Hilbert's Problems — Definition, Formula & Examples
Hilbert's Problems is a list of 23 unsolved mathematical problems presented by German mathematician David Hilbert at the International Congress of Mathematicians in Paris in 1900. They span number theory, algebra, geometry, analysis, and mathematical physics, and have guided major research directions for over a century.
Hilbert's Problems constitute a collection of 23 open questions posed by David Hilbert on August 8, 1900, intended to outline the most important challenges facing mathematics at the turn of the 20th century. The problems range from foundational questions about the consistency of arithmetic (Problem 2) to specific conjectures in number theory (Problem 8, which includes the Riemann Hypothesis), and their resolution — or continued openness — has profoundly influenced the development of modern mathematical disciplines.
How It Works
Hilbert selected problems he believed would drive mathematical progress if solved. Some, like Problem 3 (on decomposing polyhedra), were resolved within a few years. Others, like Problem 1 (the Continuum Hypothesis), led to deep foundational insights — Gödel and Cohen showed it is independent of standard set theory, meaning it can be neither proved nor disproved from the ZFC axioms. Several problems remain open or only partially resolved, including aspects of Problem 8 (the Riemann Hypothesis). The status of each problem is typically classified as resolved, partially resolved, or open.
Example
Problem: Hilbert's Problem 3 asks: given two polyhedra of equal volume, can one always be cut into finitely many polyhedral pieces and reassembled into the other?
The Question: Hilbert conjectured the answer is no — that scissors congruence (cutting and rearranging) does not always work in three dimensions, unlike in two dimensions where any two equal-area polygons are scissors congruent.
The Resolution: In 1900, Max Dehn (Hilbert's student) defined an invariant now called the Dehn invariant. He showed that a cube and a regular tetrahedron of equal volume have different Dehn invariants, so one cannot be cut into pieces and reassembled into the other.
Significance: This was the first of Hilbert's 23 problems to be solved. The Dehn invariant remains a key concept in geometric topology.
Answer: The answer to Problem 3 is no: a cube and a regular tetrahedron of equal volume are not scissors congruent, as proved by Dehn in 1900 using a new geometric invariant.
Visualization
Why It Matters
Hilbert's Problems set the research agenda for much of 20th-century mathematics. Solving or partially solving these problems led to entirely new fields, including mathematical logic (Problems 1 and 2), algebraic number theory (Problem 9), and modern foundations of physics (Problem 6). Graduate students in pure mathematics routinely encounter these problems as motivation for the theories they study.
Common Mistakes
Mistake: Assuming all 23 problems have been definitively solved or definitively remain open.
Correction: The status varies widely. Some are fully resolved (Problem 3), some are shown to be undecidable within standard axioms (Problem 1), some are partially resolved, and some — notably the Riemann Hypothesis in Problem 8 — remain open. Several problems were stated broadly enough that their "resolution" is debated.