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III Moscow Mathematics and Philosophy 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 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 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 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. 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 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. 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
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