Gösta Mittag-Leffler and Vito Volterra. 40 Years of Correspondence

I learned about Gösta Mittag-Leffler in a long-ago course in complex analysis.  It must have been a graduate course, since Mittag-Leffler's theorem about meromorphic functions with prescribed principal parts wasn't undergraduate fare.  I owe Mittag-Leffler a personal debt, I guess, since I had occasion a couple years back to employ this lovely result in an article I wrote with two friends. Mittag-Leffler was Weierstrass’s Ph.D. student at the University of Berlin; according to the book under review, he was among the master’s favorite students. He became a major complex analyst in his own right, as the theorem just mentioned certainly illustrates.

 

Beyond this, Mittag-Leffler played a major organizational role in the European mathematical community in the late 19th and early 20th centuries. A sizable part of his influence was based on his founding the journal Acta Mathematica. And then there is the famous gossip about why there’s no Nobel Prize in mathematics, i.e., the alleged romance between Mittag-Leffler and Mrs. Nobel, earning the former the enmity of the cuckold dynamite tycoon.

 

The other protagonist in the present book, Vito Volterra, is not as well-known, but he also was a major player in the mathematics of his day. We learn that he was something of a protégé of Mittag-Leffler, continually sought out by him to publish in Acta Mathematica. In due course the two became friends, as their correspondence bears out. Volterra worked in mathematical physics, differential equations (which should indeed ring a bell), and functional analysis. Evidently Volterra was responsible for the introduction of linear functionals into the subject, this name being given to them later by Jacques Hadamard.

 

The book under review focuses on the professional and personal relationship between these two scholars, fitting the discussion into the framework of turn-of-the-century European history and the concomitant evolution of mathematics. Thus, in the excellent Introduction to the book, itself well worth the price of admission, we encounter a number of other mathematicians whose names are familiar, e.g. Ulisse Dini, Enrico Betti, and Henri Poincaré.

 

There is a lot of mathematics available in this Part I of the book, including, besides discussions of the analysis proper that was Mittag-Leffler’s bailiwick, two fascinating essays titled, “Abel’s manuscripts” and “1891: Sonya Kovalevskaya’s error.”

 

Abel had died of tuberculosis in 1829, at age 27, but his legacy was, and is, profound. Mittag-Leffler, together with Sophus Lie, were instrumental in bringing Abel’s work into the limelight. On the occasion of the centennial of Abel’s birth, Volume 26 of Acta Mathematica was dedicated to this great Norwegian, with Hilbert and Poincaré among the contributors. The line-up for this volume of the journal reads like something of an all-star team, sporting, besides Hilbert and Poincaré, such historical figures as Frobenius, Minkowski, Fuchs, M. Noether, Hurwitz, Darboux, Appel, Picard, and Stokes. Mittag-Leffler himself contributed, too, and there is also Abel’s historical landmark on elliptic functions.

 

Regarding Kovalevskaya’s error, the crux of the matter was that in 1885 she published a paper in Acta dealing with the refraction of light in a crystal in which she used a method, due to Weierstrass, that did not apply to the functions she was using. It was Volterra who found the error.  He communicated the problem to the editor, Mittag-Leffler, and an interesting bit of history was precipitated. In point of fact, reading the minutiae surrounding this episode is a wonderful exercise in the history of mathematics in its own right, and this is one of the specific strengths of the present book.

 

It is worth noting that Mittag-Leffler was heavily involved in trying to get Nobel Prizes for mathematicians who worked in theoretical physics, especially Poincaré. He failed, of course. Here is a very tantalizing quote from a letter from Mittag-Lefffer to Volterra dating to 1912 (cf. p. 43 of the book under review): “The stupidity once again got the victory. They are afraid of mathematicians because they fear everything they are too stupid to understand…” And there it is: this, by itself, should add to the appeal of the letters featured in this book — no holding back between friends.

 

The first part of the book, around 60 pages, is devoted to relevant history, biography, and mathematics, with Gösta Mittag-Leffler and Vito Volterra as the foci, the remainder of the volume (over 300 additional pages) contains their voluminous correspondence, carried out in French. There are many footnotes, in English, and there is a good deal of mathematics available, in that universal language, mathematicalese. All this having been said, if you are a mathematical history buff who reads French with relative ease, and who enjoys old pictures and classic European longhand, this book is for you.

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