David Hilbert’s 23 problems, presented in 1900, are a famous set of problems that have guided much of 20th-century mathematics. These problems were translated and published in the Bulletin of the American Mathematical Society in 1902. Here’s a brief overview of each:
- Cantor’s Problem of the Cardinal Number of the Continuum: This deals with understanding the different sizes of infinity, particularly the size of the set of real numbers compared to the set of natural numbers.
- The Compatibility of the Arithmetical Axioms: This is about proving the consistency of arithmetic axioms, a foundational question in mathematics.
- The Equality of the Volumes of Two Tetrahedra of Equal Bases and Equal Altitudes: A geometric problem regarding volume comparison of tetrahedra.
- Problem of the Straight Line as the Shortest Distance Between Two Points: This involves the foundations of geometry and understanding the concept of a line.
- Lie’s Concept of a Continuous Group of Transformations Without the Assumption of the Differentiability of the Functions Defining the Group: A problem in the field of group theory, particularly in the analysis of transformation groups.
- Mathematical Treatment of the Axioms of Physics: This problem seeks a more rigorous mathematical foundation for the laws of physics.
- Irrationality and Transcendence of Certain Numbers: It involves proving the irrationality and transcendence of specific types of numbers.
- Problems of Prime Numbers (The “Riemann Hypothesis”): Focused on the distribution of prime numbers, it’s one of the most famous unsolved problems in mathematics.
- Proof of the Most General Law of Reciprocity in Any Number Field: This is a problem in number theory, involving the concept of reciprocity in mathematics.
- Determination of the Solvability of a Diophantine Equation: It’s about finding a general method to determine whether a given Diophantine equation has a solution.
- Quadratic Forms with Any Algebraic Numerical Coefficients: This concerns the theory of quadratic forms and their classification.
- Extensions of Kronecker’s Theorem on Abelian Fields to Any Algebraic Realm of Rationality: A problem in algebraic number theory.
- Impossibility of the Solution of the General Equation of 7th Degree by Means of Functions of Only Two Arguments: This problem deals with the solvability of certain algebraic equations.
- Proof of the Finiteness of Certain Complete Systems of Functions: A problem in the field of mathematical analysis.
- Rigorous Foundation of Schubert’s Enumerative Calculus: This is about providing a firm mathematical foundation for Schubert’s calculus methods.
- Problem of the Topology of Algebraic Curves and Surfaces: A problem in algebraic topology, focusing on the properties of curves and surfaces.
- Expression of Definite Forms by Squares: This involves the representation of certain forms using squares, a problem in algebra.
- Building up of Space from Congruent Polyhedra: A geometric problem about constructing space using polyhedra.
- Are the Solutions of Regular Problems in the Calculus of Variations Always Necessarily Analytic?: This is a problem in the calculus of variations, a field of mathematical analysis.
- The General Problem of Boundary Values (Boundary Value Problems in PDE’s): This concerns boundary value problems in partial differential equations, a significant topic in mathematical physics.
- Proof of the Existence of Linear Differential Equations Having a Prescribed Monodromy Group: A problem in the theory of differential equations.
- Uniformization of Analytic Relations by Means of Automorphic Functions: This involves the theory of automorphic functions and complex analysis.
- Further Development of the Methods of the Calculus of Variations: An extension of problem 19, focusing on the calculus of variations.
Each of these problems has profoundly influenced mathematical research, with some still unsolved to this day.
As of April 2023, the status of Hilbert’s 23 problems is as follows:
- Cantor’s Problem of the Cardinal Number of the Continuum: Not completely resolved. It led to the development of the Continuum Hypothesis, which was later shown to be independent of the standard axiomatic set theory (ZFC) by Paul Cohen and Kurt Gödel.
- The Compatibility of the Arithmetical Axioms: Partially resolved. Gödel’s incompleteness theorems show that within any sufficiently rich axiomatic system, there are true statements that cannot be proven within the system.
- The Equality of the Volumes of Two Tetrahedra of Equal Bases and Equal Altitudes: Solved by Max Dehn in 1900.
- Problem of the Straight Line as the Shortest Distance Between Two Points: This problem was more about developing a rigorous foundation for geometry. It has been extensively explored, though not in the form of a single definitive solution.
- Lie’s Concept of a Continuous Group of Transformations Without the Assumption of the Differentiability of the Functions Defining the Group: Solved in various forms, particularly through the work on topological groups.
- Mathematical Treatment of the Axioms of Physics: Ongoing. This is an extremely broad problem that intersects with many areas of modern theoretical physics.
- Irrationality and Transcendence of Certain Numbers: Partially solved. There have been numerous advances, but many specific cases remain open.
- Problems of Prime Numbers (The “Riemann Hypothesis”): Unsolved and remains one of the most famous open problems in mathematics.
- Proof of the Most General Law of Reciprocity in Any Number Field: Largely solved through the development of class field theory.
- Determination of the Solvability of a Diophantine Equation: Partially solved. Matiyasevich’s theorem (1970) showed that there is no algorithm to determine whether an arbitrary Diophantine equation has a solution.
- Quadratic Forms with Any Algebraic Numerical Coefficients: Solved, with substantial progress made by several mathematicians including Hasse and Minkowski.
- Extensions of Kronecker’s Theorem on Abelian Fields to Any Algebraic Realm of Rationality: Largely solved through the development of class field theory.
- Impossibility of the Solution of the General Equation of 7th Degree by Means of Functions of Only Two Arguments: Solved, with contributions from various mathematicians, including Felix Klein and others.
- Proof of the Finiteness of Certain Complete Systems of Functions: Solved for several cases, but the problem is broad and various aspects continue to be a subject of research.
- Rigorous Foundation of Schubert’s Enumerative Calculus: Solved, with substantial progress in algebraic geometry, particularly through intersection theory.
- Problem of the Topology of Algebraic Curves and Surfaces: Solved, with major contributions by many mathematicians over the 20th century.
- Expression of Definite Forms by Squares: Solved through advancements in the theory of quadratic forms.
- Building up of Space from Congruent Polyhedra: Partially solved, with various advancements in the understanding of tiling and space filling.
- Are the Solutions of Regular Problems in the Calculus of Variations Always Necessarily Analytic?: Solved, with positive answers provided in certain cases but the problem is broad and encompasses various aspects.
- The General Problem of Boundary Values (Boundary Value Problems in PDE’s): Ongoing, as it covers a wide range of problems in partial differential equations.
- Proof of the Existence of Linear Differential Equations Having a Prescribed Monodromy Group: Largely solved through the development of the theory of linear differential equations.
- Uniformization of Analytic Relations by Means of Automorphic Functions: Solved, with significant progress made in complex analysis and related fields.
- Further Development of the Methods of the Calculus of Variations: Ongoing, as it is a broad area with many applications and open questions.
These problems have had a profound impact on the development of mathematics, with some solutions leading to the creation of entirely new fields of study.