Chapter 9
THE CREATION AND PERSISTENCE OF NATIONAL SCHOOLS: THE CASE OF ITALIAN ALGEBRAIC GEOMETRY*
Aldo Brigaglia
Algebraic geometry, in spite of its beauty and importance, has long been held in disrepute by many mathematicians as lacking of proper foundations. The mathematician who first explores a promising new field is privileged to take a good deal for granted that a critical investigator would feel bound to justify step by step; at times when vast territories are being opened up, nothing could be more harmful to the progress of mathematics than a literal observance of strict standards of rigor. Nor should one forget, when discussing such subjects as algebraic geometry, and in particular the work of the Italian school, that the so-called ‘intuition’ of earlier mathematicians, reckless as their use of it may sometimes appear to us, often rested on a painstaking study of numerous special examples, from which they gained an insight not always found among modern exponents of the axiomatic creed … [But] In this field the work of consolidation has so long been overdue that the delay is now seriously hampering progress in this and other branches of mathematics … Our chief object here must be to conserve and complete the edifice bequeathed to us by our predecessors. ‘From the Paradise created for us by Cantor, no one shall drive us forth’ was the motto of Hilbert's work on foundations of mathematics. Similarly, however grateful we algebraic geometers should be to the modern algebraic school for lending us temporary accommodation, makeshift constructions full of rings, ideals and valuations, in which some of us feel in constant danger of getting lost, our wish and aim must be to return at the earliest possible moment to the palaces which are ours by birthright, to consolidate shaky foundations, to provide proofs where they are missing, to finish, in harmony with the portions already existing, what has been left undone [Weil, 1946, p. vii].
Andre Weil's discussion of his predecessors in algebraic geometry raises many questions. In what sense was ‘the work of the Italian school’ actually rooted in a national tradition? Was ‘intuition’ a specific attribute of the ‘Italian’ school of algebraic geometry? How did this Italian tradition interact with the developing structural style in mathematics which was spreading out mostly from Germany? The following will briefly sketch a few tentative answers to these questions.
Luigi Cremona (1830–1903) is generally considered as the founder of the Italian school of algebraic geometry. Highly projective, his methods were heavily influenced by the French tradition (Chasles, Poncelet) and the German one (Möbius, Plücker, Staudt, Clebsch). Cremona has been credited with the first extensive use, by a geometer, of the idea of birational transformations (also called Cremona transformations). Among his students, one may cite E. Bertini (1846–1933), G. Veronese (1854–1917), and C. Segre (1863–1924). In Pisa, Padua, and Turin, respectively, they were the true architects of a flourishing Italian school. Graduating from Padua, Castelnuovo and Severi—as well as Enriques who received a degree from Pisa—all spent some time in Turin with Segre, a mathematician who exerted a strong influence on all of them.
In the following, the word ‘school’ will be used in a free way, with no specific sociological meaning. In a sense, one of the most important features of the Italian ‘school’ of algebraic geometry was that everyone, in the scientific milieu, could understand perfectly the meaning of being part of this school. In this context, it is important to underscore that, beginning with Cremona's students, Italian algebraic geometers were conscious of belonging to a definite scientific ‘school.’ In his obituary of Cremona, Castelnuovo, for example, wrote [1930, p. 614]:
In order to give life to a school, neither the founder's qualifications nor his ability of outlining a research program that exceeds his own capacity suffices. It is moreover necessary that he succeeds in communicating his passion and his faith to his disciples, knows their demands, and directs their collaboration. Luigi Cremona was eminently endowed with these qualities.
Another important feature of the Italian school of algebraic geometry was that its growth was strictly tied to a major problem in the development of the discipline. At the end of the last century, algebraic geometry was advancing in a chaotic way—recall that Dieudonné [1974,1, ch. VI] labeled this period ‘développement et chaos'—and was completely split up into mutually non-interacting tendencies. There was an arithmetic school (Dedekind, Weber), a geometrical one (the Italian school), and a ‘transcendental’ one (the French school of Picard, Humbert, and Poincaré). Communication between these different schools was made difficult by the lack of a common language; only with the creation of new and powerful technical tools in the 1960s (Grothendieck's theory of schemes) this was eventually achieved. In his influential History of Mathematics in the 19th Century, Felix Klein was therefore entitled to write [1926, I, p. 315]:
Algebraic geometry had developed like a tower of Babel: quite quickly it became clear that the various languages were not being mutually understood.
In this context, the very fact that the appendix to Picard and Simart's book on surfaces (1906) was eventually written by Enriques and Castelnuovo may be an indication of the French scholars’ difficulty of truly grasping the Italians’ results.1
Moreover, the Italian school was not strictly a national ‘school,’ but rather a working style and a methodology, principally based in Italy, but with representatives to be found elsewhere in the world. In this regard, the words used by Mumford are typical: ‘The Italian school, and notably Severi, Todd, Eger and B. Segre developed a general theory of Chern classes in the algebraic case.’ In other words, Mumford used (rightly from my point of view) the phrase ‘Italian school’ as a label for, not a strictly national community, but rather a more general concept. Similarly, algebraic geometers coming from Belgium (e.g. Godeaux), or the Unites States (e.g. Coolidge) could also be counted as members of the ‘Italian school.’
Lastly, we may observe that it is especially appropriate to speak of a ‘school,’ here, in view of its long persistence (at least four generations spanning a whole century) and the large number of its members. An appendix lists the school's main members who had important (and sometimes decisive) contributions to the discipline.
SEGRE AND PEANO
It was during the years 1885–1891, in Turin, that many important features of an Italian style in geometry were established through the work of Segre, Castelnuovo, and, later, Enriques (who, although he only stayed in Turin for a few months, always remained in a close touch with the former two scholars). Obviously an intuitive, not very rigorous approach to algebraic geometry was already used by Cremona, Veronese, and nearly all scholars during the 1860s and 1870s. But this was, in some sense, a naive use of intuition; while, on the contrary, in the 1880s, the Turin geometers stressed their right to use intuition in a theoretical fashion. In 1891, G. Peano (1858–1932) clashed with Segre in a fierce debate about rigor versus intuition, which made this distinction in the use of intuition explicit.
At that moment, the two young scholars (Segre was then 28, and Peano 33) were at the peak of their creative strength. Segre had just completed his program of using hyperspatial geometry in order to achieve a sound basis for the so-called ‘geometry on a curve,’ and he had begun his collaboration with Castelnuovo (two years his junior). On his part, Peano had just published his most influential foundational work (Calcolo geometrico, 1888; Arithmetices princìpia, 1889; Principii di geometria, 1889). Both were thus eager to defend their respective practice in geometry: intuitive and creative for Segre, rigorous and formally unexceptionable for Peano. Before we examine briefly their important debate, let us turn to young Segre's first steps in his mathematical career.
SEGRE'S METHODS
On May 29,1883, Corrado Segre took his degree in Turin under Enrico D'Ovidio's guidance. At this time, the very beginning of Italian geometers’ methods, Segre was only 20 years old and Peano 25. The latter was deeply engaged in the rigorization of analysis, and his Aggiunte to his teacher Genocchi's book Calcolo differenziale ed integrale were going to be published in the same year—1884—as Segre's dissertation. Moreover both were engaged in the study of Grassmann's Ausdehnungslehre, and in 1888 Peano would publish his fundamental Calcolo geometrico secondo I'Ausdehnungslehre di H. Grassmann preceduto dalle operazioni della logica deduttiva.
Segre's thesis [1883a,b] was published a year after his defence in the journal of the local Academy of Science, and soon became a fundamental starting point for the development of Italian projective n-dimensional geometry. This paper was devoted to the general study, by geometrical tools, of ra-dimensional vector spaces (projective spaces) and in particular bilinear forms defined on them (quadrics).
At the beginning of the 1880s, as Segre [1891] later pointed out, different ways to give meaning to the idea of n-dimensional spaces existed. A purely analytical, ‘naive’ interpretation considered the space R” in an analytical way, simply covering it up with a geometrical language; in Plücker's style, n-dimensional objects were objects in usual space defined through n complex parameters (in our language, it amounts to saying that this space was not defined to be R”, but isomorphic to it); finally, following a Euclidean style, Veronese had used an axiomatic approach to try and define multidimensional spaces, in a partially unsuccessful way in 1882, and more completely in his later influential treatise [Veronese, 1891].
Many foundational problems were left completely unresolved. First of all, an ‘abstract,’ general, framework in which it would have been possible to consider any n-dimensional space, remained to be found. One needed a precise definition of the idea of moving from one space to another, while giving different meanings to the names (point, line, etc.) of the objects of the abstract space (isomorphism). Beside these problems, there was that of connecting the various points of view, and finding a set of independent axioms from which it would have been possible to deduce the Cartesian representation (coordinatization).
On this matter, Segre wrote in his dissertation:
The geometry of spaces with an arbitrary number n of dimensions has already taken a well-deserved place among mathematical disciplines. Even when we consider it outside the important applications to ordinary geometry that it may give us, i.e. when the element or point of such a space is not considered as a geometrical object of the ordinary 3-dimensional space … but as an object whose intimate essence has not been determined, we cannot refuse to consider it as a science in which every proposition is rigorous, because it can be proved by essentially mathematical reasonings; the lack of a sense representation of the objects we study has not much meaning for a pure mathematician.
Any continuous set of objects, whose number is m times infinity (i.e. such that there generally are a finite number among them that satisfy m simple arbitrary conditions) is said to be an w-dimensional space, of which these objects are elements.
Any w-dimensional space is said to be linear when it is possible to attach to each element m numerical values (real or complex) in such a way that, with no exception, to any set of values there corresponds one and only one element. The values of these quantities corresponding to the element are said to be its coordinates. If we characterize them by the ratios of m quantities with an (m + l)th, they will constitute the m + 1 homogeneous co-ordinates of the element of the space, so that any element, with no exception, will be characterized by the mutual ratios of these homogeneous co-ordinates and, vice versa, will characterize their ratios.
Besides being confusing, the definition is clearly wrong (e.g., the correspondence is not one to one and so on because the element (0, 0, 0, …, 0) does not have any correspondent and the word ‘ratio’ has no meaning when the second number is 0), as Peano was to reproach Segre during the controversy.
But, notwithstanding the obvious lack of rigor, Segre's definition was an important working tool, which could effectively allow him to achieve a great number of new, more advanced, algebraic results that he was always able to interpret in an arbitrary linear space. This capacity required an idea, albeit a still confused one, of isomorphism between two linear spaces, an idea he expressed as follows:
Any two linear spaces with the same number of dimensions, independently from the nature of their elements, can be considered as identical to one another, because, as we already noted, when we study them we do not pay attention to the nature of these elements, but only consider the property of linearity and the dimension of the space formed by the elements themselves. It follows that, being already known, the theory of linear forms of first, second, and third kind, for instance, of the line, of the plane, and of the space considered as dotted, may be used for any linear space of dimension 1, 2, or 3 contained in the linear space of dimension n-1 that one wants to study in general. Therefore one can use, for example, the theory of projectivity, of harmonic groups, of involution, etc., in the forms of the first kind. [Segre 1883a, p. 46].
Segre's method was fundamentally based on the so-called hyperspatial projective geometry. He fully applied his method to the complete and exhaustive study of quadrics (bilinear forms) in an n-dimensional projective space, and used his results to classify the quartic intersections in 3-dimensional space. In two years, he achieved many important results on second-degree complexes (1883–1884), line geometry (1883), Lie's sphere geometry (1884), the classification of collineations and correlations in a linear space of finite dimensions (1884), the surfaces of 4th degree with a double conic (1884), the sheaves of cones in «-linear spaces (1884), and Grassmann's varieties (1886).
Segre's methods depended on a massive use of the most recent results of linear algebra, particularly those obtained by Weierstrass (1868), Frobenius (1878), and Kronecker (1878) on matrices and bilinear forms. In particular, they required a general definition of linear spaces which was, in 1883, completely lacking (Grassmann's Ausdehnung-slehre contained many ideas on the subject, but was almost completely unknown in these years).
It is worth noting that in his book on Calcolo Geometrico, Peano, who worked in the same mathematical department as Segre, was able to give the first complete definition of a linear (real) vector space [Brigaglia, 1996], a definition which is completely analogous to the contemporary one. We cannot understand why the two young scholars, who were working on similar problems, did not have any mutual influence, without looking more carefully at their different mathematical attitudes.
Segre's and Peano's respective approaches might be described using Hilbert's words:
The building of science is not raised like a dwelling, in which foundations are first firmly laid and only then one proceeds to construct and enlarge the rooms. Science prefers to secure as soon as possible comfortable spaces to wander around and only subsequently, when signs appear here and there and there are that the loose foundations are not able to sustain the expansion of the rooms, it sets to support and fortify them, (quoted in [Corry, 1996, p. 162]).
Segre was working precisely on the building of ‘comfortable spaces’ for the ever growing needs of algebraic geometry; only as far as to ‘sustain the expansion of the rooms’ was he ever interested in foundations. Peano instead held the idea that ‘the axiomatic approach may be used only to organize well established theories, as classical mathematics’ [Lolli, 1985] and not to build new ones (as in Hilbertian mathematics).
In any case, in spite of its lack of rigor, Segre's point of view opened up new vistas towards further developments, even ones of a formal algebraic kind :
It has been noted … that for instance the general projective geometry that we build in this way, is not anything other than the algebra of linear transformations. This is only a difference not a fault. Provided that we do mathematics! [Segre, 1891, p. 405].
A closer look at Segre's arguments will clearly underscore the usefulness of ‘hyperspatial’ geometry, as illustrated by a statement made many years later by an American adept of the Italian school, C. Coolidge:
Higher spaces have two claims on our attention. One is that of giving to our formulae the greatest possible extension to afford our theorems the widest generality, the other is to provide a suggestive and powerful technique for studying the properties of figures in lower space, which depend simply on a number of explicit parameters [Coolidge, 1940, p. x]
This was exactly the direction in which Segre moved. His ideas were based on a powerful synthesis of several mathematical theories: Plücker's idea of hyperspaces generated by an element of ordinary space, Klein's Erlangen Programm, Clebsch's and Noether's results on birational geometry, and Weierstrass's, Frobenius's, and Kronecker's algebraic results. He succeeded in making a complete and orderly theory which would provide a sound foundation for Italian algebraic geometers. His ideas completely pervaded the famous book [Bertini, 1907]. His methods and results have now become so commonly used and natural that it is difficult for us to understand the novelty of his work.
At the core of Segre's ideas, was the concept that a translation of new algebraic results into geometrical language could be used to build (using the same methods) a coherent, organic theory of homographies in a projective space, conies and their pencils, correlations and polarities with respect to a quadric in any dimension and with any concrete interpretation of the word ‘point'. In a word, his aim was: ‘to build a general theory of projective n-spaces'. The idea of ‘translation’ was very deeply rooted in his way of looking at mathematics. For example, when citing Weierstrass's [1868] important theorem, on elementary divisors, he wrote:
This theorem was proved by Weierstrass in an analytical form, and it gives a complete answer, from a modern algebraic point of view, to any question on the different situations which may arise from a system of two quadratic forms and its invariants, giving a complete classification, from this algebraic point of view, of these systems of forms. We will try to translate them into a geometrical setting. [Segre, 1883a, p. 94].
In a framework in which the main algebraic ideas were far from clear, and modern abstract linear algebra not yet well established, Segre and his students immediately began to make extensive use of mathematical tools then only crudely developed. More and more, geometrical intuition was requested to overcome algebraic and analytical difficulties, which lay largely outside the possibilities of their times, as was expressed by the young Castelnuovo [1889b, p. 65]:
We must acknowledge that, in establishing this result we base ourselves more on intuition (and on many verifications), than on a true mathematical reasoning … But we let ourselves use a result not yet proven in order to solve a difficult problem, because we think that it is possible even with such attempts to be useful to the development of science, provided that we explicitly declare what we assume and what we prove.
Castelnuovo's results in this work were almost immediately picked up by Segre [1889] with these words: ‘The ingenious demonstration that this geometer gave of this important formula might cast doubt on its absolute validity, which would be reflected on the present number and below … ; however, confirmations found of these results lead me to think that they are absolutely true.’2 The necessity of having recourse to intuition was being turned into a method. Due to the many great results obtained in the next few years through the use of this method, it would soon become the seal of an Italian style in mathematics.
PEANO'S CRITICISMS
The systematization of Segre and his students’ recourse to intuition had Peano really worried. In his ‘Observations’ appended to Segre's article [Peano, 1891, p. 67], he ignited a fiery debate by quoting Segre's words—just cited above—adding the following comments:
We believe that it is very strange that in authoritative journals, occupied with the purest kind of mathematics, one may write [such words] … Be it only for an instant, we would like to have a voice influential enough among our colleagues so as to convince them of this truth: works lacking in rigor cannot make mathematics progress even by one step.
Later on in the same text, Peano explained:
We maintain that when it is possible to find a single exception, a proposition is false; and that we cannot consider a result as being truly achieved, even if no exception to it is known, if it has not been rigorously proved, [p. 68].
This drastic attitude went hand in hand with distrust for unrestrained abstraction:
Any writer may assume those experimental laws he likes, and he can make the hypotheses he prefers. The correct choice of these hypotheses has a great importance for the theory one wants to develop; but this choice is made by induction and does not belong to mathematics … If an author assumes hypotheses opposite to experience, or hypotheses which are not verifiable by experience, nor are their consequences; he will be able, I grant, to prove some wonderful theories, such wonderful ones as to have us cry: what a profit [for science this would have been] had he used his capacity to the service of useful hypotheses! [p. 68].
Young mathematicians, like Castelnuovo, Enriques, and Severi, entered the debate with the goal of loosening what they held as the unduly strict requirements of rigor in mathematics. They shared an attitude according to which, hampering the younger scholars’ creative strengths, formalism and excessive rigor were old mathematics. New mathematics was abstract but geometrically intuitive.
THE ACTORS’ VIEWPOINTS
Based on geometrical intuition, the ‘new’ method was to be increasingly discussed by leading Italian scholars, and soon held as the principal source of many of their important results. Here only a few examples will be provided, which the reader may compare with Weil's words cited earlier. The crucial point to stress is that the leading scholars of the Italian school clearly thought of themselves as explorers of a new land and in no way, as ‘mathematical architects’ in Bourbaki's sense. This attitude becomes particularly striking when looking at their treatises (see Enriques’ citation below): much more than to provide students with the ‘logical structure’ of mathematical results, these were written with the goal of giving a ‘perspective of their coming into being'—an aim very different from, say, van der Waerden's in his almost contemporary Moderne Algebra.
In his address to the International Congress, Castelnuovo was thus quite explicit about his method [1928, p. 194]:
We built, in an abstract sense of course, a large number of models of surfaces in our space or in higher spaces; and we arranged these models into two display cases. One contained the regular surfaces, for which everything proceeded as in the best of possible worlds; by analogy, the most salient properties of plane curves were conveyed to these surfaces. But when we tried to verify these properties for the surfaces in the other display case, the one containing irregular surfaces, then, the trouble began, and exceptions of every sorts turned out. At the end, an assiduous study of our models led us to guess some properties which had to be true, with appropriate modifications, for surfaces in both cases; we then put these properties to the test by constructing new models. If they stood up to the test, we looked—this was the ultimate phase—for a logical justification.
Similarly, according to Severi, analogy and experimentation fully belonged to the mathematician's practice [1937, p. 58]:
Analogies are often precious; but in many other cases they are prisons where for lack of courage the spirit remains fettered. In any branch of science, however, the most useful ideas are those that born orphans later find adoptive parents.
New mathematical constructions do not actually proceed exclusively from logical deduction, but rather from a sort of experimental procedure making room for tests and inductions—these constitute the real ability necessary for exploration and construction. And thus is it natural that theory development entails some modifications of, and adjustments to, initial conceptions and the language expressing these conceptions; and that one can reach a definite systematization only by successive approximations.
As for the exposition of results, Enriques [1949, p. x]3 observed:
Not only to give to the exposited theories a logical structure, but also … to provide an historical perspective of their coming into being. In this way one wants to offer the reader, not just the gift of something perfect at which one is allowed to look from the outside, but rather the vision of an acquisition and an advancement, the reasons for which one must understand and which the reader is invited to re-earn by, and for, himself, finding in the book a working tool … To the model of a rational science logically ordered as a deductive theory, which must appear everywhere complete and perfect, and which, going down from more general concepts to particular applications, drives off uncertain, changeable appearances of reality—all of which recalling the dark past of the search or uncovering new difficulties, breaking the harmony of the system, … we prefer the general philosophy of science resulting from modern critiques … which goes beyond the opposition between deductive and inductive methods, to reach the consideration of deduction itself as a phase in a unique process that rises from the particular to the general in order to go down again to the particular.
Even more dramatic was the following of Enriques's claims: ‘We aristocrats do not need proofs. Proofs are for commoners (quoted by Zariski in [Parikh, 1990]).’
Eventually, it is also worth quoting a passage from Castelnuovo's preface to this same book of Enriques's [1949, p. vn]:
Will someone to continue the work of the Italian and French schools come soon, who will finally succeed in giving the theory of surfaces the same perfection as the theory of algebraic curves has reached? I hope so, but I doubt it. The observation that mathematics has in this century taken quite a different direction from the predominating one in the last century, feeds my doubts. Imagination and intuition, which, then, used to guide research, are today regarded with suspicion because of the terror of the errors to which they can lead. Theories, then, arose in response to the need felt by the mathematician for making precise various objects of his thought that already in a vague form were present in his mind. It was the exploration of a vast territory seen from a faraway peak … Today, more than the land to be explored, the road leading there is of interest, and this road now is strewn with artificial obstacles, now makes its way free among the clouds.
In contrast to the ideas quoted above, it is striking to note that the formal achievements in algebraic geometry of the van der Waerdens, Zariskis, and Weils formed one of the pillars of a rising tide of structural ideas sweeping twentieth century mathematics as a whole. One may wonder how the above ideas coexisted with structuralist ones. In the following, I will limit myself to a few quotations and brief comments.4
A good starting point for the understanding of a structuralist appreciation of the work and method of the Italian school might be Fulton's following remark [1984, p. 26]:
It would be unfortunate if Severi's pioneering work in this area were forgotten; and if incompleteness or the presence of errors are grounds for ignoring Severi's work, few of subsequent papers on rational equivalence would survive.
Well aware of recent results of the German school in algebraic geometry, Severi was sharply ironic [1940, p. 224 of Memorie Scelte]:
That algebraic geometry can also be seen from the viewpoint of modern algebra—as van der Waerden for example does in his interesting works—is undoubtedly a good thing for algebra and for geometry. One also hopes that the penetrating methods of Moderne Algebra are soon to be used in an attack of essentially new problems, rather than only in the reconstruction of results already discovered by geometric means.
With barely hidden satisfaction, Severi added that van der Waerden's [1939, 202] treatise was more easily reconciled with ‘the methods of the Italian school than with those of Moderne Algebra [Severi 1940, p. 225 of Memorie Scelte].’
As one may see, Severi's criticism attacked the main objectives of the new algebraic methods. In fact, these methods were precisely intended more for ‘the reconstruction of results already discovered by geometric means’ than for ‘an attack on essentially new problems.’ In Severi's mind, the former goal was a minor one as opposed to the latter, which was obviously the purpose of ‘aristocrats’ in mathematics. It must be stressed however that fundamentally new results in algebraic geometry (and not, using Severi's words, mere ‘reconstructions') were only obtained by the ‘structural’ methods at the end of the 1950s with Hironaka's results on singularities.
Appearing almost in the same years (1932–1934), the following two quotations from books written on the same argument (resolution of singularities) exemplify (in my opinion) the strong contrast between the two different styles of writing about algebraic geometry. A result that Enriques commented on only in a few confident lines—'That in fact the indicated transformation process ends up by completely resolving all singularities was rigorously proved by B. Levi and O. Chisini [Enriques 1932, p. 10]'—received, in Zariski's hands, a rather more complex appreciation [1934, p. 18]:
The proofs of these theorems are very elaborate and involve a mass of details which it would be impossible to reproduce in a condensed form. It is important, however, to bear in mind that in the theory of singularities the details of the proofs acquire a special importance and make all the difference between theorems which are rigorously proved and those which are only rendered highly plausible.
And Zariski then went on with many pages of detailed reasoning. In this context, young Italian scholars were completely discouraged from following new ‘structural’ trends of German school—a discouragement recalled by Zariski, who studied in Italy until the end of the twenties:
It was a pity that my Italian teachers never told me that there was such a tremendous development of the algebra which is connected with algebraic geometry. I only discovered this much later, when I came to the United States. [Quoted in Parikh, 1990, p. 36]
Although the conflict between the German structural, and the Italian intuitive, practices in algebraic geometry has been stressed, we must also recognize that the two different styles were, in many a sense, strictly interrelated. In the thirties, van der Waerden wrote his many papers on algebraic geometry and developed his powerful results while always bearing in mind the great results of the Italian school (and mainly Seven's), even if he considered them as often incomplete and sometimes also wrong. While he developed his methods of commutative algebra and topology, Zariski never forgot the problems that aroused his interests when he was a student in Rome, a context which shed light on an interesting testimony of his [Parikh, 1990, p. 76]:
I wouldn't underestimate the influence of algebra, but I wouldn't exaggerate the influence of Emmy Nöther. I'm a very faithful man … also in my mathematical tastes. I was always interested in the algebra which throws light on geometry and I never did develop a sense for pure algebra. Never. I'm not mentally made for purely formal algebra, formal mathematics. I have too much contact with real life, and that's geometry. Geometry is the real life.
This kind of statement might well have been uttered by Zariski's Italian teachers. Similarly Solomon Lefschetz, the first to succeed in putting ‘the harpoon of algebraic topology into the body of the whale of algebraic geometry’ [to use his own words, 1968, p. 854], never hid his admiration for the Italian scholars’ results nor his debts to them.
CONCLUSION
To conclude, I would like to stress that (counterfactually, of course) it might indeed have been possible, starting from inside the Italian tradition, to develop a structural view of mathematics. One can hint at this simply by providing a very short account of Gaetano Scorza's work, which clearly moved in this direction (Scorza was the author of Corpi numerici e algebre [1921], one of the first books written in a modern algebraic style).
The theory of Abelian functions … and some of the more elevated theories of algebraic geometry present such frequent analogies and such remarkable affinities that whoever examines them a bit more closely is led almost spontaneously to presume that they must all be embedded in a single general theory, and that this must provide the best explanation for the numerous and close ties that link them. And since in every research on this subject there is always … a certain ‘period table’ [matrix], … one is immediately led to think that the properties of exactly this matrix must play an essential role in those theories … Through more or less recent works, … these properties have become so numerous by now and have acquired such prominence that it seemed to me that… the necessity arose of composing an organic and ordered treatment of them in a manner completely independent of the various concrete interpretations they are susceptible of. [Scorza, 1916, p. 127 of Opere Scelte].
This kind of attitude prompted Scorza to explore the deep links between ‘structural’ abstract algebra (particularly linear algebra) and the theory of Abelian matrices. I would like to provide here one example of his use of structural algebraic properties, while ‘translating’ them into geometrical ones. After defining a Riemannian matrix (R. M.) as the matrix of periods of an Abelian function, he attached to any R. M. an algebra A (the algebra of multiplications) and finally introduced this vocabulary:
The algebraic dimension of A corresponds (diminished by 1) to its multiplicability index.
A is isomorphic to A' if and only if their corresponding matrices are isomorphic.
A is reducible if and only if its corresponding matrix is impure with complementary axes.
A is a division algebra if and only if its corresponding matrix is pure.
A is simple if and only if its corresponding matrix is has no complementary axes.
This usage of an algebraic language was very close to the contemporary works of A. Albert in the United States. It moreover deeply influenced Lefschetz (who was in Italy in the years 1920–21 and got in touch with Scorza) who acknowledged:
We propose now to recall some concepts and definitions incipient in the works and only fully developed in recent writings of Scorza and Rosati. The nomenclature which we shall use is Scorza's [Lefschetz, 1921, p. 78 of Selected Papers].
A few years later, Lefschetz was even more explicit:
The theory of Abelian functions no doubt is one of the most important to have occupied mathematicians. It is the more so striking that our knowledge of matrices of period per se has until very recently remained rather fragmentary. A few fundamental theorems for an arbitrary genus p were well established and, more than anyone else, the late George Humbert studied the case p = 2 in depth. But, for this case, albeit not easy, direct calculations at least are possible; so, his methods scarcely are easily extendible. Above all, we owe to M. Scorza to have largely lifted the veil. No doubt his works would have spurred a considerable attention had they not appeared during the war, had the method he used [be different]. Belonging to projective geometry, and for the occasion applied to a question important for analysts, [this method] requires from the latter a special training which one can hardly expect them to possess. It would therefore seem highly desirable that M. Scorza's results be one day achieved analytically. I wish to testify that in the course of some of my researches his method whose elegance is assured was of considerable help to me. I thus believe to be doing something useful in summarizing here the main lines of his work [Lefschetz, 1923, p. 120].5
In this way, we may see that inside the Italian geometrical language and tradition, tendencies existed that could, in an original way, be merged with rising international trends. In conclusion, we may ask why these efforts were not successful. I have no definitive answer and may only suggest two tentative ones.
Firstly, there is a general answer. National tradition may evolve from a motive power in the development of science to an academic power thwarting the development of new ideas and methods. While useful to understand some trends in mathematical development, phrases like ‘national school’ or ‘national tradition’ are not without danger. The academic milieu often acts as individuals who, while looking for ancestors, choose among the many branches of their family tree the ones they prefer, while completely neglecting others. In our case, with their unmistakable stress on their intuitive and geometrical capability of overcoming technical difficulties (and, I may add, with the brilliant results they were able to achieve), Italian algebraic geometers had a clear awareness of partaking in a mathematical community deeply rooted in the cultural national terrain (which we may call a national school). But they (and above all Severi) often completely forgot that this school had emerged while maintaining both a deep understanding of the most recent algebraic results and firm links with the leading mathematical schools in foreign countries. Eventually, this national arrogance (perhaps encouraged for political reasons by the fascist regime) seriously hampered the further development of Italian algebraic geometry, until a point was reached when it was ‘held in disrepute’ (to put it with Weil's words) by the international community.
Secondly, I may suggest a more internal reason. As successful as they may have been in the previous period, Italian geometers’ methods were steadily reaching the limits of their creative strengths in the face of the ever growing complexity of the problems involved. Moreover, while (as Enriques claimed) aristocrats can dispense with proofs, their methods were hardly adapted to the demands of a period when exciting victories were to be followed by a weary trench warfare. Hampering new advances, the retention of traditional methods would throw a new generation of Italian algebraic geometers into deep crisis.
In this situation, any young Italian mathematician (for example beginning his studies at the end of the 1960s) who wanted to study modern algebraic geometry was compelled to sever any tie with the national tradition in order to merge completely in the international milieu. From this situation, a very unfair opinion has resulted, even from an historical point of view, about the relevance of the contributions to twentieth-century geometry by Italian geometers. Only in recent years (starting in the 1980s) have young scholars regained a consciousness of the debts owed by new mathematics to these pioneering works; and the debate on intuition and rigor in mathematics has picked up momentum once again [Jaffe and Quinn, 1993]. The example of the Italian ‘school’ of algebraic geometry may become an important test case for the evaluation of different methodological assumptions when looking concretely, for how, mathematical research develops and new ideas grow.
NOTES
* English version revised by David Aubin.
1. see the correspondence Enriques-Castelnuovo in [Bottazzini et ai. ii, 1996].
2. La demonstration mgénieuse, que ce géomètre y donne de cette importante formule, pourrait laisser sur sa valídité absoíue des doutes, qui se réfléchiraient sur le n° present et plus loin … ; cependant les confirmations qu'on trouve de ces résultats me portent à penser qu'ils sont absolument vrais.
3. It is possible to find many analogous concepts in [Enriques-Chisini, 1915].
4. For a more complete description of this issue see [Brigaglia-Ciliberto, 1995].
5. 'La théorie des fonctions abéliennes est sans contredit une des plus importantes qui aient occupé les géomètres. II est d'autant plus remarquable que notre connaissance des matrices aux périodes per se soit restée jusqu'à tout récente date plutôt fragmentaire. On possédait bien quelques théorèmes fondamentaux pour le genre p quelconque et le regretté Georges Humbert, surtout, avait étudié a fond le cas p — 2. Mais pour ce cas le calcul direct, sans être facile, est tout au moins abordable; aussi ses méthodes ne se prêtent-elles guère à une extension facile. C'est surtout à M. Scorza que !'on doit d'avoir largement levé le voile. Ses travaux auraient sans nul doute stiscité une attention plus considerable, n'était d'abord Ieur apparition en pleine guerre, ensuite la méthode dont il s'est servi. Relevant de la géometríe projective, appliquée en 1'occasion à une question très importante pour les analystes, elle exige de ces derniers un entraínement special que í'on ne peut guère s'attendre à le voir posséder. II semblerait done fort desirable qu'on arrive un jour aux résultats de M. Scorza par voie analytique. Je tiens à rappeler qu'au cours de certaines recherches, sa méthode, dont Félégance est hors de doute, me fut d'une aide considerable. Je crois done faire oeuvre utile en résumant ici ses travaux dans leurs grands traits.’
APPENDIX
I give a tentative list of the main contributors to Italian school of algebraic geometry. For every name I give the name of his teachers ânâ the places where he taught.
First generation:
Luigi Cremona (1830–1903); taught in Bologna, Milano, Roma;
Giuseppe Battagliní (1826–1894); taught in Napoli and Roma;
Second generation:
Riccardo De Paolis (1854–1892); student of Cremona; taught in Pisa;
Eugenio Bertini (1846–1933); student of Cremona; taught in Pavia and Pisa;
Enrico D'Ovidio (1843–1933); student of Battaglini; taught in Torino;
Giuseppe Veronese (1857–1917); student of Cremona; taught in Padova;
Corrado Segre (1863–1924); student of D'Ovidio; taught in Torino;
Mario Pieri (1860–1913); student of Peano and Segre; taught in Catania;
Third generation:
Guido Castelnuovo (1865–1952); student of Veronese and Segre; taught in Roma;
Federigo Enriques (1871–1946); student of De Paolis; taught in Bologna and Roma;
Francesco Severi (1879–1961); student of Veronese and Segre; taught in Padova and Roma;
Gino Fano (1871–1952); student of Segre; taught in Messina and Torino;
Michele De Franchis (1875–1946); taught in Catania and Palermo;
Gaetano Scorza (1876–1939); student of Bertini; taught in Catania, Palermo, Napoli and Roma;
Carlo Rosati (1876–1929); student of Bertini; taught in Pisa;
Beppo Levi (1875–1961); student of Segre and Peano; taught in Bologna and Torino;
Guido Fubini (1879–1945); student of Bertini and Bianchi; taught in Catania and Torino;
Giovanni Giambelli (1879–1953); student of Segre; taught in Messina;
Luigi Brusotti (1877–1959); student of Berzolari; taught in Pavia;
Fourth generation:
Annibale Comessatti (1886–1945); student of Severi; taught in Padova;
Alessandro Terractm (1889–1968); student of Segre and Fubini; taught in Torino;
Eugenío Togliatti (1890–1977); student of Segre and Fubini; taught in Genova;
Oscar Chisini (1889–1967); student of Enriques; taught in Milano;
Giacomo Albanese (1890–1947); student of Bertini; taught in Pisa and Palermo;
Ruggiero Torelli (1884–1915); student of Bertini;
Beniamino Segre (1903–1977); student of Segre and Severi; taught in Roma;
Luigi Campedelli (1903–1978); student of Enriques; taught in Firenze;
Fabio Conforto {1909—1954); student of Enriques and Severi; taught in Roma;
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