## 9.11. Weierstrass, Karl (1815-1897)

Karl Weierstrass was one of the leaders in rigor in analysis and was known as the "father of modern analysis." In addition, he is considered one of the greatest mathematics teachers of all-time.Karl Wilhelm Theodor Weierstrass was born October 31, 1815, in Ostenfelde, Westphalia, Germany. He was the first of four children of a customs official under Napoleon. His father would later enter the Prussian taxation service and would move his family often. The father himself was the stereotypical overbearing father who attempted to dominate the lives of all his children. Karl was on the receiving end of lectures well past the age of forty. Curiously, none of his children ever married. Despite the many schools and inept parenting, Karl still managed to excel in school and held down a part-time job as a bookkeeper in his spare time.

Unfortunately, this was the start of his trouble. Karl's father saw that his son was intelligent because of all the prizes he brought home. He also deduced that his son must be good bookkeeper. From these two facts, he figured his son could be a great accountant and the best accounting jobs were in the government. Therefore, his son would study commerce and law to prepare for a government career. The only problem was that his son was much more interested in mathematics. However, Karl bowed to his father's wishes and entered the University of Bonn in 1834 to pursue a career as an accountant. That was about as far as he followed his father's advice. Sick of his lectures, he simply stopped attending them and spent most of his time fencing (with swords), drinking beer, partying and reading mathematics books. After four years, he returned home without a degree in anything.

His father was not pleased. However, a friend of the father suggested that Karl enroll at the Theological and Philosophical Academy at Munster, where he could obtain a secondary schools teaching degree. In 1839, Karl was admitted after promising to the school's authorities he would not follow in his old ways. There he met his mathematical inspiration Christof Gudermann. After scaring away all his students except for Karl after the first lecture, Gudermann was able to give personal training to his prize pupil. Especially important, the professor introduced to Weierstrass the idea of the power series, which he would use as the basis of his work. Throughout the rest of his life, Karl always expressed his gratitude to his teacher. As part of his written examination, he presented a revolutionary essay on elliptic functions. The paper, which was the starting point of Weierstrass's discoveries, was so well received by Gudermann that he compared his student to "discoverers who were crowned with glory." However, Karl did not see the praise or publish the paper. He received his teacher's certificate in 1841.

For the next fifteen years, Weierstrass was employed as a secondary school teacher, teaching subjects such as mathematics, physics, German, geography, gym and penmanship(?). He spent most of his free time split between socializing at the local beer hall and working on his mathematics. For this entire period, he was practically cut off from the mathematical community. With no fellow mathematicians to talk to, no mathematical libraries to visit and no money to sustain an exchange of ideas through the mail system, he was basically left alone to completely explore his ideas. One story relates one night, he was so caught up on working on a problem that he worked through the morning and refused to teach his class until he was finished. However, he still had time to have some fun. In 1848, revolution was in the air and the local censor had his hands full keeping the seditious stuff out of the paper. Since the censor hated poetry, he gave all the poems and songs to Weierstrass to examine. He, of course, made sure that all the most extreme items were published behind the censor's back. Not all was good though. In addition to being bored with his job, he began to suffer seizures that would plague him for over a decade.

However, Karl's life was about to change. In 1854, he sent a paper on Abelian functions to the famous Crelle mathematical journal for publication. He had published only once before in the school journal but no one had noticed it (or should have). When this paper was published in 1854, it startled the mathematical community who could only wonder how this genius had been stuck teaching children. Almost immediately, he received an honorary doctorate from the University of Konigsberg and the mathematical community scrambled to find him a proper position. In 1856, he accepted an associate professorship at the University of Berlin.

Weierstrass was understandably overwhelmed by the entire chain of events. Overloaded by the new responsibilities of being a professor, he suffered a nervous breakdown and collapsed in the middle of a lecture in 1861. He would not return for two years. After his return, he never trusted himself to write at the blackboard again, delegating a student to write for him. Despite this, in 1864 he was promoted to full professor, a position he would hold for the rest of his life.

As a professor, Weierstrass slowly gained the reputation as a master lecturer. Originally, despite the fact that his lectures were generally disorganized, students flocked to his classes because he was the only person offering his advanced subjects. As the years progressed, he acquired the famous teaching skills that have given him the label as one of the all-time great lecturers. In addition, he was also well known for his constant availability, his productive leads for future study and his insistence on paying for his students at the local tavern. Eventually, his classes swelled with over 250 pupils from around the world. One of his students, Sonja Kovalevsky, became one of his closest friends and the two met or corresponded for the rest of their lives. She was the closest any woman ever came to claiming the heart of this lifelong bachelor. He was fundamental in her attaining the position of professor at the University of Stockholm in 1883.

Unfortunately, the end of Weierstrass' life was filled with tragedy. First, his friend and colleague, Leopold Kronecker, publicly attacked him for his support of the revolutionary theories of Georg Cantor and generally made his life miserable until Kronecker's death in 1891. Second, after his critic's death, Sonja died at the age of 41. The death mentally crushed Karl and he burned everything he owned that reminded him of her. He spent the last few years of his life confined to a wheelchair and died from pneumonia on February 19, 1897.

Weierstrass is famous in mathematics for numerous accomplishments. He was the first person to create a continuous function that is not differentiable at any point. He developed a general theory of Abelian integrals and Abelian functions, which he considered his lifetime work. He used the power series as the basis of functions, an idea that was key in the development of much of mathematical physics, and created the sequential definition of irrational numbers based on convergent series. However, he is most famous for his insistence on rigor in all his works, especially analysis. Karl demanded that mathematics be based on clear and correct proofs. For this reason, he rarely published because he would not release his work until he was sure it was on a firm mathematical foundation. This was generally a monumental task considering that he started most of his work basically from scratch.

Ironically, he understood his own limitations. He once said: "It is true that a mathematician who is not also something of a poet will never be a perfect mathematician." He understood that mathematical perfection, just like poetic perfection, is impossible, though there is nothing wrong in trying.

For related information on Weierstrass, see: Niels Abel, Georg Cantor, Bolzano-Weierstrass theorem.

**Sources**

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- Bell, E.T.
*Men of Mathematics.*New York: Simon and Schuster, Inc., 1937. - Biermann, Kurt-R. "Weierstrass, Karl Theodor Wilhelm."
*Dictionary of Scientific Biography*. New York: Charles Scribner's Sons, 1970. vol. ?, pp. 219-224.

**Paul Golba**