Mathematics, Communication, and Community

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Mathematics, Communication, and Community

Overview

Perhaps more than any other scientific subject, mathematics seems to depend upon individual genius and moments of inspiration. Mathematical theorems and concepts are even named after their discoverers. But in fact, mathematical research did not begin to flourish until the sixteenth century, when reliable communication networks helped support an international community of like-minded scholars who stimulated each other's work through the exchange of ideas and the spirit of competition. A dynamic balance between individual discovery and communal validation characterizes mathematics to this day.

Background

The invention of the printing press in the fifteenth century gave an enormous boost to the field of mathematics. While the great mathematical works of antiquity had been preserved in Europe, they were often studied through incomplete manuscripts, and advances to algebra, trigonometry, and geometry made in Islamic countries during the Middle Ages were little known. There was little enthusiasm for innovative ideas, and few opportunities for collaboration or exchange. But with the advent of printing, mathematical communication became newly convenient. Regiomontanus (1436-1476), one of the most important early printers, was also probably the most important mathematician in Europe during the fifteenth century. Although he died prematurely, Regiomontanus did begin an important era in the publication of the major texts of classical science and mathematics, and of textbooks based upon them. The first printed edition of Euclid's (330?-260? b.c.) famous book on geometry appeared in 1482. During the next century, more than one hundred different editions of Euclid alone were published, along with scores of other mathematical treatises.

The proliferation of mathematical books helped to stimulate mathematical education, not only in universities but also in schools that helped to prepare men to work in areas such as surveying, which required the use of arithmetic and geometry. Mathematics began to be applied to new fields and to older subjects in new ways. Bookkeeping, mechanics, surveying, art, architecture, cartography, optics, and music were all transformed by the application of mathematical techniques, and mathematics was in turn influenced by the demands of these real-world activities.

The availability of somewhat standardized printed versions of texts by Euclid and Archimedes (287?-212? b.c.), the increase in interest in mathematical education, and the expansion of the application of mathematics to important new areas established promising conditions for mathematical activity. The printed texts provided a kind of uniform starting point for those interested in addressing problems posed by the classical authors, and the use of mathematics to solve practical problems provided another outlet for mathematical novelty. The desire to exchange mathematical ideas with others interested in the still-esoteric subject drove far-flung investigators to begin correspondence with each other.

Communication among scholars living in different countries throughout Europe was greatly facilitated by the widespread use of Latin in scholarly work. Well into the seventeenth century, most scientific and mathematical books were published in Latin, and virtually every educated person would read and write Latin fluently. While vernacular languages were used to write and publish some elementary texts, Galileo (1564-1642) was in the 1630s the first prominent scientist to publish important scientific texts in his native language. But throughout the sixteenth and seventeenth centuries, the use of Latin as a universal language for European scholars made correspondence between mathematicians much simpler.

But how did these scholars find one another? Prior to the existence of scientific societies or scholarly journals, identification of others who shared common research interests or mathematical talent depended upon individual encounters and connections. Some mathematicians were affiliated with universities or monasteries and so had institutional relationships that put them in touch with others at similar posts, but many active mathematicians in this era were more or less independent. It fell to enthusiastic individuals to bring other mathematicians into contact. One of the most important of the mathematical impresarios was a Jesuit priest by the name of Marin Mersenne (1588-1648).

Mersenne's duties to his Jesuit order were for the most part strictly intellectual. He published several books of his own, beginning in 1623, on a variety of topics including theology, ancient and modern science, and mathematics. These publications often included the first published accounts of the work of other mathematicians with whom Mersenne corresponded. This was one of many ways that Mersenne encouraged the exchange of mathematical ideas.

Mersenne became the coordinator of mathematical activity for scholars based in Paris, as well as for interested foreigners who visited the French capital. He held frequent salons at his convent that brought together leading thinkers in a number of other scholarly fields as well as mathematics. These evolved into something resembling formal scientific conferences by the mid-1630s. Another significant encounter that Mersenne brought about was the first meeting of the great French mathematicians Blaise Pascal (1623-1662) and René Descartes (1596-1650). In addition to these important personal meetings and exchanges, Mersenne built up a correspondence network that came to include most of the important mathematicians in Europe. It was through Mersenne that Galileo and his followers maintained contact with scientists elsewhere in Europe. Mersenne relayed correspondence between mathematicians interested in common problems, and he often smoothed relationships among various personalities, such as those between the notoriously difficult Descartes and France's other leading mathematicians Pierre de Fermat (1601-1665), Gilles Personne de Roberval (1602-1675), and Pascal.

But Mersenne was something more than a secretary. In addition to establishing and maintaining contact among Europe's mathematicians, Mersenne also directly influenced mathematical work. He often criticized or challenged the mathematical work of others, setting one mathematician against another to advance work in an area where problems seemed to fester. He also set out mathematical problems for others to solve, thus contributing to the era's spirit of competition among mathematicians eager to claim new solutions and techniques for themselves. Mersenne's extensive correspondence, covering more than 20 years, from the early 1620s up to his death in 1648, was finally itself published in the twentieth century and stands as a tangible monument to his great influence on the development of mathematics and the international mathematics community. While Mersenne was undoubtedly the most significant mathematical correspondent of his century, others such as John Collins of London performed a similar role in their own countries.

Impact

Many conditions came together to bring about the flowering of mathematics in the sixteenth century. The dissemination of published mathematical books, the expansion of education in mathematics, new areas of application, and the establishment of an international community of scholars connected via correspondence, transformed mathematics from a limited academic subject to an actively evolving scientific endeavor. Historians agree that by the seventeenth century, the practice of mathematics had come to fully resemble the modern mathematics of later centuries.

From the start they were given by Mersenne, European mathematicians established ever-closer ties among one another. Detailed exchanges of problems and proofs, counterexamples, and challenges took place across national boundaries and between (mostly) men of very different social and professional standing. New ideas, techniques, and applications began to develop quickly as enthusiasts stirred each other to work harder and more rigorously on problem after problem. While this often had the positive effect of stimulating fresh discoveries, it also set the stage for a number of bitter disputes over who first came upon an important idea. Most notorious among these "priority disputes" was the bitter battle waged between Isaac Newton (1642-1727) and Gottfried Leibniz (1646-1716) over who had discovered the calculus.

Communication among mathematicians grew faster and easier as technology improved. Letters moved more and more quickly thanks to new roads and later railroads and improved ships. The telephone, telegraph, and eventually computers sped communication further. At the same time the mathematical community fostered institutions to aid communication among researchers. Mathematicians joined the earliest scientific societies as they formed in the late seventeenth century, and starting in the nineteenth century began to form societies and publications that were specialized to their interests alone. As mathematics became more and more specialized, so did its institutions. By the end of the twentieth century, hundreds of mathematical journals and dozens of mathematical societies reflected the diversity of the field and its profound degree of international cooperation. English is treated as a common, although not universal, language of exchange, and the highly symbolic nature of mathematics itself helps scholars from different nations to easily understand one another's work.

Mathematics developed a public face in the sixteenth century that has lasted to the present: a new mathematical concept can be accepted only after the mathematical community has been persuaded of its truth. A mathematician with a new discovery or proof must convince his peers of his accomplishment, otherwise it is no accomplishment at all. The importance of a community to modern mathematics, then, cannot be exaggerated—mathematics is at its very core a social activity, no matter how essential the contribution of individual work might be.

LOREN BUTLER FEFFER