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David Hilbert

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   CAPTION: David Hilbert

   David Hilbert (1912)
   David Hilbert (1912)
         Born        January 23, 1862
                     Wehlau, East Prussia
         Died        February 14, 1943
                     Göttingen, Germany
       Residence     Germany
      Nationality    German
         Field       Mathematician
      Institution    University of Königsberg
                     Göttingen University
      Alma Mater     University of Königsberg
   Doctoral Advisor  Ferdinand von Lindemann
   Doctoral Students Otto Blumenthal
                     Richard Courant
                     Max Dehn
                     Erich Hecke
                     Hellmuth Kneser
                     Robert König
                     Erhard Schmidt
                     Hugo Steinhaus
                     Emanuel Lasker
                     Hermann Weyl
                     Ernst Zermelo
       Known for     Hilbert's basis theorem
                     Hilbert's axioms
                     Hilbert's problems
                     Hilbert's program
                     Einstein-Hilbert action
                     Hilbert space

   David Hilbert ( January 23, 1862, Wehlau, East Prussia – February 14,
   1943, Göttingen, Germany) was a German mathematician, recognized as one
   of the most influential and universal mathematicians of the 19th and
   early 20th centuries. He invented or developed a broad range of
   fundamental ideas, in invariant theory, the axiomatization of geometry,
   and with the notion of Hilbert space, one of the foundations of
   functional analysis.

   He adopted and warmly defended Cantor's set theory and transfinite
   numbers. A famous example of his leadership in mathematics is his 1900
   presentation of a collection of problems that set the course for much
   of the mathematical research of the 20th century.

   Hilbert and his students supplied significant portions of the
   mathematical infrastructure required for quantum mechanics and general
   relativity. He is also known as one of the founders of proof theory,
   mathematical logic and the distinction between mathematics and
   metamathematics.

Life

   Hilbert was born in Wehlau, near Königsberg, East Prussia (now
   Znamensk, near Kaliningrad, Russia). He graduated from the lyceum of
   his native city and registered at the University of Königsberg. He
   obtained his doctorate in 1885, with a dissertation, written under
   Ferdinand von Lindemann, titled Über invariante Eigenschaften
   specieller binärer Formen, insbesondere der Kugelfunctionen ("On the
   invariant properties of special binary forms, in particular the
   circular functions"). Hermann Minkowski was also a doctoral candidate
   at the same university and time, and he and Hilbert became close
   friends, the two exercising a reciprocal influence over each other at
   various times in their scientific careers.

   Hilbert remained at the University of Königsberg as a professor from
   1886 to 1895, when, as a result of intervention on his behalf by Felix
   Klein he obtained the position of Chairman of Mathematics at the
   University of Göttingen, at that time the best research centre for
   mathematics in the world and where he remained for the rest of his
   life.

   In 1892, he married Käthe Jerosch (1864-1945) and had one child Franz
   Hilbert (1893-1969).

The finiteness theorem

   Hilbert's first work on invariant functions led him to the
   demonstration in 1888 of his famous finiteness theorem. Twenty years
   earlier, Paul Gordan had demonstrated the theorem of the finiteness of
   generators for binary forms using a complex computational approach. The
   attempts to generalize his method to functions with more than two
   variables failed because of the enormous difficulty of the calculations
   involved. Hilbert realized that it was necessary to take a completely
   different path. As a result, he demonstrated Hilbert's basis theorem:
   showing the existence of a finite set of generators, for the invariants
   of quantics in any number of variables, but in an abstract form. That
   is, while demonstrating the existence of such a set, it was not
   algorithmic but an existence theorem.

   Hilbert sent his results to the Mathematische Annalen. Gordan, the
   house expert on the theory of invariants for the Mathematische Annalen,
   was not able to appreciate the revolutionary nature of Hilbert's
   theorem and rejected the article, criticizing the exposition because it
   was insufficiently comprehensive. His comment was:

          This is Theology, not Mathematics!

   Klein, on the other hand, recognized the importance of the work, and
   guaranteed that it would be published without any alterations.
   Encouraged by Klein and by the comments of Gordan, Hilbert in a second
   article extended his method, providing estimations on the maximum
   degree of the minimum set of generators, and he sent it once more to
   the Annalen. After having read the manuscript, Klein wrote to him,
   saying:

          Without doubt this is the most important work on general algebra
          that the Annalen has ever published.

   Later, after the usefulness of Hilbert's method was universally
   recognized, Gordan himself would say:

          I must admit that even theology has its merits.

Axiomatization of geometry

          See supplement, Hilbert's axioms.

   The text Grundlagen der Geometrie (tr.: Foundations of Geometry)
   published by Hilbert in 1899 substitutes a formal set, comprised of 21
   axioms, for the traditional axioms of Euclid. They avoid weaknesses
   identified in those of Euclid, whose works at the time were still used
   textbook-fashion. Independently and contemporaneously, a 19-year-old
   American student named Robert Lee Moore published an equivalent set of
   axioms. Some of the axioms coincide, while some of the axioms in
   Moore's system are theorems in Hilbert's and vice-versa.

   Hilbert's approach signalled the shift to the modern axiomatic method.
   Axioms are not taken as self-evident truths. Geometry may treat things,
   about which we have powerful intuitions, but it is not necessary to
   assign any explicit meaning to the undefined concepts. The elements,
   such as point, line, plane, and others, could be substituted, as
   Hilbert says, by tables, chairs, glasses of beer and other such
   objects. It is their defined relationships that are discussed.

   Hilbert first enumerates the undefined concepts: point, line, plane,
   lying on (a relation between points and planes), betweenness,
   congruence of pairs of points, and congruence of angles. The axioms
   unify both the plane geometry and solid geometry of Euclid in a single
   system.

The 23 Problems

   He put forth a most influential list of 23 unsolved problems at the
   International Congress of Mathematicians in Paris in 1900. This is
   generally reckoned the most successful and deeply considered
   compilation of open problems ever to be produced by an individual
   mathematician.

   After re-working the foundations of classical geometry, Hilbert could
   have extrapolated to the rest of mathematics. His approach differed,
   however, from the later 'foundationalist' Russell-Whitehead or
   'encyclopedist' Nicolas Bourbaki, and from his contemporary Giuseppe
   Peano. The mathematical community as a whole could enlist in problems,
   which he had identified as crucial aspects of the areas of mathematics
   he took to be key.

   The problem set was launched as a talk "The Problems of Mathematics"
   presented during the course of the Second International Congress of
   Mathematicians held in Paris. Here is the introduction of the speech
   that Hilbert gave:

          Who among us would not be happy to lift the veil behind which is
          hidden the future; to gaze at the coming developments of our
          science and at the secrets of its development in the centuries
          to come? What will be the ends toward which the spirit of future
          generations of mathematicians will tend? What methods, what new
          facts will the new century reveal in the vast and rich field of
          mathematical thought?

   He presented fewer than half the problems at the Congress, which were
   published in the acts of the Congress. In a subsequent publication, he
   extended the panorama, and arrived at the formulation of the
   now-canonical 23 Problems of Hilbert. The full text is important, since
   the exegesis of the questions still can be a matter of inevitable
   debate, whenever it is asked how many have been solved.

   Some of these were solved within a short time. Others have been
   discussed throughout the 20th century, with a few now taken to be
   unsuitably open-ended to come to closure. Some even continue to this
   day to remain a challenge for mathematicians.

Formalism

   In an account that had become standard by the mid-century, Hilbert's
   problem set was also a kind of manifesto, that opened the way for the
   development of the formalist school, one of three major schools of
   mathematics of the 20th century. According to the formalist,
   mathematics is a game devoid of meaning in which one plays with symbols
   devoid of meaning according to formal rules which are agreed upon in
   advance. It is therefore an autonomous activity of thought. There is,
   however, room to doubt whether Hilbert's own views were simplistically
   formalist in this sense.

Hilbert's program

   In 1920 he proposed explicitly a research project (in metamathematics,
   as it was then termed) that became known as Hilbert's program. He
   wanted mathematics to be formulated on a solid and complete logical
   foundation. He believed that in principle this could be done, by
   showing that:
    1. all of mathematics follows from a correctly-chosen finite system of
       axioms; and
    2. that some such axiom system is provably consistent.

   He seems to have had both technical and philosophical reasons for
   formulating this proposal. It affirmed his dislike of what had become
   known as the ignorabimus, still an active issue in his time in German
   thought, and traced back in that formulation to Emil du Bois-Reymond.

   This program is still recognizable in the most popular philosophy of
   mathematics, where it is usually called formalism. For example, the
   Bourbaki group adopted a watered-down and selective version of it as
   adequate to the requirements of their twin projects of (a) writing
   encyclopedic foundational works, and (b) supporting the axiomatic
   method as a research tool. This approach has been successful and
   influential in relation with Hilbert's work in algebra and functional
   analysis, but has failed to engage in the same way with his interests
   in physics and logic.

Gödel's work

   Hilbert and the talented mathematicians who worked with him in his
   enterprise were committed to the project. His attempt to support
   axiomatized mathematics with definitive principles, which could banish
   theoretical uncertainties, was however to end in failure.

   Gödel demonstrated that any non-contradictory formal system, which was
   comprehensive enough to include at least arithmetic, cannot demonstrate
   its completeness by way of its own axioms. In 1931 his incompleteness
   theorem showed that Hilbert's grand plan was impossible as stated. The
   second point cannot in any reasonable way be combined with the first
   point, as long as the axiom system is genuinely finitary.

   Nevertheless, the incompleteness theorem says nothing with regard to
   the demonstration by way of a different formal system of the
   completeness of mathematics. The subsequent achievements of proof
   theory at the very least clarified consistency as it relates to
   theories of central concern to mathematicians. Hilbert's work had
   started logic on this course of clarification; the need to understand
   Gödel's work then led to the development of recursion theory and then
   mathematical logic as an autonomous discipline in the decade 1930-1940.
   The basis for later theoretical computer science, in Alonzo Church and
   Alan Turing also grew directly out of this 'debate'.

The Göttingen school

   Among the students of Hilbert, there were Hermann Weyl, the champion of
   chess Emanuel Lasker, Ernst Zermelo, and Carl Gustav Hempel. John von
   Neumann was his assistant. At the University of Göttingen, Hilbert was
   surrounded by a social circle of some of the most important
   mathematicians of the 20th century, such as Emmy Noether and Alonzo
   Church.

Functional analysis

   Around 1909, Hilbert dedicated himself to the study of differential and
   integral equations; his work had direct consequences for important
   parts of modern functional analysis. In order to carry out these
   studies, Hilbert introduced the concept of an infinite dimensional
   Euclidean space, later called Hilbert space. His work in this part of
   analysis provided the basis for important contributions to the
   mathematics of physics in the next two decades, though from an
   unanticipated direction. Later on, Stefan Banach amplified the concept,
   defining Banach spaces. Hilbert space is the most important single idea
   in the area of functional analysis that grew up around it during the
   20th century.

Physics

   Until 1912, Hilbert was almost exclusively a "pure" mathematician. When
   planning a visit from Bonn, where he was immersed in studying physics,
   his fellow mathematician and friend Hermann Minkowski joked he had to
   spend 10 days in quarantine before being able to visit Hilbert. In
   fact, Minkowski seems responsible for most of Hilbert's physics
   investigations prior to 1912, including their joint seminar in the
   subject in 1905.

   In 1912, three years after his friend's death, Hilbert turned his focus
   to the subject almost exclusively. He arranged to have a "physics
   tutor" for himself. He started studying kinetic gas theory and moved on
   to elementary radiation theory and the molecular theory of matter. Even
   after the war started in 1914, he continued seminars and classes where
   the works of Einstein and others were followed closely.

   Hilbert invited Einstein to Göttingen to deliver a week of lectures in
   June-July 1915 on general relativity and his developing theory of
   gravity (Sauer 1999, Folsing 1998). The exchange of ideas led to the
   final form of the field equations of General Relativity, namely the
   Einstein field equations and the Einstein-Hilbert action. In spite of
   the fact that Einstein and Hilbert never engaged in a public priority
   dispute, there has been some dispute about the discovery of the field
   equations.

   Additionally, Hilbert's work anticipated and assisted several advances
   in the mathematical formulation of quantum mechanics. His work was a
   key aspect of Hermann Weyl and John von Neuman's work on the
   mathematical equivalence of Werner Heisenberg's matrix mechanics and
   Erwin Schrödinger's wave equation and his namesake Hilbert space plays
   an important part in quantum theory. In 1926 von Neuman showed that if
   atomic states were understood as vectors in Hilbert space, then they
   would correspond with both Schrodinger's wave function theory and
   Heisenberg's matrices.

   Throughout this immersion in physics, Hilbert worked on putting rigor
   into the mathematics of physics. While highly dependent on higher math,
   the physicist tended to be "sloppy" with it. To a "pure" mathematician
   like Hilbert, this was both "ugly" and difficult to understand. As he
   began to understand the physics and how the physicists were using
   mathematics, he developed a coherent mathematical theory for what he
   found, most importantly in the area of integral equations. When his
   colleague Richard Courant wrote the now classic Methods of Mathematical
   Physics including some of Hilbert's ideas, he added Hilbert's name as
   author even though Hilbert had not directly contributed to the writing.
   Hilbert said "Physics is too hard for physicists", implying that the
   necessary mathematics was generally beyond them; the Courant-Hilbert
   book made it easier for them.

Number theory

   Hilbert unified the field of algebraic number theory with his 1897
   treatise Zahlbericht (literally "report on numbers"). He disposed of
   Waring's problem in the wide sense. He then had little more to publish
   on the subject; but the emergence of Hilbert modular forms in the
   dissertation of a student means his name is further attached to a major
   area.

   He made a series of conjectures on class field theory. The concepts
   were highly influential, and his own contribution is seen in the names
   of the Hilbert class field and the Hilbert symbol of local class field
   theory. Results on them were mostly proved by 1930, after breakthrough
   work by Teiji Takagi that established him as Japan's first
   mathematician of international stature.

   Hilbert did not work in the central areas of analytic number theory,
   but his name has become known for the Hilbert-Pólya conjecture, for
   reasons that are anecdotal.

Later years

   Hilbert lived to see the Nazis purge many of the prominent faculty
   members at University of Göttingen, in 1933. . Among those forced out
   were Hermann Weyl, who had taken Hilbert's chair when he retired in
   1930, Emmy Noether and Edmund Landau. One of those who had to leave
   Germany was Paul Bernays, Hilbert's collaborator in mathematical logic,
   and co-author with him of the important book Grundlagen der Mathematik
   (which eventually appeared in two volumes, in 1934 and 1939). This was
   a sequel to the Hilbert- Ackermann book Principles of Theoretical Logic
   from 1928.

   About a year later, he attended a banquet, and was seated next to the
   new Minister of Education, Bernhard Rust. Rust asked, "How is
   mathematics in Göttingen now that it has been freed of the Jewish
   influence?" Hilbert replied, "Mathematics in Göttingen? There is really
   none any more".

   By the time Hilbert died in 1943, the Nazis had nearly completely
   restructured the university, many of the former faculty being either
   Jewish or married to Jews. Hilbert's funeral was attended by fewer than
   a dozen people, only two of whom were fellow academics.

   On his tombstone, at Göttingen, one can read his epitaph:

          Wir müssen wissen, wir werden wissen - We must know, we will
          know.

   Ironically, the day before Hilbert pronounced this phrase, Kurt Gödel
   had presented his thesis, containing the famous incompleteness theorem.
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