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HILBERT



David Hilbert (1862 – 1943) was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of Hilbert spaces, one of the foundations of functional analysis. Hilbert adopted and warmly defended Georg 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 contributed significantly to establishing rigor and developed important tools used in modern mathematical physics. Hilbert is known as one of the founders of proof theory and mathematical logic, as well as for being among the first to distinguish between mathematics and metamathematics.
Hilbert was born in the Province of Prussia - either in Konigsberg (according to Hilbert's own statement) or in Wehlau (known since 1946 as Znamensk) near Konigsberg where his father worked at the time of his birth. In the fall of 1872 he entered the Friedrichskolleg Gymnasium (the same school that Immanuel Kant had attended 140 years before), but after an unhappy period he transferred (fall 1879) to and graduated from (spring 1880) the more science-oriented Wilhelm Gymnasium. Upon graduation he enrolled (autumn 1880) at the University of Konigsberg, the "Albertina". In the spring of 1882, Hermann Minkowski (two years younger than Hilbert and also a native of Konigsberg but so talented he had graduated early from his gymnasium and gone to Berlin for three semesters), returned to Konigsberg and entered the university. "Hilbert knew his luck when he saw it. In spite of his father's disapproval, he soon became friends with the shy, gifted Minkowski". In 1884, Adolf Hurwitz arrived from Gottingen as an Extraordinarius, i.e., an associate professor. An intense and fruitful scientific exchange between the three began and especially Minkowski and Hilbert would exercise a reciprocal influence over each other at various times in their scientific careers. Hilbert obtained his doctorate in 1885, with a dissertation, written under Ferdinand von Lindemann, titled "On the invariant properties of special binary forms, in particular the spherical harmonic functions".
Hilbert remained at the University of Konigsberg as a professor from 1886 to 1895. Among the students of Hilbert were: Hermann Weyl, chess champion Emanuel Lasker, Ernst Zermelo, and Carl Gustav Hempel. John von Neumann was his assistant. At the University of Gottingen, 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. Among his 69 Ph.D. students in Gottingen were many who later became famous mathematicians, including (with date of thesis): Otto Blumenthal (1898), Felix Bernstein (1901), Hermann Weyl (1908), Richard Courant (1910), Erich Hecke (1910), Hugo Steinhaus (1911) and Wilhelm Ackermann (1925). Between 1902 and 1939 Hilbert was editor of the Mathematische Annalen, the leading mathematical journal of the time. "Good, he did not have enough imagination to become a mathematician" – Hilbert's response upon hearing that one of his students had dropped out to study poetry.
Hilbert lived to see the Nazis purge many of the prominent faculty members at University of Gottingen in 1933. Those forced out included Hermann Weyl (who had taken Hilbert's chair when he retired in 1930), Emmy Noether and Edmund Landau. One who had to leave Germany, Paul Bernays, had collaborated with Hilbert in mathematical logic, and co-authored with him 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 Mathematical Logic from 1928.
About a year later, Hilbert attended a banquet and was seated next to the new Minister of Education, Bernhard Rust. Rust asked, "How is mathematics in Gottingen now that it has been freed of the Jewish influence?" Hilbert replied, "Mathematics in Gottingen? There is really none anymore."
By the time Hilbert died in 1943, the Nazis had nearly completely restaffed the university, in as much as many of the former faculty had either been Jewish or married to Jews. Hilbert's funeral was attended by fewer than a dozen people, only two of whom were fellow academics, among them Arnold Sommerfeld, a theoretical physicist and also a native of Konigsberg. News of his death only became known to the wider world six months after he had died.
On his religious views, he was said to be an agnostic. He also argued that mathematical truth was independent of the existence of God or other a priori assumptions. The epitaph on his tombstone in Gottingen are the famous lines he spoke at the conclusion of his retirement address to the Society of German Scientists and Physicians in the fall of 1930. The words were given in response to the Latin maxim: "Ignoramus et ignorabimus" or "We do not know, we cannot know": "We must know. We will know".
The day before Hilbert pronounced these phrases at the 1930 annual meeting of the Society of German Scientists and Physicians, Kurt Godel – in a roundtable discussion during the Conference on Epistemology held jointly with the Society meetings – tentatively announced the first expression of his incompleteness 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. Attempts to generalize his method to functions with more than two variables failed because of the enormous difficulty of the calculations involved. In order to solve what had become known in some circles as Gordan's Problem, 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 a constructive proof – it did not display "an object" – but rather, it was an existence proof and relied on use of the Law of Excluded Middle in an infinite extension. 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 not Mathematics. This is Theology. Klein, on the other hand, recognized the importance of the work, and guaranteed that it would be published without any alterations. Encouraged by Klein, 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 have convinced myself that even theology has its merits".
For all his successes, the nature of his proof stirred up more trouble than Hilbert could have imagined at the time. Although Kronecker had conceded, Hilbert would later respond to others' similar criticisms that "many different constructions are subsumed under one fundamental idea" – in other words (to quote Reid): "Through a proof of existence, Hilbert had been able to obtain a construction"; "the proof" (i.e. the symbols on the page) was "the object". Not all were convinced. While Kronecker would die soon after, his constructivist philosophy would continue with the young Brouwer and his developing intuitionist "school", much to Hilbert's torment in his later years. Indeed Hilbert would lose his "gifted pupil" Weyl to intuitionism – "Hilbert was disturbed by his former student's fascination with the ideas of Brouwer, which aroused in Hilbert the memory of Kronecker". Brouwer the intuitionist in particular opposed the use of the Law of Excluded Middle over infinite sets (as Hilbert had used it). Hilbert would respond: "The text Foundations of Geometry published by Hilbert in 1899 proposes a formal set, the Hilbert's axioms, substituting 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 signaled the shift to the modern axiomatic method. In this, Hilbert was anticipated by Peano's work from 1889. 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.
Hilbert 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 'foundation list' 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.
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 through some means such as the epsilon calculus.
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. a 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 spaces are an important class of objects in the area of functional analysis, particularly of the spectral theory of self-adjoint linear operators that grew up around it during the 20th century.
Hilbert unified the field of algebraic number theory with his 1897 treatise Zahlbericht ("report on numbers"). He also resolved a significant number-theory problem formulated by Waring in 1770. As with the finiteness theorem, he used an existence proof that shows there must be solutions for the problem rather than providing a mechanism to produce the answers. 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 lives on in the names of the Hilbert class field and of the Hilbert symbol of local class field theory. Results on them were mostly proved by 1930, after work by Teiji Takagi.
Hermann Weyl so has estimated David Hilbert's role in the mathematician: "Our generation has not put forward any mathematician who could be compared to it … Trying to make out through a veil of time, what future is prepared to us, Hilbert has put and has considered twenty three solved problems which … have really played key role in progress of mathematics during the subsequent forty with extra years. Any mathematician, who has solved one of them, borrowed a place of honor in mathematical community".
Contemporaries recall Hilbert as the person cheerful, extremely sociable and benevolent, mark its outstanding diligence and scientific enthusiasm.


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