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Naming Infinity

Loren Graham and Jean-Michel Kantor
Belknap Press
Publication Date: 
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Dennis Lomas
, on

Naming Infinity recounts dramatic historical events which include, along with much else, the birth of the Name Worshipping current within the Russian Orthodox Church in the early twentieth century and its influence on some Russian mathematicians, an influence which helped them resolve set-theoretic paradoxes and develop descriptive set theory. (Detailed knowledge of descriptive set theory is not required to read or appreciate this book. In fact, the book contains very little set theory.)

In the early twentieth century on Mt. Athos, a peninsula in the Aegean sea, Orthodox Christian monks, many of them Russians, took sides in a dispute between Name Worshippers and anti-Name Worshippers, ending in a Russian invasion of the island and dispersal of Name Worshippers throughout Russia, where they continued to practice and reach out.

Name Worshippers, who traced their view back to the first verse of the Gospel according to John (“In the beginning was the Word, and the Word was with God, and the Word was God.”), recited a Jesus Prayer continuously for long periods of time. (A prayer of this type refers to Jesus. It sometimes consists solely of the word ‘Jesus’.) Naming Jesus brought him into existence, according to Name Worshippers. They were mystics in the sense that knowledge of Jesus was attained through “immediate insight or illumination, rather than through ordinary sense perception or rational analysis.” (p. 92)

Skhimonakh Ilarion, an authoritative advocate of Naming Worshipping who heavily influenced the Mt. Athos monks, held that (pp. 13–14): “learning to recite the Jesus Prayer in the right way was a process requiring much practice that could last for years. The communion with God that the prayer allegedly brought involved three stages of immersion. First there was the ‘oral prayer’ in which the spoken names of God and Jesus were the main concern of the worshipper. Then, if the person praying was sufficiently devout and concentrated on the task, the prayer would move to the ‘mental’ stage, when ‘the mind starts to cling to the words of the prayer, seeing in them the Lord’s presence.’ Last came the ‘Prayer of the Heart’ when the heart gains ‘spiritual élan’ and ‘illumination’ and the person achieves a ‘oneness’ with God.” Done correctly, Name Worshipping could bring practitioners closer to God. (However, Ilarion warned (p. 14), done incorrectly Name Worshipping could lead to sin, in particular, attempting to hasten the final stage could result in sexual arousal. Licentious Rasputin, the authors note, came to be counted among the supporters of Name Worshipping.)

Florensky, a theologian and Name Worshipper who heavily influenced Russian mathematicians “was particularly devoted to the relevance of Name Worshipping to mathematics” He had a mathematical background. “Florensky saw a relationship between the naming of ‘God’ and the naming of sets in set theory: both God and sets were made real by their naming.” (p. 18) Naming infinities brought them into existence (in much the same way as naming God brought him into existence). This view allowed the development of descriptive set theory to proceed. In particular, it allowed, according to the authors, Russian mathematicians to avoid an impasse created by the rationalism of French mathematicians: “[U]nder the influence of their ultra-rationalistic traditions, [French mathematicians] lost their nerve. The Russian mathematicians (Egorov and Luzin) at first studied at the feet of the French, and then, influenced by their own philosophical and religious traditions, pushed forward toward the creation of descriptive set theory, working with exceptionally talented students.” (p. 189)

In this way, a religious mysticism played an instrumental role in the development of mathematical thought. As the authors observe: “The Russians who developed descriptive set theory and assigned new names to subsets of the continuum posed the possibility of the existence of new entities in the mathematical universe, and they went on to prove a program for future research which resulted in substantial agreement of mathematicians all over the world about the new entities. That achievement might have occurred without the inspiration of a religious heresy, but, as researchers loyal to the historical record, we maintain that the way it actually occurred was within a context of mystical, Name Worshipping stimulation.” (p. 192)

The Russian mathematicians had access — so they thought — to a way of understanding not embraced by other mathematicians. In the words of Luzin (p. 93):

In addition to discussing understanding through the senses (“physics,” “natural science”) and understanding through the mind (“mathematics,” “logic”) Florensky has given equal right to another kind of understanding, which you never hear about at the university, naming “intuitive-mystical understanding.”

“A new form of mathematics was being born, said Florensky, and it would rescue mathematics from the materialist, deterministic modes of analysis so common in the nineteenth century.” (p. 97)

The authors draw a lesson: “In contrast to the French leaders in set theory, the Russians were much bolder in embracing such concepts as non-denumerable transfinite numbers. While the French were constrained by their rationalism, the Russians were energized by their mystical faith. Just as the Russian Name Worshippers could “name God,” they could also “name infinities,” and they saw a strong analogy in the ways in which both operations were accomplished. A comparison of the predominant French and Russian attitudes toward set theory illustrates an interesting aspect of science: if science becomes too cut-and-dried, too rationalistic, this can slow down it adherents, impeding imaginative leaps.” (p. 190)

The authors write of much else including the fate of Name Worshipping within Russian mathematics, the development of the Moscow school of mathematics, Stalinist repression of Name Worshipping, of religion in general, and of academic freedom, and biographical sketches of key Russian mathematicians who were influenced by Naming Worshipping. (We find out, e.g., that Luzin had a poor memory.)

The book deserves considered comment. I confine myself to one remark, which concerns the claim that “religion is not the same as mysticism, which is usually defined as the belief that direct knowledge of reality or God comes through immediate insight or illumination, rather than through ordinary sense perception or rational analysis.” (p. 92)  (It is not clear to me whether, for the authors, the expression ‘immediate insight or illumination’ means anything different from what ‘immediate insight’ would mean for them. I assume that there is no difference in meaning.) The authors, who seem to be following Florensky here, appear to oppose ordinary sense perception to immediate insight. Of course, rational analysis is opposed to immediate insight. It may be that the authors have in mind scientific studies which start with data provided by sense observation. In any case, the role of sense perception in contributing to immediate insight is left out of the authors’ picture. Ordinary perception can provide immediate insight of an ordinary kind. In visual perception, e.g., a world of objects in space reaches our consciousness when we open our eyes. Additionally, visual perception contributes to visual intuition (in the sense of a vision-related cognitive capacity for immediate insight). Also, visual perception contributes to mysticisms associated with images. So opposing immediate insight to ordinary sense perception seems problematic. At least such a division does not seem straightforward.

Dennis Lomas has studied computer science, mathematics, and philosophy.

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