Mathematics of Western Europe

Summary: Western Europe has been home to many of the important astronomical and mathematical discoveries of the early modern age.

Historically, the term “western Europe” has had cultural and political definitions. For example, during the Cold War it was often used to designate a collection of noncommunist countries allied in some way with the United States. In the early twenty-first century, the United Nations Statistics Division for western Europe contains Austria, Belgium, France, Germany, Liechtenstein, Luxembourg, Monaco, the Netherlands, and Switzerland. There is a rich history of mathematics scholarship, education, and achievement in western Europe. Important work in a diverse array of mathematical areas like calculus, analytical geometry, probability, statistics, functional analysis, graph theory, logic, and number theory was produced by people from this geographic region, as well as many mathematical contributions to related disciplines like physics, astronomy, optics, engineering, and surveying.

Historical Contributions

Western European mathematicians have made major contributions to the development of mathematics and the application of mathematical theory to practical problems, from German mathematician and astronomer Johannes Kepler, who worked with Danish astronomer Tycho Brahe and helped established the laws of planetary motion, to French mathematician René Thom, who founded the study of catastrophe theory.

Much of modern science and mathematics has its roots in work done in Europe in the seventeenth century. Johannes Kepler studied at the University of Tubingen, where he learned both the geocentric model of astronomy (the view that Earth is the center of the universe, with the other planets revolving around it) and the heliocentric model of German astronomer Nicolaus Copernicus (the view that the sun is the center of the universe and the planets, including Earth, revolve around it). He later worked with Brahe and established the laws of planetary motion in several influential publications: Astronomia Nova, Harmonices Mundi, and The Epitome of Copernican Astronomy. Also in Germany, mathematician Gottfried Leibniz developed the field of calculus independent of Sir Isaac Newton in England.

In France, mathematician and philosopher René Descartes developed analytical geometry, including the development of Cartesian coordinates, did important work in optics, and was also one of the fathers of modern Western philosophy with influential books such as Meditations on First Philosophy, Discourse on the Method (which contains the oft-quoted statement cogito ergo sum, or “I think, therefore I am”), and Principles of Philosophy. Also in France, the basics of probability theory were developed by mathematicians Pierre de Fermat and Blaise Pascal, while Fermat also did important work in number theory, analytic geometry, and optics. Fermat’s Last Theorem, mentioned but not proved by Fermat in 1637 in the margin of a book, was among the unsolved problems in mathematics until British mathematician Andrew Wiles proved it in 1994. Pascal invented the mechanical calculator and the hydraulic press and is well known among middle school students for Pascal’s Triangle, a presentation of binomial coefficients.

In the eighteenth century, Swiss mathematician and physicist Leonhard Euler spent much of his adult life working at the Russian Academy of the Sciences in St. Petersburg. He developed the concept of the function and the notation f(x), one of several notation conventions he developed that are still used in the early twenty-first century (others include using the letter e for the natural logarithm, i for an imaginary unit, and the Greek letter sigma (Σ) for summation). He also made important contributions to calculus, number theory, graph theory (he solved the famous Seven Bridges of Konigsberg problem), and applied mathematics. French and Italian astronomer and mathematician Joseph-Louis Lagrange, who was born in Italy but worked primarily in France and Prussia, created the calculus of variations, developed a method of solving differential equations and transformed Newtonian mechanics into a branch of analysis, which facilitated the development of mathematical physics. He was also the first professor of analysis at the École Polytechnique, an elite engineering school founded in France in 1794. Also in France, mathematician and astronomer Pierre-Simon LaPlace played a key role in the development of Bayesian statistics, named for English minister and mathematician Thomas Bayes, and mathematical astronomy. He also posited the existence of black holes and gravitational collapse in the solar system.

In the nineteenth century, mathematician German Carl Friedrich Gauss made important contributions to several mathematical and physics fields including statistics, number theory, astronomy, surveying (he invented the heliotrope), and optics. The well-known normal distribution is sometimes referred to as the “Gaussian distribution” because he is often credited with discovering it. In France, Augustin-Louis Cauchy not only worked as an engineer but also pursued mathematical studies in his spare time and was appointed to the Académie des Sciences in 1816. He made numerous contributions to mathematics and physics, including his development of complex function theory, clarification of the principle of calculus, and development of the argument principle. In France, mathematician Evariste Galois proved, in parallel with the work of Norwegian mathematician Niels Henrik Abel, that there was no general method for solving polynomial equations of degree of greater than degree four.

In 1900, German mathematician David Hilbert gave an influential talk at the International Congress of Mathematicians in which he identified 23 unsolved problems in mathematics, which served as a spur for other mathematicians to focus on those problems (10 have been solved as of 2010). Hilbert is also well known for formulating the theory of Hilbert spaces, which are key to functional analysis, and did important work in mathematical logic and proof theory. Austrian mathematician Kurt Gödel, best known for his two incompleteness theorems, immigrated to the United States to escape World War II and spent his later years at Princeton University. A group of primarily French mathematicians, including Jean Dieudonne and André Weil, began publishing anonymously under the pseudonym “Nicolas Bourbaki.” They are now known as the “Bourbaki Group” or “Association des collaborateurs de Nicolas Bourbaki” and have published several books in which they attempt to ground different areas of mathematics in set theory.

Awards and Honors

There is no Nobel Prize for mathematics but several different international awards are offered that have been termed the “Mathematics Nobel Prize” because of their prestige. The Fields Medal is awarded every four years to one or more mathematicians of age 40 or younger by the International Mathematical Union. Winners of the Fields Medal from western Europe include Laurent Schwartz of France (1950), Jean-Pierre Serre of France (1954), Rene Thom of France (1958), Pierre Deligne of Belgium (1978), Alain Connes of France (1982), Gerd Faltings of Germany (1986), Jean Bourgainof Belgium (1994), Pierre-Louis Lions of France (1994), Jean-Christophe Yoccoz of France (1994), Laurent Lafforgue of France (2002), Wendelin Werner of France (2006), Ngo Bao Chau of Vietnam and France (2010), and Cedric Villani of France (2010).

The Abel Prize, named after Norwegian mathematician Niels Henrik Abel, is awarded annually by the Norwegian Academy of Science and Letters. Western European winners include Jean-Pierre Serre of France (2003), Jacques Tits of Belgium and France (2008), and Mikhail Gromov of Russia and France (2009).

The Wolf Prize is awarded in several fields, including mathematics, by the Wolf Foundation. The first prizes were given in 1978 and it is awarded almost annually, with the possibility of more than one winner in a field in a given year. Western European winners include Carl L. Siegel of Germany (1978), Jean Leray of France (197), André Weil of France and the United States (1979), Henri Cartan of France (1980), Friedrich Hirzebruch of Germany (1988), Mikhail Gromov of Russia and France (1993), Jacques Tits of Belgium and France (1993), Jurgen Moser of Germany and the United States (1994/1995), Jean-Pierre Serre of France (2000), and Pierre Deligne of Belgium (2008).

Western European countries have been regular competitors in the International Mathematical Olympiad, held annually for students younger than 20 who have not yet begun tertiary education. There is both an individual and a team competition. Each country sends six students who are assigned six questions to solve. Countries are compared based on the total score for their team, while individual students may be awarded gold, silver, and bronze medals depending on how many problems they solve correctly. Germany has twice hosted the International Mathematical Olympiad and has participated since 1977.

East Germany also twice hosted the Olympiad and first participated in 1959, the year the Olympiad began. France began competing in 1967 and hosted the competition once. Belgium began participating in 1969. Austria began competing in 1970 and has served once as host. The Netherlands hosted the Olympiad in 2011 and has been competing since 1969. Luxembourg began competing in 1970, Switzerland began competing in 1991, and Liechtenstein began competing in 2005.

Bibliography

Bradley, Robert E., Lawrence A. D’Antonio, and C. Edward Sandifer, eds. Euler at 300: An Appreciation. Washington, DC: Mathematical Association of America, 2007.

Hahn, Robert. Pierre Simon Laplace, 1749-1827: A Determined Scientist. Cambridge, MA: Harvard University Press, 2005.

Joyce, David E. “The Mathematical Problems of David Hilbert.” http://aleph0.clarku.edu/~djoyce/hilbert.

Krantz, Steven G. An Episodic History of Mathematics: Mathematical Culture Through Problem Solving. Washington, DC: Mathematical Association of America, 2010.

Mashaal, Maurice. Bourbaki: A Secret Society of Mathematicians. Translated by Anna Pierrehumbert. Providence, RI: American Mathematical Society, 2006.

Repcheck, Jack. Copernicus’ Secret: How the Scientific Revolution Began. New York: Simon & Schuster, 2008.

School of Mathematics and Statistics, St. Andrews University. “The MacTutor History of Mathematics.” http://www-groups.dcs.st-andrews.ac.uk/~history.

Segal, Sanford L. Mathematicians Under the Nazis. Princeton, NJ: Princeton University Press, 2003.