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.
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.