Hilbert's 23 Problems: The Blueprint for 20th-Century Mathematics

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Hilbert's 23 Problems: The Blueprint for 20th-Century Mathematics | Art & Curiosity | Tobias Weiss

Introduction

On the morning of August 8, 1900, the German mathematician David Hilbert stood before the International Congress of Mathematicians at the Sorbonne in Paris and delivered what would become the most influential lecture in the history of mathematics. Rather than celebrating past achievements, he looked resolutely forward:

"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 the future centuries?"

With those words, Hilbert unveiled a list of 23 unsolved problems, chosen as a roadmap for the mathematics of the coming century. The gamble succeeded beyond all measure: Hilbert's problems became a guiding star for generations of mathematicians, and their (partial or complete) solutions shaped virtually every branch of modern mathematics.

The Man and the Moment

David Hilbert (1862–1943) was already one of the most respected mathematicians in the world by 1900. His work on invariant theory, algebraic number theory (the Zahlbericht), and the foundations of geometry had established his reputation. But Hilbert was also a visionary who understood that mathematics stood at a crossroads.

The 19th century had been a golden age: Gauss, Riemann, Dedekind, Cantor, and Poincaré had transformed the mathematical landscape. Yet many foundational questions remained unsettled. The paradoxes of set theory had shaken confidence in the very underpinnings of mathematics. Hilbert believed that the right set of problems could focus the discipline and drive it forward.

He prepared the list in close consultation with his friends Hermann Minkowski and Adolf Hurwitz, who provided suggestions and critical feedback. Hilbert originally intended to present all 23 problems in his lecture but time constraints forced him to present only ten (numbers 1, 2, 6, 7, 8, 13, 16, 19, 21, and 22). The complete list was published later that year.

Among the remarkable aspects of Hilbert's problems is their extraordinary range. They span mathematical logic, set theory, number theory, algebra, geometry, topology, differential equations, the calculus of variations, and even the axiomatisation of physics. No single problem set since has matched this breadth.

The 23 Problems

The problems are summarised below, grouped by their disposition at the time of writing (2026):

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Note: This classification reflects contemporary mathematical consensus. Some problems, like #8 (the Riemann hypothesis) and #12 (Kronecker's theorem extensions), remain as open today as they were in 1900, while others were resolved with surprising speed.

The Problems in Detail

The following table presents the complete list. The status column reflects the best current understanding of each problem's resolution as of 2026.

#	Problem	Field	Status	Key Contributor(s)
1	Cantor's continuum hypothesis: is there a set whose cardinality lies strictly between the integers and the reals?	Set theory / Logic	Disputed	Gödel (1940), Cohen (1963)
2	Consistency of arithmetic: prove that the axioms of arithmetic are free of contradiction	Logic	Disputed	Gödel (1931), Gentzen (1936)
3	Scissor congruence of polyhedra: can two polyhedra of equal volume always be cut into congruent pieces?	Geometry	Solved	Dehn (1900)
4	Geometry of geodesics: construct all metric spaces where lines are shortest paths	Geometry	Vague	Hamel, Busemann
5	Continuous transformation groups: are continuous groups automatically differentiable (Lie groups)?	Topology / Algebra	Partial	Von Neumann (1933), Gleason (1953)
6	Axiomatisation of physics: treat physics via the axiomatic method	Physics / Foundations	Partial	Kolmogorov (1933), recent (2025)
7	Transcendence of $a^b$ : for algebraic $a \neq 0,1$ and irrational algebraic , is transcendental?

Landmark Solutions

Problem 3: The First to Fall (1900)

Hilbert's third problem asked whether two polyhedra of equal volume are always scissor-congruent — that is, can one be cut into finitely many pieces and reassembled to form the other? In the plane, the analogous statement is true (the Bolyai–Gerwien theorem), so it was natural to ask about the third dimension.

Max Dehn, a student of Hilbert, solved the problem in the very same year (1900) — making it the first Hilbert problem to be resolved. He introduced what is now called the Dehn invariant, a real number associated with every polyhedron. If two polyhedra are scissor-congruent, their Dehn invariants must be equal. Dehn showed that a regular tetrahedron and a cube of equal volume have distinct Dehn invariants, so they cannot be cut and reassembled into each other.

Problem 7: The Gelfond–Schneider Theorem (1934)

Hilbert asked: if $a$ is algebraic ( $a \neq 0, 1$ ) and $b$ is algebraic and irrational, is $a^b$ necessarily transcendental? This generalises the ancient question of whether $2^{\sqrt{2}}$ is transcendental.

In 1934, the Soviet mathematician Alexander Gelfond and, independently, Theodor Schneider proved that the answer is yes. The theorem now bears both their names:

Gelfond–Schneider Theorem: If $\alpha$ and $\beta$ are algebraic numbers with $\alpha \notin \{0,1\}$ and $\beta$ irrational, then $\alpha^\beta$ is transcendental.

This result famously implies that $2^{\sqrt{2}}$ and $e^\pi = (-1)^{-i}$ (by taking , , noting ) are transcendental.

Problem 10: The Algorithm That Could Not Exist (1970)

Hilbert's tenth problem asked for an algorithm to decide whether any given Diophantine equation (a polynomial equation with integer coefficients) has an integer solution. This was a fundamentally optimistic question: Hilbert believed that every mathematical problem is solvable.

The resolution, completed in 1970 by Yuri Matiyasevich (building on foundational work by Martin Davis, Hilary Putnam, and Julia Robinson), delivered a stunning negative answer: no such algorithm exists. Matiyasevich's theorem, also known as the Matiyasevich–Robinson–Davis–Putnam theorem, proves that the set of solvable Diophantine equations is recursively enumerable but not recursive — there is no general procedure that can determine solvability in finitely many steps.

This result effectively refuted Hilbert's philosophical position that mathematics has no ignorabimus (things we can never know). It was also a landmark in the theory of computation, connecting number theory to the halting problem.

The Philosophical Legacy: Hilbert's Program and Gödel

Hilbert's second problem asked for a proof of the consistency of the axioms of arithmetic — a cornerstone of what became known as Hilbert's program. The idea was to establish all of mathematics on a firm, finitistic axiomatic foundation, thereby banishing the paradoxes that had plagued set theory.

In 1931, Kurt Gödel delivered a devastating blow. His second incompleteness theorem proved that any consistent formal system strong enough to express arithmetic cannot prove its own consistency. Hilbert's program was impossible in its original form — unless one was willing to accept methods (like transfinite induction up to $\varepsilon_0$ , as Gerhard Gentzen showed in 1936) that went beyond the finitistic framework Hilbert had envisioned.

Hilbert lived for another 12 years after Gödel's publication, but he never wrote a formal response to the incompleteness theorems. His motto remained defiant:

"Wir müssen wissen — wir werden wissen." (We must know — we shall know.)

This famous statement, delivered in 1930 — just one year before Gödel's paper — was inscribed on Hilbert's tombstone in Göttingen.

The 24th Problem

In 2000, the German historian Rüdiger Thiele discovered an additional problem among Hilbert's original manuscript notes: a 24th problem on criteria for the simplicity of proofs and the development of a general proof theory. Hilbert had decided to omit it from the final list, but its existence reveals that he was already thinking about the concept of mathematical elegance and the structure of proofs themselves.

The Millennium Problems: A 21st-Century Sequel

In 2000, exactly one century after Hilbert's lecture, the Clay Mathematics Institute announced a new set of seven Millennium Prize Problems, each carrying a $1 million award for its solution. Several of the Millennium Problems overlap directly with Hilbert's:

Millennium Problem	Relation to Hilbert
Riemann Hypothesis	Hilbert's Problem 8
Poincaré Conjecture	Solved by Perelman (2003) — not in Hilbert
P vs NP	Not in Hilbert (computational complexity)
Navier–Stokes Existence	Distantly related to Problem 6
Yang–Mills Existence	Related to Problem 6
Hodge Conjecture	Not directly in Hilbert
Birch–Swinnerton-Dyer Conjecture	Distantly related to Problem 10

To date, only the Poincaré conjecture (proved by Grigori Perelman in 2003) has been solved among the Millennium Problems. The Riemann hypothesis, Hilbert's eighth problem, remains open for both lists.

Conclusion

More than 125 years after Hilbert's lecture, his problems continue to shape mathematics. Some have been solved and have become standard textbook material; others remain as elusive as ever; and a few were rendered obsolete or ill-posed by the very developments they inspired.

What made Hilbert's list uniquely powerful was not the individual problems themselves — many of them were already being actively studied — but the fact that they were presented together as a coherent vision for the future of mathematics. Hilbert told his colleagues: here is where we are going. And remarkably, the mathematical community followed.

As Hilbert himself expressed it in his 1900 lecture:

"The mathematical problems have been solved in many different senses, as we know. A mathematical problem, as we conceive it, is a question the answer to which is either a proof or a demonstration that the problem in question is incapable of solution. This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignorabimus."

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