UPPSALA UNIVERSITY BACHELOR THESIS Department of Modern Languages Russian language D-level Hjalmar Eriksson Spring semester 2009 Flogstavägen 25B 752 73 Uppsala 073-0681523 [email protected]

Умом науку не понять1 An ideo-historical study of the rise of the Moscow mathematical school

Supervisor: Fabian Linde Department of Modern Languages

1 “Science cannot be grasped by the intellect”.

1 Contents I Introduction...... 3 I.1 Preface...... 3 I.2 Objectives...... 4 I.3 Methods and material...... 5 I.3.1 Limitations...... 5 I.3.2 Material...... 5 I.3.3 Methods...... 5 II The development of mathematics in Moscow...... 6 II.1 Mathematics in before 1900...... 6 II.1.1 Nikolaĭ Bugaev...... 7 II.2 The introduction of the new mathematics...... 8 II.2.1 Dmitriĭ Egorov...... 8 II.2.2 Pavel Florenskiĭ...... 9 II.2.3 Nikolaĭ Luzin...... 9 II.3 A school emerges...... 11 III Moscow mathematics and ...... 11 III.1 A crash course in the history of Russian thought...... 11 III.2 An alternative ...... 14 III.2.1 ...... 15 III.2.2 ...... 18 III.2.3 Indeterminism...... 20 III.2.4 ...... 22 IV A paradigm shift in mathematics...... 24 IV.1 The development of mathematics...... 24 IV.1.1 Cantor and infinity...... 24 IV.1.2 The deficiencies of set theory...... 25 IV.1.3 The foundational crisis...... 26 IV.2 Comparative analysis of the paradigm shift in the West and in Russia...... 27 IV.2.1 Exemplars of the new paradigm...... 27 IV.2.2 The philosophical discrepancy...... 29 IV.2.3 Luzin's philosophical transformation...... 30 IV.2.4 Concluding words...... 31 V Conclusion...... 31 V.1 Summary ...... 31 V.2 Evaluation of sources...... 32 V.3 Acknowledgements...... 33 VI Bibliography...... 34

2 I Introduction

I.1 Preface One of the most common things that the average person “knows” about Russia is that it is one of the mathematical superpowers of the world. This preconception is perhaps more of a prejudice, frequently coupled with the idea that all Russian mathematicians are grave old men with large beards. From studies of, and repeated visits to, Russia, I have realized that large beards are not necessarily more popular in Russia than in other parts of Europe. It is however true that, regardless of the beards, the are especially successful in mathematics. This is widely recognized among mathematicians and can for instance be seen in the number of Fields Medals, the most prestigious prize in mathematics, awarded to Russian/Soviet mathematicians. I have found this fact intriguing since I am fascinated by the cultural aspects of science. This is why I have decided to dedicate this essay to conducting an ideo-historical study of Russian mathematics. My initial review of literature on the history of science in Russia indicated it was not until the 20th century that Russia/USSR reached the status as one of the most important centers for the development of mathematics. The influential mathematical school did not, as one could expect, emerge in St. Petersburg, where mathematics had been a priority since the days when Leonhard Euler worked at the St. Petersburg Academy of Sciences. On the contrary, the school seemed to have its roots in Moscow and to have formed around the time when Moscow mathematics started to receive world-wide attention in the early 20th century. From the Moscow school sprung a number of well-known mathematicians, who became deeply involved in the development of modern mathematics, for example Andreǐ N. Kolmogorov and Pavel S. Aleksandrov2. I do not wish to trivialize the contributions to Russian mathematics by mathematicians in St. Petersburg and elsewhere, but it is this early Moscow school that is the scope of the essay.

Since the mid 1980s there has been a comparably large scholarly interest in the early development of the Moscow mathematical school. As the Soviet system crumbled, information about the first decades of the 20th century resurfaced. A connection between philosophers of the silver age, a religious renaissance of the early 20th century, and the mathematicians in Moscow was established [14, 15, 22]3. A number of Russian and foreign scholars (S. S. Demidov, S. S. Polovinkin, A. N. Parshin, C. E. Ford, A. Shields, L. Graham) have, since then, continued to examine the events and people of this period. In this research attention has been given to the connection between, on the one hand, philosophy and and, on the other hand, mathematics. While the research done covers most of the events and relations of this period, what is lacking is a general outline of the ideo-historical context, of which the mathematicians in question were part, and a critical review of how it manifested itself in their work. Demidov [14] and Ford [23] touch upon this and Graham & Kantor [25, 26], with a, lamentably, very narrow view of the cultural context, attempts to analyse the connection between culture and mathematics. This is exactly what I want to remedy with my essay. I attempt to show how the intellectual climate in Russia influenced the mathematics and mathematicians of the early period of the Moscow school of mathematics. I believe my essay is important as it diversifies our views on the development of science. From a western perspective the of science is all to often seen only in the context, and as a product, of western society and culture. I , however, argue that the intellectual climate among Moscow mathematicians early in the 20th century, in the terms of and , was, 2 For the transliteration of Russian I use the ALA-LC romanization system for Russian with all diacritics but omitting two-letter tie characters. 3 The source numbering can be found in the bibliography under VI. Sometimes a page reference is supplied as well.

3 firstly, different from the main western tradition and, secondly, a positive force with regards to the reception and further development of certain new discoveries in mathematics4.

I.2 Objectives The ontology and epistemology of the mathematical circle from which the Moscow school sprung was different from the predominant philosophy of Western scientists. This can be illuminated by the tendencies that I identify within the Russian tradition in my analysis of the philosophical climate of Moscow mathematics in section III.2. The tendencies I have identified are Platonism, mysticism, indeterminism and holism. They can be traced back to the traditions of Orthodox Christianity and slavophilism and have strong ties to the philosophical renaissance following Vladimir Solov'ëv's work in philosophy in the late 19th century. At the same time, around the turn of the 19th century, new discoveries in mathematics opened new fields of research. I will argue that the philosophical climate in Moscow mathematics was more compatible with the new discoveries in mathematics. This increased compatibility enabled the Moscow mathematicians to absorb the developments in the new areas of mathematics and contribute with novel research after a comparatively short period of exposure to these new ideas. The work in Moscow can be contrasted with the “foundational crisis” which at the same time persisted in much of the rest of the mathematical community. In analyzing the development of mathematics I use a reduced version of Kuhn's theory of scientific revolutions, presented in The Structure of Scientific Revolutions (1962), as an interpretive framework. I say a reduced version because I will use Kuhn's terminology and a general understanding of it, without going into the details of his theories. I do this since I need a vocabulary to discuss and understand the development in and of mathematics. Thus, I will consider the evolution of the new mathematics as a Kuhnian scientific revolution using the terms paradigm, exemplar, crisis, revolutionary science, normal science and paradigm shift to describe it. The terms are used as follows: a 'paradigm' is the ”theoretical beliefs, values, instruments and techniques, and even ” [2] shared by the scientific community during a specific period; the 'exemplars' of a paradigm are the instances of concrete science over which there is a consensus and that characterize the paradigm, for example key terminology, methods, theories, and examples; a paradigm 'crisis' is the process when the governing paradigm is questioned as a result of the massing of discoveries contradicting it; during a paradigm crisis 'revolutionary science' is made, it questions the preceding paradigm and offers suggestions for new directions; the opposite of revolutionary science is 'normal science', it is done within a paradigm to confirm it and extend its reach; 'paradigm shift' is the term which denotes the overall process which begins with the crisis and ends when the new paradigm is established. I will specifically consider the development as a paradigm shift. A key understanding in Kuhn's theories that I use is that what characterizes a paradigm is consensus around theories, examples, methods and even around epistemology and ontology. Hence, the paradigm shift is complete when such a consensus has developed, something that can be mirrored by a corresponding shift of generations. I will not use Kuhn's conceptions about the incommensurability of different paradigms and will leave it without comment.5 The first objective is to show that the conceptual framework in Moscow was different from the dominant philosophy in natural science in general, that is, in the words of Kuhn's theory of scientific revolutions, that the paradigm in Moscow mathematics was different from the western paradigm. The second objective, in turn, is to show that around the turn of the 19th century new discoveries in mathematics provoked a crisis and that the philosophy of the Moscow paradigm was

4 The development of mathematics is described in section IV.1. For now I will just let it be known that I use the term the new (discoveries in) mathematics to signify the developments that had a basis in Georg Cantor's set theory and the controversy around it. Today these discoveries are mainly included in the areas of real analysis and mathematical . In Moscow the new mathematics took the form of measure theory, descriptive set theory, and set theory based function theory. That is, in Moscow the emphasis was mainly on real analysis. 5 The information on Kuhn's theories is mainly based on [1].

4 more compatible with the exemplars of the new mathematics. Thus the paradigm shift was completed in Moscow before it could be completed in the west and the mathematicians in Moscow started doing normal science before it could be done in the west. This enabled them to become world leading scientists.

I.3 Methods and material

I.3.1 Limitations I have limited myself to the study of four of the central figures in Moscow mathematics around the turn of the 19th century. They are the mathematician Nikolaĭ Nikolaevich Luzin (1883-1950), widely acknowledged as the founder of the Moscow mathematical school [16, 32, 37, 28]; Nikolaĭ Vasil'evich Bugaev (1837-1903), mathematician and philosopher; Dmitriĭ Fedorovich Egorov (1869-1931), mathematician and co-founder of the Moscow school; and Pavel Aleksandrovich Florenskiĭ (1882-1937), priest, theologian, philosopher, mathematician and scientist.

I.3.2 Material The material I have studied is varied. Mainly it can be put into four categories: recent (published mostly within the last twenty years) articles specifically regarding the early Moscow mathematical school; some works by the four central figures (primary sources); overviews in the history of ideas, of science, and of mathematics of both Western Europe and of Russia; biographical works.

I.3.3 Methods The core of the essay is made up of three parts. The first part (II) is a presentation of the central figures combined with a brief review of the development of mathematics in Russia based on studies of secondary literature. The second part (III) has two sections of which the first is a short review of the history of Russian thought, also based on secondary literature. The second section holds my analysis of the philosophy in Moscow mathematics. Russian philosophy, as should be evident from III.1, had been primarily aimed at the theological, moral, practical and political spectrum, as opposed to the general western direction towards , logic and epistemology. As science is concerned with the production of sound and in a sense “objective” about the physical world and, as such, is a child of the western tradition, Russian philosophical currents were not directly applicable to it. My goal in the section is therefore to identify the philosophical climate of the emerging Moscow mathematical school as a continuation of typically Russian philosophy and to point at how it was adjusted to a philosophy of science. As mentioned in Objectives, I will do this by identifying the four tendencies Platonism, mysticism, indeterminism and holism. I will trace their history in Russian thinking and give examples of where they appear among the Moscow mathematicians. In the analysis I use both primary and secondary sources but the conclusions and classification are mine. I realize that the characterization that I make is not the only possible one. Some might think that the tendencies I choose are improper. I want to stress however, that I do not try to paint a complete picture of a coherent philosophical system. I have identified said tendencies by contrasting the views of the Moscow mathematicians with the predominant philosophy of their contemporary western counterparts. However, not satisfied with simply describing the Russians as anti-this or non-that, I aim to find the philosophical motives behind their criticism of contemporary currents. Thus, my analysis is a relative philosophical classification of the Moscow mathematicians. The contrasting views of the majority of other contemporary mathematicians will be treated in part IV. The third part (IV) of the essay is also divided into two sections. The first is a short review of the contemporary developments in mathematics based on secondary literature. The second section is an analysis of the concerned period in mathematics as a paradigm shift. With the use of Kuhn's terminology as an interpretive framework I attempt to explain how the philosophical standpoints of Moscow mathematics facilitated the assimilation of the new mathematics. This will

5 be done by showing how the exemplars of the new science were relatively compatible with the preceding Russian philosophy. To support the analysis, I also make a comparison with the situation in Western Europe. The interpretation is mine and based on the preceding material and conclusions. The analysis in section IV.2 is conceptually very close to the one in section III.2. It may be read as an effect of the application of the analysis made in III.2 to the development of mathematics, but it should also be considered as further illumination of the conclusions of that section.

II The development of mathematics in Moscow

II.1 Mathematics in Russia before 1900 Surveys of Russian history usually start out in Kievan Rus', a medieval realm which spanned most of modern day Ukraine, Belarus and western Russia. For several reasons, but perhaps most noticeably because Russia did not the Renaissance that supplied Western Europe with the Aristotelian heritage of and logic, Russia was for a long period very underdeveloped. Science and mathematics were almost unheard of in Russia before the 18th century. Because of this, it is possible to speak of an introduction of science in Russia and regard the period before it as relatively irrelevant. The process of modernization was already slowly under way but I will begin when it accelerated greatly as the urgent need for development was ultimately realized by the emperor. The first attempts to introduce formal education in Russia were made by Peter I at the beginning of the 18th century. He grasped Russia's precarious situation and became determined to modernize his country by reforming the government and society. This brought with it a demand for the development of practical sciences such as shipbuilding, cartography and gunnery. Peter realized that these arts depended on mathematics and put specific emphasis on the teaching of mathematics [42:161]. Furthermore, his intentions for the uses of the subject gave rise to a tradition of applied mathematics in Peter's city, St. Petersburg, that lasted uninterrupted for centuries. Vucinich [43:175- 177] also points out that mathematics was essentially ideologically neutral and not subject to religious and mystic dogma, as were many of the other sciences. Because of this neutrality it could evolve without the interference of reactionary forces, which have been very influential during the course of Russian history. When mathematics was finally introduced as a science in Russia, it set off from a good start with Leonhard Euler (1707-1783) enrolling in the new St. Petersburg Academy of Sciences in 1727. He was no doubt the greatest mathematician of the 18th century and his work was to, through his disciples, influence mathematics in St. Petersburg even into the 20th century [42:169, 174]. Regardless of its theoretical depth the common denominator in Euler's investigations was the application of mathematics to natural phenomena [43:93-95], just as Peter had wanted. This was the tradition that thereafter thrived in St. Petersburg. In the 19th century there appeared within the Petersburg tradition a number of mathematicians, who became widely recognized. The first of these was Mikhail Ostrogradskiĭ (1801-1862) who, in the tradition of Euler and the French mathematicians Legendre, Fourier and Cauchy, was concerned with the development of mathematical analysis and its application to physics [43:309]. The next to reach fame was Pafnutiĭ Chebyshëv (1821-1894) who followed the tradition of mathematical physics but also made contributions in probability theory and number theory [43:327-328]. From Chebyshëv grew a school of mathematicians concerned mainly with applied mathematics and who were quite conservative. Their conservative stance can be seen in their stubborn use of classical analysis and in their critical assessments of unconventional ideas and methods [37:278]. Quite apart from the tradition in St. Petersburg, mathematics was initially not in high regard in Moscow. Not until the middle of the 1830s was the teaching of mathematics at Moscow

6 university expanded to include such areas as differential and integral calculus and the integration of differential equations [43:328; 19]! Thanks to the expansion of the curriculum, mathematics was, by the 1860s, one of the leading fields of investigations in Moscow [40:294]. In 1864 the Moscow mathematical society was founded [40:295], an organization that came to have profound impact on mathematics in Moscow. The society was the “center of all mathematical life in Moscow” [28:317]. The Moscow mathematicians, and especially the society, was, however, controversial in the eyes of the St. Petersburg mathematicians, a contrast which was still evident as late as in the 1920s [37:288, 294-295]. The views of the society during this period are described on the society's own homepage as ”violent antipositivism, attraction to idealist and even religious philosophy, orthodox tendencies and monarchism”6. These views were held by the society's first ever secretary and subsequently vice president and president N. Ya. Tsinger [44:350; 13:117] and also by P. A. Nekrasov, another long time member of the presidium [28:311]. However, most of all, this inclination was due to the leading mathematician in Moscow at this time, the Nikolaĭ Bugaev mentioned in I.3.1.

II.1.1 Nikolaĭ Bugaev Nikolaĭ Vasilevich Bugaev was born in Dusheti, Georgia, in 1837. At the age of ten he was sent to Moscow to go to school. In 1855 he was enrolled in the University to study mathematics. His teachers were the men who later founded the Moscow mathematical society. In 1863 he defended his Master's thesis and was sent abroad for two and a half years to prepare for a position as professor. After his return he defended his doctoral dissertation and was appointed professor at Moscow University. Bugaev was elected to the presidium of the mathematical society as secretary in 1869. Later (1886) he was elected vice president and finally president (1891), which he stayed until his death in 1903. Bugaev's research interests were within, among others, the fields of number theory, elliptical functions, differential equations, and algebra.7 Bugaev was not only interested in mathematics, he was a philosopher as well and is, because of this, significant for this essay. Bugaev started out as a positivist but as Demidov [13:117] writes, during the 1870s the philosophy within the Moscow mathematical society was changing. The aforementioned Tsinger at Moscow university held a speech where he presented rationalist as an alternative to [13:117]. Bugaev soon picked up on this and created a philosophy of his own called Evolutionary monadology, based off of Leibniz'8 monadology. Evolutionary monadology is an idealist philosophy where the basic constituents of are individual spiritual entities called “monads” [39]. Bugaev also attempted to create a mathematical theory and philosophy of discontinuity called arithmology. He created arithmology because he considered the predominant “analytic” world view to be unacceptable. Bugaev reasoned that the analytic world view, the materialist conception of the world where all change occurs continuously and according to pre-determined laws, was not complete. He argued that there are processes which are discontinuous and subject to chance and finality (as opposed to ) [39]. One example of such a process is , another is social revolution. Bugaev was also a founder and very active member of the Moscow psychological society [13:116], to which he contributed papers about his evolutionary monadology and the freedom of will [44:352]. He was concerned with the application of mathematics to all fields of thinking, something that was meant to be accomplished with the help of arithmology. He was also a well-known public figure and friends with, among others, the writers I. S. Turgenev and L. N. Tolstoĭ, and with the philosopher Vladimir Solov'ëv [13:117]. He was furthermore the father of Boris Nikolaevich Bugaev, better known as Andreĭ Bely, a famous symbolist poet and novelist [28:484]. I allow this much space for Bugaev since his ideas made a lasting impression on the

6 “воинствующий антипозитивизм, увлеченность идеалистической и даже религиозной философией, православные настроения и монархизм“. [19] 7 The information in Bugaev's biography is based on [40:297-299, 28:483-485]. 8 Gottfried Leibniz (1646-1716) was a mathematician and philosopher. He formulated the foundations of calculus independently of Isaac Newton. His notation is still in use in calculus today.

7 mathematical community in Moscow. As the leading Moscow mathematician and a very active participant in Moscow intellectual life, he, along with Tsinger and Nekrasov, caused the intellectual climate in the mathematical society to lean towards idealist philosophy. In mathematics he gave rise to a significant interest in function theory and especially in discontinuous functions [13:122-123]. Bugaev's students Egorov and Boleslav Kornelievich Mlodzeevskiĭ (1858-1923) were influenced in this direction [13; 14; 30:171-172]. They were the ones who came to bring the new mathematics to Moscow. Bugaev's philosophical views also had a strong influence on Florenskiĭ [16:30-31; 18:598- 599].

II.2 The introduction of the new mathematics Bugaev died in 1903, late enough to have encountered the new mathematics. It is however doubtful whether he realized its importance for his program, but he had followers who in turn did just this. We will meet them in this section. They are all considerably younger than Bugaev but were directly or indirectly influenced by him.

II.2.1 Dmitriĭ Egorov Dmitriĭ Fedorovich Egorov was born in 1869 in Moscow. He finished school in 1887 and was enrolled in the faculty of physics and mathematics at Moscow university. He studied for Bugaev and wrote his first scientific work under his supervision. As Demidov [13:130] writes, Egorov probably realized the futility of Bugaev's approach to a theory for discontinuous functions, arithmology as a mathematical theory, but that he, as a result of Bugaev's influence, acquired a firm interest in discontinuity. Egorov completed his university studies in 1891 with a thesis in differential geometry that was highly praised by Tsinger [30]. On the recommendation of Tsinger and Nekrasov, Egorov was admitted to the university to prepare for a position as professor. He became private docent in 1894 and finished his doctoral thesis in 1901, after which he was sent abroad to Paris, Berlin and Göttingen. At his return in 1903 he was made extraordinary professor and, in 1904, professor. Egorov was a splendid teacher and, although he was strict, took a personal interest in his students, especially in Luzin, as we shall see later. Together with Mlodzeevskiĭ he also reformed and developed the teaching of mathematics in Moscow. They were the ones who introduced the new mathematics in Moscow. Mlodzeevskiĭ, ten years older than Egorov, upon his return from an academic mission during which he had studied the new mathematics held the first course, which was in the theory of functions of a real variable, in 1900. Egorov appropriated the new mathematics during his academic mission in 1902/03 and Kuznetsov [30:189] claims that he was one of the first mathematicians in Europe to realize the impact that Lebesgues new theory of integration9 would have on mathematics. For the remainder of the decade Egorov worked continuously for the complete introduction of the new mathematics in Moscow. Demidov [13:113-115] suggests Mlodzeevskiĭ's and Egorov's search for new areas was provoked by the difficult relations between the mathematical circles of St. Petersburg and Moscow. He writes that the St. Petersburg mathematicians failed to appreciate the new mathematics since they based their evaluation of new discoveries on whether it was possible to directly apply them to physics, mechanics or astronomy. This is reiterated by Phillips [37] and Iushkevich [28]. We can conclude that Egorov was paramount for the distribution of new mathematical ideas in Moscow [40:425], but he also reformed the pedagogical work by introducing advanced seminars. The seminars, which began in 1910, became the starting point for the subsequent Moscow school of mathematics with basically all outstanding mathematicians of the new generation participating in them. Active participation by students in meetings of this kind was, however, not completely new. Already at the beginning of the decade, a student circle was organized within the mathematical

9 This theory enabled mathematical analysis to handle a much wider class of functions, including classes of discontinuous functions. It was a very important result in the mathematics of the previous century.

8 society by the young student Pavel Florenskiĭ, who will be treated shortly. To complete the presentation of Egorov I must mention his philosophical and religious interests. Egorov was a devote believer and refused to compromise his beliefs. Of course, this eventually made him a target for the Soviet authorities. Contemporary accounts relate how he was visited in his home by priests who acted with respect towards him and even kissed his hand, that on his desk religious literature lay beside mathematical books. He is also reported to have been interested in philosophical questions and to have held Kant in high regard [17:137]. After the revolution Egorov became involved in Name worshipping10, a current in Russian Orthodoxy that will be mentioned in III.1. A group of Name worshippers sometimes met at Egorov's apartment in the early 1920s [26:70]. This involvement was one of the main reasons for Egorov's arrest in 1930. It was Florenskiĭ, next in line to be presented, that attracted Egorov to Name worshipping.

II.2.2 Pavel Florenskiĭ Pavel Aleksandrovich Florenskiĭ was born 1882 in a town called Yevlakh in what is now Azerbaijan. Demidov & Ford [18:597] write that he was brought up in an atmosphere of atheist , views that were held by a majority of the intelligentsia. He went to school in Tbilisi and, after his graduation in 1900, began studying mathematics in Moscow. He went into mathematics on his parents insistence, even though he was not inclined to continue his studies. A profound spiritual crisis in 1899 had erased his faith in the scientific world-view and had provoked his conversion to Orthodox Christianity. In school, Florenskiĭ had shown talent for mathematics and natural sciences but when he entered university he had already decided to formulate a world-view of his own to replace the common scientific one [18:598]. In Moscow, Florenskiĭ encountered Bugaev's ideas about discontinuity and his alternative philosophy. Florenskiĭ was intrigued by these, as can be seen by the direction of his investigations which was formulated as “the idea of discontinuity as an element of a world view” [24:26]. This is precisely in line with Bugaev's ideas. Florenskiĭ, however, incorporated the new mathematics in his work, which is one of the first examples of its application in Russia. Florenskiĭ did not only use the new mathematics in his own work but, as mentioned above, organized a student circle within the mathematical society, where he held talks about the new discoveries [37:282]. The meetings would even be visited by some of the staff, noticeably Egorov [Fel: Det gick inte att hitta referenskällan:566] and Mlodzeevskiĭ. Florenskiĭ also wrote the first article about Cantorian set theory in Russian: “On symbols of the infinite”, published in Novy Put in 1904, a journal of the Religious-Philosophical society of writers and symbolists.11 Florenskiĭ graduated from Moscow university in 1904 but declined the offer of a fellowship in mathematics to instead study theology and philosophy. He would move on to become a priest and one of the most important philosophers of the Silver age, the religious renaissance of the late 19th/early 20th centuries. His significance for this essay is due to his very early interest in the new mathematics and his influence mainly on the final character to be presented, Nikolaĭ Luzin. Correspondence between Florenskiĭ and Luzin spanning almost 20 years was recovered in the early 1980s and published in Istoriko-matematicheskie issledovaniia in 1989 (no.31). I will use this correspondence later, to establish Luzin's scientific interests and philosophical and religious views. Some of Florenskiĭ's work is also interesting and I will use it to extrapolate the philosophy of the central figures.

II.2.3 Nikolaĭ Luzin Nikolaĭ Nikolaevich Luzin is the main character, if any, of this essay. He is recognized as the founder of the Moscow mathematical school and his immediate and secondary disciples constitute a large portion of the well-known Russian mathematicians of the 20th century. This is widely

10 Имяславие. 11 This paragraph is mainly based on [14; 22]

9 acknowledged12, as is the fact that this school grew out of the “Luzitaniia”, which was the close-knit group of Luzin's students. Most biographical accounts about Luzin accordingly starts off from the point of Luzin's first publication and the subsequent formation of the Luzitaniia13. New discoveries during the 1980s and 90s, for example the letters of Florenskiĭ and Luzin mentioned above, have completed the picture, especially in regard to Luzin's emotional development14. I use this information in my presentation. Luzin was born in either Tomsk or Irkutsk, the source material differs on this point, in 1883. It is definitive that the family was living in Tomsk at the time when Luzin started to go to school. Many sources comment on his poor performance in mathematics and Kuznetsov [31] instead points at a broad interest in philosophy. To remedy his son's inability in mathematics, Luzin's father hired him a private tutor who noticed that Luzin was not adept at memorizing methods and formulas while he actually possessed originality and great problem solving skills. This was a very fortunate for the development of mathematics as the family, once Luzin finished school in 1901, moved to Moscow so Luzin would be able to enter the faculty of physics and mathematics at Moscow university. At this time it was not clear that Luzin would pursue a career in mathematics, an issue that would stay unresolved for eight more years. Luzin first went into mathematics to build a sound foundation for an engineering career but he was soon captivated by the subject in itself. Phillips [37:282] attributes this to Mlodzeevskiĭ's course in the theory of functions but Luzin was already at this time coming under the influence of Florenskiĭ as well. In his letter of August 4 1915 [15:177-179] he recalls the great impression that Florenskiĭ made on him at the meetings of the student circle of the mathematical society. Meetings that, as we have seen, dealt with the new discoveries in mathematics. Luzin in the letter also notes how proud he was that Florenskiĭ suggested him as his successor as secretary of the circle in 1904 when Florenskiĭ graduated. In the letter Luzin tells of how he idolized Florenskiĭ and of a wish to get closer to him, but also of the feeling that his intellectual powers were inadequate and that he would only shame himself. It was also in 1904 that the correspondence between Luzin and Florenskiĭ, that was to continue for almost twenty years, began. The following year Luzin started to experience the spiritual difficulties that would prevent him from committing to mathematics for several years to come. In the letter of May 1 1906 [15:135-139] Luzin lays out his heart to Florenskiĭ. Triggered by seeing the hardships of poor women, having to resort to prostitution, while he himself was “not only studying, but enjoying, science” (as opposed to something more useful) Luzin's materialist world-view collapsed. His faith in science disappeared. Luzin had not only attracted the attention of Florenskiĭ though, but also of Egorov. Luzin writes in the same letter that, seeing him in such a state, Egorov sent him abroad to Paris, where the letter is written. Phillips [37] instead claims this was because Egorov didn't like to see Luzin's education interrupted by the revolution. Either way, Egorov was indisputably involved. Following the return to Moscow Luzin completed his exams and, again on Egorov's initiative, decided to contiune on the academic path. But this path was not an easy one as one can gather from Ford's [22] and Phillips' [37] accounts. Ford's article relates how Luzin's spiritual crisis evolved and how Luzin continuously sought the advice of Florenskiĭ. Phillips on the other hand shows how heavily Egorov sponsored Luzin by twice intervening to procure him an extension of the time allowed to complete his studies, despite the fact that Luzin during this period considered pursuing a career in medicine and followed courses in philosophy. The crisis was not resolved until Luzin received a copy of Florenskiĭ's dissertation “On Religious ” in 1908. After this he could completely commit to mathematics and thus, in 1909, finally completed his masters exam. At this time Egorov again intervened to grant Luzin a travel stipend for another academic mission. In 1910, 12 See [31; 32; 40; 28]. 13 A notable exception is Phillips [37], who does give an account of Luzin's life during the first decade of the century, based on Bari, N K & Golubev, V V 1959 Biografiia N. N. Luzina Sobranie Sochineniĭ [Collected Works] vol. III, 468-483, a publication which, lamentably, has not been at my disposal during this work. 14 See for example [15; 23; 24].

10 according to Phillips [37:283] following Egorov's instructions, Luzin departed for Göttingen.

II.3 A school emerges The event most commonly associated with the birth of the Moscow school is Egorov's publication of an article in Comptes Rendus of the Paris Academy of Sciences in 1911. The article contained what is now known as Egorov's theorem, a theorem which, quoting Demidov [16:35], states that “any convergent sequence of measurable functions can be transformed into an uniformly convergent one by ignoring a set of arbitrarily small measure”. Finishing the sentence Demidov concludes, “the theorem which later became classic”. The theorem is significant since the area of research is the new mathematics. It is the first significant contribution to this area made in Russia. The following year Luzin published an article, also in Comptes Rendus, which, building on Egorov's result, defined the C-property of measurable functions. The C-property is, again quoting Demidov [16:35], the property that “each measurable function can be converted into a continuous one, if its values within a set of an arbitrarily small measure are suitably altered”. Another seminal event for the formation of the new school were the seminars, mentioned above, that were organized by Egorov from 1910 on. In these seminars, topics from contemporary research in analysis was introduced to students, starting at a basic level and then advancing. Each year there was a new topic. The influence and importance of these seminars is conveyed by for example Kuznetsov [30], Iushkevich [28:564] and in Istoriia otechestvennoĭ matematiki [40:425]. It was not until Luzin returned from his academic mission, however, that they began to produce results. A number of discoveries within the area of the new mathematics (metric theory of functions, descriptive set theory) were published by Luzin and his students in the years prior to the Russian revolution. Noticeable contributions were made by A. Ia. Khinchin, P. S. Aleksandrov and M. Ia. Suslin. Luzin's doctoral dissertation, “The integral and trigonometric series”, was published in 1915. Phillips [37] describes the dissertation as filled with conjectures and questions that provided research material for years to come. To conclude the story of the emergence of the Moscow school we must move to the period after the revolution and the difficult years following it. According to Demidov [16], thanks to Egorov's efforts to preserve the academic tradition through the tough years, and to Luzin's ability to attract young promising students, the budding school survived and expanded. Especially after 1920 when Luzin moved back to Moscow and the group that, already at that time was called the “Luzitaniia”, was formed. It consisted of a group of mathematicians who would make significant contributions to many fields of mathematics and form the core and first generation of the Moscow school of mathematics15.

III Moscow mathematics and philosophy My introduction of the central figures is now complete and I move on to philosophy. First I give a compact summary of the history of Russian thought. The second section holds the analysis of the philosophical tendencies of the central figures.

15 Demidov [16:40] lists D. E. Menshov (1892-1988), A. Ia. Khinchin (1894-1959), P. S. Aleksandrov (1896-1982), I. I. Privalov (1891-1941), V. I. Veniaminov (1895-1932), P. S. Uryson (1898-1924), V. V. Stepanov (1889-195?), A. N. Kolmogorov (1903-1987), L. G. Shnirel'man (1905-1938), N. K. Bari (1901-1961), V. I. Glivenko (1896-1940), M. A. Lavrent'ev (1900-1980), P. S. Novikov (1901-1975), L. V. Keldysh (1904-1976), Iu. A. Rozhanskaia (1901- 1967), N. A. Selivanov, E. A. Leontovich (1905-19??), I. N. Khlodovskiĭ (1903-1951), G. A. Seliverstov (1905- 1944) as belonging to the Luzitaniia in the 1920s. Mentioned in the context are also V. V. Golubev (1887-1954), and M. Ia. Suslin (1894-1919), who died tragically young during the years of the Civil war.

11 III.1 A crash course in the history of Russian thought My history of Russian thought begins, as it should, in Kievan Rus'. The first and greatest influence on early Russian thinking was Orthodox Christianity, which can be traced back to the Christianization of Kievan Rus'. In the Primary Chronicle (early 12th century) it is told that Vladimir the Great (10th century), ruler of Rus', sent out envoys to study which religion to accept. Islam and Catholicism were rejected on account of unfavorable reports, while the envoy returning from Constantinople told of the practice in an Orthodox church, that one could not tell whether one was in heaven or on earth [8:11-12]. This conveys some of the characteristics of the Orthodox church. The liturgics are very important as the formal aspects of worshipping are also divine. Icons and rituals are thought to convey divine truth just like the text in the bible. This is also evident from the fact that the reasons for the schism within the Russian church in mid 17th century were only minor changes in liturgics. The Orthodox Church also has a tendency towards mystical practices as is evident from the practice of Hesychasm, which became important towards the end of the middle ages. Hesychasm is a sort of meditation accompanied with repeated repetition of a short Jesus prayer that leads to the actual experience of God [8:77]. It entered into the Russian church in the 14th century through the monks on Mount Athos, Greece. Simplifying, one can describe the Orthodox Church as ritualistic, symbolic and mystical. In comparison, the Catholic and Protestant Churches are language-centered and rationalistic. Another characteristic of an early precursor of the Russian state, Muscovite Russia, was xenophobia. Medieval Russia's main cultural exchange was with the Byzantine empire. As the Byzantine empire declined, and ultimately crumbled, Russia became increasingly isolated. There was widespread suspicion against foreigners, as well as against foreign religion. This had left Russia poor and underdeveloped at the time when Peter the Great ascended the throne, in 1682. He initiated a series of reforms to modernize Russia with Western Europe as model, as is mentioned in II.1. The process of westernization, however forcefully enacted by the government, was slow. As Walicki [45:1] writes, the ideas of the enlightenment did not start to spread until the reign of Catherine II (1762-1796). With the spreading of enlightenment philosophy and education in Russia came a reaction. The enlightenment was compatible neither with the Russian authoritarian system let alone with the preceding mystical and, if not anti-intellectual, then, at least, non-intellectual culture. Thus, as would be expected, Russian philosophy was strongly influenced by romanticism [45:ch4], in science represented by the Naturphilosophie of Schelling [43:209, 336], an influence that became lasting. The first to recognize the conflict between the Russian tradition and the process of westernization was Pëtr Chaadaev (1794-1856). He wrote down his views on the situation in Russia in the first of his Philosophical Letters, published in 1836. In short, Chaadaev recognized that Russia had no history, all progress had been imported from the West. This was why the Russians were born without a national identity and did not feel at home in any tradition [45:85-90]. Contemporary intellectual Aleksandr Herzen (1812-1870) conveyed that whatever the Chaadaev's letter signified, it meant that “one had to wake up” [45:88]. Indeed, the heated debate that was sparked by the letter and crystallized during the 1840s, is still today vivid in Russian society. This is the conflict between slavophiles and westernizers. The westernizers generally accepted Chaadaev's analysis and their recipe for developing Russia was a continued assimilation of western culture. The slavophiles, on the other hand, recognized precisely the western influence as that, which had separated from its historical roots: while Western Europe is based on the rational organization of individuals, the basic structure for a true Russian society, as drafted from the society supposedly preceding the reforms of Peter the Great, is a collective community, organized by free choice and centered around the Orthodox faith. This conflict was reflected in the contrast between the two capitals of Russia, St. Petersburg and Moscow. Walicki [45:75-76] writes: “Semi-patriarchal Moscow, with its old noble families, was the capital of ancient Muscovy and the center of Russian religious life; it was also the main stronghold of

12 , mysticism and resistance to rationalist, revolutionary and even liberal thought. [...] in the nineteenth [century] it was to give birth to the Slavophile movement. St. Petersburg, on the other hand, was a town without a past and at that time the only modern city in Russia; it was the cradle of the […] uprooted intelligentsia, and the main center for liberal, democratic and socialist thought.”

The ideas of the westernizers developed from a general enlightenment of development and education through and towards more radical standpoints. The 1860s is usually recognized as a period of accelerated radicalization of the left-wing opposition. Perhaps best known from this period is Nikolaĭ Chernyshevskiĭ (1828-1889) who can be classified as a rationalist and materialist [45:189]. He became one of the chief ideologues for different branches of radicals and democrats that demanded a transformation of Russian society. Directing the development would be science and the new rational man, as Chernyshevskiĭ has it in his incredibly influential novel What is to be done? During this period Comtian positivism and its vulgarization into general scientism also gained supporters in Russia [45:357; 44:234]. Both materialists and positivists campaigned against the traditions of metaphysics and mysticism in Russia. As we have seen in I.1, the philosophy of Bugaev and his allies was a reaction against this, but they were not alone. The great writers, Dostoevskiĭ and Tolstoĭ, were also part of this reaction, and, as I have noted in I.1, Bugaev and Tolstoĭ were friends. However, I will allow another friend of Bugaev's some more space: Vladimir Solov'ëv (1853-1900). Solov'ëv was arguably the most important philosopher of the time, in terms of lasting influence. Walicki [45:371] traces his importance to the fact that he managed to create a philosophical system which transcended the traditional reluctance in Russia to deal with philosophical problems of purely theoretical nature. There is of course no room to treat all of Solov'ëvs work and I will only present the elements essential for this essay. Solov'ëv's main contribution to the theory of knowledge is the concept of integral wholeness, "that was to counteract the destructive effects of rationalism" [45:375]. He argued that, in the primitive state of human history, science, philosophy and theology had been merged. As human culture evolved they were separated as were, analogously, , rationalism and (Orthodox) mysticism. Solov'ëv suggested that empiricism and rationalism on their own led to a denial of the objective of the external world and the person experiencing it. This, in Solov'ëv's understanding, absurd conclusion proved that there was some deficiency in those approaches to knowledge [45:378]. Integral wholeness meant the reintegration of the different means of acquiring knowledge into a new unity, yielding the path to absolute truth. The ontological concept all-unity is something similar. It is the concept of the unity of the divine and the physical world through man. In connection with this it is also appropriate to mention Sophia, what Solov'ëv called the female principle, the "world made flesh" [45:381]: all-unity meant the integration of Sophia with God through mankind. Solov'ëv based many of his theories on the work of the earlier slavophile philosophers. It was from them that he got the ideas about the integration of people, by voluntary participation, into an Orthodox community and also about the integration of the different aspects of knowledge beneath a guiding Christian principle. Towards the end of the 19th century developments in the philosophy of science were initiated. Bugaev's and Florenskiĭ's work was definitely part of this but they were also original. Generalizing one can say there were three main trends, , positivism and Solov'ëv's followers. Regarding materialism I can just mention that Vucinich [44:261] comments on “the dogmatic adherence of most contemporary scientists to ontological materialism”. Granted, the comment is made in connection with thinkers wishing to make a revision of the materialist philosophy of science to diversify it. Nevertheless, most contemporary scientists were materialists and rejected metaphysics and idealist notions. The positivists were not strict adherents to Comtian positivism but still accepted the label. They were proponents of the scientific philosophy of Richard

13 Avenarius (1843-1896) which recognizes science and philosophy as non-separable since their basic unit is impressions [44:251]. Thus, they transcended the traditional conflict between idealism and materialism, but thoroughly denied metaphysics, and most certainly mysticism. In turn, the group following Solov'ëv's intellectual heritage were mainly critical of science and its status in modern society. Representatives of this group were for example Nikolaĭ Berdiaev (1874-1948) and Sergeǐ Trubetskoǐ (1862-1905) [44:ch8]. They were inspired by and built on Solov'ëv's work. Some were theologians and some were philosophers. Yet others did not so much have a professional program but simply shared a disdain for marxism. Throughout Vucinich's [44:ch8] account it is however clear that they considered science to be secondary, as religious/mystical knowledge was given precedence. Walicki [45:393] classifies Florenskiĭ as beloning this group but I beg to differ. From my point of view there is a significant difference between Florenskiĭ and others of the same tradition. Namely that he did not trivialize science, but actually made use of it in his philosophical work. My interpretation is that this can be, in part, attributed to his background, and recurrent work, in mathematics and in science. Finally, another example of the rise of mysticism and religion in early 19th century was the surge of interest in Name worshipping. From the monks on Mount Athos, which was also the pathway for Hesychasm into Russia, came a mystical doctrine that stated that “in the name of God is God himself” [17:123]. Ilarion, a monk of Mount Athos and Name worshipper, wrote the book In the Mountains of the Caucasus (1907), that described how one, through the constant repetition of the Jesus prayer, would enter a religious ecstasy that was actually a union with God. The book became very influential in Moscow intellectual circles and many were drawn to the ideas of the Name worshippers. The similarities with Hesychasm are striking but Name worshipping was branded as heresy by the official Russian Church since it was considered as idolatry.

III.2 An alternative philosophy of science There is a difficulty in trying to characterize the philosophy in Moscow mathematics of the time: there is no complete program or doctrine to analyse. Florenskiĭ never created a philosophical system for science, he was more interested in religious questions, and Bugaev was not successful in elaborating his system. This is why I have singled out the four central figures and attempt to reconstruct a pattern of tendencies dominant in their circles. What I do is somewhat similar to, but more extensive than, what Graham & Kantor [26] attempt. If I may be so bold as to criticize their attempt I must point out a few deficiencies. Graham & Kantor limit themselves to the treatment of three of the four characters that I have identified as significant. Bugaev is left out. In this way the mathematical roots of the tradition are cut. Graham & Kantor even seem to be unaware of Bugaev’s contributions, as Bugaev is not mentioned at all in the discussion about continuity and discontinuity in Florenskiĭ's work or even in the passage on page 70 where they mention “continuous and discontinuous phenomena”, arithmology “(which strongly impressed Egorov, Luzin, and Florenskii and their followers)”, and discontinuous functions that “became hallmarks of the Moscow School of Mathematics”. Furthermore, in the article, Name worshipping is the only influence on the mathematicians that is considered. This is problematic since this results in a very narrow scope. The scope becomes narrower still since Name worshipping as an intellectual influence is mixed up with Name worshipping as a sect. As I have understood it Name worshipping as an extensive phenomenon during the silver age was not a prevalent sect but a doctrine that became part of the larger religious and philosophical intellectual atmosphere. As Demidov [17] points out, the charges by the secret police against religious leaders during the 1930s were strengthened by the idea of an organized underground church, of which the Name worshipping “movement” supposedly was a significant part. This has promoted the illusion that a number of persons were part of Name worshipping as a formal organization. I don't think that the ideas of the Name worshippers didn't have an impact on the

14 mathematicians. Egorov and Florenskiĭ were evidently captivated by them. However, I would not take the mathematicians' involvement with Name worshipping as them part of a sect. I would rather use it to argue that they participated in a broad Russian intellectual tradition. In that sense it’s not problematic whether Luzin was a “member” of the “movement” or not, which is debatable. Furthermore, placing the mathematicians as members of an obscure religious sect easily provokes exotism in one's treatment of them. Not to mention that Name worshipping can be seen as a popularized development of the, already at that time, 600 years old tradition of Hesychasm. I have used this critique of Graham & Kantor [26] as an introduction to my analysis since it illustrates my approach of looking at the wider pattern of tendencies that affected the Moscow mathematicians. Before going into the specific tendencies I want to make an overall description of the philosophy of the Moscow mathematicians, since their views on the particulars are sometimes diverse, sometimes unclear. The main themes of the philosophy of the Moscow mathematicians was their ”antipositivism, attraction to idealist and even religious philosophy, [and] orthodox tendencies“ as quoted above, leaving aside the issue of monarchism as my central figures are not mainly proponents of this form of government, and since it is of marginal importance for the philosophy of science. Common to all of them was the opposition against what they viewed as the predominant philosophy of the intelligentsia, and, in extension, of “science” and ultimately of the whole of Western Europe. They spoke out against positivism and materialism, sometimes in general terms, as in Luzin’s letters to Florenskiĭ, sometimes in a specific manner, as in Bugaev’s talks before the Philosophical society. At least the three younger mathematicians16 were devout believers during their adult years and Florenskiĭ and Egorov eventually died because of their beliefs. From this one can conclude that the Moscow mathematicians shared a disdain for positivism and an inclination towards philosophical idealism and Orthodox Christianity. But what were their grounds for denying positivism, and what, exactly, was the nature of their idealism? In what ways did their views influence their work? These are questions that I will address next, as I move on to a more detailed analysis of the philosophy in Moscow mathematics.

III.2.1 Platonism As I have already made clear I have identified Platonism as a philosophical tendency of the central figures. With Platonism is meant the understanding that “abstract objects, such as those of mathematics, or concepts such as the concept of number or , are real17, independent, timeless, and objective entities” [6], the philosophy of ideas or forms. Thus in this sense, Platonism is, in my understanding, a question of ontology. Hence, this will mainly be a discussion of the ontology of the central figures and especially an examination of their relation to Platonism. Bugaev's influence on the younger generation of mathematicians has already been established. He developed his ontology as a reaction against the materialist onslaught of the 1860s, as was mentioned in III.1. He created the system Evolutionary monadology, which is clearly an idealist ontology. What is interesting, is that Bugaev can be seen as quite close to Platonism. In Mathematics and the scientific, philosophical world view [9] he writes that “good and evil, beauty, fairness and freedom are not simply illusions created by human imagination [...] their roots lie in the essence of things, in the very nature of worldly phenomena […] they have, not a fictive, but a real basis.”18 From that thought, only a small step is required to arrive at the recognition of these concepts as eternal ideas. Bugaev died in 1903 and was replaced by Egorov, Luzin and Florenskiĭ. The general philosophical climate had changed. As per III.1 positivist and materialist philosophers alike were

16 I have not found any sources that clarify whether Bugaev was religious or not. 17 In this section I henceforth use the word real to denote . Realism is the view that objects or phenomena have an absolute/objective existence independently and outside of the human mind. 18 “[...] добро и зло, красота, справедливость н свобода не суть только иллюзии, созданные воображением человека […] корни их лежат в самой сущности вещей, в самой природе мировых явлений, [...] они имеют не фиктивную, а реальную подкладку.” [9]

15 realizing that a more balanced approach to the nature of reality was required. I propose that one can view the transition from Bugaev's philosophical system to that of Florenskiĭ in this same light. Florenskiĭ was clearly a Platonist. He has written that “the discrete nature of reality leads to the confirmation of forms or ideas (in a Platonic and Aristotelian sense), as complete and uniform, 'before their parts' and defining them, not as being composed by them.”19 I am, however, hesitant about classifying him as an idealist. It is true that Solov'ëv and his followers can be regarded as such since they give precedence to the spiritual and mystical aspects of reality and knowledge. The difference between Florenskiĭ and the others of the same tradition, however, is that the natural sciences are not foreign to Florenskiĭ. In their criticism of reason Solov'ëv and the others like him downgraded science and, with it, the study of the physical world. Perhaps this can be explained by that empirical experience was thought to be subject to rational reasoning through science and thus was a secondary form of truth. We recall that the older generation of mathematicians were also idealists but that they promoted reason as the universal source of knowledge, as Bugaev, who thought mathematics was the highest form of reason, definitely did. Florenskiĭ however, being deeply religious, realized the limits of rationalist idealism but, also being familiar with and sympathetic towards the natural sciences and especially mathematics, as well avoided falling into the one-sidedness of mystical idealism. While other philosophers of science tried to mediate the experience of immediate knowledge with the understanding of subjectivity and the cognitive aspects of acquired knowledge, Florenskiĭ went from the, in the context of contemporary science, radical understanding of abstract or spiritual objects as real to the affirmation that material objects have the same ontological status. In his will to give equal rights to the senses, reason and intuition, following Solov'ëv, he arrived at the confirmation of both spirit and matter as real and absolute categories. This position appears in some of Florenskiĭ's mature work; he presented his theory of the material and spiritual spheres as the opposite sides of one surface in Imaginary Values in Geometry (1922) (where he uses the complex numbers as an element in his philosophical analysis, as the title suggests) [18:603]. The result is that the two aspects of reality are in immediate contact and completely correspond. Florenskiĭ's appreciation for the study of nature is expressed in a summary of his own views with the statement that “the impulses of mathematics must necessarily come, on the one hand, from the general world view, and, on the other hand, from the empirical study of the world and of technology”20. Granted, both the above quoted sources appeared quite late (in the 1920s), but I believe these ontological views were present earlier in Florenskiĭ's thinking. For instance in On a premise of a world view (1904), Florenskiĭ states that, because of the transformation of assumptions and axioms into dogma, there “appeared the imaginary 'antinomy' between the area of contemplation (scientific and philosophical thought) and the area of mystical experience (religion). Both these areas are equally necessary to man, equally valuable and sacred [...] The one sanctity cannot, must not, contradict the other, one truth completely rule out the other!”21 This quote perhaps pertains more to a discussion of epistemology, but it lends support to my interpretation of his ontology. The views of the central figures on the nature of truth also illuminate the issue at hand. To introduce Luzin into the discussion, it appears as if he shared Florenskiĭ's faith in the philosophy of ideas. In an early (1906) letter, commenting on the spiritual crisis he experienced, Luzin wrote ”The individual, and even life itself is held in such low esteem, that one asks oneself: 'Do these things really exist at all in the world? Can they really exist? Isn't this what the 'idealists' dream, isn't it their

19 “[...] дискретность реальности ведет к утверждению формы или идеи (в платоно-аристотелевском смысле), как единого целого, которое «прежде своих частей» и их собою определяет, а не из них слагается.” [21:41] 20 “[...] направляющие импульсы математике необходима получать, с одной стороны - от общего миропонимания, а с другой - от опытного изучения мира и от техники”. [21:41] 21 ”[...] возникла мнимая «антиномия» между областью созерцания (научно-философского мышления) и областью мистических переживаний (религией). Обе эти области равно необходимы человеку, равно ценны и святы [...] Не может, не должна одна святость противоречить другой, одна истина абсолютно исключать другую!” [21:71]

16 idea? ..' If I was certain that there really is not and cannot be, in the world, absolute respect for another's soul, I would immediately kill myself […] The world view I have known so far (a materialist world view) is completely unsatisfactory.”22 Here one notices how Luzin the religious need to confirm the real existence of the spiritual world. Other than this it is difficult to extract anything about Luzin's ontology, but the epistemological comments can provide some clues to it. After reading the first draft of what would become Florenskiĭ's seminal work The Pillar and Ground of the Truth (1914) [20], Luzin wrote about it in a letter to his wife [15:146-148 ]. In the letter he states that Florenskiĭ has torn down the intellectual stronghold that the intelligentsia hides within, and that, what is left, is not “subjective impressionability”, i.e. the neo-positivist standpoint. He goes on to write about chapter two, entiteled Doubt, that “[t]his chapter is scandalous for university philosophy. Here, [Florenskiĭ], alongside the act of acquiring knowledge through the senses (“Physics”, ”Natural science”) and through the intellect (“Mathematics”, ”Logic”), on equal terms adds yet another means of acquiring knowledge, about which the university has never heard, namely the “intuitive-mystical”(Hindu).”23 In the letter, one clearly notes that Luzin is overwhelmed by Florenskiĭ's work. He writes in an incoherent, seemingly ecstatic manner with pompous statements like the one about the intellectual stronghold of the intelligentsia or another stating that the Pillar is a world wide tragedy for life and reason, and how he was “STUNNED”24 the entire time reading it. Another example of Luzin's dependence on Florenskiĭ in spiritual matters is the next letter, where Luzin states: “Two times I was very close to suicide—then I came here [...] looking to talk with you, and both times I felt as if I had leaned on a ‘pillar’ and with this feeling of support I returned home.”25 In the last paragraph of the same letter Luzin explains that only Florenskiĭ tells him what he needs to hear to escape the “inner tension” that plagues him [15:149]. Clearly Luzin found spiritual support in Florenskiĭ's philosophy. Now, coming to his understanding of truth, a major source of unrest for Luzin was how to reach absolute truth. His concerns are similar to the conclusions that Solov'ëv drew about the outcome of one-sided empiricism or rationalism. Luzin returns to the question of truth many times in the correspondence with Florenskiĭ and I will treat it further in III.2.2. For now I just need his epistemological recognition of absolute truth for its ontological consequences for his views on . As Luzin says when he comments on the Pillar, it is not a work that expresses the subjectivity of knowledge (subjective impressionability), which is the neo-positivist's standpoint. No, for, as I argue, Florenskiĭ has taken another path to the acknowledgement of both mental and physical information, the path of objectivity. Florenskiĭ affirmed the senses and intuition as legitimate sources of knowledge as they receive information from real objects, both material objects and ideas. An example of such indisputable are the “foundations of geometry and arithmetics” as Luzin hints in his letter of April 12 1909. There he concludes that the “results of the criticism of the foundations of geometry and arithmetics” is the “philosophy of chaos” in accordance with which “there is no indisputable truth”. Finally, regarding Luzin, I want to give an example which implies that he was not a one-sided idealist either. In two letters [15:142, 159] he mentions studying the theory of electrons, indicating that he did so at least during the course of

22 “Ни личность, ни даже простая жизнь так не уважается, что спрашиваешь себя: «полно, есть ли, на самом деле, эти вещи в мире? могут ли они существовать? Не мечта ли это «идеалистов», их выдумка?...» И если бы я был уверен, что действительно, нет и нет может быть в мире абсолютное уважение души другого, я немедленно убил бы себя. [...] Те миросозерцания, которые я до сих пор знал (материалистические миросозерцания) меня абсолютно не удовлетворяют.” [15:135-136] 23 “Это глава скандальная для университетской философии. Ибо тут к познанию через чувство («Физика», «Естествознание») и к позанию умом («Математика», «Логика») прибавляется на равных правах еще один род познания, о котором в университете не слышали <...>, именно «познание интуитивно-мистическое» (индусское).” [15:147] 24 “ОГЛУШАЕМ”. [15:146] 25 “Два раза я был очень близок к самоубийству - я тогда приезжал сюда [...] ища беседы с Вами и оба раза я чувствовал, что опираюсь как на «столб», и с этим чувством опоры я возвращался” [15:148]. The translation belongs to Ford [24:338].

17 nearly two years, simultaneously with the very culmination of his spiritual crisis. This suggests he was looking for material, as well as spiritual, answers to his philosophical questions. To conclude the discussion about the ontology of the central figures, I would argue that it tends towards Platonism, but a Platonism that is somewhat different from the contemporary idealists'. The texts left behind convey a high status given to matter and scientific method, in comparison with the status they were awarded by other Russian Platonists. I believe this is a result of the central figures being mathematicians and intimitely acquainted with science from an internal perspective. If Plato's original philosophy and the main Russian revival of it can be seen as hierarchical with material objects on the bottom and ideas on the top, I picture the philosophy of the central figures as equileveled with the same status for different categories. This was not precisely Bugaev's philosophy but I believe it was the main direction for the thoughts of Florenskiĭ and Luzin. Remaining is Egorov and the question of what his standpoints were with regards to ontology. This is very difficult to determine and I will only speculate somewhat as to a possible answer. Egorov was perhaps closest to the Name worshippers of the four central figures. He hosted meetings for religious personalities and well-known Name worshippers. He always kept philosophical and religious literature on his desk next to the mathematical literature he was studying. He valued Kant highly and found Florenskiĭ's work interesting [15:175-176]. What does these facts amount to? I would say that Egorov probably shared the Platonist philosophy of ideas. It is very close to the Name worshipping doctrine quoted above that states that “in the name of God is God himself”. The quote suggests that the name of God is not arbitrary but real and thus has an absolute and unconditional existence. Although it is impossible to tell how Egorov stood towards the question of the nature of the material reality perhaps he was so much part of the older generation that he should be classified as a rational idealist or maybe his interest in Kant really places him among the neo-Kantians. Either way, he was probably a philosophical idealist.

III.2.2 Mysticism With mysticism is meant “ in the power of spiritual access to ultimate reality, or to domains of knowledge closed off to ordinary thought” [4]. More specific, in this section, mysticism is the belief in immediate knowledge based on faith which is not accessible through reasoning or logic. This section is called mysticism, as this is the most controversial element in the epistemology of the central figures. I, for one, share the view that a scientific epistemology should aim to describe all phenomena that are subject to the theory. The key word here, as I see it, is describe. In saying that, I tacitly assume a rational description, as supposedly being the only description that can be transferred unaltered to another mind. In the views shared by the central figures, mysticism, however, is, as we shall see, concerned with things that are foreign to reason. But I will begin with the background for the mystical elements in the epistemology of the central figures. Bugaev had no mystical beliefs. Though critical towards the rationalism of his contemporaries, he nevertheless did not consider this as the failure of reason itself, but only as a failure of the reason of those contemporaries. As we have seen, Bugaev was certain that, through his arithmology, the theory and philosophy of discontinuous functions and probability, one could scientifically treat areas of knowledge earlier not open to scientific study. Bugaev was also certain that such a development could avoid the mechanist and reductionist pitfalls that positivism and materialism necessarily fell in. Polovinkin [38:12-19] paints the picture of Florenskiĭ's attempts to synthesize mathematics and philosophy as a continuation of Bugaev's work and it is true that Florenskiĭ was greatly inspired by Bugaev's arithmology. He completely shared Bugaev's analysis of continuity as the source of the one-sided “western” approach to science, as will be treated in the next section. Polovinkin, however, notes that Florenskiĭ moved away from the task of synthesizing mathematics and philosophy and suggests a possible model for this deviation [38:15]. The explanation for Florenskiĭ abandoning the plans of a unification is taken to be that he gives precedence to his adherence to “Christian personalism”. The problem, as I see it, with this analysis

18 is that the influence of Solov'ëv on Florenskiĭ's attempts to reconcile the different sources of knowledge is left out. Granted, Polovinkin states in the introduction of his article that he realizes that the perspective he has chosen is just one of many possible, but in discussing Florenskiĭ's direction regarding the possible synthesis of philosophy and mathematics one must examine Solov'ëv's influence to not arrive at a distorted picture. Clearly inspired by Plato's distinction26 between the three different kinds of knowledge, Solov'ëv worked out his concept of integral wholeness, the unification of science, philosophy and religion, or, correspondingly, of empiricism, rationalism and mysticism. We see this echoed in Florenskiĭ's Pillar. In the words of Luzin: “[Florenskiĭ], alongside the act of acquiring knowledge through the senses (“Physics”, ”Natural science”) and through the intellect (“Mathematics”, ”Logic”), on equal terms adds yet another means of acquiring knowledge, about which the university has never heard, namely the “intuitive-mystical”27. Florenskiĭ never intended for a unification of mathematics and philosophy under the cover of reason, but rather, as I understand it, used mathematics to complete the unification of science, philosophy and religion. He was influenced by the ideas of a unification of both Bugaev and Solov'ëv, but he never intended to leave out religion and mysticism. What he inherited from Bugaev was mainly the mathematical approach and the concept of discontinuity. He repeatedly used mathematics to illuminate problems that defy rational treatment. A recurring use of mathematics is the comparison between concepts beyond the grasp of reason and infinite sets. For instance, in chapter XVII of the Pillar, Florenskiĭ likens the relation between an irrational number and a rational number in the infinite sequence defining it with the relation between God and Man. Polovinkin [38:14] gives an example of Florenskiĭ's use of mathematics in Macrocosm and microcosm, where he, affirming that the human being is infinite, states that, as such, it can be said to be equal in size to the macrocosm, and thus can be said to contain it, in analogue with how an infinite set A, part of another infinite set B, still can have the same “size” as B. In this manner Florenskiĭ used mathematics to justify mystical knowledge and the central position of faith when acquiring knowledge, since, just as faith is necessary to believe in God, it is an act of faith to assimilate the irrational numbers into the mind. Luzin was clearly drawn to mysticism, provoked by a thirst for absolute truth as is seen in for instance the following quote: “Lord, what is this torment... Oh, how painful to see injustice, and not know what justice is, to see, to experience with all your being falsity, and not see the light of truth, absolute truth...”28 The emphasis is Luzin's own. He was very clear about what he was seeking. He also returns several times to the fact that absolute truth cannot be found in science. In 1906, Luzin wrote to Florenskiĭ that “now I understand that 'science' is, in essence, metaphysical and not founded on anything.”29 Six years later he echoed the same affirmation in saying that “There is no unconditional necessity in the development of science, just as there, in science, is no absolute truth. Yes, even if the last did exist in science, then science could not do anything with it. Hence, science doesn't even need it.”30 The remedy for Luzin's spiritual crisis appeared when he read the first edition of the Pillar in 1908 and after repeated talks with Florenskiĭ that summer. We remember Luzin's comment that Florenskiĭ in it had justified intuitive-mystical knowledge. Ford [23; 24] also notices that after this, Luzin's spiritual crisis seems to be resolved and he completely committed himself to mathematics. The religious greetings that Luzin repeatedly uses to finish his letters to Florenskiĭ tells of his religiosity. His interest in mysticism is expressed explicitly in the

26 “Objects of sense are known by pistis or opinion; mathematical and scientific objects by or reasoning; and at the summit the forms are known by noēsis.” [3] 27 See note 23. 28 “Господи, что это за мука... Ах, какая боль видеть неправду, и не знать, в чем же правда, видеть, чувствовать всем своим существом ложь, и не видеть света истины, абсолютной истины...” [15:136-137] 29 “[...] теперь мне понятно, что «наука», в сущности, метафизична и не обоснована ни на чем.” [15:136] 30 “Безусловной необходимость в развертывании науки нет, так же как нет в науке и абсолютной истины. Да если бы последняя и была в ней, то науке нечего было бы с ней делать. Она, таким образом, и не нужна науке.” [15:165-166]

19 letter of April 12 1909 where he asks Florenskiĭ of advice about reading Plotinus31. He states that his goal is “to become acquainted with the world view of the mystic, the strong man, who is no stranger to the deep logical work required for a real world view.”32 I suggest that real should be interpreted in the philosophical sense as an expression of realism and according to my analysis of the ontology of the central figures. I would also like to point out the fact that Luzin talks of the deep logical work required for a realist world view. This is a crucial point since it explains his choice of profession, pure mathematics, which is based on reason, in the face of his critical assessment of the power of reason. Luzin's view on knowledge is not that reason is useless, indeed he notices how Florenskiĭ in the Pillar “deals with the most fundamental questions of life, not by accepting anything on faith, but, on the contrary, by showing the limits of the intellect, and then going further, beyond logic, intuitively, but based on reason. In no religious work have I ever encountered so little fantasy and so much logic”.33 This view, that reason is still useful, although it has limits and will never reach absolute truth is also stated by Luzin in saying that “although there is no absolute truth in science, there will always be truth, and never will the latest truth contradict the preceding, but will cover it, like of two concentric circles, the one with larger radius will always cover the smaller.”34 To sum up, Luzin was clearly influenced by, and interested in, religious philosophy. But he did not necessarily apply it directly to his work in mathematics. After the return in 1914 from his second academic mission it appears as if he was affected by the currents in France and, perhaps, matured somewhat during the stay abroad. He soon came to the conclusion that he did not believe in absolute infinity, one of Florenskiĭ's examples of knowledge based on faith [15:178 ]. It was not a complete break with Florenskiĭ's philosophy though, as he explains in the same letter that he fears the infinite while Florenskiĭ “boldly searches for the unwavering heart of indisputable truth”. But I will comment more on this difference of opinion in the last part of the essay. Just as in the case of ontology discussed above, Egorov's views on mysticism are difficult to determine. Considering his firm religiosity and interest in Name worshipping it is probable that he entertained some mystical beliefs. However, since he saw a person's faith and outlook on life as intimate and not subjects for discussion, and regarded all ideological and religious discussion as not fitting in mathematical publications [17:135], it is very difficult to verify any claims.

III.2.3 Indeterminism Indeterminism is the opposite of , the idea that everything that happens is in some way predetermined. An ancient version of determinism is the belief in fate. It appears as though the mathematical approach to indeterminism was first recognized by Bugaev. He was, as we have noted, part of the reaction against the rise of materialism and positivism in intellectual circles towards the end of the 19th century. One cornerstone in the opposition against these , was the understanding that they inevitably led to a mechanist world view. In such a world view all events are seen simply the totality of aggregates of particles, acting according to the laws of nature. Since, at that time, all known laws were deterministic, this implied that if one knew the configuration of the universe at one instant in time, one could theoretically calculate the course of history. Philosophically this leads to the denial of the freedom of human action, in a way that is very foreign to everyday experience. On a theological level, such a predetermined course for history 31 Plotinus c.205-270 A.D. “The founder of .” [6] 32 “[...] познакомиться с миросозерцанием мистика, сильного мужа, для которого не тайна глубины логической работы реального созерцания” [15:158]. The translation belongs to Ford [23]. 33 “[...] трактует о самых фундаментальных вопросах жизни, не принимая ничего на веру, а наоборот, показывая пределы ума и идя далее за-логично, инуитивно, но опираясь на разум. Ни в какой работе по религиозным вопросам я не встречал так мало фантазии и так много логики” [15:147]. The translation is a modification of the one found in [24]. 34 “[...] хотя абсолютной истины и нет в науке, истина всегда будет в науке, и никогда позднейшая истина не будет противоречива прежней, но всегда будет ее покрывать, как из двух концентрических кругов круг большего радиуса покрывает меньший.” [15:165-166]

20 rules out the posssibility of free choice, which is so important for the Christian concept of salvation. From the understanding that human will is actually free, Bugaev concluded that this, the analytic world view, as he called it, was faulty. He recognized mathematical analysis as the culprit, hence his denotation “analytic” for the scientific world view. He reasoned that, since mathematical analysis made its entrance in the 17th century, it had proven to be incredibly successful in providing answers, mainly in physics. The subjects of (classical) mathematical analysis are continuous functions, that can describe continuous processes or objects. That is, processes that occur as smooth transformations without leaps from one stage to another and objects that are homogeneous or without abrupt transitions. Now, as a result of the success of the analytic approach, the implicit assumption that all events whatsoever are continuous was made. An example of this is Darwin's theory of evolution, where species are thought to evolve in minimal steps over a great span of time. Bugaev, however, claimed that discontinuous and indeterminable events were just as common, they were simply overlooked. As examples he gave, as mentioned, free will, but also the division of the human body into individual cells or a revolutionary perspective on history as opposed to an evolutionary, among others [9]. Bugaev's solution for the deficits of the analytic world view was arithmology. Arithmology is what he called the theory of discontinuous functions and probability. His main area of research was the theory of numbers and he attempted to generalize the methods of analysis to be able to introduce them in this area [13:119-120]. With the resulting mathematical tools he envisioned a systematic mathematization of areas earlier not open to mathematical treatment, especially social sciences, where human action was concerned. Here, it's interesting to note the influence of Bugaev's early period as a positivist, which was mentioned briefly in II.1, and which manifests itself in his great faith in mathematics as the basis for all sciences. Florenskiĭ showed considerable interest in Bugaev's ideas and both the idea of discontinuity as a fundamental concept and the idea that all areas of study should be subject to mathematical theoretization were taken up by him. These influences became lasting elements in Florenskiĭ's philosophy, as can be seen in a summary of his standpoints written in 1925-26. There, he writes: “What prevails in the world is, with regards to relations, discontinuity, and, with regards to reality itself, discreteness”35 and that he “sees mathematics as the indispensable and primary premise for a world view”36. As was quoted in the section Platonism, Florenskiĭ even used the idea that the nature of reality is discrete as an argument for the existence of ideas (as a philosophical concept). Polovinkin [38:3] also notes this, quoting Florenskiĭ saying “The principle of continuity brings 'the banishment of the philosophy of forms... it's impossible to go from one extreme to another without an intermediate element, such is the principle of continuity […]'”37 This background explains Florenskiĭ's interest in the new discoveries in mathematics. In set theory he found the affirmation that continuity was just a special, restricted case of discontinuity, which meant that “we don't have any grounds to accept 'continuity' as the basic characteristic of reality”38. As the developments in set theory were applied to function theory Florenskiĭ recognized that also in this area “prevails discontinuity and only through a combination of very intricate and artificial requirements, which place a number of conditions on the function, does it turn out to be continuous”39. Subsequently, Florenskiĭ quotes Cantor who also applied mathematical considerations to philosophical problems [21:77-78]. By noting that a set does not need to be

35 “В мире господствует прерывность в отношении связей и дискретность в отношении самой реальности.” [21:41] 36 “[...] видит в математике необходимую и первую предпосылку мировоззрения”. [21:41] 37 “Принцип непрерывности влечет за собой «изгнание понятия формы… невозможно от одного крайнего перейти к другому без промежуточного—таков принцип непрерывности […]»” 38 “[...] мы не имеем никаких оснований останавливаться на «непрерывности» как на основном признаке бытия”. [21:75] 39 “[...] господствует прерывность и только при соединении очень хитрых и искусственных требований, налагающих множество условий на функцию, она окажется непрерывной.” [21:75]

21 continuous for one to be able to construct a continuous curve between two arbitrary points in the set, following Cantor he argues that space does not need to be continuous for continuous movement to be possible. A corresponding argument can be applied to Egorov's interest in the same areas. He was a student of Bugaev's and his first published work was within Bugaev's arithmology project. Bugaev's influence can also be seen in Egorov's textbook on the theory of numbers [30:171-172]. Egorov also, most probably, shared the analysis that the analytic world view, with its obsession with continuity, was not able to account for the concepts central to his Christian faith. Consequently, he took to studying the same developments in mathematics that Florenskiĭ was interested in, and worked continuously for the complete introduction of the new mathematics in Moscow. Luzin never explicitly comments on the determinism/indeterminism antinomy in any of the material that I have studied. However, just as with Egorov, his faith suggests that he could not have been a determinist. This is further corroborated by his comments about materialism and the prospects for reaching truth through the application of science. Another interesting source is an article of Luzin's, which will be discussed again in IV.2, about the evolution of the concept of function [33]. The criticism of continuous functions spurred an interest in discontinuity within Moscow mathematics but simply discontinuous functions were all but completely accepted in mathematics already at this time. More pathological functions with even stranger and more difficult behaviour can be taken as having inherited the earlier status of discontinuous functions as unwanted. Luzin in the article comments on how Moscow mathematicians had taken a more allowing stance towards these stranger function. That he even is aware of this controversy and on top of that identifies the stance of Moscow mathematicians indicates something of the impact of the earlier philosophical interests of Bugaev, Egorov and Florenskiĭ on Luzin.

III.2.4 Holism Holism signifies an epistemological emphasis on the whole rather than on its parts. This is both a guiding principle, that one can discern in the work of the central figures and in Russian culture in general, and a well-defined and articulated premise or point of view. One of the main expressions of holism as a general perspective is the criticism of the reductionist tendencies in western science. Bugaev recognized as a non-desirable element in the contemporary scientific world view. In describing the scientific world view, he not only identified its bias towards continuity, but also that it assumed that all phenomena could be understood as the totality of its elementary parts. This meant that simple phenomena could simply be added together to produce the complete understanding of a complex phenomenon, which, according to him, was a simplification [9]. Another expression of Bugaev's holist perspective is a certain element in his monadology. It becomes even more interesting since this element is the most significant departure from Leibniz' original philosophy [39]. Where Leibniz stated that the monads were single and complete units without any possibility of interaction, Bugaev instead emphasized the monads' capability for mutual exchange. He stated that monads were connected and appeared to each other in different aspects. What united the monads was solidarity and mutual love and the interaction was aimed at their common perfection. In Bugaev's system monads could in this strife for perfection be united into a new single and whole monad. The final goal of the perfection of the monads was the complete unity of the whole world in one single monad. As Polovonkin [39] states, Bugaev was in this inspired by the concept of соборность (mystical collectivism) of the slavophiles. Florenskiĭ also expressed criticism against reductionism. In the chapter called The understanding of identities in mathematical logic of the Pillar (chapter XIX) Florenskiĭ begins in mathematics but as usual moves on to philosophical questions. In this case, the outcome is a critique of nominalism40, where Florenskiĭ argues that the concrete can never be considered as identical to a

40 The belief that individuals in a certain category, for example the category “horse”, have “only the name” in common. More generally that all universal characteristics are generalizations based on experience, as opposed to

22 collection of abstract characteristics. He argues that the concrete individual constitutes a completed whole, which transcends any collection of abstractions. It is interesting to note that he once again makes use of the new mathematics in stating: “If it is even possible to investigate the concrete formally and rationally, then, without a doubt, it could not be introduced […] in any other way than as a limit, that is, as an absolute maximum.”41 That is, it could only be introduced as a transfinite quantity. The argument for this claim is that human creativity is able to invent a new abstraction beyond any predetermined list. Florenskiĭ also recognized the question whether a transfinite set of abstractions could transcend the concrete, although he did not speculate about a possible answer. Another chapter in the Pillar is also of interest. In the section on mysticism, I mentioned chapter XVII of the Pillar where Florenskiĭ discusses the irrational numbers. Irrational numbers are the numbers on the real line which cannot be expressed as fractions, they can instead be defined as the limit of an infinite sequence of rational numbers (the rational numbers are the numbers that can be expressed as fractions). Florenskiĭ's argument is that that the irrational numbers are something completely new, they transcend the properties of the rational numbers. He makes the analogy between the rational numbers in a sequence defining an irrational number and the crystals of marble in a statue. The crystals do not have anything in common with the idea of the completed statue, just as the rational numbers do not have anything in common with the idea of the irrational number as whole and complete. The irrational number is to be seen as the complete infinite sequence of rational numbers taken as a whole and as such transcends the finite nature of rational numbers. Apart from these specific examples when a holist perspective is expressed directly, the overall philosophical goal of the central figures is also part of a holist program. Bugaev was concerned with completing the scientific world view with a theory to include the spiritual aspects of human life. He wanted to unite the natural sciences with the social sciences and even with and , clearly a holist approach to the problem of knowledge. Solov'ëv followed the same general outline, even articulating that his goal was to unify science, philosophy and theology into the ultimate source of knowledge. Florenskiĭ picked up inspiration from both Bugaev and Solov'ëv and made attempts to formally treat the problems of religion and mysticism with the help of mathematics. The principle of unifying the different sources of knowledge was echoed in ontology. As was mentioned in III.2.1, Florenskiĭ, in Imaginary Values in Geometry, presented the unity of matter and spirit as the different but completely corresponding sides of reality. Being a general principle, only explicit in the philosophical works of Bugaev, Florenskiĭ and Solov'ëv, I have not found any concrete trace of holism in Luzin's or Egorov's works. However, that they shared the belief that science in itself is not enough to explain the human condition and that religion had a significant part to play in human life, can be seen as expressions of a tendency towards a holist world view. Luzin was inspired by the prospects of unifying the different areas of human knowledge, and shared the belief that balance was necessary in the relation between the material and spiritual spheres, as we could see in the sections above. My argument is, rather than saying that they regarded holism as a basic principle in their scientific work, that their cultural context contained a tendency towards holistic thinking. Most notable are the philosophical works of Solov'ëv and Florenskiĭ, but this tendency is also exemplified by the organic element in Russian culture, cosmism and Orthodox practices. The slavophiles, partly inspired by romantic ideals, shared an organic view of society and nature, where an organism constitutes a unity greater than simply the sum of its parts. This tendency lived on, Bugaev reiterated it in stating that ”Nature is not only a mechanism, but an organism”42. Russian cosmism, a contemporary movement in the philosophy of nature, also emphasized the organic and cosmic perspectives, in sharp contrast with the that was budding in the west. The Orthodox church as well represents a more objective ideas with real existence (Platonism). 41 “Если конкретность и может быть рассматриваема под формально-рассудочным углом зрения, то она, несомненно, может быть введена [...] не иначе, как под видом предела, т. е. как абсолютный максимум.” [20:528-529 ] 42 “Природа не есть только механизм, а организм”. [9]

23 holist perspective than its western counterparts. It concentrates on the complete aesthetic experience of the divine through all senses, as was relayed by the envoys of Vladimir the Great in the 10th century but is still today the guiding principle for the religious experience. All in all the expressions of holism in Russian culture as well as in the philosophical works of the central figures paints the picture of an intellectual environment where holism is an important principle. I first identified holism through the apparent disdain for reductionism shared by the central figures but, as can be seen above, it is not simply a negative standpoint but a positive affirmation of complexity as inherently different from the simple summing-up of parts. In IV.2 we will see how holism is an important feature in the reception of the new mathematics.

IV A paradigm shift in mathematics We now shift our attention back to mathematics and its development. To make an analysis of this development a background in the general history of mathematics is needed. The next section will supply this background and the subsequent section holds the actual analysis.

IV.1 The development of mathematics There is no space available for a review of the complete history of mathematics. Of course, the events concerned have deep roots in history but the development towards the end of the 19th century can be sen as a major transformation. I will take this approach and begin as these developments accelerate. Throughout the 19th century, mathematics underwent a continuous development of increasing rigor. The boundaries of mathematics had been pushed so far that it was no longer sufficient with an intuitive understanding of underlying concepts such as numbers, continuity and convergence. Another development of the 19th century was the appearance of mathematical objects which did not correspond to any concrete phenomenon in the physical world. The development of mathematics had for a long time been driven by discoveries mainly in physics and, as a result, mathematics was, in a sense, founded on physical phenomena. The appearance of for instance non- Euclidean geometries and the considerations of n-dimensional spaces (where n > 3) during the 19th century upset this order. This was controversial since the philosophy of mathematics had been based on the view that mathematics was a representation of the physical reality.43 From the 1870s Georg Cantor (1845-1918) made important discoveries that would lead this controversy to a culmination during the first decades of the 20th century. It was no longer possible to avoid open conflict when the uncertainties came to undermine the foundations of mathematics.

IV.1.1 Cantor and infinity Georg Cantor was a German mathematician. Incidentally, he was born in St. Petersburg but the family relocated to Wiesbaden in Germany when he was still a child. As we shall see his contributions came to be of profound importance for the development of mathematics. It is quite possible to take his work as the starting point for modern mathematics, the product of the transformation mentioned in the introduction to this section. In Cantor's early work the movement towards an increase in rigor and the use of completely artificial mathematical objects combined. Cantor started out studying the representation of functions by trigonometric series but ended up constructing a theory for infinite numbers. Through considerations of the continuum of the real line, necessary for the examination of different types of functions, Cantor realized there were different sizes of infinite sets44 [11]. With this discovery as a basis, Cantor produced a theory for the real numbers, and was able to prove that they were, in a sense, more numerous than the natural numbers. Cantor, however, did 43 This paragraph is mainly based on [29:ch43]. 44 A set is the mathematical term for a collection of objects.

24 not stop at this. He developed a theory of an infinite sequence of increasingly large infinite numbers that he called the transfinite numbers. Cantor's general theory is called set theory since it deals with collections of points, numbers or more general objects. Cantorian set theory was important because it offered a basic framework from which investigations of problems in analysis regarding continuity, differentiability and integrability could set out. Although some of Cantor's initial work on sets of points of the real line soon received attention [11:47], as he developed his theory of transfinite numbers opposition grew. In Germany Leopold Kronecker (1823-1891) stubbornly opposed Cantor's work [29:995]. He used his power and influence to attempt to stop Cantor's publications [11:69-70, 135] and as a result of his opposition Cantor could never hope for a position at one of the better universities in Germany [12:68; 29:995]. In France there was similar opposition towards Cantor's set theory, for example from Henri Poincaré (1854-1912) [11:218; 29:1003], perhaps the greatest mathematician of the time. Thus, despite its merits it was not until the last years of the century, following the publication by Cantor of a summary of all his work45 [11ch8-9], that set theory was applied to other areas of mathematics, first in the works of Emile Borel (1875–1956), on measure theory, and of René Baire (1874–1932), on function theory [11:247; 26]. They were soon joined by Henri Lebesgue (1875-1941) whose work was “one of the great contributions of this [the 20th] century”46 [29:1049]. In 1900, David Hilbert (1862-1943), together with Poincaré regarded as the greatest living mathematician [12:90], indirectly gave his support to Cantorian set theory. He composed a well known list of twenty three topical problems for the coming century. The first problem on the list was a problem which Cantor had been trying to solve for 20 years [11:107-108], the continuum hypothesis47.

IV.1.2 The deficiencies of set theory As set theory started to gain recognition among mathematicians, although the opposition was still strong, a number of paradoxes were discovered that further exacerbated the conflict. They put doubts on set theory and antagonists took them as signs that it was incorrect. Poincaré, for instance, regarded the paradoxes as fatal to Cantorian set theory and maintained that set theory itself would be regarded by later generations “as a disease from which one has recovered” [29:1003]. The paradoxes of set theory arise if one considers, for instance, the set of all sets, or the set containing all sets which do not contain themselves (Russell's paradox). Paradoxes can also be derived from the properties of the continuum as Cantor defined it [10]. Paradoxes like these do not necessarily pertain to set theory but can be attributed to logic itself, for example the liar paradox: “This sentence is false”, “I am lying”. Bertrand Russell (1872-1970), for one, ascribed to this view and intended to resolve this problem by putting mathematics on a sound logical foundation [11:261- 262]. Not regarding the contradictions, there were other problems with Cantorian set theory as well. The continuum hypothesis was, as I mentioned, not resolved. And a key theorem for the theory to hold, that every set can be well-ordered, had not been proven. In 1904 Ernst Zermelo (1871-1953) supplied a proof of the well-ordering of the continuum that immediately became widely discussed. As Lebesgue has written “Zermelo arrived and the fight began!”[26:61]. One can especially note a debate by mail, which was published in 1905, between the French mathematicians Borel, Baire, Lebesgue and Jacques Hadamard (1865-1963), in which only the last was positive towards set theory [11:253-259; 26:61]. The debate is interesting because it conveys the philosophy

45 Beiträge zur Begründung der transfiniten Mengenlehre. 46 Not even this work of Lebesgue's, however, escaped a similar critique as that which was directed towards Cantor [29:1049-1050]. ℵ 47 The continuum hypothesis states that the “size” of the continuum is equal to the second transfinite number 1 (“aleph-one”).

25 of the partaking mathematicians. Zermelo's proof was questioned on two separate points, both of importance to this essay. Firstly, he made use of the so called axiom of choice and secondly, the proof was non-constructive, i.e. purely existential [11:252-253]. The axiom of choice is the assertion that for every set, there is a function which uniquely associates all non-empty subsets with a unique element of that subset. The controversy appears when the set to be well-ordered is transfinite. Cantor had assumed that it was possible to simply, as for a finite set, pick one element at a time until the set was empty. However, this argument is not rigorous enough for mathematical purposes. Formulating the axiom of choice, Zermelo identified the basic assumption that had to be true for one to be able to make a transfinite number of choices. The axiom was vehemently criticized because no distinction was seen between the method Zermelo applied and the “intuitive” successive transfinite number of choices that Cantor had presented [11:ch11; 34:ch2.3]. Indeed, the French “pioneers”, Borel, Baire and Lebesgue had made implicit use of the assumption in their earlier work [34:ch1.7] but as their use of the axiom became explicitly understood, they became increasingly reluctant to accept the it. Borel eventually even stated that he needed to abandon set theory. Baire had already in 1900 experienced psychological problems and given up on mathematical investigations [26:60]. Both simply regressed to a position against the use of transfinite numbers [11:254-255]. Overall, the reception of the axiom was ambivalent. Moore [34:118] writes that between 1904 and 1908 five German mathematicians published their views on the axiom. Of them, only two were outright positive and this was the most positive reception in any country. According to Moore [34:3], this situation did not change until 1918, when the polish mathematician Wacław Sierpiński (1882-1969) founded the Warsaw school of mathematics and began to seriously investigate the axiom's “relationship to many branches of mathematics”. As a continuation of the criticism of the axiom of choice Zermelo's proof was criticized for being non-constructive. Zermelo had merely shown that the well-ordering of a set was always possible, he did not supply a method for constructing such a well-ordering. The critics claimed that one could not demonstrate the existence of a mathematical object without defining it. This was essentially the criticism that Kronecker had immediately given to Cantor's work. His stance had been that it was imperative to explicitly construct all mathematical objects, otherwise they did not exist [36]. This idea was now echoed by Borel, Baire and Lebesgue, the ones who had initially made use of set theory.

IV.1.3 The foundational crisis A result of the debate over set theory was that a number of prominent mathematicians felt that mathematics had to be properly founded on a basic consistent theory. There emerged several different programs that tried to accomplish this. The main camps in this conflict are personified by Hilbert, Russell and L E J Brouwer (1881-1966), each one with a program of their own. Common to all these was, however, that they wanted a non-platonic foundation for mathematics [27]. Russell and Hilbert shared a view that mathematics had to be reduced into a basic system. Russell's idea was to reduce mathematics to logic while Hilbert wanted to construct an axiomatic system from which all mathematical truths could be deduced with the use of a small collection of intuitive methods. Brouwer's program, on the other hand, was a complete reworking of all of mathematics on an “intuitionist” principle. This principle states that mathematics is a mental construction and thus that all mathematical objects must be directly constructible in the mind [27]. Poincaré can be seen as an earlier proponent of this approach, claiming that that every aspect of Cantor's work should be purged from mathematics [11:266] In the end Russell abandoned his project without having succeeded. Brouwer's intuitionist mathematics survived but only in the backwaters of mathematics. Hilbert's program suffered the perhaps hardest blow when Kurt Gödel (1906-1978) in 1931 was able to prove that set theory, and even worse, common arithmetic, could never be proven consistent within an axiomatic system that

26 is sufficient to describe it. After this the battle subsided and most mathematicians have adopted some sort of [29:1209], while the foundational issues have become the concern of a small number of specialists. Set theory, in the form of Zermelo-Fraenkel's axiomatic system, eventually became canon in mathematics.

IV.2 Comparative analysis of the paradigm shift in the West and in Russia Let us now return to the characterization of a paradigm shift that was given in I.2. I hope that it is clear from the section above that the period under investigation was a period of revolutionary science. There was no consensus as to which methods were allowed, as to what was regarded as “scientific”, perhaps best illustrated by how Borel decided that parts of Zermelo's approach lay “outside mathematics” [34:93]. I also think it is telling that the conflict has been called the “foundational crisis”, a most appropriate terminology from a Kuhnian perspective. The situation was different in Moscow. Moscow mathematics was comparatively isolated, partly because of the frosty relations to the better known Petersburg school and because results were mainly published in Russian after the founding of the Moscow Mathematical Society in the 1860s [37:279]. Most of all, however, it was because Moscow was a young and comparatively small mathematical center. The new paradigm appears to have been established in Moscow at the time when the Luzitaniia formed since there was a consensus around the new methods of mathematical analysis. Phillips [37:294-295] quotes how “among the students there appeared a scornful attitude toward classical analysis” and that “For their part, some of the younger Moscow mathematicians were obviously contemptuous of the conservative interests of the mathematicians in Petersburg”. They also began to produce results within the new paradigm, first in the new area of descriptive set theory and, later, also in other areas of mathematics [28:576-577]. Perhaps most notable of these were Aleksandrov's and Kolmogorov's fundamental contributions to topology and probability theory respectively. This successful work is also a sign that Moscow mathematics had successfully adopted the burgeoning paradigm.

IV.2.1 Exemplars of the new paradigm To make a more formal analysis of the paradigm shift I have identified three possible exemplars of the new paradigm. As stated, a paradigm shift is characterized by a transformation of the consensus of the scientific community and I will discuss the philosophical aspects of the process when a consensus on the exemplars I have recognized was developed. In doing this I will identify the philosophical grounds of the Russian mathematicians, Egorov, Florenskiĭ, and Luzin for accepting the exemplars and I will specifically use the philosophical classifications made in III. This will illustrate how the Moscow intellectual climate was more compatible with the exemplars. The first exemplar I have identified is the transfinite numbers. They are the outcome of Cantor's set theory and are indispensable for the development of analysis and the understanding of the real numbers. Philosophically they pose the ancient question of whether actual infinity exists. One can trace the principle objection against actual infinity back to Aristotle who denied its existence. This can be taken as based on the subtle difference between Plato's ideas, that exist independently of the things, and Aristotle's forms, that exists in things [7]. The Aristotelian view and its development into can be seen as the precursor to western empiricism and ultimately materialism. On these grounds actual infinity is objected against since one cannot experience it, as it exists neither as a quality nor as an object. This is important when considering the mathematical development, as classical analysis traditionally had evolved though application to various problems in physics. Because of this mathematics was closely linked with the materialist and empiricist standpoints and for mathematicians working in that tradition it was natural to reject actual infinity for being an absurdity. As we have seen, Borel and Baire eventually did just that. Saying that psychological and philosophical speculation had no place in mathematics they could not accept actual infinity. Whereas, as we have seen, Florenskiĭ used actual infinity as an example of

27 knowledge reached through an effort of faith. We see how Orthodox Christianity's Platonic, rather than Aristotelian, heritage can be seen to facilitate the acceptance of actual infinity. Another aspect of the transfinite numbers is the paradoxes that appeared as a result of their introduction. I will take the set of all sets as an example. The paradox appears as the set would have to contain itself and thereby be larger than itself. Cantor himself treated this problem simply by not recognizing such sets that provoke paradoxes as sets. He called them instead inconsistent sets. This was completely in accordance with Cantor's definition of a set as “composed of any definite, separate objects of thought which could be comprehended as a whole” [11:228]. We saw in the section on holism how Florenskiĭ echoed these words of Cantor's (who's works Florenskiĭ had studied) by saying how it was essential to comprehend an infinite set as a completed whole. It is easy to put this holist conception of actual infinity against a possible reductionist reaction to actual infinity, namely that it cannot be simultaneously presented in all its parts and thus is intelligible. I will later return to argue why western mathematicians can be regarded as reductionists. The second exemplar I have identified is the axiom of choice. It has been of profound importance for the development of mathematics, not in the least for measure theory, Lebesgue's legacy. The controversy regarding the axiom stems from the early implicit uses of it as an infinity of arbitrary choices [34:ch1.7]. This is related to a debate about how to define a (mathematical) function, which Luzin himself has summarized in an article about the evolution of the function concept [33] as I mentioned in III.2.3 . The debate traces back to the Dirichlet definition of a function which states that y is a function of x if for every value for x there is a corresponding definite value for y. Appended is the understanding that “it is irrelevant in what way this correspondence is established” [33:264]. This last point is key to understanding the connection between the axiom of choice and the definition of what a function is for, as Luzin writes, “the arguments for and against this point linked up with the arguments for and against the so-called axiom of choice” [33:265]. The connection can be described as the fact that whenever there's an infinity of possible values for x, one is allowed, by the Dirichlet definition, to make an infinite number of arbitrary choices48. Borel and Baire wanted to restrict the definition while Lebesgue took a more subtle approach. According to Luzin [33] the work in Moscow followed Lebesgue's direction, which was less restrictive than that of Borel and Baire and allowed a larger class of functions while still not completely accepting the axiom. Moore [34:206], however, notes that both approaches rested on the use of the axiom! These developments in function theory were closely connected to measure theory and allowed the study of unconventional functions. Together they make up a big portion of the new mathematics. I have made this detour into the history of mathematics since a bit of background is needed for the discussion of the philosophical connotations of the axiom of choice. It is interesting to note the connection which Luzin drew between the development of the function concept and the axiom. We remember that in the section on indeterminism, Bugaev's, Florenskiĭ's and Egorov's interest in discontinuity was established. The quotes from Florenskiĭ on continuity as a special case of discontinuity can be directly connected to the discussion over the axiom and the definition of a function since the axiom was needed for it to be possible to analyse many discontinuous functions and even essential for the possibility of constructing some of the more pathological functions. Given this, it is not surprising that the Moscow mathematicians took an interest in the axiom and the mathematics created with it. Accordingly, the Sierpinski mentioned in IV.1.2 was a polish mathematician who cooperated with Egorov and Luzin in Moscow during the years of the first world war. After this Luzin and Sierpinski together penned a number of articles regarding the axiom of choice and Sierpinski's work and the Warsaw school was greatly influenced by Moscow mathematics [37]. Another point worth mentioning is that the axiom allows the human creativity complete freedom in creating functions, something that should have appealed greatly to someone pondering the nature of human freedom in

48 To non-mathematicians I must point out that this definition gives no precedence whatsoever to continuous functions but allows the creation of all kinds of discontinuous and more pathological functions.

28 connection with discontinuity as a phenomenon49. The last exemplar is so basic it has no label. One could call it the existential approach to mathematics. Moore [34:310] writes that the opposite viewpoint, constructivism, acquired a name, whereas the existential approach simply came to be called modern mathematics, since it became the established approach. The existential standpoint in mathematics is the acceptance of the existence of mathematical objects solely based on the proof that this would be consistent. The possibility of an explicit construction of the object is not necessary. I believe it's clear that mathematical “” and Platonism are quite close. It makes sense to accept that a mathematical object exists without construction if one shares the understanding that mathematical objects exist externally, outside the human mind. An empiricist viewpoint, on the other hand, would be more compatible with the constructive approach, since only in this case would one have an actual object to investigate. The Platonist inclination of the central figures has been established but equally true is the main western intellectual inclination towards positive knowledge. Graham & Kantor [26] shows this about France and in most of Western Europe it's evident, for instance, from the profound influence of logical positivism in the early 20th century.

IV.2.2 The philosophical discrepancy While the analysis of the exemplars was mainly aimed at the philosophical situation in Moscow I will now make a more thorough comparison with Western Europe. This discussion should further highlight the, in my view, profound differences between the Moscow and the Western philosophical approaches to science in general and to mathematics in particular. There are a number of points at which they differ greatly. First of all, there was a considerable split between the Moscow and the St. Petersburg mathematicians. As we have seen, classical analysis with strong connections to applications was preferred in St. Petersburg while the new discoveries in mathematics were seen with suspicion since they were too theoretical and philosophical. The conflict mirrors the traditional opposition, mentioned in III.1, between “two capitals of Russia”, Moscow representing the old religious Russia and St. Petersburg representing the rational West. Another difference also evident between the two capitals, but mainly with the west was the positivist claims on truth in the name of science. Hilbert, for instance, was convinced that all mathematical problems were solvable [12:90], which can be illustrated with a quote of his from 1930: “We must know, We will know” [12;103]. This, of course, is in stark contrast with Luzin's views, presented in his letters, about the truth content and possibilities of science. Worth mentioning is also a conjecture made by Luzin in 1925 that “One does not know, and one will never know, whether the projection of the complement of an analytic set (supposed uncountable) has the cardinality of the continuum, ... nor whether it is measurable” [41:292] prohpesying the coming of Gödel's incompleteness theorem. Yet another current in western mathematics was made up of the reductionist programs to build a complete axiomatic or logical basis for mathematics. Hilbert's and Russel's programs can be taken as attempts by a rationalist, logicist reaction to “deal successfully with the problem of mathematical consistency” and in that way remedy the uncertainties in the fundamentals of mathematics, as Dauben [11:241] writes. For, as he continues, “if mathematics could not be made certain, then where could anyone turn and hope for absolute knowledge about anything?” an additional illustration of the faith in science and reductionist approach to knowledge of the western paradigm. As mentioned in IV.1.3, both programs failed to complete their goals and mathematics stands without a solid rational basis. In Moscow, it seems these programs were all but ignored and, indeed, after a period of interest in symbolic logic Luzin in 1909 says that he has “lost both interest and faith”50 in it. Not to mention that the aims of the programs rule out all possible mystical aspects

49 We recall that Florenskiĭ's views on the power of human creativity were mentioned in the section on holism. 50 “[...] интерес и вера [...] пропали”. [15:159]

29 of mathematics and make claims of universal truth. Last, I want to lend my support to the analysis made by the Moscow mathematicians regarding the role of continuity in western culture. Graham & Kantor [26] do not especially highlight this mode of thought but still mention the prominent mathematician Emile Picard who completely rejected the use of discontinuous functions on the basis that “nature does not make jumps; we have the feeling, one can even say the belief, that in nature there is no place for discontinuity” [26:59]. Kline [29:1049-1050] relates how Lebesgue was criticized because his work dealt with non-differentiable functions, an even more restrictive condition than continuity! Kline even quotes how Lebesgue has written that “the fear and horror [of functions without derivatives] which Hermite showed was felt by almost everybody” [29:1050]. To illustrate how strong a grip continuity has had on (Western) science I would like to mention the Science article EVOLUTION: Ecology Returns to Speciation Studies [35] which states that contemporary scientists are surprised by the great importance big beneficial mutations play in the evolution of species as opposed to the old (continuous) idea of small consecutive mutations. The article was published in 1999! This is an example from biology but we recall that Bugaev took just Darwin's theory of evolution as an example of the bias towards continuity. In all of these examples the western views take either the form of conservatism or of reductionism bordering on simplification. Just consider the opinions about unconventional functions or the idea of a complete reduction of all of mathematics. Thus, in the west many mathematicians either shunned or simply ignored the new mathematics, since it challenged their faith in contemporary scientific method, or, like Hilbert, had their vision obscured by an even greater hubris. The Moscow mathematicians, on the other hand, were, as a result of their different philosophical standpoint, fascinated by the new mathematics and delighted and pleased with the complexity and diversity that it embodied. As a side note, it appears as if Sierpinski, being another of the forerunners of the new paradigm, was not only influenced professionally by Luzin and Egorov, but that he, just like them, was also very interested in Florenskiĭ's philosophical work [23:253]!

IV.2.3 Luzin's philosophical transformation In this section I must also make a comment about Luzin's standpoints in regard to the philosophy of mathematics since they can be regarded as contradictory to my arguments. In III.2.2 it was mentioned that Luzin lost faith in actual infinity, and, as stated above, his work followed the lines of Lebesgue's and he came to adopt a constructivist standpoint. This is of course interesting, but is actually of minor importance for my argument that Platonism and mysticism played parts in the acceptance of the new mathematics in Moscow. Luzin treated the axiom of choice and actual infinity as heuristic devices and regardless of his stated philosophical views definitely made use of them. Moore [34:288-289] writes that “[Luzin's] philosophical views on the Axiom and the actual infinite deviated from much of his research on projective sets”. As I have mentioned I interpret this as Luzin maturing as a mathematician and as a person. Upon the return in 1914 from his second academic mission he had changed his views on these fundamental issues, seemingly influenced by the contact with the French mathematicians. These views were, however, “professional” and he retained his religious faith. Furthermore, Luzin's interests and the basis of his mathematical knowledge were formed when he was the most under Florenskiĭ's influence, which can be seen in how he expresses his interest in foundational mathematics in letters of May 1 1906, April 12 1909, and December 24 1909. During this period, Luzin was also drawn to more general philosophical question and it entered his work, as his teacher, Mlodzeevskiǐ, noted with dislike [23:245]. Writing to Florenskiĭ on September 22 1910 even Luzin himself noticed that he was only proficient in set theory and lagging behind in other areas. Hence, his philosophical inclinations during this early period would have considerable impact on his mathematical knowledge and interests. Later both A. N. Krylov and Lebesgue made note of Luzin's (successful) philosophical approach to mathematics

30 [14:33; 26:73-74].

IV.2.4 Concluding words Finally, to conclude this section, I will allow Cantor and Gödel the last words to support my analysis of the paradigm shift. They were highly original mathematicians and their respective work can be taken as the starting and end points of the crisis in mathematics. They shared a belief in philosophical idealism and a religious faith. Thus, they were quite similar to the Moscow mathematicians. Accordingly, Cantor in 1887 identified “an academic-positivistic skepticism that powerfully dominates the scene. This skepticism has inevitably extended its reach even to arithmetic, in which domain it has led to its most fateful conclusions. Ultimately this may turn out most damaging to this positivistic skepticism itself.” [12:80] No doubt Cantor had in mind the opposition against actual infinity, his transfinite numbers and set theory. As we have seen he was correct in many ways as the continued criticism of his and his followers' work obstructed the paradigm shift for another 40 years or so. Gödel also commented on the limitations of positivism and Davis [12:108] relates that Gödel believed “it was precisely by rejecting those ideas that it became possible for him to see connections that other logicians had overlooked, making his momentous discoveries possible”. Gödel recognized a blindness among logicians that he found surprising but that “the explanation is not hard to find. It lies in a widespread lack […] of the required epistemological attitude” [12:114]. Thus, what ultimately killed the opposition to the new paradigm, Gödel's results of incompleteness in the 1930s, was only found by a mathematician sharing the philosophical standpoints of the early Moscow school.

V Conclusion

V.1 Summary I believe this essay quite convincingly argues that Moscow mathematics was an outsider in the mathematical, and even the whole scientific, community. Philosophically the central figures of this essay were even close to theology, yet nothing I have found in their philosophical work contradicts the findings of the new scientific paradigm. In my opinion this shows just how subtle and considered their standpoints were. In a time when science had basically become the new religion and unbound optimism characterized the scientific community they urged for moderation. Paradoxically, in the given environment, this meant they became revolutionaries and not reactionaries as one could expect. Criticism of this point could include the objection that Moscow mathematics was much larger than just my four central figures. Here, I rely on the accounts of Iushkevich, Demidov and others for the significance of Egorov and Luzin for the subsequent development of mathematics in Moscow. Furthermore I believe that the argument that the Moscow mathematicians' philosophy was closer to the new mathematics is quite convincing as well. There is a stark contrast between the Russian and Western reception of the new discoveries. As I point out immediately above, a parallel can be drawn with Gödel and Cantor who shared similar beliefs and who because of this could make the seminal discoveries of the epoch. Something that is interesting to note in connection with this is the influence of German romanticism. Borel has said Cantor “brought to the study of mathematics that romantic spirit which is one of the most attractive aspects of the German soul” [26:60]. Gödel was also, in the context of logical positivism, an outsider. He visited the meetings of the but was without sympathy for their ideas [12:111] and as mentioned above was both religious and an idealist. Thus, both Gödel and Cantor in a sense follow the tradition of romanticism. The attentive reader will also remember that romanticism was mentioned both in III.1

31 and III.2.4 where its considerable influence on Russian culture, and especially the aspects that came to influence the Moscow school, was touched upon. It would certainly be interesting to investigate this philosophical connection, especially as Bugaev, Luzin and Egorov all spent time in Germany. Bugaev was even staying in Berlin while Cantor, who, considering his upbringing, probably spoke Russian, also in Berlin, was making his first contributions under Kronecker's supervision. But that is another story altogether. I started out this essay explaining that I was interested in finding the reasons for the success of Russian mathematics. I don't claim to have given the ultimate answer to this question although I believe I have given a plausible ideo-historical interpretation. The success of Moscow mathematics of course had structural reasons as well. I presume the revolution brought with it a possibility for students from a wider range of social backgrounds to study at the university, of course while yet others were excluded. In this view it would be interesting to investigate the social status of the members of the Luzitaniia. Another external factor that probably boosted the Moscow school was the moving of the Academy of Sciences from Leningrad to Moscow in 1934. However, notwithstanding the structural changes I find it hard to believe that the success of the Luzitaniia was due to an abnormal collection of young prodigies and not the fact that they had successfully assimilated a new approach to mathematics. Another point which I imagine can be questioned is my Khunian analysis of the development of mathematics. That it indeed was a revolutionary change appears however to be widely supported, for instance by Dauben [11], Davis [12], Kline [29] and Moore [34]. It may also be argued that the changes were to narrow to constitute a complete paradigm shift. In my opinion, though, it's possible to expand the view of the development to an all-embracing paradigm shift not unlike the Copernican revolution. It seems to me like the discoveries in mathematics with the advent of uncertainty and relativity of truth is mirrored in the developments of Einstein's relativity and of quantum physics. This is yet another connection which I would find it interesting to examine, especially in connection with Russian cosmism. Working with this essay has been highly interesting and stimulating and it has been difficult to limit my investigations. I hope that the result relays a coherent picture and would like to think that anyone who investigates the material would be further convinced. I can especially recommend Ford's articles and Demidov's English publications to someone interested in more information but not proficient in the Russian language. For the complete understanding of the subject I would say that knowledge of both mathematics and its history and Russia in general is indispensable. This also applies to myself as my understanding of these areas has deepened significantly during this work, although I would still not consider my knowledge to be sufficient. There is, luckily, a lot left to learn and to investigate.

V.2 Evaluation of sources Except for a few remarks about contradicting facts and my critique of Graham & Kantor [26] I have not presented any assessments of my sources. In general I have found the older Soviet material (before 1980) to be biased against acknowledging the philosophical aspects of mathematics, something which can hardly be considered as surprising. Two examples are how Egorov's death is treated in [30] (it is only casually mentioned in one line) and the non-existence of Florenskiĭ in any such works. It is of course because of this that an interest in this period has surged since the fall of the USSR. This interest is however not completely unproblematic either. In contemporary Russia there is a current which can be called neo-slavophilism that aims to reevaluate genuinely Russian culture and downplay western and communist contributions. I would never dream of accusing the scholars I have read of undue bias and I have not found any such indications whatsoever. In fact, the publications I cite correspond very well to my picture of the primary source material. It is however with a grain of doubt that I read [25], a popular version of [26], where Graham tells of his meetings with a contemporary Russian mathematician and Name worshipper who is the

32 source of Graham's introduction to Name worshipping. I have encountered speculation that said mathematician could be Igor Shafarevich (1923-) or his former student Alekseĭ Parshin (1942-) (actually co-editor of [15]). Shafarevich is a prominent mathematician but a highly controversial person ideologically. However, regardless of these speculations, a contemporary Name worshipper and mathematician would probably be biased in his views of the development of mathematics, which could possibly account for the direction of Graham's presentation.

V.3 Acknowledgements Finally, I would like to express my gratitude to Sergeĭ Demidov and Sergeĭ Polovinkin for sending me the material I have been unable to obtain in Sweden. I would also like to thank Gunnar Berg, who has assisted me with expertise in mathematics and the history of mathematics. I owe great gratitude to my supervisor Fabian Linde as well, who has been a great source of information on philosophy in general and Russia in particular. He has also provided much needed enthusiasm and encouragement. Last but not least I would also like thank my friends and family for listening to my endless stories about long gone Russian mathematicians.

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