# David Hilbert

**Summary: **What kind of a man was David Hilbert that started life seemingly innocuous in his education, yet fierce in his career? He was a man with one focus in life and sought to have is work impact that area…mathematics…well into the 20^{th} Century.

**Early Years**

On January 23, 1862, David Hilbert was born in Konigsberg, Prussia (now in what is Russia). He was the son of a local judge, Otto Hilbert and Maria Therese Erdtmann. David, it seems, was destined to have a life that comprised science and mathematics as his mother had a deep interest in aspects of mathematics and in astronomy. It would be accurate to say that his mother would have a deep impact upon his early education.

When he was of six years of age, his parents would have one more child…his sister Elsie. Young Hilbert’s father, believed in living according to his faith and a strong standard of sobriety in thought. As a judge, Oto Hilbert did not deviate from the routine of his life…either in work as a Privy judge or in the home.

David’s education was homeschooling for the first six years of his life. Although the family could afford sending him onto a local school, Maria Hilbert was the lad’s main instructor during this time. Along with her love of philosophy, she spent considerable time on math (she apparently had a keen interest in prime numbers) as well as the other disciplines necessary. It also would be reasonable to assume that David’s move from home education to a formal setting was in part based upon his gaining a new sister, which would take up his needed attention.

By age eight, young Hilbert began attending Royal Friedrichskolleg. He would perform his education here, both in the junior section and then in transferring to the gymnasium (not simply physical education as known today) portion. Languages, such as Greek and Latin would comprise much of his education there. Uniquely the study of sciences and mathematics was not as rich a focused discipline as David would have appreciated.

Another aspect of his education during time at Royal Friedrichskolleg, that was not to his favor involved the way studies were handled. Having come from an early home education that did not involve the discipline of memorization, David found himself at a disadvantage. The school placed a high value on the student’s ability to retain vast amounts of what they were exposed to; which was contrary to his study methods. This caused the young Hilbert to not be one of the school’s star pupils. However, it did not seem to diminish the future accolades that were to become David Hilbert’s.

By age 17, the young man transferred over to a school that was more to his way of study and subjects to his liking. He entered the Wilhelm Gymnasium. It was here that David flourished, even for only one year that he would attend. Science and math were the greater focus of this educational facility. He was also encouraged to develop critical and original thinking…not simply rote memorization. Because of this approach that sparked his mind, David became a top-flight student in grades and deportment. It seems the once self-described “dull and silly” lad was becoming bright and envisioning. The last report card of his education at Wilhelm stated;

*“…he mastered all the material taught in the school in a very pleasing manner and was able to apply it with sureness and ingenuity.”*

David Hilbert decided that to attend university at a distance from home made no sense. He chose to enter the University of Konigsberg in 1880. The type of courses that David took were not simple matters of freshman introduction. Because he would attend lectures at a distance in his second semester, David spent his first semester focused heavily on mathematics. The emphasis upon integral calculus and surface curvature. The second semester allowed for Hilbert to visit Heidelberg, where he would listen to lectures on differential equations by Leonard Fuchs. His first year alone placed Hilbert under the experience and intellectual faculties of great mathematicians.

By his next year of university, Hilbert was to meet and become close to Hermann Minkowski. At the age of 17, this young German student won the noted Grand Prix des Sciences Mathmatiques of the Paris Academy. His connection with the shy mathematician would influence his studies while in college. Another friend that Hilbert would gain while at Konigsberg was Adolf Hurwitz. Hurwitz had become an assistant professor at university in a position known as Extraordinary. The three men spent daily walks discussing their favorite topics; including math, philosophy and the ladies. Theirs would be a lifetime friendship.

Other professors that Hilbert studied under were Heinrich Weber and later Ferdinand von Lindemann, as Weber’s successor. Lindemann oversaw the thesis work that Hilbert completed on a specialized study invariant properties of algebraic forms and his oral test. His friend Minkowski read his study on invariants and wrote to his friend (as found in C Reid’s “*Hilbert”)*

*I studied your work with great interest and rejoiced over all the processes which the poor invariants had to pass through before they manage to disappear. I would not have supposed that such a good mathematical theorem could have been obtained in Königsberg.*

In 1885, Hilbert went on to receive his Doctorate of Philosophy…all within five years. From a self-avowed dullard in school to a full Ph.D., Hilbert was progressing at an incredible rate in his pursuit of the study and practice in mathematics.

Upon graduation, David Hilbert left Konigsberg and began a study travel of Europe. He first went to Leipzig where he spent time under Felix Klein; where his visit was less than enlightening. While in Leipzig, David met Eduard Study, a mathematician whom also had an interest in the invariant properties of certain algebraic forms. Although the studies were not of his taste under Klein; the man did give Hilbert and Study the advice to go to Erlangen and listen to lectures by Paul Gordan who had amassed a great amount of time studying invariants.

After Erlangen, the two travelled to Paris to listen to the work of Henri Poincare. In spite of the fact that the ever-opinionated Hilbert found Poincare to be average in his thinking; the young man made the most of his time by listening with intense attempts of seeming interested. This was happening in the early spring of 1886. Since it was Klein who had advised the men to visit with Poincare, they took opportunity to share by letters their assessment of the Parisian mathematician to Klein.

While still in Paris, Hilbert and Study had opportunity to meet other French mathematicians (such as Amedee Mannheim and George-Henri Halphen) at a gathering in the Germans honor. Although the conversation was ripe with pleasantries and good will; the two men wrote Klein feeling the time had been rather boring. They felt the talk of their mutual interest in math was rather trite to say the least the Frenchmen’s attempt at discussion German was broken and not very in depth. However, in FrenchmanCharles Hermite he found a kinship in the discussion of invariants. In fact, they were so impressed with the elderly mathematician that made a return visit with Hermite at his home.

By late spring of 1886, Hilbert was needing to make a return visit to Konigsberg for an opportunity to rest after his intense journey through Europe to meet the men of mathematics. However, he took the time to stop in Gottingen to once again visit Klein and give him an update of his travels, even though David Hilbert had been apprising Klein along the way by letter. In Gottingen, Hilbert was encouraged to visit Leopold Kronecker.

A turning point for Hilbert was occurring when he went onto listen to the work of Leopold Kronecker in Berlin. David Hilbert had been warned to find an off-putting attitude in Kronecker. Thus far his visits to varied scholars in mathematics had colored David Hilbert to feel the sessions had been lackluster. But with Kronecker he found the man to be out and out hard to take in. The older German had a propensity for staying totally focused on one aspect of mathematics; to deviate was not in his nature. “”God made the natural numbers, all else is the work of man” was a line that Kronecker hung onto throughout his career. In the man, Hilbert found him friendly, but unwilling to accept new insights into the realm of mathematics. Because of Kronecker’s rigidness on the subject, as well as others, young Hilbert altered his mindset about things and became even more impassioned about being open to new ideas in his field of study.

Whereas other scholars in the realm of mathematics seemed to be impassioned about other fields of study in their careers; David Hilbert was a purist with mathematics. His life was evolving and establishing around the subject.

Now was his opportunity finally return home and rest. While back at university in Konigsberg, Hilbert planned on completing and offering up his study on invariants. Yet, even though this was his expertise; he was asked to give a lecture that had nothing to do with his field, the *most general periodic functions. *

From C Reid’s book, *Hilbert, *David apparently writes to his old friend Klein about his desire to complete his sabbatical in Konigsberg. He says,

“*I am content and full of joy to have decided myself for Königsberg. The constant association with Professor Lindemann and, above all, with Hurwitz is not less interesting than it is advantageous to myself and stimulating. The bad part about Königsberg being so far away from things I hope I will be able to overcome by making some trips again next year, and perhaps then I will get to meet Herr Gordan.”*

* *From the years 1886 to 1895, Hilbert was on staff at university in Konigsberg. After filling a couple of lesser teaching positions, David Hilbert made full professor in 1893. During these years as an Extraordinaire professor, then finally full position, Hilbert married Kathe Jerosch (a second cousin) in October of 1892. By August of the next year, Kathe gave birth to their son, Franz.

During his period of time as a professor, Hilbert took time to once again travel places to visit with and listen to lectures by well-spoken men in the area of mathematics. David Hilbert continued to live and breathe mathematics.

When a vacancy occurred in Gottingen for a chair position, Felix Klein thought of David Hilbert to come visit and take over the position. But others preferred the position for Heinrich Weber and was given the spot. When Weber moved on to a chair position in Strasbourg, Hilbert was once again considered and this time given the post in Gottingen in 1895. While in Gottingen, where he would spend his remaining years, David Hilbert was able to bring his old friend Minkowski to be on staff. The two had remained close friends since their days in college together.

Later Paul Gordan and David Hilbert would disagree with Hilbert’s solving of certain invariants because Gordan believed Hilbert’s approach to his conclusions stepped away from known and proven approaches; otherwise too out of the box. However, creative and open-minded approaches to issues in mathematics is what David Hilbert had followed as his method since his first travels in Europe.

**Achievements**

David Hilbert throughout his life continued to work on the invariants of algebraic forms. Through his study, he radically altered the mathematics of invariants and developed his Basis Theorem (a finite number can represent invariants in algebraic). One book he had published in 1897, *Zahlbericht *contained his work on number theory in algebraic. This book had an impact on how the subject was dealt with in the years to follow. Hilbert also developed a set of axioms in dealing with Euclidean geometry.

Another area that marks the legacy of David Hilbert is what he listed as the 23 problems in mathematics that existed in his day. In a speech given to the International Mathematical Congress in Paris in 1900, Hilbert outlines what he considered to be the most notable issues, or problems, that were resonating in the math of his day. These were what he said would be the areas of focus that mathematicians needed to resolve in the new century. To date, most of those that he stipulated as problematic have been researched and resolved.

His work inspired others to become more innovative in their way of researching mathematics.

David Hilbert died in 1943 alone from many of his old friends. Nazi Germany had impacted many of his colleagues by causing many to leave as they were of Jewish ancestry.