Mathematician David Hilbert is shown in this colorized portrait wearing a distinctive white hat with a black band, glasses, and a dark suit. Renowned for his contributions to mathematical foundations, Hilbert is best known for his work in formalism, algebraic number theory, and for proposing the famous "Hilbert's problems," which have influenced mathematical research for decades.
Mathematician David Hilbert is shown in this colorized portrait wearing a distinctive white hat with a black band, glasses, and a dark suit. Renowned for his contributions to mathematical foundations, Hilbert is best known for his work in formalism, algebraic number theory, and for proposing the famous "Hilbert's problems," which have influenced mathematical research for decades.

David Hilbert

Historical

Historical

Jan 23, 1862

-

Feb 14, 1943

Mathematician David Hilbert is shown in this colorized portrait wearing a distinctive white hat with a black band, glasses, and a dark suit. Renowned for his contributions to mathematical foundations, Hilbert is best known for his work in formalism, algebraic number theory, and for proposing the famous "Hilbert's problems," which have influenced mathematical research for decades.

David Hilbert

Historical

Historical

Jan 23, 1862

-

Feb 14, 1943

Biography

FAQ

Quotes

Biography

David Hilbert was one of the most influential mathematicians of the final part of the 19th and beginning of the 20th centuries. He was born in Königsberg, Prussia (now Kaliningrad, Russia), and being from a scholarly family, he became interested in mathematics from an early age. He continued his studies at the University of Königsberg, and during those years, he became close friends with another mathematician, Hermann Minkowski. Hilbert received his doctorate in 1885 from Ferdinand von Lindemann, and his thesis was on invariant theory. Throughout his lifetime, he contributed to various disciplines, from geometry to number theory to mathematical physics.

Hilbert may have laid down some of the most important of his work in the foundations of geometry. In his work published in 1899 entitled Grundlagen der Geometrie (Foundations of Geometry), he came up with a new set of hypotheses that have formed the basis of modern geometry. In Hilbert's axioms, mathematicians had a more precise way of approaching problems than the classical Euclidean method, and this was one of the most critical changes mathematicians had to embrace in their work. His work proved that geometry could be treated axiomatically, and his method became a model for future mathematical formalism.

Hilbert's works are also concerned with the theory of integral equations and functional analysis. His formulation of what is now known as Hilbert spaces, the infinite-dimensional vector spaces, proved very important in quantum mechanics, and physicists later used it to explain the behavior of quantum entities. In number theory, Hilbert extended the concept of quadratic forms. He helped to build a theoretical framework for class fields that paved the way for the creation of the modern algebraic number theory.

In 1900, Hilbert gave a list of twenty-three mathematical problems to be solved at the International Congress of Mathematicians in Paris. These problems were the basis for most mathematicians' research in the twentieth century. As with many other mathematicians, not all of the issues that Euler sought to solve were solved in his lifetime, but many are still challenging mathematicians. His second difficulty, concerning the nature of arithmetic, was solved by Kurt Gödel's theorem, stating that some mathematical systems improve their consistency.

Apart from his contributions to techniques, Hilbert was famous for his conviction that reason and intelligence can overcome any mathematical problem. As for mathematics, he once said, "We must know, we shall know," meaning that all mathematical truths could be found. Even though Gödel's work demonstrated that some of Hilbert's objectives were unattainable, his work was marked by incredible innovation and a thirst for knowledge. Therefore, he is an essential figure in the history of mathematics.

Quotes

"We must know. We shall know."

"Mathematics is a science which has no presuppositions."

"Physics is too complex for physics."

"A mathematical theorem is incomplete until one has clarified that the man on the street can understand it."

"Mathematics is not the manipulation of abstract symbols but the discovery of that particular case, which embraces all the possibilities of generality."

"The infinite! There is not, perhaps, another question which has ever stirred in such a way the soul of man."

"If its knowledge has only reached the clarity of its findings, every type of science becomes capable of being expressed mathematically."

"One can judge the significance of a scientific contribution by the number of earlier publications that it makes redundant."

"The axiomatic method will not fail to shed light on any scientific question like math as it has done with math."

"The more the mathematical development goes, the more it becomes more in harmony and in one tune."

"In my opinion, mathematical science is a unified subject."

"It is not the things that we do not know that constrains us but the things that we know."

"He who seeks methods without having a definite problem in mind seeks, for the most part, in vain."

"Mathematics is colorless and has no nationality; to Mathematics, the whole world is a single nation."

"There are problems the solution of which you would never suspect."

FAQ

What can we attribute to the memory of the great mathematician David Hilbert?

David Hilbert made a great contribution to geometry, algebra, and number theory and is also credited with creating the Hilbert spaces, which are crucial in quantum mechanics.

What are Hilbert spaces?

Hilbert spaces are infinite-dimensional vector spaces used in functional analysis and quantum mechanics to describe a system's state, as seen in modern physics.

What are Hilbert's 23 problems?

In 1900, Hilbert played down twenty-three mathematical problems that mathematicians should solve in the twentieth century. These problems were in different areas of mathematics, such as algebra, number theory, and mathematical physics.

What did Hilbert do for geometry?

Hilbert reformulated geometry with his Grundlagen der Geometrie, considered the starting point of modern geometry.

What is Hilbert's second problem?

The second problem Hilbert stated is that there is proof that the arithmetic is consistent. Kurt Gödel's incompleteness theorems later proved that such proof is not only impossible but also impossible within arithmetic itself.

What was Hilbert's role in the development of quantum mechanics?

As is well known, Hilbert was convinced that any mathematical problem could, in principle, be solved; as he said, "We must see, we shall know."As is well known, Hilbert was convinced that any mathematical problem could, in principle, be solved; as he said, "We must see, we shall know."

What did Hilbert think of the possibility of solving mathematical problems?

As is well known, Hilbert was convinced that any mathematical problem could, in principle, be solved; as he said, "We must see, we shall know."As is well known, Hilbert was convinced that any mathematical problem could, in principle, be solved; as he said, "We must see, we shall know."

What do you think was Hilbert's contribution to the field of number theory?

Hilbert works in the number theory, significantly impacting algebraic number fields, quadratic forms, and class field theory.

Did Hilbert do anything for physics?

Indeed, Hilbert contributed to theoretical physics by providing mathematical foundations for quantum mechanics and general relativity and working with physicists such as Albert Einstein.

Which of Hilbert's problems are solved?

Of the cleanly formulated Hilbert problems, numbers 3, 7, 10, 14, 17, 18, 19, and 20 have resolutions that are accepted by consensus of the mathematical community.

Biography

FAQ

Quotes

Biography

David Hilbert was one of the most influential mathematicians of the final part of the 19th and beginning of the 20th centuries. He was born in Königsberg, Prussia (now Kaliningrad, Russia), and being from a scholarly family, he became interested in mathematics from an early age. He continued his studies at the University of Königsberg, and during those years, he became close friends with another mathematician, Hermann Minkowski. Hilbert received his doctorate in 1885 from Ferdinand von Lindemann, and his thesis was on invariant theory. Throughout his lifetime, he contributed to various disciplines, from geometry to number theory to mathematical physics.

Hilbert may have laid down some of the most important of his work in the foundations of geometry. In his work published in 1899 entitled Grundlagen der Geometrie (Foundations of Geometry), he came up with a new set of hypotheses that have formed the basis of modern geometry. In Hilbert's axioms, mathematicians had a more precise way of approaching problems than the classical Euclidean method, and this was one of the most critical changes mathematicians had to embrace in their work. His work proved that geometry could be treated axiomatically, and his method became a model for future mathematical formalism.

Hilbert's works are also concerned with the theory of integral equations and functional analysis. His formulation of what is now known as Hilbert spaces, the infinite-dimensional vector spaces, proved very important in quantum mechanics, and physicists later used it to explain the behavior of quantum entities. In number theory, Hilbert extended the concept of quadratic forms. He helped to build a theoretical framework for class fields that paved the way for the creation of the modern algebraic number theory.

In 1900, Hilbert gave a list of twenty-three mathematical problems to be solved at the International Congress of Mathematicians in Paris. These problems were the basis for most mathematicians' research in the twentieth century. As with many other mathematicians, not all of the issues that Euler sought to solve were solved in his lifetime, but many are still challenging mathematicians. His second difficulty, concerning the nature of arithmetic, was solved by Kurt Gödel's theorem, stating that some mathematical systems improve their consistency.

Apart from his contributions to techniques, Hilbert was famous for his conviction that reason and intelligence can overcome any mathematical problem. As for mathematics, he once said, "We must know, we shall know," meaning that all mathematical truths could be found. Even though Gödel's work demonstrated that some of Hilbert's objectives were unattainable, his work was marked by incredible innovation and a thirst for knowledge. Therefore, he is an essential figure in the history of mathematics.

Quotes

"We must know. We shall know."

"Mathematics is a science which has no presuppositions."

"Physics is too complex for physics."

"A mathematical theorem is incomplete until one has clarified that the man on the street can understand it."

"Mathematics is not the manipulation of abstract symbols but the discovery of that particular case, which embraces all the possibilities of generality."

"The infinite! There is not, perhaps, another question which has ever stirred in such a way the soul of man."

"If its knowledge has only reached the clarity of its findings, every type of science becomes capable of being expressed mathematically."

"One can judge the significance of a scientific contribution by the number of earlier publications that it makes redundant."

"The axiomatic method will not fail to shed light on any scientific question like math as it has done with math."

"The more the mathematical development goes, the more it becomes more in harmony and in one tune."

"In my opinion, mathematical science is a unified subject."

"It is not the things that we do not know that constrains us but the things that we know."

"He who seeks methods without having a definite problem in mind seeks, for the most part, in vain."

"Mathematics is colorless and has no nationality; to Mathematics, the whole world is a single nation."

"There are problems the solution of which you would never suspect."

FAQ

What can we attribute to the memory of the great mathematician David Hilbert?

David Hilbert made a great contribution to geometry, algebra, and number theory and is also credited with creating the Hilbert spaces, which are crucial in quantum mechanics.

What are Hilbert spaces?

Hilbert spaces are infinite-dimensional vector spaces used in functional analysis and quantum mechanics to describe a system's state, as seen in modern physics.

What are Hilbert's 23 problems?

In 1900, Hilbert played down twenty-three mathematical problems that mathematicians should solve in the twentieth century. These problems were in different areas of mathematics, such as algebra, number theory, and mathematical physics.

What did Hilbert do for geometry?

Hilbert reformulated geometry with his Grundlagen der Geometrie, considered the starting point of modern geometry.

What is Hilbert's second problem?

The second problem Hilbert stated is that there is proof that the arithmetic is consistent. Kurt Gödel's incompleteness theorems later proved that such proof is not only impossible but also impossible within arithmetic itself.

What was Hilbert's role in the development of quantum mechanics?

As is well known, Hilbert was convinced that any mathematical problem could, in principle, be solved; as he said, "We must see, we shall know."As is well known, Hilbert was convinced that any mathematical problem could, in principle, be solved; as he said, "We must see, we shall know."

What did Hilbert think of the possibility of solving mathematical problems?

As is well known, Hilbert was convinced that any mathematical problem could, in principle, be solved; as he said, "We must see, we shall know."As is well known, Hilbert was convinced that any mathematical problem could, in principle, be solved; as he said, "We must see, we shall know."

What do you think was Hilbert's contribution to the field of number theory?

Hilbert works in the number theory, significantly impacting algebraic number fields, quadratic forms, and class field theory.

Did Hilbert do anything for physics?

Indeed, Hilbert contributed to theoretical physics by providing mathematical foundations for quantum mechanics and general relativity and working with physicists such as Albert Einstein.

Which of Hilbert's problems are solved?

Of the cleanly formulated Hilbert problems, numbers 3, 7, 10, 14, 17, 18, 19, and 20 have resolutions that are accepted by consensus of the mathematical community.

Biography

FAQ

Quotes

Biography

David Hilbert was one of the most influential mathematicians of the final part of the 19th and beginning of the 20th centuries. He was born in Königsberg, Prussia (now Kaliningrad, Russia), and being from a scholarly family, he became interested in mathematics from an early age. He continued his studies at the University of Königsberg, and during those years, he became close friends with another mathematician, Hermann Minkowski. Hilbert received his doctorate in 1885 from Ferdinand von Lindemann, and his thesis was on invariant theory. Throughout his lifetime, he contributed to various disciplines, from geometry to number theory to mathematical physics.

Hilbert may have laid down some of the most important of his work in the foundations of geometry. In his work published in 1899 entitled Grundlagen der Geometrie (Foundations of Geometry), he came up with a new set of hypotheses that have formed the basis of modern geometry. In Hilbert's axioms, mathematicians had a more precise way of approaching problems than the classical Euclidean method, and this was one of the most critical changes mathematicians had to embrace in their work. His work proved that geometry could be treated axiomatically, and his method became a model for future mathematical formalism.

Hilbert's works are also concerned with the theory of integral equations and functional analysis. His formulation of what is now known as Hilbert spaces, the infinite-dimensional vector spaces, proved very important in quantum mechanics, and physicists later used it to explain the behavior of quantum entities. In number theory, Hilbert extended the concept of quadratic forms. He helped to build a theoretical framework for class fields that paved the way for the creation of the modern algebraic number theory.

In 1900, Hilbert gave a list of twenty-three mathematical problems to be solved at the International Congress of Mathematicians in Paris. These problems were the basis for most mathematicians' research in the twentieth century. As with many other mathematicians, not all of the issues that Euler sought to solve were solved in his lifetime, but many are still challenging mathematicians. His second difficulty, concerning the nature of arithmetic, was solved by Kurt Gödel's theorem, stating that some mathematical systems improve their consistency.

Apart from his contributions to techniques, Hilbert was famous for his conviction that reason and intelligence can overcome any mathematical problem. As for mathematics, he once said, "We must know, we shall know," meaning that all mathematical truths could be found. Even though Gödel's work demonstrated that some of Hilbert's objectives were unattainable, his work was marked by incredible innovation and a thirst for knowledge. Therefore, he is an essential figure in the history of mathematics.

Quotes

"We must know. We shall know."

"Mathematics is a science which has no presuppositions."

"Physics is too complex for physics."

"A mathematical theorem is incomplete until one has clarified that the man on the street can understand it."

"Mathematics is not the manipulation of abstract symbols but the discovery of that particular case, which embraces all the possibilities of generality."

"The infinite! There is not, perhaps, another question which has ever stirred in such a way the soul of man."

"If its knowledge has only reached the clarity of its findings, every type of science becomes capable of being expressed mathematically."

"One can judge the significance of a scientific contribution by the number of earlier publications that it makes redundant."

"The axiomatic method will not fail to shed light on any scientific question like math as it has done with math."

"The more the mathematical development goes, the more it becomes more in harmony and in one tune."

"In my opinion, mathematical science is a unified subject."

"It is not the things that we do not know that constrains us but the things that we know."

"He who seeks methods without having a definite problem in mind seeks, for the most part, in vain."

"Mathematics is colorless and has no nationality; to Mathematics, the whole world is a single nation."

"There are problems the solution of which you would never suspect."

FAQ

What can we attribute to the memory of the great mathematician David Hilbert?

David Hilbert made a great contribution to geometry, algebra, and number theory and is also credited with creating the Hilbert spaces, which are crucial in quantum mechanics.

What are Hilbert spaces?

Hilbert spaces are infinite-dimensional vector spaces used in functional analysis and quantum mechanics to describe a system's state, as seen in modern physics.

What are Hilbert's 23 problems?

In 1900, Hilbert played down twenty-three mathematical problems that mathematicians should solve in the twentieth century. These problems were in different areas of mathematics, such as algebra, number theory, and mathematical physics.

What did Hilbert do for geometry?

Hilbert reformulated geometry with his Grundlagen der Geometrie, considered the starting point of modern geometry.

What is Hilbert's second problem?

The second problem Hilbert stated is that there is proof that the arithmetic is consistent. Kurt Gödel's incompleteness theorems later proved that such proof is not only impossible but also impossible within arithmetic itself.

What was Hilbert's role in the development of quantum mechanics?

As is well known, Hilbert was convinced that any mathematical problem could, in principle, be solved; as he said, "We must see, we shall know."As is well known, Hilbert was convinced that any mathematical problem could, in principle, be solved; as he said, "We must see, we shall know."

What did Hilbert think of the possibility of solving mathematical problems?

As is well known, Hilbert was convinced that any mathematical problem could, in principle, be solved; as he said, "We must see, we shall know."As is well known, Hilbert was convinced that any mathematical problem could, in principle, be solved; as he said, "We must see, we shall know."

What do you think was Hilbert's contribution to the field of number theory?

Hilbert works in the number theory, significantly impacting algebraic number fields, quadratic forms, and class field theory.

Did Hilbert do anything for physics?

Indeed, Hilbert contributed to theoretical physics by providing mathematical foundations for quantum mechanics and general relativity and working with physicists such as Albert Einstein.

Which of Hilbert's problems are solved?

Of the cleanly formulated Hilbert problems, numbers 3, 7, 10, 14, 17, 18, 19, and 20 have resolutions that are accepted by consensus of the mathematical community.

Life and achievements

Early life

David Hilbert was born in 1862 in Königsberg, Prussia, in an intellectually charged home. His father was a judge, and his mother, Maria Therese Erdtmann, had a passion for astronomy and philosophy and was passionate about inclined Hilbert to think at his tender age. He was enrolled late in school and started his education at the Friedrichskolleg Gymnasium at 10, then transferred to Wilhelm Gymnasium, where he was a bright student in mathematics and science.

In 1880, Hilbert joined the University of Königsberg, where he was considered one of the most promising mathematicians. While in university, he developed camaraderie with other young and talented mathematicians such as David Hilbert and Hermann Minkowski, who in the future would help Einstein formulate the theory of relativity. Specializing in number theory, Hilbert received his doctorate in 1885 under the supervision of Ferdinand von Lindemann with a thesis entitled On the Theory of Algebraic Invariants: The Invariant Properties of Special Binary Forms.

In the early part of his career, Hilbert focused on invariant theory, for which he produced a pioneering work. His famous Basis Theorem was solving one of the oldest problems as it proved the existence of the finite basis of invariants of a polynomial. This particular result greatly influenced the progress of algebra and the creation of contemporary abstract algebra. He made significant contributions to algebraic number theory and produced the Zahlbericht in 1897, which was considered an essential publication in the area.

Legacy

David Hilbert's work is extensive and can be applied to numerous disciplines ranging from Mathematics to Physics. His axiomatic method described in Grundlagen der Geometrie changed the course of geometry by giving the axiomatics of the Euclidean and non-Euclidean geometries. As with many other works of Euler, this publication can be viewed as one of the foundations of modern mathematics, which served as a point of reference for generations of mathematicians.

As we have seen, one of Hilbert's most significant legacies was his influence on the research program of twentieth-century mathematics. His list of 23 problems presented in the year 1900 has influenced most of the research carried out in mathematics for the last century. Such issues were varied and covered a broad spectrum of mathematical specialties, including number theory, algebra, and mathematical physics. Specific problems, such as the Riemann Hypothesis, still exist up to this date and still pose a question to many mathematicians.

In quantum mechanics, Hilbert's spaces, which Hilbert developed, offered the much-needed mathematical foundation for quantum systems. He also connected mathematics and physics to quantum theory, particularly in the spectral theory of operators and relativity. Scientists such as Werner Heisenberg and John von Neumann assumed this mathematical structure to be important in developing quantum mechanics.

Hilbert also contributed to formalism, a philosophy of mathematics that assumed that all mathematical statements are theorems of formalism. While Gödel proved that Hilbert's dream of having proof of the consistency of all mathematics is impossible, the program that Hilbert set into motion led to many advancements in mathematical logic and proof theory.

However, besides his brilliant academic results, Hilbert became a tutor to some of the brightest mathematicians of the twentieth century, such as John von Neumann and Hermann Weyl. His work still influences modern mathematics, from algebra and the theory of numbers to geometry and quantum mechanics fundamentals.

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Milestone moments

Jul 1, 1885

Doctorate in Invariant Theory
It is worth noting that in 1885, David Hilbert defended his doctoral thesis at the University of Königsberg.
His work was devoted to invariant theory – the branch of mathematics that deals with objects that remain the same after some transformations.
In his dissertation, Hilbert discussed the properties of particular binary forms, which was one of the significant questions of the algebra of that period.
His way of solving these issues was so unique that he was recognized as a mathematician who could solve abstract matters.
Hilbert's work in invariant theory culminated in his proof of the now-known Hilbert's Basis Theorem, asserting that a finite basis exists for polynomial invariants.
This result solved a long-standing mathematical problem and paved the way for the formation of modern abstract algebra.
It could be said that his dissertation signified the start of a successful career.

Apr 17, 1899

Foundations of Geometry
In 1899, David Hilbert published a book named Grundlagen der Geometrie, which revolutionized geometry.
This text developed a new set of axioms, which gave a better way of stating Euclidean geometry than the previous attempts.
Hilbert's axiomatic system was significant because it gave geometry a formal, abstract analysis method.
His work provided the basis for the formation of modern mathematical logic and influenced other fields of mathematics, such as topology and set theory.
Grundlagen der Geometrie is a work that has the potential to be considered one of the most influential works in the history of mathematics.

Aug 13, 1900

Hilbert's 23 Problems
At the International Congress of Mathematicians in Paris in 1900, David Hilbert gave his list of twenty-three unsolved mathematics problems.
These problems solve some of the most significant questions of mathematics, such as number theory, algebra, and mathematical physics.
This list of twenty-four problems guided mathematical research for much of the twentieth century.
Some questions have been answered in a few decades, but some, for instance, the Riemann Hypothesis, are still with mathematicians.
The problems illustrated the idea in Hilbert's mind regarding the future of mathematics, making him among the greatest mathematicians the world has ever produced.

Jan 18, 1912

Change to Physics and Quantum Mechanics
In 1912, Hilbert changed his research interest to theoretical physics, which was in its infancy.
Some of the topics he worked on were integral equations and functional analysis, the results of which were essential in formulating quantum mechanics.
Hilbert's idea of Hilbert spaces, which are infinite dimensional vector spaces, proved to be the mathematical tools that were required to explain quantum behavior.
In addition to mathematics, Hilbert worked on physics, particularly general relativity, with Albert Einstein.
Hilbert's mathematical precision extended the theories of Einstein and contributed to the connection between mathematics and physics.
He is still respected for his contribution to the early stages of quantum mechanics.

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