Hilbert's first work on invariant functions led him to the demonstration in 1888 of his famous finiteness theorem.

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.

Additionally, Hilbert's work anticipated and assisted several advances in the mathematical formulation of quantum mechanics.

Cantor's work also attracted favorable notice beyond Hilbert's celebrated encomium.

He started with the " betweenness " of Hilbert's axioms to characterize space without coordinatizing it, and then added extra relations between points to do the work formerly done by vector fields.

In 1957, he solved a particular interpretation of Hilbert's thirteenth problem ( a joint work with his student V. I. Arnold ).

But doubtless the significance of Gödel's work to mathematics as a whole ( and not just to formal logic ) was amply and dramatically illustrated by its applicability to one of Hilbert's problems.

Since 1900, other mathematicians and mathematical organizations have announced problem lists, but, with few exceptions, these collections have not had nearly as much influence nor generated as much work as Hilbert's problems.

Although the formalisation of logic was much advanced by the work of such figures as Gottlob Frege, Giuseppe Peano, Bertrand Russell, and Richard Dedekind, the story of modern proof theory is often seen as being established by David Hilbert, who initiated what is called Hilbert's program in the foundations of mathematics.

Kurt Gödel's seminal work on proof theory first advanced, then refuted this program: his completeness theorem initially seemed to bode well for Hilbert's aim of reducing all mathematics to a finitist formal system ; then his incompleteness theorems showed that this is unattainable.

Many axiomatic systems were developed in the nineteenth century, including non-Euclidean geometry, the foundations of real analysis, Cantor's set theory and Frege's work on foundations, and Hilbert's ' new ' use of axiomatic method as a research tool.

Another important milestone was the work of Hilbert's student Emanuel Lasker, who introduced primary ideals and proved the first version of the Lasker – Noether theorem.

David Hilbert's work on the question of the finite generation of the algebra of invariants ( 1890 ) resulted in the creation of a new mathematical discipline, abstract algebra.

The Hilbert transform arose in Hilbert's 1905 work on a problem posed by Riemann concerning analytic functions (; ) which has come to be known as the Riemann – Hilbert problem.

Hilbert's work was mainly concerned with the Hilbert transform for functions defined on the circle (; ).

Julia Hall Bowman Robinson ( December 8, 1919 – July 30, 1985 ) was an American mathematician best known for her work on decision problems and Hilbert's Tenth Problem.

Together with the work of Teiji Takagi and Helmut Hasse ( who established the more general Hasse reciprocity law ), this led to the development of the class field theory, realizing Hilbert's program in an abstract fashion.

The work of Lubotzky and Mann, combined with Michel Lazard's solution to Hilbert's fifth problem over the p-adic numbers, shows that a pro-p group is p-adic analytic if and only if it has finite rank, i. e. there exists a positive integer such that any closed subgroup has a topological generating set with no more than elements.

She was known for her work on partial differential equations ( especially Hilbert's 19th problem ) and fluid dynamics.

Let A = K be the ring of polynomials in n variables over a field K. Then the global dimension of A is equal to n. This statement goes back to David Hilbert's foundational work on homological properties of polynomial rings, see Hilbert's syzygy theorem.

Arnold then expanded on this work to show that only two-variable functions were in fact required, thus answering Hilbert's question.

In announcing the prize, CMI drew a parallel to Hilbert's problems, which were proposed in 1900, and had a substantial impact on 20th century mathematics.

Those forced out included Hermann Weyl ( who had taken Hilbert's chair when he retired in 1930 ), Emmy Noether and Edmund Landau.

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.

By early summer 1915, Hilbert's interest in physics had focused on general relativity, and he invited Einstein to Göttingen to deliver a week of lectures on the subject.

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's axioms: Hilbert's axioms had the goal of identifying a simple and complete set of independent axioms from which the most important geometric theorems could be deduced.

There was some irony that in the pushing through of David Hilbert's long-range programme a natural home for intuitionistic logic's central ideas was found: Hilbert had detested the school of L. E. J. Brouwer.

This became known as the Kronecker Jugendtraum ; and was certainly what had prompted Hilbert's remark above, since it makes explicit class field theory in the way the roots of unity do for abelian extensions of the rational number field, via Shimura's reciprocity law.

The ideal theory in question had been based on elimination theory, but in line with David Hilbert's taste moved away from algorithmic methods.

They were able to show the solutions had first derivatives that were Hölder continuous, which by previous results implied that the solutions are analytic whenever the differential equation has analytic coefficients, thus completing the solution of Hilbert's nineteenth problem.

In logic, the second problem on David Hilbert's list of open problems presented in 1900 was to prove that the axioms of arithmetic are consistent.

* Tarski's axioms: Alfred Tarski ( 1902 – 1983 ) and his students defined elementary Euclidean geometry as the geometry that can be expressed in first-order logic and does not depend on set theory for its logical basis, in contrast to Hilbert's axioms, which involve point sets.

The process of abstract axiomatization as exemplified by Hilbert's axioms reduces geometry to theorem proving or predicate logic.

He also published papers on mathematical logic, and solved a special case of Hilbert's fifth problem.

While the theorems of Gödel and Gentzen are now well understood by the mathematical logic community, no consensus has formed on whether ( or in what way ) these theorems answer Hilbert's second problem.

* it motivated the currently prevalent philosophical position that all of mathematics should be derivable from logic and set theory, ultimately leading to Hilbert's program, Gödel's theorems and non-standard analysis.

Many current lines of research in mathematical logic, proof theory and reverse mathematics can be viewed as natural continuations of Hilbert's original program.

Hilbert's axioms, unlike Tarski's axioms, do not constitute a first-order theory because the axioms V. 1 – 2 cannot be expressed in first-order logic.

From Brouwer to Hilbert, Oxford University Press, 1998, 165 – 167 ( on Hilbert's formalism ), 277 – 282 ( on intuitionistic logic ).

Principles of Mathematical Logic is the 1950 American translation of the 1938 second edition of David Hilbert's and Wilhelm Ackermann's classic text Grundzüge der theoretischen Logik, on elementary mathematical logic.

Hilbert's axioms do not constitute a first-order theory because his continuity axioms require second-order logic.