Note: This is a paper which I prepared for a class in number theory. I have left the footnotes out in this version since they are too difficult to reproduce on the web. Most of the quotes in this paper are from the 1975 AMS report on the 23 Hilbert Problems and Ian Stewart's book. See the bibliography for a list of everyone quoted. Most of the mathematical notation and accent marks are lost in HTML. Problems nineteen through twenty-three are only described briefly since I got writer's cramp after the sphere-packing problem.

III. The Paris Problems

In the summer of 1900, David Hilbert addressed the second International Congress of Mathematicians in Paris. Hilbert opened his address by mentioning two famous unsolved problems: Fermat's Last Theorem and the three-body problem. Fermat's Last Theorem was finally proved in the 1990's by Andrew Wiles, but others' attempts led to many interesting advances in number theory and algebra, specifically Kummer's discovery of ideal theory. The 3-body problem has been solved, although the N-body problem still needs a closed form, but the attempts also led to advances, such as leading Poincare to develop celestial mechanics. Hoping to inspire mathematicians in a similar manner, Hilbert presented a set of 23 unsolved problems in mathematics. Many of these problems remain unsolved. Most of them have been restated. All of them have inspired mathematicians for over one hundred years. In 1950, Hermann Weyl was asked by the American Mathematical Society to describe the history of mathematics since 1900. Weyl stated that one could simply look at which of the Paris Problems had been solved as " a chart by which we mathematicians have often measured our progress. How much better he [Hilbert] predicted the future of mathematics than any politician foresaw the gifts of war and terror that the new century was about to lavish on mankind!" Weyl also held that solving one of the Paris Problems put the mathematician in "the honors class of the mathematical community." Some of these problems, or, as Ian Stewart calls them, "Hilbert's hit-list" , and the progress made pertaining to them are described briefly below.

1.) Hilbert's first problem is the verification of the Continuum Hypothesis in Cantor set theory. Georg Cantor developed a curious number aleph-zero, denoted here N0, which is the cardinal, or number of elements in the set of natural numbers {0,1,2,...}. Let N1 denote the cardinal of the set of real numbers. The Continuum Hypothesis asserts that 2^N0 = N1. Hence, this hypothesis asserts that one infinity is bigger than another. Unsurprisingly, when Cantor proposed this conjecture, he received much criticism, particularly from Leopold Kronecker. Kronecker's public attacks on Cantor led to Cantor's nervous breakdown. Hilbert, on the other hand, praised Cantor's ideas and proclaimed that "no one shall expel us from the paradise which Cantor created." Hilbert once explained the concept of infinity as a hotel with an infinite number of rooms numbered 1,2,3.... If a bus full of an infinite number of travelers arrives at the hotel when the hotel is already full, the manager can move the guest already in room 1 to room 2, room 2 to room 4, ... room n to 2n. This frees up every odd-numbered room and, hence, each traveler on the bus can have a room. Even if an infinite number of buses each with an infinite number of guests arrives, each person can be accommodated. Hence, it was Hilbert's belief that the Continuum Hypothesis is correct, but he was unable to prove it. In 1938, Kurt Godel proved that according to the standard Zermelo-Fraenkel axioms of set theory, the Continuum Hypothesis cannot be refuted. In 1963, P.J. Cohen showed that the Zermelo-Fraenkel axioms are not sufficient to prove the Continuum Hypothesis. Hence, mathematicians have proved that the problem is undecidable given the standard view of set theory. Cohen's work has led mathematicians to develop a new set of axioms for set theory. It remains to be seen whether non-Cantorian set theory can prove the Continuum Hypothesis. If you accept random variables as standard mathematical tools, then Christopher Freiling has a neat 1986 proof for you that actually disproves the Continuum Hypothesis.

2.) Hilbert's second problem is to prove the compatibility of the axioms of arithmetic. That is, "that a finite number of logical steps based upon them can never lead to contradictory results." As mentioned earlier, David Hilbert developed a theory of proofs called Hilbert's Program to try to solve this problem. As mentioned earlier, Kurt Godel's work on logic all but demolished Hilbert's Program. G. Kreisel argues that Hilbert's Program is valid in most branches of mathematics and elementary logic. However, he contends that Hilbert's Program fails for arithmetic and metamathematics and, hence, Hilbert's second problem remains unsolved. Mathematicians, such as Detlefsen, still debate the validity of Hilbert's Program, Godel's theorems, and the compatibility of the arithmetic axioms. Despite its undecided state, Hilbert's Second Problem has provoked great changes in the theories of logic.

3.) The third problem asks whether any polyhedron can be dissected into pieces and reassembled to form a cube of the same volume. Hilbert phrased this problem in terms of the volume of a pyramid. To calculate the volume of a pyramid, one must use a rather complicated limiting process known as the "devil's staircase." However, this problem can be avoided in two dimensions. Hilbert asked whether the "devil's staircase" was indeed necessary. Hilbert personally believed that the process is necessary and any polyhedron cannot be dissected to form a cube. Max Dehn, one of the founders of topology, confirmed Hilbert's conjecture in 1900, the same year the problem was first posed. Dehn showed that a regular tetrahedron cannot be decomposed into a cube of equal volume. In 1832, F. Bolyai had determined that in two dimensions, any two equidecomposable polygons (shapes that can be dissected and reassembled into each other) have equal areas. However, this rule does not hold for higher dimensions. In 1896, Bricard developed conditions under which equidecomposable polyhedra have equal volumes. Another interesting offshoot of this problem is the 1924 Banach-Tarski Paradox. Stefan Banach and Alfred Tarski proved that it is possible to dissect a solid sphere into six pieces and reassemble the pieces to form two solid spheres each the same volume as the original. The trick is to cut the sphere into pieces so complicated that the pieces do not have volumes. This is impossible to do in reality, otherwise Banach and Tarski would have dissected and reassembled spheres of gold to make themselves rich. Despite the rather quick answer to this question, much work has been done on this problem regarding geometry. An open problem called the Hodge Conjecture can be viewed as an offshoot of Hilbert's Third Problem.

4.) Hilbert described his fourth problem as the "problem of the straight line as the shortest distance between two points." Despite the apparent simplicity of the problem, it is still an open problem involving the foundations of geometry, the calculus of variations, and differential geometry. Specifically, he asks in what geometries are the ordinary lines the shortest curves. Since differential geometry was a relatively new science in 1900, Hilbert's knowledge of metrics and individual geometries was limited. The only known metrics in 1900 were the three Riemannian metrics. It has been shown since then that there are infinite number of such metrics, so Hilbert's problem has only been proven true in certain special cases. This problem has evolved to the problem of creating Desarguesian spaces, those spaces in which Hilbert's problem is verified. Works by Pogorelov and Hamel have provided methods by which Desarguesian spaces can be constructed.

5.) Hilbert's fifth problem concerns Lie's groups of transformations. Hilbert's exact wording is as follows: "How far Lie's concept of continuous groups of transformations of manifolds is approachable in our investigation without the assumption of differentiability." This problem has been reinterpreted as: "Is every locally Euclidean group a Lie group?" The answer to this question is "yes."

6.) Hilbert's sixth problem calls for axiomitization of physics. As noted earlier, Hilbert made significant steps in this area. He believed that axiomitization would give physics the mathematical rigor that it lacked. Hilbert himself made significant contributions to the axiomatization of the kinetic theory of gases. Hilbert's work has led to progress in the theory of dynamical systems, ergodic theory, and statistical mechanics. Hilbert concentrated much of his effort in the mathematical interpretation of the statistical theory of radiation, but since radiation theory was relatively new in Hilbert's time, he had little effect on modern radiation theory. His work, Die Grundlangen der Physik, made significant process towards developing far-reaching axioms in the field of general relativity. Since 1900, physics has undergone two major revolutions, so, according to A.S. Wightman, Hilbert's Sixth Problem is "by now primarily of archaeological interest." Perhaps physics was not meant to be axiomatized. Hermann Weyl stated that "the maze of experimental facts which the physicist has to take into account is too manifold, their expansion too fast, and their aspect and relative weight too changeable for the axiomatic method to find a firm enough foothold." Since Hilbert's time, significant steps towards axiomatizing quantum mechanics and quantum field theory have been taken.

7.) Hilbert's seventh problem calls for the proof of the existence of certain transcendental numbers. Specifically, he called for a proof of the transcendence of 2^(sqrt2). As mentioned earlier, Hilbert produced simple, direct proofs of the transcendence of e and pi. In a lecture in 1919, Hilbert claimed that some of the students in the lecture hall might live to see the proof of Fermat's Last Theorem, but no one would live long enough to see the proof of the transcendence of 2^(sqrt2). Less than ten years later, the Russian mathematician A.O. Gelfond established the transcendence of 2^(sqrt-2). Building on this, Carl Ludwig Siegel, one of Hilbert's students who was present in the lecture hall when Hilbert made his announcement, proved the transcendence of 2^(sqrt2). The Gelfond-Baker method can be used to determine the transcendence of certain numbers.

8.) Hilbert's eighth problem is concerned with Euler's zeta function, which is defined as zeta(s) = Summation(n^-s). The Riemann hypothesis states that zeroes of the zeta function all have real part of one-half. It has been shown by J. van de Lune and Herman te Riele that the first 1.5 billion zeroes of the zeta function all have real part 1/2. However, this experimental data does not constitute a proof. Despite announcements of proofs, no one has proved the Riemann Hypothesis to date. The Riemann Hypothesis has been proven for curves over finite fields by Pierre Deligne and Stepanov. The Clay Mathematics Institute has placed a $1,000,000 bounty on this problem. That's a spicy meatball!

9.) The ninth problem calls for a proof of the general reciprocity law. In other words, "for any field of numbers the law of reciprocity is to be proved for the residues of the lth power, where l denotes an odd prime." This problem uses Hilbert's adaptation of the Legendre symbol described earlier. Building on the work of Hilbert and Weber, Tejii Takagi a general theory of abelian extensions of number fields. Using Takagi's work, Emil Artin was able to develop and prove a general reciprocity law.

10.) Hilbert's tenth problem asks if there is a general algorithm to decide the solubility of Diophantine equations, or equations whose solutions are required to be integers. Hilbert believed that no such algorithm existed. Using the work of Martin Davis, Hilary Putnam, and Julia Robinson, Yuri Matijasevich was able to answer this question with a resounding "no" in 1970. James Jones found an example that could not be solved by any algorithm: a system of eighteen equations of maximum degree 560 in thirty-three variables. In 1972, Carl Ludwig Siegel found an algorithm for equations of degree 2. It has been proven that no algorithm exists for equations of degree 4. Finding an algorithm for cubic equations is still an open problem. Matijasevich's work produced some interesting results on Turing machines and "a polynomial formula whose position values are precisely the primes." An open problem known as the Birch and Swinnerton-Dyer Conjecture is an offshoot of Hilbert's Tenth Problem.

11.) The eleventh problem is "to attack successfully the theory of quadratic forms with any number of variables and with any algebraic numerical coefficients." In 1924, Helmut Hasse devised a theory of quadratic forms over the rational numbers and later extended this solution to any algebraic number field.

12.) Hilbert's twelfth problem is the "extension of Kronecker's Theorem on abelian fields to any algebraic realm of rationality." Kronecker's Theorem states that "every abelian number field arises from the realm of rational numbers by the composition of fields of roots of unity." This problem and related contemporary problems are under investigation by Shimura and Shih.

13.) The thirteenth problem is to prove the "impossibility of the solution of the general equation of the 7th degree by means of functions of only two arguments." To date, Hilbert's conjecture has been proved if one assumes that all functions are continuously differentiable. A more general proof has not been presented yet, but it is believed that "linear superposition" and "the notions of capacity and entropy" will lead to a final solution.

14.) Hilbert's fourteenth problem asks whether certain subrings are finitely generated. The subring is generated as follows: Let k be a field. Let K be a subfield of the rational functions in n variables (x1, ..., xn) over k. Let the ring be K (intersect) k[x1, ..., xn]. Hilbert proved that these subrings are finitely generated in the SL(m)-invariant case. This is known as Hilbert's Basis Theorem. Hence, Hilbert optimistically conjectured that all such subrings are finitely generated. In 1959, M. Nagata provided a counterexample to Hilbert's conjecture. So the answer to the fourteenth problem is "no."

15.) The fifteenth problem is "to establish rigorously and with an exact determination of the limits of their validity those geometrical numbers which Schubert especially has determined on the basis of the so-called principle of special position, or conservation of number, by mean of the enumerative calculus developed by him." Schubert was Adolf Hurwitz's high school math teacher. This problem has been solved for certain special cases but no general theory has been developed.

16.) The sixteenth problem calls for a general investigation into the "topology of algebraic curves and surfaces." This is really more of an assignment than a question. Algebraic topology has developed as a field significantly since Hilbert's time. In 2000, Steve Smale restated this problem as an investigation of bounds on the number of limit cycles of differential equations.

17.) Hilbert's seventeenth problem asks for a proof that any positive definite form can be expressed as the sum of squares. It has been proven that any positive definite form in n variables with real coefficients is the sum of at most 2n squares. An existence proof was provided by Artin in 1926 and a constructive proof by Habicht followed in 1940.

18.) The eighteenth problem concerns "the building up of space from congruent polyhedra." Specifically, it calls for a proof of the famous Kepler Sphere-Packing Problem. In 1611, Johannes Kepler, of gravitation theory fame, became fascinated by the formation of snowflakes. He believed (correctly) that snowflakes started as tiny hexagonal seeds and received their unique appearance from atmospheric conditions, yet every snowflake retains its original hexagonal symmetry. He further conjectured that snowflakes take on hexagonal shapes because the "tightest pack" is that of a face-centered cubic lattice. That is, the face-centered cubic lattice, the stacking used by greengrocers, will take up less total volume than any other formation of spherical objects. For evidence, Kepler pointed to the pomegranate, whose seeds start out round but end up rhombic dodecahedron, as one would expect from a face-centered cubic lattice packing. In the 1830's, Gauss proved that the face-centered cubic lattice is the densest packing of spheres among all lattices, but this does not prove that it is denser than non-lattice or even random packings. In 1940, L. Fejes Toth proved Kepler's Problem true in the two-dimensional case. In the 1950's, various physicists such as J.D. Bernal and G.D. Scott experimented with compressing ball bearings in various packings. They determined that the face-centered cubic lattice produces the tightest pack. This is a case which adds credit to Hilbert's opinion that physicists were less mathematically rigorous than desired. In 1990, Prof. Wu-Yi Hsiang at Berkeley used classical geometry to produce a one hundred page proof of Kepler's Problem. Unless a fatal flaw in Hsiang's proof is found, Hilbert's eighteenth problem has been solved.

19.)Are the solutions of problems in the calculus of variations always necessarily analytic? Does every Lagrangian PDE of a regular variation have the property of admitting analytic integrals exclusively?

20.)On the solvability of boundary value problems. Thanks Hilbert. That's a big help.

21.)Proof of the existence of differential equations with regular singular points and prescribed monodromy. Solved by Deligne in 1970.

22.)Uniformization of analytic functions by means of automorphic relations. Solved many times by lots of people like Koebe.

23.)Developments in the methods of the calculus of variations. Mostly related to control problems, dynamics, and optimization nowadays.

Each of these problems is significant in that it has significant applications to several branches of mathematics. Of course, problems of great importance have developed since Hilbert's time, such as the Poincare Conjecture and the solvability of the Navier-Stokes equations. One problem is that is noticeably absent is a call for a proof that P does not equal NP. But Dave can hardly be blamed for the omission, since 1900 was well before the advent of the computer. An updated hit-list can be found at the Clay Mathematics Institute's site, which offers a one million dollar reward for each of the six problems presented. In 2000, the American Mathematical Society published a book, "Mathematics: Frontiers and Perspectives," that discusses the future of mathematics. In it, various authors propose open problems. Steve Smale and Peter Sarnak both offer very good problem lists, but no one was able to produce a single list as pointed and comprehensive as David's. Perhaps it is beyond one person's capablities to recreate Hilbert's hit-list for a new century because mathematics developed so rapidly and split off into many branches in the 20th century. It was Hilbert's goal in presenting these problems to help unify and focus the efforts of future mathematicians. David Hilbert concluded his Paris address with an optimistic wish for the future of mathematics:

The problems mentioned are merely samples of problems; yet they are sufficient to show how rich , how manifold and how extensive the mathematical science is today; and the question is urged upon us whether mathematics is doomed to the fate of those other sciences that have split up into separate branches, whose representatives scarcely understand one another and whose connection becomes ever more loose. I do not believe this nor do I wish it. ...The organic unity of mathematics is inherent in the nature of this science, for mathematics is the foundation of all exact knowledge of natural phenomena. That it may completely fulfill this high destiny, may the new century bring it gifted prophets and many zealots and enthusiastic disciples!

Amen!