Hilbert's problems


Hilbert's problems are twenty-three problems in mathematics published by German mathematician David Hilbert in 1900. The problems were all unsolved at the time, and several of them were very influential for 20th-century mathematics. Hilbert presented ten of the problems at the Paris conference of the International Congress of Mathematicians, speaking on August 8 in the Sorbonne. The complete list of 23 problems was published later, most notably in English translation in 1902 by Mary Frances Winston Newson in the Bulletin of the American Mathematical Society.

Nature and influence of the problems

Hilbert's problems ranged greatly in topic and precision. Some of them are propounded precisely enough to enable a clear affirmative or negative answer, like the 3rd problem, which was the first to be solved, or the 8th problem. For other problems, such as the 5th, experts have traditionally agreed on a single interpretation, and a solution to the accepted interpretation has been given, but closely related unsolved problems exist. Sometimes Hilbert's statements were not precise enough to specify a particular problem but were suggestive enough so that certain problems of more contemporary origin seem to apply; for example, most modern number theorists would probably see the 9th problem as referring to the conjectural Langlands correspondence on representations of the absolute Galois group of a number field. Still other problems, such as the 11th and the 16th, concern what are now flourishing mathematical subdisciplines, like the theories of quadratic forms and real algebraic curves.
There are two problems that are not only unresolved but may in fact be unresolvable by modern standards. The 6th problem concerns the axiomatization of physics, a goal that twentieth-century developments of physics seem to render both more remote and less important than in Hilbert's time. Also, the 4th problem concerns the foundations of geometry, in a manner that is now generally judged to be too vague to enable a definitive answer.
The other twenty-one problems have all received significant attention, and late into the twentieth century work on these problems was still considered to be of the greatest importance. Paul Cohen received the Fields Medal during 1966 for his work on the first problem, and the negative solution of the tenth problem during 1970 by Yuri Matiyasevich generated similar acclaim. Aspects of these problems are still of great interest today.

Ignorabimus

Following Gottlob Frege and Bertrand Russell, Hilbert sought to define mathematics logically using the method of formal systems, i.e., finitistic proofs from an agreed-upon set of axioms. One of the main goals of Hilbert's program was a finitistic proof of the consistency of the axioms of arithmetic: that is his second problem.
However, Gödel's second incompleteness theorem gives a precise sense in which such a finitistic proof of the consistency of arithmetic is provably impossible. Hilbert lived for 12 years after Kurt Gödel published his theorem, but does not seem to have written any formal response to Gödel's work.
Hilbert's tenth problem does not ask whether there exists an algorithm for deciding the solvability of Diophantine equations, but rather asks for the construction of such an algorithm: "to devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers." That this problem was solved by showing that there cannot be any such algorithm contradicted Hilbert's philosophy of mathematics.
In discussing his opinion that every mathematical problem should have a solution, Hilbert allows for the possibility that the solution could be a proof that the original problem is impossible. He stated that the point is to know one way or the other what the solution is, and he believed that we always can know this, that in mathematics there is not any "ignorabimus". It seems unclear whether he would have regarded the solution of the tenth problem as an instance of ignorabimus: what is proved not to exist is not the integer solution, but the ability to discern in a specific way whether a solution exists.
On the other hand, the status of the first and second problems is even more complicated: there is not any clear mathematical consensus as to whether the results of Gödel, or Gödel and Cohen give definitive negative solutions or not, since these solutions apply to a certain formalization of the problems, which is not necessarily the only possible one.

The 24th problem

Hilbert originally included 24 problems on his list, but decided against including one of them in the published list. The "24th problem" was rediscovered in Hilbert's original manuscript notes by German historian in 2000.

Sequels

Since 1900, mathematicians and mathematical organizations have announced problem lists, but, with few exceptions, these collections have not had nearly as much influence nor generated as much work as Hilbert's problems.
One of the exceptions is furnished by three conjectures made by André Weil during the late 1940s. In the fields of algebraic geometry, number theory and the links between the two, the Weil conjectures were very important. The first of the Weil conjectures was proved by Bernard Dwork, and a completely different proof of the first two conjectures via ℓ-adic cohomology was given by Alexander Grothendieck. The last and deepest of the Weil conjectures was proven by Pierre Deligne. Both Grothendieck and Deligne were awarded the Fields medal. However, the Weil conjectures in their scope are more like a single Hilbert problem, and Weil never intended them as a programme for all mathematics. This is somewhat ironic, since arguably Weil was the mathematician of the 1940s and 1950s who best played the Hilbert role, being conversant with nearly all areas of mathematics and having figured importantly in the development of many of them.
Paul Erdős posed hundreds, if not thousands, of mathematical problems, many of them profound. Erdős often offered monetary rewards; the size of the reward depended on the perceived difficulty of the problem.
The end of the millennium, being also the centennial of Hilbert's announcement of his problems, was a natural occasion to propose "a new set of Hilbert problems." Several mathematicians accepted the challenge, notably Fields Medalist Steve Smale, who responded to a request of Vladimir Arnold by proposing a list of 18 problems. Smale's problems have thus far not received much attention from the media, and it is unclear how much serious attention they are getting from the mathematical community.
At least in the mainstream media, the de facto 21st century analogue of Hilbert's problems is the list of seven Millennium Prize Problems chosen during 2000 by the Clay Mathematics Institute. Unlike the Hilbert problems, where the primary award was the admiration of Hilbert in particular and mathematicians in general, each prize problem includes a million dollar bounty. As with the Hilbert problems, one of the prize problems was solved relatively soon after the problems were announced.
The Riemann hypothesis is noteworthy for its appearance on the list of Hilbert problems, Smale's list, the list of Millennium Prize Problems, and even in the Weil conjectures, in its geometric guise. Notwithstanding some famous recent assaults from major mathematicians of our day, many experts believe that the Riemann hypothesis will be included in problem lists for many centuries. Hilbert himself declared: "If I were to awaken after having slept for a thousand years, my first question would be: Has the Riemann hypothesis been proven?"
In 2008, DARPA announced its own list of 23 problems that it hoped could cause major mathematical breakthroughs, "thereby strengthening the scientific and technological capabilities of DoD."

Summary

Of the cleanly formulated Hilbert problems, problems 3, 7, 10, 14, 17, 18, 19, and 20 have a resolution that is accepted by consensus of the mathematical community. On the other hand, problems 1, 2, 5, 6, 9, 11, 15, 21, and 22 have solutions that have partial acceptance, but there exists some controversy as to whether they resolve the problems.
That leaves 8, 12, 13 and 16 unresolved, and 4 and 23 as too vague to ever be described as solved. The withdrawn 24 would also be in this class. Number 6 is deferred as a problem in physics rather than in mathematics.

Table of problems

Hilbert's twenty-three problems are :
ProblemBrief explanationStatusYear Solved
1stThe continuum hypothesis 1940, 1963
2ndProve that the axioms of arithmetic are consistent.1931, 1936
3rdGiven any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces that can be reassembled to yield the second?1900
4thConstruct all metrics where lines are geodesics.—
5thAre continuous groups automatically differential groups?1953?
6thMathematical treatment of the axioms of physics

axiomatic treatment of probability with limit theorems for foundation of statistical physics

the rigorous theory of limiting processes "which lead from the atomistic view to the laws of motion of continua"		1933–2002?
7thIs ab transcendental, for algebraic a ≠ 0,1 and irrational algebraic b ?1934
8thThe Riemann hypothesis

and other prime number problems, among them Goldbach's conjecture and the twin prime conjecture		—
9thFind the most general law of the reciprocity theorem in any algebraic number field.—
10thFind an algorithm to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution.1970
11thSolving quadratic forms with algebraic numerical coefficients.—
12thExtend the Kronecker–Weber theorem on Abelian extensions of the rational numbers to any base number field.—
13thSolve 7th degree equation using algebraic functions of two parameters.—
14thIs the ring of invariants of an algebraic group acting on a polynomial ring always finitely generated?1959
15thRigorous foundation of Schubert's enumerative calculus.—
16thDescribe relative positions of ovals originating from a real algebraic curve and as limit cycles of a polynomial vector field on the plane.—
17thExpress a nonnegative rational function as quotient of sums of squares.1927
18th Is there a polyhedron that admits only an anisohedral tiling in three dimensions?

What is the densest sphere packing?
19thAre the solutions of regular problems in the calculus of variations always necessarily analytic?1957
20thDo all variational problems with certain boundary conditions have solutions??
21stProof of the existence of linear differential equations having a prescribed monodromic group?
22ndUniformization of analytic relations by means of automorphic functions?
23rdFurther development of the calculus of variations—