(highly borrowed from Naming Infinity)
What’s distinctive about Russian mathematics of early 20th century is perhaps to see one’s work and interests as closely connected to issues of philosophical, spiritual, and ideological nature. In simple words, to see knowledge as an interlinked, united whole. There is a sense of obligation to practice such a mindset, which is to say that, a mathematical theorem or a physical process, though can be analysed and understood insularly, can not be communicated to the world without the philosophical or artistic ramifications of it. The school of Lusitania in Moscow could be regarded as pioneer in their courageous and bold attempt in this regard.
Nikolai Luzin, a Soviet-Russian set theorist, alongside Dmitri Egorov, founded the Moscow Mathematics School, which was an intellectual powerhouse and birthplace to many distinguished mathematicians of 20th century; to give a few example, Sobolev, Kolmogorov, and Shnirel’man. While the school hosted a highly heterogeneous set of young scientists for research, the founders – Luzin and Egorov – were guys with a distinctive personality which made the school successful. While Luzin was social, extrovert, and theatrical in nature, Egorov was quite formal, quiet, and reticent in personality. It was Luzin and Egorov who shaped styles of the incoming students. Luzin’s style of teaching involved giving a lecture in an incomplete form, to awaken desire among students to partake in its development. Sometimes, Luzin would begin a proof at the blackboard and will acknowledge that he cannot recall it. One of the student would attempt it on the blackboard and fail; another student, on the other hand, will successfully write it down. In fact, Luzin once assigned an unsolved problem to a student – Alexandrov, due to which the student got disappointed by the failure and temporarily abandoned the field. Once Luzin met a student Shnirel’man, and the student showed him his notebook where he attempted solutions to some of the difficult problems. Luzin impressed by this, said, “I have been waiting for you for a long time; I just did not know what your name would be”. Luzin confessed to have a dream in which a youth came to him who would solve the Continuum Hypothesis. Later on, Shniler’man got permission to get enrolled in Lusitania, and his first result on Goldbach’s conjecture was that any natural number greater than 1 can be written as sum of not more than (computable) C prime numbers. Egorov, on the other hand, was strict and aloof, and at the same time, attending to practical needs of students, such as, finding jobs, fellowships, and stipends for them [1].
In effect, Lusitania, with an imposing structure, turned out to be a great blend of professors and students who were gifted and at the same time, had a slight sense of mysticism [2] in their approach to research. To give an example, in the first ICM held at Zurich in 1897, Bugaev, while giving talk on the discontinuous functions, linked his talk to his defense on freedom of will, and regarded “discontinuity” as a “manifestation of independent individuality and autonomy” [1] (though I could not find his exact speech online, please let me know if someone can). It is almost impossible to be done in the current times, unless there is some kind of geopolitical urgency about it.
Uniting rigour with a deeply philosophical attitude is a hard task, these days. Back in the days of Lusitania, Luzin was inspired by the ideas of Neoplatonism, to consider three central constituents of the complex world we live: the One, the Intelligence, and the Soul. The One transcends all beings and gives existence to all beings, including intelligence and soul. The view, precisely is that: through Intelligence and the contemplation of the “One”, new forms, whether artistic, intellectual, or philosophical, arise which serve as referential basis of all other existents [1].
[1] Graham, L., & Kantor, J. M. (2009). Naming infinity: A true story of religious mysticism and mathematical creativity. Harvard University Press.
[2] To prevent any confusion, mysticism is referred to as the belief that knowledge of reality comes through immediate insight or illumination, which would then be followed by rational and rigorous analysis. Mystics are not interested in proving or assessing the ontological One. In modern terms, it would be called, intuition followed by examination.
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