__Hilbert__:

I. Introduction

It is easy to see the lasting effect David Hilbert has
left upon mathematics by looking in the index of an advanced mathematics
textbook. An advanced analysis text will discuss Hilbert transforms in Hilbert
space. Hilbert's discoveries in the fields of kinetic gas theory, radiation, and
relativity should be noted in an upper-level physics book. A text in numerical
methods should discuss the Hilbert matrix in its section on approximation
theory. A text in number theory should mention Hilbert's numerous contributions
to the field of algebraic number theory. Any abstract algebra textbook worth the
paper it was printed on will list Hilbert's Basis Theorem, Hilbert's
Nullstellensatz, Hilbert's Satz 90, and Hilbert's Specialization Theorem.
Arguably the greatest mathematician of the twentieth century, David Hilbert had
a profound organizing effect on mathematics. Hilbert had a vast knowledge that
spread to many branches of mathematics, ranging from number theory to geometry
to logic to mathematical physics. His sincere optimism in the solvability (or at
least the proof of the impossibility) of every mathematical problem inspired
great mathematicians and still inspires mathematicians. As a teacher and mentor,
he helped shape the collective mathematical genius of a generation of
mathematical giants. His goal to axiomatize and organize the various branches of
mathematics and physics is still a work in progress, having been taken up by
scientists across the world. He was a great man.

II. Biography

David Hilbert was born on January 23, 1862 in Konigsberg,
Prussia, now Kaliningrad, Russia. His father, Otto Hilbert, was a city judge, a
very respectable position in a small city. Constance Reid indicates that Hilbert
largest influence came from his mother Maria, "an unusual woman... interested in
philosophy and astronomy and fascinated by prime numbers." As a young boy,
Hilbert quickly found that mathematics came very easily to him. Since the
gymnasium he attended emphasized language, particularly Latin, over math and
science, he put aside his love for mathematics temporarily and concentrated on
his weaker subjects, vowing to return to mathematics as soon as possible.

He attended the university in Konigsberg, studying under Heinrich Weber, the only full professor of mathematics in Konigsberg. He visited the university in Heidelberg for a semester to hear lectures on differential equations by Leonard Fuchs. In 1882, Hermann Minkowski, a fellow student at Konigsberg, won the prestigious Grand Prix des Sciences Mathmatiques of the Paris Academy at the age of 17. Hearing of this unprecedented accomplishment, Hilbert quickly became friends with the shy Minkowski. In 1884, Adolf Hurwitz became an Extraordinarius, or assistant professor, at Konigsberg. They developed the habit of taking daily walks "to the apple tree... precisely at five" to discuss philosophy, literature, women, and, above all, mathematics. The three had formed a friendship that would last to their graves.

In 1884, Hilbert completed his oral examination and completed his thesis on invariant properties for certain algebraic forms under the supervision of Ferdinand Lindemann. In 1885, he publicly defended two theses and received his degree of Doctor of Philosophy. At the suggestion of Hurwitz, Hilbert toured Europe to meet the great mathematicians of the time. After an uninspiring visit with Felix Klein in Leipzig, Hilbert went to Paris to learn from the great, although somewhat overrated in Hilbert's opinion, Henri Poincar. He then went to Berlin to study under Leopold Kronecker. Hilbert found Kronecker very disagreeable, since the older mathematician was very rigid in his beliefs. Kronecker is famous for his statement that "God made the natural numbers, all else is the work of man." Kronecker's inflexibility and refusal to accept new ideas had a profound effect on Hilbert, inspiring the young German to be more open-minded and creative in his studies.

Hilbert returned to Konigsberg and, after passing another public examination, became a privatdozent, or unpaid lecturer, at the university. Although Minkowski was serving in the army at the time, Hilbert was delighted to renew the daily walks with Hurwitz. It was difficult for him to make money and also keep himself occupied at the small university. He did not receive a salary from the university, so he had to depend on fees paid by students that attended his lectures. He complained that Konigsberg had "eleven docents depending on about the same number of students." To relieve his boredom, Hilbert went on another study trip, this time carefully planning to meet the world's 21 greatest mathematicians. He first went to Erlangen to meet Paul Gordan, "the king of invariants." Hilbert was introduced to Gordan's Problem, which asks whether there exists a finite basis to express an infinite system of invariants. This solution to this problem would become Hilbert's first mathematical triumph. He then went on to Gottingen and then Berlin, meeting Klein, Weierstrass, Schwartz, and once again the formidable Kronecker. He returned to Konigsberg and immediately set to work on Gordan's Problem. In 1888, he produced an existence proof for the problem. A mathematical feat of this magnitude was remarkable for one so young. Kronecker and Gordan, however, were unimpressed. These old mathematicians were too rigid in their beliefs and refused to accept anything less than a constructive proof of a finite basis. Gordan said about Hilbert's work, "Das ist nicht Mathematik. Das ist Theologie." However, Hilbert's work caught the eye of Felix Klein and the elder mathematician became determined to bring Hilbert to Gottingen to teach. In 1892, Hilbert produced a construction proof of Gordan's problem that met Gordan's standards.

The next few years were a time of great change for David Hilbert. In June of 1892, Hurwitz became a full professor at the Swiss Federal Institute of Technology in Zurich and Hilbert took Hurwitz's place by becoming an assistant professor at Konigsberg. On October 12, Hilbert married his second cousin, Kthe Jerosch. On August 11, 1893, their son Franz was born. A few weeks later, Hilbert and Minkowski were commissioned by the German Mathematical Society to write a comprehensive survey of number theory. This honor was probably bestowed on Hilbert because of his recent, more direct proofs of the transcendence of e (first proved by Charles Hermite in 1873) and pi (first proved by Ferdinand Lindemann in 1882). It was decided that Hilbert would tackle algebraic number theory while Minkowski would work on the geometric aspects of number theory. Although Minkowski never completed his portion of the survey, Hilbert's book, Zahlbericht, was, according to one reviewer, "a veritable jewel of mathematical literature." Before the Zahlbericht was published, however, David Hilbert received an important telegram from Felix Klein. Hilbert was promoted to full professor at Gottingen, the university which had shaped the great number theorist Carl Friedrich Gauss. The motto on the Rathaus, or town hall, of Gottingen states: "Away from Gottingen there is no life." For a mathematician in the early twentieth century, this was not far from the truth. The greatest mathematical minds of the time, both faculty and students, had gathered at Gottingen and Klein knew that Hilbert would make a splendid addition.

David Hilbert and his wife Kthe would spend the rest of their lives in Gottingen. At Gottingen, Hilbert found the tranquillity as well as the scientific stimulation he needed to become a prolific mathematician. A testament to his genius and versatility, Hilbert had a far-reaching effect on many diverse branches of mathematics. Hermann Weyl, a colleague of Hilbert at Gottingen, has classified Hilbert's work into five major areas: invariant theory, algebraic number field theory, foundations of geometry and mathematics, integral equations, and physics.

As mentioned earlier, Hilbert's thesis was on the theory of invariants. His two proofs of Gordan's Problem established him as a first-class mathematician. These proofs led him to "one of the most fundamental theorems of algebra," namely that every subset of a polynomial ring of independent variables has a finite ideal basis. Weyl claims that this theorem is "the foundation stone of the general theory of algebraic manifolds." Another famous theorem by Hilbert, the Nullstellensatz, states that for every polynomial g "vanishing" on the ideal's "set of vanishing points," some power of g is contained in the ideal. Weyl claims that the Nullstellensatz is "clearly" at the heart of the theory of algebraic manifolds. David Hilbert made other contributions to modern algebra, including a theorem under which one may substitute variables in an irreducible polynomial to obtain another irreducible polynomial and a solution of ninth degree equations. Weyl emphasizes the significance of Hilbert's discoveries: "After the formal investigations from Cayley and Sylvester to Gordan, Hilbert inaugurated a new epoch in the theory of invariants." Oddly enough, Weyl was merely paraphrasing Hilbert's own words. Characteristic of Hilbert's unabated, albeit justified, egotism, one of Hilbert's papers on invariants called Sylvester and Cayley "the representatives of the naive period" and himself the champion of "the critical period" of invariant theory. In 1892, he wrote to Minkowski that "I shall definitely quit the field of invariants."

With his move from Konigsberg to Gottingen, Hilbert's interest moved away from invariants and into algebraic number fields. The Zahlbericht was a great leap forward in algebraic number theory. Hilbert was greatly pleased at his assignment from the German Mathematical Society. Hilbert stated that "the theory of number fields is an edifice of rare beauty and harmony." One important contribution from this report is his Satz 90, which is a theorem on relative cyclic fields. Hilbert also developed a form of the Legendre symbol and the idea of a p-adic norm. Hilbert defined a p-adic norm as an integer in the quadratic field K that is congruent to the norm of a suitable integer in K modulo any power of p. The Legendre symbol (a/p) has the value +1 if x2 is congruent to a modulo an odd prime p is solvable (quadratic residue) and -1 if it is not solvable (quadratic nonresidue). Hilbert generalized this symbol to (a,K/p), which has value +1 if a is a p-adic norm and -1 if it is not. Hilbert found that this symbol is multiplicative, meaning that (a,K/p)*(b,K/p) = (ab,K/p). He then established certain reciprocity laws in terms of these norm residues. He also delved considerably in the theory of ideals and Abelian fields. Hilbert also began exploring the relations between number theory and modular functions, but, "indicative of the fertility of Hilbert's mind," he found other areas of mathematics to interest him. As mentioned before, Hilbert also provided simple and direct proofs of the transcendence of e and pi. Another breakthrough in algebraic number theory was his proof of Waring's Problem in 1909. In 1707, Edmund Waring proposed, without proof, that every positive integer can be written as the sum of 4 squares, 9 cubes, 19 fourth powers, and so on. More generally, for any k>1, there is a positive integer s such that every positive integer can be expressed as the sum of s nonnegative kth powers. However, Hilbert's proof is an existence proof and does not explain how to determine s given k. Even though he departed from algebraic number theory, after he went into retirement, Hilbert confided in his friend Olga Taussky that "much as he admired all branches of mathematics, he considered number theory the most beautiful."

David Hilbert next turned his attention to axiomatics, which is the process of laying down axioms, or laws, for geometry and mathematics in general. Hilbert's view on the foundations of geometry is summarized in his famous statement at the Berlin railway station: "One must be able to say at all times -- instead of points, straight lines, and planes -- tables, beer mugs, and chairs." The meaning of these words is that one may only use the established geometric axioms in a proof instead of the real interpretation of geometrical objects. Hilbert did much to axiomatize geometry, as found in his 1899 book, Grundlangen der Geometrie. Hilbert produced a new set of geometric axioms that were both consistent (no axiom overlapped with another) and complete (the collection of axioms enabled one to express all of geometry). This axiomatization had changed the face of geometry more than any individual since Euclid. He then turned his attention to the foundations of mathematics in general. He developed a research program, now known as Hilbert's Program, by which he hoped to rigorously prove the consistency of logic and set theory. Hilbert successfully defended his program against critics such as L.E.J. Brouwer, who wished to expose paradoxes in set theory that would render Cantorian set theory and proof by contradiction meaningless. To Hilbert, Brouwer's arguments sounded like the stifling rigidity of Leopold Kronecker. However, in 1930, a 25-year old logician named Kurt Godel published a paper that dismantled Hilbert's Program. According to rumors from Hilbert's assistant, Hilbert was furious when he learned of Godel's work. Ian Stewart summarizes the content of Godel's work as follows: "Godel showed that there are true statements in arithmetic that can never be proved, and that if anyone finds a proof that arithmetic is consistent, then it isn't!" One modern philosopher, Michael Detlefsen, has argued that Godel's second theorem is flawed and consequently Hilbert's Program is valid. Even though the success of Hilbert's Program is still being debated, Hilbert has done much to organize mathematics. As Weyl wrote after Hilbert's death, "Hilbert is the champion of axiomatics."

There was a gap of twenty years between the periods Hilbert concentrated on the foundations of geometry and the foundations of mathematics in general. During this time, he focused on integral equations and mathematical physics. Hilbert's first contributions to analysis involved homogenous integral equations and the problem of determining eigenvalues of an integral equation. Hilbert also developed a method of analysis using infinitely many vectors in an infinitely-dimensioned space. This space is now known as Hilbert space and is crucial to functional analysis. The study of transforms in Hilbert space has become very important to the studies of integral and differential equations, partial differential equations, quantum mechanics, optimization problems, bifurcation theory, approximation theory, stability problems, variational inequalities, and control problems for dynamical systems. Hilbert's work in integral equations also had a more immediate effect, namely it encouraged his student Richard Courant to delve into analysis. The study of integral equations naturally led into Hilbert's fascination with mathematical physics.

In 1910, Hilbert received the prestigious Bolyai Prize, which acknowledged him as the world's second greatest mathematician after Poincare. The award committee praised his work in mathematics, but it was at this time that Hilbert turned his attention to physics. Despising the lack of rigor that physicists displayed in mathematical proofs, Hilbert declared that "physics is much too hard for physicists." As with geometry, Hilbert hoped to axiomatize physics. He started with kinetic gas theory, employing the methods he had developed in integral equations. He turned his attention to radiation theory and later to gravitation and general relativity theory. Hilbert hired assistants from among the student body to tutor him on physics and, despite his initial awkwardness in comprehending physics, his mathematical talent gave some credit to his slanderous declaration about physicists. Constance Reid remarks that "while Einstein was attempting to solve the binding laws for the 10 coefficients of the differential form which determines gravitation, Hilbert independently solved the problem in a different, more direct way." Hilbert always gave full credit to Einstein, however. He once remarked, "Every boy in the streets of Gottingen understands more about four-dimensional geometry than Einstein. Yet, in spite of that, Einstein did the work and not the mathematicians."

David Hilbert also had a more indirect on the future of mathematics by revealing new questions and areas of research for future generations of mathematicians. This is best evidenced by Hilbert's speech at the second International Congress of Mathematicians in Paris in the summer of 1900. Hilbert presented 23 open problems that spanned the various branches of mathematics. These so-called Paris Problems continue to inspire mathematicians and most of these problems remain unsolved.

During his time at Gottingen, Hilbert was a colleague to some of the greatest mathematical minds in history. The administrator who oversaw Hilbert's work for most of his life was the great algebraist, Felix Klein. After Klein's death, Richard Courant took over as administrator. Courant would later make significant contributions to analysis and found the Courant Institute in New York. Hilbert worked with Carl Runge, a leading figure in approximation theory. The great number theorist, Edmund Landau, also taught at Gottingen. Hilbert developed his theories on the foundations of mathematics with Ernst Zermelo, the famous logician. For questions in physics, Hilbert could always turn to the great physicists Max Born, James Franck, Arnold Sommerfeld, or Niels Bohr. He kept in constant correspondence with Minkowski and Hurwitz. When Hilbert received an offer to be the department chair in Berlin in 1902, he used this offer as bargaining power to bring his friend Hermann Minkowski to Gottingen. Once established there, Hilbert resumed his habit of long daily walks with his friend.

Hilbert would have a great influence on a generation of mathematicians by becoming a mentor. Hilbert was known for his simple and clear lectures. He was generally unprepared for his lectures and bicycled or, during the winter, skied into the lecture hall. He would often derive a theorem or proof in class, so that many students were astounded to see the work of genius in progress. He taught calculus to the Nobel Prize winner Max von Laue. Hilbert's Zahlbericht inspired Tejii Takagi to leave Japan, come to Gottingen, and make great contributions to algebraic number theory. A leader in mathematical physics, Hermann Weyl came to Gottingen to study under David Hilbert. Paul Bernays, the great logician, was an assistant to David Hilbert. In a time when female mathematicians were virtually nonexistent, Hilbert fought for Emmy Noether's acceptance into the doctoral program and later her right to be a privatdozent at Gottingen. Emmy Noether later emigrated to America and made impressive achievements in group and ring theory.

When Hilbert reached the age of 68 in 1930, he was forced to retire from teaching. In 1932, Adolf Hitler became the chancellor of Germany and a law was passed forbidding full-blooded Jews from teaching positions. This ban applied to Courant, Noether, Landau, Bernays, Born, and Franck. At a banquet, the minister of education asked Hilbert, "And how is mathematics in Gottingen now that it has been freed of the Jewish influence?" Hilbert replied, "Mathematics in Gottingen? There is really none any more." The Nazi regime ended Gottingen's position as the center of the mathematical world. David Hilbert died from on February 14, 1943 in a Gottingen torn apart by World War II. In 1962, Richard Courant gave an address on the importance of Hilbert's work. Courant was unable to decide in which area of mathematics Hilbert had contributed. Courant was sure that Hilbert's belief in the solvability of every problem was his greatest strength. "I am therefore convinced," Courant stated, "that Hilbert's contagious optimism even today retains its vitality for mathematics, which will succeed only through the spirit of David Hilbert."