Mathematical Challenges and Contests

Overview

During the sixteenth century, mathematics was transformed from the traditional study of classical texts and problems to a dynamic science characterized by active research in problems both abstract and applied. Such research depended on the lively exchange of ideas and techniques, which fostered a spirit of competition among investigators. The practice of offering challenges and contests characterized sixteenth- and seventeenth-century mathematics, and left a permanent legacy of mathematical competitions.

Background

Sixteenth- and seventeenth-century European mathematicians were keenly aware of one another's work, thanks to efficient correspondence networks and the rise of printed mathematical books. This awareness helped speed the pace of research, but it also gave mathematical work a competitive spirit. Mathematicians became anxious to identify and solve important problems, and to be the first to do so. The rewards for such competition were primarily personal—a triumphant solution would bring the author prestige among his mathematical peers—but were sometimes more tangible. Mathematical teachers who lost competitions or challenges could find their jobs to be in jeopardy. On the other hand, improved posts or access to patronage could come to victorious mathematical debaters, and some mathematical contests later in the seventeenth and eighteenth centuries even offered prize money to the successful author, augmenting the spoils of intellectual victory.

Two of the most famous mathematical challenges of the sixteenth century involved the Italian mathematician Niccolo Tartaglia (1499?-1557). Tartaglia was a largely self-taught scholar who overcame poverty and a traumatic childhood disfigurement—he was stabbed in the head during a military assault on his hometown of Brescia, and formally adopted his nickname Tartaglia, or "stammerer." He made his living as a mathematics teacher in Verona and Venice, and became well known in mathematical circles early in his career by his successful participation in a number of mathematical debates, and later by publications in pure mathematics and the application of mathematics to problems of warfare.

One of the most popular fields of mathematical study in the sixteenth century was algebra; of particular interest was the search for methods to solve third (and higher) order equations. A solution to the basic problem of solving cubic equations was found sometime in the first two decades of the sixteenth century by Scipione del Ferro (1465-1526), a mathematics lecturer at the University of Bologna. He did not publish his work, but he did share it with a few disciples. In 1535 one of these disciples, Antonio Fiore, sought to exploit his master's secret and issued a challenge to Tartaglia, by then known as a master debater. The two exchanged 30 mathematical problems for each other to solve; Tartaglia's challenge included a range of mathematical problems, but all of Fiore's problems for Tartaglia were cubic equations. Under the pressure of competition, Tartaglia figured out the method for a general solution of such problems (not knowing that Fiore was in possession of such a method), and was able to dispatch all 30 of Fiore's problems in less than two hours. Fiore made little progress solving any of Tartaglia's problems, and Tartaglia emerged from the encounter with his reputation greatly enhanced.

Tartaglia did not publish the method of solving cubic equations either, perhaps hoping to keep it secret for use in future challenges or debates. This decision came to cause him grief. In 1539 Tartaglia was approached by a wealthy and well-connected Milanese mathematician and physician named Girolamo Cardano (1501-1576). Cardano was anxious to have the method for solving cubic equations, and he tried several strategies to make Tartaglia share his secret. First, he asked Tartaglia to include his method in a book Cardano was publishing; when that failed, he challenged Tartaglia to a debate. Although Tartaglia did not agree to debate Cardano, he was finally persuaded by hints of patronage opportunities to come to Milan. While he was in Milan as Cardano's guest, Tartaglia did finally share the formula—disguised as a poem—after extracting a pledge of secrecy from his persistent host.

While Cardano initially honored his promise to keep Tartaglia's formula a secret, he used it as a basis for further discoveries of his own. When he found out some years later that Ferro had discovered a method for solving cubics prior to Tartaglia, Cardano felt justified in including Ferro's method in a book he published in 1545. This infuriated Tartaglia, and led to an angry feud and finally a debate between Tartaglia and Cardano's disciple Lodovico Ferrari (1522-1565); Cardano himself refused to enter into a challenge with Tartaglia. Tartaglia traveled to Milan again, hoping that he could vanquish Ferrari in a debate and thereby secure a superior teaching post. But Ferrari quickly showed a better grasp of the problems offered than Tartaglia had, and Tartaglia fled the city after the first day of their contest, thinking it better to leave the contest unresolved than to lose outright to Ferrari. But news of his poor performance in Milan dogged Tartaglia, and his career suffered accordingly.

Tartaglia's two debates are perhaps the most famous in mathematics history, but they were by no means unusual events for their time. Challenges were a vital part of mathematical practice. They helped to identify superior mathematicians, to spur work on the solution of particular problems, and to display for mathematical audiences the range of active problems at a given time. For example, the challenge between Tartaglia and Ferrari including problems not only associated with cubic and quartic equations, but also ranged through topics including astronomy, optics, architecture, and cartography. Prior to the establishment of mathematical societies and periodical journals, challenges and the correspondence and books they inspired were one of the primary means of identifying and disseminating mathematical progress.

Impact

Tartaglia's debates were like mathematical duels —one mathematician would issue a challenge to another, and the dispute would usually be resolved publicly in a face-to-face exchange. Mathematical challenges evolved into another form in the seventeenth century, when mathematicians set out important unsolved problems for general solution by their peers. Skilled investigators throughout Europe would take up such problems, and compete with one another to have the first, and the best, solution.

Some of the best-known mathematical challenges of the seventeenth century came from Pierre de Fermat (1601-1665). Fermat is remembered today for his extraordinary contributions to mathematics, especially number theory, but he was a lawyer by profession and mathematics was just one of his several avocations. These other interests may explain Fermat's lack of publications, and his practice of presenting his mathematical ideas rather casually in letters to friends. In 1657 Fermat issued a series of problems as challenges to other mathematicians. These problems were stated in the form of theorems to be proved. Because of Fermat's failure to publish or present formally his own work, historians are not sure how many of these theorems Fermat had solved himself, or what his methods might have been. The challenge problems reflected Fermat's interest in prime numbers and divisibility, and in producing general solutions based upon a single paradigmatic solution. The final one of these problems (to show there is no solution for xⁿ + yⁿ = zⁿ for n > 2) resisted conquest until the late 1990s. The solution by Andrew Wile (1953-) of what had come to be known as Fermat's Last Theorem received enormous attention from even nonmathematicians, and is considered one of the greatest mathematical achievements of the twentieth century.

By the eighteenth century, the practice of mathematical contests had been well established. Scientific societies large and small issued challenges to draw attention to themselves and to particular mathematical problems, but perhaps most important for the advance of research have been challenges like those of Fermat issued by prominent mathematicians themselves. By singling out particular theorems or problems for attention, mathematicians such as David Hilbert (1862-1943), who issued 23 important problems for study in 1900, have helped shape research agendas for the entire mathematical community. Challenges may have lost the personal vitriol that colored Tartaglia's debate, but the spirit of competition continues to surge through research mathematics. Young mathematics students routinely enter mathematical competitions, ranging from local contests to international olympiads. Intense pressure to produce the first proof of an unsolved theorem, or to find a counterexample to discredit it, is an essential feature of mathematical practice. Careers are still made or lost on the basis of priority of solution, and the entire international mathematical community can be caught up in the evaluation of the reported solution of important unsolved problems such as Fermat's Last Theorem. In this way, research mathematics retains some of the spirit of an age when many kinds of disputes could be settled by a duel.

LOREN BUTLER FEFFER