David Hilbert

Contributions to mathematics and physics

Solving Gordan's Problem

Hilbert's first work on invariant functions led him to the demonstration in 1888 of his famous finiteness theorem. Twenty years earlier, Paul Gordan had demonstrated the theorem of the finiteness of generators for binary forms using a complex computational approach. Attempts to generalize his method to functions with more than two variables failed because of the enormous difficulty of the calculations involved. To solve what had become known in some circles as Gordan's Problem, Hilbert realized that it was necessary to take a completely different path. As a result, he demonstrated Hilbert's basis theorem, showing the existence of a finite set of generators, for the invariants of quantics in any number of variables, but in an abstract form. That is, while demonstrating the existence of such a set, it was not a constructive proof—it did not display "an object"—but rather, it was an existence proof and relied on use of the law of excluded middle in an infinite extension.

Hilbert sent his results to the Mathematische Annalen. Gordan, the house expert on the theory of invariants for the Mathematische Annalen, could not appreciate the revolutionary nature of Hilbert's theorem and rejected the article, criticizing the exposition because it was insufficiently comprehensive. His comment was:

Das ist nicht Mathematik. Das ist Theologie.

Translation:

This is not Mathematics. This is Theology.

Klein, on the other hand, recognized the importance of the work, and guaranteed that it would be published without any alterations. Encouraged by Klein, Hilbert extended his method in a second article, providing estimations on the maximum degree of the minimum set of generators, and he sent it once more to the Annalen. After having read the manuscript, Klein wrote to him, saying:

Without doubt this is the most important work on general algebra that the Annalen has ever published.

Later, after the usefulness of Hilbert's method was universally recognized, Gordan himself would say:

I have convinced myself that even theology has its merits.

For all his successes, the nature of his proof created more trouble than Hilbert could have imagined. Although Kronecker had conceded, Hilbert would later respond to others' similar criticisms that "many different constructions are subsumed under one fundamental idea"—in other words (to quote Reid): "Through a proof of existence, Hilbert had been able to obtain a construction"; "the proof" (i.e. the symbols on the page) was "the object". Not all were convinced. While Kronecker would die soon afterwards, his constructivist philosophy would continue with the young Brouwer and his developing intuitionist "school", much to Hilbert's torment in his later years. Indeed, Hilbert would lose his "gifted pupil" Weyl to intuitionism—"Hilbert was disturbed by his former student's fascination with the ideas of Brouwer, which aroused in Hilbert the memory of Kronecker". Brouwer the intuitionist in particular opposed the use of the Law of Excluded Middle over infinite sets (as Hilbert had used it). Hilbert responded:

Taking the Principle of the Excluded Middle from the mathematician ... is the same as ... prohibiting the boxer the use of his fists.

Nullstellensatz

In the subject of algebra, a field is called algebraically closed if and only if every polynomial over it has a root in it. Under this condition, Hilbert gave a criterion for when a collection of polynomials ${\displaystyle (p_{\lambda }{)}_{\lambda \in \mathrm{\Lambda }}}$ of ${\displaystyle n}$ variables has a common root: This is the case if and only if there do not exist polynomials ${\displaystyle q_1,\dots ,q_k}$ and indices ${\displaystyle {\lambda }_1,\dots ,{\lambda }_k}$ such that

${\displaystyle 1=\sum _{j=1}^k{p}_{{\lambda }_j}(\overrightarrow{x})q_j(\overrightarrow{x})}$.

This result is known as the Hilbert root theorem, or "Hilberts Nullstellensatz" in German. He also proved that the correspondence between vanishing ideals and their vanishing sets is bijective between affine varieties and radical ideals in ${\displaystyle \mathbb{C}[x_1,\dots ,x_n]}$.

Curve

In March 1890, Giuseppe Peano had published an article in the Mathematische Annalen describing the historically first space-filling curve. In response, Hilbert designed his own construction of such a curve, which is now called the Hilbert curve. Approximations to this curve are constructed iteratively according to the replacement rules in the first picture of this section. The curve itself is then the pointwise limit.

Axiomatization of geometry

The text Grundlagen der Geometrie (tr.: Foundations of Geometry) published by Hilbert in 1899 proposes a formal set, called Hilbert's axioms, substituting for the traditional axioms of Euclid. They avoid weaknesses identified in those of Euclid, whose works at the time were still used textbook-fashion. It is difficult to specify the axioms used by Hilbert 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. New editions followed the 7th, but the main text was essentially not revised.

Hilbert's approach signaled the shift to the modern axiomatic method. In this, Hilbert was anticipated by Moritz Pasch's work from 1882. Axioms are not taken as self-evident truths. Geometry may treat things, about which we have powerful intuitions, but it is not necessary to assign any explicit meaning to the undefined concepts. The elements, such as point, line, plane, and others, could be substituted, as Hilbert is reported to have said to Schoenflies and Kötter, by tables, chairs, glasses of beer and other such objects. It is their defined relationships that are discussed.

Hilbert first enumerates 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. The axioms unify both the plane geometry and solid geometry of Euclid in a single system.

23 problems

Hilbert put forth a list of 23 unsolved problems at the International Congress of Mathematicians in Paris in 1900. This list, including the Riemann hypothesis, is generally reckoned as the most successful and deeply considered compilation of open problems ever to be produced by an individual mathematician.

After reworking the foundations of classical geometry, Hilbert could have extrapolated to the rest of mathematics. His approach differed from the later "foundationalist" Russell–Whitehead or "encyclopedist" Nicolas Bourbaki, and from his contemporary Giuseppe Peano. The mathematical community as a whole could engage in problems of which he had identified as crucial aspects of important areas of mathematics.

The problem set was launched as a talk, "The Problems of Mathematics", presented during the course of the Second International Congress of Mathematicians, held in Paris. The introduction of the speech that Hilbert gave said:

Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries ? What particular goals will there be toward which the leading mathematical spirits of coming generations will strive ? What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose?

He presented fewer than half the problems at the Congress, which were published in the acts of the Congress. In a subsequent publication, he extended the panorama, and arrived at the formulation of the now-canonical 23 Problems of Hilbert (see also Hilbert's twenty-fourth problem). The full text is important, since the exegesis of the questions still can be a matter of debate when it is asked how many have been solved.

Some of these were solved within a short time. Others have been discussed throughout the 20th century, with a few now taken to be unsuitably open-ended to come to closure. Some continue to remain challenges.

The following are the headers for Hilbert's 23 problems as they appeared in the 1902 translation in the Bulletin of the American Mathematical Society.

1. Cantor's problem of the cardinal number of the continuum.

2. The compatibility of the arithmetical axioms.

3. The equality of the volumes of two tetrahedra of equal bases and equal altitudes.

4. Problem of the straight line as the shortest distance between two points.

5. Lie's concept of a continuous group of transformations without the assumption of the differentiability of the functions defining the group.

6. Mathematical treatment of the axioms of physics.

7. Irrationality and transcendence of certain numbers.

8. Problems of prime numbers (The "Riemann Hypothesis").

9. Proof of the most general law of reciprocity in any number field.

10. Determination of the solvability of a Diophantine equation.

11. Quadratic forms with any algebraic numerical coefficients

12. Extensions of Kronecker's theorem on Abelian fields to any algebraic realm of rationality

13. Impossibility of the solution of the general equation of 7th degree by means of functions of only two arguments.

14. Proof of the finiteness of certain complete systems of functions.

15. Rigorous foundation of Schubert's enumerative calculus.

16. Problem of the topology of algebraic curves and surfaces.

17. Expression of definite forms by squares.

18. Building up of space from congruent polyhedra.

19. Are the solutions of regular problems in the calculus of variations always necessarily analytic?

20. The general problem of boundary values (Boundary value problems in PDE's).

21. Proof of the existence of linear differential equations having a prescribed monodromy group.

22. Uniformization of analytic relations by means of automorphic functions.

23. Further development of the methods of the calculus of variations.

Formalism

In an account that had become standard by the mid-century, Hilbert's problem set was also a kind of manifesto that opened the way for the development of the formalist school, one of three major schools of mathematics of the 20th century. According to the formalist, mathematics is manipulation of symbols according to agreed upon formal rules. It is therefore an autonomous activity of thought.

Program

In 1920, Hilbert proposed a research project in metamathematics that became known as Hilbert's program. He wanted mathematics to be formulated on a solid and complete logical foundation. He believed that in principle this could be done by showing that:

1. all of mathematics follows from a correctly chosen finite system of axioms; and
2. that some such axiom system is provably consistent through some means such as the epsilon calculus.

He seems to have had both technical and philosophical reasons for formulating this proposal. It affirmed his dislike of what had become known as the ignorabimus, still an active issue in his time in German thought, and traced back in that formulation to Emil du Bois-Reymond.

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. This approach has been successful and influential in relation with Hilbert's work in algebra and functional analysis, but has failed to engage in the same way with his interests in physics and logic.

Hilbert wrote in 1919:

We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise.

Hilbert published his views on the foundations of mathematics in the 2-volume work, Grundlagen der Mathematik.

Gödel's work

Hilbert and the mathematicians who worked with him in his enterprise were committed to the project. His attempt to support axiomatized mathematics with definitive principles, which could banish theoretical uncertainties, ended in failure.

Gödel demonstrated that any consistent formal system that is sufficiently powerful to express basic arithmetic cannot prove its own completeness using only its own axioms and rules of inference. In 1931, his incompleteness theorem showed that Hilbert's grand plan was impossible as stated. The second point cannot in any reasonable way be combined with the first point, as long as the axiom system is genuinely finitary.

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. The basis for later theoretical computer science, in the work of Alonzo Church and Alan Turing, also grew directly out of this "debate".

Functional analysis

Around 1909, Hilbert dedicated himself to the study of differential and integral equations; his work had direct consequences for important parts of modern functional analysis. In order to carry out these studies, Hilbert introduced the concept of an infinite-dimensional Euclidean space, later called Hilbert space. His work in this part of analysis provided the basis for important contributions to the mathematics of physics in the next two decades, though from an unanticipated direction. Later on, Stefan Banach amplified the concept, defining Banach spaces. Hilbert spaces are an important class of objects in the area of functional analysis, particularly of the spectral theory of self-adjoint linear operators, that grew up around it during the 20th century.

Physics

Until 1912, Hilbert was almost exclusively a pure mathematician. When planning a visit from Bonn, where he was immersed in studying physics, his fellow mathematician and friend Hermann Minkowski joked he had to spend 10 days in quarantine before being able to visit Hilbert. In fact, Minkowski seems responsible for most of Hilbert's physics investigations prior to 1912, including their joint seminar on the subject in 1905.

In 1912, three years after his friend's death, Hilbert turned his focus to the subject almost exclusively. He arranged to have a "physics tutor" for himself. He started studying kinetic gas theory and moved on to elementary radiation theory and the molecular theory of matter. Even after the war started in 1914, he continued seminars and classes where the works of Albert Einstein and others were followed closely.

By 1907, Einstein had framed the fundamentals of the theory of gravity, but then struggled for nearly 8 years to put the theory into its final form. 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 at Göttingen. 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 (see Einstein field equations). Nearly simultaneously, Hilbert published "The Foundations of Physics", an axiomatic derivation of the field equations (see Einstein–Hilbert action). Hilbert fully credited Einstein as the originator of the theory and no public priority dispute concerning the field equations ever arose between the two men during their lives.

Additionally, Hilbert's work anticipated and assisted several advances in the mathematical formulation of quantum mechanics. His work was a key aspect of Hermann Weyl and John von Neumann's work on the mathematical equivalence of Werner Heisenberg's matrix mechanics and Erwin Schrödinger's wave equation, and his namesake Hilbert space plays an important part in quantum theory. In 1926, von Neumann showed that, if quantum states were understood as vectors in Hilbert space, they would correspond with both Schrödinger's wave function theory and Heisenberg's matrices.

Throughout this immersion in physics, Hilbert worked on putting rigor into the mathematics of physics. While highly dependent on higher mathematics, physicists tended to be "sloppy" with it. To a pure mathematician like Hilbert, this was both ugly and difficult to understand. As he began to understand physics and how physicists were using mathematics, he developed a coherent mathematical theory for what he found – most importantly in the area of integral equations. When his colleague Richard Courant wrote the now classic Methoden der mathematischen Physik (Methods of Mathematical Physics) including some of Hilbert's ideas, he added Hilbert's name as author even though Hilbert had not directly contributed to the writing. Hilbert said "Physics is too hard for physicists", implying that the necessary mathematics was generally beyond them; the Courant–Hilbert book made it easier for them.

Number theory

Hilbert unified the field of algebraic number theory with his 1897 treatise Zahlbericht (literally "report on numbers"). He also resolved a significant number-theory problem formulated by Waring in 1770. As with the finiteness theorem, he used an existence proof that shows there must be solutions for the problem rather than providing a mechanism to produce the answers. He then had little more to publish on the subject; but the emergence of Hilbert modular forms in the dissertation of a student means his name is further attached to a major area.

He made a series of conjectures on class field theory. The concepts were highly influential, and his own contribution lives on in the names of the Hilbert class field and of the Hilbert symbol of local class field theory. Results were mostly proved by 1930, after work by Teiji Takagi.

Hilbert did not work in the central areas of analytic number theory, but his name has become known for the Hilbert–Pólya conjecture, for reasons that are anecdotal. Ernst Hellinger, a student of Hilbert, once told André Weil that Hilbert had announced in his seminar in the early 1900s that he expected the proof of the Riemann Hypothesis would be a consequence of Fredholm's work on integral equations with a symmetric kernel.