Good afternoon, dear readers! Today we would like to tell you about the life and work of David Hilbert, one of the most prominent mathematicians of the XIX-XX centuries. David Hilbert was born on January 23, 1862 in Königsberg, Prussia (now Kaliningrad, Russia). He was 1.75 meters tall. He worked in the fields of mathematics, physics, and philosophy, and was the author of many fundamental theories and problems that influenced the development of these sciences. Some of his most famous projects are Hilbert’s axiomatics, Hilbert space, the Einstein-Hilbert action, and Hilbert’s fundamental theorem. If you want to learn more about this extraordinary personality, we invite you to read our article.
David Hilbert was educated at the University of Königsberg, where he studied mathematics under Heinrich Weber and Ferdinand Lindemann. He defended his doctoral dissertation on the theory of invariants in 1885. After that, he visited several European universities where he met with prominent mathematicians such as Felix Klein, Henri Poincaré, Leopold Kronecker, and Carl Weierstrass. He also studied physics, astronomy, and philosophy.
David Hilbert began his career as a private assistant professor at the University of Königsberg in 1886. He worked there until 1895, when he received an offer from Felix Klein to move to the position of professor of mathematics at the University of Göttingen. Hilbert accepted the offer and moved to Göttingen, where he worked for the rest of his life. He became one of the leaders of the Göttingen School of Mathematics, which was the most prestigious in the world at the time. He was also an active participant and organizer of various scientific conferences, symposia, and congresses, where he gave presentations and discussions. He collaborated and competed with many prominent scientists, such as Albert Einstein, Hermann Weyl, Erwin Schrodinger, Kurt Gödel, and others. He has received many awards and recognition for his achievements, such as the Lobachevsky Prize, the Lobachevsky Medal, the Copley Medal, honorary membership in many academies of science, and honorary doctorates from various universities.
David Hilbert was one of the most versatile and influential mathematicians of his time. He made significant contributions to many areas of mathematics, such as algebra, number theory, geometry, analysis, topology, proof theory, mathematical logic, and foundations of mathematics. He also worked on the application of mathematics to physics, astronomy, and philosophy. Some of his most important achievements are:
Creation of the first complete axiomatics of Euclidean geometry, which included 20 axioms covering the concepts of point, line, plane, angle, equality, parallelism, perpendicularity, distance, area, volume, etc. This axiomatics was recognized as the best among the existing ones and became the basis for further research on non-Euclidean geometries.
Introduction of the concept of a Hilbert space, which is an abstract vector space that has a definite norm and a scalar product. Hilbert spaces allow us to consider functions as elements of space and apply geometric methods to them. Hilbert spaces are an important tool in analysis, operator theory, quantum mechanics, and other fields of mathematics and physics.
Development of the Einstein-Hilbert action, which is a variational principle for obtaining the Einstein field equations in general relativity.
The Einstein-Hilbert action contains two terms: a gravitational term, which depends on the Ricci scalar and the determinant of the metric, and a material term, which depends on the energy-momentum distribution in space-time. The action has the form:
where G is the gravitational constant, g is the determinant of the metric tensor, R is the Ricci scalar, and Sm – the action of matter.
The principle of least action states that physical processes occur in such a way that the action is stationary, that is, its variation is zero. By applying this principle to the Einstein-Hilbert action, we can obtain the Einstein equations that describe the dynamics of spacetime in the presence of the gravitational field and matter:
where Rμν is the Ricci tensor, gμν is a metric tensor, and Tμν is the energy-momentum tensor.
The Einstein-Hilbert action was proposed by David Hilbert in 1915, after he learned about general relativity from Albert Einstein. Hilbert was a prominent mathematician who worked in various branches of mathematics, such as algebra, number theory, geometry, analysis, topology, proof theory, and foundations of mathematics. He also applied mathematics to physics, astronomy, and philosophy. Some of his most important achievements are Hilbert’s axiomatics, Hilbert space, the Einstein-Hilbert action, and Hilbert’s main theorem.
The formulation of Hilbert’s main theorem, which states that any consistent theory that contains arithmetic is incomplete, i.e. there are propositions that can be neither proved nor disproved within the framework of this theory. This theorem was proved by Kurt Gödel in 1931 and showed that Hilbert’s ambitious project to find a complete and rigorous axiomatics of mathematics was impossible. Hilbert’s theorem has profound philosophical implications for the concepts of truth, proof, computability, and infinity.
The proposal of 23 Hilbert problems, which were formulated by him in his famous lecture at the International Mathematical Congress in Paris in 1900. These problems were chosen by Hilbert as the most important and interesting for further research in mathematics. They covered various branches of mathematics, such as number theory, algebra, geometry, analysis, topology, proof theory, and foundations of mathematics. Some of these problems have been solved, some remain unsolved, and some have been recognized as impractical or incorrect. Hilbert’s problems have been a source of inspiration and challenge for many generations of mathematicians.
I hope you found this article useful and educational. David Hilbert was an extraordinary, talented, courageous, and influential person who made a significant contribution to the development of mathematics and physics. Here are some quotes by David Hilbert that reflect his views on life, art, politics, and society:
“We should not think of mathematics as an abstract science, but as a science of all possibilities.”
“Nothing is impossible for the one who has faith.”
“The art of mathematics is to find that special case that contains all the hermetic ideas.”
“Everything that can be said can be said clearly.”
“No one can drive us out of the paradise that our own math has created.”