Book Review

Naming Infinity:A True Story ofReligious Mysticism andMathematical CreativityReviewed by Alexey Glutsyuk

Naming Infinity: A True Story of ReligiousMysticism and Mathematical CreativityLoren Graham and Jean-Michel Kantor,Belknap Press of Harvard University Press, 2009.ISBN-13: 978-06740-329-34.

During the nineteenth century, a foundationalcrisis in mathematics led to signal events offundamental importance. The first was the creationof set theory by Georg Cantor at the end of thenineteenth century. The second was the creationof the theory of functions and of measure theoryand integration theory by the French trio EmileBorel, René Baire, and Henri Lebesgue. Their worksrelied heavily on Cantor’s set theory. A majorcontribution to the further development of settheory, function theory, and topology was made byRussian mathematicians: Dmitry Egorov, NikolaiLuzin, and their school, the famous Lusitania.

The book of Jean-Michel Kantor and LorenGraham presents the history of this importantperiod of mathematics through vivid portraits thatbring to life the personalities of the mathematicians.The main heros of the book are Cantor, theabove-mentioned French trio, and a Russian trioconsisting of Egorov, Luzin, and their close friendPavel Florensky, an extremely talented scientistand engineer and a priest of the Russian OrthodoxChurch. The book intertwines and links theirmathematical research with their cultural andreligious backgrounds.

The authors describe the continuous develop-ment of mathematics from Cantor to the Russians.

Alexey Glutsyuk is chargé de recherche in the Cen-tre National de la Recherche Scientifique at the ÉcoleNormale Supérieure de Lyon. His email address isaglutsyu@ens-lyon.fr.

DOI: http://dx.doi.org/10.1090/noti1071

Cantor, who was the first to compare differentkinds of infinities and prove key results aboutthem, was a Protestant Christian believer and aphilosopher of “free mathematics”. With Cantor’sset theory as a basis, the young French mathemati-cians Borel, Baire, and Lebesgue created modernmeasure theory and function theory. But whensome difficulties and paradoxes were discovered inthe foundations of set theory, they retreated fromresearch in the subject. After that, further researchwas carried out by the Russians Egorov and Luzin,who were Orthodox Christian believers,1 and bytheir students. They were men of great spirit andcourage, inspired by their Christian faith, and theyattacked difficult classical problems directly. Theauthors of the book claim that the mathematicalresearch of the Russian trio was inspired by NameWorshipping, which was a heretical current in theRussian Orthodox Church at the beginning of thetwentieth century.

More than half of the book is devoted to theRussian mathematicians, who worked during adramatic period of Russian history: the Revolutionsof 1905 and 1917, the Civil War, the Bolsheviks’rise to power, Stalin’s terror, . . .. The book showshow, in these very difficult conditions, Egorov andLuzin managed not only to obtain their famous

1As shown by the lives of Cantor and the Russian trio—andby the book under review—a religion does not contradict sci-ence; the two can complement each other in a harmoniousway. The same point of view is expressed in the book Sci-ence and Religion (Moscow, Obraz, 2007) by ArchbishopLuka (Voyno-Yasenetsky) of Crimea (1877–1961), a famousRussian and Soviet surgeon. He shows (with detailed his-torical analysis and citations) that a majority of the mostfamous scientists were believers. (Archbishop Luka was per-secuted by the Bolsheviks for his Christian faith, beforehe was awarded the Stalin Prize for his achievements insurgery. He was recently canonized by the Russian OrthodoxChurch.)

62 Notices of the AMS Volume 61, Number 1

results, but also to create the outstanding Moscowmathematical school “Lusitania”. This school putMoscow on the mathematical map of the world andmade it one of the world centers with a maximalconcentration of outstanding mathematicians. Themajority of famous Moscow mathematicians aredescendants of Lusitania.

The authors describe the dramatic personalfates of the Russian trio and the Lusitania studentsafter the Revolution of 1917. During Stalin’s terror,Egorov, Florensky, and Luzin were persecuted fortheir Christian faith. All three are highly admirable,especially Egorov and Florensky, who showed greatpersonal bravery. When the Bolsheviks cruellypersecuted Christian believers, these two menremained believers and did not change theirhabitudes at all. In fact, Florensky’s courage onlyincreased: he caused sensations by always wearinghis priest’s robe at scientific and engineeringmeetings. Egorov and Florensky were arrested,and their lives ended tragically: Egorov died indetention, and Florensky was executed. Luzin, abeliever and a professor of the old generation,barely escaped a similar destiny after he wasaccused, publicly and wrongly, of being a traitor.He was more productive scientifically than Egorovand Florensky, though more unstable and lessbrave. The book describes in an honest way howsome of the famous Lusitania students werecontradictory people, with good and bad aspects.

I liked very much the authors’ choice of thepicture on the book’s cover, a reproduction ofthe painting Philosophers by the famous Russianpainter Mikhail Nesterov. The painting representstwo great Russian philosophers and priests, PavelFlorensky and Sergei Bulgakov, whose fates werecompletely different. Bulgakov was expelled fromRussia by the Bolsheviks on the Philosophers’Ship, along with many other philosophers. Afterhis expulsion, he remained extremely active asa philosopher and theologist and published atremendous number of works. He was one of thekey creators of the famous Saint Serge Ortho-dox Institute in Paris. Many other scientists andphilosophers decided to leave Russia after theRevolution of 1917. Florensky was one of the veryfew theologists and philosophers who decidedto stay.2 He was quite aware of the new politicalsituation in Russia and of what his fate would

2While in prison after his second arrest, Florensky had theopportunity to emigrate to the Czech Republic together withhis family. He refused. Florensky’s grandson and biogra-pher, Igumen (Father Superior) Andronik (Trubachev) saysthat he does not know of any other case in which a Gulagcamp prisoner refused to leave the camp. The biography,published in Moscow in 2007, would be interesting to thosewho would like to learn more about Pavel Florensky.

be. Nevertheless, he decided to stay and to serveRussia with all his energy.

It is very impressive that the authors, not beingof Russian origin, know the history of Russia, itsmathematics, and its church so deeply. They havedone an unimaginable amount of work, includingmany trips to and across Russia, many interviews,and extensive reading in many archives. I would liketo add that the author Jean-Michel Kantor greatlyhelped young mathematicians from the FormerSoviet Union during a very difficult period in the1990s. At that time, in order to have something toeat, many mathematicians from the Former SovietUnion chose either to leave the country or to leavemathematics in order to earn money. Jean-Michelmiraculously organized financial assistance fromthe French government, thereby saving many youngmathematicians, including myself, by allowingthem to do only mathematics while staying in theircountry. I wish to take this occasion to thank hima lot once more.

Returning to the book, I would like to make aremark related to my own preferences. I wouldhave been glad if the book had put less emphasison details about personal lives and philosophicalreasonings about inspiration, and more emphasison mathematics (explained in a way understand-able by nonmathematicians) and on history. Thereasoning behind my preference is that, while onecan check whether a scientist was inspired bysome other scientific work, one cannot check in arational way whether the inspiration for a person’sscientific creativity came from outside of science.

I will now describe in more detail the content ofthe book. The first chapter presents the origin andhistory of Name Worshipping, which, according tothe authors, was a source of inspiration for theRussian mathematical trio. The main part of thechapter is devoted to a dramatic event of February1913: the storming of St. Pantaleimon Monastery atMount Athos by the army of Russian Tsar Nikolai IIand the cruel expulsion of the Name Worshippingmonks from the monastery.

The second chapter describes the life andmathematical achievements of Georg Cantor andthe reception of his theory by other famousmathematicians of his time, including a detailedhistory of the development of Cantor’s set theoryand of his famous continuum hypothesis (CH),with links to his philosophy.

The third chapter is devoted to the receptionof Cantor’s theory in France and the French trioof Borel, Baire, and Lebesgue. It starts with animportant event in the history of mathematics, theInternational Congress in Paris in 1900. At thecongress, Hilbert made clear that Cantor’s theorywould play a major role in the future development

January 2014 Notices of the AMS 63

of mathematics and placed the continuum hypoth-esis at the top of his famous problem list. The bookdescribes some of the French trio’s contributionsthat were heavily based on Cantor’s set theory:the Heine–Borel theorem, the basis of the future“Borel measure”; the introduction of Borelian andmeasurable sets; the introduction by Baire of thenotion of semicontinuity and his classificationof discontinuous limits of continuous functions;and the construction of the “Lebesgue integral”.The authors intertwine the development of mathe-matics in France with descriptions of the culturalspirit and important historical events in Franceat the beginning of the twentieth century. One isthe tragic Dreyfus Affair, in which leading Frenchmathematicians, including Henri Poincaré, activelydefended Dreyfus. The authors also describe thelives of the members of the French trio, such asthe extremely rich and intense life of Borel, who,besides being a mathematician, played many otherroles: Navy minister, mayor of his home town, andparticipant in the Résistance.

The rest of the chapter focuses on contradictionsand paradoxes that appeared in the foundationsof Cantor’s set theory at the beginning of thetwentieth century, such as the difficulties foundby Cantor3 himself in 1895 and various paradoxes,including that of Russell. There is also a discussionof Zermelo’s Axiom of Choice and the famousexchange of five letters about it by Borel, Baire,Lebesgue, and Hadamard. This exchange confirmedthe critical state of the foundations of mathematicsand raised important problems that were partiallysolved later, including famous incompletenessresults by Gödel and Cohen. Even now, not all isresolved.

Chapter four is devoted to the Russian trio:Dmitry Egorov, Nikolai Luzin, and Pavel Florensky.At the end of the nineteenth and the beginningof the twentieth century, Russian mathematicswas closely related to philosophy and religion,and the chapter describes the spirit of this timein a remarkable way. The authors discuss thecreation of Markov chains, which appeared asa result of a philosophical debate between P. A.Nekrasov and A. A. Markov. Nekrasov, who was aChristian believer and a supporter of the Tsar’spower, drew motivation from philosophy relatedto the question of free will and was thereby ledto make overly strong claims about probabilities.Markov, an atheist and a critic of both Tsaristpower and the Russian church, constructed hisfamous chains as a counterexample to Nekrasov’sstatement. Nikolai Bugaev, the teacher of the three

3As is mentioned in the book, Cantor escaped from con-tradictions by naming the objects “too big to be sets” as“Absolute”.

members of the Russian trio and the president ofthe Moscow Mathematical Society, defended freewill and connected it to mathematics. While manymathematicians were frightened by discontinuousfunctions and called them “monsters”, Bugaevcalled them beautiful and morally strengtheningbecause they freed the human being from “fatalism”.The opinion of his student Florensky was that thenineteenth century was intellectually a disasterand that one of its main origins was the “governingprinciple of continuity”, which “was cementingeverything in one gigantic monolith”.

The book describes the lives of the Russian triobefore the Revolution of 1917, their mathematicalworks, and their personal qualities. It brieflydiscusses Egorov’s first famous achievement indifferential geometry, after which “Egorov surfaces”appeared. Egorov is described as a deep Christianbeliever whose modesty mixed in a remarkableway with his courage to express his disagreementon matters of principle. For example, he signed apetition protesting the 1903 pogrom against Jewsin Kishinev even though he had not been politicallyactive. The authors describe Luzin’s mental crisisand depression after he saw bloody events inthe Revolution of 1905, and they discuss howcorrespondence with Florensky helped Luzin torecover, become a Christian believer, and return tomathematics. This shows that, for Luzin and for thewhole Russian trio, Christian belief was the Pillarand Ground of the Truth (to use words from the titleof Florensky’s book). Florensky converted to theChristian orthodox faith at the age of seventeen.Eventually, after successfully graduating fromMoscow University, he left mathematics, studiedat the Theological Academy at Sergiev Posad, andbecame a priest. Florensky protested the executionof Peter Schmidt, a revolutionary lieutenant of theTsar’s army. He did not share Schmidt’s politicalopinions; he simply opposed capital punishment.After that, Florensky was arrested and held in jailfor a week, where he wrote one of his mathematicalworks.

Chapter five describes the relations between Rus-sian mathematics and “mysticism”. Henri Lebesguespoke of “naming a set”. Luzin emphasized thesignificance of naming in his mathematical work.As already mentioned above, the authors of thebook relate the creativity of the Russian trio toName Worshipping. This was a heretical current inthe Russian church. Its supporters practiced theJesus Prayer and claimed that, after repeating itcorrectly many times, a person achieves a unitywith God: roughly speaking, the name of God isGod himself. The authors explain the influence ofName Worshipping on the mathematical creativityof the Russian trio in set theory, basically, bynoting the importance of naming in both of them.

64 Notices of the AMS Volume 61, Number 1

Remark. This is the point of view of the authorsof the book under review. From my own point ofview, a claim that Name Worshipping was a majorinspiration for the Russian trio would seem a bittoo strong.

Chapter six gives an impressive descriptionof the spirit and life of the mathematical schoolfounded by Egorov and Luzin, the famous Lusitania.The professors created an atmosphere of opennessand closeness. Sometimes Luzin’s classes finishedin his apartment, with discussions about math-ematics, culture, arts, religion, etc., that wouldcontinue into the night. Most of the students wereyoung, having joined the Lusitania when they werearound seventeen years old. The book describestwo key achievements of Lusitanians, namely, theproof of the continuum hypothesis for Boreliansets by Pavel Alexandrov (1915) and the creation ofdescriptive set theory (1916) by Mikhail Suslin andNikolai Luzin, after Suslin found a fundamentalmistake in Lebesgue’s seminal paper of 1905.

Chapter seven describes the dramatic fatesof the members of the Russian trio after theRevolution of 1917. Egorov and Florensky werepersecuted for having courageously confessed theirChristian belief. The authors describe the attacksagainst Egorov, his arrest and imprisonment, hishunger strike in detention, his hospitalization,and finally his death. A highly admirable personappearing in the book is Nikolai Chebatorev, afamous mathematician though not a Lusitanian.Chebatorev was an atheist and a former RedArmy soldier. He and his wife tried, at huge riskto themselves, to save the believer Egorov. Theauthors discuss the fate of Florensky, who wasfirst arrested in 1928 and sent into exile. After hissecond arrest in 1933, he never came back. It isabsolutely remarkable that, even while in detention,Florensky remained very active and made manyimportant scientific and engineering achievements.He spent the last period of his life as a prisonerat the infamous Solovetsky Gulag camp, where hecreated a famous iodine enterprise. Much later,after Stalin’s death, it was found that he had beenexecuted in 1937.

Luzin was much more cautious than Egorovand Florensky: he became a “secret believer”. Atsome point, he even stopped going to church andrestarted only at the end of the Second WorldWar. However, his caution did not save him, asthe authorities knew he was a believer and aprofessor of the old generation. The authors ofthe book describe the “Luzin Affair”, initiatedby a communist mathematician, Ernst Kolman.Tragically, many of Luzin’s former students andfriends, including some famous mathematicians,were against him in the Luzin Affair and agreed that

Luzin was a traitor. Luckily, Luzin was saved fromimprisonment and death by a letter of supportfrom the famous physicist Peter Kapitsa to Stalin.

Chapter 8 describes the fates of the best-knownmembers of the Lusitania school. It starts withthe impressive genealogical tree of Luzin’s school,his students, grandstudents, etc., which includesthe most famous Russian mathematicians. Tradi-tionally, in Soviet times, the Moscow mathematicalschool was called the “Luzin school”. The nameof Egorov as one of its founding fathers wasnot mentioned at all, because of his arrest andsubsequent death. I wish to thank the authorsfor mentioning this fact and for noting that, evenafterthe collapse of the Soviet Union, Egorov wasnot given the credit he deserved. Moscow mathe-maticians have an obligation to correct this. Theauthors also present portraits of some of Luzin’sfamous former students, with an emphasis onAndrei Kolmogorov, Pavel Alexandrov, and PavelUrysohn. Descriptions of some of their mathemati-cal works are intertwined with information abouttheir personal lives. The friendship of Alexandrovand Urysohn included a very productive collab-oration in topology as well as swimming, tripsabroad, etc. Urysohn wrote one of his famousmathematical papers on the beach at Batz-sur-mer,just a few days before he drowned while swimming.Alexandrov and Kolmogorov were also friendsand collaborators, and they both were among theaccusers of their former teacher Luzin in the LuzinAffair. Both of them were asked by the police towrite a condemnation of Alexander Solzhenitsyn,calling him a traitor, and both did so. Shortlybefore his death, Kolmogorov confessed that hewould fear the secret police to his last day.

Chapter 9 presents the authors’ conclusionsabout, in particular, the relationship betweenscientific creativity and religion. There is also a dis-cussion of the history of the further developmentof the descriptive set theory that Luzin and Suslincreated.

This book under review weaves mathematics,history, religion, philosophy, and human dramain a remarkable story that will appeal to a wideaudience. It is accessible to nonmathematicians andis also well structured, so that readers interestedin specific topics can read parts of the bookindependently of the rest. I highly recommend thisunusual and compelling book.

January 2014 Notices of the AMS 65