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

UPPSALA UNIVERSITY

Department of Modern Languages Hjalmar Eriksson

Flogstavägen 25B 752 73 Uppsala 073-0681523

hjer0099@student.uu.se

BACHELOR THESIS

Russian language D-level Spring semester 2009

Умом науку не понять

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”.

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 Russia 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 philosophy...11

III.1 A crash course in the history of Russian thought...11

III.2 An alternative philosophy of science...14

III.2.1 Platonism...15

III.2.2 Mysticism...18

III.2.3 Indeterminism...20

III.2.4 Holism...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

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 Russians 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. Aleksandrov

. 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]

. 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 religion 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 evolution of science is all to often seen only in the context, and as a product, of western society and culture. I will, however, argue that the intellectual climate among Moscow mathematicians early in the 20th century, in the terms of ontology and epistemology, 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.

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 mathematics

.

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 difference 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 metaphysics” [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.

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 logic. 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].

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 rationalism, logic and epistemology. As science is concerned with the production of sound and in a sense “objective” knowledge 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

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 experience the Renaissance that supplied Western Europe with the Aristotelian heritage of natural philosophy 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

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”

. 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.

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 idealism as an alternative to positivism [13:117]. Bugaev soon picked up on this and created a philosophy of his own called Evolutionary monadology, based off of Leibniz'

monadology. Evolutionary monadology is an idealist philosophy where the basic constituents of reality 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 causality) [39]. One example of such a process is free will, 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 human 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.

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 integration

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.

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 worshipping

, 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 scientism, 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.

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]

acknowledged

, 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 Luzitaniia

. 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 development

. 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 event 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 Truth” 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].

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 mathematics

.

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.

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 Russian culture 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

conservatism, 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 ideology of development and education through socialism 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, empiricism, rationalism and (Orthodox) mysticism. Solov'ëv suggested that empiricism and rationalism on their own led to a denial of the objective existence 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, materialism, 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

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

mathematicians. Egorov and Florenskiĭ were evidently captivated by them. However, I would not take the mathematicians' involvement with Name worshipping as them being 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 mathematicians

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 justice, are real

, 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.”

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 philosophical realism. 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]

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.”

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”

. 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!”

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]

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.”

Here one notices how Luzin experiences 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).”

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”

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.”

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 objectivity. 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 truths 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].