Pursuing the Infinite the Mathematicians and the Heretical Sect Is Not in Doubt

NATURE|Vol 458|23 April 2009 OPINION

population are sustained for the rest of this the best we can hope is that we can manage that some infinite sets were larger than others. century, but at a huge cost — the complete them ourselves, taking over the heavy respon- A group of French mathematicians at the turn loss of all the natural ecosystems of the world. sibility for keeping Earth habitable, which Gaia of the century — including Emile Borel, Henri Most of us, living in cities and insulated from once did for us automatically. Lebesgue and René Baire — were searching the natural environment, would barely notice The more likely outcome is that we would for a systematic way to determine the size of until it was too late to do anything about it. barely manage them at all. In that case, we these infinite sets. This aroused the scepticism This is what many politicians, economists and would face a sequence of global environmental of colleagues such as Henri Poincaré, who industrialists seem to want — their mantra of crises and a steady degradation of the planetary claimed that Cantor’s hierarchy of infinities unceasing economic growth implies that we environment that would eventually kill just as had “a whiff of form without matter, which should take for ourselves all Gaia’s resources many of us as a sudden collapse. Given that, is repugnant to the French spirit”. According and squeeze from them the maximum short- perhaps we had better hope that Lovelock is to the authors, the French researchers found term gain, leaving nothing for the future. right, and Gaia does for us — or most of us — themselves at the edge of an “intellectual Following this vision, we will need to trans- before we do for her. ■ abyss” where, “under the influence of their form the entire planet into a factory farm to Andrew Watson is a professor at the School of ultra-rationalistic traditions, they lost their feed our 10 billion or 15 billion mouths. There Environmental Sciences, University of East Anglia, nerve” and abandoned their work. will be no room on this giant spherical feedlot Norwich NR4 7TJ, UK. The situation was different in Moscow, for anything but ourselves and our half-dozen e-mail: [email protected] where the Russian mathematicians took up species of domestic plants and animals. Gaia, the same problems with zeal and eventually the natural Earth system, will have disappeared. Watch Oliver Morton’s interview with James resolved them, advancing the far-reaching As for the underpinning biogeochemical cycles, Lovelock at www.nature.com/nature/videoarchive. field of measure theory and launching descriptive set theory. Graham and Kantor argue that the spiritual views of these mathematicians were crucial to their scholarly work. That there were ties between some of Pursuing the infinite the mathematicians and the heretical sect is not in doubt. The geometer Dmitri Egorov Naming Infinity: A True Story of Religious a number of Russian intellectuals. Among believed in name worshipping. His student Mysticism and Mathematical Creativity them was a handful of mathematicians in Pavel Florensky, a mathematician turned by Loren Graham and Jean-Michel Kantor Moscow who, working in the young field theologian, held that the ‘set of all sets’ might Belknap/Harvard University Press: 2009. of set theory, also found themselves dealing be God himself. The eminent mathematician 256 pp. $25.95, £19.95 with the infinite. Nikolai Luzin was privately sympathetic to These Russian mathematicians had been the sect. racing their French colleagues to take the None of this illuminates a substantive con- Religious mystics have a long history of measure of infinite sets of real numbers. In nection between the ideas of the monks and borrowing from mathematics. It is less com- 1891 the German mathematician Georg Can- the mathematicians. These Russian scholars mon for mathematicians to draw on religion. tor made a crucial advance when he proved did push forward where the French would In Naming Infinity, historian of science Loren not, so it is reasonable to ask whether their Graham and mathematician Jean-Michel Kan- religion gave them an edge: did their belief tor argue that an esoteric Christian sect con- that both God and sets could be named into tributed to advances in set theory in Russia existence help them deal more creatively in the first decades of the twentieth century. with the infinite? The authors do not settle In pursuing their claim, they reveal a much this question, and never fully explain why the larger drama: the flourishing of mathemat- work of the Russians should have required a ics under the repression of the early Soviet belief in name worshipping as opposed to regime. another spiritual belief. In the end, they back- Graham and Kantor pedal to say they are “not claiming a unique begin in 1913, when the or necessary relationship” between mysticism Imperial Russian Navy and mathematics but are merely saying that stormed a monastery on the heresy of name worshipping “played a role a Greek peninsula where in their conceptions”. They don’t, however, say a sect of Russian Orthodox what that role was. monks had fled to pursue a Whatever their ties, the mathematicians 24–30; 1991/SPRINGER SCI. & BUS. MEDIA; N. MALYSHEV/ITAR-TASS 24–30; 13, mystical practice known as and the heretics suffered similar fates under name worshipping. Holding the Soviet regime. For a time both escaped the heretical view that God the worst treatment. The name-worshippers comes into existence when named, these hid in the shadows as Vladimir Lenin went monks believed that repeating the name of after the mainstream Orthodox church.

MATH. INTELLIGENCER MATH. Jesus while controlling their breath and heart- Mathematicians survived longer than other beat would bring them closer to the infinite. academics because, unlike physicists or Their persecution at the hands of the Tsar in Secret-police archives document the execution of chemists, they did not need special equip- C. E. FORD C. E. FORD the ensuing years aroused the sympathy of mathematician Pavel Florensky by the Soviet regime. ment, and unlike historians or philosophers,

their findings did not immediately fall foul useful to the government. It is not clear why ‘right’ size places it correctly in its — and our of Soviet dogma. Stalin listened, but his whim ensured the future — physical universe. Eventually the Stalinist state caught up with of a discipline. How We Live and Why We Die is a translation everyone. Egorov was detained in 1930 for It will be hard for the uninitiated to follow from another language — biology. Years ago, “mixing mathematics and religion” and died in Naming Infinity, owing to the book’s uneven Bob Burchfield, then the editor of the Oxford prison. Florensky confessed under torture and exposition and narrow biographical focus. English Dictionary, told me that biology had was sent to the Gulag, where he studied perma- The connection between mathematics and more words of its own than any other area of frost and seaweed before his execution in 1937. mysticism is tenuous. The real drama appears knowledge, 60,000 or so — a greater number The case of Luzin is a miraculous exception. around the edges, as the researchers survive than most of the world’s languages. Thus the In 1936 he was accused of collaborating with famine, repression and war long enough to set scale of biological language is a measure of foreigners by the Marxist mathematician Ernst the direction for a century of mathematics. It how biology has overwhelmed its history. Kolman, who proclaimed, “Soviet science will is a story of the persistence of intellectual life The meanings of these words — the things rip away your mask!” He was saved by a let- against the wrecking tide of history. ■ and ideas that lie behind them — have to be ter to Joseph Stalin from the physicist Peter Jascha Hoffman is a writer based in New York. negotiated into the semantic space of our eve- Kapista, who argued that Luzin might yet be e-mail: [email protected] ryday, more familiar world. Indeed, the book shows just how unfamiliar, eclectic, mongrel, or simply borrowed and recycled, the biological vocabulary is: HOX genes, sonic hedgehog, The hidden language of cells spindle, apoptosis, aster, telomere, P53, French flag model, and so on. How We Live and Why We Die: The Secret colour is false. What we don’t know so well is Translation from biology is harder than that Lives of Cells how cells and molecules actually behave, and from a national language where much in its by Lewis Wolpert how their behaviour explains ours. culture would be recognizable and the nego- Faber and Faber: 2009. 256 pp. £14.99 Wolpert’s book has no pictures and few tiation of meaning easier. It is a difficulty all too numbers. So, if you do not know about the size apparent in How We Live and Why We Die. And and appearance of cells and their mol ecules, it is a difficulty emphasized by the telegraphic The greatest implication of evolutionary you won’t discover it here. This matters because style Wolpert has adopted. The book says, for theory is the common kinship of living things. in biology, size is so often the key to under- instance, that “Animal cells like ours generate It is expressed no better than by Henry Harris standing both anatomy and physiology. A cell’s energy from the breakdown of their food when in The Birth of the Cell (Yale University Press, size is conceptual, not merely a fact. Being the combined with oxygen, while plants make use 2000) as “the doctrine that all plants, of sunlight”. A teenage student would animals, or whatever, are composed not get away with that. Both animal SPL of independent but co-operative and plant cells break down ‘food’ units we call cells”. And, whatever using oxygen to produce energy. It is life’s origin, “the universal solution to simply that plants make that food in the problems confronting its further the first place, using sunlight, whereas evolution was the progressive assem- animals do not — they have to for- bly of the cell”. age for it. Sloppy language adds to the Lewis Wolpert’s latest book confused picture in sentences like this: attempts to relate just how far that “The factory for producing energy in ‘progressive assembly’ has gone in animal cells comprises special struc- our own evolution. And, more con- tures known as mitochondria”. “Fac- cretely, he asks what insights the ‘cell tories” don’t produce energy, they use doctrine’ gives us into the processes it. A power station is the right anal- through which, and by which, we are ogy. “Comprises” is the wrong word conceived, develop, function, mature, and “special” tells us nothing. grow old — and die. Rendering science into under- It is a book in a tradition, not standable everyday language without limited to science, of explaining the losing its point may be hard. “Lan- visible in terms of the invisible. But guage is the dress of thought,” wrote however invisible the world of cells Samuel Johnson. How well do the new and their molecules once was, it is linguistic clothes fit the ideas? The not now. Through scanning electron example above — only too typical of microscopy and X-ray diffraction, the book — shows that sometimes the coupled with modern information answer is, badly. The author seems to technology, people have become have been lost for words. And regret- acquainted with images of cells and tably, so am I. ■ ‘living’ biological molecules. We have John Galloway is at the Eastman seen what the world of the very small Dental Hospital, 256 Gray’s Inn Road, looks like in three dimensions and London WC1X 8LD, UK. in dramatic full colour, even if that Modern microscopy reveals the many mitochondria (red) in a heart cell. e-mail: [email protected]