*‘Perhaps in time, the so-called Dark Ages will be thought of as including our own’*

G.C. Lichtenberg (mathematician/astronomer), 1782.

In 1632, Galileo was summoned before the Inquisition, having aroused the anger of the Church by means of his heretical astronomical findings. In 1932, Vito Volterra was forced to leave the University of Rome following his refusal to sign an oath of allegiance to the Facist regime. Six years later, thirteen Italian mathematicians, of international repute, were removed from their teaching positions, not due to their mathematical views, but simply because they happened to be Jewish.

History is littered with examples of State suppression of cultural and intellectual developments. Individuals have often been persecuted simply on the grounds of religion or race. Tyrants have been threatened by the emergence of ideas beyond their comprehension, as in the case of Stalin’s persecution of writers and composers. But, apart from Galileo, how many mathematicians have been victimised because of their mathematical ideas? How many dictators, apart from Napoleon, have had any knowledge of mathematics in the first place? In keeping with such themes, we now have a book that investigates the extent to which Facism caused a ‘decline’ in Italian mathematics during the inter-war years.

The story begins with a fascinating description of the years 1865 to 1920, referred to as the ‘Golden Age’ of Italian mathematics, when it was considered to be third in terms of its importance, with only France and Germany ranked above it. This ‘boom’ in Italian mathematics took place following the period when Italy was founded as a modern unified state around 1860, when many Italian mathematicians took part in the wars of independence (against Austria). Among such patriotic figures were Enrico Betti, Francesco Brioschi, Felice Casorati, Luigi Cremona, Eugenio Beltrami and so on. Moreover, in 1858, Betti, Brioschi and Casorati visited France and Germany making contact with mathematicians of the calibre of Riemann, Dedekind, Weierstrass and Dirichlet . We then read the of the resulting Golden Ages of Italian Algebraic Geometry, Analysis and Mathematical Physics, with descriptions of many of the mathematicians at the heart of such developments.

And so to those darker ages, commencing with the First World War and its aftermath. Pre-war optimism subsided, academic connections with German colleagues were severed and ten years elapsed before Italy would again host an international congress (Bologna, 1928). In 1922, matters worsened with the onset of Facist control of Italy and the resulting diminution of academic autonomy. As mentioned, academics were required to swear an oath of allegiance to the Facist regime and Jews were ultimately debarred from employment in the academic sphere, meaning that splendid talents were denied participation in the mathematical life of Italy .

There were, however, other factors that led to changes of direction in Italian mathematics. For example, demands from the growing life assurance industry and from Mussolini’s newly formed public statistics service led to growth in courses on statistics and actuarial mathematics, so that, by1936, Rome then had the first European faculty of statistical, actuarial and demographic sciences. And, for reasons unspecified, Castelnuovo turned his attention from algebraic geometry to probability and ‘social mathematics’. But mathematical biology also emerged as a new discipline in Italy , typified by the Lotka-Volterra partnership. Large-scale public engineering projects required the training of engineers and applied mathematicians, and the Bank of Naples financed the development of the National Institute of Applied Calculus.

All of this, of course, seems to indicate that, rather than Italian mathematics being in a period of *decline,* it was actually expanding in a variety of directions — a belief given greater credence by the four-fold increase in the output of mathematics graduates during the inter-war years. Yet, despite ongoing activity in the field of pure mathematics, with the likes of Severi still involved with algebraic geometry, there was, by 1938, a flattening of the creativity of former days, which may have happened anyway, since ‘Golden Ages’ are, by definition, ephemeral phenomena. The authors of this book, however, point to a failure of Italian mathematics to build upon its former achievements and they refer to its position of relative isolation compared to 1914. They also observe the repressive ethos that enshrouded academic life in the late 1930s, which discouraged the recruitment of new talent to mathematical research.

Another of the book’s major themes concerns the rise of the various mathematical institutions of Italy, and the internecine tensions between them. There is also the central question of ‘leadership’, which was of importance in Italian mathematics. Firstly, Volterra is portrayed as *the* central figure up to the time of his dismissal in 1932. Taking his place was Francesco Severi — an altogether different character, who, in earlier times, was a socialist and one of the liberal elite. But, with the deteriorating political ethos, he gradually veered towards the Facist position and assumed leadership of many of the mathematical institutions of importance, and he was the only mathematician to be appointed to Mussolini’s Reale Accademia Italia. Opportunist or pragmatist, he was nevertheless still a mathematician of repute, and he confronted Enriques about his supposed ‘over-intuitive’ approach to algebraic geometry.

Overall, this book deals with many themes that continuously interweave across its chronologically structured narrative, but the many ‘flashbacks’ often left me wondering what particular period was under discussion. There are, however, additional impediments to the interpretation of its contents. For a start, the quality of translation from Italian is very uneven, with some of the sentences being almost inscrutable. There are also misspellings, which places queries over the quality of the proof reading. Another of my reservations concerns the very many passages written in Italian (mainly written correspondence between mathematicians). These appear in the main body of the text, whilst the translated versions exist as footnotes in very small print. Why the need for both? On the other hand, there are also many passages written in French, with no translation at all!

Readers are also likely to experience navigational difficulties with this book, because, although the name index is very helpful, there is no subject index. And there are very many acronyms to which the authors refer repeatedly (UMI, PNF, INAC, SIPS, CNR, INDAM etc). These represent the main mathematical and cultural associations and a glossary of such terms would be of great help.

Discussion of the mathematical ideas is dispersed across the eight chapters of the book. Some of this goes into fair detail, whilst some developments are described in more general terms. What the book conveys, however, is the richness of Italian mathematics throughout the period and, despite being expert in none of these fields, the authors captured my imagination and make me want to know more of such mathematical ideas. Not an easy book to read — and even more difficult to review, but immensely enjoyable for all that.

Peter Ruane has now escaped the bureaucratic confines of higher education, where he spent a working life training primary and secondary mathematics teachers.

## Comments

## Another review of Guerraggio & Nastasi2005

I wrote a comparable review of this book for the Mexican journal Mathesis, volume III, issue 2-2, 2007, pages 485-492. That journal seems to have disappeared entirely. My review begins with the words “An important and effective book...” and considers many aspects of the book’s contents. The review closes with very critical comments about production matters, including corrections for many flagrant errors in the book's language. Here is a link to my review:

http://math.sfsu.edu/smith/Documents/ReviewOfGuerraggio&Nastasi2005.pdf

James T. Smith

Professor Emeritus, San Francisco State University