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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 [email protected].
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
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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
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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.
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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