Hilbert's problems

In 1900, the mathematician David Hilbert published a list of 23 unsolved mathematical problems. The list of problems turned out to be very influential.

After Hilbert's death, another problem was found in his writings; this is sometimes known as Hilbert's 24th problem today. This problem is about finding criteria to show that a solution to a problem is the simplest possible.[1]

Of the 23 problems, three were unresolved in 2012, three were too vague to be resolved, and six could be partially solved. Given the influence of the problems, the Clay Mathematics Institute formulated a similar list, called the Millennium Prize Problems in 2000.

Overview

The formulation of certain problems is better than that of others. Of the cleanly-formulated Hilbert problems, problems 3, 7, 10, 11, 13, 14, 17, 19, 20, and 21 have a resolution that is accepted by consensus. On the other hand, problems 1, 2, 5, 9, 15, 18+, and 22 have solutions that have partial acceptance, but there exists some controversy as to whether it resolves the problem.[source?]

The solution for problem 18, the Kepler conjecture, uses a computer-assisted proof. This is controversial, because a human reader is unable to verify the proof in reasonable time.[source?]

That leaves 16, 8 – the Riemann hypothesis – and 12 unresolved. On this classification 4, 16, and 23 are too vague to ever be described as solved. The withdrawn 24 would also be in this class. 6 is considered as a problem in physics rather than in mathematics.[source?]

Table of problems

Hilbert's twenty-three problems are:

Problem Brief explanation Status Year Solved
1st The continuum hypothesis (that is, there is no set whose cardinality is strictly between that of the integers and that of the real numbers) Proven to be impossible to prove or disprove within the Zermelo–Fraenkel set theory with or without the Axiom of Choice (provided the Zermelo–Fraenkel set theory with or without the Axiom of Choice is consistent, i.e., contains no two theorems such that one is a negation of the other). There is no consensus on whether this is a solution to the problem. 1963
2nd Prove that the axioms of arithmetic are consistent. There is no consensus on whether results of Gödel and Gentzen give a solution to the problem as stated by Hilbert. Gödel's second incompleteness theorem, proved in 1931, shows that no proof of its consistency can be carried out within arithmetic itself. Gentzen's consistency proof (1936) shows that the consistency of arithmetic follows from the well-foundedness of the ordinal ε0. 1936?
3rd Given any two polyhedra of equal volume, is it always possible to cut the first into finitely many polyhedral pieces which can be reassembled to yield the second? Resolved. Result: no, proved using Dehn invariants. 1900
4th Construct all metrics where lines are geodesics. Too vague to be stated resolved or not.[n 1] –
5th Are continuous groups automatically differential groups? Resolved by Andrew Gleason or Hidehiko Yamabe, depending on how the original statement is interpreted. If, however, it is understood as an equivalent of the Hilbert–Smith conjecture, it is still unsolved. 1953?
6th Axiomatize all of physics Partially resolved.

[2] [3] [4] [n 2]

–
7th Is a b transcendental, for algebraic a ≠ 0,1 and irrational algebraic b ? Resolved. Result: yes, illustrated by Gelfond's theorem or the Gelfond–Schneider theorem. 1934[5]
8th The Riemann hypothesis ("the real part of any non-trivial zero of the Riemann zeta function is ½") and other prime number problems, among them Goldbach's conjecture and the twin prime conjecture Unresolved. –
9th Find most general law of the reciprocity theorem in any algebraic number field Partially resolved.[n 3] –
10th Find an algorithm to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution. Resolved. Result: impossible, Matiyasevich's theorem implies that there is no such algorithm. 1970
11th Solving quadratic forms with algebraic numerical coefficients. Partially resolved.[source?] –
12th Extend the Kronecker–Weber theorem on abelian extensions of the rational numbers to any base number field. Partly resolved by class field theory, though the solution is not as explicit as the Kronecker–Weber theorem. –
13th Solving 7-th degree equations using continuous functions of two parameters. Unresolved. The problem was partially solved by Vladimir Arnold based on work by Andrey Kolmogorov. [n 5] 1957
14th Is the ring of invariants of an algebraic group acting on a polynomial ring always finitely generated? Resolved. Result: no, counterexample was constructed by Masayoshi Nagata. 1959
15th Rigorous foundation of Schubert's enumerative calculus. Partially resolved.[source?] –
16th Describe relative positions of ovals originating from a real algebraic curve and as limit cycles of a polynomial vector field on the plane. Unresolved. –
17th Expression of definite rational function as quotient of sums of squares Resolved by Emil Artin and Charles Delzell. Result: An upper limit was established for the number of square terms necessary. Finding a lower limit is still an open problem. 1927
18th (a) Is there a polyhedron which admits only an anisohedral tiling in three dimensions?
(b) What is the densest sphere packing? 		(a) Resolved. Result: yes (by Karl Reinhardt).
(b) Resolved by Thomas Callister Hales using computer-aided proof. Result: cubic close packing and hexagonal close packing, both of which have a density of approximately 74%.[n 6] 		(a) 1928
(b) 1998
19th Are the solutions of Lagrangians always analytic? Resolved. Result: yes, proven by Ennio de Giorgi and, independently and using different methods, by John Forbes Nash. 1957
20th Do all variational problems with certain boundary conditions have solutions? Resolved. A significant topic of research throughout the 20th century, culminating in solutions[source?] for the non-linear case. –
21st Proof of the existence of linear differential equations having a prescribed monodromic group Resolved. Result: Yes or no, depending on more exact formulations of the problem.[source?] –
22nd Uniformization of analytic relations by means of automorphic functions Resolved.[source?] –
23rd Further development of the calculus of variations Unresolved. –

Other websites

 Wikilivres:Mathematische Probleme – source texts, documents, translations, media

References

About the problems

  1. ↑ According to Gray, most of the problems have been solved. Some were not defined completely, but enough progress has been made to consider them "solved"; Gray lists the fourth problem as too vague to say whether it has been solved.
  2. ↑ The search for an axiomatic description of fundamental physics can be seen as an older wording for the search for the theory of everything.
  3. ↑ Problem 9 has been solved by Emil Artin in 1927 for abelian extensions of the rational numbers during the development of class field theory; the non-abelian case remains unsolved, if one interprets that as meaning non-abelian class field theory.
  4. ↑ D. Hilbert, "¨Uber die Gleichung neunten Grades", Math. Ann. 97 (1927), 243–250
  5. ↑ It is not difficult to show that the problem has a partial solution within the space of single-valued analytic functions (Raudenbush). Some authors argue that Hilbert intended for a solution within the space of (multi-valued) algebraic functions, thus continuing his own work on algebraic functions and being a question about a possible extension of the Galois theory (see, for example, Abhyankar, Shreeram S. Abhyankar: Hilbert's Thirteenth Problem, Vitushkin, A. G. Vitushkin: On Hilbert's thirteenth problem and related questions, Chebotarev (N. G. Chebotarev, "On certain questions of the problem of resolvents") and others). It appears from one of Hilbert's papers [n 4] that this was his original intention for the problem. The language of Hilbert there is "...Existenz von algebraischen Funktionen...", i.e., "...existence of algebraic functions...". As such, the problem is still unresolved.
  6. ↑ Gray also lists the 18th problem as "open" in his 2000 book, because the sphere-packing problem (also known as the Kepler conjecture) was unsolved, but a solution to it has now been claimed (see reference below).

Other references

  1. ↑ Hilbert’s twenty-fourth problem Rüdiger Thiele, American Mathematical Monthly, January 2003
  2. ↑ Koji Nagata "There is no axiomatic system for the quantum theory" International Journal of Theoretical Physics, Volume 48, Issue 12 (2009), Page 3532--3536, DOI 10.1007/s10773-009-0158-z.
  3. ↑ Koji Nagata and Tadao Nakamura "Can von Neumann's theory meet the Deutsch-Jozsa algorithm?" International Journal of Theoretical Physics, Volume 49, Issue 1 (2010), Page 162--170, DOI 10.1007/s10773-009-0189-5.
  4. ↑ Koji Nagata, Chang-Liang Ren, and Tadao Nakamura "Whether quantum computation can be almighty" Advanced Studies in Theoretical Physics, Volume 5, Number 1, (2011), Page 1--14.
  5. ↑ Waldschmidt, Michel (2001), "G/g130020", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4[permanent dead link]