Karl Weierstrass

German mathematician

• Born: October 31, 1815; Ostenfelde, Bavaria (now Germany)
• Died: February 19, 1897; Berlin, Germany

A pioneer in the field of mathematics, Weierstrass is often called the “father of modern analysis” for having provided a solid arithmetical foundation for calculus.

Primary field: Mathematics

Specialty: Calculus

Early Life

Born in 1815, Karl Theodor Wilhelm Weierstrass was the oldest of four children. His father was a customs official whose employment necessitated many family relocations during Weierstrass’s childhood. His mother Theodora died in 1827, an event that strengthened his father’s already powerful influence on the boy. When Wilhelm secured employment in the Paderborn tax office, the family’s frequent movement came to an end. In 1829, young Karl entered the Catholic Theodorianum, where he proved an excellent student, especially in mathematics. As he worked on his education, the young pupil found part-time employment as a bookkeeper to assist in paying his family’s expenses.

Despite Weierstrass’s interest in developing his talents in math, his father had a different path chosen for him. Wilhelm desired that his son follow in his footsteps and chose a conservative career path as a civil servant. His father’s influence was such that the nineteen-year-old Weierstrass enrolled in the University of Bonn in 1834 to study public finance and law. Although he continued with the interest in mathematics that he had demonstrated in his youth, he struggled with his education on the college-level not from lack of talent but from lack of interest. Fencing and beer dominated his days, while studies took a back seat. Much to the dismay of his demanding father, Weierstrass returned home without a degree after four years at the university.

With few options available for the seemingly disinterested student, Weierstrass, with the help of a family contact, was admitted into Münster’s Theological and Philosophical Academy in 1839, where he received a teaching certificate in secondary education. Münster proved an agreeable sojourn for Weierstrass; there, he came into contact with the German mathematician Christof Gudermann. Gudermann introduced Weierstrass to the revolutionary field of elliptic functions. In 1841, Weierstrass received his teaching certificate and began a fifteen-year career as a secondary school teacher.

Life’s Work

Weierstrass’s extended career as a high school teacher brought with it much professional isolation. The distance from academic mathematicians afforded him the opportunity to develop his own thinking in the field of calculus, free from the prejudices and constraints typically encountered in professional fields. Weierstrass blossomed as an intellectual, while, at the same time, he taught a dizzying array of subjects that included history, botany, German, and mathematics. His days were filled tending to the needs of his students, while his evenings were spent in pursuit of mathematics.

Every night, despite bouts of dizziness that appeared intermittently from 1850 onward, Weierstrass explored the implications of abelian functions (named for Norwegian mathematician Niels Abel), which offered a generalized understanding of elliptic functions. These nocturnal studies ultimately paved the way for his own pioneering work in the field. Until 1848, Weierstrass’s work in mathematics occurred in complete isolation, but he was soon to share his findings with the world. His first publication was part of a recruitment prospectus for the Collegium Hoseanum in Braunsberg, where he worked, and it attracted little attention save for convincing a few parents to send their children to the school. He made his first major impact on the mathematics community with the 1854 publication of “Zur Theorie der Abelschen Funktionen” (On the theory of abelian functions), originally published in the Journal for Pure and Applied Mathematics. In its examination of abelian functions as a convergent power series, the work underscores Weierstrass’s rigorous and pioneering methodological approach. His study attracted much attention and finally earned the mathematician the opportunity for university employment that he had been looking for.

In 1854, the University of Königsberg granted Weierstrass an honorary doctoral degree. After several failed attempts to secure a prestigious university post, Weierstrass settled on an appointment at the Industry Institute of Berlin in June 1856. Several months later he was offered and accepted a post at the University of Berlin—a position he truly coveted—but could not actually fill the chair for several years due to contractual obligations incumbent upon his earlier acceptance of the post at the Industry Institute. In the same year that he became a university professor, he published another groundbreaking article that expanded upon his earlier work on abelian functions, this time examining the inversion of hyperelliptic integrals. It was this work that made him an even more prized commodity in the German university community.

In Berlin, Weierstrass finally found himself in the presence of fellow mathematicians. He threw himself into his new position and found his passion for mathematics heightened by the presence of such esteemed peer mathematicians as Ernst Eduard Kummer and Leopold Kronecker, both of whom challenged and supported Weierstrass. The three educators helped to make the University of Berlin the premier institution for the study of mathematics in Europe. Weierstrass regularly taught before packed classrooms, as he developed offerings in calculus with an emphasis on analytic functions.

Despite his professional success, Weierstrass found his health failing. In 1861, his growing problem with vertigo led to his collapse and year-long hiatus from the university. Upon his return, the bouts of dizziness largely disappeared, only to be replaced by chronic chest pains. No longer able to lecture in the traditional fashion, Weierstrass addressed his students from a seated position, while advanced students wrote on the board for him. He published little but commanded a profound influence. Over the course of his work at the University of Berlin, several of his students published their lecture notes from Weierstrass’s class, which brought the professor international acclaim. From these lectures, collected in Gesammelte Abhandlungen (Collected works, 1894–1927), Weierstrass’s role in developing modern analysis becomes apparent. He carefully defined calculus functions in terms of inequalities, which allowed mathematicians to approach problems in a far more rigorous manner than was permitted by the theoretical geometric approach popular in the mid-nineteenth century.

The collegial culture of the University of Berlin continued into the 1880s, but a schism between Weierstrass and his colleague Leopold Kronecker over the concept of infinity produced much acrimony and nearly resulted in Weierstrass seeking an appointment in Switzerland. He opted instead to stay in Berlin, where he would oversee the publication of his complete works. Failing health and near immobility marked the final three years of his life. He died in Berlin from pneumonia in 1897.

Impact

Weierstrass was widely recognized as one of the preeminent mathematicians of his era, and he is now remembered as the “father of modern analysis.” His rigorous approach to calculus created a solid arithmetical foundation for the discipline, which had previously been governed by vague, poorly defined theories that lacked the rigor of other mathematical fields. Modern calculus owes much of its theoretical foundation to his work. Some of his discoveries, such as the nondifferentiable function, proved highly controversial when first introduced, while others provided a solid basis for preexisting, yet tenuously stated, concepts. Many questioned the validity of the construction, but time has proven the soundness of his theory, which is today known as the Weierstrass function.

He is also famous for the Weierstrass M-test for detecting the uniform convergence of a series of functions, as well as advances in abelian functions. What remains most remarkable about Weierstrass’s career is that he spent fifteen years of his adult life laboring as a high school educator. Through diligent effort, and despite having failed in college, Weierstrass worked nightly in isolation to develop the foundation of the mathematical work that would make him famous and inspire the research of a generation of scholars. Once established at the University of Berlin, Weierstrass and his colleagues helped to train some of the most influential mathematicians of the late nineteenth century. Weierstrass’s students included Georg Cantor, Sofia Kovalevskaya, Ludwig Boltzmann, and Max Planck, all of whom went on to revolutionize mathematics.

Bibliography

Aczel, Amir D. A Strange Wilderness: The Lives of Great Mathematicians. New York: Sterling, 2011. Print. Explores the lifestyles of mathematicians from the ancient to the modern world, including that of Weierstrass.

Bartle, Robert G., and Donald R. Sherbert. Introduction to Real Analysis. Fourth Ed. New York: Wiley, 2011. Print. Standard modern textbook underscores the seminal role that Weierstrass played in the development of calculus. Features his ideas and advances on nearly every page.

Hawking, Stephen, ed. God Created the Integers: The Mathematical Breakthroughs that Changed History. Philadelphia: Running, 2007. Print. Tracks the evolution of modern mathematics, posing Weierstrass as a figure who demands attention. Offers a thorough treatment of the German mathematician.

Watson, Peter. The German Genius: Europe’s Third Renaissance, the Second Scientific Revolution, and the Twentieth Century. New York: Harper, 2010. Print. Explores the vibrant milieu of German intellectual thought inspired by trailblazing nineteenth-century thinkers such as Weierstrass.