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David Hilbert

David Hilbert was one of the last universal mathematicians, who contributed greatly to many areas of mathematics. After joining the faculty at Göttingen, in Germany, he became a major reason for that university’s surpassing Berlin as the preeminent university for mathematics in Germany, and probably the world, through the first third of the twentieth century.(1) The mathematics historian Victor Katz believes that “(Hilbert) was . . .probably the premier mathematician of the late nineteenth and early twentieth centuries.”(2) Of the attempts to set up a complete set of axioms from which Euclidean geometry could be derived, the most successful was probably that of Hilbert. His work in geometry had the greatest influence since Euclid.(3) In 1899, Hilbert published his Foundations of Geometry. His aim was to choose for geometry a simple and complete set of independent axioms and to deduce from these the most important geometrical theorems in such a manner as to bring out as clearly as possible the significance of the different groups of axioms and the scope of the conclusions to be derived from the individual axioms.

The importance of Hilbert’s work lay not so much in his answering the various objections to parts of Euclid’s deductive scheme, but in reinforcing the notion that any mathematical field must begin with undefined terms and axioms specifying the relationships among the terms. There were many axiom schemes developed in the late nineteenth century to clarify various areas of mathematics. Hilbert’s work can be considered the culmination of this process, because he was able to take the oldest such scheme and show that, with a bit of tinkering, it had stood the test of time. Thus, the mathematical ideas of Euclid and Aristotle were reconfirmed at the end of the nineteenth century as still the model for pure mathematics.(4) A century later, these ideas continue to prevail.

In 1900, Hilbert presented 10 central open problems in mathematics to be targeted in the 20th century, at the International Mathematical Congress in Paris. Later that year he published his complete list of 23 problems. The mathematician John Casti stated: “Without a doubt the most influential list of problems ever put together. . .”(5) The 23 problems encompassed virtually all branches of mathematics. For example, from the foundations of mathematics, Hilbert asked for a proof of the continuum hypothesis as well as an investigation of the consistency of the axioms of arithmetic. From analysis came the question of whether all complex zeros of the Riemann zeta function have real part ½ (the Riemann hypothesis), and whether one can always solve boundary value problems in the theory of partial differential equations. Hilbert’s problems had in fact proved to be central in twentieth-century mathematics. Finally, his problems were designed to serve as examples for the kinds of problems whose solutions would lead to the furthering of disciplines in mathematics. As such, some were areas for investigation and therefore not strictly "problems."

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Hilbert's 23 Problems can be divided into five types:

1. Foundation Problems: 1, 2, and 10

2. Foundations of Specific Areas: 3, 4, 5, 6

3. Number Theory: 7, 8, 9, 11, and 12

4. Algebra and Geometry: 14, 15, 16, 17, and 18

5. Analysis Problems: 13, 19, 20, 21, 22, and 23

Hilbert's 23 Problems (6):

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1. Continuum Hypothesis. "Cantor's problem of the cardinal number of the continuum." The question is if there is a transfinite number between that of a denumerable set and the numbers of the continuum was answered by Gödel and Cohen in their solution to the continuum hypothesis to the effect that the answer depends on the particular version (axioms) of set theory assumed. The question of if the continuum of numbers be considered a well ordered set is related to Zermelo's axiom of choice. In 1963, the axiom of choice was demonstrated by Cohen to be independent of all other axioms in set theory. 

2. Logical Consistency of Arithmetic.  Prove that the standard axioms of arithmetic can never lead to a contradiction. Solved in 1931 by Kurt Gödel’s second incompleteness theorem which indicated that it was impossible with the usual axioms for set theory. 

3. Equality of Volumes of Tetrahedrons. If two tetrahedrons have the same volume, can you always cut one into finitely many polygonal pieces and reassemble them to form the other? In 1901 Max Dehn showed that you cannot.

4. Find geometries whose axioms are closest to those of Euclidean geometry if the ordering and incidence axioms are retained, the congruence axioms weakened, and the equivalent of the parallel postulate omitted. This problem was solved by G. Hamel.

5. Can the assumption of differentiability for functions defining a continuous transformation group be avoided? (This is a generalization of the Cauchy functional equation.) It was solved by John von Neumann in 1930 for bicompact groups. Also solved for the Abelian case, and for the solvable case in 1952 with complementary results by Montgomery and Zippin (subsequently combined by Yamabe in 1953). Andrew Gleason showed in 1952 that the answer is also "yes" for all locally bicompact groups.

6. Can physics be axiomatized? Develop a rigorous system of axioms for mathematical areas of physics, such as probability and mechanics. Andrei Kolmogorov axiomatized probability in 1933.

7. Irrational and Transcendental Numbers. Prove that certain numbers are irrational or transcendental. Solved by Aleksandr Gelfond and Theodor Schneider in 1934.

8. Prove the Riemann hypothesis. Prove that all nontrivial zeros of Riemann's zeta-function lie on the critical line. The conjecture has still been neither proved nor disproved. See a discussion under Riemann and near the end of the conclusion of the mathematics section where the seven millennium problems are discussed.

9. Laws of Reciprocity in Number Fields. Generalise the classical law of quadratic reciprocity, about squares to some modulus, to higher powers. Partially solved. 

10. Determine When a Diophantine Equation Has Solutions. Does there exist a universal algorithm for solving Diophantine equations (i.e. polynomial equations in many variables)? The impossibility of obtaining a general solution was proven by Yuri Matiyasevich in 1970. 

11. Quadratic Forms with Algebraic Numbers as Coefficients. Technical issues about the solution of many-variable Diophantine equations. Partially solved.

12. Kronecker's Theorem on Abelian fields. Extend a theorem of Kronecker to arbitrary algebraic fields by explicitly constructing Hilbert class fields using special values. This calls for the construction of holomorphic functions in several variables which have properties analogous to the exponential function and elliptic modular functions (Holzapfel 1995). Still unsolved.

13. Solving Seventh-Degree Equations using Special Functions. Prove the impossibility of solving the general seventh degree equation using functions of two variables. One interpretation disproved by Andrei Kolmogorov and Vladimir Arnold.

14. Finiteness of Complete Systems of Functions. Extend a theorem of Hilbert about algebraic invariants to all transformation groups. Proved false by Masayoshi Nagata in 1959.

15. Schubert's Enumerative Calculus. Hermann Schubert found a non-rigorous method for counting various geometric configurations. Make the method rigorous. No complete solution yet.

16. Topology of Curves and Surfaces. How many connected components can an algebraic curve of given degree have? How many distinct periodic cycles can an algebraic differential equation of given degree have? Limited progress. 

17. Expressing Definite Forms by Squares. If a rational function always takes non-negative values, must it be a sum of squares? Solved by Emil Artin, D.W. Dubois, and Albrecht Pfister. True over the real numbers, false in some other number systems.  

18. Build spaces with congruent polyhedra. General issues about filling space with congruent polyhedrons. Also mentions the Kepler conjecture, now proved. 

19. Analyze the analytic character of solutions in calculus of variations. The calculus of variations answers questions like: 'Find the shortest curve with the following properties.' If such a problem is defined by nice functions, must the solution also be nice? Proved by Ennio de Giorgi in 1957 and by John Nash.

20. Boundary Value Problems. Understand the solutions of the differential equations of physics, inside some region of space, when properties of the solution on the boundary of that region are prescribed. Essentially solved, by numerous mathematicians.

21. Solve differential equations given a monodromy group. More technically, prove that there always exists a Fuchsian system with given singularities and a given monodromy group. Several special cases had been solved, but a negative solution was found in 1989 by B. Bolibruch (Anasov and Bolibruch 1994).

22. Uniformization using Automorphic Functions. Technical question about simplifying equations. Solved by Paul Koebe soon after 1900.

23. Extend the methods of calculus of variations. Hilbert appealed for fresh ideas in the calculus of variations. Much work done; question too vague to be considered solved.

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Two other problems, the Fermat Conjecture and the N-Body Problem are mentioned by Hilbert in his actual lecture and in the full version of the published paper in his preamble but omitted from the list of problems itself. So are there really 25 problems in all? Fermat’s Conjecture or Fermat’s Last Theorem as you recall, was solved by Andrew Wiles in 1995 as discussed under Fermat in this section.

The N-Body problem involves predicting the individual motions of a group of celestial objects interacting with each other gravitationally. Solving this problem has been motivated by the desire to understand the motions of the Sun, Moon, planets, and visible stars.

Poincaré’s efforts on the N-Body problem, even though they failed to actually solve the problem, laid the foundations of the modern qualitative theory of differential equations, which in turn has evolved into what is today termed dynamical system theory. Modern work in dynamical system theory by Don Saari and Jeff Xia at Northwestern University in the United States has all but closed out the N-Body Problem by producing examples of specific systems for which particles do escape off to infinity.

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Footnotes:

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(1) Victor J. Katz, A History of Mathematics -- An Introduction, 2nd Edition (Reading, 1998), p. 868.

(2) Ibid., p. 867.

(3) Donald R. Franceschetti (editor), Biographical Encyclopedia of Mathematicians (New York, 1999), p. 277.

(4) Katz, p. 870.

(5) John Casti, Mathematical Mountaintops – The Five Most Famous Problems of All Time (Oxford, 2001), p. 4.

(6) Ian Stewart, Visions of Infinity: The Great Mathematical Problems (New York, 2013), pgs. 307-309.

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Key References: Hilbert by Constance Reid, 1996; The Honor’s Class: Hilbert’s Problems and Their Solvers by A.K. Peters, 2001.