# Hungarian mathematics

Hungarian mathematics refers to the mathematical tradition and achievements associated with Hungary, particularly during its golden age in the early 20th century. Hungary has produced a disproportionately large number of influential mathematicians relative to its population size, leading to what has been called the Hungarian phenomenon in mathematics.

## History
### Early history

Hungarian mathematics began its rise to prominence in the early 1800s with János Bolyai, one of the creators of non-Euclidean geometry, and his father Farkas Bolyai. Though they were largely ignored during their lifetimes, János Bolyai's groundbreaking work on hyperbolic geometry would later be recognized as foundational to modern mathematics. Farkas Bolyai had studied at Göttingen from 1796 to 1799, where he formed a lasting friendship with Carl Friedrich Gauss. Both were interested in the "problem of parallels"—the independence of Euclid's fifth postulate. When Farkas sent his son's discoveries to Gauss, he wrote back with "I cannot praise this work too highly, for to do so would be to praise myself." Gauss had apparently anticipated the discovery but never published his findings, inadvertently denying János the recognition he deserved during his lifetime. The Hungarian Academy of Sciences later established the Bolyai Prize in János's honor in 1905, though it was only awarded twice—to Henri Poincaré and David Hilbert—before being discontinued due to World War I. The Bolyai Prize was re-established nearly a century later in 2000.
The emergence of Hungary as a mathematical powerhouse occurred in the early 20th century through a combination of social, educational and institutional developments. A crucial turning point came with the Austro-Hungarian Compromise of 1867, which granted Hungary significant autonomy within the Habsburg Empire. This was followed by the emancipation of Hungarian Jews, who made up about 5% of the population. For the first time, Jews were permitted to work for the state and teach in its schools.
Two significant institutions emerged in 1894 that would shape generations of Hungarian mathematicians. The first was the Középiskolai Matematikai és Fizikai Lapok (KöMaL), founded by Arany Dániel. This monthly publication presented challenging mathematical problems and published solutions submitted by students, which created a culture of mathematical problem-solving and helped identify promising young talent. The second was the Eötvös Competition, named after the physicist Baron Loránd Eötvös. This prestigious competition would become legendary for its ability to identify mathematical genius—a remarkable number of its winners went on to become world-renowned mathematicians, including John von Neumann, Paul Erdős, and Peter Lax.

### The Golden Age

The early 20th century saw Hungarian mathematics flourish under several towering figures. Lipót Fejér created what is considered the first coherent mathematical school in Hungary. His 1900 work on Fourier series, completed while still a student, revolutionized the field of mathematical analysis. Fejér was known not only for his mathematical achievements but for his role as a mentor and teacher. Frigyes Riesz established the University of Szeged as an important mathematical center through the Bolyai Institute. His book on functional analysis, co-authored with Béla Szőkefalvi-Nagy, became a classic in the field. Riesz was known for his elegant approach to mathematics, achieving far-reaching results using elementary methods—a characteristic that would become associated with the Hungarian style of mathematics.
A third influx in Hungarian mathematics emerged in the 1930s with the rise of discrete mathematics, including combinatorics, graph theory, and number theory. This development began with Dénes Kőnig, who wrote the first book on graph theory in 1936. However, it was Paul Erdős who would become the field's most influential figure. Erdős, working with collaborators around the world, helped establish discrete mathematics as a major mathematical discipline.

## Hungarian phenomenon

The Hungarian phenomenon refers to Hungary's remarkable ability to produce leading mathematicians far out of proportion to its population. This was particularly evident in the period between the World Wars, when a small, relatively poor country of fewer than 10 million people produced such figures as John von Neumann, Paul Erdős, George Pólya, Theodore von Kármán, and many others who profoundly influenced 20th-century mathematics. When asked to explain this phenomenon, Hungarian mathematicians typically cite both internal factors like the country's unique educational institutions and external factors like broader socioeconomic conditions that encouraged intellectual pursuits.