On the Infinite was Hilbert ’ s most important paper on the foundations of mathematics, serving as the heart of Hilbert's program to secure the foundation of transfinite numbers by basing them on finite methods.

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.

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.

In continuation of his " program " with which he challenged the mathematics community in 1900, at a 1928 international conference David Hilbert asked three questions, the third of which became known as " Hilbert's ".

In mathematics, specifically commutative algebra, Hilbert's basis theorem states that every ideal in the ring of multivariate polynomials over a Noetherian ring is finitely generated.

In addition, from at least the time of Hilbert's program at the turn of the twentieth century to the proof of Gödel's incompleteness theorems and the development of the Church-Turing thesis in the early part of that century, true statements in mathematics were generally assumed to be those statements which are provable in a formal axiomatic system.

Hilbert's goals of creating a system of mathematics that is both complete and consistent were dealt a fatal blow by the second of Gödel's incompleteness theorems, which states that sufficiently expressive consistent axiom systems can never prove their own consistency.

The two results are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible, giving a negative answer to Hilbert's second problem.

Many logicians believe that Gödel's incompleteness theorems struck a fatal blow to David Hilbert's second problem, which asked for a finitary consistency proof for mathematics.

The early development of mathematical proof theory was driven by the desire to provide finitary consistency proofs for all of mathematics as part of Hilbert's program.

In mathematics, Hilbert's second problem was posed by David Hilbert in 1900 as one of his 23 problems.

Hilbert's problems form a list of twenty-three problems in mathematics published by German mathematician David Hilbert in 1900.

The 6th problem concerns the axiomatization of physics, a goal that twentieth century developments of physics ( including its recognition as a discipline independent from mathematics ) seem to render both more remote and less important than in Hilbert's time.

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.

He contributed to Hilbert's program in the foundations of mathematics by providing a constructive consistency proof for a weak system of arithmetic.

The leading school was that of the formalist approach, of which David Hilbert was the foremost proponent, culminating in what is known as Hilbert's program, which thought to ground mathematics on a small basis of a logical system proved sound by metamathematical finitistic means.

Hermann Weyl: " Comments on Hilbert's second lecture on the foundations of mathematics ," 480-484.

But in this address Weyl " while defending Brouwer against some of Hilbert's criticisms ... attempts to bring out the significance of Hilbert's approach to the problems of the foundations of mathematics.

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.

The notion arose from the theory of magnitudes of Ancient Greece ; it still plays an important role in modern mathematics such as David Hilbert's axioms for geometry, and the theories of ordered groups, ordered fields, and local fields.

* 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.

In 1900 David Hilbert included it in his list of twenty three unsolved problems of mathematics — it forms part of Hilbert's eighteenth problem.

This is an unsolved problem which probably has never been seriously investigated, although one frequently hears the comment that we have insufficient specialists of the kind who can compete with the Germans or Swiss, for example, in precision machinery and mathematics, or the Finns in geochemistry.

A variant of counting-out game, known as Josephus problem, represents a famous theoretical problem in mathematics and computer science.

The problem with the psychological approach to mathematics and logic is that it fails to account for the fact that this approach is about formal categories, and not simply about abstractions from sensibility alone.

In mathematics and computer science, the (, German for ' decision problem ') is a challenge posed by David Hilbert in 1928.

Bombieri is also known for his pro bono service on behalf of the mathematics profession, e. g. for serving on external review boards and for peer-reviewing extraordinarily complicated manuscripts ( like the papers of John Nash on embedding Riemannian manifolds and of Per Enflo on the invariant subspace problem ).

If an isomorphism can be found from a relatively unknown part of mathematics into some well studied division of mathematics, where many theorems are already proved, and many methods are already available to find answers, then the function can be used to map whole problems out of unfamiliar territory over to " solid ground " where the problem is easier to understand and work with.

Many areas of mathematics and computer science have been brought to bear on the problem, including elliptic curves, algebraic number theory, and quantum computing.

The problem often arises in resource allocation where there are financial constraints and is studied in fields such as combinatorics, computer science, complexity theory, cryptography and applied mathematics.

He was appointed a lecturer in mathematics at Cambridge in 1927, where his 1935 lectures on the Foundations of Mathematics and Gödel's Theorem inspired Alan Turing to embark on his pioneering work on the Entscheidungsproblem ( decision problem ) using a hypothetical computing machine.

These formal terms are manipulated by the rules of mathematics and logic, and any results are interpreted or translated back into the problem domain.

Descartes ' work provided the basis for the calculus developed by Newton and Leibniz, who applied infinitesimal calculus to the tangent line problem, thus permitting the evolution of that branch of modern mathematics.

In the OECD's international assessment of student performance, PISA, Finland has consistently been among the highest scorers worldwide ; in 2003, Finnish 15-year-olds came first in reading literacy, science, and mathematics ; and second in problem solving, worldwide.

In the following decades, the problem was studied by many researchers from mathematics, computer science, chemistry, physics, and other sciences.

Internationally, the university is known for research relating to the genome of the Populus tree ( Life sciences ), contributions to the Gleason problem and function spaces on fractals ( mathematics ) and its school of industrial design which gives degree programs in English open to students from all of the world.

In various branches of mathematics, a useful construction is often viewed as the “ most efficient solution ” to a certain problem.

In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group G is the algorithmic problem of deciding whether two words in the generators represent the same element.

The word problem was one of the first examples of an unsolvable problem to be found not in mathematical logic or the theory of algorithms, but in one of the central branches of classical mathematics, algebra.

* Word problem ( mathematics )

The Fermi – Pasta – Ulam problem is credited not only as " the birth of experimental mathematics ", but also as inspiration for the vast field of Nonlinear Science.

The falsifiability of this claim is perhaps the central problem in the cognitive science of mathematics, a field that attempts to establish a foundation ontology based on the human cognitive and scientific process.