Hilbertsche Probleme
Die hilbertschen Probleme sind eine Liste von 23 Problemen der Mathematik. Sie wurden von dem deutschen Mathematiker David Hilbert am 8. August 1900 beim Internationalen Mathematiker-Kongress in Paris vorgestellt und waren zu diesem Zeitpunkt ungelöst.
希尔伯特问题（德语：Hilbertsche Probleme）是德国数学家大卫·希尔伯特于1900年在巴黎举行的第二届国际数学家大会上作了题为《数学问题》的演讲，所提出23道最重要的数学问题。希尔伯特问题对推动20世纪数学的发展起了积极的推动作用。在许多数学家努力下，希尔伯特问题中的大多数在20世纪中得到了解决。
希尔伯特问题中未能包括拓扑学、微分几何等领域，除数学物理外很少涉及应用数学，更不曾预料到电脑的发展将对数学产生重大影响。20世纪数学的发展实际上远远超出了希尔伯特所预示的范围。
希尔伯特问题中的1-6是数学基础问题，7-12是数论问题，13-18属于代数和几何问题，19-23属于数学分析。
以下列出希尔伯特的23个问题，各问题的解答状况可参见各问题条目。
| # | 主旨 | 进展 | 说明 |
|---|---|---|---|
| 第1题 | 连续统假设 | 部分解决 | 1963年美国数学家保罗·柯恩以力迫法证明连续统假设不能由策梅洛-弗兰克尔集合论（无论是否含选择公理）推导。也就是说，连续统假设成立与否无法由ZF/ZFC确定。 |
| 第2题 | 算术公理之相容性 | 部分解决 | 库尔特·哥德尔在1931年证明了哥德尔不完备定理，但此定理是否已回答了希尔伯特的原始问题，数学界没有共识。 |
| 第3题 | 两四面体有相同体积之证明法 | 已解决 | 答案：否。1900年，希尔伯特的学生马克斯·登以一反例证明了是不可以的。 |
| 第4题 | 建立所有度量空间使得所有线段为测地线 | 已解决 | 满足此性质的几何很多，因而需要加以某些限制条件。在对称距离情况下，问题获解决。 |
| 第5题 | 所有连续群是否皆为可微群 | 已解决 | 1953年日本数学家山边英彦证明在无“小的子群”情况下，答案是肯定的[1]；但此定理是否已回答了希尔伯特的原始问题，数学界仍有争论。 |
| 第6题 | 公理化物理 | 部分解决 | 希尔伯特后来对这个问题进一步解释，而他自己也进一步研究这个问题。柯尔莫哥洛夫对此也有贡献。然而，尽管公理化已经开始渗透到物理当中，量子力学中仍有至今不能逻辑自洽的部分（如量子场论），故该问题未完全解决。 |
| 第7题 | 若b是无理数、a是除0、1之外的代数数，那么ab是否超越数 | 已解决 | 答案：是。分别于1934年、1935年由苏联数学家亚历山大·格尔丰德与德国数学家特奥多尔·施耐德独立地解决。 |
| 第8题 | 黎曼猜想及哥德巴赫猜想和孪生素数猜想 | 未解决 | 虽然分别有比较重要的突破和被解决的弱化情况，三个问题均仍未被解决。 |
| 第9题 | 任意代数数域的一般互反律 | 部分解决 | 1927年德国的埃米尔·阿廷证明在阿贝尔扩张的情况下答案是肯定的；此外的情况则尚未证明。 |
| 第10题 | 不定方程可解性 | 已解决 | 答案：否。1970年由苏联数学家尤里·马季亚谢维奇证明。 |
| 第11题 | 代数系数之二次形式 | 部分解决 | 有理数的部分由哈塞于1923年解决。 |
| 第12题 | 一般代数数域的阿贝尔扩张 | 未解决 | 埃里希·赫克于1912年用希尔伯特模形式研究了实二次域的情形。虚二次域的情形用复乘理论已基本解决。一般情况下则尚未解决。 |
| 第13题 | 以二元函数解任意七次方程 | 部分解决 | 1957年苏联数学家柯尔莫哥洛夫和弗拉基米尔·阿诺尔德证明对于单值解析函数，答案是否定的；然而希尔伯特原本可能希望证明的是代数函数的情形，因此该问题未获得完全解答。 |
| 第14题 | 证明一些函数完全系统（Complete system of functions）之有限性 | 已解决 | 答案：否。1962年日本人永田雅宜提出反例。 |
| 第15题 | 舒伯特演算之严格基础 | 部分解决 | 一部分在1938年由范德瓦尔登得到严谨的证明。 |
| 第16题 | 代数曲线及表面之拓扑结构 | 未解决 | 此问题进展缓慢，即使对于度为8的代数曲线也没有证明。 |
| 第17题 | 把有理函数写成平方和分式 | 已解决 | 答案：是。1927年埃米尔·阿廷解决此问题，并提出实封闭域。[2][3] |
| 第18题 | 非正多面体能否密铺空间、球体最紧密的排列 | 已解决 | 1911年比伯巴赫做出“n维欧氏几何空间只允许有限多种两两不等价的空间群”；莱因哈特证明不规则多面体亦可填满空间；托马斯·黑尔斯于1998年提出了初步证明，并于2014年8月10日用计算机完成了开普勒猜想的形式化证明，证明球体最紧密的排列是面心立方和六方最密两种方式。 |
| 第19题 | 拉格朗日系统（Lagrangian）之解是否皆可解析 | 已解决 | 答案：是。1956年至1958年恩尼奥·德乔吉和约翰·福布斯·纳什分别用不同方法证明。 |
| 第20题 | 所有边值问题是否都有解 | 已解决 | 实际上工程和科研中遇到的边值问题都是适定的，因而都可以确定是否有解。[4] |
| 第21题 | 证明有线性微分方程有给定的单值群（monodromy group） | 已解决 | 此问题的答案取决于问题的表述：部分情况下是肯定的，部分情况下则是否定的。 |
| 第22题 | 将解析关系（analytic relations）以自守函数一致化 | 部分解决 | 1904年由保罗·克伯和庞加莱取得部分解决。详见单值化定理。 |
| 第23题 | 变分法的长远发展 | 开放性问题 | 包括希尔伯特本人、昂利·勒贝格、雅克·阿达马等数学家皆投身于此。理查德·贝尔曼提出的动态规划可作为变分法的替代。 |
| 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 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. | 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 proved in 1936 that the consistency of arithmetic follows from the well-foundedness of the ordinal ε0. | 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, assuming one interpretation of the original statement. 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.[14] Items (a) and (b) were two specific problems given by Hilbert in a later explanation.[1] 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".[15] | 1933–2002? |
| 7th | Is ab transcendental, for algebraic a ≠ 0,1 and irrational algebraic b ? | Resolved. Result: Yes, illustrated by the Gelfond–Schneider theorem. | 1934 |
| 8th | The Riemann hypothesis ("the real part of any non-trivial zero of the Riemann zeta function is 1/2") 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.[16] | — |
| 12th | Extend the Kronecker–Weber theorem on Abelian extensions of the rational numbers to any base number field. | Partially resolved.[17] | — |
| 13th | Solve 7th-degree equation using algebraic (variant: continuous) functions of two parameters. | Unresolved. The continuous variant of this problem was solved by Vladimir Arnold in 1957 based on work by Andrei Kolmogorov, but the algebraic variant is unresolved.[j] | — |
| 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.[citation needed] | — |
| 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) Are there only finitely many essentially different space groups in n-dimensional Euclidean space? | Resolved. Result: Yes (by Ludwig Bieberbach) | 1910 |
| (b) Is there a polyhedron that admits only an anisohedral tiling in three dimensions? | Resolved. Result: Yes (by Karl Reinhardt). | 1928 | |
| (c) What is the densest sphere packing? | 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] | 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 | Partially resolved. Uniformization theorem | ? |
| 23rd | Further development of the calculus of variations | Too vague to be stated resolved or not. | — |