Hilbert

Hilbert's problems are twenty-three problems in mathematics published by German mathematician David Hilbert in The problems were all unsolved at the.
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Hilbert decided to stay close to home: While studying for his degrees, Hilbert made friends with two other exceptionally talented mathematicians, Hermann Minkowski, a fellow student, and Adolf Hurwitz, an associate professor. The three pushed one another to ever greater mathematical heights — they would continue to exchange ideas for the rest of their careers. Hilbert was a pure mathematician. His knowledge of mathematics was unusually broad as well as deep, and he contributed to several areas of mathematics and also physics. The mathematics he did is often at a level that can stretch the best of us, so here are brief summaries of some of his most famous achievements.

In Hilbert proved the finite basis theorem for any number of variables.


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In , Paul Gordan had been able to prove the theorem, but for only two variables: Hilbert used an entirely new abstract strategy for his proof, establishing that the theorem was true for an arbitrary number of variables. This was a major advance in algebraic number theory. In Hilbert took a sweeping overview of mathematics, defining his famous 23 problems.

In doing so, he had a greater effect in shaping mathematics in the 20th century than any other person. Hilbert outlined 23 problems or questions he thought, if answered correctly, would carry mathematics to a new level. The list, he said, was not meant to exclude other problems, it was merely a sample of problems.

Sample or not, since Hilbert first posed the 23 Problems, a huge amount of work has been done seeking the answers. Mathematicians solved some of the problems within a few years, and others later, but some remain unsolved. Over a hundred years since Hilbert first listed them, a bright new light would shine on mathematics if the remaining problems could be solved. The great unsolved problems Hilbert identified are:.

There is little if any separation between applied mathematics and mathematical physics. Hilbert sometimes dabbled in this area, often as a result of discussions with Hermann Minkowski, his old friend from his student days. After graduating, Minkowski went on to teach Albert Einstein in Zurich. Gradually a fascination with mathematical physics grew in Hilbert, and he spent increasing amounts of time thinking about the subject. By it had become his primary research field. He believed most physicists approached problems with insufficient mathematical rigor. As mentioned earlier, Hilbert's thesis was on the theory of invariants.

His two proofs of Gordan's Problem established him as a first-class mathematician. These proofs led him to "one of the most fundamental theorems of algebra," namely that every subset of a polynomial ring of independent variables has a finite ideal basis. Weyl claims that this theorem is "the foundation stone of the general theory of algebraic manifolds. Weyl claims that the Nullstellensatz is "clearly" at the heart of the theory of algebraic manifolds.

David Hilbert made other contributions to modern algebra, including a theorem under which one may substitute variables in an irreducible polynomial to obtain another irreducible polynomial and a solution of ninth degree equations. Weyl emphasizes the significance of Hilbert's discoveries: Characteristic of Hilbert's unabated, albeit justified, egotism, one of Hilbert's papers on invariants called Sylvester and Cayley "the representatives of the naive period" and himself the champion of "the critical period" of invariant theory.

In , he wrote to Minkowski that "I shall definitely quit the field of invariants. With his move from Konigsberg to Gottingen, Hilbert's interest moved away from invariants and into algebraic number fields. The Zahlbericht was a great leap forward in algebraic number theory. Hilbert was greatly pleased at his assignment from the German Mathematical Society. Hilbert stated that "the theory of number fields is an edifice of rare beauty and harmony.

Hilbert also developed a form of the Legendre symbol and the idea of a p-adic norm. Hilbert defined a p-adic norm as an integer in the quadratic field K that is congruent to the norm of a suitable integer in K modulo any power of p. He then established certain reciprocity laws in terms of these norm residues. He also delved considerably in the theory of ideals and Abelian fields. Hilbert also began exploring the relations between number theory and modular functions, but, "indicative of the fertility of Hilbert's mind," he found other areas of mathematics to interest him.

As mentioned before, Hilbert also provided simple and direct proofs of the transcendence of e and pi. Another breakthrough in algebraic number theory was his proof of Waring's Problem in In , Edmund Waring proposed, without proof, that every positive integer can be written as the sum of 4 squares, 9 cubes, 19 fourth powers, and so on. However, Hilbert's proof is an existence proof and does not explain how to determine s given k. Even though he departed from algebraic number theory, after he went into retirement, Hilbert confided in his friend Olga Taussky that "much as he admired all branches of mathematics, he considered number theory the most beautiful.

David Hilbert next turned his attention to axiomatics, which is the process of laying down axioms, or laws, for geometry and mathematics in general. Hilbert's view on the foundations of geometry is summarized in his famous statement at the Berlin railway station: Hilbert did much to axiomatize geometry, as found in his book, Grundlangen der Geometrie.

Hilbert produced a new set of geometric axioms that were both consistent no axiom overlapped with another and complete the collection of axioms enabled one to express all of geometry. This axiomatization had changed the face of geometry more than any individual since Euclid. He then turned his attention to the foundations of mathematics in general.

David Hilbert

He developed a research program, now known as Hilbert's Program, by which he hoped to rigorously prove the consistency of logic and set theory. Hilbert successfully defended his program against critics such as L. Brouwer, who wished to expose paradoxes in set theory that would render Cantorian set theory and proof by contradiction meaningless. To Hilbert, Brouwer's arguments sounded like the stifling rigidity of Leopold Kronecker. However, in , a year old logician named Kurt Godel published a paper that dismantled Hilbert's Program. According to rumors from Hilbert's assistant, Hilbert was furious when he learned of Godel's work.

Ian Stewart summarizes the content of Godel's work as follows: Even though the success of Hilbert's Program is still being debated, Hilbert has done much to organize mathematics. As Weyl wrote after Hilbert's death, "Hilbert is the champion of axiomatics.


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    • There was a gap of twenty years between the periods Hilbert concentrated on the foundations of geometry and the foundations of mathematics in general. During this time, he focused on integral equations and mathematical physics. Hilbert's first contributions to analysis involved homogenous integral equations and the problem of determining eigenvalues of an integral equation. Hilbert also developed a method of analysis using infinitely many vectors in an infinitely-dimensioned space. This space is now known as Hilbert space and is crucial to functional analysis.

    David Hilbert - Biography, Facts and Pictures

    The study of transforms in Hilbert space has become very important to the studies of integral and differential equations, partial differential equations, quantum mechanics, optimization problems, bifurcation theory, approximation theory, stability problems, variational inequalities, and control problems for dynamical systems. 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.

    Hilbert's problems

    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. Hilbert published his views on the foundations of mathematics in the 2-volume work Grundlagen der Mathematik. Hilbert and the 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, ended in failure. In 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 subsequent achievements of proof theory at the very least clarified consistency as it relates to theories of central concern to mathematicians.

    The basis for later theoretical computer science , in the work of Alonzo Church and Alan Turing , also grew directly out of this 'debate'. Around , 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. Until , 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 , including their joint seminar in the subject in In , 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. Even after the war started in , he continued seminars and classes where the works of Albert Einstein and others were followed closely. By Einstein had framed the fundamentals of the theory of gravity, but then struggled for nearly 8 years with a confounding problem of putting the theory into final form. During November Einstein published several papers culminating in "The Field Equations of Gravitation" see Einstein field equations. Hilbert fully credited Einstein as the originator of the theory, and no public priority dispute concerning the field equations ever arose between the two men during their lives.

    Mathematical Physics

    Additionally, Hilbert's work anticipated and assisted several advances in the mathematical formulation of quantum mechanics. Throughout this immersion in physics, Hilbert worked on putting rigor into the mathematics of physics.

    While highly dependent on higher mathematics, physicists 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 physics and how 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 Methoden der mathematischen Physik 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.

    Hilbert unified the field of algebraic number theory with his treatise Zahlbericht literally "report on numbers". He also resolved a significant number-theory problem formulated by Waring in 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 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 were mostly proved by , after work by Teiji Takagi.

    His collected works Gesammelte Abhandlungen have been published several times. The original versions of his papers contained "many technical errors of varying degree"; [50] when the collection was first published, the errors were corrected and it was found that this could be done without major changes in the statements of the theorems, with one exception—a claimed proof of the continuum hypothesis. The Hilberts had by this time [around ] left the Reformed Protestant Church in which they had been baptized and married. From Wikipedia, the free encyclopedia. For other uses, see Hilbert disambiguation.

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