David Hilbert is a German Mathématicien born the January 23rd 1862 with Königsberg in Eastern Prussia and dead the February 14th 1943 with Göttingen in Germany. He is often regarded as one of the largest mathematicians of the 20th century, as well as Henri Poincaré. He created or developed a large range of fundamental ideas, that it is the Théorie of the invariants, the axiomatization of the geometry or the bases of the analyzes functional (with the spaces of Hilbert).

One of the examples the best known ones of its position of leader is his presentation, in 1900, of its famous problems which durably influenced mathematical research of the 20th century. Hilbert and its students provided a significant portion of the mathematical infrastructure necessary to the blossoming of the quantum Mécanique and General relativity.

It adopted and defended with strength the ideas of Georg Cantor in Set theory and the transfinite numbers. It is also known like one of the founders of the Théorie of the demonstration, the Logique mathematics and clearly distinguished the Mathématiques from the Métamathématique S.

Biography

Hilbert is born with Königsberg, in Eastern Prussia. Graduate of the College of this city, it supplements his doctorate with the Université of Königsberg under the supervision of Ferdinand von Lindemann. In 1885, it gives its thesis entitled Über invariant Eigenschaften specieller binärer Formen, insbesondere der Kugelfunctionen (which one can translate by On the invariant properties of the special binary formats, particularly the circular functions ). At the same period, Hermann Minkowski studies at the same university. Both become good friends and each one will have an influence marked at one time or another on the scientific career of the other.

Of 1886 with 1895, Hilbert is professor at the University of Königsberg. In 1892, it marries Käthe Jerosch (1864 - 1945) and they will have a fore-mentioned son Franz Hilbert (1893 - 1969).

In 1895, on the recommendation of Felix Klein, it is named with the pulpit of mathematics of the Université of Göttingen. At this time, Göttingen is regarded as the best research center in mathematics in the world. Hilbert will remain there until its retirement in 1930, in spite of other offers. At the beginning of the 20th century, it is badly seen for a woman to teach at the university level in Prussia. Towards 1910, Hilbert supports the efforts of Emmy Noether, mathematician of first order, which wishes to teach at the University of Göttingen. To make bend the opponents, he affirms whereas the university is not a Turkish bath. To thwart the established system, Hilbert lends its name to Noether which can thus announce the schedule of its courses without sullying the reputation with the university.

In 1930, the year of its retirement of Göttingen, Hilbert makes a short speech with the radio, denouncing a certain pessimism in the German thought, carrying the ignorabimus of Emil of Wood-Reymond. It is on this occasion that he pronounces “ Wir müssen wissen, to wir werden wissen ” (“We must know, we will know”). Ironically, a day before he pronounces this sentence, Kurt Gödel gives his thesis which contains its theorem of incomplétude, theorem which makes useless the Programme of Hilbert.

In 1933, Hilbert sees the Nazis dismissing several eminent members of the University of Göttingen. Among those, let us quote Hermann Weyl, which replaces Hilbert with the pulpit of mathematics after its retirement in 1930, Emmy Noether and Edmund Landau. Paul Bernays, collaborator of Hilbert in Logical mathematics and joint author with him of Grundlagen der Mathematik , an important book published in two volumes in 1934 and 1939, leaves Germany following the pressures of the Nazis. Their work was the continuation of the book published by Hilbert and Ackermann: Principles off Theoretical Logic (1928).

Approximately a year later, Hilbert, guest with a banquet, sat beside the Minister for Education Bernhard Rust. With the question of Rust: “How is mathematics with Göttingen now that it is free Jewish influence? ”, Hilbert retorted: “Of mathematics with Göttingen? There is more hardly. ”

When Hilbert dies in 1943, the Nazis completely restructured the university, all the Jews and united Jews forced to leave, some having succeeded in fleeing Germany, others off-set. Approximately a dozen people witness its funeral, two only being ex-colleagues. On his tomb with Göttingen, one can read this epitaph: “ Wir müssen wissen, to wir werden wissen ” Among the students of Hilbert, let us quote Hermann Weyl, Emanuel Lasker, Ernst Zermelo and Carl Gustav Hempel. John von Neumann was his assistant. At the University of Göttingen, the circle of friends of Hilbert was composed of the best mathematicians of the 20th century, such Emmy Noether and Alonzo Church.

Work

One retains of him in particular his list of the 23 problems, of which some are still not solved in 2007, qu ' it presented in 1900 to the international Congrès of mathematics to Paris.

Its contributions to mathematics are numerous:

• Consolidation of the Theory of the invariants, which was the subject of its thesis.
• the axiomatization of the Euclidean Geometry to return, it Coherent, appeared in its Grundlagen DER Geometry ( Base of the geometry ).
• Work on the Algebraic theory of the numbers, beginning again and simplifying, with the assistance of Minkowski, work of Kummer, Kronecker, Dirichlet and Dedekind, and publishing them in its Zahlbericht ( Report/ratio on the numbers).
• Contribution of the spaces bearing its name, during its work in analyzes on the integral equations.
• Contribution on the basis of mathematical the General relativity of Einstein, in particular the derivation of its equation starting from the Action of Einstein-Hilbert.

The theorem of the bases

The first work of Hilbert on the invariant functions leads it to show in 1888 its theorem of the bases. Twenty years earlier, using a complex method of calculating, Paul Gordan shows the theorem on the finitude of the generators of the binary formats. The attempts to generalize its method with the functions with several variables fail because of the complexity of calculations. Hilbert decides to borrow another way. It thus shows the theorem of the bases, which affirms the existence of a finished whole of generators for the invariants of the algebraic forms for any number of variables. It does not build indeed such a base nor does not indicate of means of building some. It proves the existence formally by showing that to reject this existence leads to a contradiction.

Hilbert sends its results to the Mathematische Annalen . Gordan, the expert house on the Theory of the invariants, is unable to appreciate the revolutionary nature of work of Hilbert. It rejects the article, affirming that it is incomprehensible: “It is theology, not mathematics! ”

Felix Klein, on its side, recognizes the importance of work and guarantees that it will be published without modification, in spite of its friendship for Gordan. Stimulated by Klein and the comments of Gordan, Hilbert, in a second article, prolongs its results, giving an estimate on the maximum degree of the minimal whole of the generators. After reading, Klein writes to him: “Without any doubt, it is about most important work on the general Algèbre ever published by the Annalen ”.

Later, once the methods of Hilbert largely recognized, Gordan itself affirms: “I must admit that even theology has merits. ”

Axiomatization of the geometry

See also: Axioms of Hilbert

Hilbert publishes Grundlagen der Geometrie ( Fondations of the geometry ) in 1899. It replaces the five Axiome S usual of the Euclidean Géométrie by 21 axioms. Its system eliminates the weaknesses from the geometry of Euclide, only taught at this time.

Its approach is precursory changes for the axiomatic methods subsequent. The axioms are not immutable any more. The geometry can codify the intuition that we have in connection with the “objects”, but it is not necessary all to codify. The elements, a such not, a right and a plan, can be substituted by beer glass, a chair and a table, for example. It is rather necessary to concentrate on their relations.

Hilbert axiomatizes the plane geometry according to five great groups:

1. Axioms of membership: eight axioms express the bond between the concepts of point, right-hand side and plan.
2. Axioms of order: four axioms define the term “enters” and make it possible to define the order of the aligned points, Coplanaire S or in space
3. Axiomes of congruence: five axioms define the concept of Congruence and Déplacement.
4. Axiom of the parallels: it is primarily about the Fifth axiom of Euclide.
5. Axioms of continuity: it contains the Axiome of Archimedes and that of the linear integrity.

These axioms unify in only one system the plane Geometry and the Solid geometry, both Euclidean.

23 problems

See also: Problems of Hilbert

At the time of an international congress mathematicians held in 1900 with Paris, it proposes its famous list of the 23 problems. Even with the 21e century, she is regarded as compilation having had the most influence in mathematics, in front of the Three major problems of Antiquity. Some estimate that it is about the best list of open problems ever produced by only one mathematician.

After having proposed new foundations with the traditional geometry, Hilbert could have attempted to extrapolate for the remainder of mathematics. It rather decides to determine the fundamental problems to which the mathematicians must attack themselves to make mathematics more coherent. Its approach is opposed to those of the logicists Russell and Whitehead, of the “encyclopedists” Bourbaki and of the metamathematician Giuseppe Peano. Its list puts at the challenge the community complete mathematicians, it does not matter its interests.

At the time of the congress, its speech starts as follows:

Who among us would not be happy to raise the veil which masks the future, to observe the future developments of our science like his secrecies during the centuries to come? Which will be the fine ones towards which will tenderont the spirit of the future generations of mathematicians? Which methods, which new facts, the new century will reveal in the rich person and vast field of the mathematical thought? (free translation of Wer von ones würde nicht gerne den Schleier lüften, unter dem die Zukunft verborgen liegt, um einen Blick zu werfen auf die bevorstehenden Fortschritte unserer Wissenschaft und in die Geheimnisse ihrer Entwicklung während DER künftigen Jahrhunderte! Are Welche besonderen Ziele werden center, denen die führenden mathematischen Geister DER kommenden Geschlechter nachstreben? Welche neuen Methoden und neuen Tatsachen werden die neuen Jahrhunderte entdecken - auf dem weiten und reichen Felde mathematischen Denkens? )

To the suggestion of Minkowsky, it presents approximately ten problems to the room. The complete listing will be published in the acts of the congress. In another publication, he proposes a version increased, and final, of his list, which account 23 problems.

Some problems were quickly solved. Others were discussed during the 20th century, some are now regarded as being too vague to give an definitive answer. Even today, there remain some well defined problems which defy the mathematicians.

Formalism

The problems of Hilbert are also a kind of Manifeste which allows the blossoming of the formal school , one of the three major schools of the 20th century in mathematics. According to this school, mathematics exists apart from any intention and of very thought. It are symbols which require to be handled according to formal rules. However, it is not certain that Hilbert has such a simple and mechanical sight mathematics.

The program of Hilbert

In 1920, it explicitly proposes a research program in Métamathématique which will be possibly the Programme of Hilbert. It wishes that mathematics firmly and be completely formulated while being based on logic. Hilbert believes that it is possible, because:

1. All mathematics rise from a finished whole of Axiome S correctly selected.
2. It can be shown that this unit is coherent.

It seems that Hilbert is based on arguments at the same time technical and philosophical to propose such a program. He affirms that he hates the Ignorabimus relatively running in the German thought of the time (which one can recall the formulation with Emil of Wood-Reymond).

This program now left the Formalisme. Bourbaki adopted a version pruned and less formal for its projects (A) to write an encyclopedic foundation and of (b) to support the axiomatic Méthode as research tools. Although this approach obtained success in Algèbre and in functional analyzes, she knew little success elsewhere.

Impact of Gödel

Hilbert and the other mathematicians who work with the company want to succeed. However, their work was to finish in an abrupt way.

In 1931, Kurt Gödel shows that any not-contradictory and sufficiently complete formal system to include at least the Arithmétique, cannot show its complétude while being based on its axioms. As formulated, the great design of Hilbert is thus dedicated to the failure.

The Théorème of incomplétude of Gödel does not say that it is impossible to carry out such a system according to the spirit of the program of Hilbert. The completion of the Théorie of the demonstration made it possible to clarify the concept of coherence, which is central in modern mathematics. The program of Hilbert launched the Logique on a way of clarification. The desire to better include/understand the theorem of Gödel allowed the development of the Théorie of recursion and the clarification of logic. The latter became a discipline with whole share in the decades of 1930 and 1940. It forms the starting point of what is today called the theoretical Informatique, developed by Alonzo Church and Alan Turing.

Analyzes functional

As of 1909, Hilbert studies in a methodical way the differential equations and integral. This work has an incidence marked on the analyzes functional modern.

With an aim of concluding its task, it introduces the concept of Euclidean spaces of infinite size, called later the spaces of Hilbert. In an unexpected way, this work will be resumed in theoretical physics during the two subsequent decades.

Later, Stefan Banach will generalize the concept to make of it the Espace of Banach.

Physics

Minkowski seems responsible for the majority of the searchs for Hilbert in physics before 1912, including their seminar united on the subject in 1905. Indeed, until in 1912, Hilbert make pure Mathématiques exclusively.

This year, it pays its attention on the Physique. It has even committed a “tutor in physics”. It starts by studying the kinetic Théorie of the gases, then continuous with the theory of the Radiation S and complete with the molecular Théorie. Even during the First World War, it proposes seminars and course where is presented work of Albert Einstein and other physicists.

Hilbert invites Einstein with Göttingen to pronounce a series of readings there on the General relativity in June and July 1915. The exchanges between the two scientists lead to the creation of the equation of Einstein of general relativity (i.e. the equation of the field of Einstein and the Action of Einstein-Hilbert). Even if Hilbert and Einstein never disputed in connection with the paternity of the equation, some wanted to call into question this one (see Controverse on the paternity of relativity).

Moreover, the work of Hilbert anticipates and supports the projections in the mathematical formulation of the quantum Mécanique. Its spaces of Hilbert is essential with work of Hermann Weyl and John von Neumann on the mathematical equivalence of the Matrice of Heisenberg and the equation of Schrödinger, like with the general formulation of quantum mechanics.

In 1926, von Neumann shows that if the atomic states are regarded as vectors in the space of Hilbert, then they correspond to the Fonction of wave of Schrödinger and to the matrix of Heisenberg.

Within the framework of its work in physics, Hilbert is baited to make more rigorous the use of mathematics. Whereas their work depends entirely on higher mathematics, the physicists are negligent when they handle the mathematical objects. For a mathematician of the gauge of Hilbert, this situation is difficult to include/understand, going until qualifying it the “ugly one”.

When he manages to be made a portrait of the use of mathematics in physics, he develops a coherent mathematical theory with the use of the physicists, especially with regard to the integral equations. When Richard Running publishes off Methods Mathematical Physics by including some ideas of Hilbert, he adds the name of Hilbert as author, even if this last did not take part in its drafting. Hilbert wrote: “Physics is too difficult for the physicists”, wanting to draw the attention to the difficulty inherent in the use of higher mathematics. The work of Current and Hilbert tries to level these difficulties.

Theory of the numbers

Hilbert unifies the Algebraic theory of the numbers with its Rapport on the numbers ( Zahlbericht , published in 1897). It résoud the Problem of Waring in an almost complete way. Its treaty exhausts the subject, but the emergence of the modular Forme of Hilbert means that its name is attached once again to a major part of mathematics.

It made several Conjecture S about the Théorie of the bodies of classes. The concepts have important remarkable, and its own contributions appear in the Corps of Hilbert and the Symbole of Hilbert of the theory of the local body of classes. The results of these theories all are almost proven in 1930, after a major opening of Teiji Takagi, which establishes it like the first mathematician Japan board of international gauge.

Hilbert did not work on the principal parts of the analytical Théorie of the numbers, but its name remains attached to the Conjecture of Hilbert-Pólya for anecdotic reasons.

Anecdotes

At the time of a reception at the family home, his wife intimates to him to change shirt. It is carried out, but does not return. Worry, she seeks it and discovers it deadened. Indeed, the fact of taking off its shirt meant for him to go to sleep!

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