Mathematics: Its Content, Methods and Meaning

Mathematics: Its Content, Methods, and Meaning (Three Volumes Bound as One [!]), by (Can you believe it?) Aleksandrov, Kolmogorov, and Lavrent’ev, is a titanic work, weighing in at 356 + 374 + 351 = 1081 pages. In Part I (in two parts) a it follows a trajectory from general themes in analysis to analytic geometry, the theory of algebraic equations, and ODE. In Part II (also in two parts) we go from PDE through differential geometry, calculus of variations, and complex analysis, then through some prime number theory, probability, to (the perhaps somewhat dissonant theme of) “electronic and computing machines.” Finally, in Part III (yes, in two parts), the parth goes from real variables through linear algebra, non-Euclidean geometry, to (and this is by itself worth the price of admission:) topology (by Aleksandrov), functional analysis (by Gel’fand), and groups and then some (by Mal’cev). Well, the whole book is peppered with Russian (or Soviet — see below) luminaries: Delone, Sobolev, Ladyzenskaya, Postnikov, and Faddeev, to name a few (in addition to the aforementioned).

Thus, the book under review is a compendium of rather beefy survey articles written by long-ball hitters in their fields, playing in the Soviet Union in the late 1950s, well before the empire crumbled. So it is that in the Preface to the Russian Edition the editors state that “the authors have kept in mind the goal of acquainting a sufficiently wide circle of the Soviet intelligentsia with the various mathematical disciplines, their content and methods, the foundations on which they are based, and the paths along which they have developed.” As such it covers a load of serious mathematics, accessible to, say, a strong senior undergraduate with adequate Sitzfleisch — well, presumably the reader can pick and choose: the sections are autonomous and the book’s sections can easily be read excursionally, to coin a phrase.

(About the intended audience of Soviet intelligentsia, is it possible to ignore the image of Krushschev or Kosygin curling up before a warm fire in their dacha’s living room, at the end of another rough day at the Politburo, with a well-thumbed copy of the book under review in hand and a bottle of vodka at the ready? I’m willing to believe the part about the vodka …)

In any event, Party leaders and apparatchiks aside, as far as the book proper is concerned, my favorite part is Part III, where, on, page 30, a cool discussion by Steckin of the Lebesgue integral is given, replete with the y-axis being sliced up, and where (on pp. 227–261) we find I. M. Gel’fand’s essay on functional analysis. Even in a treatment pitched at the present level, the master’s artistic touch cannot be hidden. Perhaps it is precisely under such circumstances that mastery shines through most remarkably. En passant, speaking of Gel’fand’s mastery, I’d recommend to the reader the gorgeous book by Gel’fand, Graev, and Piatetskii-Shapiro, Representation Theory and Automorphic Forms, just for sheer elegance.

Of course, Gel’fand eventually left Russia, as the thaw set in, spending the last decades of his life at Rutgers, and Piatetskii-Shapiro eventually left for Yale and Tel-Aviv. They both died free men in 2009, Gel’fand at 96, Piatetskii-Shapiro at almost 80; God rest their souls. With both of them also being Jews, I am reminded of my own dear senior colleague, Lev Abolnikov, who is also an expatriate Russian Jew. Lev escaped from the USSR many decades ago, when the darkness was severe, and he has horror stories to tell of the regime. He tells me, also, that scientists were obliged in those days to start off their publications with special encomia to the Party and dialectical materialism. And, to be sure, we find on page iii of the book under review the ideological allusion that “[the] abstract character of mathematics gave birth even in antiquity to idealistic notions about its independence of the material world.” Even if that reads as something of an anachronism today, to read anything resembling a paean to the regime that gave us the Gulag is, to put it mildly, jarring. But turning the page, or turning over a new leaf, as one prays Russia has done too, takes us safely to Mathematics proper, and the ensuing thousand pages are, by and large, a treat and a marvelous achievement: even professionals will find a lot in these pages to enjoy and to learn. The authors have provided lists at the end of chapters of suggested further reading: an autodidact’s dream.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.