*Jacques Rabie makes the case for collaboration in Mathematics by comparing the approaches of two mathematicians, 16th century Niccolo Fontana and 20th century Paul Erdős, and looking at the impact each one’s approach made on the progress of Mathematics.*

In 1535, the mathematician Niccolo Fontana ([3]) won a duel against fellow mathematician, Antonio Fior. This was no ordinary duel: instead of a battle of physical strength, the mathematicians were testing each other’s mathematical abilities. Such duels were not uncommon in 16th century Renaissance Italy; in particular, Bologna University, where this public competition was held.

The focus of this particular competition was solving cubic equations. At this time, a general solution to a cubic equation (ax3 + bx2 + cx + d = 0) had come to be seen as impossible. Fior had made some progress by finding a formula for cubic equations of the form x3 + ax + b = 0. Fontana, having heard of Fior’s discovery, and fearing defeat in the competition, also developed a formula for such cubic equations, as well as a formula for equations of the form x3 + ax2 + b = 0 — a discovery that ensured his success in the duel. In the highly competitive and cutthroat environment of 16th century Italy, Fontana even encoded his solution in the form of a poem in an attempt to make it more difficult for other mathematicians to steal it.

In 1539, the mathematician Gerolamo Cardano ([1]) invited Fontana to Milan and convinced him to reveal the solution to the special cubic equations. Fontana agreed, but made Cardano take an oath not to share the secrets of his formula. Using Fontana’s work, Cardano then found a way to reduce any cubic equation to one of the form x3 + ax + b = 0. In 1545, six years after Fontana shared his secrets with Cardano, Cardano published a paper with this result. Despite the fact that he gave credit to Fontana in his work, Fontana was furious. This resulted in a decade-long fight over the publication. Undoubtedly, however, the result paved the way for many new mathematical discoveries. Cardano’s student, Lodovico Ferrari, used similar techniques to that of Fontana to find a formula for the roots of a quartic polynomial. Almost 300 years later, the young Évariste Galois showed that there was no formula to find the roots of a general quintic polynomial, settling a centuries-long question.

Fast forward to the 20th century, when one of the world’s most renowned mathematicians, the Hungarian Paul Erdős, was born ([2]). Over his career of more than 60 years, Erdős published more than 1500 papers in various mathematical fields, surpassing the 18th century Swiss mathematician Leonhard Euler. Most of his work was collaborative — in his lifetime, Erdős had 511 different collaborators. Until his death in 1996, he regularly travelled back and forth between mathematical institutions. Due to his prolific collaborative output, the Erdős number was created to describe an individual’s degree of separation from Erdős, based on collaboration. For example, Albert Einstein’s Erdős number is 2, since he did not collaborate with Paul Erdős, but he did publish joint research with Ernst Straus, who was one of Erdős’ major collaborators. The median Erdős number for published mathematicians is 5, while the median Erdős number for Fields medalists is 3.

The legacies of Erdős and Fontana stand in stark contrast with one another. While Fontana obsessed over keeping his discoveries secret, Erdős took pride in working with other mathematicians. Whereas Fontana’s work took years to develop, Erdős was a main force in revolutionising modern mathematics. While Fontana gained fame in the mathematical community through competitive success, Erdős hardly ever won awards for his work: Erdős himself never won a Fields medal, nor did he collaborate with anybody who did. Fontana prided himself in solving mathematical problems on his own, while Erdős famously offered cash prizes for problems that he felt were just out of the reach of the current mathematical thinking (both his and others).

The contrast between these two mathematicians illustrates my view about competition and collaboration in mathematics: that it is collaboration, not competition, which most effectively leads to mathematical advancements. Fontana’s methods provided a stepping stone towards a rich body of work, but had Cardano not gone against his wishes, this body of work may have taken many more years to develop. Conversely, Erdős’ collaborative approach to mathematics was critical to the development of mathematics in the 20th century.

The failure of competitive mathematics in this regard, I believe, is for three main reasons. Firstly, competition does not incentivise those who succeed to continue pushing the boundaries of known mathematics: either Fontana wasn’t bothered to try and solve a general cubic equation, or he couldn’t do it by himself. Either way, his competitive nature hindered him from further developing mathematical knowledge. Secondly, competition discourages those who fail from trying again. It is often the case that a struggling student gives up hope when comparing themselves to their peers. Thirdly, and probably most importantly, competition shifts focus away from the goal of pushing the boundaries of mathematics, and instead focuses on the desire to be the best in a field, even if it means that progress is stifled in the process.

Collaboration, on the other hand, has the potential to unify great minds at any given time towards a common goal. It is for this reason that I believe that collaboration should be favoured above competition in the study of abstract mathematics.

### Jacques Rabie

Currently doing Masters in Mathematics,

STELLENBOSCH UNIVERSITY

[1] Cardano, G. and Stoner, J. 1931. The Book of My Life: (De Vita Propria Liber). London: J.M. Dent.

[2] Hoffman, P. 1999. The Man Who Loved Only Numbers. London: Fourth Estate.

[3] Katz, V., 1999. A History Of Mathematics: An Introduction. 2nd ed. Reading, Mass: Addison-Wesley.