Bored by a homework assignment set by his tutor that involved computing the squares of numbers from 1 to 30, Federigo, then age eleven, figured out that he could generate the squares by adding successive odd numbers: 1, 1 + 3, 1 + 3 + 5, and so on. Buoyed by his discovery, he went on to calculate the squares of numbers from 1 to 1,000, publishing his results in a small pamphlet, which cost him his entire savings (seven lira). When Enriques's daughter asked him, many years later, whether his parents had been pleased with his enterprise, he flashed a smile and replied, "They never knew about it." The eleven-year-old had shown even then a streak of independence he never lost.Two years later, when he was thirteen, Federigo met geometry for the first time and became very enthusiastic about the subject. His mother wrote to her sister at that time  (see also ):-
Elbina studies with Mr Rodolfo and gives us much satisfaction. Ghigo [Federigo] has just turned thirteen. His studies also go well. Today, just imagine, he started a geometry course. You know how this boy is, every day his head has a new idea that lasts for about the space of a morning.Of course, she was wrong about Enriques's new found love for geometry - it lasted his lifetime. He became fascinated by other subjects while at the high school, such as logic, epistemology, pedagogy, and the history of science. It is clear that his later interest in philosophy originated during this period in his life. Mr Rodolfo, who is mentioned in the quote above, was employed by the Enriques family to tutor the three children. Although it is not clear exactly what he taught Federigo, it is reasonable to assume that he made a highly significant contribution to the young boy's education teaching him topics not covered in the school syllabus.
Federigo took his final school examinations in the summer of 1887. He entered the University of Pisa, also studying at the Scuola Normale in Pisa, and he was awarded his degree in June 1891. He was fortunate to have been taught by Enrico Betti, Luigi Bianchi, Ulisse Dini, and Vito Volterra in Pisa. His thesis advisor had been Riccardo De Paolis (1854-1892), who had been appointed professor of higher geometry at the University of Pisa in 1881. After his laurea degree, Enriques continued to study at Pisa for a year. In 1892 he asked Guido Castelnuovo, who was in Rome, for advice on which direction his research should take. He took Castelnuovo's advice and worked on algebraic surfaces, sometimes collaborating with Castelnuovo. before moving to Rome to work with him. He spent a year studying in Rome before moving again, this time to Turin where he worked with Corrado Segre. In 1893 he published Ricerche di geometria sulle superficie algebriche , an important contribution to the theory of algebraic surfaces. This was the same year in which he began to seek a university chair.
Giuseppe Bruno died early in 1893 and later that year a competition was announced to fill his chair of projective and descriptive geometry at the University of Turin. Ferdinando Aschieri, Eugenio Bertini, Enrico D'Ovidio, Corrado Segre, and Giuseppe Veronese were appointed as the referees and they examined six eligible candidates. In addition to Enriques, these were Luigi Berzolari, Mario Pieri, Alfonso Del Re, Fererico Amodeo, and Edgardo Ciani. Berzolari was appointed to the chair with Enriques ranked in fourth equal position behind Berzolari, Pieri and Del Re :-
... it is apparent that Enriques, younger than most candidates, felt perhaps abnormally insecure about finding a stable position in a difficult world. He evidently also trusted personal lobbying more than the competition process. He described his experiences and plans and poured out his feelings in frequent - sometimes daily - long letters to Castelnuovo: 668 letters between November 1892 and December 1906.An extraordinary professorship (called professor straordinario) in descriptive and projective geometry became vacant at the University of Bologna when Domenico Montesano left Bologna to take up a chair at Naples in 1893. Given the ranking in the previous competition, it was expected that Pieri or
Enriques had been considering the situation and concluded that (1) should there be a competition, the faculty would fill the instructional need temporarily by hiring a professor incaricato; or (2) should the faculty call Pieri with no competition, Pieri's positions at Turin would become vacant. In either case, Enriques would be a stronger candidate than he would be that year for straordinario at Bologna. Enriques had written to Volterra, his former teacher at Pisa, for advice. Volterra replied that Arzelà had written to him from Bologna referring to a possible incaricato position; and Volterra suggested that Enriques go to Bologna to talk with Pincherle and Arzelà. Enriques delayed, apparently at Arzelà's suggestion, but did go, probably on the day of the faculty meeting, Friday or Saturday, 17 or 18 November.The Bologna faculty agreed to appoint Pieri without a competition and it appeared that it would be a formality that the minister of education would take the advice of the Faculty of Bologna and confirm Pieri's appointment. However the government became embroiled in a scandal and the minister of education resigned. The new minister of education did not approve Pieri's appointment, telling the Faculty at Bologna that they had to hold a competition and make a temporary appointment while this was taking place. Enriques was appointed to the temporary post in January 1894. The competition for the permanent Bologna post did not take place until October 1896; Enriques was appointed to the chair of descriptive and projective geometry with Pieri coming a close second. The long saga surrounding this appointment is described in detail in .
Enriques made important contributions to geometry and to the history and philosophy of mathematics. He produced a series of papers over a period of 20 years which, together with Castelnuovo, finally produced a classification of algebraic surfaces :-
In the 1890s Enriques, a mathematician who once quipped that "intuition is the aristocratic way of discovery, rigour the plebian way", and his colleague and future brother-in-law Castelnuovo began their monumental work on the birational theory of algebraic surfaces over the complex numbers. Severi joined them in this effort a few years later.The key to the classification was contained in a remarkable paper Sulla proprietà caratteristica delle superficie algebriche irregolari published by Enriques in 1905. It contained what is now called the "Completeness theorem". His work on algebraic surfaces gained world-wide recognition when it was highlighted by H F Baker in his presidential address to the International Congress of Mathematicians in Cambridge in 1912. Another topic which Enriques worked on was differential geometry. In this area he also won fame with the joint award of the Bordin prize to him and Severi in 1907 for work on hyperelliptic surfaces. Around this time he contributed three articles to the Encyklopädie der mathematischen Wissenschaften , one on the foundations of mathematics and the other two, co-authored with Castelnuovo, on algebraic surfaces and birational transformations.
The foundations of mathematics had always interested Enriques, and, at Klein's request, he wrote an article on the foundations of geometry. His interest also extended to psychology when he asked questions such as:-
What leads a mathematician to make a conjecture?For example, in a letter to Guido Castelnuovo written in May 1896, he writes (see ):-
I have been working for several days on another point that mathematics takes only as a pretext: hearing the name you will feel more horror than astonishment. This is the philosophical problem of space. Books on psychology and logic, physiology, and comparative psychology, a critique of knowledge etc., sit on my coffee table where I savour them with delight trying to extract the essence for what concerns my problem ... Read Wundt's 'Logik', at least that part which concerns the method of mathematics, and think that he is a physiologist who writes it: a physiologist who is not afraid to climb the steep slope of Kant's conception to illuminate from above the great progress of all sciences.Giorgio Israel and Marta Menghini write in :-
Enriques himself recalled how his interest in mathematics was due to a "philosophical infection caught at school." His interest in science problems (and geometry in particular) was never simply of a technical nature but was always motivated by questions of "general culture" and lively reflection on the role of scientific thought in human activity. His mathematical research was interwoven with the intention of finding an answer to the great philosophical question on which science is founded. It is, after all, impossible to separate Enriques, the philosopher of science, from Enriques, the mathematician, without running the risk of failing to understand both of them.His book Problemi della scienza written in 1906, stressed the unifying aspect of scientific theories, the association of ideas and of scientific representation. He writes:-
It is plainly seen that scientific questions include something essential, apart from the special way in which they are conceived in a particular epoch by the scholars who study such problems. ... In the formulation of concepts, we shall see not only economy of thought ... but also a somewhat determinate mental process ... .Jeremy Gray writes in :-
The mathematical community has evolved sophisticated ways of reading Enriques's work in algebraic geometry, and we see most of it as either correct or easy to put right. It is harder for us today to accommodate his writing as a philosopher or populariser. He held a subtle position, according to which knowledge is inseparable from the means of knowing, logic from psychology. This has long been unfashionable in the sciences. It may be that cognitive psychology will reopen the avenues Enriques explored; there are signs that it has reached at least the philosophy of mathematics. ... Enriques offered a position on the nature of knowledge that was original and sophisticated. His readers found a rare grasp of modern science, traditional philosophy, and contemporary psychology.
In addition to his research work, Enriques also wrote textbooks for schools. He co-founded Mathesis, becoming president of the Society. He founded a number of journals including Scientia and Periodico di Matematiche. He also founded the Italian Philosophical Society and was president of it from 1907 until 1913. During that time he organised the fourth international congress of philosophy in Bologna in 1911.
He remained at Bologna until 1922 when he was called to Rome to teach complementary mathematics, a new course designed for high school mathematics teachers. In the following year he accepted the chair of higher geometry at the University of Rome where he founded the National Institute for the History of Science and the School of the History of Science. Since he was Jewish, he was affected by the Manifesto della razza (Manifesto of Race) enacted by Mussolini in July 1938. This stripped Jews of Italian citizenship and banned them from positions in banking, government, and education. Enriques had to resign from teaching in 1938. The fact that he had tried to avoid political problems by joining the Fascist Party was of no help in letting him avoid the difficulties caused by the Manifesto of Race. He was forced into hiding during the war years but, from 1941, he was able to participate in the illegal school organised by Castelnuovo in Rome to give special courses to instruct Jewish students disadvantaged by anti-Semitic government policies. He also continued writing but, unable to publish under his own name, published some works under the names of his loyal students. After the fall of Fascism in 1944, he was able to return to teaching at the University of Rome.
Many of Enriques' students recalled their experiences :-
As a teacher, Enriques loved nothing better than to engage in his own leisurely peripatetic conversations with students, in the public gardens in Bologna or under its arcades after class. When he moved to Rome, the labyrinthine network of paths in the Villa Borghese became his favourite destination; he would stop there every so often, one student at that time recalls, "to trace mysterious figures on the ground, with the tip of his inseparable walking stick".Enriques received many honours. He was awarded an honorary degree by the University of St Andrews. He was elected to the Reale Accademia dei Lincei in 1906 and, in the following year, was awarded (together with Levi-Civita) the Royal Prize in Mathematics. He was also elected to the National Academy of Sciences of Italy (the "Academy of Forty"), the Académie des Sciences Morales et Politiques (1937) and many other academies.
Article by: J J O'Connor and E F Robertson