David Hilbert: Education and a Monumental Mathematical Career

David Hilbert (1862-1943) was a towering figure in the world of mathematics, leaving an indelible mark on numerous fields. His work spanned invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators, integral equations, mathematical physics, and the very foundations of mathematics itself, particularly proof theory. He championed Georg Cantor's set theory and transfinite numbers, and his students played a key role in establishing rigor and developing tools essential to modern mathematical physics.

Early Life and Education

David Hilbert was born on January 23, 1862, in the Province of Prussia, either in Königsberg (modern-day Kaliningrad) or in Wehlau (now Znamensk), a town near Königsberg. His parents were Otto Hilbert, a county judge, and Maria Therese Hilbert (née Erdtmann), whose father was a merchant. Hilbert had one sibling, a sister named Elise, born when he was six years old. His paternal grandfather, also named David Hilbert, was a judge and Geheimrat. His mother had an interest in philosophy, astronomy and prime numbers, while his father Otto taught him Prussian virtues. After his father became a city judge, the family moved to Königsberg.

Hilbert's early education began at the Friedrichskolleg Gymnasium, the same school Immanuel Kant had attended. However, he found the experience unsatisfactory and later transferred to the Wilhelm Gymnasium, which had a stronger focus on science. He graduated from there in early 1880. In the autumn of 1880, Hilbert enrolled at the University of Königsberg, known as the "Albertina".

The Königsberg Years: Collaboration and Early Career

The University of Königsberg proved to be a fertile ground for Hilbert's mathematical development. In 1882, Hermann Minkowski, two years younger than Hilbert, returned to Königsberg and entered the university. In 1884, Adolf Hurwitz arrived from Göttingen as an Extraordinarius (associate professor). This confluence of talent sparked an intense and fruitful scientific exchange among the three mathematicians. Minkowski and Hilbert, in particular, developed a reciprocal influence that would continue throughout their careers.

Hilbert remained at the University of Königsberg as a Privatdozent (senior lecturer) from 1886 to 1895. During this period, he laid the groundwork for his future groundbreaking work.

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Göttingen: A Professor at a Mathematical Mecca

In 1895, thanks to the intervention of Felix Klein, Hilbert secured a position as Professor of Mathematics at the University of Göttingen. This marked a turning point in his career and in the history of mathematics. Göttingen was then becoming the world's leading center for mathematical research, and Hilbert's presence further solidified its reputation. The Mathematical Institute in Göttingen became a hub of innovation and collaboration under his leadership.

Hilbert attracted a remarkable group of students, including Hermann Weyl, chess champion Emanuel Lasker, Ernst Zermelo, and Carl Gustav Hempel. John von Neumann served as his assistant. Among his 69 Ph.D. students in Göttingen were many who later became famous mathematicians, including Otto Blumenthal (1898), Felix Bernstein (1901), Hermann Weyl (1908), Richard Courant (1910), Erich Hecke (1910), Hugo Steinhaus (1911), and Wilhelm Ackermann (1925).

From 1902 to 1939, Hilbert served as an editor of the Mathematische Annalen, the leading mathematical journal of the time, further shaping the direction of mathematical research.

Hilbert's Contributions to Mathematics

Hilbert's contributions to mathematics are vast and multifaceted. Some of his most important achievements include:

Invariant Theory and Hilbert's Basis Theorem

Hilbert's early work on invariant functions culminated in his famous finiteness theorem in 1888. This theorem addressed a problem posed by Paul Gordan, who had previously demonstrated the finiteness of generators for binary forms using a complex computational approach. Attempts to generalize Gordan's method to functions with more than two variables had failed due to the complexity of the calculations.

Hilbert took a completely different approach. He demonstrated Hilbert's basis theorem, proving the existence of a finite set of generators for the invariants of quantics in any number of variables, but in an abstract form.

When Hilbert submitted his results to the Mathematische Annalen, Gordan, the journal's expert on invariant theory, rejected the article, deeming it insufficiently comprehensive and declaring, "Das ist nicht Mathematik" ("This is not mathematics"). However, Felix Klein recognized the significance of Hilbert's work and ensured its publication without alterations.

Encouraged by Klein, Hilbert extended his method in a second article, providing estimations on the maximum degree of the minimum set of generators. Despite his success, the abstract nature of his proof sparked controversy. While Kronecker eventually conceded its validity, Hilbert later responded to criticisms by stating that "many different constructions are subsumed under one fundamental idea." He argued that through a proof of existence, he had obtained a construction, and that the proof itself was the object.

Hilbert's Nullstellensatz

In the field of algebra, Hilbert made a fundamental contribution with his Nullstellensatz (German for "zero-locus theorem"). This theorem states that a field is algebraically closed if and only if every polynomial over it has a root in it.

Space-Filling Curves: The Hilbert Curve

In 1890, Giuseppe Peano published an article in the Mathematische Annalen describing the first historically documented space-filling curve. Hilbert, in response, designed his own construction of such a curve, now known as the Hilbert curve. This curve is constructed iteratively, with approximations generated by applying specific replacement rules.

Foundations of Geometry: Hilbert's Axioms

Hilbert's Grundlagen der Geometrie (Foundations of Geometry), published in 1899, presented a formal set of axioms to replace Euclid's traditional axioms. These axioms addressed weaknesses identified in Euclid's system, which was still in use at the time. It is difficult to specify the axioms used by Hilbert exactly without referring to the publication history of the Grundlagen since Hilbert changed and modified them several times. The original monograph was quickly followed by a French translation, in which Hilbert added V.2, the Completeness Axiom. An English translation, authorized by Hilbert, was made by E.J. Townsend and copyrighted in 1902. This translation incorporated the changes made in the French translation and so is considered to be a translation of the 2nd edition. Hilbert continued to make changes in the text and several editions appeared in German. The 7th edition was the last to appear in Hilbert's lifetime.

Hilbert's approach marked a shift to the modern axiomatic method, anticipated by Moritz Pasch's work in 1882. In this approach, axioms are not considered self-evident truths. Geometry may deal with objects about which we have strong intuitions, but it is not necessary to assign any explicit meaning to the undefined concepts. Hilbert began by enumerating the undefined concepts: point, line, plane, lying on (a relation between points and lines, points and planes, and lines and planes), betweenness, congruence of pairs of points (line segments), and congruence of angles. Hilbert brings together Euclid’s plane and solid geometry into one system.

Hilbert's Problems

At the International Congress of Mathematicians in Paris in 1900, Hilbert presented a list of 23 unsolved problems that he believed would guide mathematical research in the 20th century. This list has had a profound impact on the development of mathematics.

Hilbert's problem set can be seen as a manifesto that paved the way for the formalist school, one of the three major schools of mathematics in the 20th century. Formalists believe that mathematics is the manipulation of symbols according to agreed-upon formal rules.

Hilbert's Program

In 1920, Hilbert proposed a research project in metamathematics known as Hilbert's program. His goal was to formulate mathematics on a solid and complete logical foundation. He seems to have had both technical and philosophical reasons for formulating this proposal. This program is still recognizable in the most popular philosophy of mathematics, where it is usually called formalism. For example, the Bourbaki group adopted a watered-down and selective version of it as adequate to the requirements of their twin projects of (a) writing encyclopedic foundational works, and (b) supporting the axiomatic method as a research tool.

In 1931, Gödel's incompleteness theorem demonstrated that Hilbert's grand plan was impossible as stated. Nevertheless, the subsequent achievements of proof theory at the very least clarified consistency as it relates to theories of central concern to mathematicians. Hilbert's work had started logic on this course of clarification; the need to understand Gödel's work then led to the development of recursion theory and then mathematical logic as an autonomous discipline in the 1930s.

Hilbert Space

Around 1909, Hilbert dedicated himself to the study of differential and integral equations, leading to significant consequences for modern functional analysis. He introduced the concept of an infinite-dimensional Euclidean space, later called Hilbert space. This work provided the basis for important contributions to the mathematics of physics in the following decades. Later on, Stefan Banach amplified the concept, defining Banach spaces.

Shift to Physics and General Relativity

Until 1912, Hilbert was almost exclusively a pure mathematician. His friend Hermann Minkowski, immersed in studying physics in Bonn, joked that he had to spend 10 days in quarantine before visiting Hilbert. However, in 1912, three years after Minkowski's death, Hilbert turned his attention to physics. He arranged for a "physics tutor" and began studying kinetic gas theory, elementary radiation theory, and the molecular theory of matter.

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. Einstein received an enthusiastic reception. Over the summer, Einstein learned that Hilbert was also working on the field equations and redoubled his own efforts.

During November 1915, Einstein published several papers culminating in "The Field Equations of Gravitation." Nearly simultaneously, Hilbert published "The Foundations of Physics", an axiomatic derivation of the field equations.

Personal Life and Later Years

Hilbert married Käthe Jerosch in 1892, and they had one son, Franz Hilbert. He was raised a Calvinist in the Prussian Evangelical Church but later left the Church and became an agnostic. He argued that mathematical truth was independent of the existence of God or other a priori assumptions.

Hilbert lived to see the Nazis purge many prominent faculty members at the University of Göttingen in 1933, including Hermann Weyl, Emmy Noether, and Edmund Landau. Paul Bernays, who had collaborated with Hilbert in mathematical logic and co-authored the important book Grundlagen der Mathematik, was also forced to leave Germany.

About a year after the purge, Hilbert attended a banquet and was seated next to the new Minister of Education, Bernhard Rust. Rust asked whether "the Mathematical Institute really suffered so much because of the departure of the Jews". Hilbert replied: "Suffered? It doesn't exist anymore, does it?"

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