Infinity

BOOK REVIEW

Naming Infinity, by Loren Graham and Jean-Michel Kantor, Harvard University Press, Cambridge, MA, 2009, 239 pp., hardcover, ISBN 978-0-674-03293-4 Neque solum rhetorica, sed omnes, quotcunque sunt, honeste artes, ipsa quoque philosophia et theologia, scientiarum omnium regina, tibi servient. (And not only rhetoric, but all branches of knowledge whatsoever, even philosophy itself and theology, the queen of all the sciences, are your faithful servants.) Francesco Petrarca, Contra medicum quendam, III (1355). Die Mathematik ist die K¨ onigin der Wissenschaften, und die Arithmetik ist die K¨ onigin der Mathematik. (Mathematics is the queen of the sciences and number theory is the queen of mathematics.) Carl Friedrich Gauss, quoted in Gauss zum Ged¨ achtnis by Wolfgang Sartorius, Baron von Waltershausen (1856). To´υ ς τ ε γ `αρ ‘αγιωτ ´ατ oυς θρ´oνoυς κα`ι τ `α πoλυ ´oµµατ α κα`ι πoλ´υ πτ ερα τ ´αγµατ α Xερoυβ `ιµ , , ‘Eβρα´ιων ϕων ηιˆ κα` ι Σεραϕ` ιµ ωνoµασµ´ ενα κατ ` α τ` ην π´ αντ ων υπερκειµ ‘ ´ενην εγγ ´υ τ ητ α περ`ι θε`oν , , αµ´ εσως ‘ιδρˆ υ σθα´ ι ϕησι. . . δευτ ´ εραν [‘ιεραρχ´ ιαν ] δ ’ εˆ ι’να´ ι ϕησιν τ ` η ν υπ ‘ `o τ ω ˆ ν εξoυσιˆ ω ν κα` ι κυριo, , , τ´ η τ ων κα` ι δυν ´ αµεων συµπληρoυµ´ ενην κα` ι τ ρ´ιτ ην επ ’ εσχ´ ατ ων τ ω ˆ ν oυραν ´ ιων ι‘εραρχιˆ ων τ ` ην τ ω ˆν , , , αγγ ´ ελων τ ε κα` ι αρχαγγ ´ ελων κα` ι αρχˆ ω ν διακ´ oσµησιν . (It is said that the most holy Thrones and many-eyed and many-winged orders, called Cherubim and Seraphim in Hebrew, are seated above all others in the highest rank, directly beside God. . . . The second [hierarchy]. . . is said to be made up of the Powers, Dominions, and Virtues; while the Angels, Archangels, and Principalities are arrayed in the third and most remote hierarchy.) Dionysius the Areopagite, The Celestial Hierarchy, Ch. VI,

§ 2 (ca. 400 CE).

Theorem. Every finite simple group is one of 26 sporadic simple groups or belongs (up to isomorphism) to at least one of the following families: (1) A cyclic group of prime order; (2) An alternating group of degree at least 5; (3) A simple group of Lie type, including both the classical Lie groups, namely the groups

(4)

of projective special linear, unitary, symplectic, or orthogonal transformations over a finite field; The exceptional and twisted groups of Lie type (including the Tits group).

Result of years of labor by dozens of group-theorists spread over thousands of pages of journals (ca. 1975). Das ist keine Mathematik. Das ist Theologie. (That is not mathematics; it is theology.) Paul Gordan, on seeing Hilbert’s first proof of the Hilbert Basis Theorem, quoted in Max Noether’s obituary of Gordan in the 1914 Mathematische Annalen (1888). 1

2

BOOK REVIEW

1. Introduction The reason for the unusually long string of epigrams just given is the unique nature of the book under review. This book is about the personal side of mathematics, the inner thoughts and life struggles of two mathematical “trinities,” one (Borel, Lebesgue, Baire) in a politically free and liberal society, the other (Egorov, Florenskii, Luzin) in a closed and authoritarian one. It is particularly about the role, if any, that religion plays in mathematical creativity. The quotations were chosen in order to point out some parallels I have long noticed between mathematics in the modern world and theology in medieval times. A few years ago, I gave a public lecture on these parallels. Even as a graduate student in the mid-1960s, living high (relative to most graduate students) on the post-Sputnik benefactions of the National Science Foundation, I could not help wondering why my country was supporting me comfortably while I studied σ-rings and Banach algebras. What, I wondered, did my studies have to do with catching up with the USSR in the race to the moon? Why are mathematicians, most of whom spend their time thinking about such impractical things as topological semigroups, esteemed as scholars and intellectuals? The similarities to theology were all too apparent. Is the distinction between logicism and intuitionism really more important than that between transubstantiation and consubstantiation? Is it more real? I think so, but I can understand that others may not. The riddle of why mathematicians working on arcane, obscure, and inbred problems are economically supported by society continues to occupy me.1 In Naming Infinity Loren Graham and Jean-Michel Kantor explore a neighborhood of that “point at infinity” in some human minds where the two parallel worlds of mathematics and theology intersect. They admit that they approach this subject from a rationalist perspective, and I must admit that I share that perspective. Nevertheless, they have a very strong sympathy for the spiritual struggles of their Russian trinity, as do I. Before I get down to details about all this, I need to confess my own deep engagement with this book. I too have looked at some of the archival documents in Moscow that the authors have studied and read the reports of Russian scholars who have investigated the origins and fate of the Moscow school of mathematics. As I told the authors and encouraged them to quote me in their advertising, in all my work, I saw only the ten percent of the iceberg that was visible above the surface. This book reveals the 90 percent that lay beneath. Throughout a quarter century of research in the history of mathematics, I have disciplined myself to focus on mathematical results and say relatively little about the personal side of things. Since this book frankly focuses on the personal, and I have long been involved with the matters it discusses, I welcome the opportunity that writing this review provides to cast off my normal restraints and frankly reveal my own personal “take” on these matters. Both the mathematical and the personal story should be told, and they really are interwoven in this case. Let me summarize the mathematical story first up to the point where the Russians enter it, then tell how the personal story of these three Russians is interwoven with the rest of the story. 1I am afraid I became rather tiresome on this subject as a graduate student, so much so that

one of my fellow students once asked me, “If you feel such doubts, why do you continue to accept your fellowship? Are you really interested in mathematics, or is this just your way of getting even with the government?”

BOOK REVIEW

3

2. The mathematical side of the story From the point of view of an analyst, the main mathematical subject involved— set theory—has traversed approximately a full circle in a little over a century, starting as an attempt to clarify certain uniqueness theorems for trigonometric series representations, a problem that was soon abandoned in favor of set theory for its own sake. In the 1980s, descriptive set theorists once again returned to this problem and produced metatheorems showing the inherent limitations of such uniqueness theorems (see [10], [4]). Cantor began to be interested in set theory during the 1870s. At that time, he was working to extend Riemann’s uniqueness theorem, which asserts that the coefficients of a trigonometric series that converges to zero at all points except for a finite number are all zero, so that in fact there are no exceptions. Cantor was able to weaken the assumption to allow infinitely many exceptional points, provided the set of exceptional points had only a finite number of points of accumulation (in modern terminology, the derived set was finite). Then, it became possible to do the same if the number of points of accumulation of the derived set was finite, and so on, for all exceptional sets of what we now call finite Cantor–Bendixson rank. Since the successive derived sets are nested, the derived set of order ∞ has a natural definition as the intersection of all derived sets of finite order. Once that is done, it becomes possible to start over with the next derived set, of order ∞ + 1, and—voil`a!—transfinite ordinal numbers are created as the indices for all possible derived sets. The connection with trigonometric series was severed by this time, and Cantor began to explore set theory, ordinal numbers, and topology on the real line. 2.1. The French school. Although set theory would have had some success because of its applications in complex analysis (championed by Mittag-Leffler, among others), it certainly got an enormous boost from the work of the French “trinity” of Lebesgue, Borel, and Baire, who were the alpha, beta, and gamma stars in a large constellation that included others like Fatou, Fr´echet, and Denjoy. In a rather intriguing way, set theory had to bifurcate into measure theory and descriptive set theory in order to grow into its full stature. Measure theory needs some description of sets on which a measure can be defined. Borel defined such a class by requiring it to contain all intervals and to be finitely subtractive (if A and B are measurable, so is A \ B) and countably additive. However, he made little attempt to visualize the class (now called Borel sets) that he had defined, saying [3] that “It may be noticed that any set we can effectively form is obtained by carrying out the operations represented by the symbols (E1 +E2 +· · · ) [countable union] and (E1 , E2 , . . . ) [countable intersection] a finite number of times.” This was sufficient for the purposes of measure theory anyway, since for any Lebesgue-measurable set E there is an Fσ set F and a Gδ set G such that F ⊆ E ⊆ G and m(G \ F ) = 0. Borel’s use of the term effective here reflects the reason why the French did not forge ahead with the development of descriptive set theory. By 1905, set theory had developed some antinomies, and it had become necessary to explore the validity of various axioms and principles of inference. The French preferred processes that were more constructive, less like conjuring entities into existence by fiat. In particular, they liked processes that could be defined unambiguously, without the arbitrary features that always occur when one uses the axiom of choice. Borel, as already noted, drew back from the abyss of

4

BOOK REVIEW

transfinite ordinal numbers with its limit ordinals and inaccessible cardinals and the Borel hierarchy of sets that was indexed by transfinite ordinal numbers. Lebesgue tried to deal with functions that were analytic in a new sense: not differentiable or even continuous, but obtainable from continuous functions by well-defined countable processes. These turn out to be precisely what we now call Borel-measurable functions. Lebesgue liked objects that were nameable (nommable), that is, definable without using the axiom of choice. In 1905 ([11], p. 112, footnote), he wrote, “I do not know if it is possible to name even one function that is not B-measurable.” Only Baire plunged willingly into the maelstrom of set theory. His 1899 thesis had been favorably reviewed by Picard, who singled it out for praise in an address at Clark University on 5 July that year, especially noting that some continuity assumptions normally made in analysis and differential equations would have to be revisited on the basis of the new, more general concept of a function. Picard offered the following prescient opinion ([14], p. 24): Some questions are of purely philosophical interest and will probably never have the least utility for mathematics, for example, to know whether priority belongs to cardinal or ordinal numbers. . . But the matter is different in other cases. Thus, it is probable that M. Cantor’s theory of sets. . . is about to play a useful role in problems that were not posed expressly to be an application of the theory. And so it turned out, too late for poor Baire. But although Baire did not get the success he hoped for, the problems he investigated were not forgotten. The seeds of set theory fell on soil especially favorable to their development, in Russia. 2.2. The Moscow school. The first Russian textbook on set theory was Structure and Measure (Stroenie i Mera), published in 1907, by Vladimir Leonidovich Nekrasov (1864–1922), a graduate of Kazan University and a professor at the Tomsk Technological Institute, not to be confused with Pavel Alekseevich Nekrasov (1858– 1924), head of the Moscow Mathematical Society from 1903 to 1905, and a fierce antagonist of Andrei Andreevich Markov (1856–1922) in a polemic over the proof of the central limit theorem that continued for years. (This was a regrettable incident in the lives of two very brilliant mathematicians.) In contrast to the French and the Petersburg mathematicians (Chebyshev, Sonin, Markov), who must have been suspicious of the long passages of philosophy in Cantor’s early papers, the Moscow mathematicians had a tradition that was more receptive to philosophy and were consequently more willing to delve into this new area. P. A. Nekrasov described this characteristic of the Moscow school in a speech entitled “The Moscow PhilosophicalMathematical Society and its Founders,” given on 16 March 1904 in memory of Nikolai Vasilevich Bugaev (1837–1903), who had been one of the founders of the Moscow Mathematical Society forty years earlier. According to Nekrasov, While they ascribe great importance to facts, experiment, and the experimental sciences, the founders of the Mathematical Society are opponents of the slavish worship of facts by certain scholars. They were among the first to protest this enslavement of modern scientific thought and clearly explained the value of imagination and will equipped with the prerequisite objective and subjective (authoritative and nonauthoritative) world-views and the more or less exact theories that consciousness, living by its own pure

BOOK REVIEW

5

process and internal experience, combines with the phenomena of external facts in motivating actions to be taken. Further, The profoundly penetrating human logos 2 is always enriched by subtly developed mathematical elements. The judgment of what is essential depends on these elements. Therefore the profoundly penetrating logos is not merely dialectical (corresponding to a word only) nor dialectical-empirical (corresponding to both word and deed), but mathematical-dialectical-empirical. A more or less developed sense of measure that unifies the internal and external judgments and which may be called an inner eye judging in a measured manner is characteristic of the awareness of this logos. The language of this logos and its syllogism, by preserving inward beauty, group things into various ranks using different intuitive and empirical standards and work out an adequate foundation for the action of thought and will. Nekrasov carried on in this vein for hundreds of pages spread over several volumes of the Matematicheskii Sbornik, which may explain why the tough-minded Markov ungenerously described his later work as “an abuse,” despite its very significant mathematical merit. Nekrasov was perhaps an extreme example and did not speak for every mathematician in Moscow—in 1905, he was succeeded as head of the Moscow Mathematical Society by the very down-to-earth applied mathematician Zhukovskii—but he was not at all out of place there. Bugaev adhered to a metaphysical system he called arithmology and tried to develop a theory of discontinuous functions in parallel with the theory of continuous functions.3 Such was the world that Dmitrii Fedorovich Egorov (1869–1930) entered as a student in 1887, Pavel Aleksandrovich Florenskii (1882–1937) in 1899, and Nikolai Nikolaevich Luzin (1883–1950) in 1901. One could easily continue to write just the mathematical part of this story without mentioning the personal. It makes a sufficiently fascinating narrative all on its own. All three of these men wrote some rather standard mathematical papers, Egorov on “Bending on a principal basis” (Izgibanie na glavnom osnovanii) and Luzin on “Integration and the trigonometric series” (Integral i trigonometriqeski$ i rd). Florenskii wrote an undergraduate thesis on “The idea of discontinuity as an element of a world-view,” (Ide razryvnosti kak lement mirosozercani), which sounds like philosophy, although its first part is actually about singularities of algebraic curves. (Comparisons with Ren´e Thom’s Structural Stability and Morphogenesis come to mind.) However, continuing the mathematical 2As far as I know, no Russian has made any comment on the similarity of this Greek word with the Russian word golos (golos), meaning voice or vote. They probably are not related. I suspect the Russian word is cognate with the Greek word glossa (γλˆ ω σσα or γλˆ ω τ τ α), meaning tongue or language. 3 Lest that seem like developing a theory of non-bananas in parallel with the theory of bananas, it should be noted that one can make perfectly good sense of it by associating the discontinuous functions with functions on a discrete space or at least one with a very strong topology, and this was eventually what Luzin was to do. His generic Baire space NN , which is topologically equivalent to the irrational numbers in (0, 1) via continued-fraction expansions, led Ernst Kol’man to attack him on the dialectical grounds that this monstrosity was neither discrete nor continuous and therefore a contradiction, of no use in the real world.

6

BOOK REVIEW

story would make this review almost as long as the book itself and thereby defeat the main purpose of writing a review, which is to help the reader decide whether to read the book. While reading the book, I made so many notes of things I would like to say in the review that I have had to ignore the vast majority of them. I therefore turn to the personal side of the story. 3. Russian religious tradition and set theory Even as Charles Ford ([8], [9]), Sergei Demidov ([7], [6]), and others unearthed more and more information about the connection between mathematics and religion in the minds of these men, I continued to see these two areas of interest as separate aspects of them. While I was researching the story of descriptive set theory in Moscow, I kept encountering Luzin’s futile attempts to prove the continuum hypothesis, almost always focused on his attempt to enumerate the transfinite ordinal numbers (as far as the first uncountable one) “effectively” (ffektivno), to “name” (nazvat~) them. I assumed these words were borrowed from Borel and Lebesgue, as indeed they were. But there was much more to the story than I was seeing. At last the whole story has been assembled and told in a fascinating way by Loren Graham and Jean-Michel Kantor. The key to the roman ` a clef that I was reading without fully understanding lies in a movement within the Orthodox Church known , in Greek as onomatodoxy (oνoµατ oδ´oξα) and in Russian as imeslavie (imeslavie). Both words mean name glorification or name worshipping, in the sense suggested by the phrase hallowed be thy name. According to the authors, this movement was frowned upon by the Russian Orthodox hierarchy and brutally suppressed in 1913, when the Tsar sent a Russian ship to storm the monastery on Mount Athos and bring back to Moscow for trial those monks who refused to renounce the “heresy.” But, as heresies tend to do, this one continued to live, and apparently still has adherents today, who are being harassed by the Russian Orthodox authorities with the cooperation of the state. This apparent coincidence of the supreme importance of names (meaning in mathematics, proper definitions) is the common thread in the tapestries of mathematics and religion. Although Egorov had an apparently typical upbringing as a member of the Russian Orthodox Church, his two younger prot´eg´es Luzin and Florenskii did not. Both were secularists who converted after some spiritual struggles. In Florenskii’s case, this conversion led to a life within the religious community. Luzin remained a layman and observed religious rituals as well as he could after the Communists began to rule Russia. Both Florenskii and Egorov attempted to work within the new system without compromising their beliefs or principles. That was not to be. Both were arrested, Egorov in 1930, Florenskii in 1928 and again in 1933. Egorov’s health was not good. He refused to eat after his sentence was imposed, and he died shortly thereafter in Kazan. (He is buried not far from the great Russian mathematician Lobachevskii.) Florenskii was tortured and made to implicate Luzin in a fantastic plot against the Soviet Union. He was executed on 8 December 1937. Luzin came near to the same fate in July of 1936, but through some mysterious caprice of the Soviet system not fully explained, the denunciations of him were quietly forgotten, and he was never formally charged in a Soviet court. Although severely frightened, he lived on to die, in the Russian phrase, “his own death” in 1950.

BOOK REVIEW

7

All that is of course exciting political, social, and personal drama. But what does it have to do with mathematics? Philosophically, onomatodoxy resonates with the Greek word logos (λ´oγoς), which means not only word but also the human faculty of reason and the process of presenting one’s reasoning as an argument. Words matter, the imeslavists believed, not just as an essential tool in thinking; the very use of them affects the universe. One thinks of Moses asking God for his name, and being told simply that God is “the one who is,” too sacred to be given an ordinary name. And that name is not to be taken in vain, but used in the proper way. In the introduction to his book Onomatodoxy as a Philosophical Premise, Florenskii wrote the following: I. The word is human energy, the energy of both the human race and the individual person, manifesting itself under the form of the energy of humanity. But one cannot assert that this energy itself is the object of a word or its content in the precise sense. The word, as an act of consciousness, leads the mind beyond subjectivity and brings it into contact with a world outside our own psychological states. Although it is psychophysiological, the word does not fly through the world like smoke; rather, it brings us face to face with reality. Consequently, in coming into contact with its object, it can equally refer to both the revelation of its object to us, and the revelation of us to it and before it. In this way, we arrive at a conviction that is inseparable from an idea common to humanity. In all times and among all peoples, it lies at the foundation of our understanding of the world. But philosophically there has been a schism in its expression: as ancient idealism, then as neoplatonism, and later as medieval realism. It was profoundly stated by the Eastern Church in the fourteenth century in connection with the theological controversy over the Name of God, the Tabor Light.4 Still later, it nourished Goethe5 and can be detected, though dimly, in the writing of Mach.6 Finally, in our time it has abruptly broken through as a fiery protest against philosophical and theological illusionism and subjectivism in the Athos controversy over the Name of God. In an appendix, Florenskii showed how onomatodoxy affects theology: 4This is a reference to the Transfiguration of Christ (Matthew 17; Mark 9; Luke 9). The mountain where it occurred is not identified in the Gospels, but according to a tradition that originated in the fifth century, it was Mount Tabor in Lower Galilee. 5This reference is somewhat obscure (to me). I don’t know where Goethe discussed language and naming things, except in his whimsical epigram, “Die Mathematiker sind eine Art Franzosen: Redet man zu ihnen, so u ¨ bersetzen sie es in ihre Sprache, und dann ist es alsobald ganz etwas anderes.” (“Mathematicians are rather like Frenchmen. Whenever anything is said to them, they translate it into their own language, and it immediately becomes something quite different.” Maximen und Reflexionen, No. 1279.) 6This appears to be an allusion to Mach’s 1914 book Beitr¨ age zur Analyse der Empfindungen (Contributions to the Analysis of Sensations, [13]), which contains many passages referring to the naming and describing of the world, including (p. 150), “So nennt ein Kind gelegentlich die Federn des Vogels Haare, die H¨ orner der Kuh F¨ ulh¨ orner, den Bartwisch, den Bart des Vaters und den Samen ds L¨ owenzahns ohne Unterschied ‘Bartwisch,’ u. s. w.” (“Thus a child casually refers to a bird’s feathers as hair, to a cow’s horns as antennae, to a paintbrush, his father’s beard, and dandelion seeds all equally as brush, and so forth.”)

8

BOOK REVIEW

Of a book one may say either “This is paper” or “This is a great work of art.” One may say either thing, but it is more correct to point out the spiritual meaning of a book rather than the condition by which it is observed. One may say that a book of Gospels is a pound of paper or even that the Name of God is a sound; and although, on the one hand (from one point of view) it is correct to talk in this way, for example, of the weight of a book of Gospels when sending it through the mail, still it is more correct to point out its most important characteristic, the spirit of the symbol. A physicist can say that the Name of God is a set of sounds. Yes, but it is more than sounds. And moving a lower-order truth to the foreground this way, setting a part in place of the whole, is a lie. The Name of God is God, but God is more than a name. The Divine Essence is greater than His energy, although this energy expresses the essence of the Name of God. What I see when looking at the sun, is indeed the sun, but the sun itself is more than the action it exerts on me. Again, when I hear a familiar voice, I may say “That’s N. N.”7 But of course that is only his voice. He himself is incomparably greater than his voice, having a multitude of other individual characteristics, and by no means consists of his voice alone. Or, “That’s N. N,” but in fact, it is only a picture post card of him, and he is present in it only in the form of his energy. So much for the connection of name-glorification with theology, which seems natural. What is the connection with mathematics? To relate this kind of thinking to mathematics by strict logic is impossible, but a psychological link is very easy to see. Supposing that a mathematician adheres to this belief in the sacred importance of names, how would such a person approach the antinomies of set theory that were being debated in the first decade of the twentieth century? Would it not be natural to insist on clear definitions that are unambiguous and free of circular references? That is what Luzin did. There are huge philosophical problems to be overcome here. In the light of such startling results as the Banach–Tarski paradox, for example, what exactly does the word existence mean in mathematics? Luzin was willing to allow mathematical entities to be conjured into existence by definition, provided the definitions were “analytic” or “effective” in the sense used by Lebesgue and Borel, that is, completely unambiguous and therefore made without relying on the axiom of choice. However, in 1918 Sierpi´ nski had compiled a list of important results in analysis whose proofs used that axiom, and its length horrified Luzin, causing him insomnia. He realized that analysis was not going to get along without this axiom, and that he needed to uncover the meaning of the word existence as used in the axiom of choice. In one of the most inspiring paragraphs I read among his many notes, he wrote (see pp. 210–211 of Naming Infinity): Each definition is a piece of secret ripped from Nature by the human spirit. I insist on this: any complicated thing, being illumined by definition, being laid out in them, being broken up into pieces, will be separated into pieces completely transparent even to a child, 7Russians often use the letter N to indicate an unspecified person or place. It is unlikely that the “N. N.” in this passage refers to Luzin. Still, it is an interesting coincidence that Florensky chose to express himself that way.

BOOK REVIEW

9

excluding the murky and mysterious things that our intuition whispers to us while acting, separating into logical pieces. Then only can we move ahead to new successes due to definitions. . . The passages of Luzin’s notebooks in this vein now make sense to me in a way that they did not before I read Naming Infinity. Even so, there is much that I still cannot appreciate. In the correspondence between Luzin and Florenskii [7], for example, there is speculation that God is a trinity because three is the maximum number of points that can be joined each to each in a plane without crossing. This idea (to which, I am happy to report, the correspondents did not attach much importance) struck me as utterly bizarre. Did they think of the persons of the trinity as being like a telephone network? And, granted that the persons of the deity must be connected as a complete graph, why must it be planar? Reading this book has explained some things to me that had completely passed below my consciousness before, for example, why Luzin referred to Florenskii as “Pyotr Afanas’evich” in this correspondence. As the authors explain (pp. 83–84), Florenskii’s belief in the sacredness of names led him to formulate the idea of renaming a person who has undergone a religious conversion, perhaps analogous to the saint’s name taken at confirmation by Western Christians. Although I remain suspicious of any claim of a mystical way of knowing (see p. 93 for Luzin’s interest in this), largely because the people who claim such knowledge frequently contradict one another, and I still do not see any logical connection between the religious activity of name-glorification and set theory, I can understand how a mathematician might be guided by a particular mind-set into a certain way of looking at problems, and that that way might be very fruitful, as it was for Luzin. The authors reach a similar conclusion, comparing Luzin to Grothendieck (who was living in the Pyrenees in the latest account that I read and apparently very deeply involved in mysticism, believing that the speed of light has mysteriously changed.) Their conclusion is that, while Grothendieck may be correct in saying that mathematicians do not need religion, nevertheless “sometimes it can help.” 4. The Luzin File With all that praise, can I find anything to criticize? Very little, actually. On p. 62, the authors mention that Borel gave an address at the International Congress of Philosophy of Science in 1951, “just a few months before his death.” (Borel died in 1956. I don’t know when the Congress was held.) That is the only factual correction I can make. There is, however, one point where I would like to offer a somewhat different interpretation of facts that are not in dispute, namely Aleksandrov’s aggressive attacks on Luzin during the hearings held by a special committee of the Academy of Sciences in July 1936 and an inconsistency the authors point out between what Aleksandrov admitted at those hearings and what he said in his later attempt at autobiography [1]. While I do not claim I can exonerate Aleksandrov, I believe I can at least introduce some ambiguity into the indictment against him. Both the authors in this book and I in earlier writing (see [5], p. 458) have taken our view of these hearings from the Russian experts, and rightly so. As the Russians say, “It is not for the egg to teach the chicken.” According to that view, Aleksandrov had been coached by the people who concocted the charges on the approach to be taken, although the coaching had not been sufficiently complete ([6],

10

BOOK REVIEW

pp. 27, 31), since an article in Pravda on the second day of the hearing attacked both Luzin’s defenders and his attackers, including Aleksandrov, for groveling before the West and publishing their works abroad. Aleksandrov responded to this criticism by confessing his own sins and redoubling his attack on Luzin. Reading what Aleksandrov actually said, however, I am now inclined to judge him less harshly. At bottom, he wished to get Luzin out of his way for professional reasons and because he sincerely thought Luzin had become a hindrance to mathematical progress with his adamant rejection of Aleksandrov’s own school of topology. That in no way excuses Aleksandrov for joining in the denunciation of Luzin at such a time, of course. Luzin came within a hair’s-breadth of losing his life in this affair. It is setting a high value indeed on one’s own advancement, or even the disinterested advancement of science, to sacrifice a fellow human being for it, even a person with whom one has quarreled. Apart from that context, which does make Aleksandrov deserving of censure, what he actually said does not seem to me to be intrinsically either unfair or selfish. He desperately wanted to keep the accusations against Luzin on the academic level, not the political level, possibly believing that by avoiding political accusations, he could avoid utterly destroying Luzin, yet still be rid of him. He wanted the special committee of the Academy of Sciences that was hearing the case to confine itself to what he saw as Luzin’s professional sins: taking credit for work done by his students and rewarding mediocre work. Except for those accusations, which Aleksandrov made on behalf of Novikov and Suslin rather than himself, he was appropriately grateful and admiring of Luzin and clearly tried to protect Luzin from any accusations of a political nature. As he said on the first day of the hearings ([6], pp. 60–61):

In 1920 I returned to Moscow and again came into contact with Nikolai Nikolaevich. I fell among his students. I must say that in this Soviet period I became witness to an extraordinary picture, a picture that I had never seen anywhere in the world, a picture of outstanding scholarly enthusiasm and an outstanding scholarly upsurge that surrounded this man, a huge human collective of more than 36 people all concentrated around the personality of Nikolai Nikolaevich. These were the years from 1920 to 1922. I believe Nikolai Nikolaevich played a larger role than that of founder of a school. He was actually an organizer of young people, a person who knew how to imbue that youth, now Soviet youth, with an extraordinary scholarly enthusiasm. So of course, to speak of anything anti-Soviet seems to me absolutely impossible, because Nikolai Nikolaevich was a responsible man. Holding absolute sway over all of us, to such an extent that every word he spoke was taken as the absolute truth, if he had wished to turn us in an anti-Soviet direction, he would have had every chance to do so, opportunities such as no other professor had, because he was surrounded by people who literally prayed to him. . . and I know Nikolai Nikolaevich’s life at this period very well. I categorically deny that any antiSoviet tendencies could have been present in Nikolai Nikolaevich, given his complete frankness with me.

BOOK REVIEW

11

This picture of Luzin is consistent with what I have read in other sources. I have always pictured the young Luzin in the role of George Passant, the protagonist of C. P. Snow’s second novel in the Strangers and Brothers ennealogy, named after its hero. Passant’s relation to his students very much resembles that described here by Aleksandrov. Passant reappears years later, in the ninth of these novels, The Sleep of Reason, to confront the effects of the ideals he had advocated as a young man. The older Luzin also bears some resemblance to the older Passant, in my opinion. The authors (p. 120) criticize Aleksandrov for having, years later [1], claimed all the credit for the discovery of analytic sets. I researched this topic some 20 years ago ([4], p. 311). At that time, having only Aleksandrov’s paper [2] as a source, I concluded that Aleksandrov had discovered what was later called the A-operation, which Suslin learned of and used to create the concept of an analytic set. That view seemed to be shared by Tikhomirov ([15], p. 134), who noted the following: (1) Suslin was the first to construct the A-operation, although the idea of such an operation is actually contained in the work of Aleksandrov and Hausdorff (co-discoverer with Aleksandrov of the proof that every uncountable Borel set contains a non-empty perfect subset). (2) The new class of A-sets introduced by Suslin turned out to be larger than the Borel sets. (3) The work of both Aleksandrov and Suslin was carried out under the direction and influence of Luzin, and Luzin later enriched the theory of A-sets by brilliant results and ideas. However, it must be remembered that neither Tikhomirov nor I had seen the transcript of the hearing when we came to those conclusions. Aleksandrov’s paper was the paper of a student who had been advised by his professor. He may himself not have realized how much of it was due to Luzin. But as the paper of a student, it goes into the record book as his result, even if Luzin actually gave him such broad hints as to make the idea completely obvious. What is clear from the record is that the idea of the A-operation is in Aleksandrov’s paper, and he is formally entitled to the credit for that, even if Luzin was the real genius behind it. Under the stress of defending himself in a hearing that could easily have led to a formal trial and execution, Luzin violated some of the protocols usually observed by mentors and tried to get Aleksandrov to admit that the latter hadn’t done as much by himself as he claimed he had done. Luzin claimed ([6], p. 161) that Aleksandrov had repeatedly come to his dacha in the summer of 1915 with confusing, muddled arguments which he, Luzin, had to straighten out. And Aleksandrov gave a reply that seems to grant this claim. On these grounds, Lorentz [12] convicts Aleksandrov of falsely claiming the A-operation as his own. But appearances may be deceiving. Aleksandrov had clearly shown by his generous words quoted above that he did not wish to destroy Luzin. I think he may have taken pity on Luzin and allowed him to wriggle out of the charge of plagiarism. (A more conspiratorial view would be that he had been ordered to “throw the game” by those who concocted the scheme in the first place.) There is no doubt that in his old age, Aleksandrov allowed vanity to usurp his memory. When he recounted the history of the summer of 1915 at the age of 93, he had a different version entirely from the one presented by Luzin, to which he had assented in 1936. Luzin, said Aleksandrov ([1], pp. 285–286),

12

BOOK REVIEW

was completely skeptical and advised me to scrap my plan of solving the problem by contradiction and by trying to construct a Borel set of power ℵ1 [that is, uncountable but smaller than the cardinality of the continuum, although by the continuum hypothesis itself, ℵ1 is the cardinality of the continuum]. Fortunately, I did not follow my teacher’s advice on this occasion. By the end of the summer I had succeeded in carrying out my proof in full generality and in all its details. The first person to whom I explained it at length and who probed it with all rigorous severity, was Stepanov, who was capable like no one else of critically examining proofs, in any then known field of mathematics. Then I presented the proof to Privalov and only after that to Luzin, who was thus forced to abandon his initial skepticism. . . I told [Suslin] in great detail about my results. We began to talk endlessly about related problems. . . It was then that Suslin proposed that my new set-theoretical operation should be called the A-operation and that the sets obtained by applying it to closed sets should be called A-sets. He emphasized that he was proposing this terminology in my honor, by analogy with Borel sets, which by then were usually called B-sets. . . I spent the whole winter of 1915–16 trying to prove [that every A-set is a B-set]. My extremely persistent speculations only ceased when it became known early in the autumn of 1916 that Suslin had constructed during that summer an example of an A-set that is not a B-set. I do not infer from reading this that Aleksandrov was claiming to have formulated the concept of an A-set, although he did claim (justly, by the rules of mathematical publication) to have created the A-operation. The major issue was whether that operation leads to any sets besides B-sets. Aleksandrov properly credits Suslin with identifying the class of A-sets at a time when neither of them knew whether there was any difference between A-sets and B-sets, and he gives full credit to Suslin for proving that there is a difference. To me, this is anything but “taking all the credit to himself.” It is true that he claims Suslin named the A-sets in his honor. Here indeed, he is caught in a contradiction with his earlier testimony, as I shall show below. With that exception (surely a forgivably convenient “memory lapse” in an old man and a very minor point of vanity, not a priority claim), I see this all as being completely consistent with what he wrote in his original 1916 paper, that “this problem was posed to me by Mr. N. Luzin, and I have obtained the result below through his support. Certain parts of the proof are also due to him,” and with what he said at the hearing in 1936. At the hearing, he gave more details ([6], pp. 89–90): The problem I solved in that paper was posed by Luzin. If he had not posed it, I could not have begun thinking about it, and consequently could not have solved it. The actual solution of the problem is entirely mine. . . However, I gave the proof in a very complicated and clumsy form. Luzin greatly simplified the proof I had given, bringing it into an elegant and compact form. He did not in any way change any of the ideas of the proof. My paper was published on 20 February 1916, but it had been presented at the

BOOK REVIEW

13

student mathematics club on 13 October 1915. . . After the work had been published, Suslin came to me and said that not all points of the proof were clear to him and asked me to go into more detail. In a conversation with him that lasted several hours, I expounded the entire proof to Suslin. This was in the spring, March or April of 1916. In the summer Suslin constructed his brilliant theory, almost, one might say, a work of genius. . . I had proved that certain classes of sets are formally a special case of a certain class of sets A. I had not proved that there exists any A-set that is not a B-set. For that reason, I believe I did not discover this class of A-sets. Suslin proved that these A-sets do indeed form a new class. Consequently, he and in no way I, must be regarded as the one who discovered the class of A-sets. . . I do not claim that Suslin named them in my honor, because I never heard anything of the sort from Suslin himself. That, in my opinion, is the maximum provable extent of Aleksandrov’s academic sin. He was a bit vain and wanted something named after him to which he had some claim. I hope that the mere exposure of this fib is sufficient punishment to his memory, and that his huge contributions to mathematics will not be seriously obscured by it. As for the vast discrepancy between his and Luzin’s account of the year 1915, both had good reason to stretch the truth, and I do not think the evidence is conclusive enough to call either of them a liar. 5. Conclusion This is not only a unique book. It is a splendid book, a beautiful book. If you buy it, you will want to keep the dust cover, adorned with a magnificent painting by the Itinerant artist Mikhail Vasilevich Nesterov, showing Florenskii walking in company with the philosopher Sergei Nikolaevich Bulgakov, painted in 1917. And mainly, you will want to read it. It brings to life the immense empire of mathematics in the USSR during the early twentieth century, an empire that has planted “colonies” all over Europe, the United States, Canada, and Israel since the end of the Soviet Union, all without exhausting the vast resources of talent in the mother country. References [1] P. A. Aleksandrov. Pages from an autobiography, part 1. Russian Mathematical Surveys, 34(6):267–302, 1979. [2] P. Alexandroff (P. A. Aleksandrov). Sur la puissance des ensembles mesurables B. Comptes Rendus de l’Acad´ emie des Sciences, 162(28 F´ evrier):323–325, 1916. ´ [3] Emile Borel. Le¸cons sur les fonctions de variables r´ eelles et les d´ eveloppements en s´ eries de polynˆ omes. Gauthier-Villars, Paris, 1905. [4] Roger Cooke. Uniqueness of trigonometric series and descriptive set theory, 1870–1985. Archive for History of Exact Sciences, 45(4):281–334, 1993. [5] Roger Cooke. The History of Mathematics: A Brief Course. John Wiley & Sons, New York, 1997. [6] S. S. Demidov (S. S. Demidov) and B. V. Levshin (B. V. Levxin), editors. The file on Academician Nikolai Nikolaevich Luzin (Delo Akademika Nikola Nikolaeviqa Luzina). Russian Christian Humanities Institute (Russki$ i Hristianski$ i Gumanitarny$ i Institut), Saint Petersburg (Sankt-Peterburg), 1999. [7] S. S. Demidov (S. S. Demidov), A. S. Parshin (A. S. Parxin), and S. M. Polovinkin (S. M. Polovinkin). The correspondence between N. N. Luzin and P. A. Florensky (Perepis~ka

14

[8] [9]

[10] [11] [12] [13] [14] [15]

BOOK REVIEW

N. N. Luzina i P. A. Florenskogo). Istoriko-Matematicheskie Issledovaniya (IstorikoMatematiqeskie Issledovani), XXXI:116–191, 1989. Charles Ford. Dmitrii Egorov: Mathematics and religion in Moscow. The Mathematical Intelligencer, 13(2):24–33, 1991. Charles Ford (Qarl~z Ford). Dmitrii Fedorovich Egorov: Archival materials at the University of Moscow (Dmitri$ i Fedoroviq Egorov: Materialy iz arhiva Moskovskogo universiteta). Istoriko-Matematicheskie Issledovaniya (Istoriko-Matematiqeskie Issledovani), 36(2):146–165, 1996. Alexander S. Kechris and Alain Louveau. Descriptive Set Theory and the Structure of Sets of Uniqueness. Cambridge University Press, 1987. Henri Lebesgue. Sur les fonctions repr´ esentables analytiquement. Journal des math´ ematiques pures et appliqu´ ees (6), 1:139–216, 1905. Reprinted in Œuvres scientifiques, Vol. 3, 103–180. George G. Lorentz. Who discovered analytic sets? The Mathematical Intelligencer, 23(4):28– 32, 2001. Ernst Mach. Beitr¨ age zur Analyse der Empfindungen. G. Fischer, Jena, 1886. ´ Emile Picard. Sur le d´ eveloppement de l’analyse et ses rapports avec diverses sciences. Gauthier-Villars, Paris, 1905. V. M. Tikhomirov (V. M. Tihomirov). The discovery of A-sets (Otkrytie A-mnoestv). Istoriko-Matematicheskie Issledovaniya (Istoriko-Matematiqeskie Issledovani), XXXIV:129–139, 1993.

Roger Cooke University of Vermont E-mail address: [email protected]