David Hilbert

David Hilbert
	David Hilbert, German mathematician
Tradition	Mathematics, Foundations of mathematics, Hilbert's program
Influenced by	Carl Friedrich Gauss, Leopold Kronecker, Felix Klein
Lifespan	1862–1943
Notable ideas	Hilbert's problems; Hilbert space; formalism in mathematics; contributions to algebraic number theory, invariant theory, and proof theory
Occupation	Mathematician
Influenced	Kurt Gödel, John von Neumann, Alan Turing, Modern mathematics
Wikidata	Q41585

David Hilbert (1862–1943) was a German mathematician whose work reshaped many areas of mathematics in the late 19th and early 20th centuries. He was one of the leading figures of his time, famous for bringing a high level of rigor and abstract thinking to mathematics. Hilbert introduced new methods (such as formal axiomatic systems) and solved problems in geometry, number theory, algebra, analysis and physics. He is perhaps best known today for his list of 23 open problems presented in 1900, which guided much of 20th-century mathematics. His students and colleagues included many future stars (such as Hermann Weyl and John von Neumann) and his influence extended from the foundations of geometry to the concepts underlying quantum mechanics.

Early life and education

Hilbert was born on January 23, 1862, in Königsberg, then part of Prussia (now Kaliningrad, Russia). His father, Otto Hilbert, was a judge, and his mother, Maria Therese Hilbert, had a strong interest in mathematics and the sciences. In fact, Maria was an amateur astronomer and loved discussing numbers and philosophy, and she taught young David at home in his early years. David did not start formal schooling until age eight, and he first attended the Royal Friedrichskollegium Gymnasium in Königsberg. This school emphasized classical subjects (Latin and Greek) and heavy memorization, rather than science or mathematics, so Hilbert found the schooling somewhat at odds with his interests. He then transferred to the Wilhelm-Gymnasium (also in Königsberg), which offered a stronger mathematical curriculum. He graduated from high school in Königsberg well enough to enter university in 1880.

In autumn 1880 Hilbert enrolled at the University of Königsberg to study mathematics and physics. In his first semester he took courses in calculus, determinants, and the curvature of surfaces. Pursuing the German academic custom of the time, he spent the second semester in Heidelberg listening to the lectures of Lazarus Fuchs (a noted algebraist). Returning to Königsberg, he attended courses by Heinrich Weber (on number theory and analysis) and met fellow students Hermann Minkowski and Adolf Hurwitz, who became close friends. Minkowski and Hurwitz were also talented young mathematicians, and their friendship and collaboration helped sharpen Hilbert’s own ideas. Under the guidance of Ferdinand von Lindemann (Hilbert’s doctoral advisor), Hilbert worked on invariant theory, and on December 11, 1884, he passed his oral examination and submitted his doctoral dissertation on "invariant properties of certain quadratic forms." He received his PhD in 1885.

After earning his doctorate, Hilbert stayed in Königsberg. He completed the German Habilitation (a qualification for lecturing) soon after and became a Privatdozent (lecturer). In 1886–1892 he lectured in Königsberg in mathematics. In 1892 he was promoted to associate professor and that same year he married Käthe Jerosch. They had one son, Franz, born in 1894. Hilbert became a full professor (Ordentlicher Professor) at Königsberg in 1893. In 1895 he accepted a professorship at the University of Göttingen, which was already famous for the legacy of Gauss, Dirichlet and Riemann. Hilbert would remain in Göttingen for the rest of his life, helping to build its mathematical institute into the world’s leading center of mathematics in the early 20th century. He turned down offers from other universities (including Leipzig, Berlin, and Heidelberg) to stay in Göttingen amidst its lively academic community.

Major works and ideas

Hilbert’s contributions spanned many fields. He had a particularly profound impact on geometry, algebra, analysis and the foundations of mathematics. A few of his key achievements include:

Axiomatic geometry: In the second half of the 19th century mathematicians recognized hidden assumptions in Euclid’s Elements. Hilbert addressed this by giving a new, rigorous list of axioms for Euclidean geometry. In his influential book Grundlagen der Geometrie (“Foundations of Geometry”, 1899, English translation 1902), he proposed a complete set of axioms for points, lines and planes. Hilbert’s axioms made explicit all assumptions needed for plane and solid geometry. He studied how these axioms related to each other (showing, for example, they were independent and consistent if Euclid’s field of real numbers was consistent). This work turned geometry into a purely logical system and became a model for all axiomatic mathematics.

Algebra and number theory: Hilbert made major advances in algebraic invariant theory and algebraic number theory. In 1890 he proved the basis theorem, which implies that polynomial equations can be simplified to a finite set of generating equations. Loosely speaking, he showed that problems in algebraic geometry (properties that remain unchanged under transformations) have finitely many basic invariants. In simpler terms, Hilbert proved that if a family of polynomial equations has infinitely many consequences, they can all be generated by finitely many of them. This solved a long-standing problem in invariant theory.

In number theory, his 1897 Zahlbericht (“Report on Numbers”) was a landmark summary of algebraic number theory. It consolidated all existing work on number fields and class field theory up to that time into a unified framework. Hilbert’s report served as the starting point for much later work in number theory. For instance, he introduced the idea of Hilbert class field and stated what now are called “Hilbert’s 9th and 12th problems” in class field theory.

He also solved several specific problems: for example, he completely resolved Waring’s problem on sums of powers. In 1909 he proved that for any exponent $n$, every positive integer can be expressed as a sum of a fixed number of $n$th powers (for example, he showed every integer is a sum of four squares, nine cubes, etc.). This general result, known as Hilbert’s solution of Waring’s problem, was a milestone in additive number theory.

Functional analysis and physics: Hilbert was a foundational figure in analysis. Around 1909 he studied integral equations (equations involving integrals of an unknown function), and his methods led to the birth of modern functional analysis. He introduced the concept of an abstract, infinite-dimensional space of functions, now called a Hilbert space. In simple terms, a Hilbert space is like the usual geometry of vectors but with infinitely many dimensions (think of "vectors" whose “coordinates” form an infinite sequence or function). This idea became central in quantum mechanics and partial differential equations. Hilbert proved the existence of solutions to many problems in analysis using these ideas.

Hilbert also applied mathematical methods to physics. He worked on problems in theoretical physics using his strong analytic techniques. Notably, in 1915–1916 he worked on the mathematical formulations of Albert Einstein’s General Relativity theory. Using a variational (action) approach, Hilbert derived equations equivalent to Einstein’s gravitational field equations. (Later historical analysis found that Hilbert’s first draft was not fully correct, and Einstein’s final 1915 paper actually arrived at the equations independently, so Einstein is credited with priority.) Additionally, Hilbert wrote important papers on gas theory and radiation (problems in statistical and quantum physics). His work on the kinetic theory of gases and blackbody radiation helped advance early quantum ideas.

Foundations of mathematics: Hilbert’s philosophy was that all of mathematics should be based on a consistent set of axioms. He believed that mathematical reasoning should be completely rigorous and that any mathematical statement should be provable (or disprovable) from a clear system of rules. In the 1920s he put forward what became known as Hilbert’s program. The core idea was to formalize all of mathematics by writing it in axiomatic form, and then to prove that these axioms were free from contradictions by using only elementary "finitary" reasoning. (Here finitary meant only concrete operations: direct counting or computation steps, avoiding any appeal to actual infinity. For example, rather than invoking an infinite set, one might only argue by direct enumeration or finite construction whenever possible.) If successful, Hilbert’s program would establish mathematics on a completely secure base.

In practice, Hilbert and his collaborators worked on axiomatizing arithmetic and analysis and on showing these systems consistent. Hilbert insisted that the consistency of a mathematical theory (the assurance that no contradictions can be derived from its axioms) was the highest requirement. This reflected his formalist viewpoint: mathematical truths are exactly those formally deduced according to axiom rules, not relying on any informal intuition. He famously said that Cantor’s set theory (with its actual infinite sets) was a mathematical “paradise” that no one should leave, defending the use of infinite sets despite criticism (famously quoting, “No one shall expel us from the paradise that Cantor has created.”).

Though Hilbert fully developed his program by the late 1920s, it later ran into fundamental obstacles: Kurt Gödel’s incompleteness theorems (1931) showed that any sufficiently powerful axiomatic system (like arithmetic) cannot prove its own consistency using only those finitary methods. In other words, Hilbert’s goal of a complete, self-verifiable foundation was unattainable in its original form. Even so, Hilbert’s foundational work deeply influenced modern logic and proof theory. His insistence on precise axioms and proof standards also became a hallmark of mathematics going forward.

Hilbert’s problems: One of Hilbert’s most famous contributions was his list of 23 problems, presented at the International Congress of Mathematicians in Paris in 1900. In his address “Mathematical Problems,” Hilbert surveyed many fields of mathematics and singled out 23 specific challenges he believed would be pivotal for future research. These problems touched on number theory, geometry, analysis, algebra, and more. Solving many of these questions became a major focus of 20th-century mathematics. For example, his first problem (the continuum hypothesis on the size of infinite sets) was later shown to be undecidable from standard axioms (Gödel and Cohen showed it can neither be proved nor refuted from the usual set theory axioms). His eighth problem included the famous Riemann Hypothesis, which remains unsolved and is now often seen as the single most important open problem in mathematics. Other problems (like his 23rd, which was to encourage further development of variational methods) set research directions. Overall, the list of problems cemented Hilbert’s reputation as a visionary: it inspired decades of work, and many of the problems were eventually solved in the later 20th and 21st centuries.

These examples illustrate Hilbert’s broad influence: from establishing the rules of geometry, to solving classical puzzles in algebra and number theory, to laying the groundwork for modern analysis and mathematical physics.

Method and philosophy

Hilbert’s approach to mathematics was distinctive and came to symbolize a new style of rigorous, systematic reasoning. He believed that mathematics should be developed by first stating clear axioms (basic assumptions) and then proving all results from them. This axiomatic method meant that any mathematical theory would stand on a solid logical foundation. For Hilbert, even laws of arithmetic or geometry were not just intuitive truths but rules to be checked for consistency. For example, in his geometry he went beyond Euclid by examining which axioms were independent or necessary, eventually adding a completeness axiom to ensure no additional points could be added without violating the system.

More broadly, Hilbert adopted a formalist view of mathematics. Formalism holds that mathematical statements get their meaning from the formal rules and symbols, rather than from some external reality. In this spirit, Hilbert insisted that questions of mathematical existence or truth should be decided by logical proof within an axiomatic system, not by philosophical or intuitive arguments. In practice, this meant that he saw problems like “Does this object exist?” as equivalent to “Can we prove, from the axioms, that such an object must exist?” Formal manipulation of symbols and formulas was legitimate as long as it followed clear rules. Hilbert famously contrasted his view with that of the intuitionist school (led by L.E.J. Brouwer), which argued that mathematics must always be constructive (built from explicit constructions). Hilbert responded to Brouwer: “By no means may the principle of excluded middle be abandoned from mathematics, for to us, in mathematics, there is no such thing as an experimental justification.” In other words, Hilbert held that every mathematical problem has a solution that could be reached by logic, even if it involved infinite or abstract methods, whereas Brouwer restricted math to what could be constructed step by step. Hilbert’s famous retort was that modern set theory and the abstract infinity of Cantor’s work form a “paradise” of mathematics from which he would not be expelled.

Central to Hilbert’s philosophy was the idea of proving consistency. He argued that it was not enough to assume axioms seemed reasonable: one should ideally prove they could not yield a contradiction. In his program, the aim was to use only finitary reasoning for such proofs. Finitary means concrete, finite, explicit arguments (such as counting and simple manipulations) as opposed to arguments that assume the existence of an infinite totality. For instance, if one had an axiom asserting that some set is infinite, a finitary proof would avoid appealing to the set’s actual infinite nature and would instead simulate arguments with any arbitrary large finite part. Hilbert planned to show that arithmetic and analysis are consistent by reducing them to very basic operations whose consistency is “obvious.” He even posed the task of giving such a consistency proof as his second problem in 1900.

However, Gödel’s results in 1931 showed that this grand vision could not achieve everything Hilbert hoped if one limits oneself only to those finitary methods. But by forcing these questions into precise mathematical form, Hilbert had already spurred the creation of proof theory and mathematical logic. His own perspective can be summed up by his unwavering optimism in the power of reason: late in life he proclaimed "Wir müssen wissen, wir werden wissen." This German motto, usually rendered “We must know, we will know,” expresses Hilbert’s belief that any mathematical question is, in principle, solvable by human intelligence.

Influence and reception

David Hilbert’s time at Göttingen coincided with one of the most intellectually fertile periods in mathematics. Under his leadership, the Göttingen Mathematical Institute became a world center that attracted students and visitors from across Europe and beyond. The institute’s prestige was famed: for example, three future Nobel Prize winners in physics – Max von Laue, James Franck, and Werner Heisenberg – all spent significant years at Hilbert’s Göttingen during this era, benefiting from its rich scientific environment. Many mathematicians sought to study under Hilbert or to work in Göttingen; he turned Göttingen into the mathematical capital of the world. In the words of mathematician Abraham Fraenkel, Hilbert was “correctly considered the scientific head of German mathematics” and acknowledged globally. Students came from the United States, Britain, and other countries to learn modern techniques from him and his colleagues.

Hilbert’s reputation was such that he received numerous honors. He declined offers to leave Göttingen for other universities, feeling great loyalty to the city. In 1910 he won the Bolyai Prize of the Hungarian Academy (a prestigious award in mathematics), for which Henri Poincaré himself wrote a congratulatory citation. When he retired in 1930, his native city Königsberg made him an honorary citizen, and he gave a famous farewell lecture there (“The Understanding of Nature and Logic”) closing with the motto “We must know – we will know.” In 1939 he was awarded the first Mittag-Leffler Prize of the Swedish Academy (one of the most prestigious international honors at the time), shared with the French mathematician Émile Picard.

In terms of mathematics itself, Hilbert’s influence pervades the curriculum. After Hilbert, it became standard to teach geometry from an axiomatic perspective; one can say that virtually all modern geometries have Hilbert as a forebear. His work on number theory and algebra provided tools and concepts still in use (for example, the modern view of polynomial equations and ideals stems from his basis theorem). The notion of a Hilbert space is now a basic object studied by every mathematician and physicist working in analysis or quantum theory.

Hilbert’s famous list of problems also guided the development of the field. For decades, mathematicians measured progress by its relation to these problems – some of which have only been solved in the late 20th century (and one, the Riemann Hypothesis, remains a central challenge). His style of posing big-picture problems inspired future generations of mathematicians to think systematically about open questions in their fields. In the broader culture, Hilbert became something of a symbol of mathematical rigor and optimism. His personality – genial, determined, and sometimes eccentric – and his broad array of accomplishments made him one of the most celebrated scientific figures of his day.

Critiques and debates

Hilbert’s ideas were not without controversy, especially in the addled domain of a still-new foundations of mathematics. In the 1910s and 1920s, a debate known as the “Foundational Crisis” gripped mathematicians and philosophers. On one side stood Hilbert and his formalist program; on the other were intuitionists like L.E.J. Brouwer and others who questioned the uncritical use of actual infinity and non-constructive proofs in mathematics. Brouwer objected to Hilbert’s approach, arguing that a proof of existence must be accompanied by a construction. Hilbert and his colleagues replied that formal proof from axioms was what mattered, and that mathematics could include abstract existence reliably if the underlying system was consistent. This came to a head at the 1928 International Congress of Mathematicians, where Hilbert and Brouwer famously squared off over such issues, with Hilbert defending classical logic and Brouwer advocating intuitionism.

Another critique came from Henri Poincaré in 1906. Poincaré pointed out that one of Hilbert’s early proposals for proving the consistency of arithmetic relied on the very principle of mathematical induction that needed justification, leading to a kind of circular reasoning. Hilbert took this seriously and worked to make his methods more robust, ultimately abandoning the flawed sketch and seeking better finitary arguments. In general, mathematicians recognized that Hilbert’s great drive for axiomatic rigor was only part of the story: it needed careful handling to avoid hidden pitfalls or unjustified assumptions.

A later intellectual blow came from Kurt Gödel in 1931, whose incompleteness theorems showed that Hilbert’s program as originally conceived could not be fully completed. Gödel proved that any sufficiently rich axiomatic system (such as arithmetic) is either inconsistent or cannot prove all true statements about the natural numbers; in particular it cannot prove its own consistency by purely finitary means. This resolved the foundational debate by showing a natural limit to what Hilbert’s axiomatic program could achieve. Although Gödel’s result was discouraging to Hilbert’s specific goals, it did not invalidate the axiomatic approach entirely; rather, it clarified its limitations and gave rise to new fields (proof theory and mathematical logic). Hilbert himself understood and accepted Gödel’s result, continuing to work on mathematics even as this debate settled.

Elsewhere, Hilbert became involved in a famous historical controversy in physics. In 1915 both Hilbert and Einstein were working on the equations of general relativity. Hilbert submitted a paper on November 20, 1915, and Einstein submitted his key paper five days later. For decades it was debated who derived the equations first. Modern historians, after studying the archives, found that Hilbert’s original submission did not contain the correct generally covariant equations; they appeared only in later revisions. This means Einstein independently discovered the field equations first. The episode briefly stirred tension, but today scholars generally credit Einstein with the priority on physical grounds while acknowledging Hilbert’s parallel work as historically significant.

On a practical note, some critics felt that Hilbert’s broad style was too abstract and formal for some branches of mathematics and physics. For instance, Hilbert’s own work in physics (like gas kinetic theory) was later viewed as less fruitful than his abstract mathematics. And some in algebra or number theory preferred more concrete approaches over Hilbert’s high-level algebraic methods. Nonetheless, these are more matters of taste than scandal; Hilbert’s approach became one of the two main schools (and arguably the dominant one) in modern mathematics, with intuitionism and logicism being the others that ultimately had less influence.

Legacy

David Hilbert’s legacy is profound and enduring. Many fundamental concepts and results carry his name, reflecting how central his ideas have become. Perhaps the most famous is the notion of a Hilbert space: a complete infinite-dimensional space of functions or sequences with an inner product. This concept is now the language of quantum mechanics and much of modern analysis. Other examples include the Hilbert basis theorem (that ideals in polynomial rings are finitely generated) which underlies algebraic geometry, Hilbert’s Nullstellensatz connecting algebra and equation-solving, the Hilbert matrix, Hilbert transform (in harmonic analysis), Hilbert’s girl (epsilon symbol) in logic, and many more.

Hilbert’s problems continue to influence mathematics. Each year, textbooks and researchers refer to “solving Hilbert’s #n” or working on variants of it. Some unsolved problems from his list (like parts of the continuum hypothesis and the Riemann hypothesis) are part of the contemporary agenda. Others have shaped entire fields; for example, his sixth problem (axiomatize physics) helped inspire the development of probability theory foundations and physical theories.

The axiomatic and formalist approach Hilbert championed permeates mathematics and science today. Modern algebra and topology are built on Hilbert’s ethos of abstract structures defined by axioms. The idea that a mathematical theory must be consistent and complete is now foundational (even if sometimes an ideal rather than always attainable). In particular, computer science and logic grew out of Hilbert’s program: the work of Gödel, Church, Tarski and others in logic and computation theory can be seen as heirs to the questions Hilbert asked.

Hilbert’s influence also lives on in mathematics culture. The International Mathematical Union (IMU) awards the Hilbert Medal for outstanding mathematical achievement, and his name appears on the Göttingen mathematics institute building. A famous quote attributed to him is that “we must know, we will know,” reflecting his faith in rational inquiry; it often serves as a motto for scientific determination. In the social memory of mathematics, Hilbert is routinely named among the greatest mathematicians of all time (sometimes compared to Gauss or Euler in fame and impact). His style of problem-setting and rigor set a standard that modern mathematicians still strive for.

Most of all, Hilbert’s work showed the power of clear axioms and logical reasoning. He transformed geometry into a precise science, unified threads of algebra and number theory, and helped lay the modern intellectual structure in which mathematics operates. As such, Hilbert’s legacy is the very structure of mathematics as we know it – a structured, formal system in which any result must be deduced from agreed assumptions, and open questions drive progress. His passion for knowledge and confidence in human intellect (“we will know”) remain inspirational to scientists and mathematicians even today.

Selected works

- Zahlbericht: Bericht über die Theorie der algebraischen Zahlkörper (1897) — A comprehensive report on algebraic number theory that unified many results from that field.

- Grundlagen der Geometrie (The Foundations of Geometry, 1899; English 1902) — Hilbert’s classic axiomatic treatment of Euclidean geometry, introducing his full system of axioms.

- Mathematical Problems (lecture at the International Congress of Mathematicians, Paris, 1900) — Published statement of the famous list of 23 problems guiding 20th-century research.

- Vorlesungen über Integralgleichungen (Lectures on Integral Equations, 1912) — A text on Hilbert’s work in analysis and integral equations, which influenced the development of functional analysis.

- Die Grundlagen der Physik (Foundations of Physics, vols. 1–2, 1915–1917) — Two volumes on the mathematical principles of physics, including Einstein’s theory of gravitation.

- Grundlagen der Mathematik (Foundations of Mathematics, vols. I–II, 1928–1930, with Paul Bernays) — A two-volume work setting out Hilbert’s formalist program and logic of mathematics.

Timeline

1862: Born January 23 in Königsberg, Prussia.

1880: Entered the University of Königsberg to study mathematics.

1885: Earned Ph.D. with a thesis on invariant theory.

1892: Completed habilitation in Königsberg and became associate professor; married Käthe Jerosch.

1895: Appointed Professor at the University of Göttingen.

1897: Published Zahlbericht on algebraic number theory.

1899: Published Grundlagen der Geometrie (Foundations of Geometry).

1900: Presented his 23 famous mathematical problems at the Paris Congress.

1909: Published major work on integral equations; introduced concepts of functional analysis (Hilbert spaces).

1910: Awarded the Bolyai Prize (Hungarian Academy of Sciences).

1915–1916: Published Foundations of Physics (including relativity theory work).

1928–1930: Published Grundlagen der Mathematik (with Bernays), advancing axiomatic logic.

1930: Retired from teaching; gave his famous lecture “The Understanding of Nature and Logic” in Königsberg.

1939: Awarded the first Mittag-Leffler Prize.

1943: Died February 14 in Göttingen, Germany.

Despite his passing in 1943, David Hilbert’s work continues to live on in the foundations and vocabulary of modern mathematics. His clear insistence on axioms and proof, his belief in the solvability of problems, and his many concrete contributions ensure that his legacy remains central to the mathematical sciences.