David Hilbert

How to use this course

David Hilbert worked in one field at a time, mastered it, reshaped it, and moved on. His career falls into clean phases: algebraic forms until about 1893, algebraic number theory until 1899, the foundations of geometry, analysis and integral equations from about 1904 to 1912, physics until the end of the First World War, and the foundations of mathematics from 1917 onward. Few mathematicians have left a lasting mark on so many subjects. This course follows those phases, with one technical lesson for each.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, proof sketches and exercises with folded solutions. The lessons can be read independently. Lesson 09 on the Paris problems reaches into every other area and works well as a capstone.

Timeline

Königsberg, 1862–1885

David Hilbert was born on 23 January 1862 into a Protestant family of lawyers and officials in East Prussia. His own account placed his birth in Königsberg, the provincial capital; some records name Wehlau, a small town nearby where his father was working at the time. His father, Otto Hilbert, became a judge in Königsberg. His mother, Maria Therese Erdtmann, was the daughter of a Königsberg merchant.

Königsberg was the city of Immanuel Kant, and of the seven bridges whose puzzle Leonhard Euler solved. It also had a strong mathematical tradition: Carl Gustav Jacobi had taught there, and the university remained a serious centre. Hilbert entered the Friedrichskolleg, Kant’s old school, in 1872. Its emphasis on classical languages and memorization did not suit him. In 1879 he moved to the Wilhelm Gymnasium, which taught more mathematics, and he graduated in 1880 with a report noting his strong mathematical ability.

University

In autumn 1880 he enrolled at the University of Königsberg. Apart from a semester at Heidelberg, he studied at home. His teachers included Heinrich Weber and, from 1883, Ferdinand von Lindemann, who had recently proved that \(\pi\) is transcendental. Lindemann suggested a thesis topic in invariant theory, the most fashionable algebra of the day.

Hilbert received his doctorate in 1885 for the dissertation Über invariante Eigenschaften spezieller binärer Formen, insbesondere der Kugelfunktionen, and passed the state teaching examination the same year. He then travelled to Leipzig to attend Felix Klein’s seminar and, on Klein’s advice, to Paris, where he met Charles Hermite and Henri Poincaré. In June 1886 he completed his habilitation at Königsberg and became a Privatdozent.

Three friends and Gordan’s problem, 1882–1893

Two friendships shaped Hilbert more than any course. In 1882 he met Hermann Minkowski, two years younger, who in 1883, aged 18, shared the Paris Academy’s Grand Prix for work on quadratic forms. In 1884 Adolf Hurwitz, only three years older than Hilbert but already accomplished, arrived as extraordinary professor. The three met almost daily and walked together through the town, discussing mathematics. Hilbert later said that in these conversations he learned more than from any book. Minkowski remained, in Hilbert’s words, his best and truest friend until his death in 1909.

Gordan’s problem

As a young lecturer Hilbert worked in invariant theory. Its central question, called Gordan’s problem, asked whether the invariants of binary forms of every degree are generated by finitely many of them. Paul Gordan at Erlangen had proved this in 1868 by explicit and laborious computation. Extending the result to forms in more variables seemed hopeless by his methods.

In 1888, after a tour of German universities on which he met Gordan, Hilbert found a new proof of finiteness for invariants of binary forms. Over the next two years he generalized it to forms in any number of variables. His approach was existential: he proved that a finite generating set exists without saying how to find it. The key ingredient was his basis theorem, published in Mathematische Annalen in 1890. Gordan’s reaction became famous: Das ist nicht Mathematik. Das ist Theologie. Klein backed Hilbert, and the paper appeared unchanged. In 1893 Hilbert gave a second paper with more constructive methods, including the Nullstellensatz. Lesson 04 presents this work.

Promotion and marriage

In 1892 Hurwitz left for the ETH in Zürich and Hilbert succeeded him as extraordinary professor. The same year he married Käthe Jerosch, a Königsberg merchant’s daughter who shared his interests and later copied out many of his lecture notes. Their only child, Franz, was born on 11 August 1893. In 1893 Lindemann left for Munich, and Hilbert became full professor in his place.

The call to Göttingen, 1893–1900

In 1893 the recently founded German Mathematical Society (DMV) asked Hilbert and Minkowski to prepare a report on the state of number theory. Minkowski was to cover the more elementary parts, and Hilbert algebraic number theory. Minkowski eventually withdrew, and the whole task fell to Hilbert.

Felix Klein, rebuilding Göttingen as a mathematical centre, wanted Hilbert. When Heinrich Weber left Göttingen in 1895, Klein secured the chair for him over some opposition from colleagues who feared a difficult young man. Hilbert arrived in 1895 and remained in Göttingen for the rest of his life.

The next five years produced three landmarks. The Zahlbericht, dated 10 April 1897 and published in the DMV’s Jahresbericht, organized all of algebraic number theory into a single theory (Lesson 05). The Grundlagen der Geometrie of 1899, written for a Göttingen celebration of Carl Friedrich Gauss and Wilhelm Weber, made geometry an axiomatic system of a new kind (Lesson 06). And on 8 August 1900, at the Second International Congress of Mathematicians at the Sorbonne in Paris, Hilbert presented a list of problems for the new century (Lesson 09). By 1900 he was, with Poincaré, one of the two leading mathematicians in the world.

Invariant theory and the basis theorem

For the general reader. A quantity is invariant if it does not change when you change coordinates. The discriminant of a quadratic is an example. In the nineteenth century mathematicians tried to list all invariants of a given kind. Hilbert proved that a finite list always exists, without writing it down, using a short argument about ideals that became one of the foundations of modern algebra.

The basis theorem

Proof sketch. Let \(I\subseteq R[x]\) be an ideal and suppose it is not finitely generated. Choose \(f_1\in I\) of least degree, then inductively \(f_{k+1}\in I\setminus(f_1,\dots,f_k)\) of least degree. Let \(a_k\) be the leading coefficient of \(f_k\) and \(d_k=\deg f_k\), so \(d_1\le d_2\le\cdots\). The ideal \((a_1,a_2,\dots)\subseteq R\) is finitely generated, so some \(a_{N+1}=\sum_{i\le N}r_ia_i\). Then

lies in \(I\setminus(f_1,\dots,f_N)\) and has degree less than \(d_{N+1}\), contradicting the choice of \(f_{N+1}\). Induction on \(n\) gives the statement for \(k[x_1,\dots,x_n]\). Emmy Noether later took the conclusion of this theorem as an axiom; rings in which every ideal is finitely generated are now called Noetherian.

Finiteness of invariants

Let a group \(G\) act linearly on \(V=k^n\), with \(k\) of characteristic 0, and hence on \(S=k[x_1,\dots,x_n]\). Hilbert treated the special linear group acting on forms. What the argument needs is a Reynolds operator: a \(k\)-linear projection \(\rho:S\to S^G\) that preserves degree and satisfies \(\rho(af)=a\,\rho(f)\) for invariant \(a\). For a finite group, \(\rho\) is averaging over \(G\). For \(\mathrm{SL}_n\), Hilbert built such an operator from Cayley’s \(\Omega\)-process; today one averages over a compact real form using the Haar measure.

Proof. Let \(I\subseteq S\) be the ideal generated by all homogeneous invariants of positive degree. By the basis theorem, \(I=(f_1,\dots,f_r)\) for finitely many such invariants. We show by induction on degree that every homogeneous invariant \(f\) of positive degree lies in \(k[f_1,\dots,f_r]\). Since \(f\in I\), write \(f=\sum a_if_i\) with \(a_i\) homogeneous of degree \(\deg f-\deg f_i<\deg f\). Apply \(\rho\):

Each \(\rho(a_i)\) is an invariant of lower degree, so by induction it is a polynomial in the \(f_j\). Hence so is \(f\). The proof gives no bound and no algorithm, which is what offended Gordan.

The 1893 paper

In Über die vollen Invariantensysteme Hilbert answered the constructive objection. Its tools have become standard in algebraic geometry and commutative algebra.

The degree of the Hilbert polynomial of \(S/I\) is the dimension of the projective variety cut out by \(I\), and its leading coefficient, times \((\deg P)!\), is the degree of the variety. In 1893 Hilbert also showed that the invariants \(f_1,\dots,f_r\) can be replaced by algebraically independent invariants over which the whole invariant ring is integral, a forerunner of Noether normalization.

Number fields: the Zahlbericht

For the general reader. Number theorists of the nineteenth century, Ernst Kummer, Leopold Kronecker and Richard Dedekind among them, had extended arithmetic to new systems of numbers, each in their own way. Hilbert’s report of 1897 reorganized the whole subject into one theory. It then pointed beyond itself to a conjectured class field theory that occupied number theory for the next half century.

The setting

A number field \(K\) is a finite extension of \(\mathbb Q\); its ring of integers \(\mathcal O_K\) is the set of elements satisfying a monic polynomial with integer coefficients. Following Dedekind, every non-zero ideal of \(\mathcal O_K\) factors uniquely into prime ideals. The ideal class group \(\mathrm{Cl}(K)\), fractional ideals modulo principal ones, is finite; its order \(h_K\) measures the failure of unique factorization of elements. The Zahlbericht has five parts: general number fields, Galois fields, quadratic fields, cyclotomic fields, and Kummer fields, ending with Kummer’s proof of Fermat’s Last Theorem for regular primes.

Splitting of primes

Let \(L/K\) be a Galois extension of degree \(n\) with group \(G\), and \(\mathfrak p\) a prime of \(K\). Then

where \(e\) is the ramification index and \(f\) the residue degree. Hilbert introduced the decomposition group \(D_{\mathfrak P}=\{\sigma:\sigma\mathfrak P=\mathfrak P\}\), of order \(ef\), and the inertia group \(I_{\mathfrak P}\subseteq D_{\mathfrak P}\), of order \(e\), with \(D_{\mathfrak P}/I_{\mathfrak P}\) cyclic of order \(f\). The generator of this quotient that induces \(x\mapsto x^{N\mathfrak p}\) on residue fields is the Frobenius element. He also defined the higher ramification groups that measure wild ramification.

Theorem 90

Proof. For \(\theta\in L\) set

The automorphisms \(1,\sigma,\dots,\sigma^{n-1}\) are linearly independent over \(L\) (Dedekind’s lemma), so some \(\theta\) gives \(\beta\neq0\). Applying \(\sigma\) and multiplying by \(\alpha\) shifts each term to the next; the last term becomes \(N_{L/K}(\alpha)\,\sigma^n(\theta)=\theta\). Hence \(\alpha\,\sigma(\beta)=\beta\). In cohomological language, \(H^1(G,L^\times)=0\) for cyclic \(G\), and Emmy Noether extended this to all finite Galois extensions.

The Hilbert symbol and reciprocity

For \(a,b\in K^\times\) and a place \(v\) of \(K\), with completion \(K_v\), define \((a,b)_v=1\) if \(ax^2+by^2=z^2\) has a solution in \(K_v\) other than \((0,0,0)\), and \((a,b)_v=-1\) otherwise. Hilbert reformulated quadratic reciprocity as a product formula over all places, the archimedean ones included.

Over \(\mathbb Q\), the product formula for odd primes \(a=p,\ b=q\) is equivalent to Gauss’s law of quadratic reciprocity, with its two supplementary laws coming from \(v=2\) and \(v=\infty\).

The class field

In papers of 1898–1902 Hilbert conjectured the existence of an extension that now bears his name. Philipp Furtwängler proved existence in 1907; Emil Artin’s reciprocity law of 1927 gave the isomorphism; Furtwängler proved the principal ideal theorem in 1930.

Foundations of geometry

For the general reader. For two thousand years Euclid’s Elements was the model of rigour, yet its proofs quietly used facts that were never stated, such as which side of a line a point lies on. Hilbert rebuilt geometry from a complete list of axioms, and then asked new questions about the axioms themselves: are they consistent, and is any of them a consequence of the others?

The axioms

Hilbert starts from three systems of objects, points, lines and planes, and three undefined relations: incidence, betweenness, and congruence. Nothing is assumed about what the objects are. Hilbert is reported to have said that one must be able to say tables, chairs and beer mugs in place of points, lines and planes. The axioms fall into five groups:

1. Incidence (8 axioms): two points determine a line, three non-collinear points a plane, and so on.
2. Order (4 axioms), including Pasch’s axiom: a line in the plane of a triangle that enters through one side, and avoids the vertices, leaves through another side.
3. Congruence (5 axioms) for segments and angles, ending with side-angle-side.
4. Parallels: through a point not on a line there is at most one parallel.
5. Continuity: the Archimedean axiom and, from the second edition onward, an axiom of completeness for lines.

Consistency and independence

Hilbert proved consistency by building a model inside arithmetic. Let \(\Omega\) be the smallest field of real numbers that contains 1 and is closed under the operation \(\omega\mapsto\sqrt{1+\omega^2}\). Points are triples in \(\Omega^3\), lines and planes are given by linear equations, and congruence is defined by Euclidean distance. All the axioms except completeness hold, and \(\Omega\) is countable. So a contradiction in the axioms would yield a contradiction in the arithmetic of \(\Omega\).

Independence of the parallel axiom follows from a model of hyperbolic geometry built in the same way. Independence of the Archimedean axiom uses analytic geometry over a non-Archimedean ordered field, such as a field of algebraic functions of a parameter \(t\) ordered so that \(t\) is a positive infinitesimal.

Segment arithmetic and the theorems of Desargues and Pascal

Without the congruence axioms, Hilbert defined addition and multiplication of segments on a line by constructions with parallels. Which laws of algebra this arithmetic obeys depends on which configuration theorems hold.

Hilbert also constructed a non-Archimedean ordered division ring that is not commutative, giving a Desarguesian geometry in which Pascal’s theorem fails. These results founded the study of projective planes by their coordinate algebras. The book went through many editions; the seventh appeared in 1930.

Integral equations and Hilbert space

For the general reader. A vibrating string has a list of natural frequencies, and any motion is a sum of its basic modes. Hilbert found the general mathematics behind such lists. He replaced functions by infinite sequences of coordinates and built a geometry with infinitely many dimensions. Twenty years later this space became the setting of quantum mechanics.

From Fredholm to Hilbert

In 1900–1903 Ivar Fredholm solved the integral equation \(\varphi(s)-\lambda\int_a^bK(s,t)\varphi(t)\,dt=f(s)\) by treating it as the limit of a linear system of \(n\) equations as \(n\to\infty\). Hilbert heard of the work in 1901 and took up the subject in six communications to the Göttingen Society (1904–1910), collected in 1912 as Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen. He concentrated on symmetric kernels, \(K(s,t)=K(t,s)\), and on the eigenvalue problem \(\varphi=\lambda K\varphi\), where \((K\varphi)(s)=\int_a^b K(s,t)\varphi(t)\,dt\).

Infinitely many variables

Hilbert’s decisive step, in the fourth communication (1906), was to fix an orthonormal basis \(\{\omega_p\}\) and replace a function \(f\) by its coordinates \(x_p=(f,\omega_p)\). By Bessel’s inequality \(\sum x_p^2<\infty\). The integral equation becomes a problem about a quadratic form \(\sum k_{pq}x_px_q\) in infinitely many variables, on the space of sequences with \(\sum x_p^2<\infty\), now called \(\ell^2\). He called a form completely continuous when it is continuous for weak convergence, and for these he proved that the form is a sum of squares, exactly as for a symmetric matrix. For general bounded forms he found that a continuous spectrum, a Streckenspektrum, may appear. The word spectrum in this sense is his. In modern terms:

Proof idea. The largest eigenvalue comes from a variational problem. Maximize \(|\langle Ax,x\rangle|\) over \(\|x\|=1\). Compactness lets a maximizing sequence converge (after passing to a subsequence and using weak compactness of the ball), and the maximizer is an eigenvector with eigenvalue \(\pm\|A\|\). Restrict to the orthogonal complement and repeat. Each \(\mu_n\) is the reciprocal of Hilbert’s \(\lambda_n\).

After Hilbert

Erhard Schmidt (1907) gave the geometric language of orthogonality and projection. Frigyes Riesz and Ernst Fischer showed in 1907 that \(L^2[a,b]\) is complete and isometric to \(\ell^2\). In 1927–1929 John von Neumann, then Hilbert’s assistant, defined abstract Hilbert space by axioms and proved the spectral theorem for unbounded self-adjoint operators, giving quantum mechanics its mathematical form. The 1928 paper of Hilbert, von Neumann and Lothar Nordheim on the foundations of quantum mechanics marks the transition.

Variations, Dirichlet, and gravitation

For the general reader. Many laws of nature say that something is as small as possible: light takes the quickest path, a soap film the least area. Bernhard Riemann used such a principle to prove existence theorems, but Karl Weierstrass showed that a minimum need not exist. Hilbert repaired the method. Later he applied the same variational idea to gravity and, in the same weeks as Einstein, wrote down the action from which the field equations of general relativity follow.

The Dirichlet principle

To solve \(\Delta u=0\) in a domain \(\Omega\) with \(u=g\) on the boundary, Riemann minimized the Dirichlet integral \(D[u]=\int_\Omega|\nabla u|^2\) among functions with the given boundary values. He assumed a minimizer exists because \(D\ge0\). In 1870 Weierstrass showed that a functional bounded below need not attain its infimum (Exercise 8.1). Hilbert, in 1900 and 1904, showed how to take a minimizing sequence and extract a convergent subsequence whose limit is the minimizer. This is the direct method of the calculus of variations. In modern form:

The proof combines weak compactness of bounded sets in the Hilbert space \(H^1_0\) with lower semicontinuity of \(D\). Weyl’s lemma then upgrades the weak solution to a smooth harmonic function. Hilbert placed two related questions, problems 19 and 20, on his Paris list, and problem 23 asked for the further development of the calculus of variations. There he presented his invariant integral, which gives a sufficient condition for a minimum.

The Einstein–Hilbert action

From about 1912 Hilbert worked on physics. In June–July 1915 Einstein lectured on his unfinished theory of gravitation at Göttingen. Hilbert then sought a unified theory of gravitation and electromagnetism, based on the electrodynamics of Gustav Mie, deriving everything from one variational principle. On 20 November 1915 he presented Die Grundlagen der Physik to the Göttingen Society. His action was the integral of \(R\sqrt{-g}\) plus a matter term. In modern notation, with signature \((-,+,+,+)\) and \(\kappa=8\pi G/c^4\),

Derivation. Vary the inverse metric \(g^{\mu\nu}\). Three facts are needed:

The last term is a divergence and integrates to zero for variations of compact support. Defining \(T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\,\mathcal L_M)}{\delta g^{\mu\nu}}\), we get

Because the action is invariant under coordinate changes, the left side satisfies \(\nabla^\mu G_{\mu\nu}\equiv0\) identically, which forces \(\nabla^\mu T_{\mu\nu}=0\). Hilbert noticed that energy conservation in this theory behaves unlike in earlier physics, and he and Klein asked Emmy Noether to explain why. Her 1918 theorems were the answer.

The priority question

Einstein presented the final field equations to the Prussian Academy on 25 November 1915, five days after Hilbert’s lecture. Hilbert’s paper was printed only in March 1916, in a revised form. In 1997 Leo Corry, Jürgen Renn and John Stachel published Hilbert’s surviving printer’s proofs, dated 6 December 1915. These proofs do not contain the explicit field equations, and Hilbert’s original theory was not generally covariant. Part of one page has been cut out, and some authors have argued about what it held. The published version credited Einstein with the essential ideas, and the two men did not dispute priority in public. Today the field equations carry Einstein’s name, and the action carries both.

The 23 problems

For the general reader. In August 1900 Hilbert addressed the International Congress in Paris. Instead of surveying what had been done, he described what should be done next. His list of problems, ranging from the foundations of mathematics to physics, guided research for decades. Solving one became a mark of the highest achievement.

Because of time, Hilbert presented only ten problems orally (numbers 1, 2, 6, 7, 8, 13, 16, 19, 21 and 22). The full text appeared in the Göttinger Nachrichten in 1900 and in the Archiv der Mathematik und Physik in 1901; Mary Winston Newson’s English translation appeared in the Bulletin of the American Mathematical Society in 1902. Hilbert opened with the conviction that every mathematical problem can be solved, a theme he returned to in 1930.

Problem 3 and the Dehn invariant

In the plane, two polygons of equal area can be cut into finitely many pieces and reassembled into each other (Bolyai–Gerwien). Hilbert asked whether this holds for polyhedra. Dehn, Hilbert’s student, answered within months. For a polyhedron \(P\) with edges \(e\) of length \(\ell(e)\) and dihedral angles \(\theta(e)\), define

Cutting a piece along a plane creates new edges, but their contributions cancel: an edge cut in two adds \(\ell_1\otimes\theta+\ell_2\otimes\theta=(\ell_1+\ell_2)\otimes\theta\), and a new edge interior to a face or to the solid contributes dihedral angles summing to \(\pi\) or \(2\pi\), which are zero modulo \(\pi\). Jean-Pierre Sydler proved the converse in 1965: equal volume and equal Dehn invariant imply scissors congruence.

Problems 7, 10 and 17

Hilbert himself had shown in 1888 that rational functions are necessary: there are non-negative polynomials that are not sums of squares of polynomials. Artin’s proof used the theory of real closed fields he had developed with Otto Schreier.

Proof theory and the formalist programme

For the general reader. Around 1900 paradoxes appeared in set theory, and some mathematicians, notably L. E. J. Brouwer and for a time Hermann Weyl, proposed to give up parts of classical mathematics. Hilbert refused. He proposed to turn proofs themselves into finite objects that could be studied mathematically, and to prove by safe, finite reasoning that no contradiction can ever arise. Kurt Gödel showed that this cannot be done as planned. What survived is proof theory, a central branch of logic.

The programme

Hilbert first addressed foundations at the Heidelberg congress of 1904, then returned to them with his 1917 Zürich lecture Axiomatisches Denken and with Paul Bernays as collaborator. The programme took shape in lectures of 1922 in Hamburg and in Über das Unendliche, given in Münster in 1925. Its outline:

1. Formalize mathematics: each theory becomes a formal system whose proofs are finite strings of symbols.
2. Distinguish real statements, finitely checkable claims about concrete symbols, from ideal statements involving completed infinities. Ideal elements are justified, as \(i=\sqrt{-1}\) is in algebra, if they lead to no false real statements.
3. Prove consistency, that no formal proof ends in \(0\neq0\), by finitary reasoning about proofs alone, a metamathematics.

In 1925 Hilbert put the aim memorably: no one would drive mathematicians out of the paradise that Georg Cantor had created.

The epsilon calculus

Hilbert’s formal tool was a term-forming operator. For each formula \(A(x)\), \(\varepsilon_xA(x)\) is a term, read as some object satisfying \(A\) if any does. The only new axiom is the critical formula

Quantifiers become definable: \(\exists x\,A(x)\) is \(A(\varepsilon_xA(x))\), and \(\forall x\,A(x)\) is \(A(\varepsilon_x\neg A(x))\). A consistency proof then needs to show that the critical formulas used in any proof can be satisfied by a finite assignment of values to epsilon terms. Wilhelm Ackermann’s dissertation (1924–25) and work of von Neumann (1927) carried out this substitution method for weak systems, and for a time Hilbert believed the consistency of arithmetic had been established.

Completeness, decidability and Gödel

In the Hilbert–Ackermann textbook of 1928 and at the Bologna congress the same year, Hilbert posed two sharp questions: is first-order logic complete, and is there an algorithm to decide whether a formula is valid (the Entscheidungsproblem)? Gödel proved completeness in 1929. Alonzo Church and Alan Turing answered the second question negatively in 1936. Gödel announced the decisive result at a conference in Königsberg on 7 September 1930, the day before Hilbert’s address in the same city.

The first full proof of the second theorem, with the derivability conditions that bear their names, appeared in the second volume of Hilbert and Bernays’ Grundlagen der Mathematik (1939).

Gentzen’s proof is finitary except for one principle, induction along a specific, concretely describable well-ordering. Ordinal analysis, which assigns such an ordinal to each theory, is the direct descendant of Hilbert’s programme. Other descendants include reverse mathematics, which measures exactly which axioms a theorem needs, and proof mining, which extracts explicit bounds from non-constructive proofs.

The Göttingen school, 1900–1930

After Paris, Göttingen became the place where mathematics students from everywhere wanted to study. Klein supplied the organization and Hilbert the ideas. In 1902 Klein and Hilbert persuaded the Prussian ministry to create a third chair for Minkowski. The two friends ran joint seminars, from 1905 on the electrodynamics of moving bodies, and Minkowski’s 1908 lecture on space and time came out of this work. His death from appendicitis in January 1909, at 44, was the hardest loss of Hilbert’s life. Hilbert dedicated his proof of Waring’s conjecture, that every positive integer is a sum of a bounded number of \(k\)-th powers, to Minkowski’s memory.

Hilbert supervised 69 doctoral dissertations. Among the students were Otto Blumenthal, Max Dehn, Felix Bernstein, Erhard Schmidt, Ernst Hellinger, Hermann Weyl, Alfréd Haar, Erich Hecke, Richard Courant, Hugo Steinhaus, Wilhelm Ackermann and Haskell Curry. His lectures were famous for working through ideas in front of the audience. He walked with students in the countryside and around his garden, where a long blackboard hung for discussions. He edited Mathematische Annalen from 1902 to 1939.

War and its aftermath

Hilbert did not sign the nationalist Manifesto of the Ninety-Three of October 1914, which many German scholars did sign, though he endorsed a later declaration of German university teachers. In 1915 he and Klein brought Emmy Noether to Göttingen, and when the faculty refused her habilitation he defended her with the famous remark that a faculty is not a bathhouse. After the war he worked to restore international cooperation. When German mathematicians were readmitted to the International Congress in Bologna in 1928, some German nationalists called for a boycott. Hilbert led a large German delegation and spoke for the unity of mathematics.

In 1925 he fell ill with pernicious anaemia, then usually fatal. A liver-based treatment developed in the United States saved him, but he never regained his full working strength. Bernays and Ackermann did much of the detailed work of the foundations programme. In 1929 the new Mathematical Institute building, funded by the Rockefeller Foundation and organized by Courant, opened in Göttingen.

Retirement, 1933, death and legacy

Hilbert retired in 1930, and Hermann Weyl succeeded to his chair. That September the Society of German Natural Scientists and Physicians met in Königsberg, and his home city made him an honorary citizen, an honour he valued above all others. On 8 September 1930 he gave the address Naturerkennen und Logik, and read its ending for a radio broadcast. He rejected the old slogan that there are things we will never know, and closed with the words now on his grave: Wir müssen wissen. Wir werden wissen.

In 1933 the Nazi civil-service laws destroyed the Göttingen institute within months. Noether, Courant, Bernays and Landau lost their posts, and Weyl, whose wife was of Jewish descent, left for Princeton. According to a well-known anecdote, about a year later the minister of education Bernhard Rust asked Hilbert whether mathematics in Göttingen had suffered from the departure of the Jews. Hilbert answered that it no longer existed. The two volumes of Grundlagen der Mathematik with Bernays appeared in 1934 and 1939.

Hilbert died in Göttingen on 14 February 1943, aged 81. In the middle of the war, fewer than a dozen people attended his funeral, among them Arnold Sommerfeld, and news of his death reached many colleagues abroad only months later. He is buried in the Stadtfriedhof in Göttingen.

Legacy

• Algebra and geometry: the basis theorem, the Nullstellensatz, syzygies and the Hilbert polynomial are the foundation of commutative algebra and algebraic geometry. The Hilbert scheme parametrizes subschemes by their Hilbert polynomials.
• Number theory: the Zahlbericht and the class field conjectures framed algebraic number theory up to Artin and Hasse. Theorem 90 and the Hilbert symbol remain basic tools.
• Analysis and physics: Hilbert space and spectral theory are the language of quantum mechanics and much of modern analysis.
• Foundations: the axiomatic method, metamathematics, proof theory and the Entscheidungsproblem, which led Turing to the theory of computation.
• Honours: the Lobachevsky Prize (1903), the Bolyai Prize (1910), International Member of the US National Academy of Sciences (1907), Foreign Member of the Royal Society (1928). A lunar crater and an asteroid bear his name.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, David Hilbert, MacTutor History of Mathematics; and David Hilbert’s doctoral students.
2. H. Freudenthal, Hilbert, David, Dictionary of Scientific Biography.
3. Hilbert, David, Neue Deutsche Biographie.
4. David Hilbert, Wikipedia; German edition.
5. H. Weyl, David Hilbert and his mathematical work, Bull. Amer. Math. Soc. 50 (1944), 612–654.
6. C. Reid, Hilbert, Springer, 1970.
7. J. T. Smith, David Hilbert’s radio address, Convergence, Mathematical Association of America.

Mathematics

1. D. Hilbert, Über die Theorie der algebraischen Formen, Math. Ann. 36 (1890); Über die vollen Invariantensysteme, Math. Ann. 42 (1893).
2. D. Hilbert, Die Theorie der algebraischen Zahlkörper, Jahresber. DMV 4 (1897). English: The Theory of Algebraic Number Fields, Springer, 1998.
3. D. Hilbert, Grundlagen der Geometrie, 1899. English translation by E. J. Townsend, Project Gutenberg.
4. D. Hilbert, Mathematical problems, trans. M. W. Newson, Bull. Amer. Math. Soc. 8 (1902).
5. D. Hilbert, Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen, Teubner, 1912.
6. D. Hilbert, Die Grundlagen der Physik (Erste Mitteilung), Nachr. Ges. Wiss. Göttingen (1915, published 1916).
7. L. Corry, J. Renn, J. Stachel, Belated decision in the Hilbert–Einstein priority dispute, Science 278 (1997); see also General relativity priority dispute.
8. D. Hilbert, Über das Unendliche, Math. Ann. 95 (1926).
9. D. Hilbert and W. Ackermann, Grundzüge der theoretischen Logik, Springer, 1928; D. Hilbert and P. Bernays, Grundlagen der Mathematik, 2 vols., Springer, 1934, 1939.
10. Hilbert’s problems, Wikipedia (status table with references); B. H. Yandell, The Honors Class, A K Peters, 2002.
11. R. Zach, Hilbert’s Program, Stanford Encyclopedia of Philosophy.