Mathematics of Eastern Europe

Summary: Eastern Europe has a long tradition of both mathematics research and education.

Throughout history, the countries of Europe have had shifting political and social boundaries. Eastern European mathematics evolved within the context of many mathematics traditions, including Soviet Union mathematics, over the past centuries. Historically, gifted young scholars from regions around the world completed their mathematical studies at Europe’s well-known and respected universities. Studies of mathematicians’ letters and scientific papers show that they often maintained connections with people in other countries who shared their fields of interest. The Soviet Union exercised broad social and political influence over most of eastern Europe and also impacted U.S. mathematics in the twentieth century. Within the Soviet Union, students from the far reaches of the nations within its boundaries were often brought to Russia for work or education, as well as sent to other parts of the Soviet Union to teach or to establish research centers. In the twenty-first century, students in the United States and around the worked attend study abroad programs, such as the Budapest Semesters in Mathematics. In the twenty-first century, the United Nations Statistics Division classified the following countries belonging to eastern Europe: Belarus, Bulgaria, Czech Republic, Hungary, Moldova, Poland, Romania, Russia, Slovakia, and Ukraine. The CIA World Factbook adds Estonia, Latvia, and Lithuania, which were among the member nations of the Soviet Union, though the United Nations classifies them as belonging to northern Europe. Geographical boundaries continued to change in the twentieth century because of post-World War II structures and, later, the breakup of the “Eastern Bloc” nations, which were once under the Soviet Union’s political influence. Therefore, mathematics contributions of some people from eastern Europe may be included within the histories of other regions or countries.

History of Russian and Soviet Mathematics Education

When examining past and present states of mathematics in Belarus, Moldova, Russia, Ukraine, Estonia, Latvia, and Lithuania, it is pertinent to acknowledge that they share a common sociopolitical root: they are all former member states of the Soviet Union. Further, the broader Eastern Bloc of Soviet Union allies included Bulgaria, Romania, Hungary, East Germany, Poland, Albania (until the early 1960s), and Czechoslovakia (which later split into the Czech Republic and Slovakia). The Eastern Bloc is sometimes known historically as “eastern Europe,” versus the “western Europe” countries allied with the United States, a rival of the Soviet Union. During its several decades of existence in the twentieth century, the Soviet Union included many mathematicians who made significant contributions to the body of modern mathematical knowledge. Further, Russian and Soviet mathematicians were influential on many other countries.

One important landmark in mathematics education in Russia is the creation in 1701 of the School of Mathematical and Navigational Sciences in Moscow. Peter the Great, who had traveled widely in other parts of Europe to study the state of mathematics and science as part of his effort to modernize Russia and expand the empire, founded this school. It educated students in basic mathematics as well as more specialized subjects, such as astronomy and navigation. Notably, students from all social classes except serfs were admitted, and financial assistance was available. Graduates worked in the navy, as engineers, and as teachers in a variety of settings, so the school had a multiplier effect in terms of spreading mathematics education throughout Russia. Peter the Great also founded the Saint Petersburg Academy of Sciences in 1724, influenced in part by correspondence with mathematician Gottfried Leibniz, who also purportedly recommended a three-tiered educational system of schools, universities, and academies. Many eminent foreign mathematicians, such as Leonhard Euler, Christian Goldbach, and Daniel Bernoulli, worked at the Saint Petersburg Academy.

As part of her goal of modernizing Russia in the European style, Empress Catherine the Great, who was born in Germany, established the first gymnasiums in Russia. These gymnasiums were schools meant to prepare students for higher education and were created in most major Russian cities in the nineteenth century. Nicolai Ivanovich Lobachevsky, one of the first Russian mathematicians to achieve international recognition, was a beneficiary of this expanded educational opportunity. He graduated from Kazan Gymnasium and Kazan University (in Tatarstan) and is most noted for his work in hyperbolic geometry, a form of non-Euclidean geometry. However, despite this considerable expansion, access to education was far from universal until the Soviet era. The Soviet Union was founded by revolution in 1917, when the monarchy of the Russian Empire was overthrown, but was not made official until 1922. The Saint Petersburg Academy of the Sciences evolved into the Russian and then Union of Soviet Socialist Republics (USSR) Academy of the Sciences. It reverted to the Russian Academy of Sciences following the dissolution of the Soviet Union, and remains an influential organization in the twenty-first century. Academies of sciences were also founded in most of the states of the Soviet Union. Universal compulsory education was established in 1919. Soviet schools had both political and educational goals but the expectation that all children would attend school rapidly increased literacy and played a key role in modernizing and industrializing the country.

In the Soviet Union, the study of mathematics and the sciences was emphasized, a choice that not only fostered rapid economic growth but also became a point of national pride, as by mid-century the Soviet Union was frequently seen to rival or even surpass the United States in scientific and applied research. When the Soviet Union successfully launched the satellite Sputnik in 1957, it raised concern in the United States not only because of the possibility that the Soviet Union was developing weapons for which the United States had no counter but also because it put into question the common assumption that the United States was the world leader in mathematics and science. One result of Sputnik in the United States was a substantial increase in federal funding for scientific education and research in the hope of catching up and surpassing the Soviet Union in the “space race.”

As part of this concern that the Soviet Union was surpassing the United States, many studies were commissioned of the Soviet educational system and how it differed from the American system. Among the differences noted by researchers were the facts that in Soviet schools, specialists taught mathematics from the fourth grade onward, a uniform curriculum was used across the entire country, and much greater emphasis was placed on developing the talents of students who were identified as gifted in mathematics. The Soviet Union had “special schools,” which were free boarding schools at high school level for gifted students and specialized in particular subjects. Four such schools were devoted to mathematics. Correspondence courses in advanced mathematics were also available to increase the number of students studying those subjects. American observers noted that the level of mathematics required for university admittance during the Soviet period was much higher than what would be expected for entering freshmen in the United States. At the same time, other authors have noted that English-language sources often do not reflect the full scope and influence of Russian and Soviet mathematics. These omissions may be because of Cold War influences and a period of Soviet isolationism from the United States and much of Europe, a policy that contrasts strongly with earlier Russian connections and the growing collaborations following the Soviet era.

Notable Soviet and Russian Mathematicians

Andrey Kolmogorov (1903-1987) is known for his work in the fields of probability theory and topology, including the Kolmogorov axioms, Kolmogorov’s zero-one law, and Kolmogorov space.

Stefan E. Warschawski (1904-1989) studied at the University of Königsberg and Göttingen. His doctoral thesis was on the boundary behavior of conformal mappings.

Sergei Lvovich Sobolev (1908-1989) worked in mathematical analysis and partial differential equations. Sobolev spaces (named after him) can be defined by growth conditions on Fourier transforms.

Israel Moiseevich Gelfand (1913-2009) worked in the field of functional analysis. He is known for the Gelfand representation in Banach algebra theory; the representation theory of the complex classical Lie groups; contributions to distribution theory and measures on infinite-dimensional spaces; integral geometry; and generalized hypergeometric series. His name is linked to the development of mathematical education.

Igor Shafarevich (1923-) is the founder of the major school of algebraic number theory and algebraic geometry in the Soviet Union. He has also written well-known textbooks.

Grigori Perelman (1966-) declined the Fields medal, a prestigious award in mathematics often equated to the Nobel Prize, for his work on the Poincaré conjecture, named for Henri Poincaré. He cited inequities and reportedly noted, “If the proof is correct then no other recognition is needed.”

Other well-known Soviet or Russian twentieth-century mathematicians include Boris Pavlovich Demidovich, who worked on problems in mathematical analysis, and Yakov Isidorovich Perelman, who was a science writer and author of many popular science books.

Czech Republic and Slovakian Mathematicians

Kurt Gödel (1906-1978) proved fundamental results about axiomatic systems. Gödel’s Incompleteness Theorems are named for him.

Stefan Schwarz (1914-1996) studied semigroups, number theory, and finite fields and founded the Mathematico-Physical Journal of the Slovak Academy of Sciences in 1950.

Hungarian Mathematicians

Hungarian mathematicians of the twentieth century are well known in the mathematical world. Many of them immigrated to the United States after World War II.

Frigyes Riesz (1880-1956) was a founder of functional analysis. He produced representation theorems for functional on quadratic Lebesgue integrable functions, named for Henri Lebesgue, then introduced the space of q-fold Lebesgue integrable functions. He also studied orthonormal series and topology.

George Pólya (1887-1985) worked in probability, analysis, number theory, geometry, combinatorics, and mathematical physics. He wrote books about problem-solving methods, complex analysis, mathematical physics, probability theory, geometry, and combinatorics. He was regarded by many as a great teacher and influenced many mathematicians.

Cornelius Lanczos (1893-1974) worked on relativity and mathematical physics. He invented what is now called the Fast Fourier Transform, named for Joseph Fourier. He published more than 120 papers and books.

John von Neumann (1903-1957) worked in quantum mechanics, game theory, and applied mathematics, as well as helping pioneer computer science. His doctoral thesis was on set theory. His definition of ordinal numbers is the one commonly used in the early twenty-first century.

Rózsa Péter (1905-1977) is known for teaching, for her books on the history of mathematics, and for her series of theorems about primitive recursive functions.

Paul Erdos (1913-1996) is well known among mathematicians for his insatiable ability to pose and solve problems. It is often said that he lived on mathematics and coffee, touring the circle of his friends and pupils and giving lectures on combinatorics, graph theory, and number theory. He advocated for elegant and elementary proof. One of the most prolific mathematicians in history, he wrote more than 1500 papers.

Paul Richard Halmos (1916-2006) is known for his contributions to operator theory, ergodic theory, functional analysis (in particular Hilbert spaces, named for David Hillbert), and for his textbooks.

Alfréd Rényi (1921-1970) worked on probability theory, statistics, information theory, combinatorics, graph theory, number theory, and analysis.

László Lovász (1948-) published his first paper called On graphs not containing independent circuits when he was only 17 years old. He is a prominent figure of post-World War II mathematicians.

Notable Polish Mathematicians

Stefan Banach (1892-1945) worked on the theory of topological vector spaces, measure theory, integration, and orthogonal series. His doctoral thesis “On Operations on Abstract Sets and their Application to Integral Equations” (1920) marks the birth of modern functional analysis. He defined the “Banach space.”

Benoit Mandelbrot (1924-2010) is known as the father of fractal geometry. The Mandelbrot set, a connected set of points in the complex plane, is named after him.

Mathematicians From Romania

János Bolyai (1802-1860) is perhaps the most famous Romanian mathematician because of his treatise on a complete system of non-Euclidean geometry in his book Appendix. In his own words, he created a new world out of nothing.

Caius Iacob (1912-1992) worked in the fields of analytic geometry, descriptive geometry, analysis, and complex functions.

Grigore C. Moisil (1906-1973) worked on differential equations, the theory of functions, and mechanics. He set up the first Romanian computer science course. Moisil was appreciated for his philosophy and humor.

Other important Romanian mathematicians include Dimitrie Pompeiu, Ferenc Radó, Isaac Jacob Schoenberg, Simion Stoilow, Gheorghe Titeica, Gheorghe Vranceanu, Octav Onicescu, Ion Colojoara, and Dan Barbilian.

Competitions and Contests

Building on eastern Europe’s strong mathematics traditions, many mathematical contests are hosted frequently or entirely within the region, such as International Mathematical Olympiad, Romanian Master of Sciences (formerly called the Romanian Masters in Mathematics it was expanded to include physics), Czech-Polish-Slovak Match, Bulgarian Competition in Mathematics and Informatics, Romanian National Olympiad, and the International Kangaroo Mathematics Contest (often called “Math Kangaroo”) among others. Individuals from all over the world participate regularly in these competitions. There are also several winners of the Fields Medal who were born or worked in eastern Europe.

Bibliography

Davis, Robert B. “An Analysis of Mathematics Education in the Union of Soviet Socialist Republics.” Report for the National Institute of Education. December 1979. http://www.eric.ed.gov/PDFS/ED182141.pdf.

Dickson, Paul. Sputnik: The Shock of the Century. New York: Walker Publishing, 2001.

Sinai, Iakov. Russian Mathematicians in the 20th Century. Singapore: World Scientific Publishing, 2003.

Vogeli, Bruce R. Soviet Secondary Schools for the Mathematically Inclined. Washington, DC: National Council of Teachers of Mathematics, 1968.