David hilbert

David Hilbert (Königsberg, East Prussia, January 23, 1862-Göttingen, Germany, February 14, 1943) was a German mathematician, recognized as one of the most influential of the XIX and early XX. He established his reputation as a great mathematician and scientist by inventing and developing a wide range of ideas, such as the theory of invariants, the axiomatization of geometry, and the notion of Hilbert space, one of the foundations of functional analysis. Hilbert and his students provided significant parts of the mathematical infrastructure necessary for quantum mechanics and general relativity. He was one of the founders of proof theory, mathematical logic, and the distinction between mathematics and metamathematics. He adopted and strongly defended Cantor's set theory and transfinite numbers. A famous example of his world leadership in mathematics is his presentation in 1900 of a set of open problems that influenced the course of much of the century's mathematical research XX.

Life

Hilbert was born in Königsberg, in East Prussia (present-day Kaliningrad, Russia). He graduated from high school in his hometown and enrolled at the University of Königsberg (Albertina). In this he received his doctorate in 1885, with a dissertation, written under the supervision of Ferdinand von Lindemann, entitled Über invariante Eigenschaften specieller binärer Formen, insbesondere der Kugelfunctionen (On the invariant properties of special binary forms, particularly circular functions). Hermann Minkowski met Hilbert, at the same university and at the same time, as a doctoral candidate, and they became close friends, exerting a reciprocal influence on each other at various times in their scientific careers.

Hilbert worked as a professor at the University of Königsberg from 1886 to 1895, when, as a result of intervention on his behalf by Felix Klein, he obtained the position of Professor of Mathematics at the University of Göttingen, which at that time was the best mathematical research center in the world; he here he would remain the rest of his life.

The Finitude Theorem

David Hilbert's first work on invariant functions led him, in 1888, to the proof of his famous finiteness theorem. Twenty years earlier, Paul Gordan had proved the generator finiteness theorem for binary forms, using a complex computational approach. Attempts to generalize this method to functions with more than two variables have failed because of the enormous difficulty of the calculations involved. Hilbert realized that it was necessary to follow a completely different path. As a result, he proved Hilbert's fundamental theorem: show the existence of a finite set of generators, for quantum invariants in any number of variables, but in an abstract way. That is, he proved the existence of said set, but not algorithmically but by means of an existence theorem.

Hilbert sent his results to the Mathematische Annalen. Gordan, the Annalen's invariant theory expert, failed to appreciate the revolutionary nature of Hilbert's theorem and rejected the paper, criticizing the exposition as insufficiently comprehensive. His comment was: "This is theology, not mathematics!"

Klein, on the other hand, recognized the importance of the work and ensured that it was published without alterations. Encouraged by Klein and Gordan's comments, Hilbert extended his method in a second paper, providing estimates on the maximum degree of the minimum set of generators, and sent it once more to the Annalen. After reading the manuscript, Klein wrote to him, in these terms: "Without doubt this is the most important work in general algebra that the Annalen has ever published." Later, when the usefulness of Hilbert's method had been universally recognized, Gordan himself would say: "I have to admit that even theology has its merits."

Axiomatization of geometry

The text Grundlagen der Geometrie (Fundamentals of Geometry), published by Hilbert in 1899, replaces the traditional Euclid axioms with a formal system of 21 axioms. They avoid the weaknesses identified in those of Euclid, whose classic work Elements was still being used as a textbook at the time.

Hilbert's approach marked the shift to the modern axiomatic system. Axioms are not taken as self-evident truths. Geometry can be about things, about which we have powerful intuitions, but it is not necessary to assign an explicit meaning to indefinite concepts. As Hilbert says, elements such as the point, line, plane, and others can be substituted with tables, chairs, beer mugs, and other objects. What is discussed and developed are their defined relationships.

Hilbert begins by listing the undefined concepts: point, line, plane, incidence (a relationship between points and planes), being between, congruence of pairs of points, and congruence of angles. The axioms unify Euclid's plane and solid geometry into a single system.

The 23 problems

Hilbert proposed a comprehensive list of 23 unsolved problems at the International Congress of Mathematicians in Paris in 1900. This is generally acknowledged to be the most successful and deeply considered collection of open problems ever produced by a single mathematician.

After rewriting the foundations of classical geometry, Hilbert could have extrapolated it to the rest of mathematics. This approach differs, however, from the later "logicists" Russell-Whitehead or the "mathematical formalism" of his contemporary Giuseppe Peano and more recently from the "group of mathematicians" Nicolas Bourbaki. The entire mathematical community could embark on problems that he identified as crucial issues in the areas of mathematics that he considered to be key.

Launched the set of problems at the "The Problems of Mathematics" presented during the course of the Second International Congress of Mathematicians held in Paris. This is the introduction to Hilbert's lecture:

"Who among us would not be happy to lift the veil behind which the future hides; to observe the developments to come from our science and the secrets of its development in the centuries to come? What will be the goal to which the spirit of future generations of mathematicians will tend? What methods, what new facts will reveal the new century in the vast and rich field of mathematical thinking?"

He presented less than half of the problems in Congress, which were published in the minutes. He expanded the picture in a later publication, and with it came the current canonical formulation of Hilbert's 23 Problems. The full text is important, as the exegesis of the issues may remain the inevitable subject of debate, whenever one wonders how many have been resolved:

1. Cantor's problem on the cardinal of the continuum. What is the cardinal of the continuum?

2. The compatibility of the axioms of arithmetic. Are the axioms of arithmetic compatible?

3. The equality of the volumes of two tetrahedrons with the same base and the same height.

4. The problem of the shortest distance between two points. Is the straight line the shortest distance between two points, on any surface, in any geometry?

5. Establish the concept of Lie group, or continuous group of transformations, without assuming the differentiability of the functions that define the group.

6. Axiomatization of physics. Is it possible to create an axiomatic field for physics?

7. The irrationality and transcendence of certain numbers, such as 22{displaystyle 2^{sqrt {2}}}}etc.

8. The problem of the distribution of prime numbers.

9. Demonstration of the most general law of reciprocity in a field of any numbers.

10. Establish effective methods for solving Diophantine equations.

11. Quadratic forms with any algebraic coefficients.

12. The extension of Kronecker's theorem on abelian fields to any domain of algebraic rationality.

13. Impossibility of solving the general quadratic equation by means of functions of only two arguments.

14. Proof of the finite condition of certain complete systems of functions.

15. Rigorous foundation of Schubert's enumerative calculus or algebraic geometry.

16. Problem of the topology of algebraic curves and surfaces.

17. The expression of forms defined by sums of squares.

18. Construction of the space of congruent polyhedra.

19. Are the solutions to regular problems in the calculus of variations always analytical?

20. The general Dirichlet boundary condition problem.

21. Demonstration of the existence of linear differential equations of Fuchsian class, known their singular points and monodromic group.

22. Uniformity of analytical relations by means of automorphic functions: it is always possible to standardize any algebraic relation between two variables by means of automorphic functions of one variable.

23. Extension of the methods of calculation of variations.

Some were resolved in a short time. Others have been discussed throughout the 20th century, with a few now concluded to be irrelevant or impossible. to close. Some continue to be a challenge for mathematicians today.

Formalism

Following the trend that had become standard by mid-century, Hilbert's set of problems also constituted a kind of manifesto, which opened the way for the development of the school of Mathematical Formalism, one of the three schools of mathematics most important of the XX century. According to formalism, mathematics is a game—meaningless—in which one plays it with meaningless symbols according to pre-established formal rules. Therefore it is an autonomous thought activity. However, there is room for doubt as to whether Hilbert's own view was simplistically formalistic in this sense.

Hilbert's program

In 1920 he explicitly proposed a research project (in metamathematics, as it was then called) that came to be known as Hilbert's program. He wanted mathematics to be formulated on a solid and completely logical basis. He believed that, in principle, this could be achieved by showing that:

1. all math is followed by a finite system of axioms chosen correctly; and
2. you can prove that such axiomatic system is consistent.

He seemed to have technical and philosophical reasons for making this proposal. This affirmed his distaste for what had become known as ignorabimus, which was still an active problem in his time within German thought, and which could be traced in that formulation back to Emil du Bois- reymond.

The program is still recognizable in the more popular philosophy of mathematics, where it is commonly called formalism. For example, the Bourbaki group adopted a selective and watered-down version as adequate for the requirements of their twin projects of (a) writing seminal encyclopedic works, and (b) supporting the axiomatic system as a research tool. This approach has been successful and influential in relation to Hilbert's work in algebra and functional analysis, but has not caught on equally with his interests in physics and logic.

Gödel's work

Hilbert and the talented mathematicians who worked with him on this venture were dedicated to the project. His attempt to support axiomatized mathematics with definite principles, which would remove theoretical uncertainties, succumbed to unexpected failure.

Gödel showed that no non-contradictory formal system that was large enough to include at least arithmetic could be proved complete by its own axioms alone. In 1931 his incompleteness theorem showed that Hilbert's ambitious plan was impossible as stated. The second requirement could not reasonably be combined with the first, as long as the axiomatic system is genuinely finite.

However, the completeness theorem says nothing about the proof of the completeness of mathematics by a different formal system. The later achievements of proof theory at least clarified the relation of consistency to theories of primary interest to mathematicians. Hilbert's work had started logically on its way to clarification; the need to understand Gödel's work then led to the development of computability theory and later mathematical logic as an autonomous discipline in the 1930s–1940s. From this 'debate' The foundation for theoretical computing by Alonzo Church and Alan Turing was born directly.

The Göttingen School

Hilbert's students include Hermann Weyl, world chess champion Emanuel Lasker, Ernst Zermelo and Carl Gustav Hempel. John von Neumann was his assistant. At the University of Göttingen, Hilbert found himself surrounded by a social circle made up of some of the most important mathematicians of the 20th century, like Emmy Noether and Alonzo Church.

Functional analysis

Around 1909, Hilbert devoted himself to the study of differential equations and integral equations; His work had direct consequences on 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 the Hilbert space. His work on this part of the analysis provided the basis for important contributions to mathematical physics in the following two decades, though in directions not then anticipated. Later, Stefan Banach amplified the concept, defining Banach spaces. The Hilbert space is itself the most important idea in functional analysis, which grew up around it during the 20th century.

Physics

Until 1912, Hilbert was almost exclusively a "pure" mathematician. When he planned to pay a visit to Bonn, where he was immersed in the study of physics, his friend and fellow mathematician Hermann Minkowski joked that he had to spend 10 days in quarantine before he could visit Hilbert. In fact, Minkowski appears to be responsible for most of Hilbert's research in physics prior to 1912, including his joint seminar on the subject in 1905.

In 1912, three years after the death of his friend, he shifted his focus to this subject almost exclusively. He arranged for her to be assigned a “physics tutor.” He began by studying the kinetic theory of gases and then moved on to the elementary theory of radiation and the molecular theory of matter. Even after the outbreak of war in 1914, he continued to hold seminars and classes closely following the work of Einstein among others.

Hilbert invited Einstein to Göttingen to give 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, specifically the Einstein field equations and the Einstein-Hilbert action. Although Einstein and Hilbert never got into a public dispute over priority, there has been some discussion about the discovery of the field equations, although research on historical documentation seems to confirm that Einstein was ahead of the game, since Hilbert's work was incomplete. Hilbert, in the printed version of his article, added a reference to Einstein's conclusive role and an allowance for Einstein's priority: "The resulting differential equations of gravitation are, it seems to me, in with the magnificent theory of general relativity established by Einstein in his later works "[(3), p. 404].

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

Through this immersion in physics, he worked to give rigor to the mathematics that underpins it. Although highly dependent on advanced mathematics, the physicist tends to be "sloppy" with it. For a "pure" mathematician like Hilbert, this was "ugly" and difficult to understand. As he began to understand physics and the way physicists used mathematics, he developed a mathematically coherent theory for what he found, mainly in the area of integral equations. When his colleague Richard Courant wrote the classic Methods of Mathematical Physics she included some of Hilbert's ideas, adding his name as a co-author even though Hilbert did not actually contribute to the paper. Hilbert said that "physics is too hard for physicists", implying that the necessary mathematics was generally out of their reach; Courant-Hilbert's book made things easier for them.

Number Theory

Hilbert unified the field of algebraic number theory with his 1897 treatise Zahlbericht (literally 'report on numbers'). He brought down the Waring problem in the broad sense. Since then he had little more to say on the subject; but the emergence of Hilbert's modular forms in a student's dissertation implies that his name is more closely tied to an important area.

He proposed a series of conjectures about the field theory of classes. The concepts were highly influential, and his own contribution is evident in the names of the Hilbert class field and the Hilbert symbol of local class field theory. The results on these conjectures were mostly proven around 1930, after the important work of Teiji Takagi that established him as the first internationally recognized Japanese mathematician.

Hilbert did not work in major areas of analytic number theory, but his name became attached to the Hilbert-Pólya conjecture, for anecdotal reasons.

Talks, essays and miscellaneous contributions

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

Last years

Hilbert lived to see the Nazis purge most of the outstanding faculty members from the University of Göttingen in 1933. Among those forced to leave were Hermann Weyl, who had held Hilbert's chair upon his retirement 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 Hilbert-Ackermann's 1928 book Foundations of Theoretical Logic.

A year later, he attended a banquet and was seated next to the new Minister of Education, Bernhard Rust. Rust asked him: "How is mathematics going in Göttingen now that it has been freed from Jewish influence?" To which Hilbert replied, “The mathematics in Göttingen? There's nothing left of that anymore."

Tomb of David Hilbert in Göttingen:
Wir müssen wissen
Wir werden wissen

By the time Hilbert died in 1943, the Nazis had almost completely restructured the university, as many of the previous faculty were Jewish or married to Jews. Hilbert's funeral was attended by fewer than a dozen people, and only two of them were academic colleagues.

On his grave, in Göttingen, you can read his epitaph:

We must know, we'll know. (in German): Wir müssen wissen, wir werden wissen)

Ironically, the day before Hilbert uttered this sentence, Kurt Gödel was presenting his thesis, which contained the famous incompleteness theorem: there are things we know to be true, but we cannot prove them.

Eponymy

In addition to numerous mathematical entities and theorems that bear his last name, the designation of two astronomical elements pays homage to him:

• Moon crater Hilbert
• The asteroid (12022) Hilbert

Note and references