Unfortunately, the authors really do not deliver. They begin with the attempted suppression of Name-Worshiping by the Russian government in 1913, including lots of vivid detail (chanting monks being beaten by Russian Imperial marines) of no relevance whatsoever.
They then switch gears to the math, and retell Cantor's invention of set theory and transfinite numbers, strongly implying that Cantor's psychiatric problems were caused by his math being literally mind-bending. (How would they know?) They then switch to the French fathers of measure theory, giving mini-biographies of Borel, Baire and Lebesgue and saying a little about how they used set theory to help develop a new theory of integration. Ultimately, however, the authors implied that transfinite numbers and the axiom of choice were just too much for Cartesian rationalists, so they chickened out and backed away from fully embracing Cantor's infinities. (They even suggest that Borel did so because transfinite numbers were too abstract for someone so rooted in the bucolic French countryside. I suspect that, had he grown up in Paris, they'd have invoked gritty urban realism.) There is a strong implication that all this was a simple failure of courage; the actual arguments are barely alluded to, let alone addressed.
The scene then shifts to Russia, with the founding of the Moscow School by Egorov and his pupil Luzin; the latter was a Name-Worshiper, strongly influenced by his friend, the philosopher-monk Pavel Florensky. That mystical practices were important to Luzin and Florensky is very clear; for Egorov it is not clear at all; that any of this had the least influence on the mathematics they actually did, they provide no evidence whatsoever. They do however give a lively account of the early days of the School, and especially of how the students idolized Luzin, something he seems to have encouraged shamelessly. Rather than explain the mathematics or document its substantive connection to Name-Worship, they recount the sordid persecution of the aged Egorov by the Soviet state, the persecution of Florensky, and the yet more sordid persecution of Luzin, in which many members of the Moscow School — his students, by then eminent mathematicians in their own right — actively participated. There is also a chapter on sexual relationships within the Moscow School, emphasizing the homosexuality of Alexandrov and Kolmogorov. That clusters of creative people tend to be highly charged emotionally, and indeed sexually involved either with each other or with their partners, is something of a cliche, but the authors do nothing with it, and don't even relate it to their ostensible theme. The cumulative effect of these chapters is to make all of the Russian mathematicians seem like awful people, except for Egorov and Bari, who appear merely tragic. Of course, Russia in the 20th century was not a place calculated to bring out the best in human nature.
In the conclusion, the authors summarize: the French mathematicians backed away from fully embracing transfinite numbers because set theory was just too spooky for secular rationalists; the Russians used them because it fit with their mysticism and ideas about the power of names. At this point the authors' intellectual conscience catches up with them, and they admit that they haven't proved any of this, but they absolve themselves by invoking Hume (!) and reflecting that no one can ever prove causation at all, and anyway it's important to see mathematicians as full-bodied human beings rather than abstract reasoning machines.
Let me enumerate my main objections.
To be clear, I don't think it's always impossible to say what drove historical figures. To give an example that the authors discuss, we have a pretty good idea of why Markov developed the theory of "chain-dependent" random variables and their law of large numbers and central limit theorem. He did so because a rival mathematician, P. A. Nekrasov of Moscow, claimed that the law of large numbers only applied to independent random variables, but the real world was full of statistical dependence, therefore God and free will existed and Christianity was true. This pissed off Markov, who in keeping with the St. Petersburg academic tradition was an anti-clerical rationalist, so he set out to refute Nekrasov by means of a counterexample. We know all this because we have letters from Markov explaining his motives. (And his self-account fits with everything else we know about him and his situation.) I presume that there is no similar documentation for either the French or the Russian mathematicians' mathematical work, or else the authors would have mentioned it.
Similarly, the work of the Moscow School was immediately accepted by mathematicians all over the world, including many who would have been baffled, even repelled, by Name-Worshiping, had they known of it. This makes it implausible that mystical pre-occupations influenced the ultimate content of the mathematics. (Though it sets up a horror story where we are all unwittingly invoking one of the Great Old Ones every time we use a certain theorem; for the proper effect, it should be written by Greg Egan and Charlie Stross.) It may have influenced the direction of the research, suggesting certain problems or even approaches to solving them; they don't document this or even make it plausible. For example, were there other, non-Name-Worshiping mathematicians worked along similar lines? (Ans.: yes, including ones they mention.) Or: The other center of mathematics in Russia was St. Petersburg/Petrograd/Leningrad; the two had a long-standing difference in academic culture, exemplified by the contrast between Markov (St. Petersburg) and Nekrasov (Moscow); a natural question would have been whether early-Soviet-era Leningrad mathematicians were less receptive of set theory and transfinite numbers than the Moscow School was. The authors do not address any of this, and I lack the knowledge to say.
A final point of which I am less sure. Why study the Moscow School rather than any random collection of contemporaneous (say) school teachers or clerical workers? Those people, too, had their irreplaceable lives, their struggles and hopes and sufferings (at that time and place, probably a lot of sufferings), they too were unique, transient parts of our common story. Humane piety, compassion, solidarity, even curiosity all say it would be a good thing to perpetuate their memory. Yet Graham and Kantor are not performing such an act of piety; they specifically care about Egorov, Luzin et al. because of their mathematics. Which is legitimate! But that, I think, changes what is and is not relevant to the work.
Obviously, I find this book deeply unsatisfactory. I am not sure what has gone wrong here, since Graham has a distinguished track-record as a historian of Soviet science and its interactions with philosophy and official ideology. (Kantor I don't know about.) But something definitely has.
Mathematics / History of Science
Currently in print as a hardback, ISBN 978-0-674-03293-4 [buy from Powell's], US$25.95