Editing David Hilbert (section)

===23 problems===
{{Main|Hilbert's problems}}
Hilbert put forth a highly influential list consisting of 23 unsolved problems at the [[International Congress of Mathematicians]] in [[Paris]] in 1900. This is generally reckoned as the most successful and deeply considered compilation of open problems ever to be produced by an individual mathematician.{{By whom|date=February 2021}}

After reworking the foundations of classical geometry, Hilbert could have extrapolated to the rest of mathematics. His approach differed from the later "foundationalist" Russell–Whitehead or "encyclopedist" [[Nicolas Bourbaki]], and from his contemporary [[Giuseppe Peano]]. The mathematical community as a whole could engage in problems of which he had identified as crucial aspects of important areas of mathematics.

The problem set was launched as a talk, "The Problems of Mathematics", presented during the course of the Second International Congress of Mathematicians, held in Paris. The introduction of the speech that Hilbert gave said:

{{blockquote|Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries ? What particular goals will there be toward which the leading mathematical spirits of coming generations will strive ? What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose?<ref name="BAMSProblems">{{cite journal | last=Hilbert | first=David |translator-last1=Winston Newson |translator-first1=Mary  |translator-link1=Mary Frances Winston Newson| title=Mathematical problems | journal=Bulletin of the American Mathematical Society | volume=8 | issue=10 | date=1902 | issn=0273-0979 | doi=10.1090/S0002-9904-1902-00923-3 | doi-access=free | pages=437–479}}</ref>}}

He presented fewer than half the problems at the Congress, which were published in the acts of the Congress. In a subsequent publication, he extended the panorama, and arrived at the formulation of the now-canonical 23 Problems of Hilbert (see also [[Hilbert's twenty-fourth problem]]). The full text is important, since the exegesis of the questions still can be a matter of debate when it is asked how many have been solved.

Some of these were solved within a short time. Others have been discussed throughout the 20th century, with a few now taken to be unsuitably open-ended to come to closure. Some continue to remain challenges.

The following are the headers for Hilbert's 23 problems as they appeared in the 1902 translation in the [[Bulletin of the American Mathematical Society]].

: 1. Cantor's problem of the cardinal number of the continuum.
: 2. The compatibility of the arithmetical axioms.
: 3. The equality of the volumes of two tetrahedra of equal bases and equal altitudes.
: 4. Problem of the straight line as the shortest distance between two points.
: 5. Lie's concept of a continuous group of transformations without the assumption of the differentiability of the functions defining the group.
: 6. Mathematical treatment of the axioms of physics.
: 7. Irrationality and transcendence of certain numbers.
: 8. Problems of prime numbers (The "Riemann Hypothesis").
: 9. Proof of the most general law of reciprocity in any number field.
: 10. Determination of the solvability of a Diophantine equation.
: 11. Quadratic forms with any algebraic numerical coefficients
: 12. Extensions of Kronecker's theorem on Abelian fields to any algebraic realm of rationality
: 13. Impossibility of the solution of the general equation of 7th degree by means of functions of only two arguments.
: 14. Proof of the finiteness of certain complete systems of functions.
: 15. Rigorous foundation of Schubert's enumerative calculus.
: 16. Problem of the topology of algebraic curves and surfaces.
: 17. Expression of definite forms by squares.
: 18. Building up of space from congruent polyhedra.
: 19. Are the solutions of regular problems in the calculus of variations always necessarily analytic?
: 20. The general problem of boundary values (Boundary value problems in PDE's).
: 21. Proof of the existence of linear differential equations having a prescribed monodromy group.
: 22. Uniformization of analytic relations by means of automorphic functions.
: 23. Further development of the methods of the calculus of variations.