1. Overview
Richard Courant was a prominent German-American mathematician whose work significantly influenced the fields of real analysis, mathematical physics, the calculus of variations, and partial differential equations. He is widely recognized for co-authoring the influential textbook Methods of Mathematical Physics with David Hilbert and the popular science book What is Mathematics? with Herbert Robbins, which aimed to make higher mathematics accessible to a general audience. Beyond his research and publications, Courant made substantial contributions to mathematical education and institution-building, most notably by founding and leading the Institute for Graduate Studies in Applied Mathematics at New York University, which was later renamed the Courant Institute of Mathematical Sciences in his honor. His career was profoundly shaped by the political turmoil of Nazi Germany, from which he emigrated, seeking and establishing an environment for academic freedom and advanced mathematical research in the United States.
2. Early Life and Education
Richard Courant was born on January 8, 1888, in Lubliniec, then part of the Prussian Province of Silesia in the Kingdom of Prussia, which is now located in Poland. His parents were Siegmund Courant and Martha Freund, originally from Oels. Notably, Edith Stein, a philosopher and saint, was Courant's cousin on his maternal side. During his youth, Courant's family frequently relocated, moving to places such as Kłodzko and Wrocław, before settling in Berlin in 1905.
Courant remained in Wrocław to begin his university studies at the University of Wrocław. However, finding the lectures there unsatisfactory, he transferred to the University of Zürich and subsequently to the University of Göttingen, which at the time held the highest reputation in mathematics. In Göttingen, he became an assistant to the renowned mathematician David Hilbert, who served as his academic mentor. Under Hilbert's guidance, Courant completed his doctoral studies and was awarded his doctorate in 1910. He further qualified as a professor in 1912.
3. World War I and Early German Career
Courant's early academic career was interrupted by World War I. He was conscripted into military service but sustained an injury shortly after enlisting, leading to his dismissal from active duty. Consequently, he did not participate extensively in combat. Following the war, Courant continued his research in Göttingen. In 1921, he left the University of Münster to assume the position previously held by Erich Hecke at the University of Göttingen.
At the University of Göttingen, Courant played a pivotal role in establishing the Mathematical Institute. He served as its director from 1928 until 1933, transforming it into a leading center for mathematical research.
4. Emigration and Founding the Courant Institute
In 1933, Richard Courant made the crucial decision to leave Nazi Germany, preceding many other Jewish individuals who would later flee the regime. Although his status as a former front-line soldier exempted him from immediate dismissal on racial grounds under Nazi laws, his public membership in the Social Democratic Party of Germany, a left-wing political party, was deemed sufficient reason by the Nazis for his removal from his academic position. This political persecution forced Courant to abandon his established career in Germany and seek academic freedom elsewhere.
After his departure from Germany, Courant spent a year at Cambridge University in England. In 1936, he accepted a professorship at New York University (NYU) in New York City, United States. There, he was entrusted with the significant task of establishing an institute for graduate studies in applied mathematics. Courant's organizational talent and vision led to the remarkable success of this endeavor. The institute, initially known as the Institute for Graduate Studies in Applied Mathematics, was officially renamed the Courant Institute of Mathematical Sciences in 1964, in recognition of his foundational contributions and leadership. Today, the Courant Institute remains one of the most respected and authoritative research centers in applied mathematics globally [https://www.usnews.com/best-graduate-schools/top-science-schools/applied-mathematics-rankings US News Rankings of Applied Mathematics Programs].
5. Mathematical Contributions
Richard Courant made profound and lasting contributions across various branches of mathematics, significantly impacting both theoretical research and practical applications. His work bridged the gap between pure mathematics and its relevance to physical phenomena, and he was also a prolific author of influential textbooks and popular science works that shaped generations of students and the general public's understanding of mathematics.
5.1. Research Areas
Courant's primary research interests spanned several core areas of mathematics. He focused extensively on real analysis, which deals with real numbers and real-valued functions. His work also delved deeply into mathematical physics, exploring the mathematical structures underlying physical theories. Furthermore, he made significant advancements in the calculus of variations, a field concerned with finding functions that optimize certain integrals, and in the study of partial differential equations, which are fundamental to describing a wide range of physical processes.
5.2. Major Publications
Courant authored or co-authored several highly influential books, ranging from seminal textbooks for advanced students to works aimed at popularizing mathematics for a broader audience. These publications reflect his dedication to both rigorous mathematical exposition and accessible education.
5.2.1. "Methods of Mathematical Physics"
One of Courant's most significant contributions to mathematical literature is the influential textbook Methods of Mathematical Physics (German: Methoden der mathematischen PhysikGerman), which he co-authored with his mentor, David Hilbert. The first volume was originally published in 1924, with a second revised edition appearing in 1931. The second volume was published in 1937. This work is a cornerstone in the field, bridging theoretical mathematics with practical applications in physics. It systematically covers topics such as linear algebra, calculus of variations, integral equations, and partial differential equations, presenting a comprehensive framework for solving problems in mathematical physics. Despite being published decades ago, the book, with its revised editions, remains current and is still widely used by students and researchers in physics and mathematics due to its enduring clarity, rigor, and comprehensive scope.
5.2.2. "What is Mathematics?"
In collaboration with Herbert Robbins, Courant co-authored the popular science book What is Mathematics? An Elementary Approach to Ideas and Methods, first published in 1941. This book was specifically designed to provide a comprehensive and accessible overview of higher mathematics for the general public. It covers fundamental concepts from number theory, geometry, topology, and calculus, explaining complex ideas in an intuitive and engaging manner. The book's aim was to demystify mathematics and highlight its beauty and logical structure, making it a highly impactful work for promoting mathematical literacy and inspiring countless individuals to explore the subject further. It continues to be published and read widely today.
5.2.3. "Introduction to Calculus and Analysis"
Courant also co-authored the comprehensive two-volume work Introduction to Calculus and Analysis with Fritz John. The first volume was initially published in 1965, followed by the second volume in two parts in 1974. This textbook is known for its rigorous yet clear pedagogical approach, providing a thorough treatment of calculus and mathematical analysis. It serves as a foundational text for advanced mathematical education, guiding students through the intricacies of real analysis, vector calculus, and differential equations with detailed explanations and numerous examples.
5.3. Numerical Analysis and Applied Mathematics
Courant made significant contributions to the fields of numerical analysis and applied mathematics. His name is notably associated with the finite element method, a powerful numerical technique used for finding approximate solutions to partial differential equations and integral equations. While the method was originally conceived by engineers, Courant provided it with a solid mathematical foundation through his numerical treatment of the plain torsion problem for multiply-connected domains, published in 1943. This work was crucial in establishing the method's theoretical rigor and expanding its applicability. Today, the finite element method is one of the most important and widely used techniques for numerically solving partial differential equations across various scientific and engineering disciplines.
Additionally, Courant is a namesake of two other fundamental concepts in numerical analysis and mathematical physics: the Courant-Friedrichs-Lewy condition (CFL condition) and the Courant minimax principle. The CFL condition, developed with Kurt Otto Friedrichs and Hans Lewy, is a necessary condition for the convergence of explicit finite difference methods used to solve hyperbolic partial differential equations. The Courant minimax principle provides a way to characterize the eigenvalues of symmetric matrices and self-adjoint operators, which is fundamental in areas like quantum mechanics and vibration analysis.
6. Mathematical Philosophy
Richard Courant held distinct philosophical views on the nature of mathematical proof and its relationship with empirical evidence. He emphasized the crucial role of mathematical rigor in scientific inquiry, asserting that empirical observations, while valuable, cannot substitute for formal mathematical existence proofs.
In his analysis of experimental results obtained from in-laboratory soap film formations, Courant articulated his perspective:
"Empirical evidence can never establish mathematical existence-nor can the mathematician's demand for existence be dismissed by the physicist as useless rigor. Only a mathematical existence proof can ensure that the mathematical description of a physical phenomenon is meaningful."
This statement underscores Courant's belief that while physical phenomena can inspire mathematical questions, the validity and meaning of their mathematical descriptions ultimately rely on rigorous proof rather than mere observation. He saw mathematics as providing the fundamental framework that ensures the coherence and reliability of scientific understanding.
7. Personal Life and Family
Richard Courant's personal life was marked by both professional success and personal tragedy. In 1912, he married Nelly Neumann, who had earned her doctorate in synthetic geometry from Breslau in 1909. They resided together in Göttingen until their divorce in 1916. Tragically, Nelly Neumann was later murdered by the Nazis in 1942 because she was Jewish, a stark reminder of the horrific human cost of the Nazi regime's persecution.
In 1919, Courant married Nerina (Nina) Runge (1891-1991). Nerina was the daughter of Carl Runge, a distinguished professor of Applied Mathematics at Göttingen, known for his contributions to the Runge-Kutta methods in numerical analysis.
Richard and Nerina had four children, all of whom pursued notable careers in science and mathematics:
- Ernest Courant (1920-2020) became a distinguished particle physicist and an innovator in the field of particle accelerators.
- Gertrude Courant (1922-2014) pursued a career as a biologist and was married to the mathematician Jürgen Moser (1928-1999).
- Hans Courant (1924-2019) was a physicist who notably participated in the Manhattan Project, the top-secret research and development undertaking during World War II that produced the first nuclear weapons.
- Leonore Courant (1928-2015), known as "Lori," became a professional violist. She was married first to the mathematician Jerome Berkowitz (1928-1998) and subsequently to the mathematician Peter Lax until her death.
8. Awards and Recognition
Richard Courant received numerous accolades and was elected to prestigious academic societies in recognition of his significant contributions to mathematics and science. He was an elected member of the American Philosophical Society in 1953, a distinguished organization promoting useful knowledge in the sciences and humanities. In 1955, he was elected to the United States National Academy of Sciences, one of the highest honors for scientists and engineers in the United States. In 1965, the Mathematical Association of America honored him with their Award for Distinguished Service to Mathematics, acknowledging his profound impact on the field through his research, teaching, and institution-building efforts.
9. Legacy and Influence
Richard Courant's legacy is profound and multifaceted, extending across mathematical research, education, and institutional development. The Courant Institute of Mathematical Sciences at New York University stands as his most enduring institutional achievement. Under his leadership, it became a world-renowned center for applied mathematics, attracting top talent and fostering groundbreaking research. The institute's success is a testament to Courant's vision for a vibrant academic environment that integrates theoretical rigor with practical application.
His influential publications, particularly Methods of Mathematical Physics and What is Mathematics?, continue to shape mathematical thought and education. Methods of Mathematical Physics remains a standard reference, demonstrating the deep connections between mathematics and the physical sciences. What is Mathematics? has inspired generations of students and the general public, making complex mathematical ideas accessible and fostering a broader appreciation for the subject.
Courant's contributions to numerical analysis, including his foundational work on the finite element method and the development of the Courant-Friedrichs-Lewy condition, have had a lasting impact on how complex problems in science and engineering are solved today. His philosophical stance on the necessity of mathematical proof, even when faced with empirical evidence, reinforced the importance of rigor in scientific inquiry. Overall, Courant's work and leadership significantly shaped the landscape of modern mathematics, particularly in the realm of applied mathematics, and his influence continues to be felt in academic institutions and scientific research worldwide.
10. Death
Richard Courant died of a stroke on January 27, 1972, at the age of 84, in New Rochelle, New York, United States.