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Science Networks · Historical Studies
Founded by Erwin Hiebert and Hans Wußing
Volume 29

Edited by Eberhard Knobloch and Erhard Scholz

Editorial Board:
K. Andersen, Aarhus D. Buchwald, Pasadena H.J.M. Bos, Utrecht U. Bottazzini, Roma J.Z. Buchwald, Cambridge, Mass. K. Chemla, Paris S.S. Demidov, Moskva E.A. Fellmann, Basel M. Folkerts, München P. Galison, Cambridge, Mass. I. Grattan-Guinness, London J. Gray, Milton Keynes

R. Halleux, Liège S. Hildebrandt, Bonn Ch. Meinel, Regensburg J. Peiffer, Paris W. Purkert, Leipzig D. Rowe, Mainz A.I. Sabra, Cambridge, Mass. Ch. Sasaki, Tokyo R.H. Stuewer, Minneapolis H. Wußing, Leipzig V.P. Vizgin, Moskva

Angelo Guerraggio Pietro Nastasi
Italian Mathematics Between the Two World Wars
Birkhäuser Verlag Basel · Boston · Berlin

Authors’ addresses:
Angelo Guerraggio Centro PRISTEM-Eleusi Università Bocconi Viale Isonzo, 25 20135 Milano Italy
email: angelo.guerraggio@uni-bocconi.it

Pietro Nastasi Dipartimento di Matematica Università di Palermo Via Archirafi, 34 90123 Palermo Italy
email: nastasi@math.unipa.it

2000 Mathematics Subject Classification: 01A60, 01A72

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ISBN 3-7643-6555-2 Birkhäuser Verlag, Basel – Boston – Berlin
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ISBN-10: 3-7643-6555-2

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Preface
During the first decades of the last century Italian mathematics was considered to be the third national school due to its importance and the high level of its numerous researchers. The decision to organize the 1908 International Congress of Mathematicians in Rome (after those in Paris and Heidelberg) confirmed this position. Qualified Italian universities were permanently included in the tour organized for young mathematicians’ improvement. Even in the years after the First World War, Rome (together with Paris and Göttingen) remained an important mathematical center according to the American mathematician G. D. Birkhoff.
Now, after almost a century, we can state that the golden age of Italian mathematics reduces to the decades between the 19th and the 20th century. In the centre of interest stood the algebraic geometry school with Guido Calstelnuovo, Federico Enriques and Francesco Severi acting as key figures. Their work led to an almost complete systematization of the theory of curves to the complete classification of the surfaces and to the bases of a general theory of algebraic varieties. Other important contributions came from the Italian school of analysis. Its main representative was Vito Volterra – an outstanding analyst with a strong interest in mathematical physics – who produced important results in real analysis and the theory of integral equations and contributed to the initiation of functional analysis.
Guiseppe Vitali, Guido Fubini and Leonida Tonelli were well known in the integration theory and the calculus of variations. At the beginning of the century Tulli LeviCivita’s scientific adventure started: He became one of the most recognized and esteemed Italian mathematicians abroad. There also was a strong connection between the authority in the scientific disciplines and the role they could play for the future and the modernization of Italy. In chapter 1 we describe this thrilling season of Italian mathematics.
The golden age however, is only the prologue of our history. We will focus our attention to the years between the two World Wars. The turning point during those years was marked by the Great War – it was an epochal change. Nothing remained as it was before. The ingenuous hope that the war could simply be a gap of time and afterwards one could come back to the belle époque were illusions. In chapter 2 we analyze the changes in Italian mathematics.
From a strict mathematical point of view the twenties and the thirties were less stimulating for Italy than the previous ones, but from the context of the whole century they were attractive on other sides: The social and political situation suddenly changed with the raise of the fascist regime (chapter 4). Also structural scientific aspects changed with the creation of new institutions which should play an important role in the development of Italian science and mathematics for the rest of the century. In chapter 3 we describe the birth of UMI (Italian Mathematical Union) and of CNR (National Research Council); in the chapters 6 and 8 we deal with the consecutive presence of INAC and of Severi’s INDAM.

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Naturally the next step is to consider whether there was any link between the changes in the political and scientific spheres and if these influenced the organization of the mathematical research, its contents and its quality level.
We can describe the main problems dealt with by our analysis in a more detailed manner. In the period between the two World Wars the leading actors of Italian mathematics were rather the same as before. Perhaps the most relevant difference was the arrival of Mauro Picone on the scene. His presence was particularly noticeable in a numerical and applied perspective and also in the ideas that guided the creation and the development of INAC – an absolute novelty in the international mathematical panorama. Even if the names were rather the same, their role had changed. Volterra’s brilliant career was stopped by fascism, and so the old liberal generation was marginalized by the new government. Severi became piece by piece the head of the mathematical group. We dedicate the chapter 3 and 5 to this leadership change. Like Volterra, Severi was an outstanding mathematician and a broad-minded man, and his personality was charismatic, even if different in the coherence of his behaviour patterns.
His leadership should remain till the Second World War. There should be some tensions – see chapter 6: the alternative of the CNR – but in the thirties Italian mathematics grew with a sufficient continuity (chapter 7). It needed another external event, the tragical experience of the Second World War to induce a new discontinuity in the Italian mathematical life (chapter 8).
The mathematical research itself was always at a good level. The influence of Italian researchers on algebraic geometry was a strong one. Enriques contributed some important historical studies to his research in this field. The “old lion”, Volterra, wrote a last relevant chapter in his scientific career by analyzing population dynamics. Tonelli’s Fondamenti di Calcolo delle Variazioni were published and his esteem – about all for the use of direct methods – was high in the mathematical world. Picone’s influence has already been described. Some other young brilliant scholars joined the already acknowledged researchers: Renato Caccioppoli, Lamberti Cesari, Francesco Tricomi and others. Another young man, Bruno de Finetti, increased the suspense of the probabilistic studies and anchored a research directed towards economic and social applications. Not to forget the undoubted authority of Levi-Civita and the role he played by corresponding with Einstein and many younger colleagues.
Nevertheless this survey makes a clear statement: for Italian mathematics the golden age was on the retreat. Its potential did never return after the First World War. Not quite a crisis but rather the difficulty to maintain the previously excellent level and to continue in playing a role in originality and creativity. On the contrary the orthodox respect towards a still young tradition and the acceptance of a level just achieved seemed to prevail.
The new abstract and algebraic languages did not speak Italian any more. They were born in situations where the weight of tradition was lower and we could speak of a decline of Italian mathematics with respect to its level 30-40 years before compared to the new languages that were developing in the other countries between the wars.
This is the point where the two histories – the Italian and the mathematical one – met. The conditions and the progress of Italian mathematics are analyzed by focusing on both the inner and the external influences. Is there any link between the establishment of

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a dictatorial regime and the decline of Italian mathematics? Can we find this possible link in the most repressive fascism facts – the 1931 oath and the 1938 racial laws – or rather in its politics towards science and particularly in its attitude in favour of the applied sciences?
In the following pages we will try to give an answer to these questions by analyzing the most important works of the Italian mathematicians living in the period, the life of the Italian mathematical community, some correspondence of the most representative members of it and their positions outside the research or educational fields. But our interest goes beyond the historical facts of the period between the two World Wars and its influences on the present problems. So the previous questions have a “modern” version too. Can the scientific world accept – and at which conditions – a confrontation with the political power or is it necessary to avoid these contaminations? Which are the possibilities of the political sphere to orient the trends of the scientific developments? And in the particular case of mathematics? How can a political will overcome the constraints imposed by the economic structure? In the light of the episode of the oath and the silence of too many mathematicians at the sight of the racial laws, which are the ethic and political responsibilities of a researcher?
As one can see, the questions are numerous. We just hope to give a contribution in answering them through the analysis of Italian mathematics between the two World Wars.
Angelo Guerraggio Pietro Nastasi

Contents
Chapter 1 Prologue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. The Risorgimento generation . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. The golden age. The Italian school of algebraic geometry . . . . . . . . . . . 10 3. The golden period. The mathematical physics . . . . . . . . . . . . . . . . . 16 4. The golden age. The analysis. . . . . . . . . . . . . . . . . . . . . . . . . . 19 5. External interests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
Chapter 2 Nothing is as it was before . . . . . . . . . . . . . . . . . . . . 29 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2. Italian mathematicians take sides. . . . . . . . . . . . . . . . . . . . . . . . 31 3. Mathematicians at the front . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Chapter 3 Volterra’s leadership . . . . . . . . . . . . . . . . . . . . . . . . 55 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2. Rome, 1921 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3. The foundation of the Unione Matematica Italiana . . . . . . . . . . . . . . 67 4. The foundation of the Consiglio Nazionale delle Ricerche . . . . . . . . . . 74 5. Volterra’s scientific activity . . . . . . . . . . . . . . . . . . . . . . . . . . 76 6. Volterra and Ecology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Chapter 4 Fascism: somebody rise, others fall . . . . . . . . . . . . . . . . 83 1. The march on Rome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 2. Giovanni Gentile and school reform . . . . . . . . . . . . . . . . . . . . . . 85 3. The battle of the “manifestos” . . . . . . . . . . . . . . . . . . . . . . . . . 90 4. Enriques’ rentrée . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Chapter 5 One man alone in the lead . . . . . . . . . . . . . . . . . . . . . 101 1. The novelty of the Accademia d’Italia . . . . . . . . . . . . . . . . . . . . . 101 2. Severi as a mathematician, in the 1920s . . . . . . . . . . . . . . . . . . . . 104 3. Severi: politician . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4. The difficult presence of Algebra . . . . . . . . . . . . . . . . . . . . . . . 119 5. Enriques and his school . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 6. Castelnuovo, Probability and “social Mathematics” . . . . . . . . . . . . . . 147
Chapter 6 The CNR alternative . . . . . . . . . . . . . . . . . . . . . . . . 159 1. End of decade balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 2. Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 3. Distinguished Senator, Dear Colleague . . . . . . . . . . . . . . . . . . . . 178 4. The dualism U.M.I. – C.N.R. . . . . . . . . . . . . . . . . . . . . . . . . . . 183

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5. The oath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 6. Tullio Levi-Civita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
Chapter 7 The 1930s move forward . . . . . . . . . . . . . . . . . . . . . 215 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 2. Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 3. Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
Chapter 8 Towards disaster . . . . . . . . . . . . . . . . . . . . . . . . . . 243 1. European events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 2. The international Congress of 1936 . . . . . . . . . . . . . . . . . . . . . . 247 3. The anti-Semitic laws of 1938 . . . . . . . . . . . . . . . . . . . . . . . . . 251 4. Crisis signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268
Chapter 9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

Chapter 1
Prologue
1. The Risorgimento generation
The history of modern Italy starts in 1860. In that year the various nation-states into which the Italian peninsula had been politically and administratively divided were unified in a process called the Risorgimento. Under the leadership of Piedmont (the northwestern region of Italy on the French border, whose capital is Turin) and its hegemony, a remarkable idealistic and democratic impulse with significant popular support led to the unification of the country. Yet, a number of uprisings, two wars of independence against Austria (1848–9 and 1859, the latter of which was fought with crucial help by France), and intense diplomatic activity, were still necessary to achieve this goal.
Some of the mathematicians whom we shall shortly present, who will figure prominently in this prologue, participated in the military mobilisation for these wars of independence, particularly in the years 1848 to 18591. Enrico Betti was a volunteer in a student battalion from the University of Pisa. In 1848, Luigi Cremona participated in the defence of Venice, which had rebelled against Austrian rule and was given the rank of corporal and later that of sergeant. Francesco Brioschi participated in 1848 in the insurrection of Milan against the Austrians and in 1870 in the storming of Rome.
In 1860 the peninsula had not yet been completely unified. The Veneto region (in the north-east of Italy) was still under the sway of Austria. It would only be annexed to the new Italian state after the third war of independence (1866). In particular, Rome was still ruled by the papacy. In this case, public, political and diplomatic issues were of much greater complexity. It would only be in 1870 that the Italian government could overcome the temporal power of the papacy. On this occasion it exploited the opportunities offered by the difficulties faced by the Vatican’s erstwhile ally, France, in the aftermath of the FrancoPrussian war, the fall of Napoleon III and the end of the Second Empire. The annexation of Rome by the Italian state would usher in a long period of difficulties in its relationship with
1 As for the commitment of Italian mathematicians during the Risorgimento, see: Universitari Italiani nel Risorgimento (ed. by L. Pepe), Bologna, CLUEB, 2002.

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Enrico Betti
Brioschi-statue
Catholicism, and would cause at the same time a cooling in the Franco-Italian alliance and Italy’s entry into the sphere of influence of the central European states.
Vittorio Emanuele II thus became the first king of Italy. The capital of the newly founded state, initially established in Turin, was subsequently moved to Florence following the Italo-French agreements of 1864. Actually, this was a step towards making Rome the capital, as it was considered the historical and ideal centre of the Country. Rome was finally made the capital of Italy in 1870.
The next fifty years, before Italy’s entry into the First World War, can be characterized as full of intense efforts to weld the country into a nation, with infrastructures, standards of living and vital statistics as close as possible to the more developed nation states. The initiatives taken to modernize agriculture and to industrialize the economy

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were held back markedly by the great differences between different regions of the country. The situation in the South, the so-called southern question, was particularly critical from the social and economic viewpoint. The north of Italy benefited from a much more solid economic and social base. In the last two decades of the 19th century, a bourgeoisie began to develop here which would progressively influence city styles and customs. The consolidation of a bourgeoisie would be accompanied by a similar development of the working class, mainly concentrated in the Milan-Turin-Genoa triangle.
The mathematicians were in the front-line of this process of nation building, occupying significant political and administrative positions. Just to mention some of the names cited above, both Brioschi and Betti would become parliamentarians, senators and undersecretaries in the Ministry of Education (in the years 1861–2 and 1874–6, respectively). Cremona was appointed the Minister for Education in 1899, even if only for one month. Brioschi, in particular, was a key protagonist in establishing an education system that reflected the outlook of the new entrepreneurial bourgeoisie, which was consolidating in the north of Italy in opposition to a lazy and passive landowning class. The expectations of this emerging class that scientific progress and its technological fallout would nurture and accelerate the industrial development of Italy are paralleled in the mindset of scientists and the culture of scientific research. In particular, they were reflected in the perspectives mathematicians envisaged for their teaching and research. This common world view formed the motivating factor behind the establishment of the Polytechnic of Milan, founded in 1863 by Brioschi with the intention of creating a class of qualified technicians indispensable for the rise of Italy’s industrial initiatives.
In short, during the first half-century of its existence as a unified nation state, Italy went through a period which was in many respects similar to that of other European countries. Unlike them, however, it had to race to make up for its late start because of the backwardness and uneven progress of the vast underdeveloped areas surrounding its limited industrial base. For a time, Italy was blessed with political stability accompanied by a gradual, albeit not straightforward and not altogether peaceful, widening of its democratic base. It survived the economic crisis of the last quarter-century. Later it could not resist the siren call of colonial adventure in East Africa. Its agricultural and industrial development gathered pace over the last years of the 19th century with constantly increasing rates of production which sometimes attained considerable heights before 1908. By a remarkable coincidence the boom characterizing the decade before this date also involved mathematics, given that 1908 was to be the year of the fourth International Congress of mathematicians in Rome.
We can now introduce Italian mathematics over the first half-century more systematically by describing its structure and protagonists starting with the generation of the Risorgimento. This era precedes a period on which we will focus later. We have already mentioned Enrico Betti (1823–1892) and Francesco Brioschi (1824–1897) in terms of their participation in political and military events. Together with the young Felice Casorati (1835–1890) both these mathematicians visited the universities of Göttingen, Berlin and Paris in 1858 to learn of the most significant advances in European mathematics both from the scientific and organizational point of view. They were able to meet, amongst others, such distinguished mathematicians as R. Dedekind, P. C. L. Dirichlet,

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Felice Casorati
B. Riemann, L. Kronecker, K. Weierstrass and C. Hermite. Tradition has it that this voyage marked the birth – almost from nothing – of Italian mathematics. The theory that the Risorgimento also caused a new starting point in mathematics naturally derived from a patriotic ideology which emphasized the view that unification had set wings to the aspirations and enthusiasm of the best minds in the country, including science2. Actually, it cannot be argued that the mathematical school had sprung up from nothing (nor simply through a fact-finding mission). Even so, extreme as it may appear, this view can still be taken as a suitable starting point.
The collaboration between Betti and Brioschi can be considered the true driving force behind this rebirth of Italian mathematics, which was to be extremely fruitful both in terms of organization and quality of research. Betti’s meeting3 with Riemann in Göttingen and their intense cooperation during the latter’s stay in Pisa (from 1863 to 1865) was a turning point. Following Riemann’s death in 1866, Betti became a reference point for all European mathematicians interested in further investigating this German mathematician’s works. Betti was a physicist-mathematician and the author of significant research (which was also translated into German) into the theory of potential and elasticity.
2 In the inaugural speech of the International Congress of mathematicians in Rome, in 1908, Volterra asserted: “Hence, I would not be surprised if, following scientific development, there were a sudden transformation in the Italian thought, brought about by its quick progress and dissemination, and by the new enriching features it took in the years following the period of the political Risorgimento”.
3 About Betti, the mathematical school of Pisa, and more in general about Italian mathematics after the Unity, see U. Bottazzini, Va’ pensiero. Immagini della Matematica nell’Italia dell’Ottocento, Bologna, Il Mulino, 1994.

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In this field his most well-known contribution is the so-called reciprocity theorem. This held that, if for an elastic solid one can consider two states of equilibrium consequent on the action of two different force systems, the work carried out by the first system (with respect to the deformations involving the second) is equal to the work of the second system with respect to the deformations involving the first. He also investigated complex variable functions, elliptic functions, and had even earlier looked into a number of issues concerning algebra and algebraic topology. It is no coincidence that Poincarè would coin the expression: Betti numbers as a means of measuring the different connection orders in n- dimensional figures. Betti was amongst the first in Europe to realize the value of Galois and Abel’s research for the resolution of algebraic equations, arriving at original results which subsequently were rediscovered and praised by Hermite. Finally, Betti was appointed director of the Scuola Normale di Pisa4 (from 1865 to his death), making the first contribution to the establishment of what was to become the most important research centre in Italy.
We have already mentioned Brioschi’s5 “political” involvement and his contribution to the education of a ruling class in Italy which would step over the limits drawn by an exclusively legal – literary schooling. In his case, from a more strictly mathematical point of view, it is difficult to single out a particular discipline with which to identify him. Brioschi’s research ranged from algebra, analysis, geometry and mechanics, to mathematical physics. In analysis he made important contributions in the field of elliptic functions and differential equations, and particularly in that of differential invariants (associated with singling out the class of differential equations referable to constant coefficients equations). However, it was in algebra where he made his most lasting contributions, with innovative research into the theory of determinants and algebraic forms. By the time he embarked on his “European trip” in 1858, Brioschi was already a highly regarded mathematician. His book on La teorica dei determinanti e le sue principali applicazioni (published in 1854) had already been translated into French and German by 1856. His reputation derived in particular from his resolution of fifth and sixth degree algebraic equations (after Galois had demonstrated that it was impossible to solve for radicals equations that were greater than the fourth degree). Brioschi’s works accompanied others results by Hermite and Kronecker for the solution of general equations of the fifth degree through elliptic functions, and all three mathematicians were accorded merit for their solution of sixth degree equations through hyperelliptic functions. Finally – as in the case of Betti – one must mention Brioschi’s efforts in founding and then successively promoting the Annali di Matematica pura e applicata destined shortly to become one of the most prestigious journals in the sector.
4 The Scuola Normale, founded in 1813, prepared the future school teachers in the Napoleonic Kingdom of Italy. Napoleon’s fall caused its closing (as well as that of other Napoleonic institutions) in 1814. The Grand Duchy of Tuscany reopened it in 1846, always with the same objective. After the Unity of Italy, besides this “old vocation”, it developed as a research centre, different from the university, and as a training centre for future researchers.
5 On F. Brioschi see U. Bottazzini, Francesco Brioschi and the “Annali di Matematica”, in C.G. Lacaita, A. Silvestri (eds.), Francesco Brioschi e il suo tempo (1824–1897), Milano, Angeli, 2000, pp. 71–84; A. Brigaglia, Brioschi, Cremona e l’insegnamento della Geometria nel Politecnico, ibidem, pp. 403–418.

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During their trip to the European capitals of mathematics in 1858, Betti and Brioschi were accompanied by the young Casorati6 (who was then only 23, having been one of Brioschi’s students). In his case, it is easier to single out a specific area of research to discuss: this was complex analysis. Casorati was the one who disseminated in Italy the ideas of Cauchy, Riemann and Weierstrass, also by publishing a monograph titled Teorica delle funzioni di variabili complesse (1868), containing original results which often preceded similar discoveries usually wrongly attributed to Weierstrass, Mittag-Leffler and Picard.
Aside from Betti, Brioschi and Casorati, few other names need to be mentioned to give a fairly complete picture of the first generation of mathematicians in the recently unified Italy. Among these, the most important were Luigi Cremona (1830–1903) and Eugenio Beltrami (1836–1900).
The former is considered the founder of the Italian school of algebraic geometry7. His commitment, and the role he intended to play in the field of geometry, can already be seen in his Prolusione published in 1860 at the University of Bologna where he wrote very clearly about the absence of “modern” geometry in Italy although it was already an essential part of teaching in France, Germany and Great Britain. Cremona moved from Bologna to the Polytechnic of Milan (where he held a course of static graphics) and then to Rome, to the School of Engineering, where the appeal of his teaching among students can be considered one of the first indications that the study of mathematics was coming into its own in Italy. In particular, two of his monographs8 (published in 1861 and 1867) marked the peak of projective studies and introduced a method for the geometric treatment of numerous algebraic problems, in the belief that synthetic geometry, with its clear supremacy, was the only system that could ensure the application of a methodology both rigorous and intuitive. Cremona’s main contribution (in which he showed he could appreciate the ideas already expressed by Riemann and the German school) was the introduction of the concept of the birational transformations of planes and space. These are a generalization (later called cremonia transformations) of the classic concept of linear transformations, and can be expressed through rational functions, usually invertible with functions of the same type. It was by using this concept, as well as the analysis of algebraically invariant properties with respect to birational transformations, that the study and classification of algebraic curves and surfaces starts. This research, in particular his synthetic study of cubic surfaces, won him, together with Charles Sturm, the Steiner prize of the Academy of Sciences of Berlin in 1866 (considered at the time the most prestigious award in the field). He received this prize again in 1874 without participating in any preliminary examination, in recognition of all his publications on geometry.
6 On F. Casorati see U. Bottazzini, Alla scuola di Weierstrass, in Va’ pensiero, op. cit., pp. 195218; A. Gabba, Il carteggio Brioschi-Casorati, in C.G. Lacaita, A. Silvestri (eds.), Francesco Brioschi e il suo tempo (1824–1897), op. cit., pp. 419–429.
7 His great interest for the history of geometry and his many international relationships can be appreciated in his correspondence, being printed by a research group coordinated by G. Israel.
8 See L. Cremona, Introduzione ad una teoria geometrica delle curve piane, Mem. Accad. Sci. Bologna, 12 (1861), pp. 305–436; Preliminari di una teoria geometrica delle superficie, Mem. Accad. Sci. Bologna, n.s., 6 (1867), pp. 91–136 e 7 (1867), pp. 29–78.

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Beltrami9 is mainly remembered for his research on differential geometry, undoubtedly influenced once again by Riemann’s ideas and their dissemination during the latter’s Italian sojourn10. Riemann’s studies kindled his interests in non-Euclidean geometry and the creation of their first model on the pseudosphere. Beltrami’s research can hence be situated between differential geometry and mathematical physics. With the publication of his monographs: Saggio di interpretazione delle geometria non – euclidea (1868) and
9 As a young man, Beltrami was very active, given his Risorgimento ideals. As a result of these in 1856 he had to suspend his studies at the University of the Pavia before graduation and start working as a humble clerk. After the Kingdom of Italy was founded, Brioschi had him appointed without a public examination (on Cremona’s recommendation) as visiting professor in algebra and analytical geometry at the University of Bologna in 1862. Beltrami could at last devote himself to research and teaching, swinging for two decades between the Universities of Pisa, Rome and Pavia. He finally decided to settle in Rome, where he succeeded Brioschi as president of the Accademia nazionale dei Lincei. On Beltrami, see R. Tazzioli, Beltrami e i matematici “relativisti”. La meccanica in spazi curvi nella seconda metà dell’Ottocento, Bologna, Pitagora Editrice, 2000.
10 Due to health reasons, Riemann spent the winter of the year 1862 in Sicily. From October 1963 until July 1965 he stayed in Pisa.

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Teoria fondamentale degli spazi di curvatura costante in the following year, Beltrami’s work took its rightful place in the history of non-Euclidean geometry. These works, together with Beltrami’s proof of the coherence of Gauss, Lobatchevsky and Bolyai’s hyperbolic geometry, lend credibility to their reassessment of the privileged status which Euclidean geometry had hitherto enjoyed.
So far, we have dealt with Betti, Brioschi, Casorati, Cremona and Beltrami. We can also add Giuseppe Battaglini (1826–1894) to this group. Battaglini is essentially a geometrician, a self-educated mathematician whose main concerns were the more analytical “neo-geometry” of Plucker and the geometric theory of algebraic forms by Clebsch. We owe to Battaglini also the Italian translation of Todhunter’s classic manual on Calculus and the publication, from 1863, of the Giornale di Matematiche (known precisely as “Battaglini’s Journal”), which promoted the education of young researchers through the dissemination and explication of major research programs and their results11. However, our list of mathematicians stops here. It was this small group which worked towards the mathematical modernization of the country by taking as its model the most advanced European situations. These close links with other countries would remain a constant feature in all the programs established in this period, together with a strong public and political commitment by mathematicians, an almost inevitable consequence of the great ideals and the fervent aspirations expressed in previous decades. Hence, mathematicians can be numbered amongst the most impassioned intellectuals committed to finding solutions for the many problems which afflicted the Italian education system in the period following unification. Foremost amongst these problems were the great differences between the Italian regions.
The development of the Italian education system can be seen from the right perspective when one realizes that it was only in 1877 that the first two years of primary school became compulsory (after a long struggle against the most intransigent sectors of the Catholic church which sought to maintain family prerogatives). Indeed, at the time Italy was united, about 70% of the population was illiterate and this percentage would only decrease slowly in successive decades (from 69% to 62% in the 1871 and 1881 censuses respectively) reaching the threshold of 50% only at the beginning of the 20th century. In Europe a similar situation could be encountered only in Spain (and an even worse one in the Russian Empire). By the mid -19th century the other European countries had just under 58% illiterate people (Austrian Empire, Belgium, France) or even less (Great Britain 25%, Prussia 20%, Sweden 10%). Given that this proportion of educated people form the base of the educational pyramid, one should not be surprised by the small number of university students. Indeed, there were little more than 12,000 in 1871 and they doubled over the next 30 years, with a particularly accentuated progression in the period 1881 to 1901 also because of the prolonged economic crisis at the time (as always one of the variables with the greatest impact on the length of schooling). About one-third of university population attended the polytechnics or scientific degree courses. Here, amongst the teaching staff, the presence of mathematicians was prepon-
11 A collection of his letters, from 1854 to 1891, can be found in M. Castellana and F. Palladino (eds.), Giuseppe Battaglini, Bari, Levante ed., 1996.

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Luigi Cremona
derant for the half-century this prologue is dealing with. In 1881, for example, mathematicians held 69 positions, which was slightly less than half of the total number of positions assigned to the scientific faculties.
The boom in mathematics (in terms of students numbers and quality of the curricula mentioned above) can be appropriately explained in terms of the initial situation (at the beginning of unification), which we described as being extremely inadequate, making what happened later appear extremely positive by comparison. The same, in particular, can be said for any type of research which did not require great expense or investments and which could therefore develop rapidly even in a country with severe social problems. Also fundamental was the cohesion of the small group of mathematicians introduced above and the atmosphere in Italy during the last decades of the 19th century. At the time, Positivist thought was in the ascendancy and it informed the values of the growing bourgeoisie. The mathematical and physical sciences (not to mention economics) were seen as instruments for its affirmation, as was the development of a prevalently technical education in opposition to the literary and artistic curricula considered as antiquated and typical of a backward social organization. The mathematization of the social sciences also met with a certain measure of success because of the widespread belief in the objectivity of economic laws, contrasted with any attempt to subject economics to moral or ideological priorities.

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2. The golden age. The Italian school of algebraic geometry
The virtues of the Risorgimento generation (Betti, Brioschi, etc.) are to be seen, however, in terms of the creation of the conditions which made possible the second generation to transform Italian mathematics into a great power, second only to France and Germany. This corresponds to what we can call the golden age of Italian mathematics. It is of greater interest to us because it was during this period that some of the future protagonists of the years between two world wars began their careers. The levels of excellence that this group attained set the standards the following generation would have to measure up to.
First we deal with the school of algebraic geometry (which we already mentioned when we spoke of Cremona). The Premio Bordin of the Académie des Sciences, was awarded to Italian mathematicians12 on two occasions in 1907 and in 1909 for research in this field, and the prestige of the Italian school is reflected in the epithet: italienische Geometrie attributed to algebraic geometry.
One student of Cremona’s13 was Giuseppe Veronese14 (1854–1917) who worked in Berlin and Lipsia in 1880 and 1881 where he met Felix Klein. It was certainly an important encounter: the structural approach of the German mathematician encouraged Veronese to study the foundations of non-Archimedean geometry and the projective geometry of hyperspaces, to the extent that he would be recognized as one of the fathers of projective geometry in n-dimensional spaces. Battaglini, too had a student, Enrico D’Ovidio (1843–1933) who, after arriving in Turin, began to work with the young Corrado Segre (1863–1924). Their collaboration would bring, either by their own efforts or through those of their students, Italian algebraic geometry to full maturity. It was in this school that the study of algebraic surfaces would develop to become the greatest achievement of the Italian mathematical tradition.
Segre15 started his career with an outstanding dissertation on hyperspatial quadrics and some studies regarding their geometry, following the concepts of Veronese. Soon, these projective techniques would be placed “at the service” of other research allowing him to ‘import’ and develop A. Brill and M. Noether’s program regarding the geometry of an algebraic curve, or in other words, the study of the properties of algebraic curves which are invariant with respect to birational transformations. In addition to these studies which represented the core of his scientific efforts, Segre also investigated such fields as: the ruled surfaces in hyperspaces, enumerative geometry, algebraic topology, and the initial elements of a theory of algebraic surfaces (with the intention of rigorously demonstrating Noether’s theorem for the existence of a smooth birational
12 The prize was awarded in 1907 to Federigo Enriques and Francesco Severi and in 1909 to Giuseppe Bagnera (1865–1927) and Michele de Franchis (1875–1946).
13 Among other pupils of Cremona, we should cite at least Eugenio Bertini (1846–1933). 14 Veronese graduated in Rome in 1877. From 1897 to 1900 he was Member of Parliament, and later
town counsellor in Padua and (from 1904) Senator. 15 Part of this correspondence (in particular 270 letters and postcards exchanged with Castelnuovo from
1891 to 1898, almost all regarding his early studies on the geometry over a surface) has been published and analysed in P. Garzio, “Singolaritá e Geometria sopra una superficie nella corrispondenza di C. Segre a G. Castelnuovo”, Archive for History of Exact Sciences, 43 (1991), n. 2, pp. 145–188.

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Corrado Segre

Eugenio Beltrami

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model for every algebraic surface) and the varieties described by families of projective spaces. His research on the relations between surfaces submerged in a projective space and partial differential equations would make him the most distinguished mathematician in Italy in the fields of differential geometry of curves, surfaces or varieties submerged in a projective space without a metric structure.
Of equal import was the incisiveness of his teaching in his famous courses on “Superior geometry” held between 1888 and 1924. The radical contraposition between geometry and analysis which can be seen in Cremona’s “purism”, was in some respects overcome whilst still remaining within a framework highlighting the supremacy of synthetic methods. Their elegance and productiveness would become a model for the entire mathematical edifice. Students of Segre’s were Guido Castelnuovo (1865– 1952), Federigo Enriques (1871–1946) and Francesco Severi (1879–1961)16. The Italian school of algebraic geometry is generally identified with them. It is worthwhile having a closer look at their role and activities: they debut brilliantly at the turn-of-the-century but we will find them again – maybe in other fields of academic endeavour – also in the 1930s. Severi in particular would become one of the key figures of Italian mathematics between the two world wars.
Immediately after graduating in Padua under Veronese, Castelnuovo began postgraduate study in Rome in 1886 where he heard Cremona’s lessons. The following year he went to Turin, where he began what was to be his lasting and friendly collaboration with Segre. His research mainly concerned algebraic curves, for which he elaborated a rigorous proof of Riemann-Roch’s theorem and the formula of maximum genus, with the subsequent determinations of maximum genus curves. This result generalized a discovery made by G. Halphen and M. Noether, but previously valid only for three dimensional projective spaces.
The techniques used by Castelnuovo to elaborate his proof were original and still striking today for their simplicity and elegance. The turning point came a few years later, in 1891 when he was given a professorship in geometry at the university of Rome. Henceforth he focused on a new study of algebraic surfaces, but we should not neglect results such as those obtained in 1901 when he formulated the first rigorous proof of the theorem for which each cremonian flat transformation can be seen as the product of quadratic and linear transformations. In Rome, Castelnuovo met Enriques, with whom he was to write fundamental works in the history of the theory of algebraic surfaces. Until then the points of reference had been E. Picard’s transcendent and M. Noether’s geometric approach. The former had studied simple integrals of total differentials of the first kind annexed to an algebraic surface, coming to the result that these only existed on particular surfaces, for example hyperelliptic ones. The latter introduced the invariants constituted by the geometric genus pg, the linear genus p (1) and the numerical genus pa. It could be pa = pg , as always happens in the case of curves, or it could be q = pg– pa ≠ 0. Cayley had verified the second possibility in the case of the ruled ones. Since then it had been hypothesized that q was null, with the exception of the ruled ones. However, already in 1891, after studying certain particular types of surfaces, Castelnuovo had built
16 We should not fail to mention Gino Fano (1871–1952).

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the first example of an irregular unruled algebraic surface for which pa = pg. In 1896 he made his most important discovery17, with Enriques already working at his side, formulating a famous counterexample, where he extended the Riemann-Roch theorem of curves and the determination of the criterion of rationality. The condition valid for the curves – their rationality is linked to a null genus – was generalized: a surface is rational if and only if: q = P2 = 0, where q is the surface irregularity index and the plurigenus Pn is a new birational invariant, introduced by Enriques.
Enriques had graduated in 1891 at Pisa university. He had wanted to undertake postgraduate study in Turin with Segre. Instead, he managed to find a position in Rome. Here he immediately changed from the team led by the then elderly Cremona to the one of the promising Castelnuovo, who would direct him towards the study of algebraic surfaces. Already in 1893 and in 1896, when he had been in Bologna for two years, Enriques published two fundamental memoirs where he laid the basis for the organic theory and the classification of algebraic surfaces. Enriques would never completely abandon this field of research, unlike Castelnuovo, who would practically stop publishing on algebraic geometry in the early years of the 20th century. However, within this field he would soon dedicate significant attention to elementary mathematics (developed also thanks to his personal acquaintance with Felix Klein) and to the philosophy and history of mathematics.
His meeting with Castelnuovo, their friendship (further strengthened when Castelnuovo married Enriques’s sister) and their scientific plans have been documented by an exceptional collection of correspondence containing almost 700 letters written by Enriques to Castelnuovo between 1892 and 190618. Their personalities appeared to be complementary: Enriques was exuberant and possessed an extraordinary power of intuition. Often he would appear already certain of an outcome before securing it with successive formulation. But he was less interested in proofs and their rigour; he was impatient and often superficially read articles by colleagues. In contrast, Castelnuovo was perhaps less brilliant but original as well. He also sought nonetheless to refine and channel his brotherin-law’s genial intuitions into more suitable and productive outcomes. Their twenty year collaboration would develop a new method of formulating the theory of algebraic surfaces leading to a particularly simple classification, with the elimination of all the special cases. Consequently, the study of algebraic surfaces would involve now only those of curves lying on the surface. Amongst these, particular attention was dedicated to linear systems and to nonlinear continuous systems (existing only on irregular surfaces, for which the difference pg – pa is positive). In two notes19 written in 1914 Enriques presented almost definitive results on the theme of classifications: the surfaces were subdivided into classes of birational equivalents according to the values assumed by the plurigenera and the geometric genus. In the same year, the publication of a long article20, written together
17 G. Castelnuovo, Alcuni risultati sui sistemi lineari di curve appartenenti ad una superficie algebrica, Mem. Soc. It. Sci. XL, 10 (1896), pp. 82–102.
18 The whole correspondence is published in U. Bottazzini, A. Conte, P. Gario (eds.), Riposte Armonie. Lettere di Federigo Enriques a Guido Castelnuovo, Torino, Bollati Boringhieri, 1996.
19 F. Enriques, Sulla classificazione delle superficie algebriche e particolarmente sulle superficie di genere lineare p(1) = 1, Note I e II, Rend. Acc. Lincei, 23 (1914), pp. 206–214 e 291–297.

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with Castelnuovo, in the Enzyklopädie der Mathematischen Wissenschaften was to represent the crowning achievement of their research and the official recognition of its importance by the international mathematical community.
We shall discuss Severi in the coming pages, both to illustrate his research in algebraic geometry and to reveal his rich and complex personality, together with his cultural and philosophical interests. His political role between the two world wars as the undisputed leader of mathematicians will also be examined. Even younger than Enriques, Severi graduated in 1900 at the university of Turin under Segre, with an outstanding thesis on enumerative geometry. Two years later he was in Bologna working with Enriques, who encouraged him to investigate the theory of algebraic surfaces. Severi will win a professorship already in 1905, first in Parma and soon after in Padua. From 1903 onwards, in particular, he concentrated on irregular surfaces (after the counterexample by Castelnuovo who had proven that the conjecture according to which ruled surfaces were the only irregular surfaces was groundless). There is already a glimpse – always within the school – of a strong and original personality, with a marked attention towards topological and functional aspects.
In particular, Severi “retrieved” transcendent methods21 as a means of determining the link between irregular surfaces and surfaces endowed with total differential integrals of the first and second kind. It is thus proved – also thanks to an algebraic-geometric proof of Enriques, which Severi will not see fit to aprove, though – that irregular surfaces and those with Picard’s integrals of the first kind are the same set, and the existing relation between q and the number of integrals of the first and second kind (linearly independent) is stated. At that time, his relationship with Enriques was excellent and their collaboration continued: in 1907 both mathematicians, as mentioned above, received the Premio Bordin for their research on the classification of hyperelliptic surfaces by finishing G. Humbert’s work. In particular, Severi is awarded the prize Medaglia Guccia at the International Congress of Rome, in 1908, by a committee formed by M. Noether, E. Picard and C. Segre. He tries to extend those results and methods, that had proved so effective in the case of surfaces, to the study of varieties. In the same year he is appointed member of the Accademia dei Lincei, that in 1913 will award him the Premio Reale. In 1912, Severi and Enriques collaborate again, publishing a work on the foundations of enumerative geometry, which B. L. van der Waerden would consider of fundamental importance as a rigorous basis for algebraic geometry. In this work a solid basis was given to enumerative methods and in particular to Schubert’s principle of the conservation of number, according to which, if an enumerative problem had in the general case a finite number of solutions, then the same number of solutions (unless they become infinite) can also be found in particular cases.
We shall now leave Severi and his studies on algebraic geometry to briefly deal with differential geometry. In reality, these two fields of research are not so distinct (although for clarity’s sake we discuss them as if they were) and the protagonists in-
20 G. Castelnuovo, F. Enriques, Die algebraischen Flächen vom Gesichpunkte der birationalen Transformationen aus, in Enzyklopädie d. Math. Wissensch., III (1914), 2, 1, C, pp. 674–768.
21 One can see C. Houzel, La geómetric algebrique, ed. Blanchard, Paris, 2002.

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volved were often the same. However, two names are new to the scene described above: Luigi Bianchi (1856–1928) and Gregorio Ricci-Curbastro (1853–1925). Both graduated at the university of Pisa and spent a period of postgraduate work in Göttingen together with Klein. It is not the first time that we encounter this German mathematician; in fact, Klein played a similar role to that of Riemann with the first generation of Italian mathematicians, confirming the appeal that German mathematics and its organisational methods exercised over their Italian counterparts.
Bianchi spent his whole mathematical career in Pisa, where he was to become the director of the Scuola Normale between 1918 and 1928. He also wrote22 on subjects such as analysis, algebraic number theory and one of his most important first contributions was his activity as a writer of treatises. Whole generations of Italian mathematicians would study from his book Lezioni di geometria differenziale. Of equal merit was his teaching work in algebra, with monographs (on finite groups and the theory of Galois, on continuous groups, and on the arithmetic theory of quadratic forms) which disseminated in Italy the arithmetic techniques formulated by the German school, in particular, by L. Kronecker, R. Dedekind, H. Weber and D. Hilbert. Regarding differential geometry, in his doctoral thesis of 1879, Bianchi introduced the so-called “complementary transformation” for surfaces submerged in the ordinary space. The result was applied in the theory of partial differential equations and in particular in nonlinear equations which we today term sine – Gordon. A few years later, the Swedish mathematician A. E. Bäcklund generalized Bianchi’s transformation, and in turn Bianchi integrated Bäcklund’s theory with the so-called “permutability theorem”, which allowed their transforms to be found using only algebraic and derivative calculations (after Bäcklund’s transforms of an initial pseudo-spherical surface are all known). Other notes examined the general theory of Riemann’s spaces. In a paper published in 1898, Bianchi with greater simplicity demonstrated the result (already known to Riemann) according to which n-dimensional spaces with constant and equal curvatures can be mapped isometrically to each other. In a successive work23 (dated 1902) he obtained the famous Bianchi identities, satisfied by the covariant derivatives of Riemann’s four index curvature symbols. However, despite the use of the covariant derivatives, as L. Pizzochero observed24, Bianchi was substantially unfamiliar with the methods of absolute Calculus.
The true “Master” of this field in Italy was Ricci-Curbastro, who on his return from Göttingen, finally settled in Padua. Here, in the decade from 1885 to 1895 he studied the calculus of tensors, finding his main source of inspiration in the invariant theory of Riemann’s varieties, developed in research carried out by E. B. Christoffel, R. Lipschitz and of course, B. Riemann himself. As early as 1886 one of his notes introduced what he would later call covariant derivatives of a function (without, to tell the truth, quoting either Lipschitz or Christoffel, who had both already analysed the same operation). This expression appeared for the first time in a work published in the following year: Ricci
22 His writings, collected in Opere (10 volumes), were published in 1952 (Roma, Cremonese). 23 L. Bianchi, Sui simboli di Riemann a quattro indici e sulla curvatura di Riemann, Rend. Acc. Lincei,
11 (1902), pp. 3–7. 24 L. Pizzocchero, Geometria differenziale, in S. Di Sieno, A. Guerraggio, P. Nastasi, La Matematica
Italiana dopo l’Unità. Gli anni tra le due guerre mondiali, Milano, Marcos y Marcos, 1998, pp. 321–379.

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Curbastro studied multiple index covariant systems, applying to them the classic law of transformation after changes in coordinates. In a memoir of 1888, with the emergence of systems with many counter-variant indices, he practically announced the birth of absolute Calculus, with the change of the ordinary procedures of differential calculus proposed so that formulas and results keep the same form, whatever system of variables is used. The expressions absolute differential Calculus and absolute systems appear in the memoir, Méthodes de calcul différential absolu et leur application, written together with his student Tullio Levi- Civita (1873–1941) and published in 1900 in Mathematische Annalen on F. Klein’s invitation. The memoir expounds Riemann’s geometry, with the new terms, and physical applications (to elasticity, to electrodynamics, etc.). The usefulness of the new methods would only be realized after some time. International aknowledgment for Ricci-Curbastro would arrive only on the eve of the First World War. In 1913, Einstein would adopt absolute Calculus as the basic mathematical language for the theory of general relativity which he was developing at the time.
The infinitesimal methods of differential geometry were ‘exported’ to projective geometry. Finally Guido Fubini25 (1879–1943) dedicated some notes to the construction and the analysis of metrical structures in projective spaces and, in particular, to the description of the metrical structure induced by a hermitian form over a complex projective space (of any dimension). We shall discuss Fubini again when we deal with the Italian school of real analysis. The line element ds2 in the projective space is still today indicated with his name (together with that of the German mathematician E. Study).

3. The golden period. The mathematical physics
We should now give due recognition to one of Ricci-Curbastro’s students, Levi-Civita, one of the most creative Italian mathematicians in the first half of the century. We will mention him often in this book.
Tullio Levi-Civita26 (1873–1941) graduated in Padua, where he received his entire education, if we except a brief period of postgraduate study in Bologna (where he met Enriques, becoming his lifelong friend) and some teaching in Pavia. In 1918 he was appointed at the University of Rome as professor of Superior analysis and successively Rational mechanics. Levi-Civita was in essence a mathematical physicist whose interests ranged from electromagnetism to analytical mechanics, from celestial mechanics to Relativity, from hydrodynamics to the theory of heat. Throughout his work, as observed by L. Dell’Aglio and G. Israel27, there was a close correlation between innovation and tradition. He explored new and original perspectives without weakening his steadfast attachment to a method which oriented analytical investigation according to results emerging from the preliminary use of geometric models.
25 His writings are collected in three volumes in Opere (Roma, Cremonese, 1961–1963). 26 His writings, edited by the Accademia dei Lincei, are gathered in six volumes in Opere matematiche
(Bologna, Zanichelli, 1954–1970). 27 See the article by Dell’Aglio-Israel in La Matematica italiana tra le due guerre mondiali (A. Guer-
raggio ed.), Pitagora ed., Bologna, 1987 .

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In the first years of his career, Levi-Civita expanded the research on stability according to Liapounov, ; in 1901 he developed his theorem on stationary movement and began his study of the theory of wakes in hydrodynamics that he would later deepen more fully in Rome. In the case of celestial mechanics he focussed on the classic problem of the three bodies, starting from P. Painlevé’s results and deducing a regularization of motion equations (he was able to predict and therefore to eliminate their singularities). Levi-Civita’s first essential contribution to absolute Calculus dated back to 189628. It was also the first time Ricci’s Calculus was adopted in a context outside of metric differential geometry, to solve a problem of analytical mechanics. The memoir confronts the issue, already raised by K. Appell in 1852, of the mutual transformability of “two systems of dynamic equations with the same number of variables”. The problem, in the case of forces independent of speed, was to be re-examined by Painlevé, who “by an opportune modification” had revealed that it could be applied to the determination of all systems (called correspondents) that have common trajectories. Hence, the invariant character of the problem emerged, and it was reduced to the singling out of all the correspondents of a given system. This suggested quite naturally that Ricci’s Calculus could be applied. It was by using this Calculus that Levi-Civita came to the conclusion, for the most general pair of correspondent dynamic systems (having the same number of degrees of freedom and not stimulated by other forces) that “n perfectly determinate types” were possible.
Another, but no less significant proof of the fruitfulness of Ricci’s Calculus was provided in a memoir29 published shortly afterwards (1899), containing research on the types of potentials that can be made to depend on only two spatial coordinates. The analytical evaluation of the problem from Riemann onwards had led to differential systems, which were so complex as be intractable. Levi-Civita took as his starting point the observation that all those potentials that allow “infinitesimal transformation in themselves” were independent of one coordinate. From here Levi-Civita went on to consider the infinitesimal transformation to allow by the Laplace D2 y = 0 equation, finding five categories of infinitesimal transformations to which corresponded five types of binary potentials. Ricci’s Calculus was used at this point to show (also following advice by F. Klein) that the binary potentials found in this manner are the only ones possible. In the same year – as we already said – F. Klein invited Ricci-Curbastro to arrange a whole and systematic explanation of the calculus of tensors, to be published in Mathematische Annalen. In the writing of the article, later considered as the manifesto of tensorial algebra, Ricci-Curbastro let the young Levi-Civita, whose contribution would be fundamental especially for its applications to mathematical physics, join in. Tensorial relationships are not modified by the change in the coordinate system, therefore their language is particularly useful to express the properties that are naturally independent of the chosen reference.
The works cited above, published at the end of the century, were written by an extremely young Levi-Civita. Over the same period, the reputation of another Italian
28 T. Levi-Civita, Sulla trasformazione delle equazioni dinamiche, Ann. Mat., 24 (1896), pp. 255–300. 29 T. Levi-Civita, Tipi di potenziali che si possono far dipendere da due sole coordinate, Atti Acc. Torino,
49 (1899), pp. 105–152.

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mathematician, Vito Volterra30 (1860–1940) was nearing its zenith. Volterra was the undisputed leader of Italian mathematicians in the first decades of the new century. He graduated in Pisa in 1882 after having studied under Betti. He was then appointed professor at the University of Pisa and successively Turin. In 1901 he moved definitively to Rome to become Beltrami’s successor. This move to the capital increased Volterra’s public profile and his involvement in positions of increasing responsibility in determining the scientific and cultural policies of the nation. At the beginning of the century, Volterra was elected president of the Società Italiana di Fisica. In 1905 the Italian Prime Minister, Giolitti, appointed him to the Senate. In 1907 he founded the SIPS (Società Italiana per il Progresso delle Scienze) – becoming its first president – on the model of similar societies already existing in France, England and other industrialized countries. His objective was of establishing a meeting point among scientists from different backgrounds as well as giving them a chance to disseminate their research. We shall deal with Volterra again later in this book. His presence influenced 50 years of Italian scientific research and makes us possible to deal not only with mathematical physics and analysis, but also with mathematical economics and mathematical biology.
The age difference with Levi-Civita (who represented an interesting balance between innovation and tradition) is less than 15 years, but it was enough to place Volterra in a more classicist “19th century” perspective, where one feels the powerful pull of a strongly cohesive research, capable of describing the complexity of macroscopic physical phenomena by using only a few basic equations. As regards mathematical physics, the most important contributions, over the turn-of-the-century, regarded the propagation of light in birefractable equipment, the movements of the terrestrial poles (or, to be more precise, the movements of the Earth’s surface with respect to the Earth’s rotational axis), hereditary phenomena and what in modern terms is called dislocation theory. This last subject, which Volterra called distorsioni (distortions), constitutes part of his theory of elasticity which, according to Klein had become a “national issue” for the Italians31. In 1901, L. G. Weingarten had proven that a state of tension can exist in an elastic body without being subjected to external forces (occupying a non-simply connected dominion). The first example that comes to mind is that of a ring which after being cut transversally, removing a slice of matter, is then re-attached. Volterra’s studies, which were to have a significant impact on the theory of elasticity in non-simply connected dominions, began from this point. His findings, the classification and theory of distortions which derived from his research, were collected in a sizeable memoir dated 1907 (published in the Annales scientifiques de l’Ecole Normale Supérieur) “Sur l’équilibre des corps élastiques multiplement connexes”. Other authors would continue this research, including
30 Volterra’s Opere matematiche were issued in five volumes in 1962, edited by the Accademia dei Lincei.
31 Among the several works on elasticity, we would like to point out Introduzione alla teoria matematica dell’elasticità (Turin, Fratelli Bocca, 1894) by the Neapolitan Ernesto Cesàro (1859–1906), who died tragically at sea while trying to save his son in danger. Particularly influenced by Beltrami in his works on mathematical physics, Cesàro is still remembered today for his works on analysis and for his classic method of summation of series, and stands out also for his results in the field of intrinsicgeometry and of asymptotic arithmetic.

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Carlo Somigliana (1860–1955) who was a friend and colleague of Volterra’s, a descendant of Alessandro Volta and the author of a general theory of distortions.
Volterra deserves a final mention as a physicist and mathematician for his researches on hereditary phenomena or on systems with memory, quoted above. His studies, starting from the observation that the deformations of an elastic body depend also on previous deformations, investigated those bodies which maintained the memory of their history and whose future state subsequently depended on their present as well as previous states. Once again an interesting convergence emerged between experimental data and mathematical instrumentation: the equations are no longer differential but integral-differential equations (which would be applied in particular to electrostatics and hereditary elasticity) given that heredity is expressed by functions that are integral with respect to time, of linear combinations of deformation components.

4. The golden age. The analysis.
By briefly referring to Fubini and in particular Volterra, we have been able to bring the study of analysis into the discussion. In Italy, this third great discipline of 19th-century mathematics was developed particularly in Pisa. The leader of this school was Ulisse Dini (1845–1919), who graduated under Betti in 1864 with a thesis on differential geometry. His name32 is universally known among mathematicians and students of mathematics for his theorem of implicit functions and for the “Dini derivative”, in which the customary passage to the limit is generalized through the notion of upper or lower limits. Also deserving mention are his studies on numerical and trigonometric series, complex variable functions, and differential equations. But the greatest impact that Dini had on the Italian mathematical scene (and not only the Italian) was due to the publication of his monograph: Fondamenti per la teorica delle funzioni di variabili reali (1878), in which he developed his rigorist program. For the objective was not to discover new results so much as to place already known ones on more solid foundations by completing them and specifying the dominion of their validity.
Giuseppe Peano (1859–1932), from Turin, was another protagonist of the rigorist turning point. His contribution was to present the axioms of arithmetic, to give some counterexamples – some of which were ruthless in their simplicity, with which he ridiculed unsubstantiated hypotheses, mistakes and approximations (some contained in the most widely used manuals) and to obtain a precise and general formulation of a number of fundamental notions of analysis (limits, area of a region, Taylor’s formula, partial derivatives, maxima and minima for functions of several real variables, etc.). He is a particularly well-known mathematician33: his importance in the axiomatization of math-
32 Dini’s Opere, edited by the Unione Matematica Italiana, were published in three volumes in 1955 (Roma, Cremonese)
33 See H.C. Kennedy, Life and work of Giuseppe Peano, Dordrecht, D. Reidel Publ. Comp., 1980. Peano’s Opere scelte have been published in three volumes, edited by the Unione Matematica Italiana (Roma, Cremonese).

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ematical theories is undeniable; his non – recursive definition of a derivative of order n is still used today in some research on analysis and non-smooth optimization. His contribution (in the second half of the 1880s) to the theorem of existence for the differential equation y1 = f(x,y), proven with the sole condition of the continuity of function f, is specifically mentioned in many manuals. Equally well known is his role in devising a system of axioms for vector spaces described in his monograph dedicated to the dissemination of Grassmann’s ideas. Peano’s Curve (1890) more than 100 years on, still remains one of the most amazing and least intuitive conclusions which deductive rigor has brought to set theory and has played a truly significant role in the history of the concept of dimension: it is possible to find a curve, expressed by two continuous functions x = f (t) and y = g(t), which goes through all the points of the unity square whilst t varies over the interval [0.1]. In other words it is not always possible to enclose a continuous curve within an arbitrarily small area.
Indeed, it was on the issue of scientific rigour that Peano engaged in a lively dispute in 1891 with Segre (and Veronese). Segre had backed a less rigid and absolute position by distinguishing the period of discovery from that of rigour. Peano instead retorted tersely that a theorem can be considered as discovered only when it is proven and that in the absence of the only – absolute – rigour that mathematics comprehends, one may write poetry, but not mathematics. Peano had another, much harsher, dispute with Volterra. Mathematical content34 concerns the motion of … a cat, allowed to fall in a vacuum upside-down and more generally the internal movements of a body (and the possibility of modifying their orientation) that Volterra had analysed in specific reference to the terrestrial globe subjected to the action of internal forces. Paradoxically in this case, Peano stood accused for the lack of rigour and originality of his conclusions. The dispute increased his isolation. Given the almost forgone outcome of his battle in favour of mathematical rigour, Peano gradually left his research in analysis and began to develop his ambitious plan of reconsidering all of the propositions of classical mathematics, breaking them down and analysing them in their smallest parts so as to be certain that they contained nothing less and nothing more than what was necessary. The same propositions were rewritten using combinations of algebraic and logical signs which leave no scope for misunderstanding and allow their precise and succinct formulation.
In referring to Volterra we can return to Pisa, which we have depicted as the main centre of Italian analysis. Dini’s influence on the young Volterra can be seen in the latter’s early but famous contribution of 1881 at the young age of 21. Volterra was engaged in the process of completing the Riemann integration theory. One of the main issues of interest were the so-called two fundamental theorems of Calculus, that is, the study of the relationships between the operations of derivation and integration. It was here that Volterra devised the now classic example of a function derived in an interval, with a limited but not integrable derivative. At this point, Volterra’s research horizons widened beyond the strictures of a rigorist program. Also thanks to Betti and his competence in physics, Volterra was attracted by the possibility of applying analytical tools,
34 See A. Guerraggio, Le Memorie di Volterra e Peano sul movimento dei poli, Archive for History of Exact Sciences, 1984, pp. 97–126.

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of course in a sophisticated and adequate manner, to the exigencies of the problem to be faced.
These were the characteristics which we can find in his work on functional analysis. Volterra can rightly be considered one of the founding fathers of this discipline and its independent development, although his pioneering ideas would attract greater appreciation in other countries (in France, for example, thanks to the attention and sensitivity displayed by Hadamard) with slightly different characteristics. We have already spoken of Volterra’s ‘classicist’ outlook when dealing with his research in physical mathematics. Whilst using an abstract language in functional analysis, which was very distant from concrete applications, Volterra still kept his sights on “practical” objectives, such as real problems in physics or other mathematical issues. It would be these “practical” issues that suggested the specific abstraction to implement and which constituted a means of validating the significance of the formalization adopted. His first notes on functional analysis were published at the end of the 1880s. Within a few years, Volterra had introduced the concept of a functional with its associated calculus (up to its development using Taylor’s polynomial) and carried out his first research about linear functionals on a given functions space. Actually, he did not use the term functional (which would be suggested later by Hadamard) but the term function of a line to indicate a real number which depends on all the values taken up by a function y(x) defined over a certain interval, or the configuration of a curve. A functional can be considered as a limit case, for n Æ + •, of a function with n variables. In this manner, the first coherent research was carried out in spaces of infinite dimensions and the whole edifice of classical analysis was generalized to some specific functional spaces. The evolution of such an extension, starting from n-dimensional spaces, was highlighted and took on both an explanatory and reassuring role at the same time. Hence the derivative of a functional (defined on the set C[a,b] of continuous functions over a given interval) is what today we would call a directional derivative, or a Gâteaux-Lévy directional derivative. This is obtained by passing from an initial value f0 to an incremented one: f0 + eh, making e tend to 0 and hence reducing to the customary concept of derivative for a real function (adopting a procedure well-known to Calculus of variations). Volterra is not so interested in studying the functional properties of ‘his’ derivatives, so much as their actual calculus. And in defence of his approach he reminded those who accused him of giving a too specific definition (with respect to the ensuing “differential according to Fréchet”), such as Hadamard and especially Fréchet, that maximum generality is not the ultimate value to be sought after, but rather the most adequate generality for the problem being dealt with35. One should remember that Volterra’s first results took place at the end of the 19th-century and that M. Fréchet’s thesis is dated 1906. Although his initial works still considered specific functional spaces, they already did so from the perspective of general theory. Hence, they would enable and encourage unifying studies of metrical and topological structures.
35 Fréchet would not give up either. Still in 1965, in a letter to P. Lévy from the 30th July (published in Cahiers du Séminaire d’Histoire des Mathématiques, 1980, n. 1), he clarified that “si je considère que Volterra a réalisé un grand progrès en donnant au moins une définition de la différentielle d’une fonction dont l’argument est une fonction, d’autre part, je considère que sa définition est mauvaise”.

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Volterra’s other well-known contribution in this period were integral equations, inserted for the first time into a general theory, later taken up and developed by E. Fredholm, D. Hilbert and others. Volterra investigated integral equations of the first and second kind with a triangular kernel. Here too, the procedure for their resolution was accompanied by the formulation of the principle of the passage from discrete to continuous, for which an integral equation of the first kind is the limit case (for n Æ + •) of a system of n algebraic equations in n unknowns.
Giulio Ascoli36, Cesare Arzelà37 and Salvatore Pincherle (1853–1936) all graduated in Pisa. The first two names can be seen in every text on functional analysis for their studies on the concept of equicontinuity and the extraction of a converging subsequence from a sequence of equilimited and equicontinuous functions. After graduating, Pincherle studied in Pavia (with Casorati) and Berlin where he studied under the guidance of K. Weierstrass38. His stay in Germany is fundamental for an understanding of how his research developed. Pincherle is considered another pioneer of functional analysis thanks to his theory of analytic functions. The remark that each of these functions can be singled out from a countable infinity of parameters, which could be interpreted as its coordinates, led Pincherle to investigate functions spaces of infinite dimension and the abstract study of the linear functionals acting on these spaces. He sought to create a calculus for these functionals similar to the already well known one for the functions of a complex variable. Over the next few decades, these concepts would be developed along different pathways to an extent which was unthinkable at the turn of the century. Instead, the route taken by Pincherle would not be as well trodden, as he himself would serenely come to recognize.
After his brief stay in Berlin, Pincherle moved definitely to Bologna, that would become, together with Pisa, a new important research centre in analysis. The most representative exponent of the school in Bologna was Leonida Tonelli (1885–1946), whom we shall encounter as one of the foremost protagonists of Italian mathematics in the period between the two world wars39. He had studied at Bologna under Arzelà and Pincherle, graduating in 1907. His academic career as full professor would begin only after the war for a number of reasons (first at Bologna and later at Pisa). Nevertheless even before 1915, Tonelli had written a number of very important works, numbered among his most significant, in the field of real analysis and Calculus of variations. In 1908 he published a note40 on the length of rectifiable continuous curves with particular reference to the case in which the functions representing the curve are absolutely continuous. In the
36 G. Ascoli (1843–1896) graduated at the Normale in Pisa in 1868. Then he taught at the Polytechnics in Milan.
37 Also C. Arzelà (1847–1912) graduated in Pisa, at the Normale, in 1869. Later, he taught at the Universities of Palermo and Bologna. His Opere, in two volumes, have been issued in 1992 (Roma, Cremonese) and edited by the Unione Matematica Italiana.
38 Pincherle’s Opere scelte, in two volumes, edited by the Unione Matematica Italiana, were published in 1954 (Roma, Cremonese).
39 Tonelli’s Opere scelte were issued in 1961 (Roma, Cremonese) and edited by the Unione Matematica Italiana.
40 L. Tonelli, Sulla rettificazione delle curve, Atti Acc. Sci. Torino, 1908.

Prologue

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following year, he published a note41 where in generalizing the integration formula by parts to the functions of two variables, he provided a criterion for the integrability (according to Lebesgue) of a measurable function f(x,y) ≥ 0 which admits a pair of successive integrals. It can be affirmed that this article completed the well-known result proven by Fubini in 1907 according to which the double integral (assuming it existed) of f(x,y) can be calculated by two successive simple integrals, independently of the order of integration.
Fubini, whom we have already mentioned for the originality of his studies in projective differential geometry, was another leading figure in the Italian school of real analysis. He is mainly remembered for his theorem on double integrals but he was also the author of other important works in the theory of integration, the minimum principle, automorphic functions and integral equations.
But let us return to Tonelli. His fundamental memoirs on Calculus of variations were published in 1911, 1914 and 191542. Calculus of variations took its rightful place in functional analysis by the systematic use of direct methods, already used in particular cases by B. Riemann, D. Hilbert, J. Hadamard, H. Lebesgue, C. Arzelà etc., based on the notions of compactness and semicontinuity (generalizing the definition given by Baire for real functions). It was through direct methods that Tonelli proved some theorems of the existence for the so-called simplest problem in Calculus of variations, avoiding the passage through Euler’s equation and hence avoiding difficulties about the calculation (and the existence) of the solution of a boundary value problem, the strong limitation imposed on the functional class by the consideration of differential equations, the privilege given to the relative extrema and then the search for suitable sufficient conditions.
Giuseppe Vitali (1875–1932) was the other main exponent of the school of Bologna, even if he graduated in Pisa (after having studied in Bologna under Arzelà and Enriques)43. The year 1905, in particular was a “magical” one in terms of his scientific endeavours. After having proven the necessary and sufficient condition for Riemann integrability of a limited function over a limited interval (depending on the measure of the set of its discontinuity points), in the same year Vitali published a series of notes in which he proved the so-called Lusin’s theorem on the almost continuity of measurable functions, giving the famous example of non-measurable sets (according to Lebesgue). Moreover, he characterized the integral functions of not necessarily limited functions by inventing the term, of absolutely continuous functions (and studying the class of these functions in relation to those of bounded variation). Many of these results were more or less obtained over the same period by H. Lebesgue. Nevertheless, they were obtained
41 L. Tonelli, Sull’integrazione per parti, Rend. Acc. Lincei, 1909. 42 L. Tonelli, Sui massimi e minimi assoluti nel calcolo delle variazioni, Rend. Circolo Mat. Palermo,
1911, pp. 297–337; Sur une méthode directe du calcul des variations, C. R. Acad. Sci. Paris, 1914, pp. 1776–1778 and pp. 1983–1985; Sur une méthode directe du calcul des variations, Rend. Circolo Mat. Palermo, 1915, pp. 233–264. 43 Vitali’s Opere sull’Analisi reale e complessa, edited by the Unione Matematica Italiana, were published in 1984 (Roma, Cremonese); the publication of the letters addressed to him would follow (edited by M.T. Borgato and L. Pepe).

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wholly independently. At this junction one should remember that Vitali was unable to find a university position and for many years was forced to teach in high schools in distant locations removed from the customary channels of scientific communication. At the same time, Lebesgue could also complain that at Poitiers, where he taught from 1906 to 1910, he was not able to consult any Italian journal. This may explain the partial overlap of Vitali’s results with those of Lebesgue, without diminishing the originality and value of his research, in particular that regarding absolutely continuous functions. In this instance, Vitali’s priorities were clear, not so much because he introduced the term or for his generalization to the functions of two variables, but because of the central position he accorded to such a concept in his theory of integration.
Pisa was also where Eugenio Elia Levi studied. He was born in 1883 and died in 1917 in the war44. With him we introduce the topic of complex analysis which we touched upon when discussing Pincherle. His brother45 Beppo (1875–1961) was also a mathematician and at the same time as Vitali engaged in a brief controversy with H. Lebesgue regarding the cogency of some proofs by the latter. Nevertheless, he would mainly concentrate his efforts on algebraic geometry, number theory, logic and the foundations of geometry. In the complex analysis, Eugenio Elia’s research focused on the singular point sets of a holomorphic function of several variables. However, he also wrote on issues relating to: differential geometry, Lie’s groups, partial differential equations and the minimum principle. E. E. Levi would also demonstrate the falsity of Weierstrass’s conjecture according to which given an open A of C2 , a merophormic function will always exist in A which has essential singularities in each point of the border of A, providing further evidence in favour of the differentiation between the theory of the single complex variable and the theory of more than one complex variable. His research followed Hertogs’s theorem (1906) which signals the rise of multidimensional complex analysis as an independent research field.
This springtime in Italian mathematics at the beginning of the 20th century was not confined to geometry, mathematical physics and to analysis but also involved the “new” disciplines. We have already mentioned how Peano went on to study logic after embarking on his rigorist struggle and his search for extreme precision in definitions and proofs, also for teaching purposes. Around him and his publishing plans and the Rivista di Matematica (founded in 1891), a school of young and combative scholars would rapidly coalesce. Their presence would enliven many conferences which were still an innovation at the beginning of the century. Bertrand Russell would remember his meeting with Peano at the International Philosophy Congress in Paris in 1900 as being a particularly significant event for the formulation of his program. In partial contradiction with the
44 E. E. Levi had graduated from Pisa in 1904. He had been Dini’s assistant and then taught at the University of Genoa. His Opere, edited by the Unione Matematica Italiana, were printed in two volumes in 1959 (Roma, Cremonese).
45 B. Levi graduated from Turin in 1896. After a short period as assistant and as secondary school teacher, he taught geometry in Cagliari and then in Bologna. After the racial laws of 1938, he was forced to emigrate to Argentina, contributing to organize the mathematical activity in that country. His Opere, edited by the Unione Matematica Italiana, have been printed in two volumes in 1999 (Roma, Cremonese).

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new ideas of the period, mathematical logic for Peano did not involve the application of algebraic techniques to traditional logic (and hence was not – nor could be – an independent mathematical discipline) but a tool and a language which were essential to mathematical activity, allowing concepts and proofs to be expressed with the greatest clarity.
With even less traditions a group of young mathematical economists also formed. The “lesson” taught by Walras approach had been adopted by Vilfredo Pareto (1849– 1923) who, despite teaching in Lausanne, became the founder of the Italian school of mathematical economics and the true disseminator of the theory of general economic equilibrium. Mathematical economics was already an independent discipline but it had not yet expressed those distinguishing features typical of its full maturity. This field continued to entertain a close exchange with other areas of mathematical research and with those sectors of Italian culture and society interested in mathematizing a science which had traditionally been considered part of the social sciences. Economics was thereby endowed with quantitative and “objective” foundations46. The most active season of Italian mathematical economics was brief, very much associated with Pareto’s commitment to it. Indeed, in 1909 with the publication of the French edition of the Manuale di economia politica, Pareto would in practice cease his research in economics. This would not stop an economist and an economic historian such as Joseph Schumpeter to consider Italian economic research in 1915 (thanks to the mathematical economists) as second to none.

5. External interests
As representative of the Risorgimento generation, we have dealt with a small group of mathematicians of great ability, tempered and selected by the political and military events of the period. These mathematicians associated their research with their public lives and were inspired by the most advanced research of the time in Europe. This small group had now grown. In the next generation we have met almost all the protagonists of our history: Volterra and Levi-Civita, Enriques and Severi, Tonelli. University positions in mathematics was increasing as was the number of young students aspiring to a university career. Before a national society of mathematicians was established, a number of scientific associations and academies had already developed (and they often published their journals and “bulletins”). In 1870, with the taking of Rome, the historical Accademia dei Lincei was reorganized. In 1884 the Circolo matematico of Palermo was founded; its Rendiconti would soon draw international attention, and it would be given the task (together with the mathematical section of the Accademia dei Lincei) of organizing the fourth International Congress of mathematicians in Rome in 1908. By that date the Circolo would number 924 members, of which 618 were foreigners, and its international prestige would be universally recognized. Also in 1908,
46 On this issue, see A. Guerraggio, Economia e matematica in Italia tra Ottocento e Novecento, Scientia, 1986, pp. 13–39.

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Poincarè publicly declared that the Circolo was the most important world mathematical organization!47
Although this increase in the scientific community would make it difficult for a collective mind to formulate and carry out a single plan, the picture that we have before us at the beginning of the new century is sufficiently consistent to be considered as a whole. The efforts made for greater prominence in the international scene continued (as proof we have the exemplary story of the Circolo matematico of Palermo), and were still associated with strong engagement in public affairs. Increasing social complexity, however, meant that this engagement was not only expressed in terms of participation in government and parliament. A greater number of steps and a longer march through society became mandatory.
Such a trek for mathematics could not begin but in the school system. The grave problems in education, noted immediately after unification, would not be solved. The process of homogenization of the different regional situations would be slow, and the modernization of the country placed an added burden of tasks and objectives on the educational system. Tertiary education faced the problem of having too many universities, inherited from the various Italian states before unification, which brought to the fore the problem of the quality of teaching. In the secondary schools, the need to increase levels of education led to many calls to reduce and simplify programs (particularly and especially in mathematics).
Despite this difficult situation the teachers of mathematics would react positively by displaying strong individual commitment, founding (in 1895) a society, called Mathesis, which published the Periodico di Matematiche, and attracted the collaboration of a substantial number of university lecturers and professors. Unfortunately, results did not always match efforts, as the crises which this association would have to cope with testify to. Nevertheless, a distinguishing feature of Mathesis in this period was its great faith in active and direct involvement by members and in the establishment of a grass-roots reform movement. All the main educational issues were expressed and subjected to consultation amongst teachers in a positive fashion. From the point of view of Mathesis, the strength of this representation and logic would almost inevitably transform the resulting solutions into a reform project.
The relationship of Italian mathematics to the rest of society was not confined to establishing and disseminating scientific culture among the youngest generations. At the same time, its intention was to “export” the language and rationality it considered distinguishing features of its research, particularly by influencing traditionally closest scientific disciplines. Starting in 1895, Il nuovo cimento became the official publication of Italian physicists with Volterra as a member of the scientific committee for the journal. Two years later, the Società italiana di fisica was founded, with Volterra becoming its president, as we have already seen.
Even more surprising were the mathematical “incursions” into fields traditionally occupied by the “other” culture. It must not be forgotten that Italian mathematicians developed a strong historical consciousness and also expressed their opinions on philo-
47 See A. Brigaglia, G. Masotto, Il Circolo Matematico di Palermo, Bari, Dedalo, 1982.

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sophical issues. This happened thanks to the presence of a strongly interconnected general culture. The rigid separation channels which would bound the disciplines in 20th century thought had not yet been fully excavated.
The most prominent Italian mathematician in this sense was Enriques (although he was not the only one). The first examples of his interest in philosophy date back to the mid -1890s, if we except his first encounter with it as a secondary school student. However, it is in the 20th century that the activities of Enriques as a philosopher acquired public significance. In 1906 he published a volume titled I problemi della scienza. He began by philosophically analyzing the construction of geometrical systems and the problem of space. Enriques faced several problems which had also been studied by mathematicians such as F. Klein and H. Poincaré: what is the nature of geometrical postulates? How can the different geometries be explained from this perspective? Enriques stressed the importance of intuition and of the interaction among real space, space intuition and geometry postulates, refusing to consider the latter as a purely formal system. He saw geometry postulates as conceptual abstractions, but based on the different ways in which space is perceived. That same year Enriques founded the Società filosofica italiana (SFI), becoming its president. In 1907 he founds the review Rivista di scienza; in 1911 it would adopt the name Scientia turning into an international journal of scientific synthesis, in an attempt to counter tendencies towards excessive specialization. In 1907 he participated in the second congress of the SFI presenting a paper titled: “Il rinascimento filosofico nella scienza contemporanea”. In the next congress, he even approached Hegel in a paper titled: “La metafisica di Hegel considerata da un punto di vista scientifico.” By now it had become clear that his work could no longer be ignored by “professional” philosophers, in particular by Benedetto Croce (1866–1952) and by Giovanni Gentile (1875–1944) who at the time were the leading exponents of Italian idealism. Already at the beginning of the century, they had become exponents of a plan to extend their philosophical hegemony over the culture of the whole country. The event for the redde rationem was to be the fourth International Congress of Philosophy (1911). Since it was to be held in Italy it was organized and chaired by Enriques (in his capacity as president of SFI). The clash with Croce and Gentile began immediately, during the preparation of congress events. The congress then went smoothly. It was only once it was finished that Croce publicly attacked Enriques, in a newspaper interview, by directly accusing him, coupling ironic comment with harsh judgment, of being an amateur and for encroaching on a field which he knew nothing about. Croce’s severe criticism was emblematic: by declaring its incomprehension and hostility, ‘official culture’, or rather that more closely rooted in the traditions of the country, handed down its negative sentence (destined to “count” for many decades to come) on the enthusiastic attempt by mathematicians to link their extremely qualified professional capacities to active participation in the cultural and social life of the country.
Although the Croce – Enriques controversy is perhaps the most well- known event of the period, the most “political” incident saw the participation of Volterra, with the establishment of the already mentioned SIPS. This association was founded with a double objective, which we have already noted in regard to Mathesis. The internal objective addressed the scientific community by advocating consciousness of one’s intellectual

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role. Although specialization in academic research was considered as a positive necessity, it should not lead to fragmentation and isolation into small sectors, inspired only by technical perspectives. Consciousness of a greater mission to fulfil, as well as a more attractive image (which was to be achieved by publicising the character and work of scientists) were considered the prerequisites for applying strong pressure to combat the inertia of the political establishment, encouraging it to recognize the usefulness of science by according it a rightful place in society. This was the second objective of the SIPS: to participate in the development of a modern country, which recognized the social function of science by following in the footsteps of the more developed European countries. This message was lucid and strong. Volterra suggested that both he and the scientific community should take on a leading role in the development of the country by expressing a model of rationality and organization powerful enough to control and resolve the contradictions of its own growth. We must keep this in mind when describing Italian mathematics in the years between the two world wars.

Chapter 2
Nothing is as it was before
1. Introduction
In the Prologue we introduced Italian Mathematics as a young discipline, but certainly growing fast. At the beginning of the 20th century it was extraordinarily exuberant. Its contributions to different research fields, the level it had reached in international ranking, and, again, the quickness with which such a position had been achieved (starting from a relative obscurity), were all strongly positive elements.
Mathematics was beginning to clearly distinguish its different research areas, so we must be very careful in the difficult task of identifying a unique leader with maximum influence and authority. And yet, in Italian mathematics the figure of Volterra stands astride the 19th and the 20th centuries. His scientific authority in analysis and in mathematical physics, his international contacts, his prestige even outside national boundaries and, finally, his public activity, turned him into the main icon in the Italian mathematical world. Volterra’s work was the best expression of the so-called 19th century tradition, whose brilliant examples have illumined the story of Mathematics. His physical-mathematical approach was traditional, as was the relationship between the physical world and the mathematical formalism, but he showed as well a remarkable skill in pushing this tradition towards forms of a great modernity (we have seen this skill at work in functional analysis and in the theory of integral equations). Volterra represented the most advanced edge of tradition, both in science and in his values and cultural-political position: “enlightened” conservator, keenly fond of the Risorgimento, from which he took his faith in the scientific internationalism – he developed intense relationships mainly with the French mathematical world – and the sensibility to understand the social role of science. Of course, he was also – as we have seen – a man of power who in the years of our study would further develop his public dimension. Beside him, but independently, grew a generation of younger researchers who, at the beginning of the 20th century, left their stamp especially in the real analysis areas: mathematicians such as Tonelli, Vitali, Fubini, etc.
The other pole of Italian Mathematics at the beginning of the 20th century was algebraic geometry and the triumvirate Castelnuovo, Enriques and Severi, whose author-

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ity and scientific prestige (even at an international level) deserve recognition similar to that given to Volterra. Castelnuovo was the oldest one, but in the years considered in the Prologue he had actually not reached his fifties yet. In the period between the two World Wars he will be a researcher (and supporter of the studies) of probability and an authorial exponent of the Roman mathematical group, though in a less central position. Enriques and Severi will have a bigger role. The former, Castelnuovo’s pupil and later his acquired relative1, had already displayed his intellectual talent in those years. He was an extremely intelligent, cultivated, brilliant man, and set quite naturally a working style and manner concentrated on “great ideas”, to the detriment of what he considered simple details. After his controversy with Croce and Gentile he was also known to a wider public. In a more moderate way than Volterra, he also undertook collective enterprises – the example of Scientia is enough – but always strictly cultural ones. In contrast, Severi, who did not hide his socialist ideas from the local administration benches of Padua, was interested in politics and in more general contexts. It was easy to see in Enriques’ pupil a rising star. His relationship with the master was still good, even if already strained by an “incident” which showed that Severi was “champing at the bit”. He felt shackled not only because of the politics of Padua. Some events, such as those regarding the Associazione Nazionale Insegnanti Universitari and Severi, who became its president, following Enriques, some years before the war, could be interpreted symbolically too.
Neither is the young Levi-Civita to be forgotten, whose memoir of 1901 and whose contribution to the problem of the three bodies and to relativity theory had attracted international attention. Besides, his correspondence with Einstein confirmed the importance of his research2. Levi-Civita’s character was different from Volterra, Enriques and Severi. He came from a progressive educated bourgeois family and would never hide his socialist stance. But he would never mix the political sphere with the professional one – as almost any Italian mathematician of this generation would – neither would he add other commitments to the scientific and academic one. Levi-Civita would support his political ideas – very resolutely – but in a private sphere. He was a meek and quiet character (who would gradually show traits of a great humanity) but he could defend his own beliefs with determination.
Finally, we must remember that the liveliness of Italian Mathematics at the beginning of the 20th century existed not only within but as projections into another disciplines, indeed as “field invasions” that were characterized by their originality; Enriques’ invasion into the philosophical culture was the most clamorous of them. This phenomenon requires some further comment.
The golden age of Italian Mathematics actually ended with World War I. Some warning signals could have been seen before, perhaps. These were not just isolated and specific events, such as the controversy between Enriques and Croce where because of
1 Castelnuovo had married one of Enriques’ sisters. 2 The correspondence is reproduced in P. Nastasi, R. Tazzioli, Calendario della corrispondenza di Tul-
lio Levi-Civita (1873–1941) con appendici di documenti inediti, Palermo, Quaderni Pristem, No. 8 (1999), pp. 204–238.

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31

the reaction of the philosophical world the former was seen to have been defeated by the latter. Rather it was a question of raised expectations, lack of confidence in commitment, and few proper results from attempts to reach out. Disappointment was spreading, as well as fatigue. Commitment to and enthusiasm for intellectual progress were decreasing.
The war was, after all, a real turning point for a whole society. This was not a war between armies but between peoples. Violence and social upheaval became the first dramatic experience directly lived by millions of people. The end of the liberal age came with the October Revolution and the American intervention, which broke off the USA’s long isolation period. The tsarist regime and the German, Austro-Hungarian and Ottoman empires collapsed. Twenty-six new nations arose. Relations among the main industrialized countries changed irreversibly. New York and the dollar replaced London and the pound sterling. Stability of prices and of the value of gold and silver coins was only a memory. States looked for larger sharing in their economies; business executives saw their power strengthened, while Parliaments and other governmental bodies lost it.
After World War I, nothing was as before. It was impossible to set out again with a simple heri dicebamus.

2. Italian mathematicians take sides.
The great war broke out in summer 1914, when the Austro-Hungarian empire declared war on Serbia: on the 28th June the archduke Francis Ferdinand of Austria and his wife had been murdered in Sarajevo. On the 1st August Germany opened hostilities against Russia; two days later it invaded Belgium and “inaugurated” the French front. On the 4th of August it was England’s turn to declare war on Germany.
At the time Italy abstained. It would go to war a year later, on the 24th May 1915. In those months the debate in the country was extremely fervent, as one can easily guess. Pacifists and interventionists from different places and with different attitudes confronted each other with great vehemence.
The mathematical world was quite homogeneous, and, on the whole, sided with the Allied powers in favour of a democratic interventionism, against the “German barbarity”. The work that had aimed at developing a tighter and tighter network of relationships between Italian and French (but also English and North American) mathematics, especially interwoven by Volterra, had begun to pay off. It is not by chance that Volterra himself was the most committed one in the world of mathematics, to urge an explicit alignment with France, England and Russia. He wrote to Gaston Darboux3:
Très honoré Monsieur e cher Maître
j’ai reçu de plusieurs côtés vos nouvelles et j’ai appris de la part de M. Appell que vous êtiez dans les Pyrénées et que vous êtes rentré à Paris dès le commencement de la guerre. Permettez-moi de vous dire que ma pensée est toujours tournée, avec
3 Accademia dei Lincei, Rome; Archives Volterra.

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le plus profond attachement, vers vous, vers les maîtres, les confrères, les amis que j’ai en France. Votre noble et grande nation lutte pour la cause de la justice et de la civilisation. Tous mes voeux sont per le succès et le triomphe de la France. L’acte par lequel les deux empereurs ont déchainé la guerre et la destruction en Europe a êté regardé par moi, ainsi que par la majorité de mes compatriotes comme un crime abominable. Les innombrables actions barbares que les allemands ont accomplies pendant la guerre n’ont fait qu’accroître l’horreur et l’indignation du premier moment. A mon avis l’Italie doit prendre sa place à côté de sa soeur latine: la France, et de ses alliés contre l’Autriche et l’Allemagne. C’est son rôle et sa mission. Elle ne doit pas y manquer. J’éspère de tout mon coeur que cela arrivera. Voilà mes voeux et mes espérances. Puissent ces voeux et ces espérances, si repandues dans toutes les régions et parmi toutes les classes sociales d’Italie, être réalisées et puissent nos deux pays être unis toujours davantage pour la liberté et la civilisation.
Volterra’s letter was dated 7th September 1914. A month later, on the 4th October, 93 German intellectuals – among which were Felix Klein and Max Planck – created and spread a manifesto to defend with very resolute tones the reasons for their own patriotic commitment, against what they called the distortions of western public opinion. Volterra received the manifesto from O. E. Staude, Klein’s pupil in his teaching period in Lipsia, and soon after he received a similar opposite declaration from his French colleagues. On the 16th of October, for example, É. Borel wrote to him4:
Mon cher ami,
vous avez sans doute lu l’appel adressé par les intellectuels allemands au monde civilisé. Je désirerais publier dans la Revues du Mois quelques-une des appréciations ou réponses relatives à cet appel, dues à des neutres. Il me semble que il sérait préférable, si possible, d’avoir des textes déjà publiés dans les pays neutres, et non pas écrites spécialement à l’instigation de français ou d’anglais. Pourriez-vous me signaler et me procurer au besoin des textes de ce genre parus en Italie? Nous voyons bien ici quelques journaux italiens, mais pas tojours régulièrement et nous sommes parfois forcés de nous borner à y lire les nouvelles directes de la guerre de source allemande ou autrichienne, qu’il est toujours intéressant de connaître, mais qui ne nous enlévent pas notre confiance dans le succès de notre cause.
Volterra answered quickly (24th October5), confirming his whole support to the Allied cause.
4 Accademia dei Lincei, Rome; Archives Volterra. 5 The letter, unpublished, is kept in the Archives Volterra at the Accademia dei Lincei in Rome.

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Mon cher Ami
Je viens de recevoir votre très-chère lettre du 16 Octobre et je m’empresse à vous repondre. Je vous ai écrit vers la moitié de Septembre une lettre où je vous demandais des nouvelles de M. Gateaux6, de M. Pérès, de M. Boutroux et d’autres jeunes amis Français. Je n’ai reçu aucune réponse et, comme vous ne faites aucune allusion à cette lettre, elle doit avoir été perdue. A un mois de distance je ne peux que confirmer ce que je vous écrivais alors, c’est à dire mes félicitations pour les succès de la France, ma plus vive sympathie pour votre noble pays qui lutte pour la justice et la liberté et pour la cause de la civilisation contre la violence du plus brutal et odieux impérialisme. Je vous disais que le rôle de l’Italie est, à mon avis, celui de s’unir à la triple entente. Je puis ajouter aujourd’hui que la confiance dans cette union n’a fait qu’augmenter, car la sympathie pour la France l’Angleterre et la Russie s’est accrue chéz-nous. D’autre part la persuasion que tous nos intérêts au point de vue moral ainsi que politique sont en opposition avec l’Autriche et l’Allemagne n’a fait que se raffermir. Vous avez raison de désirer un article déjà publié. Je vous envoie un article remarquable qui a paru dans la “Tribuna” du 6 Octobre, le jour après celui où l’appel des savants allemands a paru. L’article est signé “Rastignac” c’est à dire Vincenzo Morello très-connu dans notre monde littéraire et un des meilleurs écrivains parmi les journalistes Italiens. Il est aussi très-apprécié comme auteur dramatique. Je crois que l’article de Rastignac est justement ce que vous cherchez et ce qu’il vous faut. Je chercherai cependant s’il y en a aussi d’autres qui pourraient vous convenir. Nous avons eu une foule d’articles de toute sorte qu’on a appelé la “polemica nazionale” sur la question de la guerre et beaucoup de protestations contre les barbaries des allemands, leurs violations des conventions de la Haye et des traités ainsi que sur les distructions qu’ils ont accompli en France et en Belgique. A ces protestations se sont associés des Universités des Académies des hommes politiques des savants etc. J’ai toujours adhéré à ces protestations, mais il est presque impossible de suivre et de recueillir toutes ces protestations qui sont répandues un peu partout dans les journaux de Rome et de la province et qui ont paru la plupart sous forme des télégrammes et d’ordres du jour. Vous savez sans doute que M. Richet7 a été très-fêté chez-nous. Ses conférences, auxquelles j’ai assisté avec beaucoup d’intérêt, ont eu un grand succès et un grand nombre d’auditeurs. Il a aussi très-bien réussi dans la polémique contre l’appel des savants allemands. Nous avons adhéré à ses protéstations.
6 René Gateaux died during the first months of the war. Just in February 1914 he had held a conference titled: “Une face du développement du calcul fonctionnel”, at the Seminario Matematico di Roma. Volterra would personally commemorate him on the 19th December 1914 during a session at the same Mathematical Seminar.
7 Charles Robert Richet (1850–1935), Nobel prize for Physiology in 1913. As it can be inferred from the text, his conferences (one of them titled: “Science and civilization today”) encouraged Italy’s entering into the war on the Allies’ side.

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Vous avez parfaitement raison d’être sûr du succès de la cause de la France et des alliés. Tout le monde civilisé est contre l’Autriche et l’Allemagne. Je suis parmi ceux chez-nous, qui sont les plus impatients de sortir de la neutralité, mais je ne doute pas que même ceux qui ont un peu moins d’impatience ne peuvent pas manquer d’avoir les mêmes espérances et les mêmes aspirations que moi. Mon voeux aujourd’hui est le même que celui que j’ai fait dépuis le commencement de la guerre. L’Italie, la France et ses alliés doivent être unies contre l’ennemi qui a accompli le crime de déchaîner la guerre et qui voulait asservir l’Europe.
And some days later (on the 29th October)8:
Mon cher Ami
J’espère que vous aurez reçu ma lettre du 24 Octobre en réponse à votre lettre du 16 Octobre et le journal que je vous ai envoyé dans un pli à part. Je vous envoie aujourd’hui un autre article qui est paru dans le “Messaggero” le plus populaire de nos journaux. L’article est de M. Edoardo Cimbali professeur de Droit International et il est aussi une réponse à l’appel des intellectuels allemands. J’espère vous envoyer aussi quelque autre article. En vous exprimant encore une fois mes voeux les plus chaleureux pour le succès des armées alliées et pour l’union de nos deux pays, je vous envoie l’expression de tout mon dévouement et de l’amitié la plus sincère.
English intellectuals mobilized against German propaganda too. Already on the 21st of October, 150 English scholars drafted a counter-manifesto which denounced Germany as “the common enemy of Europe and of all peoples”. Volterra confirmed his position immediately (in a letter to the physician Joseph Larmor)9.
Ho ricevuto il suo biglietto e la ringrazio dei suoi auguri che ricambio cordialmente. E vivamente li contraccambio e li estendo anche a tutto il vostro grande paese verso il quale le simpatie già così grandi presso di noi sono ancora maggiormente accresciute nelle circostanze attuali. Ho ricevuto da Sir Archibald Geikie vari opuscoli relativi alla guerra in cui si parla delle ragioni che hanno spinto l’Inghilterra nel conflitto, e della sua condotta verso il Belgio. fra essi vi è anche la lettera colla quale gli scienziati inglesi hanno risposto ai tedeschi. Ho già risposto a Sir Archibald Geikie che per parte mia divido pienamente le idee manifestate dagli scienziati inglesi nella loro risposta e che è viva in me l’ammirazione per la condotta dell’Inghilterra sia verso il Belgio sia nel voler salvare l’Europa dall’aggressione dei due imperi tedeschi. E che la guerra sia derivata da una
8 Accademia dei Lincei in Rome; Archives Volterra. 9 Accademia dei Lincei in Rome; Archives Volterra.

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aggressione voluta e preparata dalla Germania e dall’Austria lo mostrano tutti i documenti comparsi. Tali idee sono lieto che siano divise dalla gran maggioranza dei miei compatrioti e io non dubito che esse serviranno di norma alla condotta dell’Italia il cui intervento io mi auguro possa condurre a notevoli conseguenze. Sono ben lieto di inaugurare l’anno 1915 col voto che i vincoli fra i nostri due paesi si facciano sempre più stretti10.
Castelnuovo and Enriques had the same stance as Volterra, though not so overt. The discussions and controversies of the years 1914–15 about Italy’s position in the war brought Enriques to resign from Scientia, the journal he had founded in 1907 and that he directed with Eugenio Rignano. It all started with an inquiry upheld by Rignano himself, that considered it unacceptable for a scientific journal “to shut itself up in the ivory tower of the abstract synthesis” and “to remain impassive in front of the tragic reality of the present hour”. The inquiry, which was intended to be an “objective, serene research – that is a scientific one – of the war causes and factors”, slipped though towards positions that – at least so thought Enriques – risked straying from the initial objectives. An article by Rignano, I fattori della guerra ed il problema della pace, was specially worrying; in it the author – Italy had already entered the war with the Allies – had no hesitation in speaking freely about the imperialistic aims of the England, its allies, and their responsibility for the war. Enriques asked Rignano explicitly to withdraw the article, which was not an expression of a free scientific opinion but a real political act, and could offend patriotic feelings. Faced with Rignano’s refusal, Enriques precipitously left the editorship – together with many of the founders – to go back to it only in 1930, after Rignano’s death.
Enriques’ position on war did not emerge from the letters he wrote then to Rignano, but it can be easily inferred from his correspondence with Xavier Léon, director of the Revue de Métaphysique et de Morale11. At Easter 1914, Enriques and Léon had
10 I received your note and I thank you for your wishes, which I heartily return. I warmly return and extend them also to your great country, towards which our great sympathy has, in the present circumstances, further increased. I have received from Sir Archibald Geikie some pamphlets concerning the war, about the reasons which have driven England to war, and about its behaviour towards Belgium. Among them there is also the reply of English scientists to Germans. I have already told Sir Archibald Geikie that I fully share the ideas English scientists display in their reply, and that I deeply admire England’s behaviour both towards Belgium and towards its will to save Europe from the aggression of the two German empires. And the fact that the war was due to a wilful aggression prepared by Germany and Austria has been demonstrated by the shown documents. I am glad these ideas are shared by most of my compatriots and I do not doubt that they will guide the behaviour of Italy, whose intervention will lead to remarkable consequences, I hope. I am really pleased to begin the year 1915 with the wish that the bonds between our two countries become tighter and tighter.
11 The letters, which come from the Léon “papers” kept at the Bibliothèque Victor Cousin of the Sorbonne in Paris, have been published in L. Quilici, R. Ragghianti: Il carteggio Xavier Léon: corrispondenti italiani, Giornale critico della Filosofia Italiana, 1989, p. 295–368.

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organized in Paris a congress of Philosophy of Mathematics, held a latere of the Commission on the Teaching of Mathematics (presided over by F. Klein, with Henri Fehr as secretary). The congress had approved the proposal of creating an International Society of Mathematics’ Philosophy, was actually swept away by the war. On the 25th August 1914 Enriques wrote to Léon:
Cher Ami
Je ne veux pas rétarder plus longtemps à vous exprimer mes sentiments de sympathie chaleureuse pour votre pays dans cette heure tragique pour l’Europe. Je me trouvais en Suisse, à Zürich, lorsque l’orage est éclaté (…). J’ai passé des heures d’angoisse avant qu’on eut proclamé la néutralité italienne; en rentrant en

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Italie j’ai compris que toute autre décision du gouvernement aurait été impossible, puisque le sentiment de tous les Italiens, de toutes les classes et de tous les partis, est unanime contre les aggresseurs. Si vous pourriez voir avec quelle anxieté on attend ici les nouvelles de la guerre et comment le peuple entier fait de voeux pour le salut de la France, vous en seriez touché. D’ailleurs on a ici le sentiment que l’acte d’indépendance accompli visà-vis de l’Allemagne, va nous coûter cher au cas où les Allemands seraient les vaniqueurs. On est préparé à être attaqué à notre tour; mais si la paix ne peut être mantenue, que nous nous trouvions du côté de la civilté et du droit! C’est la pensée intime du peuple italien tout entier, dont la calme et le pacifisme ne cachent en somme que le propos de contribuer nous aussi – lorsque l’heure sera sonnée – à l’oeuvre de libération. Cher Ami, veuillez participer mes sentiments aux communs amis auxquels s’adresse une pensée au moment où la France offre au monde un si beau spectacle d’unité, de fermeté et de dignité. Comme vous, j’ai confiance dans le succés final, coûte ce qui coûte.
And some months later, on the 4th February 1915, he added:
Quant à l’évenement de cette guerre, personne ne sait bien à quoi s’en tenir. Vous savez quelles sont nos sympathies et nos aspirations, il n’y a peut-être qu’un petit nombre de personnes qui ne les partagent (malhereusement la phlilosophie hégélisante ne se fait pas honneur, elle est du petit nombre des sympathisant pour l’Allemagne). Mais je crois que la presse italienne ne donne pas un’idée juste de la situation lorsqu’elle semble ne s’occuper guère du traité d’alliance. Personne ne connaît bien nos engagement, mais il est à craindre que le gouvernement ne soit pas entièrement libre, sauf dans le cas où les Allemands eux-mêmes commetteraient la méprise d’attenter à nos droits ou de nous ménacer.
The reference to “hégélisante” philosophy, discredited because of its pro-German sympathies, was obviously directed against Benedetto Croce. In an interview given to the Corriere d’Italia, on the 13th of October 1914, when the journalist asked whether he had, “in Italian and foreign journals, kept up with controversies about the relationship between Italian culture and French and German thought”, the Neapolitan philosopher curtly answered that he considered those controversies “manifestations of the state of war. It is no more a matter of rational questions, but of clashes between passions; not of logic solutions, but of assertions of interests, which, even if rather high, are national, that is, particular; not of reasoning, but of fake reasoning, built by imagination”12. There was also an explicit attack on that expression – German barbarity – on which Volterra and the democratic intellectuals so much insisted:

12 Cf. B. Croce, Giudizi passionali e nostro dovere, in B. Croce, L’italia dal 1914 al 1918. Pagine sulla guerra, Laterza, Bari, 1950, pp. 11–12.

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Credo che, a guerra finita, si giudicherà che il suolo d’Europa, non solo ha tremato per più mesi o per più anni sotto il peso delle armi, ma anche sotto quello degli spropositi. E Francesi, Inglesi, Tedeschi e Italiani si vergogneranno e chiederanno venia pei giudizî che hanno pronunciati, e diranno che non erano giudizî ma espressioni di affetti. E anche più arrossiremo noi, neutrali, che molto spesso abbiamo parlato, come di cosa evidente, della “barbarie germanica”. Fra tutti gli spropositi, frutti di stagione, questo otterrà il primato, perché certo è il più grandioso13.
About the war, Severi’s ideas are similar to those of Volterra and Enriques. He could be placed in that sector of political thought which historiography called revolutionary interventionism, that joined people with interventionist positions belonging to different revolutionary movements (socialists, anarchists, trade-unionists, republicans, etc). Severi was an official socialist, as he defined himself in a speech that appeared in the journal L’Adriatico on the 9th March 1915. This public stance laid down the beginning of his moving away from the socialist party, siding increasingly with neutralist positions.
Io spero e credo che l’atteggiamento degli organi direttivi del mio partito, in questo grave momento, sia l’espressione dello stato di angoscioso dissidio in cui ogni socialista d’intelletto e di cuore si trova fra gli imperativi ideali della propria fede e la percezione delle necessità ineluttabili dell’ora presente; piuttosto che indice di un proposito d’azione maturato e metodicamente perseguito. Ma se così è, e se è pur vero, secondo io penso, che il partito socialista, come organismo politico, non potrebbe mai farsi promotore di un intervento guerresco, assai meglio parmi si provvederebbe, se la protesta socialista contro la guerra fosse, in ogni occasione, contenuta nel campo puramente ideale, riconoscendo nello stesso tempo la ineluttabilità d’una situazione che non ci è dato oggi di modificare, appunto perché deriva da condizioni sociali che il partito nostro non può cambiare di colpo. Porsi da un punto di vista di assoluta negazione di problemi che esistono e che reclamano una soluzione indifferibile, significa lasciarsi cullare dalla ingenua illusione di poter violentare lo svolgersi dei fenomeni storici, e venir quindi, in ultima analisi, a contraddire a quello che è lo spirito animatore della dottrina socialista. Un atteggiamento meno assoluto della Direzione del nostro Partito, sarebbe importantissimo anche dal punto di vista politico, giacché lascerebbe ad ogni inscritto la libertà di valutare gli elementi reali della situazione, secondo la propria coscienza di cittadino italiano, e nello stesso tempo consentirebbe ad ognuno di noi di conti-

13 I am sure that, when the war ends, it will be said that European ground has shivered during several months or several years not only under the weight of weapons, but also under that of blunders. And French, English, German and Italian people will shame and beg pardon for the judgements they gave, and will say that they were not judgements but expressions of feelings. And we neutrals, who so often have talked, as of an evidence, about the “German barbarity”, will blush even more. Of all blunders, product of the time, this will hold the record, because it is certainly the most striking one.

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nuare la propaganda socialista fra le masse, additando loro quali disastri immani conseguono dall’ordinamento capitalistico della società. Io, che sono convinto della necessità dell’intervento dell’Italia a fianco della Triplice Intesa, sento di non aver mai provato un odio così implacabile contro la guerra – la quale non crea, ma sfrutta valori morali già esistenti; – né di aver mai desiderato, con altrettanto ardore, profondi rinnovamenti sociali, come da quando assistiamo alla spaventosa ecatombe di vite umane, all’enorme distruzione di ricchezza, all’acutizzazione del disagio economico del proletariato, al dispregio del diritto e delle bellezze dell’arte che la guerra europea trascina con sé. Ma come non v’ha uomo cui la violenza ripugni, che ad essa non possa contro ogni sua voglia essere costretto; come non v’ha socialista che, vivendo e vestendo panni in questa società borghese, non s’adatti, nelle pratiche contingenze della vita, a ciò che l’ambiente gli impone senza che per questo egli rinunci a dar l’opera sua per un migliore domani, così non trovo vi possa essere contraddizione sostanziale fra la fede nei nostri ideali e l’azione che oggi cagioni storiche superiori alla nostra volontà possono prescriverci. Vi sarà è vero, per chi ami dilettarsi in così tragico momento di quisquilie dialettiche, una contraddizione formale; ma sciaguratamente le più angosciose situazioni sentimentali si sciolgono di rado alla stregua della logica pura. Eppoi il partito socialista non ha forse riconosciuto che nella pratica quotidiana conviene adattarsi ad un programma minimo e non evitare talvolta contatti con le frazioni più illuminate della borghesia, quando occorra, ad esempio, contrastare la vittoria di partiti i quali minaccino di prevalere in modo pericoloso per le libertà politiche, che costituiscono il presupposto delle conquiste economiche del proletariato? E perché dovremmo racchiuderci in una formola d’intransigente negazione, proprio in una questione che di gran lunga trascende la importanza della minuscola politica d’ogni giorno, e che è in fondo ancora una questione vitale di libertà? Giacché è ben vero che le cause di questa guerra sono giustamente capitalistiche, ma non si può disconoscere che, sia per le brutali violazioni del diritto naturale dei popoli compiute dalla Germania, sia per l’esistenza di molte questioni insolute, sia infine per l’interesse di alcuni belligeranti, e soprattutto dell’Inghilterra, affinché vengano rispettate le nazionalità minori (“L’interesse e il dovere spingono l’Inghilterra nella stessa direzione”, hanno scritto i professori dell’Università di Oxford), la guerra è andata acquistando, in modo prevalente, il carattere d’un conflitto fra due opposte concezioni dei diritti e delle forze, che debbono prevalere nel mondo moderno. Inoltre, secondo la lettera e lo spirito della dottrina marxista, il socialismo potrà e dovrà succedere agli attuali ordinamenti, soltanto allora che la civiltà sia passata per tutte le fasi del suo sviluppo, tra le quali vi è appunto la conquista delle unità e delle autonomie nazionali. Di guisa che, per dirla con una frase scritta in questi giorni nell’Avanti da Enrico Leone, la Nazione diventa la porta d’ingresso dell’Internazionale. E quando si parla della Nazione non ci si appiglia ad un “diversivo borghese”, poiché la Nazione è una formazione storica naturale, la quale vive nelle tradizioni

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The retreat (1917)
di lingua, di arte, di cultura, di ciascuna razza, e sta al disopra e al di fori delle iniquità degli ordinamenti statali. Riconosco come vi siano purtroppo molti, i quali, per le condizioni di inferiorità culturale e materiale in cui si trovano, non certo per colpa loro, non possono sentire tutto il valore spirituale dell’idea di Nazione; ma essi comunque non dovranno disconoscere che il dominio straniero rappresenta sempre un altro sfruttamento, da Nazione a Nazione, che viene ad aggiungersi allo sfruttamento del capitalista sul salariato. Eppoi in qual modo si concreterebbe la solidarietà internazionale se, fino a quando non sarà più diffusa la coscienza della disastrosa follia degli armamenti, di fronte a tentativi di sopraffazione imperialistica a danno di altri popoli, non si fosse disposti anche a sacrifici di sangue? D’altronde i socialisti, predicando l’avversione alle spese militari, hanno sempre presupposto la sincerità e l’efficacia della propaganda antimilitarista negli altri paesi, ed hanno inteso con ciò di cercar di diminuire la possibilità di conflitti armati fra i popoli, ma non già di negare le idealità nazionali. Allorché la Patria sia in pericolo, ancor più impellente sorge quindi per noi socialisti il dovere di difenderla, avvalorando agli occhi di chi ci considera utopisti, la nostra persuasione che dalla coscienza di un buon diritto possa – ove occorra – sprigionarsi la più grande delle forze. Ed io credo per certo che sul riconoscimento di questo dovere, la stragrande maggioranza dei socialisti italiani sia senza esitanze concorde, anche se qualche eccesso polemico possa a taluno far supporre il contrario. Non imprigioniamoci dunque nell’adorazione di formule assolute, giacché il pericolo per il nostro Paese

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è insito nella grave situazione internazionale, la quale potrebbe trascinarci più tardi, anche nolenti, ad una guerra disastrosa per l’Italia e più particolarmente per il Veneto.

Prof. Francesco Severi, dell’Università di Padova

P.S. – Ragioni varie hanno fatto ritardare per circa due settimane la pubblicazione di questa mia lettera. Non ho ora nulla da mutare, ma di fronte al fatto – segnalato anche ieri in queste colonne dall’amico Gino Piva – che le condizioni economiche del proletariato veneto vanno di giorno in giorno aggravandosi, in modo veramente doloroso e allarmante, desidero di aggiungere una parola di viva deplorazione per l’inerzia del Governo, il quale sembra non abbia capito e non capisca che la preparazione non deve limitarsi alle sole provvidenze militari. Come si potrebbe sperare che le masse popolari offrissero la necessaria resistenza morale e materiale, se la nostra regione dovesse essere assoggettata, dall’intervento dell’Italia nel conflitto europeo, ad altre e ben più dure prove? Provvedimenti eccezionali (lavori e sovvenzioni dello Stato ai Comuni) urgono qui nel Veneto per fronteggiare la grave crisi. Altro che proibire i comizi!

9 marzo 1915

F.S14.

14 I hope and believe that the behaviour of my party’s directive organs, in this serious moment, is the expression of the state of painful disagreement in which each socialist by intellect and by heart finds himself, between the ideal imperatives of his faith and the perception of the inescapable needs of the present moment; rather than sign of a matured and methodically persecuted intention. But if it is so, and if it is also true, as I think, that the socialist party, as a political organ, could never promote a warlike intervention. It would be better, I think, if the socialist protest against war was always limited to a purely ideal field, while acknowledging the ineluctability of a situation which can not be modified today, since it arises from social conditions that our party cannot suddenly change. The absolute denial of existent problems which claim an urgent solution means to cherish the ingenuous illusion that it may be possible to force the development of historical events, and therefore to contradict, in the end, the inspiring spirit of socialism. A less absolute attitude of our Party Direction would be very important also from a political point of view, because it would leave to each member the freedom to value the real elements of the situation, according to the conscience of each Italian citizen, and at the same time it would allow to each of us to continue the socialist propaganda with the masses, showing them to which immense disasters leads the society’s capitalistic order. Even if I am sure of the need of Italy’s intervention on the side of the Triple Entente, I have never experienced such an implacable hate towards war – which does not create moral values, but exploits the already existent ones – nor have I ever wished deep social changes so fervently as since we witness this horrible hecatomb of human lives, this enormous destruction of richness, the acuteness of the economic hardships of the proletariat, the scorn of law and of the art’s beauties that the European war drags with itself. But as there is no one who, disgusted by violence, could not be obliged to it against his will; as there is no socialist who, living and breathing in a bourgeois society, does not, in the practical circumstances of life, adapt to what the environment imposes on him, without renouncing thus to work for a better future, so I do not find any substantial contradiction between the faith in our ideals and the action that present historical reasons, superior to our will, order us.

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14 (continued)

Those who in such a tragic moment delight in dialectical trifles will find a formal contradiction, it’s

true; but unfortunately, awfully distressful sentimental situations release themselves rarely according

to the criteria of pure logic.

And then has maybe the socialist party not recognised that in daily practice it is better to follow an

abridged programme and not to avoid sometimes contacts with the most enlightened bourgeoisie, when,

for example, it is necessary to contrast the triumph of parties which threaten to prevail in a dangerous

way for political freedom, which are the premises for the economic achievements of the proletariat?

And why should we withdraw into an intransigent negation, just in a question that goes beyond the

insignificant daily politics and which after all is still a vital question of freedom?

Since it is absolutely true that the causes of the war are rightly capitalistic, but one cannot deny that,

because of Germany’s brutal violations of the natural right of peoples, because of the existence of

many unsolved questions and, lastly, because of the concern of some belligerents, specially England,

for the respect of minor nationalities (Oxford University professors have written: “Interest and duty

push England in the same direction”), war has mainly acquired the nature of a conflict between two

opposite views of the rights and forces which must prevail in modern world.

Besides, in the letter and spirit of Marxist doctrine, socialism can and must succeed the actual order,

only when civilisation has passed through all stages of its development, among which there is precisely

the achievement of national unions and autonomies. So that, to say it with a sentence written in these

days in Avanti by Enrico Leone’s, the Nation becomes the entrance door of the International.

And when we talk about Nation we are not following a “bourgeois diversionary”, since Nation is a

natural historical formation living inside the traditions of language, arts and culture of each race, and

it is above and outside the iniquities of the state order.

I know that unfortunately many people, because of the conditions of cultural and material inferior-

ity in which they find themselves, certainly not because of their fault, cannot wholly feel the spiri-

tual value of the idea of Nation; but they should not deny, though, that foreign domination is another

kind of exploitation, from Nation to Nation, which adds to the exploitation of the waged by the

capitalist.

And then, how would international solidarity be achieved if, until the consciousness of the disastrous

craziness of armaments is not further spread, we were not ready even to blood sacrifices in front of

the imperialistic attempts to overwhelm other peoples?

On the other hand, socialists, while urging to despise military expenses, have always believed in the

sincerity and efficacy of antimilitaristic propaganda in other countries, and they have tried to dimin-

ish thus the possibility of armed conflicts between countries, but not to deny national ideals. If our

country is in danger, we socialists feel even more impellent the duty to defend it, reinforcing, in front

of those who consider us utopian, our belief that, if necessary, a greatest force can be released from

the consciousness of a good right.

And I firmly believe that the vast majority of Italian socialists will without hesitation support this

duty, even if some polemic excesses may let suppose the contrary. Let’s not get caught, then, in the

veneration of absolute formulas, since the danger for our country is in the serious international situa-

tion, which could drag us later, even unwillingly, to a war disastrous for Italy and specially for Veneto.

Professor Francesco Severi, University of Padova

P.S. – Several reasons have delayed almost for two weeks the publication of this letter. I do not intend

to change anything, but – as my friend Gino Piva remarked yesterday in these columns too – seen that

economic conditions of the proletariat in Veneto grow worse day by day, in a really painful and alarm-

ing way, I wish to express my vivid disapproval for the inertia of the Government, that seems not to

have understood and not to understand that preparation does not mean only military provisions. How

could we hope that popular masses would resist morally and materially to much harder proofs, if,

following Italy’s intervention in the European war, our region were to be subdued?

Extraordinary provisions (State works and subventions to Municipalities) are urgently required here

in Veneto so as to face this serious crisis. Other than banning meetings!

9 March 1915

F.S.

Nothing is as it was before

43

There were obviously divergent positions from this majority line even among mathematicians, but they were mostly embodied in little gestures in professional everyday life or in private and personal communications. For example, in the first months of 1916 Volterra asked Somigliana to invite J. Hadamard to deliver a speech in Turin, and Somigliana was obliged to confess that C. Segre had raised doubts15.
Ne ho parlato a Segre, perché facesse lui l’invito, come preside. Ma mi ha sollevato parecchi dubbi. Intanto vuol sapere quando Hadamard potrà essere a Torino; poi sotto qual forma dovrà esser fatto l’invito. In conclusione la mia impressione è che egli è preoccupato del pensiero di dover fare una qualsiasi dimostrazione che non sia quella di una corretta accoglienza al matematico Hadamard. Purtroppo l’ambiente della nostra Facoltà è così; Segre poi lo intensifica per conto suo. Il concetto predominante è che si debba vivere come nel limbo dei Santi Padri, ignorando la guerra, privi di qualunque antipatia o simpatia per alcuno, salvo il dovuto rispetto ai tedeschi. Ora francamente io penso, che Hadamard sarà venuto in Italia per qualche cosa di più che una semplice esposizione di teorie analitiche; e che il metterlo a contatto con questi elementi potrebbe fargli riportare un’impressione del nostro paese, che non è quella che desideriamo16.
The Paduan socialist Levi-Civita, was, together with Segre, out of the “chorus” – the only real discordant note in the interventionist positions of Volterra, Enriques, Severi, etc. His pacifism would never fade during the whole war, causing a remarkable cooling in the relationship with Volterra: their usually very friendly correspondence, in the war years, took on a formality more eloquent than any speech. Levi-Civita never broke off his relationship with German scholars, and he asserted his neutralism and his pacifism every time the opportunity came. On the 23rd of August 1916, for example, he wrote to G. D. Birkhoff17:
Comme vous l’imaginez aisement, on ressent en Europe, bien plus qu’en Amérique, l’influence deprimente de la guerre sur l’activité et sur la collaboration scientifique: efforts, aspirations, jeunes énergies, et, en général, toute forme d’énergie
15 Accademia dei Lincei in Rome; Archives Volterra. 16 I have asked Segre to make the invitation, as dean. But he has raised several doubts. First, he wants
to know when could Hadamard be here in Turin; then, which form should the invitation take. In short, I think he is worried about having to display more than a right welcome to the mathematician Hadamard. Unfortunately so is the faculty’s atmosphere; Segre on his side intensifies it. The leading idea is that we have to live as in the limb of Holy Fathers, ignoring the war, with no sympathy or antipathy for anybody, except for the due respect for the Germans. Now I sincerely believe that Hadamard has come to Italy for something more than a simple exposition of analytic theories; and that to get him in touch with these elements would give him an undesired impression of our country. 17 The letter, unpublished, comes from the Harvard University Archives, Cambridge (Mass.). We seize the opportunity to thank Brian A. Sullivan, Reference Archivist, for putting at our disposal LeviCivita’s letters to Birkhoff.

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sont sensibilisés par les évênements du jour; et malheuresement aucun symptôme ne laisse pas encore soupçonner une détente prochaine, capable d’assurer la justice en rétablissant la fraternité des peuples.
Noting an excessive enthusiasm towards the Allies’ cause in a paragraph of a letter from Birkhoff dated 3rd December 1916 (probably lost), he replied gentlemanly but resolutely18.
Il m’est bien agréable d’apprendre par votre lettre du 3 Décembre dernier que vous participez au concours de l’Istituto Veneto pour les mathématiques, expirant le 31 Décembre 1917. En m’en faisant part vous ajoutez des considérations qui témoignent de la délicatesse de vos sentiments. Je me regarde autorisé d’or et déjà à les faire connaître si par hasard quelque circonstance imprévue dût m’en montrer l’opportunité. A présent une telle opportunité n’existerait pas. L’Istituto a tenu, jusqu’à l’heure actuelle, à fonctionner, pour tout ce qui se rapporte à son activité strictement scientifique, comme dans les temps normaux. Si les choses continueront à se passer ainsi, il ne me paraît pas le cas de les compliquer avec des déclarations ou des réserves de votre part, malgré la noblesse d’esprit et la sympathie pour la cause des alliés, qui les inspirent.
Just in those months the United States got ready to take part in the war – they would declare war on Germany on the 5th April 1917 – and Birkhoff felt the need to clarify his position with as much elegance and firmness19.
You refer in your letter to my sympathy for the Allies. I would be ashamed of my country if I did not believe that sympathy of the very deepest kind for their cause is felt by almost all Americans. The vote of Congress the other day is a testimony of this fact. Of all my colleagues at Harvard only two not of pure German blood and German born incline toward the other side; and even these keep very quiet. (I might say that I am entirely of Dutch descent, all of my great grand parents being born in the Netherlands.) Unless President Wilson vastly misunderstands American sentiment he will proceed at once to arm our ships and take other necessary steps to uphold our rights upon the seas which the Central Powers have so flagrantly violated. Personally I favor even more rigorous participation on our part. The Germans are a great people of course, but their success would be the defeat of civilization and the best interests of mankind.
On the 10th of April 1917, a few days after the USA entered the war, Levi-Civita closed the argument with a new profession of pacifism20:
18 Letter to Birkhoff, 3rd January 1917, in the Harvard University Archives. 19 The letter, unpublished, comes from the Archives Levi-Civita at the Accademia dei Lincei in Rome. It
is reproduced in P. Nastasi, R. Tazzioli, Aspetti scientifici e umani nella corrispondenza di Tullio Levi-Civita (1873–1941),Palermo, Quaderni Pristem, No. 12 (2000), p. 201. 20 In the Harvard University Archives.

Nothing is as it was before

45

Mon cher Collègue,
J’ai bien reçu votre lettre du 7 Mars. Elle est extrêmement intéressante au point de vue scientifique, et une profession de foi sur la guerre, qui sévit depuis trois ans, clairvoyante, élevée, équitable. Les évenements vous ont donné pleine satisfaction. Il n’est point douteux que les principes, si noblement proclamés par votre Président, trionpheront dans et après la fin de la guerre. On doit seulement souhaiter que ce soit au plus tôt.

3. Mathematicians at the front
World War I was not a technological war, at least in the sense we usually give to this expression from World War II onwards (with the use of missiles and radar technique). It was obviously also a submarine and aerial war, with problems caused by the incredible production of guns and munitions, but it was above all a war of position and of wear and tear. It did not depend as much on the acquisition of chiefly new knowledge as on the ability to organize mass production. So researchers took part in the war not as such, but rather as ordinary citizens and patriots defending their country. They often were sent to the front. As scientists, they acted at the most as referees in the several projects presented by the various “inventors”. In short, the consciousness of the utility of science and modern technology to win a war was raised in the tragic experience of 1914–1918, but not yet their first systematic application. And of course this newborn awareness was quite far from imagining the close relationship between science and defence that would be established twenty years later.
As with the rest of the world, the time of a planned involvement of scientists as such had not come for Italy either. War mobilization developed in a framework of great confusion and disorganization, with spontaneous research into a more rational use of resources. In July 1915, a Comitato Nazionale Esami invenzioni attinenti al materiale di guerra, legally recognized only in March of the following year, was established at Milan’s Polytechnic. This was one of the very few research centres linked to the military apparatus through the testing of steel, wood, cement, compressed gas recipients, projectiles, parts of aeroplanes, etc. Other centres (apart from Medical and Pharmaceutical Institutes) were only the Istituto di Chimica farmaceutica e tossicologica in Naples and the Istituto Geografico Militare in Florence. The former produced the chloropicrin (nitro-chloroform), used as asphyxiating tear gas in the retaliation against the Austrian enemy, which in June 1916 had attacked the Italian positions using poison gas. The second saw to the realization and updating of cartographic material from the war theatres and to the training of artillery officers on the trigonometric-topographic procedures for fire direction.
We mustn’t be surprised at the scarce use of mathematical knowledge. Perhaps the most relevant exception – certainly the most advertised one afterwards – was Mauro Picone (1885–1977), whom we will meet in the next chapters as one of the chief protagonists of Italian Mathematics in the years between the two World Wars. At the outburst of the first war, he was a young educated at the Scuola Normale of Pisa, where he graduated in 1907,

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having studied under Bianchi, Dini and E. E. Levi, among others. He was in Pisa until 1913, as Dini’s assistant. Then he moved to Turin, teaching later at the Universities of Cagliari, Catania, Pisa again, and Naples, where he stayed from 1925 until 1932. Then he moved to Rome, where he settled himself definitively. As we said, in the pre-war years Picone was still a young mathematician. But he already had to his credit numerous publications with important and original contributions to ordinary and partial differential equations (with a few “concessions”, also, to differential Geometry and to the strong influence that Bianchi’s personality had). His most known memoir of the time was the dissertation for his teaching qualification (1910), in which he proved the so-called Picones identity for ordinary linear differential equations of the second order depending on a parameter:

΄ ΅ d

dy

5 dx q(x, l) 5 dx + Q(x, l) y(x) = 0 ,

repeatedly quoted and appreciated because of its simplicity and of the many results to which it led in several situations. This is how Picone himself spoke about his war experience in an autobiography21 published only five years before his death.

Chiamato alle armi, con la mia classe (del 1885), nell’aprile del 1916, fui assegnato al 6° Reggimento di Artiglieria di Fortezza, il cui Deposito era a Torino, col grado di sottotenente della territoriale, senza che io avessi mai prestato, in precedenza, servizio militare e avessi mai visto, da vicino, un cannone. Nel luglio del 1916, dopo aver perso un tempo prezioso a fare la scuola a piedi, fui inviato alla fronte di combattimento e assegnato alla I Armata, operante sulle montagne del Trentino. In ciò il caso, il puro caso, fu fortunato, poiché bastava che il Comando del Deposito di Torino, anziché alla I Armata, mi avesse inviato ad una di quelle operanti in pianura, sull’Isonzo, perché, come si vedrà fra poco, le mie qualità di matematico non avessero avuto modo di rivelarsi subito utili ed io fossi rimasto, forse per sempre, nella concezione puramente speculativa della Matematica. Presentatomi al Comando d’Artiglieria della I Armata, vi fui accolto con un freddo discorso, come questo: “I depositi seguitano a mandarci ufficiali su ufficiali, dei quali non abbiamo bisogno. Non sappiamo, per ora, cosa farne di lei. Torni a presentarsi fra otto giorni. Cosa faceva da borghese?”. Io risposi che ero libero docente di Calcolo infinitesimale all’Università di Torino e me n’andai mogio e deluso. Allo spirare dell’ottavo giorno mi presentai a detto Comando e mi fu comunicato che il colonnello Federico Baistrocchi (…) si era dimostrato interessato ad avere alle sue dipendenze un ufficiale esperto in Calcolo, e che perciò ero stato assegnato a quel Raggruppamento che, con mezzi di fortuna, dovevo raggiungere in giornata. Dopo un viaggio, quanto mai fortunoso, arrivai, a notte inoltrata, al Comando al quale ero stato destinato e fui subito ricevuto dal Comandante, Colonnello Bai-

21 M. Picone, La mia vita, Roma, 1972.

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Mauro Picone in 1917 during the I World War
strocchi, che mi aspettava. (…) Io risposi al Colonnello Baistrocchi, forse anche non riuscendo a celargli il mio stupore, che non possedevo nozione alcuna di artiglieria e, tanto meno, del suo impiego tattico. Ma questi, e con ciò dimostrò di essere all’altezza della situazione, mi disse: “Si tratta di risolvere un problema di calcolo e lei deve essere in grado di farlo, si tratta di calcolare i dati da fornire alle nostre artiglierie d’assedio, per il tiro contro bersagli per i quali le tavole di tiro regolamentari, che esse possiedono, non sono sufficienti”. Ma io, aggiunsi, non ho neppure nessuna nozione di Balistica, sulla quale, suppongo, devono fondarsi quei calcoli. Allora il Colonnello tirò fuori da una cassetta d’ordinanza un ingiallito voluminoso libro e mi disse: “Qui c’è il trattato di Balistica di FRANCESCO SIACCI, le dò l’ordine di studiarlo e di ricavarne, entro un mese da oggi, il calcolo dei dati di tiro per le nostre artiglierie d’assedio, contro i capisaldi dello schieramento nemico”. E mi congedò. Mi misi febbrilmente all’opera, dedicandovi anche la notte, all’incerto lume di una candela e presto riconobbi la giustezza delle opinioni del Colonnello Baistrocchi, pervenendo anche a spiegarmi le difficoltà, nel calcolo dei dati di tiro, incontrate dai nostri artiglieri, che non potevano essere da essi superate. Ecco come stavano le cose. Per il tiro d’artiglieria in montagna era previsto, nel precedente periodo di pace, l’impiego di cannoni del più piccolo calibro, detti appunto da montagna, trasportabili a dorso di mulo sulle più alte creste montane, cannoni che tiravano senza calcolo, a puntamento diretto, laddove, per la possibilità recente di costruire rapidamente solide strade, anche nell’impervio terreno montano, e di impiegare potenti autotrattrici che potevano trainare, anche su strade di forte pendenza, pezzi d’artiglieria di qualsiasi calibro e peso, si pensò – da noi e

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dal nemico – di valersi, anche in alta montagna, del concorso del tiro di cannoni di medio e di grosso calibro e per distruggere le resistentissime fortificazioni dell’avversario ed anche, con un nutrito e preciso fuoco, per precedere, nell’ offensiva, le proprie truppe d’assalto avanzanti o per sbarrare, nella difensiva, il passo a quelle nemiche dinanzi alle nostre linee. Senonché, le tavole di tiro regolamentari, in dotazione alle artiglierie di medio e grosso calibro, fornivano i dati di tiro per bersagli posti nello stesso piano orizzontale della batteria, consentendo lievi correzioni, dei dati stessi, ove si fossero verificati dislivelli, fra batteria e bersaglio, che non dovevano però superare certi limiti. Ora fra le gole del Trentino, questi limiti erano di regola sorpassati, ed anche sovente sorpassati fino a tal punto da essere il dislivello fra batteria e bersaglio dello stesso ordine di grandezza della loro mutua distanza orizzontale. Ciò constatato, potei facilmente determinare le cause dei disastri provocati dal tiro delle nostre artiglierie, che veniva, spesso, per fatali inevitabili errori di calcolo, centrato sulle nostre difese, anziché su quelle dell’avversario. Occorreva, senza indugio, rifare, con criterii tutti diversi, le tavole di tiro per le dette artiglierie, fondandosi su taluni perfezionamenti non immediati della Balistica razionale classica, ciò che non poteva essere conseguito che da un matematico. Li ottenni nel mese prescrittomi e a cominciare dal successivo mese di settembre 1916 tutte le artiglierie del 21° Raggruppamento d’assedio tiravano correttamente con dati calcolati da me. (…) Si può immaginare, dopo questo successo della Matematica, sotto quale diversa luce questa mi apparisse. Pensavo: ma, dunque, la Matematica non è soltanto bella, può essere anche utile22.
22 Called-up with my class (1885’s) in April 1916, I was assigned as second-lieutenant of the Territorial Army to the 6th Regiment of Fortress Artillery, whose depot was in Turin. I had never served before nor seen a cannon close by. In July 1916, after having wasted a precious time going to school on foot, I was sent to the front and assigned to the I Army, working on the mountains of the Trentino. This was a happy chance, a happy sheer chance, because if the Command of Turin’s depot had sent me to one of the Armies working on the plain, over the Isonzo, instead of sending me to the I Army, my qualities as mathematician, as we will see soon, would not have had the possibility of turning out useful immediately and I would had been left, maybe forever, with a purely speculative conception of Mathematics. When I reported to the I Army Artillery Command, I was received with a cold speech, such as: “The depots go on sending us official after official, whom we don’t need. We don’t know what to do with you, now. Come back to report in eight days. What did you do as civilian?”. I answered that I lectured in Infinitesimal Calculus at the University of Turin, and went away dejected and disappointed. By the end of the eighth day I reported to the said Command and was told that Colonel Federico Baistrocchi (…) was interested in having at his service an official expert on Calculus, and that for this reason I had been assigned to that Group, which I had to reach that same day by whatever means of transport was available. After a quite eventful journey, I arrived, late at night, at the Command to which I had been posted and was immediately received by the Commandant, Colonel Baistrocchi, who was waiting for me. (…) I answered the Colonel Baistrocchi, maybe not being even able to conceal my astonishment, that I had knew nothing about artillery, let alone about its tactical use. But, proving to be able to cope with the situation, he told me: “It is a calculus problem which you should be able to solve: to calculate the

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49

Actually, other mathematicians – for example Severi, E. E. Levi, Tonelli and Volterra himself – were involved in similar experiences too. They solved several problems on external ballistics and telemetry brought about by the quick development of artillery – the rifled cannon, which permitted longer and more precise fire; the adoption of curvedfire guns; etc. – and by the adaptation of the fire tables to the specific geographical setting of some war stages. The last sentence of the quotation turns Picone’s experience into a significant one – beyond the personal and nationalistic emphasis of the narration. The sudden revelation of a mathematics which was not only beautiful but can be useful too, happened in years which were decisive for the building and completion of his scientific personality and career. The discovery of mathematics’ utility was not a parenthesis which was to be closed when returning to civilian clothes and resuming an already well thriving activity, but the feature on which Picone decided to bet so as to get a specific qualification within the mathematical world. The foundation of the Istituto Nazionale per le Applicazioni del Calcolo would realize the intuition born during the war years.
Picone’s contribution to the enhancement of the ballistic tables is to be valued as a rara avis in a war that opened up the world’s collective imagination regarding
22 (continued) data our siege artilleries need for the fire against targets for which the actual regulation fire tables are not enough.” I added that I knew nothing about ballistics either, on which, I supposed, that calculation should be built. Then the Colonel took out from a regulation box a yellowed voluminous book and told me: “Here you have FRANCESCO SIACCI’s treatise on ballistics, I order you to study it and to calculate, within a month from today, the fire data our siege artilleries need against the strongholds of the enemy formation.” And he dismissed me. I began to work feverishly, also overnight, under the changeable candlelight, and I soon admitted Colonel Baistrocchi was right, reaching even an explanation for the difficulties our artillerymen had with the calculation of the fire data, and that they could not overcome. It was like this. In the previous peacetime it was set for the artillery’s mountain fire the use of small calibre cannons, so-called mountain cannons, which could be transported on the back of a mule over the highest mountain peeks and fired with no calculation, by direct pointing. But now, thanks to the recent possibility of building solid roads quickly even in the inaccessible mountain terrain and of using powerful auto-tractors that could tow artillery pieces of every calibre and weight even in strong sloped roads, we – and the enemy too – had thought of using, even in high mountain, middle and big calibre cannons, which, thanks to their fire range, could destroy the enemy’s highly resistant fortifications and, with an intense and accurate fire, could during the offensive precede the advancing assault troops or block the way to the enemy troops before our lines in the defensive. Regulation ballistic tables, with which middle and great calibre artillery were equipped, gave fire data for targets positioned on the battery’s same horizontal level, allowing to slightly correct the data itself in case of steeps between battery and target, but only to some extent. Within the Trentino’s gorges these limits were usually overstep, often so much than the steep between battery and target was as long as their mutual horizontal distance. Once this was established, I could easily determine the reasons for the disasters caused by the artillery fire, which, because of fatal unavoidable calculation errors, often stroke not the enemy but our defensive works. The artillery ballistic tables had to be redone without delay, with quite different criteria, based on some improvements of classical rational ballistics, which were not direct and could be obtained only by a mathematician. I got them within the prescribed month and from the following month, September 1926, all Artilleries of the 21° Siege Group fired correctly using the data I had calculated. (…) After this success of mathematics, one can guess I saw things in a different light. I thought: but, then, mathematics is not only beautiful, it can be useful too.

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the future and “modernity”. A marginal experience within an event with quite a different meaning but an experience that would deeply influence his scientific personality and the organization of Italian mathematics. We find the same features in Volterra, whose leadership within the mathematical world was further strengthened by the great war.
A keen interventionist – as we have seen – when Italy went to war Volterra immediately volunteered. In 1915 he was 55! In the following months, he distinguished himself in risky deeds on Zeppelins, whose optimization he helped also from a technical point of view: he suggested the use of helium instead of inflammable hydrogen, prepared a means for photo-telemetric surveys and personally experimented with the installation and use on board of a 65 mm calibre cannon so as to fire upside downwards. His commitment can be retraced in a letter to Mittag-Leffler23 of May 191624.
Vous me parlez d’un congrès de mathématiques en Suède cette année et d’un voyage en Suisse pendant le printemps. Je vois que vous ne vous faites pas une idée de l’état d’âme en Italie. Ce n’est pas le moment de voyager. Toutes nos pensées sont tournées à la guerre que nous combattons avec le plus grand enthousiasme à côté de nos alliés et nous ne pensons qu’a rapprocher l’instant de la victoire définitive contre nos ennemis. Nous sommes sûrs de la victoire et nous espérions dans un avenir heureux pour notre patrie qui n’a hésité à se placer du côté de la justice et de la liberté. Je suis engagé dans l’armée et je suis officier du Génie. Mes occupations militaires et techniques dans le corps d’aéronautique absorbent maintenant toute mon activité. Mes connaissances de mathématiques et de physique me sont utiles dans ce moment.
A year later, he wrote to an Italian physicist25:
Chiarissimo Professore, in risposta alle sue lettere del 1 e 4 maggio, sono lieto che Ella dimostri tanta attività e spero che l’opera sua potrà essere di efficace aiuto alla difesa del Paese. In modo speciale hanno vivamente interessato gli studi che Ella ha intrapreso di un microfono subacqueo, per la segnalazione di navi e sottomarini26. Spero che la pros-

23 About his efforts to create a better climate between German and Allied mathematicians after the war, it is possible to read: W. Dauben, “Mathematics and world war I: the international diplomacy of G. H. Hardy and Gösta Mittag-Leffler as reflected in their personal correspondence”, Historia Mathematica, 1980, pp. 261–288.
24 The letter, unpublished, comes from the Archives of Mittag-Leffler Institute. 25 The letter, addressed to Michele La Rosa, has been published in P. Nastasi (ed.): Lettere a Michele La
Rosa (1903–1932), Science History Seminar of Palermo’s Faculty of Science, 1991. 26 In 1917 the problem of the localization of submarines (both with magnetic surveys and through ultra-
sounds, as the French physicist P. Langevin proposed) was on the agenda of almost all belligerent countries. The works developed on this subject by R. A. Millikan and the American physicians are well-known.

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sima chiusura dei corsi di quest’anno scolastico, possa permetterLe di proseguire questi studi che sono particolarmente importanti nelle presenti circostanze. Per quanto riguarda la proposta di un congegno per lo scoppio dei proiettili sott’acqua contro i sottomarini, la quale pure sembra notevole, la miglior cosa sarebbe che Ella mandasse senz’altro, in doppia copia, una nuova relazione ed i disegni relativi, senza tener conto dei precedenti che non hanno rapporto col nostro Ufficio. Infine per quanto riguarda lo scudo-corazza Pagano, esiste già in questo Ufficio un parere sfavorevole di uno scudo-corazza-zaino Pagano, presentato il 10 ottobre 1916 ed esperimentato dalla Scuola di Applicazione di Fanteria di Parma. Se si tratta della stessa invenzione, e non di omonimia, non potrebbe esser presa in considerazione, a meno che, nel modello costruito nel suo Istituto, non siano state apportate modificazioni e correzioni a quello già esaminato a Parma. Nella speranza che le ricerche così felicemente da Lei iniziate, possano presto portare un contributo attivo e fortunato ai lavori di questo Ufficio, Le porgo i miei distinti saluti27.
There is a good summary of this stage in Volterra’s activity in Edmund Whittaker’s commemoration (published in 1941 among the Obituary Notices of Fellows of the Royal Society).
Before describing the scientific work of the last twenty-five years of Volterra’s life, let us take up again the thread of his personal history. In March 1905 he was created a Senator of the Kingdom of Italy – a great honour for a man still comparatively young – and about this time he was appointed by the Government as Chairman of the Polytechnic School at Turin, and Royal Commissioner. The way was open for him to become a great figure in political and administrative life: but he preferred the career of a pure scientist, and took an active part in public affairs on only two occasions – the Great War of 1914–1918, and the struggle with Fascism.

27 Most distinguished Professor, replying to your letters from the 1st and 4th May, I am pleased You are so dynamic and I hope your work will effectively help to defend our country. Above all, your studies about an underwater microphone to signal ships and submarines are certainly interesting. I hope the next closure of this academic year’s courses will allow you to continue these studies, especially important in the present circumstances. Regarding the proposal of a device for the underwater detonation of projectiles against submarines, which is remarkable too, it would be better if You could send, in a double copy, a new report and the concerning designs, without considering the previous ones which have no relation with our Office. Lastly, concerning the shield-armour Pagano, this Office gave already an unfavourable opinion of a Pagano shield-armour-rucksack, presented on the 10th of October 1916 and experimented by the Infantry School of Application of Parma. If it is the same invention, and not an homonymy, it won’t be considered, unless the model built in your Institute has been modified and corrected in relation to the one already examined in Parma. Hoping that the researches You have so satisfactorily begun could soon bring an operative and successful contribution to this Office’s works, I send You my kind regards.

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In July 1914 he was, according to his custom at that time of year, at his country house at Ariccia, when the war broke out. Almost at once his mind was made up that Italy ought to join the Allies: and in concert with D’Annunzio, Bissolati, Barzilai and others, he organized meetings and propaganda which were crowned with success on the 24th of May in the following year, when Italy entered the war. As a Lieutenant in the Corps of Engineers he enlisted in the army, and, although now over fifty-five years of age, joined the Air Force. For more than two years he lived with youthful enthusiasm in the Italian skies, perfecting a new type of airship and studying the possibility of mounting guns on it. At last he inaugurated the system of firing from an airship, in spite of the general opinion that the airship would be set on fire or explode at the first shot. He also published some mathematical works relating to aerial warfare, and experimental with aeroplanes. At the end of these dangerous enterprises he was mentioned in dispatches, and decorated with the War Cross. Some days after the capitulation of Gorizia he went to this town while it was still under the fire of Austrian guns in order to test the Italian instruments for the location of enemy batteries relative by sound. At the beginning of 1917 he established in Italy the Office for War Inventions, and became its Chairman, making many journeys to France and England in order to promote scientific and technical collaboration among the Allies. He went to Toulon and Harwich in order to study the submarine war, and in May and October 1917 took part in the London discussion regarding the International Research Committee, to the executive of which he was appointed. He was the first to propose the use of helium as a substitute for hydrogen, and organized its manufacture. When in 1917 some political parties – especially the Socialist – wanted a separate peace for Italy, he strenuously opposed their proposal: after the disaster of Caporetto, he with Sonnino helped to create the parliamentary bloc which was resolved to carry on the war to ultimate victory.
Whittaker’s commemoration introduces what will be later seen as Volterra’s biggest contribution during the war years. We have seen that the relationships between science, industry and military apparatus were still slender. In the allied countries, which had more developed military and state structures, the need was felt to go towards an explicit involvement. France, for example, already had, since 1894, a Commission for the examination of militarily interesting inventions. The physicist Mascart, the chemist Moisson and the mathematician P. Appell belonged to it. In 1914, just at the beginning of the war, the Commission – transformed to Superior Commission of Inventions for National Defence – was reinforced by the presence of technicians, members of Parliament and academicians. Presided over by Paul Painlevé, who would later become Minister28 and would be replaced by E. Borel, it used the work of prestigious mathematicians (such as E. Borel, J. Hadamard, H. Lebesgue and P. Montel) and of physicists such as Cotton, Langevin,
28 Between 1915 and 1916, P. Painlevé was Minister of Education, Fine Arts and Inventions for National Defence (this Ministry was just created for him). In 1917 he was Minister for War.

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Francesco Severi (1879–1961)

Vito Volterra

Perrin and Weiss. It used as well university and industry laboratories to develop the research on: anti-gas protection systems, enhancement of fire tables, sound localization of enemy batteries, submarine detection, etc. Volterra had the merit of understanding at once the importance of such initiatives (in France, but also in England and in the United States, for the war’s outcome and a peaceful, hopefully near future) and of using many personal contacts to enter this “network”.
In January 1917 he presented to the War Ministry a project for an Ufficio Invenzioni e Ricerche, following a mission in France, where he went “to pick up information and news on the present relationships between French scientific laboratories and military administration organs”. The project was accepted two months later, but within a year Volterra would make over the Ufficio Invenzioni e Ricerche – the seed of the future Consiglio Nazionale delle Ricerche (C.N.R.) – into a scientific and technical consultancy organ directing the creation of an autonomous center of applied research29. The delay with respect to the allied countries (and their means) was sensible, but it was anyway the first achievement of a research institution on applied problems of national interest.
The first Conferenza interalleata sulla organizzazione scientifica, in which delegates of the Scientific Academies of the allied countries and of some neutral countries
29 Cf. G. Paoloni, Vito Volterra e il suo tempo (1860–1940), Roma, Accademia Nazionale dei Lincei, 1990, c. IV.

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took part, was held in London at the end of the war, from 9th to 11th October 1918. In Italy, the Accademia dei Lincei delegated Volterra, also in his capacity as director of the Ufficio Invenzioni e Ricerche. Before the Conference, the American delegate G. E. Hale – Volterra’s old friend since 1909, when he had gone to Rome to lecture on “Solar Vortices and Magnetic Fields” – had presented a proposal which aimed at transforming the several organizations that had appeared in the allied countries during the war to coordinate the research world and the industrial production world (on the model of the National Research Council). The “research national councils” thus created, should have, in turn, set up with their own delegates an “international council”. The London Conference approved the proposal and created a Commission, formed by É. Picard, the English physicist A. Schuster (secretary), Hale, Volterra and the Belgian astronomer G. Lecointe, entrusted with studying which methods could be used to realize this council. In a second meeting in Paris (26th to 29th November), a provisional “International Research Council” was founded by instituting an executive committee which among other things had the task of preparing the draft for the statutes and of organizing the following conference in Brussels, from the 18th to the 28th of July 1919. It was on this occasion that the International Council, the general organ of the International Unions concerning the different disciplines, was officially presented. In each country, adhering to a Union, a corresponding national Union or Committee was to be formed, and all these would join together in a “Research National Council”.
Volterra would tenaciously work in both directions: to constitute the Unione matematica italiana and to found the C.N.R.

Chapter 3
Volterra’s leadership
1. Introduction
Everything had changed with the war, significantly. The deep transformations which had taken place in the relations between States, in the political scene and in the social field itself, involved also the mathematical community of Europe. Jean Dieudonné offered a synthetic but most effective view1:
Jusqu’à la guerre de 1914–1918, les écoles française et allemande, dominées par leurs plus illustres représentants, H. Poincaré et Hilbert, génies universel d’une rare envergure, restent les plus nombreuses et les plus variées, et excercent en mathématiques une prépondérance incontestée. A leurs côtés, les foyers de recherche mathématique comptant les plus nombreux et les plus actifs participants sont l’Italie et l’Angleterre. La première brille surtout par ses écoles de Géométrie algébrique (Castelnuovo, Enriques, Severi), de Géométrie différentielle (Levi-Civita, E. E. Levi) et d’Analyse fonctionnelle autour de Volterra, et ne subira une éclipse (dont elle n’a commencé à sortir que récemment) qu’à partir de 1935 environ; tandis que qu’après Cayley et Sylvester, l’école anglaise, changeant de cap, se groupe à partir de 1910 environ autour de Hardy et Littlewood et va entrer pendant 30 ans dans une féconde série de découvertes sur l’Analyse classique et ses applications à la Théorie des nombres, avant de céder la place, à l’époque actuelle, à une brillante phalange d’algébristes et de topologues. Après 1918, la France, dont la jeunesse scientifique a été saignée à blanc per l’hécatombe, va se replier sur elle-même pendant 10 ans, et, à l’exception de E. Cartan et de Hadamard, l’école mathématique française se cantonnera dans le domaine restreint de la théorie des fonctions d’une variable réelle ou complexe; dont le développement considérable aux aléntours de 1900 avait d’ailleurs été surtout son
1 Cf. J. Dieudonné, Introduction to Abrégé d’histoire des Mathématiques: 1700–1900, Paris, Hermann, 1978, I–II, p. 1–17.

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oeuvre (avec Picard, Hadamard, E. Borel, Baire, Lebesgue, puis Montel, Denjoy, Julia). L’Allemagne, au contraire, qui a su mieux préserver la vie de ses savants, conserve intactes ses traditions d’universalité; en outre, elle voit éclore une remarquable école d’Algèbre et de Théorie des nombres (E. Noether, Siegel, Hecke, E. Artin, Krull, R. Brauer, Hasse, van der Waerden (d’origine hollandaise), qui inaugure la tendance axiomatique déjà en germe dans les travaux de Dedekind et de Hilbert. Entre 1920 et 1933, ces mathématiciens assurent aux Universités allemandes, où se pressent les étudiants de tous les pays (notamment les jeunes Français, venus renouer aves les traditions oubliées chez eux), un éclat et un rayonnement exceptionnels, qui seront malheuresement brisés brutalement par l’ère hitlerienne. Il faudra attendre ensuite les environs de 1950 pour que l’école allemande se reconstitue, influencée cette fois (par un curieux renversement de la situation) par les mathématiciens français de tendance «bourbachiste». Toutefois, le phénomène le plus marquant après 1914 est l’apparition sur la scène mathématique de vivaces écoles nationales dans des pays qui n’avaient guère connu jusque-là qu’un petit nombre de savants ayant atteint une renommée internationale. Dès avant la fin de la première guerre mondiale, il faut d’abord citer l’U.R.S.S. et la Pologne, d’où surgit brusquement une pléiade de mathématiciens de premier ordre (Lusin, Souslin, puis Urysohn, P. Alexandrov, Kolmogorov, Vinogradov, Pontrjagin, Pterowski, Gelfand en U.R.S.S.; Sierpinski, Janiszewski, Kuratowski, Banach, puis Hurewicz, Eilenberg, Zygmund, Schauder en Pologne); c’est à leurs efforts que l’on devra surtout le développement des fondements de la Topologie et de l’Analyse fonctionnelle modernes. En U.R.S.S., l’élan ainsi donné ne s’arrêstera ps, et a continué à produire de très nombreux mathématiciens de grande valeur; quant à la Pologne, dont la moitié des mathématiciens ont été massacrés par les nazis, elle n’a commencé que récemment à combler ses vides et reprendre sa marche en avant.
Some of the elements presented by J. Dieudonné had already been displayed by A. Denjoy in his speech at the Réunion Internationale des Mathématiciens, held in Paris in 1937 on occasion of the Exposition internationale2:
Depuis la guerre mondiale de 1914–1918, la production mathématique a cru en intensité dans de très forte proportions. Le fait a été moins sensible dans les régions appartenant à des pays constitués avant 1914 que dans celles dont les nouveaux Etats ont été formés. Dans ces derniers, un nationalisme très vif, mais de la nature la plus louable, a poussé les gouvernements et les peuples à la fondation de nombreuses universités dont le personnel professoral s’est pris d’une très noble émulation pour rivaliser avec les représentants des écoles mathématiques étrangères les plus réputées, et pour tenter, souvent avec succès, de les surpasser.
2 Cf. A. Denjoy, Aspects actuels de la pensée mathématique, Conférences de la Réunion Internationale des Mathématiciens, Gauthier-Villars, Paris, 1939, pp. 1–12.

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The “great war” brought about conspicuous changes in the mathematical world (too): an unusually rapid generational imbalance occurred in some of the most developed countries (due to the especially tragic war) and the birth of new national schools which often followed a strong and aggressive course. Fortunately, the first element affected Italy only slightly3. More than 2000 university students died in the war – we don’t know how many of them had begun studying Mathematics – but among the researchers mentioned in the Prologue, who before the war had already won public favour, only E. E. Levi died in the front, during Caporetto’s retreat (1917)4.
Mathematicians on the Italian scene after the war were essentially the same as before it, but the atmosphere had certainly changed. The end of war made the resumption of normal life possible, but in a disenchanted atmosphere in which institutions and people felt acutely the (material and moral) injuries of a lethal war. The doubt that modernity could slide into barbarity, subverting the consolidated hierarchy of custom and power, had entered society. The happy dream of the beginning of the century had ended, and now one had to reckon with a much harsher reality. That dream was of universal progress thanks to scientific achievements; in reality one woke up wondering whether or not to resume scientific meetings and relationships with colleagues from the defeated countries.
The question of relationships with German researchers blocked the mathematical community for almost ten years. International relationships in general returned to a balanced if precarious normality only in 1928, with the Congress of Bologna. There were “hawks” and “doves”. Among the former were most French mathematicians (particularly E. Picard) and some Italian ones of Volterra’s “calibre”. Picard, in his address for the closing meeting at the VI Mathematicians International Congress (Strasbourg, 23–30.9.1920), stated5.
En ce qui regarde spécialement notre Congrès, nous n’avons jamais dissimulé que nous entendions lui donner une signification particulière, en le réunissant à Strasbourg. Aussi avons-nous été extrêmement touchés de l’empressement avec lequel nos amis étrangers ont répondu à notre appel. (…) Des liens plus intimes ont été formés, qui resteront précieux. Nous continuerons ainsi, entre peuples amis, nos travaux scientifiques, apportant dans cette collaboration nos qualités diverses,
3 The Italian situation was better than the French one. According to J. J. Gray (The Hilbert Challenge, Oxford Univ. Press, Oxford, 2000) more than 40% of students in Mathematics or in Sciences were killed or blessed in France during the war.
4 E. E. Levi took part in the war as a volunteer. 5 The quotation is taken from the Comptes Rendus du Congrès International des Mathématiciens (Stras-
bourg, 22–30 September 1920), Toulouse, 1921, p. XXXI–XXXIII. Among French mathematicians, E. Picard (1856–1941) was one of the most resolute nationalists (he wrote among other things a book with the significant title L’Histoire des Sciences et les pretensions de la Science Allemande, Perrin, Paris 1916). He had lost a son in the war and took an almost racist position. Hadamard, despite having lost two sons in the war, had a more moderate position. French nationalist mathematicians decided that it would be symbolically appropriate for the disputed Strasbourg (newly passed from German into French hands) to host the International Congress of 1920, from which German mathematicians had been excluded.

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sans qu’aucun prétende exercer une insupportable hégémonie et sans nous soucier de certaines menaces, qu’avec une impudeur qui ne nous étonne pas, on a osé proférer. Quant à certaines relations, qui ont été rompues par la tragédie de ces dernières années, nos successeurs verront si un temps suffisamment long et un repentir sincère pourront permettre de les reprendre un jour, et si ceux qui se sont exclus du concert des nations civilisées sont dignes d’y rentrer. Pour nous, trop proches des événements, nous faisons encore nôtre la belle parole prononcée pendant la guerre par le cardinal Mercier, que, pardonner à certains crimes, c’est s’en faire le complice.
The view coming from the other side of the Rhine was in stark contrast with this one, as documented in the letter6 of the German mathematician R. Rothe to M. de Franchis (second president of the Circolo Matematico di Palermo, after the founder, G. B. Guccia), dated 24th March 1921.
Sono stato ripetutamente da Lei invitato di indicare il mio indirizzo. Io Le confesso francamente che sono stato per molto tempo in dubbio al riguardo, perché del Circolo Matematico sono ancora soci persone come E. Picard, Ch. de La ValléePoussin ed altri, che fino al momento attuale hanno manifestato il loro odio contro tutto ciò che è tedesco. Come esempio cito soltanto l’istituzione d’un così detto congresso internazionale di Matematica nell’autunno 1920, da cui erano intenzionalmente esclusi i Matematici tedeschi e che in segno di speciale sarcasmo contro di loro è stato tenuto nella vecchia città tedesca di Strasburgo nell’Alsazia. Però dopo che Ella mi ha ripetutamente e in maniera così amichevole invitato a pronunciarmi, non voglio più a lungo tacere, e invece voglio manifestarle i miei più sentiti ringraziamenti per la Sua gentilezza, ed esprimere la mia speranza che il Circolo Matematico possa riuscire mediante i suoi sforzi a ripristinare le antiche pacifiche relazioni fra i Matematici, per il bene della nostra scienza, che dovrebbe unirci tutti7.
The President of the Circolo Matematico di Palermo opposed the request of French mathematicians and some of the Italian ones, that German members be expelled from

6 Archives of the Circolo Matematico di Palermo. The letter is obviously written in German, but it is translated in Italian on the margins. Rothe taught at the time at the Technische Hochschule in Berlin.
7 You have often asked me to give my address. I sincerely confess You that I have hesitated for a long time, because people as E. Picard, Ch. de La Vallée-Poussin and others, who have until now manifested their hate against everything which is German, are still members of the Mathematical Circle. As an example I quote only the creation of a so-called International Mathematics Congress in Autumn 1920, which intentionally excluded German mathematicians and which, with special sarcasm against them, was held in the former German city of Strasbourg, in Alsace. But as You have frequently and friendly asked me to pronounce on the matter, I won’t keep silent anymore; I wish to warmly thank you for Your gentleness, and to express my hope that the Mathematical Circle will restore the old peaceful relationship among Mathematicians, through your effort, for the good of our science, which should join us all.

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the Circle. He also refused the resignation that M. Noether offered “in gentleness”. In the meanwhile, some – like Bianchi – proposed an intermediate solution8.
Riguardo ai Soci tedeschi, la questione è molto complessa e delicata. Intanto, per evitare attriti e proteste, abbiamo deciso di rinviare ancora l’elezione del nuovo Comitato di Redazione dei Rendiconti. Personalmente, io ritengo che, come non vorrei essere tenuto responsabile di una impresa abominevole eventualmente condotta da una minoranza di faziosi col concorso del popolo suggestionato abilmente con falsi miraggi, così non posso addossare su tutto un popolo e tanto meno su una eletta schiera di studiosi il peso degli orribili misfatti dei quali si è macchiato l’imperialismo germanico e dei quali, tra parentesi, sarebbe capace di macchiarsi qualsiasi altro imperialismo spinto alle ultime conseguenze da un manipolo di persone interessate. Per me, il principale nemico dell’umanità è l’imperialismo il quale costituisce ai nostri tempi un anacronismo e solo per il tornaconto di un esiguo gruppo di furbi che si arricchiscono e godono sui lutti e sulle miserie della grande maggioranza. E ritengo che inconsciamente i popoli che hanno lottato contro l’imperialismo siano ora scaltramente incanalati verso di esso; un indice è lo stato di inestinguibile odio che si vuole perpetuare non fra umanitari e imperialisti ma fra popoli. Converrei quindi nell’idea di radiare quei soci che avessero firmato il manifesto dei 93 intellettuali, perché li ritengo indegni di coltivare relazioni con gente civile, ma non posso convenire, per esempio, che sia giusto radiare un Hilbert, che fin dal principio della guerra ha fatto sapere di disapprovarla, solo perché Hilbert è nato in terra germanica. Insomma che non si voglia avere contatti con persone di sentimenti ignobili è giusto, ma che debba esserci anche il peccato originale del luogo di nascita, non mi pare che possa sostenersi. Ma queste sono idee mie e posso anche sbagliare. Ciò che però è fuori di dubbio è che le distinzioni che si vogliano fare tra gli scienziati a seconda del paese di origine, toglieranno per lungo tempo alla Scienza il carattere internazionale, togliendo ad una parte dell’umanità i frutti del lavoro di un’altra parte (…). E badi che dopo ciò, tra qualche anno, la collaborazione scientifica è fatale che si riattivi, ma intanto la nostra Società sarà morta9.
8 Bianchi’s letter to de Franchis, dated 1st March 1919, is kept in the Archives of the Circolo Matematico di Palermo.
9 The question about German members is quite complex and delicate. We have decided to postpone again the election of the new Reports Editing Committee, meanwhile, so as to avoid conflicts and protests. I would not like to be held responsible for an abominable deed led, if anyone, by a minority of factoids supported by the folk, which has been ably persuaded with false mirages; at the same time, I think that an entire folk, and certainly not some elected scholars, cannot be blamed for the terrible misdeeds with which Germanic imperialism has stained itself and with which, incidentally, every imperialism driven to its last consequences by venal people could stain itself. In my opinion, mankind’s main enemy is imperialism, which is in our times an anachronism and fits only a few crafty people who enrich and enjoy themselves thanks to the mourning and miseries of the vast majority. And I think that unconsciously nations which have fought against imperialism now move shrewdly towards it; the intent to perpetuate an everlasting hate not among humanitarians and imperialists but among peoples is a sign of this. I would agree then to the idea of striking those members who signed the 93 scholars manifesto off, because I think they do not deserve to have any contact to civilized people, but I don’t

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Bianchi’s proposal involved, obviously, a researcher of the value of F. Klein. De Franchis, of course, did not know the reasons that could be adduced to defend the behaviour of other German mathematician, whom André Weil remembered in his autobiography10.
To conclude with the “manifesto”, I will note here what I learned much later, in Germany. First of all, Hilbert, who always behaved with the utmost dignity throughout the war, refused to sign it – though I do not think that his name was familiar enough to me in 1922 for me to notice that it was absent from the list of signatories. Second, I have been told that many of those who signed, including Felix Klein, had not seen the text; they had simply been asked over the telephone to support what was put forth as a patriotic duty. Only those who have no inkling of how petitions, protests, and declarations of all sorts are peddled among the intelligentsia would find this surprising.
In the Prologue and at the beginning of Chapter II, Volterra has been identified as the main leader of Italian Mathematics in the pre-war time. His image was strengthened by the conflict. He had “bet” on intervention and on a “democratic interventionism”, against German “barbarity”. During the war, besides his military actions – already mentioned – and a remarkable commitment to convey and use scientific knowledge, he promoted also cultural initiatives which were to consolidate a social alliance that he thought of as much more than military. Thus, for instance, in 1916 he promoted an Associazione italiana per l’Intesa fra i Paesi alleati e amici to favour the exchange of information, teachers and students among the Universities of the allied countries. The facts and the outcomes of the battlefield proved he was right. His “philosophy” was strengthened too. It favoured development of mathematics within the boundaries of an abstract and deeply innovative research, which in the way to its formalizations, however, had to refer always to applications. He realized that the interaction between science, on one hand, and social and productive fields, on the other, would become a driving force for industrialized countries. The role of science, both on the economic and on the cultural level, would lead to defining and experimenting with new organizational forms of research and of scientific communities, with more funds, more power and more social responsibility. In 1915 Volterra, during H. Poincaré’s commemoration held at the Rice Institute, affirmed that science was no longer the exclusive product of a few highly privileged sci-
9 (continued) think it right, for example, to strike off Hilbert, who since the beginning of the war said he disapproved of it, only because Germany is his native soil. In short, I think it right to avoid having contact with ignoble people, but the idea of the birthplace original sin cannot be approved of. However, this is my opinion and I may be wrong. There is no doubt, though, about the fact that if we discriminate between scientists because of their mother country, science will be deprived of its international character for a long time, as a sector of mankind will be deprived of the results of another sector (…). And listen to me: scientific collaboration will be inevitably resumed within some years, but in the meanwhile our Society will be dead.
10 Cf. A. Weil, The Apprenticeship of a Mathematician, Birkhäuser, Basel-Boston-Berlin, 1992, p. 38.

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entists; it had become a collective enterprise which needed, besides conceptual thinkers, a host of specialists who would lead research in an organized and coordinated manner. Lastly, Volterra’s personal prestige would come out of the war further consolidated. He was, actually, the representative of Italian Mathematics in all organisms and assemblies that, during the war and in the months immediately following its end, organized Europe’s new scientific assets on an institutional level.
The first accomplishments of Italian Mathematics and science after the war – the Unione Matematica Italiana (U.M.I., 1922) and the Consiglio Nazionale delle Ricerche (C.N.R., 1923) – were published in his language and took his name. Before exploring this topic, however, we would like to dwell upon an apparently minor episode which happened in 1921 and concerned Rome University. It was a “usual” conflict of interest in the academic world but, later on, it would turn out to be anything but negligible, due to its consequences.

2. Rome, 1921
In our Prologue we had occasion to cite the leading role played by Rome on a cultural and scientific level by virtue of its political standing as the capital of Italy. With regard to mathematics, this politics began to function early on with the “call” to the capital of personalities such as Cremona, Beltrami, Castelnuovo and Volterra. At the beginning of the 20th century, the same politics prevailed, though it had its ups and downs; a few first-rate researchers did not accept the transfer because they found the city’s life (and scientific atmosphere) to have hard rhythms, and some second-rate researchers took their places. One refusal from an “excellent” choice was that of Levi-Civita, who in 1909 cold not be persuaded to leave the quiet Paduan environment and his family for what became known as the “big leap”. Anyway, it was the pair Castelnuovo and Volterra that ran the operations there and represented the largest academic power centre within the Roman Institute.
After the war, and encouraged by the nationalist and patriotic climate, the political pull “towards the capital” resumed more vigorously. For a professor, the “call” from Rome was always a moment of great prestige, often the culmination of his career. Levi-Civita himself – in the meanwhile married (1914) and temporarily in Rome after Caporetto’s defeat of 1917 – thought it over and moved there in 1918. Then, on the 29th December 1920, an analyst died11 and the race for the succession began. Castelnuovo’s letter to LeviCivita, dated 4th January 1921, that is, a few days after his colleague’s death12, reports it.
Carissimo amico,
Ghigo [Federigo Enriques] mi ha parlato del suo desiderio di venire ad occupare una delle cattedre lasciate vacanti dal povero Tonelli (l’Algebra). La notizia mi ha fatto molto piacere e mi ha fatto anche piacere che egli, risparmiandomi dei passi
11 Alberto Tonelli, not to be confused with Leonida Tonelli. 12 The letter has been published in P. Nastasi, R. Tazzioli, Aspetti scientifici e umani nella corrispon-
denza di Tullio Levi-Civita (1873–1941), Quaderni Pristem, 12, 2000, p. 263–264.

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Enriques

Francesco Severi

penosi per la mia posizione delicata, abbia discorrere discorso con vari colleghi di Facoltà. Sembra che egli abbia trovato buone disposizioni anche presso il Volterra, del quale, per ragione di scuola, si poteva temere qualche contrasto. Mi par di capire che il Volterra, il quale mi ha parlato della cosa, accoglierebbe volentieri Ghigo all’Algebra, purché accettassimo la sua proposta riguardo il Calcolo. Sai che il Volterra, anche prima che tu esprimessi il desiderio di venire a Roma, aveva pensato per l’Analisi al nome di Leonida Tonelli; ed ora, negli ultimi giorni di malattia del nostro collega, mi ha ripetuto lo stesso nome per le cattedre che si sarebbero rese vacanti. Più che alla persona il Volterra tiene all’indirizzo; egli tiene che a Roma sia rappresentata la teoria delle funzioni di una variabile reale nei suoi ultimi perfezionamenti; ed io, pur non avendo speciali simpatie per questo indirizzo, non so dargli torto nell’aspirazione che ogni importante indirizzo di matematica abbia il suo rappresentante nella nostra Facoltà. Perciò credo che egli non rinunzierebbe, se non forzato, a desistere dal suo proposito, e dubito se ci convenga forzarlo, dato pure che si riuscisse, quando, come in questo caso, egli adduce delle ragioni plausibili della sua idea. È quasi superfluo dirti quanto piacere mi farebbe la venuta di Severi, insieme a quella di Ghigo. Son legato al Severi da cordiale amicizia, mentre conosco appena il Tonelli; l’indirizzo del Severi lo giudico molto più importante di quello del Tonelli; e, a parte l’indirizzo, ritengo molto superiore il valore del primo rispetto al secondo. Per tutte queste ragioni, a cui tu pure alludi nella tua lettera, dovrei sostenere la candidatura Severi, e lo farei con tutto il cuore se non sentissi in Volterra una opposizione, per pregiudizio di scuola, che non credo opportuno di

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combattere in questo momento. Bada per pregiudizio di scuola e non per la persona, giacché, avendo detto a Volterra, prima di ricever la tua lettera e di conoscere la specifica aspirazione del Severi, che questi desiderava alla prima occasione di venire a Roma, il Volterra mi disse che sarebbe ben lieto di appoggiarlo appena l’occasione si presentasse. Ti prego dunque, nell’esprimere al Severi il mio rincrescimento per le difficoltà che incontra il suo trasferimento a Roma nel momento attuale, di dirgli che il suo nome è molto ben accetto tra i colleghi di qui, matematici e non matematici, e che, appena si renda scoperta una cattedra che ci sembri a lui adatta, terremo conto del suo desiderio che è anche il nostro. Quanto al Bianchi non credo sia il caso di pensarvi, perché a Pisa ha la direzione della Scuola Normale, e perché egli più volte mi disse che a Roma egli non si trova bene, per la vita tumultuosa e agitata che vi si conduce13.
So the “candidates” for the suddenly free chair were three. Actually, the Roman Faculty had also suggested some older authoritative colleagues like Bianchi and Pincherle. The “short-list”, though, soon shrank to Tonelli (Leonida), Enriques and Severi. The first one had already distinguished himself before the war for his works on real analysis and his important contributions to the Calculus of Variations; his candidacy was supported by
13 My dearest friend, Ghigo [Federigo Enriques] has told me about your wish to occupy one of the professorships left vacant by poor Tonelli (algebra). I was very pleased by the news and by the fact that he has talked to several colleagues of the faculty, sparing me painful steps because of my delicate position. Even Volterra, who could contrast it because of school reasons, seems favourably inclined. I think that Volterra, who told me about it, would willingly admit Ghigo in algebra, if we accept his proposal regarding calculus. You know that Volterra, even before you expressed your wish to come to Rome, had thought of Leonida Tonelli for analysis; and now, during the last illness days of our colleague, he often named him for the shortly vacant professorships. For Volterra, the branch is more important than the person; it is important that the function theory of a real variable and its last progresses are represented in Rome; though I do not have special sympathy for this branch, I cannot blame him for his ambition to have in our Faculty a representative of each important branch of mathematics. This is why I think that, if he can, he will not abandon his intentions, and I don’t think it worthwhile to force him, even if we managed to, as in this case his reasons are praiseworthy. I don’t need to tell you I would be very pleased if Severi, as Ghigo, comes. Severi is a good friend, while I hardly know Tonelli; I consider Severi’s branch much more important than Tonelli’s; and, apart from it, I think the former is quite above the second. For all these reasons, which you mention in your letter too, I should support Severi’s candidature, and I heartily would if I did not suspect that Volterra, because of a school prejudice which I do not feel like fighting right now, is against him. Mind that I say due to school prejudice and not due to the person, for when I told Volterra, before receiving your letter and knowing Severi’s exact aspiration, that he wished to come to Rome at the earliest, Volterra said he would be very glad to support him as soon as the chance came. Please inform Severi that I am sorry for the trouble he finds now in moving to Rome; tell him that he is welcomed by the colleagues here, mathematicians and not, and that, as soon as a professorship suited to him is vacant, we will consider his wish, which is also ours. As for Bianchi, we don’t need to think about it, since he is director of the Scuola Normale in Pisa, and he told me several times he does not feel good in Rome, because of the tumultuous and restless life of the city.

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Leonida Tonelli
Volterra himself, who wanted absolutely an analyst to obtain the professorship in analysis. Enriques was partially supported by Levi-Civita, as well as by his brother-in-law Castelnuovo; Levi-Civita, however, “spent” his authority above all on Severi (with whom he had an excellent relationship of esteem and friendship, consolidated in the familiar surroundings of Padua).
It was a delicate situation in which several aspects had to be considered: analysis and geometry; particular situations like Enriques’, who wanted to leave Bologna for some of the usual personal reasons; family relationships (Enriques and Castelnuovo) and school relationships (Severi had been a pupil of Enriques), in a framework rendered even more delicate by the unquestionable quality of the three competitors. These were researchers of great prestige who were being measured by an international standard; to complicate matters, one of them, Severi, had become in the meanwhile chairman of the University Association, which grouped university professors for professional and syndicalist purposes, and was about to become a member of the Superior Council of Education.
This pressing debate among the Faculty actually engaged Roman mathematicians for almost two years. Out of “gentleness” reasons, Castelnuovo did not participate openly, so that Volterra and Levi-Civita are the main actors in the “scene”. The former organized14
14 Here Volterra was helped by Giovanni Vacca, whom we’ll find in Chapter V, when talking about historical studies.

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for the first time (only he could afford it) a national lobbyist action of analysts. With several letters and other personal interventions, he denounced to colleagues the “rapacity” of geometers. He turns it into a matter of principles – e.g., teaching posts in analysis had to remain with analysts – which obviously Levi-Civita did not share15.
Al prof. Volterra segue il prof. Levi-Civita che rileva che il prof. Volterra ha impostata la questione da un punto di vista, degno indubbiamente della massima attenzione, ma che, secondo lui, non è il solo. Egli ritiene che, in linea di massima, vi sia un criterio preponderante cui ispirare la nostra condotta e le nostre decisioni. Coprire le cattedre vacanti cogli elementi di maggior valore che si possano nel momento attuale attrarre a Roma, assicurandosi soltanto che all’altezza scientifica si accompagni l’eccellenza didattica, onde rimangano soddisfatte in maniera ineccepibile le esigenze molteplici dell’insegnamento16.
In the end, Levi-Civita and Severi “succeeded”. In Rome nobody would talk about Tonelli anymore, at least for some years. Enriques instead arrived almost immediately, “reintegrated” with the “sacrifice” of Castelnuovo who voluntarily relinguished the chair in superior geometry he had held for twenty years17.
Il Preside legge una lettera inviata dal prof. Castelnuovo, il quale dichiara che non è intervenuto alla seduta giacché si tratta di una questione che lo riguarda personalmente; ma che, se fosse stato presente, avrebbe consigliato i Colleghi di valersi della cattedra di Geometria superiore per assicurare alla Facoltà uno scienziato del valore di Federigo Enriques, benché egli non si distacchi senza dolore da questo insegnamento e senza la speranza di riprenderlo un giorno18.
Volterra had lost his battle. The minutes of the Faculty Councils tell of his perseverance, but also of the many defeats he encountered, on this occasion, within his own Faculty. Severi’s “call” from Rome is a minor episode, pointed out because of the “unsuccess” of Volterra and the first manifestations of Levi-Civita’s personality, who, just arrived in Rome, did not hesitate a moment to oppose – with placating but resolute tones – a charismatic and powerful leader such as Volterra. Levi-Civita went on thinking
15 From the minutes of the Science Faculty Council of Rome University, dated 18.3.1921. 16 After Prof. Volterra comes Prof. Levi-Civita, who notes that Prof. Volterra has set out the question
from an undoubtedly very interesting point of view, which is though not the unique one, according to him. He thinks that, on the whole, there is a preponderant criteria on which to base our behaviour and our decisions. That is, to cover the vacant teaching posts with the most valued elements who can currently be attracted to Rome, making sure only that scientific quality goes with didactic excellence, so as to perfectly satisfy the multiple exigencies of teaching. 17 From the minutes of the Science Faculty Council of Rome University, dated 15.2.1923. 18 The Dean reads a letter sent by Prof. Castelnuovo, who explains that he had not participated in the meeting because it deals with a question that regards him personally; but that, if he had been present, he would have advised his colleagues to make use of the teaching post in superior geometry to assure the Faculty of a worthy scientist such as Federigo Enriques, though he detaches himself from this teaching with sadness and hopes of recovering it one day.

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to Volterra as the princeps mathematicorum – as he writes in a letter of 13.11.1918 to Volterra – but, anyway at the time Levi-Civita was a well-known researcher himself. His paper “Sur la regularisation du probléme des trois corps” received important acknowledgements. G. Mittag-Leffler stated that the article had pathed the way to a long series of studies. G. D. Birkhoff writes that “your treatment of the problem of regularization is definitive”.19 Later, Levi-Civita published the well-known memoir on parallel displacement where – as E. Cartan will say – he succeeded in bringing the fundamential notions of the tensorial calculus “jusqu’alors purement analytique dans le domaine de la Géometrie”. In a 1938 report on his works, sent to E. Borel and to E. Picard for his election as a associé étranger of the Académie des Sciences, Levi-Civita wrote:
A vrai dire cette notion de parallélisme et son application presque immediate á la dérivation des vecteurs avaient été conçues par l’auteur comme un couronnement ou mieux (si l’on peut s’exprimer ainsi) comme le vernissage du développement de la géométrie différentielle des variétés, telle qu’elle s’était constituée en corps de doctrines, depuis Gauss et Riemann (á travers Lamé, Christoffel, Darboux, Beltrami et beaucop d’autres) jusqu’á Ricci. Mais ce vernissage a été un point de départ pour des nouvelles conceptions géométriques, dont M. Cartan a su tirer des théories d’une rare élégance et des liaisons des plus fécondes avec la théorie des groupes de Lie. Je fais allusion aux transports des vecteurs le long d’une courbe, qui a été, par Weyl, Schouten, Veblen er surtout par Cartan, détaché de la méthode de l’espace ambient, et s’est développé comme géométrie différentielle des connections, l’existence d’un ds2 riemannien constituant pour ces investigations une hypothése tout á fait particuliére et souvent inessentielle.
In the same year T. Y. Thomas explained Levi-Civita’s idea as reported down below:
In the case of a two-dimensional surface S in a three-dimensional euclidean space, where the idea of Levi-Civita has its strongest appeal, the producedure, in brief, may be described as follows. Let F be a developable surface tangent to S along C (envelope ot the one-parameter family of tangent planes to S along C.) Let xP be a vector in the tangent plane to S at a point P on C. Roll F on a plane so that C becomes a curva C¢ in the plane, P a point P¢ and C¢, and xP a vector xP¢ at P¢. Displace xP¢ along C¢ by parallel displacement in the ordinary sense (parallel displacement in a euclidean plane). We thus define vectors x¢(t) along C¢ parallel in the ordinary sense. Now wrap the plane about the surface S along C to secure the original developable surface F. Thereby the vectors x¢(t) go to into vectors x(t) tangent to S along C. We define the vectors x(t) which are in the surface S, that is, in the tangent plans in S along C, to be parallel with respect to C and in fact to result from the original vector xP at P by parallel displacement along C. Levi-Civita’s definition of parallel displacement of a vector in a surface S generalizes the ordinary euclidean
19 The letter from G. D. Birkhoff to Levi-Civita (26.10.1916) is kept in the Archives of the Academia dei Lincei.

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concept of parallel displacement in the sense that if S is a plane (in general an n-dimensional euclidean space) the parallel displacement is identical with the ordinary euclidean displacement.
The concept of parallel displacement play an important role in the development and in the strengthing of some ideas on the relativity. We quote the 40 notes written by Levi-Civita just on the relativity but an equally important contribution can be found in the letters with Einstein from March to May 1915: “a such important correspondence never happened to me – Einstein wrote to Levi-Civita on 02.04.1915 – you would have to see how anxiously I wait for your letters”. It was the German physician M. Abraham (1875–1922) – who had met Levi-Civita at the International Congress of Rome in 1908 – who introduced the two scientists each other. The Italian mathematician had pointed out a mistake in an important proof of Entwurf; from this the covariant properties of the gravitational tensor would be deduced. Einstein tried many times to oppose himself to Levi-Civita’s critics but at the and he was obliged to admit that the colleague was right. Then, after a long and hard period of research, he was able to publish the note on Preussische Akademie der Wissenschaften where he proved the right version of gravitational equations. He admitted: “they represent a true triumph of Ricci’s Calculus”. The difficulties of this method were balanced by the originality of the results that it allowed to get in the theory of the relativity.
The episode we told is a minor event that would bring about important consequences, though. Severi had arrived in Rome as a valuable mathematician, educated man (even if he did not have Enriques’ prominence and cultural interests) and ex-socialist who was just beginning to enter the centres of power. Volterra had found the future alternative to his leadership at home.

3. The foundation of the Unione Matematica Italiana
There was obviously nothing which at once seemed to give to Severi’s move to Rome and to his presence in the country’s political centre any more importance than a routine university change. Volterra went on carrying through the institutional program which he had further developed during the war years.
His first achievement, at the beginning of the 1920s, was the Unione matematica italiana (U.M.I.), which appeared not so much as a drive from within the mathematical community (as it had been for other Italian professional societies, such as, the physical or chemical societies), but due to the vote expressed in Brussels by the International Research Council at the meetings held between the 18th and the 28th July 1919. As indicated at the end of Chapter II, the wish to have a Union for each discipline in every allied country had emerged in Brussels. The U.M.I. appears concretely first in a circular written by Volterra in March 1920.
È vivo desiderio di molti studiosi di costituire una unione italiana la quale raccolga i matematici, analoga a quelle già fondate per le scienze chimiche, per le scienze

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astronomiche e per l’Oceanografia. L’Unione si propone:
a) l’incoraggiamento alla scienza pura – b) il ravvicinamento tra la matematica pura e le altre scienze – c) l’orientamento ed il progresso dell’insegnamento – d) l’organizzazione, la preparazione e la partecipazione a congressi nazionali ed
internazionali.
Questa unione si propone altresì di ricongiungere e coordinare le forze delle altre associazioni nazionali esistenti20.
Volterra intended the circular to be signed by about ten mathematicians (among them Bianchi, Pincherle, Somigliana and obviously Volterra himself). At the end, some of the expected signatures were not obtained, maybe due to the same reasons appearing in a letter21 received by Levi-Civita in October 1920.
So che ti sei meravigliato di vedere il mio nome tra le firme della circolare d’invito per una unione dei matematici. In verità mi sono meravigliato anch’io. Non prestai sufficiente attenzione a quanto mi disse rapidamente il Sen. Volterra, e non pensai che l’unione potesse mirare – come ora mi si dice – ad una azione antitedesca. Aderii verbalmente senza dare importanza alla cosa e senza credere d’impegnarmi con una firma. Questo ti dico confidenzialmente e pregandoti di non farne parola al Volterra, perché infine non posso incolpare che me stesso. E te lo dico perché tu abbia direttamente la conferma di quanto avevi felicemente intuito sulla portata della mia azione. Ma io credo e spero che l’Unione resti un pio desiderio, e che la germanofobia sia, per questo lato almeno, un male passeggero22.
20 It is a strong wish of many researchers to create an Italian union which gathers mathematicians together, a union similar to those already founded for chemical sciences, astronomical sciences and Oceanography. The Union intends to: a) encourage pure science – b) draw pure mathematics and other sciences closer – c) guide and improve teaching – d) organize, prepare and participate in national and international congresses. This union also undertakes to gather the forces of other current national associations together and to coordinate them.
21 The letter, dated 5th October, and written by a mathematical physician from Bologna, Pietro Burgatti, is kept in the Archives of the Accademia dei Lincei.
22 I know you were astonished at seeing my name among the signatures in the circular inviting to a union of mathematicians. Actually, I was astonished too. I didn’t pay much attention to what Senator Volterra said to me quickly, and I didn’t think that the union could aim – as now they say – to an anti-German action. I agreed verbally without giving much importance to it and without thinking to commit myself with a signature. I say this to you confidentially, so please do not to tell Volterra, because in the end I am the only to blame. I only want to confirm you what you had rightly guessed about the importance of my action. But I believe and hope that the Union will be a pious wish, and Germanophobia a fleeting evil, at least on this side.

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The excessive political anti-German portrayal was the main difficulty the U.M.I. had to face in its gestation period. But there were also other difficulties, such as the unavoidable rivalry which arose with the present mathematical and scientific associations (especially with the Circolo matematico di Palermo) and the resulting need to show the usefulness of the new association, distinguishing its content and scope. The same could be said about the project of Bollettino, with which the Union should provide itself, within a journalistic-mathematical scene in which there were several other headlines, many of them with a common generalist tendency.
It is necessary to return to Pincherle, mentioned in the Prologue, so as to talk about the problem of the presidency. A first-class analyst and researcher in the theory of analytic functions, Pincherle was one of the “founding fathers” of functional analysis. In fact, his “golden age” were the decades straddling the two centuries. After the war, the decline in scientific activity forced him to concentrate mainly on institutional roles and organizational tasks. The underlying theme of his research remained, however the study of linear functionals through their intrinsic operational properties, independently of particular analytic expressions.
His two posthumous memoirs summarize the research he did in his last twenty years. Pincherle thought, with clear satisfaction, that a number of studies he had developed during decades of research, which had not always been really appreciated, were being reintroduced by modern Physics. In a Note of 1926, for instance, he wrote: “the triumph of the discontinuous in natural Philosophy, with statistical mechanics, with the new views of the constitution of matter, with the theory of the quanta, etc, cannot help but bring about a revaluation of that branch of analysis which stands out exactly because of its discontinuous changing of subject”.
The U.M.I.’s presidency represented another difficulty for the birth of the association because it came from above and it was unilaterally chosen by the Accademia dei Lincei, maybe so as to avoid any discussion. Volterra informed Pincherle officially about his successful appointment on the 18th March 1921.
Mi pregio di comunicarLe che si è costituita la “Unione Matematica Italiana” la quale entra così a far parte della “Unione Matematica Internazionale” che insieme alle altre Unioni Scientifiche, compone il “Conseil International de Recherches”. Sono lieto di aggiungere che la Presidenza della “Unione Matematica Italiana” è a Lei affidata; ed a Lei è pure connessa la nomina del Segretario della Unione stessa; nomina della quale, a suo tempo, Ella vorrà dare comunicazione al prof. Emilio Picard Presidente del “Conseil International”23.
Pincherle intensified his pursuit of recovering dissidents or anyway of the “halfhearted”, with particular care for Levi-Civita. The following letter, dated 12th April 1922,
23 I am glad to inform you that the “Unione Matematica Italiana” has been founded, being therefore member of the “Unione Matematica Internazionale”, which along with the other Scientific Unions forms the “Conseil International de Recherches”. I am pleased to add that the chairmanship of the “Unione Matematica Italiana” has been entrusted to you; and related to you is also the appointment of the Union’s Secretary, which, to its due time, you will communicate to Prof. Emil Picard, Chairman of the “Conseil International”.

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was addressed to him. A few days afterwards, when writing to Volterra, Pincherle would regretfully notice that “among the list of the sometimes enthusiastic adhesions, Rome is quite sparsely represented”.
E poiché ho occasione di scriverti, mi permetto di rivolgerti una preghiera; ed è di non fare cattivo viso all’iniziativa che, a preghiera d’alcuni colleghi e quasi à son corps défendant, ho dovuto prendere, di costituzione dell’Unione Matematica Italiana. Si tratterebbe di aver fra noi qualcosa d’analogo alla Deutsche Math. Vereinigung o alla Société Mathématique de France e a quanto hanno da noi i fisici, i chimici, i geodeti, ecc. La mia azione è affatto provvisoria, ma ci ho messa tutta l’anima, coll’intendimento, appena si raggiunga un numero sufficiente di soci, di lasciare l’impresa nelle mani d’una presidenza che verrà eletta dai soci stessi e dovrà essere formata d’elementi giovani e fattivi: la mia è dunque pura opera d’éclanchement, e per riuscire, conto sul ben volere e sull’appoggio dei colleghi. Spero dunque che non vorrai rifiutare la tua adesione – che va mandata alla ditta Zanichelli di qui – non solo, ma che vorrai darci, per uno dei primi numeri del futuro bollettino, un tuo scritto, sia pure brevissimo, ma che porti la tua firma. Sei così versato nell’argomento che ora interessa sopra tutti il mondo scientifico, che lo scrivere due righe su qualche problema connesso alla teoria della relatività, deve essere per te cosa di nessuna fatica, e ti saremo estremamente grati d’una risposta favorevole. Ti prego pure di volere interessare a fare parte dell’Unione i cultori della Matematica, anche applicata, fra i quali conti un così grande numero d’amici e di ammiratori. Uno degli scopi che, secondo me, l’Unione si deve proporre, è d’abbassare, per quanto è possibile, la barriera fra la scienza pura e le applicazioni. E se il promotore dell’impresa ti sembra troppo impari alla riuscita, pensa che formata che sia e posta in mani autorevoli, l’Unione Matematica potrà rappresentare un organo importante in avvenire, per la Scienza che coltiviamo24.
24 And as I have the opportunity to write to you, I take the liberty to ask you not to oppose the initiative that I have had to take, by request of some colleagues and almost à son corps défendant, of constituting the Unione matematica italiana. The point is to create here something similar to the Deutsche Math. Vereinigung or to the Société Mathématique de France, similar to what our physicists, chemists, geodeticists, etc. already have. This move is absolutely provisory, but I have given myself up to it, and, as soon as there are enough members, I intend to leave it in the hands of a presidency elected by the members themselves and composed of young and efficient people: mine is a pure work d’éclanchement, whose success relies on my colleagues’ esteem and support. I hope then that you will give us not only your support – to be sent to the company Zanichelli here – but also, for one of the first numbers of the upcoming bulletin, one essay, even if very short, but with your signature. You know so well the subject that now chiefly interests scientific world, that writing two lines on any problem related to the theory of relativity will be no effort for you, and we will be extremely thankful for a positive answer. I also ask you to drag researchers of mathematics, and applied mathematics, among which you have so many friends and fans, into becoming members of the Union. I think one of the aims of the Union is to lower, as far as possible, the barrier between pure science and applications. And if the promoter of the undertaking seems to you too unfit for the success, think that, once it has been created and put in authoritative hands, the Mathematical Union will be, in the future, an important organ for the science we cultivate.

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Salvatore Pincherle
In this letter Pincherle included an especially important passage: his determination to reach not only a simple representation in international projects, but to create a real professional association, with an autonomous structure, similar to that of other existing societies (the Société Mathématique de France, founded in 1872, the Deutsche Mathematiker-Vereinigung, founded in 1890, and the American Mathematical Society, founded in 1891). It was a decisive moment in the history of Italian Mathematics. When numbers increase, the professional association makes the functioning of well organised disciplinary communities possible, both by creating an agreement around standards of scientific quality and by regulating the distribution of resources according to scientific status. Finally, changing its direction by partially and progressively smoothing the most critical positions25, the U.M.I. set out on its voyage. Volterra had made it! On the 31st March 1922, Pincherle made up his mind and sent to his colleagues a circular introducing the programme of the proposed society.
25 Levi-Civita agreed at the end of 1922. In 1926 – perfectly coordinated with the deliberation of the International Research Council of abrogating the exclusion clause against German scientists – most Italian mathematicians were members of the U.M.I.

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The first years of the Union were not very fascinating. Pincherle handed the management of the didactic aspects over to Mathesis (and to Enriques) and receded into the background with respect to the Accademia dei Lincei, as regards the school’s reform (presided over by Giovanni Gentile). The only wide-ranging act was the organization of the Congresso Internazionale, held in Bologna from the 3rd to the 10th of September 1928. In this Congress the intransigent work for the scientific internationalism found finally a resolute affirmation.
The series of International Congresses of mathematicians was resumed (after the war) with the ones in Strasbourg (1920) and Toronto (1924), from which mathematicians of German, Bulgarian, Austrian and Hungarian nationality had been excluded. But already in Toronto – with France’s strong opposition – the delegates of the United States, Great Britain, Italy and other countries had proposed the abolition of the preclusion. Pincherle did everything he could to reach a normalization of the international situation. On the 1st September 1925 he wrote to Volterra thus:
Carissimo amico,
ricevo ora la tua gradita lettera del 29. Non ho qui il testo esatto della mozione presentata a Toronto, il 15 agosto 1924, dalla Delegazione americana, e fatta propria dalle delegazioni di altri 8 o 9 Stati, fra cui l’Italia, ma era formulata all’incirca così: “La delegazione degli U.S.A., all’unanimità, invita il Consiglio internazionale di ricerche ad esaminare se non sia giunto il momento di modificare in senso meno restrittivo l’ammissione di altri Stati al Consiglio stesso”. Ora una lettera del Polya, che è stato parecchi mesi a Cambridge e che ha conferito coi maggiori matematici inglesi, mi avverte che questi, e in particolare la London Math. Society, in accordo in ciò colla American Math. Soc., sarebbero propensi a dichiararsi ostili ad un Congresso internazionale che mantenesse le esclusioni. Specialmente se la Germania verrà ammessa alla Società delle Nazioni, l’atteggiamento del Consiglio Internazionale, più che poco simpatico, sarebbe addirittura puerile, a giudizio del pubblico anglosassone. (…) A giorni scriverò una lettera circolare a tutti i componenti del Consiglio di presidenza dell’Unione, per prendere accordi circa alla prima preparazione del futuro congresso, e per sentire se la maggioranza è d’accordo di riunirsi, a Parigi o a Ginevra, per trattare di questi accordi. (…) Ma prima sarebbe bene che la questione dell’ammissione venisse ripresa e risoluta in senso liberale, diversamente, la crisi è indubbia26.

26 My dearest friend, I have just received your letter of the 29th. I don’t have here the exact text of the motion presented in Toronto by the American Delegation on 15th August 1924, which was approved by the delegations of 8 or 9 countries, among which Italy, but it said something like: “The U.S.A. delegation, to the unanimity, invites the International Research Council to consider whether the moment has come to modify in a less restrictive sense the admission of other countries to the Council itself ”.

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At the meeting of the 26th June 1926, the International Research Council decided to abolish every restriction. The U.M.I. was given the all- clear and could invite German mathematicians to the Congress in Bologna. However, the sequels of particularly conflictual years were long lasting. On the German side, some were unwilling to accept the olive branch and, a few days before the opening of the Congress, the Dutch L. E. J. Brouwer published an open letter, recalling the words with which years before Painlevé had justified ostracism towards German scientists and polemically wondering: “according to these words, how can a mathematician think of participating in the planned congress without deriding the memory of Gauss and Riemann, the cultural nature of mathematical sciences and the independence of human spirit?” Thus the German mathematical community split into two groups: the first one around Hilbert, supporter of the participation and coherent upholder of scientific internationalism27, the other around Brouwer, diametrically opposed. With him sided mainly Berliner mathematicians (Erhard Schmidt, Ludwig Bieberbach and Richard von Mises), representing the most resolutely nationalistic tendencies and German science’s values as typical.
The Congress was regularly held, at last, as Italian mathematicians intended. The Minutes registered the participation of 836 mathematicians, among them, 76 Germans (the most numerous group after the Italians). In the German delegation, representing – with R. Courant and E. Landau – Göttingen University and its Gesellschaft der Wissenschaften, appeared also David Hilbert, who was appointed Chairman of the Congress and gave the first general lecture: Probleme der Grundlegung der Mathematik.

26 (continued) A letter from Polya, who has been in Cambridge for several months and has conferred with the greatest English mathematicians, warns me that these, and in particular the London Math. Society, agreeing with the American Math. Soc., would be inclined to oppose themselves to an International Congress that would keep the exclusions. The behaviour of the International Council, in the opinion of the Anglo-Saxon public, would then be not only unpleasant, but even childish, specially if Germany will enter the Nations’ Society. (…) In a few days I will write a circular to all members of the Union’s Presidency Council, to make arrangements for the first preparation of the upcoming congress, and to see if the majority approves to convene in Paris or in Geneva, in order to discuss about this accordance. (…) But first it would be better to take up again the question of the admission and to solve it in a liberal sense; otherwise there will certainly be a crisis.
27 During the spring of 1928, Hilbert wrote: “We are convinced that pursuing Herr Bieberbach’s way will bring misfortune to German science and will expose us to all justifiable criticism from well disposed sides. (…) The Italian colleagues have troubled themselves with the greatest idealism and expense in time and effort. (…) It appears under the present circumstances command of rectitude and the most elementary courtesy to take a friendly attitude towards the Congress” (cf. O. Lehto, Mathematics Without Borders. A History of the International Mathematical Union, Springer, New York, 1998, p. 46). One can read S. L. Segal, Mathematicians under the Nazis, Princeton University Press, Princeton, 2003, pp. 349–355, for a detailed historical reconstruction of the polemics of German mathematicians about the Bologna Congress.

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4. The foundation of the Consiglio Nazionale delle Ricerche
The C.N.R. (Consiglio Nazionale delle Ricerche) was Volterra’s second great achievement in the first half of the twenties. It was the culmination of a long activity and expressed the deep tuning existing between the still – in those years – uncontested leader of Italian mathematical world and the trends of European research.
At the end of Chapter II we mentioned the Ufficio invenzioni e ricerche (accomplished by Volterra in 1917), the inter-allied Conference on scientific organization (London, October 1918) and G. E. Hale’s proposal for the launching of the International Research Council (taken place at Brussels conference in July 1919). The foundation of the C.N.R. emerged out of this process: the C.N.R. was to be the Italian expression of the International Research Council, and its aim was to rationalize and develop the first achievements on scientific organization, which arose with the war.
The birth date was 18th November 1923, day in which the royal decree that founded the C.N.R. in Rome as a non-profit making company was published. The gestation period turned out to be particularly long, if we consider that Volterra began working on a first draft of the statute in February 1919. The causes of such a delay are evident28 at once if we consider red tape slowness, the continual changes of government and the very delicate time that Italy was going through just in those months. The statute – finally approved in October 1924 – underlined the national role of the new organism, which was not only the Italian ramification of the International Research Council: the

Middle thirties CNR building in Rome
28 The difficulties of that period were not inferior in the other countries. In France, for example, CNRS was created just in 1939.

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C.N.R. had to “coordinate and stimulate the national activity in the different branches of science and its applications; be in touch with the different state centres for scientific matters; manage and eventually establish general and special research laboratories”.29 The self-government board was represented by a Council which elected, among its members, the President and the Secretary General. The Council itself chose the Administrator.
On 12.01.1924 Volterra was elected unanimously as chairman of the C.N.R. It was his moment of greatest prominence. That same year he was elected president of the Accademia dei Lincei, of which he had been vice-president in the previous three-year period. He was the “grey eminence” of the U.M.I. (which acted as the C.N.R.’s mathematical Committee), and the most authoritative person of the S.I.P.S., without considering presences and presidencies in other scientific societies and a really considerable political weight in several international organisms.
The statutes of the C.N.R. reflected the great expectations of its promoters. The conditions of academic research had to be improved. It was nominally free, but in reality heavily constrained by scarcity of resources and the compartmentalisation imposed by the university administration, and by the personalities of its faculty. At the same time, research had to be focused on the great national problems, overcoming the modest aspirations of the few existing public boards and laboratories. During its first years, the C.N.R. certainly operated beneath the level implied by such ambitious expectations. The determining factor was the scarcity of financial resources. The support provided by the government was hardly enough to cover current expenses, which included the functions of the executive organs and payment of dues to international organizations. At least until 1925, the C.N.R. contributed especially to the basic expenses of the U.M.I., undertaking also the burden of the “missions” of Italian mathematicians involved in several international congresses. With such scarce means, the plan for an extra-university research structure could never be executed; those plans were for the creation of a great national technical-experimental laboratory that would overcome the chronic dysfunction of the existing small and dull university laboratories, which were always weighed down by scarcity and the need for didactics.
Four great Institutes had been planned: one for chemistry and the industries depending on it; a second one for physics, electronics and mechanics; a third one for biological sciences and a forth one for those sciences closely related to agriculture. But money would never be found. Most professors opposed the project; though they were obviously favourable to greater resources, they preferred the “indiscriminate distribution” system to a politics that would have seriously risked marginalizing many university institutes.
But other tiles must be introduced into the mosaic to wholly understand this progress. Volterra’s chairmanship in the C.N.R. ran from 1923 to 1926. These years – as recalled in the next chapter – saw a radical change in Italy’s political structure: the C.N.R. and its chairman would no longer be circumscribed by the cultural atmosphere and changing political choices of newly appointed executives.
29 For what concerning CNR’s history – in particular its first years – one can read: Per una storia del Consiglio Nazionale delle Ricerche (R. Simili, G. Paoloni eds.), I–II, Laterza, Bari, 2001.

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5. Volterra’s scientific activity
Volterra was born in 1860. After the war he was then in his sixties. His personal habits and his institutional activity, developed with a feverish rhythm, make almost predictable the observation that his study activity had not the intensity and the originality of the preceding decades.
During the 1920s and 1930s, Volterra abandoned gradually the research field, and engaged instead in summary conferences, expository and review articles or monographs that subsumed organically a whole series of previous studies (adding sometimes some significant element). Many were the issues of interest: Le calcul des variations, son evolution et ses progrès, son rôle dans la physique mathématique30, for instance, is a highly instructive article which reconstructed some stages of the history of calculus of variations. The methodological elements that distinguish Volterra’s style come up again, almost with obstinacy. The “old lion” did not stop having his say. He reintroduced the passage from the discrete to the continuous and from real analysis to functional analysis; he defended his definition of the derivative for a functional in opposition to Hadamard’s and Fréchet’s outlook, and above all he continually reminded his younger and more “modern” colleagues that “n’est qu’en donnant droit de cité à des éléments formels que plusieurs branches de l’Analyse ont pu avancer”. Volterra, specially in the first twenties, was mostly interested in the theory of the composition of functions:
** y
f g = ∫ f (x, z) g (z, y) dz . x
About this argument he wrote, together with J. Pérès, some notes and a monograph, that resume some pre-war studies in which the theory of composition (firstly motivated by the solution of integral equations and by the study of some questions of hereditary mechanics) was totally autonomously developed as first study of an algebra of operators.
Volterra’s minor presence on the most strictly scientific side was very natural – the “great anomaly” were his biological studies, examined in the next paragraph – and just as natural was the fall of the whole “old guard” of the analysts introduced in the Prologue. The only remarkable exception was Tonelli, who will be discussed in more detail later. The period of greater commitment and originality of Peano, disappeared in 1932, was already far away; from the beginning of the century his main interests were others. About Pincherle we have already spoken31.
Vitali was in the same situation as Peano, Pincherle and Volterra, even if through a different path. For long years clientele and insensibility of the academic world had not allowed him a university career, to the point that H. Lebesgue could be ironically satisfied about the Italian situation, so prosperous as to “forget” in high school teaching valued mathematicians such as Vitali. After the war, finally settled at University, Vitali’s scientific attention was gradually attracted by the research on absolute differential calculus and
30 It is the text of the lectures given in 1931 at Prague University “Charles” and Brno University “Masaryk”.
31 G. Ascoli and C. Arzelà died, respectively, in 1896 and 1912.

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on differential geometry. His production on analysis was some ten notes and a few didactic or popular speeches, to conclude with a monograph on real functions. It was not, as for Volterra or Pincherle, a matter of “simple” re-elaborations of previous studies, maybe requested and prompted by a younger colleague. There was commitment and originality. But Vitali’s publications in analysis usually had not the same impetus, which is confirmed by the number of publications and by the change of his research barycentre. He resumed with the usual ability and neat elegance the speeches left almost suspended at the beginning of the 20th century. In the meanwhile, though, the discipline had significantly developed and Vitali, in spite of his effort to “keep up to date”, had inevitably lost his position and centrality. As for his commitment to follow the most recent contributions, the references to some works of S. Banach and the contacts established with the Polish school stand out (in the view of the Italian analysis of the time). In the recently published32 correspondence there are letters of O. Nikodym and W. Sierpinski. Vitali became a member of the Société Polonaise de Mathématique; one of his papers appeared in the Proceedings of the I Congrés des Mathématiciens des Pays Slaves33, and he had also some articles published in Fundamenta Mathematicae and in the Annales de la Société Polonaise de Mathématique.
His most committed memoirs of the time were “Analisi delle funzioni a variazione limitata” of 1922 and “Sulle funzioni continue” of 1926. In the first one Vitali studied the structure of bounded variation functions of one variable, starting from the established result according to which such a function could always be written as the sum of its jump function (that “absorbs its discontinuities”) and a continuous bounded variation function: if the latter “is absolutely continuous, that is an integral function, the structure of f (x) can be considered to have been adequately identified”. Otherwise, Vitali introduced a discard function, “which in some way shows to what extent the given function diverges from absolute continuity”, and came to the main representation theorem: each bounded variation function can be written as the sum of its jump function, an absolutely continuous function and a linear (eventually infinite) combination of peculiar, so-called elementary, discards. The letter of M. Fréchet dated 30th March 1923 referred to this memoir: he observed how its central part “semble indiquer que vous n’avez pas eu connaissance des travaux où une telle décomposition a été déjà obtenue”, and quoted the classic book of C. de La Vallée Poussin (whose third edition was from 1914!) and one note of his from 1913. Fréchet’s request was very polite, but explicit enough: “au cas où vous jugeriez utile de publier tout ou partie de la présente lettre, je n’aurais aucune objection”. Vitali took up the invitation promptly. His answer was particularly “amiable” and Fréchet, at this point, did not skimp praises. He underlined the “historical” importance of Vitali’s example of a continuous bounded variation function but not absolutely continuous, and appraised “la décomposition d’un “scarto” en une somme dénombrable de “scarto” élémentaires que vous établissez dans votre dernière mémoire (…) nouvelle et intéressante”.
32 Lettere a Giuseppe Vitali (M.T. Borgato, L. Pepe eds.) in G. Vitali, Opere sull’Analisi reale e complessa, Cremonese, Roma, 1984.
33 Warsaw, 1929.

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A similar “incident” involved his second main memoir of the time. In this case results were so synchronized that Vitali had no special responsibility, though. He proved the remarkable proposition commonly known, still today, as a theorem of Banach-Vitali: given a real function (of one variable) which is continuous from (a, b) into (c, d) and specifying that Gr Õ (c, d) is the set of values which have at least r counter-images, the necessary and sufficient condition for f to be a bounded variation function is that it converges the series of measures of the sets Gr; the sum of this series gives the total variation of f, anyway. As far as the last terms in the series are concerned, a new definition of total variation was seen: even if it didn’t match the usual one when applied to discontinuous functions, anyway it allowed one to rewrite the definition of absolute continuity and, above all, to extend such concepts to two variable functions. For Vitali it was really a pity that W. Sierpinski, when accepting the memoir for publication in Fundamenta Mathematicae, remarked how some results were to be found also in a note by S. Banach published in the journal’s previous volume. Hence the statement opening Vitali’s Memoir:
mentre correggo le bozze ho da segnalare la recente importantissima nota di Stefan Banach (…) nella quale l’autore con analoghi intendimenti sebbene con procedimenti diversi, ha conseguito risultati che collimano con quelli del presente lavoro. Sono lieto che l’opinione dell’illustre collega dell’Università di Leopoli e la mia concordino nell’indicare la via per estendere alla superficie i noti risultati sulla rettificazione delle curve ed in particolare il bel teorema di Leonida Tonelli34.
Fubini’s case further confirmed the old guard’s gradual estrangement from the research mainstream. Its exponents, who obtained great achievements in real analysis at the beginning of the century, considered that time almost definitely closed and went towards new stimulating research (to which they could apply their mastery of tools and their highly developed expertise). Hence, in the years between the two wars, Fubini was mainly engaged in projective differential geometry. He wrote only about ten notes in analysis, mostly occasional papers derived from recent publications, encouraging a deeper or an easier proof; he planned no systematic and articulated research program. It goes without saying that Fubini was always Fubini and that, although he was “distracted” by other subjects, his works reached nonetheless the usual high quality. The discussed matters concern complex analysis and differential equations (ordinary and partial ones). He obtained, with a very simple proof, a comparison theorem for ordinary linear equations of the second order or took again, for the equation:
∂4u ∂3u ∂2u 6 ∂x4 + 6 ∂x3 + 6 ∂t 2 = 0 ,
34 As I am correcting the drafts I have to point out the recent and extremely important note by Stefan Banach (…) in which he with similar intention but through different processes, has reached results which are analogous to the present ones. I am glad that the opinion of my illustrious colleague from Leopoli University and mine agree on indicating how to extend to surfaces the widespread results on rectification of curves and specially Leonida Tonelli’s remarkable theorem.

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an unicity theorem proved by Levi-Civita. Before his “American” note of 1940 (in which, developing an article of K. Menger of the previous year, he verified a necessary and sufficient condition for a differential to be exact), he published his two most significant memoirs of the time (1937). In “Studi asintotici per alcune equazioni differenziali” he expounded a method for homogeneous linear equations (formally considered as nonhomogeneous), which was essentially based on arbitrary constants variations that turned them into integral equations of Volterra; solutions could be expressed then through absolutely and uniformly convergent series, which were specially useful in the asymptotic study of solutions, because of their nature as asymptotic series. Actually written in May 1938 (a few months before the promulgation of the racial laws, that would strike also Fubini), “Sopra una nuova classe di problemi al contorno” started from some mean theorems which E. E. Levi, Volterra, Vitali and Tonelli had proved to be distinctive features of harmonic functions. Similar theorems were true for harmonic functions in a non-Euclidean space and for the solutions of an elliptic homogeneous linear equation of the second order, so that some integral features which synthesize the theorem of the mean matched some differential features (summarized in the differential equation). The memoir at issue inverted this correspondence, proving that functions with such integral features solved problems which were similar to those accomplished by the previous solutions.
Fubini loved great studies and wide horizons rather than limited research. When reading his writings, one is pleasantly impressed by sentences which often emphasize his open and conjectural research. Fubini stood out in the Italian context also because of his real interest in applications – he, a pure analyst! – and in the engineering world he knew and in which he moved, due also to his son Gino’s studies. His natural interest and his scrupulous personality, which “took seriously” the didactic location within Turin Polytechnic, brought him to positions which differed a great deal from the bombastic rhetoric typical of the time, and which turned into a concrete sensitivity and just as concrete dissemination work. Mathematics was not to be introduced as a chess game or, better, it had to avoid confusing “the art of the chess-player with the game’s rules”. Mathematicians did not choose by chance crazy hypotheses so as to deduce useless results, but rather they developed useful tools which Fubini persisted in presenting and explaining in some lectures (on functional and on symbolic calculus) given to engineers. In the meanwhile, he did not renounce at all to the typical features of the mathematical enquiry, although they were often a “safe promise of deep bore”. As vehemently did Fubini claim the pleasure to study “for the pride of human spirit alone” and the qualities of the abstract procedure: it was not “a defect; but on the contrary a rare credit and an important richness for our doctrines”. This is – in the Italy of the 1930s, which would officially grab any chance to take advantage of any scientific speech – why he tried to avoid that polytechnic teaching be nullified and reduced to mere practice and that “the young never forget that the greatness and the economic and technical independence of a country go together with the love for knowledge, with the pleasure in scientific research”.

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6. Volterra and Ecology
So far in this chaper we have focused on Volterra’s leadership in the first part of the 1920s. Let us now expand this period by some years to introduce his most important scientific contribution in the period between the two world wars. We refer, particularly, to 1926 (when the Accademia dei Lincei published “Variazioni e fluttuazioni del numero d’individui in specie animali conviventi”35) and to 1931, when the already classical Leçons sur la théorie mathématique de la lutte pour la vie36 were printed in Paris. Volterra’s studies on the dynamics of populations, which were original applications of mathematics beyond the classical physical context and which produced important results, were a worthy milestone for the birth of the field of mathematical biology, as it was outlined in the first half of the 20th century.
The story is wellknown. One of Volterra’s sons-in-law – Umberto D’Ancona, zoologist – asked his father- in- law for a theoretical explanation of a piece of data fairly evident in the fishing statistics of the North Adriatic Italian ports, regarding the years 1905–1923; according to this data, the percentage of great fishes (predators) had considerably increased in the total amount of the fish caught during the years 1915–1918 and the following years. The exogenous elucidations – essentially based on the minor fishing activities during the war years – did not convincingly explain the different behaviour of prey and predators.
Obviously, Volterra was not a biologist, but these questions were not wholly alien to him, either. In 1901, newly transferred to Rome University, he had had the honour of giving the inaugural lecture to the academic year and he had chosen as subject Sui tentativi di applicazione delle matematiche alle scienze biologiche e sociali37. In 1911 he had been appointed vice-president of the Regio Comitato Talassografico Italiano. Later he had supported a project (linked to the Comitato Talassografico) that contemplated the creation of an “Istituto Oceanografico Nazionale”, analogous to those of Monaco and Paris. In 1916 he had inaugurated in Messina the Istituto Centrale di Biologia marina at the presence of Prince Alberto I of Monaco (creator of the oceanographic Institutes of Monaco and Paris) and Louis Joubin and Odon Bouen, directors of the oceanographic Institutes of Paris and Madrid, respectively. On these occasion he had declared himself proud of the creation of the Messina Institute, which could be “honourably compared” to “Naples Station, unquestioned property of the Neapolitan city, today at last free from German subjection and purely Italian”38. In 1923, again, he was appointed member of the Italian delegation in the “International Commission for the study of the Mediterranean”.
35 Mem. Accademia dei Lincei, 1926, p. 31–113.36 The lectures, written by M. Brelot, were published by Gauthier-Villars.
37 The text was printed separately and reissued in the Giornale degli Economisti (1906) and in V. Volterra, Saggi Scientifici, Roma, 1920.
38 Actually, the glorious Neapolitan Institute of marine biology was just short-term acquired: in 1920, after a harsh parliamentary debate, Benedetto Croce – new Minister of Education – returned it to Rinaldo Dhorn, in spite of the vibrant protests of Volterra, of some Italian biologists and scientists of the allied Countries, among which were a group of English oceanographers.

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Unaware of A. Lotka’s contributions39, Volterra began to study the problem raised by his son-in-law at the end of 1925. He opened his Leçons thus: “à la suite de conversations avec M. D’Ancona, qui me demandait s’il était possible de trouver quelque voie mathématique pour étudier les variations dans la composition des associations biologiques, j’ai commencé mes recherches sur ce sujet à la fin de 1925”. He wrote the chosen model as a system of ordinary differential equations of the first order:
Ά x¢ = ax – bxy y¢ = –cy + dxy
where x = x(t) e y = y(t) represented, respectively, the evolution in time of the population of prey and predators and a, b, c, d Œ ᑬ. From a first hypothesis on the isolated evolution of both species (in terms of constant percentage rates of their growths x¢/x and y¢/y), followed a behavioural one on the principle of the encounters, according to which predation effects depended on the possible encounters xy in the time unit. The system’s solution (with the suitable initial conditions) was set in an explicit way through a clever method that used a reference system with four axes. Out of it Volterra would derive the three laws that govern the model’s biological fluctuations: the law of the periodic cycle (which proved the endogenous character of fluctuations), the law of conservation of the mean and the law of perturbation of the mean, which answered the initial problem. A perturbation due to external causes – for example, to the fishing action or to a change in its intensity – would bring new average values and the comparison with the previous ones would justify the experimental observation according to which the diminishing of fishing activity favoured, in a sense, minor species.
Volterra would be engaged in bio-mathematical research until his final years. After having studied the mentioned biological fluctuations’ model (which would be expanded several times, starting with the consideration of n species, to the introduction of memory terms) he studied the analytic mechanics of biological associations and later logistic curves. His research on theoretical ecology would not arouse in Italy particular reactions, at least straightaway; there were only, within the social-economic field, some “resumptions” of themes of population dynamics. Then, at once, the most complete ostracism arrived. But these are the years of his leadership and his greatest prominence. This story, still to be told, has been brought forward only to explain why Volterra had to ask Borel to sponsor in the Comptes Rendus some notes on the analytic mechanics of biological associations, in the same year in which A. N. Kolmogorov published in the Giornale dell’Istituto Italiano degli Attuari his famous article “Sulla teoria di Volterra della lotta per l’esistenza”.
In the Prologue we stated that Volterra was a worthy exponent of the great 19th century tradition, so as to emphasize both his familiarity with several research fields – he was analyst, mathematical physicists, bio-mathematician, and wrote worthy papers on mathematical economics – and that his scientific personality was shaped during the period
39 For the comparison between the formulation by Lotka and by Volterra and the question of priority, see A. Millán Gasca, G. Israel The Biology of Numbers. The Correspondence of Vito Volterra on Mathematical Biology, Birhäuser, Basel, 2002 and the essay of P. Manfredi and G. Micheli in S. Di Sieno, A. Guerraggio, P. Nastasi, La Matematica italiana dopo l’Unità. Gli anin fra le due guerre, quoted, pp. 671–733.

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Cover of Volterra’s book on Mathematical Biology
of greatest influence of physical analogy, a concept considered as the basis for the new theories. Bio-mathematical research was part of this framework. We could speak of his reductionist project. It was one of the most brilliant applications of the methods of classical mechanics, which marked one of the “strong” moments of scientific thought in the 20th century. “The turning point of the rise of mathematical ecology as an autonomous science is not to be sought then in Lotka’s or Pearl’s work, but in Volterra’s, and in the wholly Eurocentric cultural humus in which the big shift of interest from man to mass nourished Mc Kendrick, Ross and Thompson’s new population mathematics and Le Bon and Sorel’s mass psychodynamics, so close to the great political upheavals of the second quarter of the century”40.
40 P. Manfredi, G. Micheli in S. Di Sieno, A. Guerraggio, P. Nastasi, La Matematica italiana dopo l’Unità. Gli anni fra le due guerre, quoted, p. 723.

Chapter 4
Fascism: somebody rise, others fall
1. The march on Rome
We have talked about the founding of the U.M.I. and the C.N.R. and about the consolidation of Volterra’s leadership. These events cannot be, however, interpreted as a sign of a normal restarting of mathematical and scientific activity after the 1915–18 break. In Italy, the restarting of a normal life was quite problematic.
The title of the second chapter, “Nothing is as it was before”, is meant to emphasize the fact that the war experience was much more than just an additional element in a changing scene. The young democracy had to face – after only 60 years from the Nation’s Unification – vaster and more complex problems. The old establishment had hoped for some sort of continuity, even if slow and difficult, but reality dashed these hopes almost immediately.
With the advent of fascism in 1922 the political-institutional scene changed. Italians would realize it on the 27th October, when a statement by the leadership of the Partito Nazionale Fascista announced that its own militants were marching on Rome to seize power. The march on Rome found no sizeable resistance, partly because of the incredible about-face of the King, who refused to sign the proclamation of martial law (previously agreed upon with the government). Benito Mussolini had founded in 1919 the Fasci italiani di combattimento, which in 1921 became the Partito Nazionale Fascista. He was entrusted with forming the new government as early as the afternoon of the 29th October. His opening speech to Parliament left no doubt about his intentions. Formally, and temporarily, he led a coalition government, but his direction was clear: “potevo fare di quest’aula sorda e grigia un bivacco di manipoli. Potevo sprangare il Parlamento e costituire un governo esclusivamente di fascisti. Potevo, ma non ho, almeno in questo primo momento, voluto”1.
1 “I could have made out of this deaf and grey hall a bivouac of squads. I could bar Parliament and establish an entirely fascist government. I could, but I have not wanted, at least initially”. This is the bivouac’s speech, pronounced before the Camera on the 16th November 1922, on occasion of the vote of confidence for the new executive.

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The march on Rome
A moral and political judgement on fascism has already been stated, so we will not expatiate upon its antidemocratic, anti-popular and repressive character. It is difficult, though, to present fascism briefly in a manner useful to the understanding of the course of Italian mathematics between the two world wars.
Fascism, is based on an ingrained malleability of beliefs and inner contradictions that defy labeling it as a well-defined political philosophy. These attributes are moreover boasted of as exalting the personal prestige of the man who has to mediate its needs and aspirations. Fascism is at one and the same time a popular movement, a political party, and an authoritarian regime, each aspect merging imperceptibly into the others, and believing economic forces to be the best cure for the kind of popular unrest that had led to such phenomena as the occupations of the factories and the “red biennium” of 1919–1920. Fascism was also considered the most suitable and safest channel through which to build industrial capability. It also offered hope to an Italy disappointed by a war that did not fulfill nationalistic aspirations that it felt it deserved and had earned on the field. A wide spectrum of middle and working classes supported Fascism for a variety of reasons: they were

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disappointed at the resumption of some negative aspects of the civil life they knew and the failure of the “red biennium”, and they saw it as the guardian of their own humble interests.
In short, fascism was not, not even in its folkloristic or tragically ridiculous traits, foreign to the Italian nation, simply imposed by landowners and industrial organizations. It was not just an interval in history. It was rather – to use another popular metaphor – an illness that incubated germs that were already present in the Italian social fabric. It is a mixture of modern and anti-modern drives, or better stated, an “old” recipe (and anyway different from the modernity that would be asserted later), often cooked up by a new political staff. We will examine the importance of the generational question for Italian mathematics. War had brought to the fore a new generation, confident in its leadership ability, in the awareness of its own strength and in a sense of comradeship which grew out of the sharing with their peers of a trauma that had divided sons from their fathers, the new from the old generation. Social and cultural changes following the war brought about a further division also in style and behaviour. Fascism was the channel through which the new generation came to the political forefront. In this sense, it became the “natural” instrument through which new protagonists asserted themselves and their perceived right to change the status quo.
The course of many events involving fascism, new generations and modernity, culture and Mathematics can be traced through the image and actions of Giovanni Gentile. As we have seen, in the first years of the century Gentile and Croce were engaged in a heated debate against mathematicians, especially Enriques. Of course, the dispute focused not on mathematicians as such, but on the consolidation of an idealist hegemony in Italian culture.

2. Giovanni Gentile and school reform
From the turn-of-the-century onwards, Gentile had further enlarged his sphere of influence. Above all, he had openly affirmed his independence in regard to Croce in 1913, with a controversy. This did not lead to conflictual behaviour, but it certainly cooled down their relationship. He was called to Rome University on the 24th October 1917, the same day of the defeat of Caporetto. This “call” consecrates his authority, which was favoured by his new studies and philosophic publications, his contacts within the academic world and his particular civil commitment. Gentile was a member of the Superior Council of Education since 1915. In the years immediately following the war, he intensified his “public appearances” as a columnist for several national journals. He supported ideas which brought him politically closer to nationalists, even if he did not share their ideology.
His ability in positioning the scholastic question as of national importance certainly helped in raising his popularity and the move toward cultural hegemony. He posed it as a political, not just a financial, question. He put forward an over-all proposal for school reform – above all for the level that Gentile called scuola media (the first three years of the secondary school) – through a series of articles, speeches in conventions and publications. The reform was based on the simple recipe of few but good (state schools). There

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would be a quality state school for a small, elite number of students, and the majority would have the chance to enter private schools. In this way he realized a pragmatic embrassons-nous with the catholic world, which was always sensible to the issue of private school and teaching “freedom”. They particularly agreed on the proposal of introducing a state exam at the end of secondary school. With it, Gentile meant to ascertain the quality and rigor provided by these schools; Catholics saw it as a further step towards the equalization of state schools to private ones, which already passed this final test.
On the 16th of June 1920, in one of the frequent government changes, Croce became minister of Education. It was the right moment to strengthen again a personal and political relationship. Gentile’s public tasks increased. He had a well-known influence on Croce, who was certainly less committed on the pedagogic side. But Croce’s ministry lasted only a few months. The “march on Rome” was only a few months away.
Gentile was minister of Education during the first Mussolini government. He immediately set to work in order to fully use the 13 months of full powers (until the end of 1923) which fascism had to carry through the proposed financial and administrative State reforms and to legislate organic school reform. Croce had not been able to start it, although he tried through the escamotage of several gradual partial measures. The reform would obviously involve mathematics and mathematicians too. But in order to follow their vicissitudes, we have to see first the development of Gentile’s political path. He embraced fascism on the 31st of May 1923, with an open letter to Mussolini2.
Caro Presidente, dando oggi la mia formale adesione al Partito Fascista, La prego di consentirmi una breve dichiarazione, per dirLe che con quest’adesione ho creduto di compiere un atto doveroso di sincerità e di onestà. Liberale per profonda e salda convinzione, in questi mesi da che ho l’onore di collaborare all’alta Sua opera di Governo e di assistere così da vicino allo sviluppo dei principi che informano la Sua politica, mi son dovuto persuadere che il liberalismo, com’io l’intendo e come lo intendevano gli uomini della gloriosa Destra che guidò l’Italia del Risorgimento, il liberalismo della libertà nella legge e perciò nello Stato forte e nello Stato concepito come una realtà etica, non è oggi rappresentato in Italia dai liberali, che sono più o meno apertamente contro di Lei, ma per l’appunto, da Lei. E perciò mi son pure persuaso che fra i liberali d’oggi e i fascisti che conoscono il pensiero del Suo fascismo, un liberale autentico che sdegni gli equivoci e ami stare al suo posto, deve schierarsi al fianco di Lei3.
2 Cf. G. Gentile, La riforma della scuola in Italia, Le Lettere, Firenze, 1989, pp. 94–95. 3 Dear President, as I formally join the Fascist Party today, I beg You to let me make a brief statement. I
would like to tell You that I see my support as a necessary act of sincerity and honesty. I am a firmly fervent liberal. In these months I have had the honour to collaborate to Your high government work and to follow at close range the development of the principles on which Your politics are based. And I have persuaded myself that liberalism, as I see it and as the men of the glorious Right that guided Italy during the Risorgimento saw it, is not represented in present Italy by liberals, who are more or less openly against You. Indeed, it is You who represent that liberalism, the liberalism of freedom in law and therefore in the strong State, in the State conceived as an ethic reality. Hence I am also persuaded that, among contemporary liberals and the fascists who know Your fascism, an authentic liberal who hates misunderstandings and knows his place must side with You.

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Mathematicians began to line up too. Some joined the fascist Party or the fascist syndicate of University professors4. Gentile’s motivations to join the fascist Party were
enthusiastically approved of by Mauro Picone, who would define himself as “a black shirt since the very beginning”5.

Illustre e venerata Eccellenza,

Catania, li 5/VI/1923

mi consenta di esprimer Le tutto il mio vivissimo intimo compiacimento per l’adesione che Vostra Eccellenza ha voluto dare al Partito nazionale fascista al quale anch’io appartengo. La nobile lettera del 31 maggio da Vostra Eccellenza diretta al Presidente del Consiglio rimarrà memorabile nella storia di questi tempi. Quest’ultima adesione al partito fascista – così cospicua – e le meditate affermazioni contenute in quella lettera, vinceranno le esitazioni di tanti colleghi e porteranno ancora nuovo purissimo sangue nelle robuste vene del partito che ricostruisce e rinnova la Patria! Viva l’Italia! Con i più rispettosi ossequii, Le invio le sincere espressioni della più profonda mia devozione.

Mauro Picone6

The reform of school and of university by Gentile was composed of a set of formal decrees adopted during the whole of 1923, on the strength of the legislative proxy given to the government. At an administrative level the Educational system was organized in a rigidly centralistic way: elective representatives were abolished; headmasters of secondary schools, rectors, Faculty deans, Institute directors and components of the academic Senate would all be appointed by the minister. On the other hand, the control powers of the Institute directors and Faculty deans over the teaching staff were increased, which reinforced the sensitivity of the hierarchic order.
The new structural design of the scholastic system rejected the democratic example of a unique scuola media. It rather increased the choices young people had at their disposal, through a precocious channeling that intended to make middle school homoge-

4 Among the first to join the fascist syndicate of the professors of Rome University there is Giovanni

Vacca, Peano’s pupil and Volterra’s collaborator, historian of Mathematics and expert sinologist.

5 The letter is quoted in A. Guerraggio, P. Nastasi, Gentile e i matematici italiani. Lettere 1907–1943,

Boringhieri, Torino, 1993, p. 185.

6 Illustrious Excellence,

Catania, the 5/VI/1923

let me express You my deepest innermost satisfaction for giving Your support to the fascist national

Party to which I belong.

The valuable letter of the 31st May that Your Excellence addressed to the Prime Minister will stand out

in the history of our time. Your major support to the fascist party, and the well-considered statements

within that letter, will overcome the hesitations of many colleagues and will bring more and new pure

blood in the strong veins of the party which is rebuilding and reorganizing our country!

Long live Italy!

With my most respectful regards, and my deepest devotion.

Mauro Picone

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neous, as much as possible. This selection at the source also helped to develop the Liceo classico7. Instead, the “modern” (scientific) one was abolished and the technical school reformed, transformed and essentially downgraded to a vocational training school. In short, the reform’s fundamental axis was the neat division of secondary education along two paths: the classical-humanistic one (which had to educate the ruling class) and the technical one which aimed at spreading professional working skills. The importance and independence of scientific education diminished everywhere. It had to conform to the double formative channel. In the first path the teachings of Mathematics and Physics were paired. This answered to a need for greater oneness and organic unity and reinforced the formative character of scientific education (but the number of hours was actually decreased). The second path underlined the entirely instrumental aspect of mathematical teaching. At a university level, the same logic divided neatly between scientific Faculties, which aimed at the education of researchers, and the Polytechnics, which was to educate engineers.
There were certainly protests and resistance to the reform by Gentile. To start with, students opposed both the increase of taxes and of exams. It was the Autumn of 1923. The spreading of the students unrest obliged Mussolini himself to take the field on the 6th of December. He then labelled the reform as “the most fascist one among those which my government has approved”.
The Mathesis expressed the first strong perplexities and discontents among mathematicians. Enriques, its president since 1918, was engaged in a process meant to further entrench the association among teachers. The results arrived soon: within five years the Mathesis would almost triple its members, and in 1924 it numbered 1161 members. The protest of the Mathesis concentrated mainly on the specific aspect of the unification of the teachings of Mathematics and Physics in secondary schools, which the reform provided for. Sometimes the amount of hours scheduled for both teachings were less than Mathematics alone previously had. The vast majority of the teachers in Mathematics denounced the pairing. The reasons were several: the competence needed for both teachings, the difficulties to realize the reform with an untrained teaching body, the unbearable didactic load and, also, the “constitutional” diversity between the mathematician and the physicist (even if since 1922, to show its own openness, the Mathesis had become Società Italiana di Scienze fisiche e matematiche). The Mathesis reacted by appointing a Commission which had among its members Enriques and Castelnuovo. The latter asked for an encounter with the Minister to make sure (in the prudent and softened language of educational politics) that the reform would not diminish “the importance of scientific teaching”. Gentile declared himself “glad” to dispel the association’s worries. But when in May 1923, two months later, Enriques insisted by handing him counter proposals, in a summarizing pro-memorandum of the Society, he just answered that the proposals put forward were “hardly compatible” with the economic situation of the country8.
The opposition of the Mathesis was soon followed by that of the Accademia dei Lincei, the maximum expression of Italian science. In 1925 it was “reinforced” by the opposition of many University Science Faculties, owing to a 30% reduction in their equip-
7 Italian secondary school which underlines classical education. 8 Cf. Periodico di Matematiche, S. IV, vol. III, No. 4 (1923), p. 339–341.

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ment for scientific laboratories. Volterra – elected President of the Lincei – wrote to a Florentine physicist, Antonio Garbasso, in July 19239.
Tutti sono allarmati dalle riforme Gentile. La nostra Facoltà ha votato un ordine del giorno, ma non così vivace come avrei voluto. Né è passata la proposta di inviarlo anche al Presidente del Consiglio. Fui tra i pochi che votai in questo senso. Nell’ultima seduta dell’Accademia dei Lincei fu proposto e accolto il voto di incaricare una commissione di riferire sulle riforme, giacché l’Accademia non poteva disinteressarsene. Nella commissione sono entrati Scialoja, Pais, Bonfante, Castelnuovo ecc. Avrei voluto mettere anche il tuo nome, ma data la urgenza, e non potendo sperare nella tua venuta a Roma, non ho potuto farlo. Dopo ampia discussione il Castelnuovo relatore ha redatto un rapporto fatto a mio avviso molto bene il quale figura come relazione della Commissione alla Accademia10.
The conclusions of the analysis carried out by the Commission of the Accademia dei Lincei were quite explicit11.
Per merito di illuminati legislatori la scuola italiana tra il 1860 e il 1880 era salita ad un alto livello e poteva competere con le migliori straniere. Deplorevoli indulgenze e rilassatezza di disciplina avevano forse negli ultimi decenni diminuita l’efficacia della scuola; ma sarebbe bastata una mano ferma, che avesse rimesso in vigore le norme più austere, per ridare alla scuola l’antico prestigio, pur tenendo conto delle nuove esigenze portate dal progresso culturale ed economico del nostro paese. Una riforma radicale, per quanto ispirata da nobili intendimenti, non sembrava necessaria12.
As we have seen, several reasons and stances were mixed up in the reaction to the reform by Gentile. The protest of the Mathesis did not concern as much the base-elements
9 The letter is quoted in R. Simili, La presidenza Volterra, in R. Simili, G. Paoloni (eds.), Per una storia del Consiglio Nazionale delle Ricerche, Laterza, Roma-Bari, 2001, 2 vol. I, p. 91.
10 Gentile’s reforms have alarmed everybody. Our Faculty has voted on an agenda, but not so lively as I would have liked. And the proposal of sending it to the Prime Minister has not passed. I was between the few who voted for it. In the last sitting of the Accademia dei Lincei, the vote of entrusting a commission to report about the reforms was proposed and accepted, since the Academy could not neglect them. Scialoja, Pais, Bonfante, Castelnuovo, and others are its members. I would have liked to add your name too, but since it is urgent, and I could not hope in your coming to Rome, I could not do it. After a long discussion, the speaker Castelnuovo has written quite a good report, in my opinion. This is the report of the Commission to the Academy
11 Sopra i problemi dell’insegnamento superiore e medio. A proposito delle attuali riforme, Accademia Nazionale dei Lincei, Roma, 1923, p. 12.
12 The Italian school between 1860 and 1880 had risen to a high level and could compete with the best foreign ones thanks to enlightened legislators. Maybe in the last decades deplorable tolerances and discipline laxity had lessened the efficacy of school; but a steady hand could have restored the old prestige of the school, by reinstating the most austere norms, while considering as well the needs that cultural and economic progress had brought to our nation. A radical reform, although suggested by laudable intentions, was not necessary.

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of the reform by Gentile as some specific aspects, which caused also the instant mobilization of the teachers. The Mathesis did not question the scientific culture’s debasement, as one could have expected. The stance of the Accademia dei Lincei was based on a defence of the status quo: the Italian school, especially after the war, had undoubtedly some cracks, but they could be rectified just by reinserting a bit of “energy” and the rigour that made possible the post-unitary take-off. Considered the cultural and social atmosphere emerging in the country, the protests of the mathematicians are certainly bold. But they result from a defensive mentality, certainly different from the confident and propulsive behaviour of the beginning of the century. In a word, fascism embraced the change, even if it was a change towards restoration. And it had the charm resulting from proposals presented as new and resolutive of the backwardness that now prevailed.
The opposition of Mathesis could be borne, but the judgement of the Accademia dei Lincei needed a reply. In the press the Academy was accused of holding itself excessively apart from the country’s life life. Gentile himself took part directly and talked about “academic environment, where the sectarians opposing the Minister think they have some followers; they still don’t want to yield and recognize that the country’s cultural rebirth is one of the issues of the Government presided over by Mussolini”13. And in an interview in Milan’s newspaper “La Sera” on the 17th August 1923 he tried to distinguish the position of the Commission from the one held by the Academy as a whole, forgetting that Volterra was member of the first and President of the second. He continued thus: “if I wished to use a matter ad hominem, in the meanwhile I would begin by questioning my critic’s competence to criticize me. Obviously, they are all renowned scientists. But how many of them have focused their spiritual work on the academic problem, as I can boast?”14.
Despite Gentile’s presumption, opposition to his projects in the academic surroundings increased. Even Severi, in his capacity as President of the Associazione nazionale dei professori universitari, rose up against the threatened purge of the university teachers stated in a decree of March 1923.

3. The battle of the “manifestos”
Anyway, it would not be Gentile who would implement his reform. The political situation plunged quickly and on the 25th January 1924 the Chamber of Deputies was dissolved. On the 6th April there was an election. The atmosphere was well represented by Mussolini’s statement: in any event, he would not leave the power, which he had won not through “paper games” but through “revolutionary right”. Fascists, though, did not want to run any risk and preferred to make sure that they would win by using pressure, intimidation and violence. They had a majority, with four and a half million votes and 356 deputies, while the opposition (with three million votes) won only 179 deputies, dispersed in several and sometimes tiny lists.
13 Cf. G. Gentile, La riforma della scuola in Italia, quoted, p. 140. 14 Cf. L’Università italiana, a. XIX (1923), No. 6, p. 73.