希尔伯特的23个数学问题

1900年，德国数学家希尔伯特向全世界数学家提出23个待解决非数学问题，给二十世纪数学发展指明了方向，指出了数学发展的重点是什么？

然而，很可惜的是，国内对希尔伯特的23个数学问题没有完整的介绍，本文附件列出全部希尔伯特23个数学问题，然后再进行分析。

袁萌陈启清 1月21日

附件：Hilbert’s 23 problems

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 Zermelo–Fraenkel set theory with or without the Axiom of Choice (provided Zermelo–Fraenkel set theory is consistent, i.e., it does not contain a contradiction). There is no consensus on whether this is a solution to the problem.

1940, 1963

2nd

Prove that the axioms of arithmetic are consistent（无矛盾性）.

There is no consensus on whether results of G?del and Gentzen give a solution to the problem as stated by Hilbert. G?del's second incompleteness theorem, proved in 1931, shows that no proof of its consistency can be carried out within arithmetic itself. Gentzen proved in 1936 that the consistency of arithmetic follows from the well-foundedness of the ordinal ε?.

1931, 1936

3rd

Given 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?

Resolved. Result: No, proved using Dehn invariants.

1900

4th

Construct all metrics where lines are geodesics.

Too vague to be stated resolved or not.[h]

—

5th

Are continuous groups automatically differential groups?

Resolved by Andrew Gleason, 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

Mathematical treatment of the axioms of physics

(a) axiomatic treatment of probability with limit theorems for foundation of statistical physics

(b) the rigorous theory of limiting processes "which lead from the atomistic view to the laws of motion of continua"

Partially resolved depending on how the original statement is interpreted.[9] Items (a) and (b) were two specific problems given by Hilbert in a later explanation.[citation needed] Kolmogorov's axiomatics (1933) is now accepted as standard. There is some success on the way from the "atomistic view to the laws of motion of continua."[10]

1933–2002?

7th

Is ab transcendental, for algebraic a ≠ 0,1 and irrational algebraic b ?

Resolved. Result: Yes, illustrated by Gelfond's theorem or the Gelfond–Schneider theorem.

1934

8th

The Riemann hypothesis

("the real part of any non-trivial zero of the Riemann zeta function is ?")

and other prime number problems, among them Goldbach's conjecture and the twin prime conjecture

Unresolved.

—

9th

Find the most general law of the reciprocity theorem in any algebraic number field.

Partially resolved.[i]

—

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.[11]

—

12th

Extend the Kronecker–Weber theorem on Abelian extensions of the rational numbers to any base number field.

Unresolved.

—

13th

Solve 7-th degree equation using algebraic (variant: continuous) functions of two parameters.

The problem was partially solved by Vladimir Arnold based on work by Andrei Kolmogorov.[j]

1957

14th

Is the ring of invariants of an algebraic group acting on a polynomial ring always finitely generated?

Resolved. Result: No, a counterexample was constructed by Masayoshi Nagata.

1959

15th

Rigorous foundation of Schubert's enumerative calculus.

Partially resolved.

—

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, even for algebraic curves of degree 8.

—

17th

Express a nonnegative rational function as quotient of sums of squares.

Resolved. Result: Yes, due to Emil Artin. Moreover, an upper limit was established for the number of square terms necessary.

1927

18th

(a) Is there a polyhedron that admits only an anisohedral tiling in three dimensions?

(b) What is the densest sphere packing?

(a) Resolved. Result: Yes (by Karl Reinhardt).

(b) Widely believed to be resolved, by computer-assisted proof (by Thomas Callister Hales). Result: Highest density achieved by close packings, each with density approximately 74%, such as face-centered cubic close packing and hexagonal close packing.[k]

(a) 1928

(b) 1998

19th

Are the solutions of regular problems in the calculus of variations always necessarily 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 for the non-linear case.

21st

Proof of the existence of linear differential equations having a prescribed monodromic group

Partially resolved. Result: Yes/No/Open depending on more exact formulations of the problem.

22nd

Uniformization of analytic relations by means of automorphic functions

Unresolved.

23rd

Further development of the calculus of variations

Too vague to be stated resolved or not.

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