**Cesare Burali-Forti**'s parents were Cosimo Burali-Forti (1834-1905) and Isoletta Guiducci. Cosimo was Chief Secretary of the Province of Arezzo. He was described as a [8]:-

After schooling in Arezzo, Cesare attended the Military College in Florence, completing his secondary level studies there in 1879. The years he was growing up were difficult years in the new Italy with widespread unrest which eased after Rome became part of Italy in 1870. He entered the University of Pisa in December 1879, where his teachers included Ulisse Dini, Luigi Bianchi, Enrico Betti, and Vito Volterra.... man of a thousand interests, a versatile genius, who took charge of legal and administrative disciplines, was a student of local history, a painter, but particularly a passionate lover of music that became his favourite activity.

Burali-Forti studied at Pisa from 1879 to 1884. Documents show that he progressed from the first to the second year of studies in the Faculty of Mathematics and Physics on 13 November 1880, and began his third year of studies on 7 November 1881, without having taken the Physics and the Chemistry examinations. He moved from the third year to the fourth in November 1882. During these years he was taught by several mathematicians, the most influential being those we mentioned above. Dini held the chair of analysis and higher geometry as well as the chair of infinitesimal analysis. Bianchi had been a student at Pisa but had studied abroad before returning to Pisa in 1881 where he taught differential geometry. Betti, although his interests had been in algebra and topology, was by this time becoming interested in mathematical physics and held the chair of celestial mechanics. For the first part of Burali-Forti's studies at Pisa, Volterra was a research student but he became Professor of Mechanics in 1883. With rapidly developing political events in the new country of Italy, several of Burali-Forti's lecturers were also involved in politics. For example after serving in local government, Dini was elected to the national Italian parliament in 1880 as a representative from Pisa. Among the courses taken by Burali-Forti we mention those on infinitesimal analysis and higher analysis by Dini, and courses on rational mechanics, celestial mechanics and mathematical physics taught by Betti. He graduated with his laurea in mathematics on 19 December 1884 after submitting his dissertation *Caratteristiche dei sistemi di coniche* Ⓣ. At his final examination he had to answer questions on mathematical physics topics that Betti had discussed with Riemann during his visits to Pisa in the 1860s.

Immediately after graduating Burali-Forti taught at the Technical Institute in Augusta, Sicily, where he was appointed on 29 August 1886, but he moved to Turin in 1887 when he was successful in the competition for an extraordinary professorship at the Military Academy of Artillery and Engineering. He could not marry before obtaining a position with an income which would allow him to support a wife and family and this appointment, starting on 1 September 1887, certainly meant that he was now financially secure. He also taught at the Sommeiller technical school in Turin until 1894 which provided him with further income. On 29 October of 1887, he married Gemma Viviani; they had a son Umberto who was born on 9 August 1889. His appointment in 1887 had been on a temporary basis but on 30 June 1900 his position was made permanent. On 30 October 1902 he was promoted to extraordinary professor first-class and then in 1906 he became a full professor, holding the chair of projective geometry. He was the only professor at the Military Academy who was not himself a military man. Burali-Forti spent the rest of his career at the Military Academy. Roberto Marcolongo [25], described him as a lecturer:-

Some of Burali-Forti's colleagues at the Military Academy were particularly important in his developing research interests. On 13 November 1887, just a few weeks after Burali-Forti was appointed, Giuseppe Peano joined the faculty of the Military Academy. The two became good friends and collaborated with each other in mathematical investigations. Through the years Burali-Forti did much to bring Peano's work to a wider audience. Peano taught mathematics there until 1901 as well as having university appointments. Also Mario Pieri was hired as a professor in the Military Academy in the same year as Burali-Forti and remained there until 1 May 1900, when he moved to Sicily, having won the competition for a professorship in projective and descriptive geometry at the University of Catania.... the teacher of many generations of military officers, the scholarly man was fondly remember as the gruff but kindly professor and they studied his elegant and rich courses of analytic and projective geometry.

A university teaching position would have been more to Burali-Forti's liking but in this he had difficulties. He was a great believer in vector methods but, at this time, these were not in favour. It is hard to believe from our present view of mathematics that vector methods would ever be less than welcomed. However, at this time many mathematicians opposed vector methods and unfortunately these views prevailed on the committee that considered Burali-Forti's submission for a *libera docenza*, the necessary qualification to hold positions in a university and to enter competitions for a university chair. He was failed on these grounds, never tried again, and as a consequence was never able to teach officially in a university, although he did give informal lecture courses there.

Something of the different opinions on the vector calculus, and in particular on Burali-Forti's approach to it are at THIS LINK.

The fact that he was working at the Military Academy and not a university put Burali-Forti in a more difficult position regarding research than other academics. For example in 1912 he could not attend the International Congress of Mathematicians held at Cambridge, England, in August of that year, despite having presented a communication *Sur les lois générales de l'algorithme des symboles de fonction et d'operation* Ⓣ, because in August he had to deliver his courses at the Academy. He refers to this in the letter sent from Turin on 22 March 1912 to Bertrand Russell, the organiser of the Philosophical Section of the Congress. See THIS LINK.

In 1893-94 Burali-Forti gave an informal series of lectures on mathematical logic at the University of Turin. After the course the lectures were written up as the book *Logica matematica* and Burali-Forti presented a copy of the book to the Academy of Sciences of Turin in June 1894. We have translated the Preface of this book into English and present our version at THIS LINK.

At the start of the 1894-95 academic session, Burali-Forti became Peano's assistant at the University of Turin. He was to hold this position until 1896.

The first International Congress of Mathematicians was held in Zürich from 9 to 11 August 1897. Burali-Forti attended the Congress and presented a paper *Postulats pour la géométrie d'Euclide et de Lobatschewsky* Ⓣ to the Geometry section of the Congress.

Burali-Forti is famed as the first discoverer of a set theory paradox in 1897 which was framed in technical terms but in essence reduces to a 'set of all sets' paradox. Cantor was to discover a similar paradox two years later. Burali-Forti's paradox appeared in his paper *Una questione sui numeri transfiniti* Ⓣ (1897). In this he considered the set of all ordinal numbers *W*. He showed that one easily obtains the contradiction

*W*+ 1 >

*W*and

*W*+ 1 ≤

*W*.

*Sulle classi ben ordinate*Ⓣ (1897). Irving Copi writes [16]:-

However, despite the fact that the corrected result clearly demonstrated a contradiction, the original mistake and the resulting confusion of many, led to his result being attributed to a 'misunderstanding'. Georg Cantor wrote, most unfairly:-He concluded the note with the observation that his result could be established on the basis of the correct definition of well-ordered set as easily as for the "perfectly ordered sets" for which it had first been obtained.

Even after Bertrand Russell had discovered his own "set of all sets" paradox in 1903, he did not seem to realise that it had any connection to Burali-Forti's paradox. Russell [16]:-What Burali-Forti has produced is thoroughly foolish. If you go back to his articles in the 'Circolo Matematico' you will remark that he has not even understood properly the concept of a well-ordered set.

Copi gives another reason for Burali-Forti's paradox being ignored [16]:-... observed that he could not accept Burali-Forti's conclusion that ordinal numbers are not necessarily comparable, preferring to accept Cantor's results in this connection as established. He urged that Burali-Forti's contradiction could be simply resolved by denying the premise that the series of all ordinal numbers is well-ordered, "... since, so far as I know, it is incapable of proof."

As well as set theory and vector analysis, Burali-Forti also worked on linear transformations and their applications to differential geometry [23]:-Burali-Forti had written his article in the logical symbolism of Peano. This new mathematical logic of Peano, which had far-reaching implications and tremendous importance, achieving its culmination in the 'Principia Mathematica' of Whitehead and Russell, was then but little known. Burali-Forti's language was new and strange, and intelligible to but few mathematicians. This circumstance undoubtedly contributed to the relative neglect of his paradoxical result.

Edgar Odell Lovett (1871-1957) reviewed Burali-Forti's bookBurali-Forti investigated the applications of vector calculus in a very wide variety of fields - from projective and differential geometry to the continuum mechanics, from optics to Lorentz transformations and to hydrodynamics - and introduced the fundamental notion of the derivative of a vector as compared with a point, which allowed him to unify and considerably simplify the foundations of vector analysis.

*Introduction à la géométrie différentielle suivant la méthode de H Grassman*(1897) in [22]. He writes:-

For a list of Burali-Forti's publications see THIS LINK.Burali-Forti's work, though by no means a pioneer in the application of Grassmann's theories to differential geometry ..., shows the elegant power and simplicity of the geometrical calculus in elementary differential geometry and points the student to a vast field of transformations and researches in higher geometry.

Another important contribution by Burali-Forti is discussed by Erika Luciano and Clara Silvia Roero in [23]:-

Not only was Burali-Forti a prolific writer, with over 200 publications, he was also very interested in how to teach mathematics. The "Mathesis" Italian Society of Mathematicians, aimed at school teachers of mathematics, was founded in 1895. Burali-Forti joined Mathesis in academic year 1897-98. He played a major role in the first congress of the Society which was held in Turin in September 1898.The sound mastery of the methods and language of ideographic logic made Burali-Forti one of the first and most active collaborators on the 'Formulaire de Mathématiques'. He wrote the chapter on arithmetics and the theory of magnitudes for the first edition of the treatise(1895); he edited the sections on curvature, torsion, relative torsion, ordinary points, inflection points, cusps, cycloid and epicycloid for the1902-03edition and in conclusion drafted those on algebra, in particular on the "producto logico"(intersection)and "summa logica"(union)for the final,1908edition. Peano also inserted in the fourth volume of the 'Formulario'(1902-03)some notions of projective geometry taken from Burali-Forti's three reports entitled 'Il metodo di Grassmann nella Geometria proiettiva'Ⓣ.

We have mentioned that Burali-Forti was a close friend of Peano's but his closest friend and mathematical collaborator was Roberto Marcolongo. Burali-Forti and Marcolongo were called the "vectorial binomial" by their friends. However this collaboration ended when they differed in their views on relativity. Burali-Forti never understood the theory of relativity and, in 1924, Burali-Forti, in collaboration with Tommaso Boggio (1877-1963), published *Éspaces Courbes Critique de la Relativité* Ⓣ. The authors write that they want:-

Marcolongo writes in [25] that their disagreement on the theory of relativity was:-... to consider Relativity under its mathematical aspect, wishing to point out how arbitrary and irrational are its foundations. ...[They state in the Preface]We wish to shake Relativity in all its apparent foundations, and we have reason for hoping that we have succeeded in doing it.[They conclude]Here then is our conclusion. Philosophy may be able to justify the space-time of Relativity, but mathematics, experimental science, and common sense can justify it NOT AT ALL.

Kennedy writes in [4] (see also [1]):-... the only time that the peace and solidity of the 'vector binomial' seemed to be compromised ... . It was not possible for them to agree, first on the new range to be given to vector methods; then, even more deeply, on the essence of the whole theory. Despite the interest he felt for all the modern physical questions and above all for those of high and absorbing philosophical interest, he remained firmly obedient to the classical systems and in attack as in defence he could not remain calm and objective.

Roberto Marcolongo [25], described Burali-Forti as:-Many of his publications were highly polemical, but in his family circle and among friends he was kind and gentle. He loved music, Bach and Beethoven being his favourite composers. He was a member of no academy. Always an independent thinker, he asked that he not be given a religious funeral.

Burali-Forti died in Turin's Mauriziano Hospital suffering from stomach cancer. As we noted above, he requested that there should be no religious funeral service.... witty and terribly caustic! A really terrible polemist, a true noble fearless knight, he cared not where and against whom his blows were struck; so that those who had never come near him formed a strange, mistaken concept of his intractable personality. Yet I have never known a better soul, so exquisitely gentlemanly, a man of few words, but of refined conversation, witty and learned. It was enough to approach him to appreciate his culture, to see what an old-style gentleman he was. ...

**Article by:** *J J O'Connor* and *E F Robertson*