David Hilbert

David Hilbert (January 23, 1862, Wehlau, East Prussia – February 14, 1943, Göttingen, Germany) was a German mathematician, recognized as one of the most influential mathematicians of the 19th and early 20th centuries. He established his reputation as a great mathematician and scientist by inventing or developing a broad range of ideas, such as invariant theory, the axiomization of geometry, and the notion of Hilbert space, one of the foundations of functional analysis. Hilbert and his students supplied significant portions of the mathematic infrastructure required for quantum mechanics and general relativity. He is one of the founders of proof theory, mathematical logic and the distinction between mathematics and metamathematics. He adopted and warmly defended Cantor's set theory and transfinite numbers. A famous example of his world leadership in mathematics is his 1900 presentation of a set of problems that set the course for much of the mathematical research of the 20th century.

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 center for mathematics in the world and where he remained for the rest of his life.

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

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, 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?

In spanish: Quién entre nosotros no estaría contento alzar el trasero del velo que está oculto el futuro; ¿para mirar fijamente a los próximos desarrollos de nuestra ciencia y a los secretos de su desarrollo por los siglos para venir? ¿Cuál será los fines hacia qué el espíritu de generaciones futuras de matemáticos tenderá? ¿Qué métodos, qué nuevos hechos el nuevo siglo revelará en el inmenso y rico campo de pensamiento matemático?

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:

all of mathematics follows from a correctly-chosen finite system of axioms; and
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 and Ernst Zermelo. John von Neumann was his assistant. At the University of Göttingen, he 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, he turned his focus to the subject almost exclusively. He arranged to have a "physics tutor"Reid p. 129. 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, he 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.

Miscellaneous talks, essays, and contributions

His paradox of the Grand Hotel, a meditation on strange properties of the infinite, is often used in popular accounts of infinite cardinal numbers.

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".Reid p. 205.

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.Redi p. 213.

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: there are things which we know to be true, but which we cannot prove to be so.

Notes, references and links

Notes

References

Primary literature in English translation:

Ewald, William B., ed., 1996. From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols. Oxford Uni. Press.
- 1918. "Axiomatic thought," 1115-14.
- 1922. "The new grounding of mathematics: First report," 1115-33.
- 1923. "The logical foundations of mathematics," 1134-47.
- 1930. "Logic and the knowledge of nature," 1157-65.
- 1931. "The grounding of elementary number theory," 1148-56.
- 1904. "On the foundations of logic and arithmetic," 129-38.
- 1925. "On the infinite," 367-92.
- 1927. "The foundations of mathematics," with comment by Weyl and Appendix by Bernays, 464-89.
Jean van Heijenoort, 1967. From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931. Harvard Univ. Press.
- an accessible set of lectures originally for the citizens of Göttingen.

Secondary:

Bottazini, Umberto, 2003. Il flauto di Hilbert. Storia della matematica. UTET, ISBN 8877508523
Corry, L., Renn, J., and Stachel, J., 1997, "Belated Decision in the Hilbert-Einstein Priority Dispute," Science 278: nn-nn.
Ivor Grattan-Guinness, 2000. The Search for Mathematical Roots 1870-1940. Princeton Uni. Press.
Gray, Jeremy, 2000. The Hilbert Challenge, ISBN 0198506511
Piergiorgio Odifreddi, 2003. Divertimento Geometrico - Da Euclide ad Hilbert. Bollati Boringhieri, ISBN 8833957144. A clear exposition of the "errors" of Euclid and of the solutions presented in the Grundlagen der Geometrie, with reference to non-Euclidean geometry.
Reid, Constance, 1996. Hilbert, Springer, ISBN 0387946748. The biography in English.
Sauer, Tilman, 1999. "The relativity of discovery: Hilbert's first note on the foundations of physics", Arch. Hist. Exact Sci., v53, pp 529-575. (Available from Cornell University Library, as a downloadable Pdf ^*)
Thorne, Kip, 1995. Black Holes and Time Warps: Einstein's Outrageous Legacy, W. W. Norton & Company; Reprint edition. ISBN 0393312763.
Folsing, Albrecht, 1998. Albert Einstein, Penguin.
Mehra, Jagdish, 1974. Einstein, Hilbert, and the Theory of Gravitation, Reidel.

External links

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