary worldview, the space where we live can always be measured and evaluated. However, the understanding of space at the level of the Aristotelian concept of space in antiquity and the natural science of scientists in Modern times ignores the metaphysical specifics of space, which already exceeds such classical properties of space as three-dimensionality, continuity, infinity, infinity, uniformity, isotropy, and absolute metric properties. With the development of science, it became clear that different real environments can be characterized by spaces with different properties. For example, the metric properties of space are not applied in the theory of relativity. The geometry of spherical spaces was formed by Riemann, which caused disputes about such issues as eternity and temporality, finiteness and infinity, etc. Subsequently, this contributes to the understanding of the fifth dimension of space, which is closely related to the spiritual hypostasis of a person. In a study on quantum cosmology, A. D. Linde also boldly suggested: "Could it be that consciousness, like space-time, has its own degrees of freedom, without which the description of the Universe would be fundamentally incomplete? Will it not turn out with the further development of science that the study of the universe and the study of consciousness are inextricably linked with each other and that final progress in one area is impossible without progress in the other?" ^{[9, p. 248]}. 

Therefore, the relational direction of understanding space is still relevant for us, which seeks to reveal the nature of its being, i.e. whether it has an objective character, or whether it stems from the peculiarities of our consciousness. As V. M. Samsonov asked the question in the article, "is space an abstract concept or a material reality?", and concluded that "it is necessary to pay special attention to the erroneous interpretation of three–dimensional and four-dimensional space as a deformable material medium similar to a solid body" ^{[10, p.18]}. The most famous answer to these questions is given by Kant, who created the well-known concept of space as an a priori (subjective) form of our contemplation, determined by the peculiarities of our consciousness, and not by objective being outside consciousness. He took one step further to the thesis that Euclidean geometry, as "embedded" in our sensory perception apparatus, is a transcendent phenomenon. Further, on the basis of the analysis of the experimental origin of non-Euclidean geometry, its philosophical meaning was revealed by G. Helmholtz in the article "On the origin and meaning of geometric axioms" published in the journal "Knowledge" in 1876. G. Helmholtz attached considerable importance to "discussing the philosophical significance of the latest research in the field of geometric axioms" and also opened the prospect of "creating analytically new systems of geometry with axioms other than those of Euclid" ^{[7, p. 122]}. In this sense, non-Euclidean and Euclidean geometry becomes the key to a deep understanding of the category of space and an important philosophical thesis

Influenced by the popularity of non-Euclidean geometry, Dostoevsky also reflects intensely on this issue, which is clearly reflected in his workbook. "Parallel lines should merge in infinity, but this infinity will never come. If it came, it would be the end of infinity, which is absurd. If parallel lines converged, then there would be an end to the world and geometric law and God, which is absurd, but only for the human mind. The real (created) world is finite, the immaterial world is infinite. If parallel lines converged, the law of this world would end. But in infinity they converge, and infinity is undoubtedly. For if there were no infinity, there would be no finiteness, it would be unthinkable. And if there is infinity, then there is God and the world is different, on different laws than the real (created) world" ^{[6, p. 43]}. This shows his special geometric understanding, which is already connected with religious search. Barsht claimed that "the idea of the infinity of space was connected in Dostoevsky's mind with the idea of eternity, he can be considered one of the first thinkers who built a model of a multi-layered space-time continuum" ^{[1, p. 136]}. K. G. Isupov joins this opinion, pointing out that "Dostoevsky stubbornly believes in "non-Euclidean" harmony, harmony in which there will be no place for suffering, doubts and torments. This faith, which he came to at the end of his creative career, is one of Dostoevsky's most valuable acquisitions" ^{[8, p. 30]}. It follows from this that for Dostoevsky, non-Euclidean geometry, in a sense, is already turning into a metaphysical spatial image. This image symbolizes the achievement of an ideal state of human existence based on a holistic perception of one's being and overcoming the limits of positivist-theoretical thinking, which simplifies reality. In Dostoevsky's novel The Brothers Karamazov, Ivan unexpectedly begins to talk about space, about its ethical and religious meanings, based on his imperfect knowledge of natural sciences, which allows us to consider the metaphysical meaning of non-Euclidean geometry in Ivan's desire to believe in God. 

Ev klidov 's dilemma of Ivan 's thinking

Ivan's phrase denotes the stated topic very well: "If there is a God and if he really created the earth, then, as we absolutely know, he created it according to Euclidean geometry, and the human mind with the concept of only three dimensions of space. Meanwhile, there have been and are even now geometers and philosophers, and even the most remarkable, who doubt that the entire universe or, even more vastly, the whole of existence was created only by Euclidean geometry, dare even dream that two parallel lines, which, according to Euclid, can never converge on earth, maybe, and would converge somewhere in infinity. My dear fellow, I decided that if I can't even understand this, then where can I understand about God? I humbly confess that I have no ability to resolve such issues, I have a Euclidean, earthly mind, and therefore, where do we decide about what is not of this world" ^{[4, p. 264]}. Thus, Ivan perceives the Earth's space on the basis of the principles of elementary, Euclidean geometry, i.e. on the basis of rational cognition, which does not pretend even to those unusual inputs that modern, "non-classical" science gives. Moreover, the inability to understand the data of science becomes an argument for him to admit his weakness in faith. He believes in God, but only formally, without acknowledging that He can realize his omnipotence in a radical transformation of earthly reality. His inability to recognize the power of God in relation to earthly reality is directly connected by Ivan with the inability to imagine non-Euclidean geometry. Ivan considers the laws of geometry to be inviolable, so nothing that goes beyond reason can happen in our world. And this leads to the fact that he perceives the good deeds of people only from the point of view of rational motives, only in their conditionality with real obligations and laws. He does not believe in the possibility of people acting directly with sincere religious love, he assumes only action "with a tear of lies, because of the love ordered by duty, because of the penance pulled on themselves" ^{[4, p. 265]}. This coincides with Gubailovsky's opinion: "At the same time, Ivan acts exactly like a mathematician — he considers the world (being, space) as a closed system that can be described by listing a set of axioms and specifying logically correct inference rules. Ivan's mind, contrary to his words, is completely non-Euclidean. The "Euclidean mind" cannot reflect on its own Euclideanness" ^{[3, p. 156]}. The foreign scientist Kimberly Young also argued that "by adhering to empirical facts in a closed space-time, Ivan eliminates contradictions and paradoxes of divine or metaphysical concepts, limiting the horizons of meaning to finite, flattened planes of (literal and symbolic) Euclidean space" ^{[11, p. 55]}.

In Ivan Dostoevsky shows the meaninglessness of dogmatic, formal faith. Ivan seems to humbly submit to dogmatic beliefs, but doubts ethical rules that seem meaningless to him, and that the world created by God can be transformed, while it is the transformation of the world that is the main thing in the true faith that the writer assumes: "I don't accept God, understand this, I don't accept the world created by him, the world of God, and I can't agree to accept it. I will make a reservation: I am convinced, like a baby, that suffering will heal and smooth out, that all the offensive comedy of human contradictions will disappear, like a pathetic mirage, like a vile invention of a weak and small, like an atom of the human Euclidean mind, that, finally, in the world finale, at the moment of eternal harmony, something precious will happen and appear, that it will be enough for all hearts, to quench all indignation, to atone for all the villainies of people, all the blood they shed, enough to not only be able to forgive, but also to justify everything that happened to people — let it all be and appear, but I don't accept it and I don't want to accept it!" ^{[4, p. 214]}.

Formally, Ivan is talking about a new world where human suffering will be redeemed, but he does not see a direct continuity between this new world and our real world, so he cannot accept the redemption of suffering as real and effective. As a result, Ivan falls into the trap of consistent rational thinking. Having a passionate desire for faith, but requires that faith be rationally justified and meet the criteria of his clear consciousness. But since faith cannot be rationally justified, he finds himself in an insoluble contradiction between the desire for faith and the impossibility of acquiring it according to the criteria that his mind requires.

In this regard, we can say that he treats the earthly world and its laws negatively from the point of view of his faith, which cannot be combined with the laws of the world that allow evil. But at the same time, he cannot recognize the possibility of another world or the transformation of the earthly, Euclidean world. 

The struggle between rational consciousness and faith

Ivan is often considered as an outspoken atheist based on his words "everything is allowed". Although in fact his doubts about God reflect his deep desire for Him, for human harmony and happiness. Therefore, respecting Alyosha for the firmness of his religious beliefs, Ivan tells him: "My little brother, it's not you I want to corrupt and move from your foundation, I might like to heal myself with you" ^{[4, p. 215]}. Ivan is a searching person who seeks to find a solution to his problem and find true faith. The main assistant in his search is not even Alyosha, but the elder Zosima, who clearly shows how Ivan's problem can be solved.

Alyosha's faith is directly conditioned by the conviction of the fidelity of the teachings of Elder Zosima. In fact, talking with Alyosha, Ivan is arguing with elder Zosima, and the main problem in their correspondence dispute is the possibility or impossibility for a person to imagine the existence of other worlds and other spaces (non-Euclidean). Ivan relies only on rationality and logical thinking, so he does not accept the idea of a plurality of worlds: "Even if parallel lines converge and I will see it myself: I will see and say that they have converged, but still I will not accept" [4. p. 215]. In this conscious belittling of his being to straightforward rationality, which proves the complete subordination of man to the laws of nature, Ivan calls himself a "bedbug": "I am a bedbug and I admit with all belittlement that I can't understand anything why everything is so arranged. It means that people themselves are to blame: they were given paradise, they wanted freedom and stole fire from heaven, knowing themselves that they would become unhappy, so there is nothing to pity them. Oh, in my opinion, according to my pathetic, earthly Euclidean mind, I only know that there is suffering, that there are no culprits, that one thing comes out of the other straight and simple, that everything flows and is balanced — but this is just Euclidean game, because I know it, because to live by I can't agree with her!" ^{[4, p. 222]}. Ivan compares himself to an insect crawling on the surface of Euclidean space and unable to feel another dimension of space, thanks to which Euclidean space can become "curved" and non-Euclidean.  

The struggle between rational consciousness and Ivan's faith is vividly manifested in his nightmare. The devil in this vision of his speaks about God and "other worlds", it turns out that he expresses exactly the idea that Ivan wants to believe in, but he cannot believe in the devil himself. Together with the devil, he recognizes both God and "other worlds" as an illusion: "You don't know, but do you see God? No, you are not by yourself, you are me, you are me and nothing more! You're trash, you're my fantasy! — That is, if you want, I am of the same philosophy with you, that would be fair. Je pense donc je suis, I know this for sure, the rest is everything around me, all these worlds, God and even Satan himself — all this is not proven for me, whether it exists by itself or there is only one of my emanations, the consistent development of my self, existing prematurely and alone..." ^{[4, p.77]}.

Nevertheless, the devil manages to force him to take a step towards accepting his story about the paradise world as the truth due to the fact that he gives a very everyday and realistic portrait of an "unbeliever philosopher" who believed in God after entering paradise and feeling heavenly pleasure. "And just as he was opened to paradise, and he entered, then, without staying two more seconds — and this by the clock, by the clock (although his watch, in my opinion, should have decomposed into its constituent elements in his pocket long ago, dear), without staying two seconds, he exclaimed, that in these two seconds, not only a quadrillion, but a quadrillion quadrillions can be passed, and even raised to a quadrillion degree!" ^{[4, p. 79]}. The devil bribes Ivan with the scientific accuracy of calculations of time and distance in the paradise "locality", and here the idea of the difference between the laws of the earthly world and that of paradise, which the devil tells about, appears: "But you're thinking all about our present land! Why, the present earth itself may have been repeated a billion times; well, it has become obsolete, froze, cracked, crumbled, decomposed into composite principles, again water, which is above the firmament, then again a comet, again the sun, again the earth from the sun - after all, this development may already be infinitely repeated, and all in the same form, down to the line. The most indecent boredom..." ^{[4, p. 79]}. It is significant how in one phrase the devil combines a purely scientific description of the natural process and a biblical phrase from the story of the creation of the world by God. The devil lures Ivan's consciousness to the area where his scientific consciousness can reconcile with religious faith.

For Ivan, this story becomes a revelation – he realizes that another space exists, and it is based on other laws than the space of the material world. Having taken one step towards the recognition of "other worlds", he finds himself on the right path, which can lead him to the final faith. 

Discovery of non-Euclidean worlds

The fact that it is the recognition of the existence of "other worlds" that exist according to laws other than the laws of the earthly world that is the main basis of faith, Dostoevsky shows by the example of elder Zosima.

Only after starting to develop the image of the elder Zosima, Dostoevsky in his workbook writes down the most important thing that he guessed in his image and in his faith: "He realized that knowledge and faith are different and opposite, but he understood ? comprehended, at least, or felt even only – that if there are others worlds and if it is true that man is immortal, that is, he himself is from other worlds, then, therefore, there is everything, there is a connection with other worlds" ^{[4, p. 201]}. Like Ivan, Zosima realized that faith and knowledge are incompatible, that they are "the opposite", but if Ivan could not combine knowledge and rationality with faith, then Zosima combined them by believing in other worlds as in addition to the earthly world. In Dostoevsky's world, faith is not found on the path of complete renunciation of knowledge, his heroes do not accept Tertullin's maxim "I believe, because it is absurd." Faith should complement and enrich knowledge, and not deny it, despite all their apparent incompatibility. The discovery of non-Euclidean geometries in science became for Dostoevsky the clearest expression of the possibility of such an "expansion" of science when it turns out to be consistent with faith. The most important point of his faith is the existence of "other worlds". When science recognized their existence, and even with laws sharply different from the laws of the earthly world, it embarked on the path of union with faith.

The devil from Ivan's vision visually connects our earthly world with "other worlds", puts them on a par when he talks about the seduction of some especially "valuable" souls: "You will forget the whole world and worlds, but you will stick to one of these, because a diamond is very precious; one such soul is worth another once the whole constellation — we have our own arithmetic. Victory is precious!" ^{[4, p. 80]}.

K. A. Barsht noted the fundamental importance of "other worlds" in Dostoevsky's understanding of the essence of religious faith and their direct connection with the diversity of people with their own shades of faith: "... Dostoevsky's heroes are sure that their author only assumed: there are as many worlds in the Universe as there are people, and each of them has its own set space-time horizons with their degree of curvature of existential zones, prospects and routes of salvation" ^{[1, p. 140]}. The most important point here is the fact that "other worlds" determine the posthumous fate of people, as follows from Svidrigailov's famous reasoning about ghosts (the novel "Crime and Punishment"): "Ghosts are, so to speak, scraps and scraps of other worlds, their beginning. A healthy person, of course, does not need to see them, because a healthy person is the most earthly person, and therefore must live one life here, for completeness and for order. Well, a little sick, a little disturbed the normal earthly order in the body, immediately begins to affect the possibility of another world, and the more sick, the more contact with the other world, so that when a person dies completely, he will go directly to another world.” I've been talking about this for a long time. If you believe in the future life, then you can believe this reasoning" ^{[4, p. 221]}.

Dostoevsky was probably greatly impressed by the discovery of non-Euclidean geometries (worlds) precisely in connection with his idea of other worlds as the most accurate expression of the religious idea of immortality. This convergence of the religious idea and the scientific idea makes it possible to accept faith even for an extremely rational consciousness. It is along this path that Ivan goes, thanks to the "scientific" reasoning that the devil leads in his vision.

The devil does not turn him away from faith, but, on the contrary, skillfully pushes him to it, using the moves permissible for Ivan's rational thinking: "A new method, sir: after all, when you completely lose faith in me, you will immediately begin to assure me in my eyes that I am not a dream, but I really am I already know you; then I'll reach my goal. And my goal is noble. I will only throw a tiny seed of faith at you, and an oak will grow out of it—and even such an oak that, sitting on an oak, you will want to join the "desert fathers and immaculate wives"; for you really, really secretly want to, you will eat acrides, you will drag yourself to the desert!" ^{[4, p. 80]}.

Barsht also pointed out the difference between the desire for "harmony" of Ivan and the desire for "paradise" of elder Zosima. Harmony can arise in the earthly reality according to the laws of this reality, it does not require the mystical abolition of the laws of nature to transition to a perfect state. Here again, Ivan's rational consciousness comes to the fore, which even perfection wants to realize according to the laws of earthly nature. Zosima, on the other hand, speaks of paradise, and he understands it precisely in the sense of a mystical complement to earthly reality, not removable from earthly reality and even invisible from it (as other dimensions of space are not visible), but constantly existing nearby. To see this addition and connect the earthly world with paradise in your life, you need a movement of faith, which should also complement, but not cancel rational consciousness; "life is paradise, we have the keys," as elder Zosima says in Dostoevsky's drafts to the Brothers Karamazov. The complementarity of man together with the earthly world, on the one hand, and paradise perfection, on the other, are dedicated to his words, recorded by Dostoevsky in drafts: "Around man is the mystery of God, the great mystery of order and harmony" ^{[4, p. 246]}. It is possible that Dostoevsky did not accidentally put the word "harmony" here, hinting that Ivan and Zosima's ideas about the ideal converge in their essence, but differ in the method of implementation. Ivan thinks only rational methods, but Zosima knows that the ideal is achievable only by a mystical alteration of the world and man, and even more of the man himself than the world.

Conclusion: Dostoevsky's "Ideal"

This thought is consonant with Dostoevsky's interpretation of the "ideal" in his diary entry of April 16, 1864: the ideal is not an abstract idea of moral perfection, but a living person, and the achievement of the ideal is the fullness of transformation, endowment "in the self of Christ", entry into His "synthetic nature", transubstantiation of being from the mortal, scattered, suffering into an immortal, all–one, full of bliss, where "we will be faces, without ceasing to merge with everything" ^{[4, p. 174]}. Such an "ideal" indicates the need for a transition to a transcendent dimension of being, which in scientific language means a transition to other worlds that exist according to non-Euclidean laws. Thus, for Dostoevsky, even the transformation of man to the ideal of Christ is indirectly connected with the idea of other worlds, hinted at by the discovery of non-Euclidean geometries.

Following the Slavophiles, Dostoevsky believed that man is whole in his essence and it is as a whole being that he is connected with God. This means that all his abilities, even if separated from the whole essence, still have an internal connection with it and can reveal in themselves the content associated with God. Dostoevsky sharply criticized scientific reason for its formality and for opposing faith. But he was sure that the mind could experience a transformation and restore its connection with the whole essence of man and with God. This process began after the discovery of non-Euclidean geometries and the introduction into science of the idea of the existence of other worlds that function according to laws that differ sharply from the laws of our world. God is not subject to the rational laws of the earthly world and cannot be found in it, but He exists in the heart of every believer. And he can and must reveal himself in everyone's heart. As A. G. Gacheva pointed out, "the truth, based on the supernatural law, in order to give humanity a real ground for salvation, cannot be introduced from the outside, on the contrary, it must arise from its very environment" ^{[2, p. 57]}. For Dostoevsky, faith in God is the belief that man himself will become like God. 

But is it possible to assume that the world is completely devoid of the divine principle? In Dostoevsky's worldview, this is impossible, for him nature also has a hidden divine principle and it can be worshipped, as Marya Lebyadkina does in the novel "Demons". God must somehow "appear" in the world around us, become manifest in this world. This presence of God in the natural world is gradually becoming visible even to science due to the fact that it has discovered "other worlds" that exist according to different laws and other geometries.

The revelation of the divine principle in man and humanity inevitably had to lead to its disclosure in nature. Man, discovering God in himself, could not but discover him in nature. Dostoevsky was the first to identify this coincidence of the path to God inside and outside, his heirs in Russian philosophy of the late XIX — early XX century brilliantly developed this revelation of his.  "The dogma of God—manhood is truly universal, cosmic, whose roots reach "to the depths of the earth and heaven, to the innermost secrets of the Holy Spirit. The Trinity and the created nature of man”, which is originally Sophia. The path from his confession to fulfillment is equal to the unction of humanity, the transformation of the world into the Kingdom of Christ" ^{[2, p. 61]}, A. G. Gacheva writes with reference to the works of the famous religious philosopher S. N. Bulgakov. Thus, the idea of God-manhood, picked up by Dostoevsky's further Russian religious philosophy, was inextricably linked with the idea of the "Mother of God" (in the words of V. S. Solovyov) and with the belief in the transformation of the world to the state of the Kingdom of Heaven. And, paradoxically, Dostoevsky saw in science itself the prediction of this mystical process through the latest theories about other worlds and non-Euclidean geometries.

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