Discover the famous 23 Hilbert's Problems ...
This is the first of some articles we will publish here, about great mathematical problems. And of course, we start from David Hilbert (January 23, 1862 – February 14, 1943), a great German mathematician who live and work Gottingen University for several years. The same University of Edmund Landau, Carl F. Gauss, Felix Klein and others.
The 8 August of 1900, at times where Tesla, Einstein and other great scientist are still alive, Hilbert list ten problems that represent the state-of-art in Mathematics. He did that in the Soborne, in the Congress of Mathematicians, in Paris.
Today, 108 years later, there are still some problems open, and some others had been solved. Minds like Godel and others get involved in the solution on in the work to find a solution.
We here list all the 23 problems using Wikipedia like a source. However, the book Benjamin H. Yandell - The Honors Class, Hilbert's Problems and Their Solvers offers a complete view.
We include here the original Hilbert's speech in German, and so we open now the door to German Language.
Hilbert's twenty-three problems are:
|Problem ||Brief explanation ||Status |
|1st ||The continuum hypothesis (that is, there is no set whose size is strictly between that of the integers and that of the real numbers) ||Proven to be impossible to prove or disprove within the Zermelo-Frankel set theory with or without the Axiom of Choice. There is no consensus on whether this is a solution to the problem. |
|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. |
|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. |
|4th ||Construct all metrics where lines are geodesics. ||Too vague to be stated resolved or not. |
|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. |
|6th ||Axiomatize all of physics ||Unresolved. |
|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. |
|8th ||The Riemann hypothesis (the real part of any non-trivial zero of the Riemann zeta function is ½) and Goldbach's conjecture (every even number greater than 2 can be written as the sum of two prime numbers). ||Unresolved. |
|9th ||Find most general law of the reciprocity theorem in any algebraic number field ||Partially resolved. |
|10th ||Find an algorithm to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution. ||Resolved. Result: no, Matiyasevich's theorem implies that there is no such algorithm. |
|11th ||Solving quadratic forms with algebraic numerical coefficients. ||Partially resolved. |
|12th ||Extend the Kronecker–Weber theorem on abelian extensions of the rational numbers to any base number field. ||Unresolved. |
|13th ||Solve all 7-th degree equations using functions of two parameters. ||A variant of this problem, looking for a solution within the universe of continuous functions, was solved (negatively) by Andrei Kolmogorov and Vladimir Arnold. It is not difficult to show that the problem has a positive 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, Abyankar , Vitushkin , Chebotarev  and others). It appears from one of the Hilbert's papers  that this was his original intention for the problem. As such, the problem is still unresolved. |
|14th ||Proof of the finiteness of certain complete systems of functions. ||Resolved. Result: no, generally, due to counterexample made by Masayoshi Nagata. |
|15th ||Rigorous foundation of Schubert's enumerative calculus. ||Partially resolved. |
|16th ||Topology of algebraic curves and surfaces. ||Unresolved. |
|17th ||Expression of definite rational function as quotient of sums of squares ||Resolved. Result: An upper limit was established for the number of square terms necessary. |
|18th ||Is there a non-regular, space-filling polyhedron? What is the densest sphere packing? ||Resolved. |
|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. |
|20th ||Do all variational problems with certain boundary conditions have solutions? ||Resolved. A significant area 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 ||Resolved. Result: Yes or no, depending on more exact formulations of the problem. |
|22nd ||Uniformization of analytic relations by means of automorphic functions ||Resolved. |
|23rd ||Further development of the calculus of variations ||Resolved. |