Hilbert’s 23 Problems at ICM 2026: Where Are We Now?

“Who among 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?” Mathematician David Hilbert asked this question on August 8, 1900, while addressing attendees of the second-ever International Congress of Mathematicians.

In his address, Hilbert laid out 10 problems that he believed represented the future of mathematics. (In the later published version, he expanded the list to 23 problems.) To Hilbert, mathematics was driven by problems. “They’re the lifeblood of mathematics,” says Jeremy Gray, a historian of mathematics at England’s Open University and author of the 2000 book The Hilbert Challenge, an account of the Hilbert Problems and their impact on the subsequent development of mathematics.

Hilbert’s address teases out the interplay between problems and theory: The right math problems spur the creation of interesting theoretical frameworks, and, in turn, fruitful mathematical theories burst with new problems to solve.

Now, 126 years and dozens of ICMs later, Hilbert’s problems are still relevant. At this July’s ICM in Philadelphia, the first held in the United States since the 1980s, two lectures will provide updates on work probing two of Hilbert’s problems. Hilbert would no doubt be astounded by the breadth of expertise represented at the conference, which will be attended by thousands of the world’s top mathematicians, and gratified to see that the problems he posed are still inspiring people in the field.

Crafting the 23 Problems

Hilbert’s 1900 address emphasized the unity of mathematics. Modern mathematics can feel like a fractured discipline: A number theorist may be completely unable to understand a paper about differential geometry, and vice versa. But in producing his 23 problems, Hilbert wished to promote unity in the discipline and the common cause for all mathematicians of pursuing rigorous truth. “The organic unity of mathematics is inherent in the nature of this science,” Hilbert wrote, “for mathematics is the foundation of all exact knowledge of natural phenomena.”

Hilbert’s problems span a wide swath of mathematics, though many have noted some puzzling (and perhaps pointed) gaps in the fields and topics he covered. He includes questions about the foundations of mathematics, number theory, geometry, analysis and algebra. The questions also vary markedly in their specificity and level of precision. At first glance, several seem too vague to be meaningful. Yet, “If I were Hilbert’s lawyer,” Gray says, “I’d say, no, no, no, they’re vague for a purpose.”

The vague, ambiguous and ill-posed questions on Hilbert’s list give mathematicians some breathing room. In 2000, the Clay Mathematics Institute issued another list of problems, known as the Millennium Problems, that they deemed the most important open questions in the field. Each problem has a $1 million bounty on its head. The statements of these problems are watertight and precise. There is no room for interpretation. With $1 million at stake for each, there can be no debate about what would count as a solution.

Hilbert’s list could almost seem amateurish in comparison. That word, amateur, comes from the Latin for “love.” An amateur pursues a hobby or activity for enjoyment, not for financial gain, and perhaps Hilbert’s problems are amateurish in that sense as well. His love for mathematics radiates from the page. If his questions are vague, they are also open ended, allowing mathematicians to interpret some of them in different ways. Many may never be considered fully resolved because, as he wrote, “The supply of problems in mathematics is inexhaustible, and as soon as one problem is solved, numerous others come forth in its place.”

Reading Hilbert’s address in the 2020s, one is struck by an almost disarming innocence and optimism. He took it as a given — in mathematical terms, as an axiom — that there could be no “ignorabimus,” the Latin word for “we will not know.” He believed that every mathematical problem could be solved. Carved on his gravestone are two German sentences: “Wir müssen wissen. Wir werden wissen.” Or, “We must know. We shall know.”

In the last years of the 19th century, Hilbert had been working on a new axiomatic foundation for the field of geometry, and he hoped to continue that work in other disciplines, seeking the perfect set of axioms that would put other fields of mathematics on the same sure footing. He was certain that with the right axioms, all of mathematics would be knowable.

The 19th century was a time of increasing mathematical rigor and a proliferation of new disciplines and techniques. In the 20th century, mathematicians discovered their limitations. Unknowability was one of the major undercurrents of mathematics over the course of the century. In 1931, Kurt Gödel’s incompleteness theorems dealt a fatal blow to Hilbert’s goal of complete knowledge by showing that any axiomatic system that can support basic arithmetic must contain statements that can be neither proved nor disproved.

The 1900 International Congress of Mathematicians

The 1900 Paris ICM was plagued with organizational failures and a disappointing turnout. Charlotte Angas Scott, an English-American mathematician whose account of the meeting was published in the Bulletin of the American Mathematical Society, notes that 1,000 attendees had been anticipated, but only 250 attended. The importance of Hilbert’s address was not immediately obvious, with only “desultory” discussion afterward, in Scott’s account. But that changed in the years following the presentation, and today Hilbert’s address is considered the meeting’s most lasting contribution.

Hilbert’s problems gained in prominence and prestige in the early decades of the 20th century. Hilbert himself was 38 years old and already one of the most respected mathematicians of his generation at the time of the second ICM. He was in the mathematics department at Göttingen University, which during the 19th century had grown into the world’s leading mathematics department. “Göttingen was the place,” Gray says. “So, it’s not just the problems, it’s the place and the person.” The luster of Hilbert and Göttingen helped cement the problems in the general mathematical consciousness. To solve one of Hilbert’s problems would promote a mathematician to the “honors class,” in the words of Hermann Weyl, a student of Hilbert who would become a leading mathematician of the first half of the 20th century.

As Hilbert hoped, his questions, both resolved and not, have spurred countless new investigations. In 1974, the American Mathematical Society held a symposium called “Mathematical developments arising from Hilbert problems.” A hefty 600-page book collects its proceedings. Headlines continue to tout new progress or resolutions to Hilbert’s problems, and a Hilbert problem solver’s reputation is assured.

ICM 2026 and the Future of Hilbert’s Problems

More than a century after he gave it, Hilbert’s 1900 address still looms large at the ICM. At this year’s meeting, which will take place at the end of July in Philadelphia, two addresses will dive into specific Hilbert problems.

David Aspero and Ralf Schindler will speak on “Hilbert’s first problem, revisited,” in a session focusing on logic. Hilbert’s first problem is about the continuum hypothesis. In 1870, Georg Cantor demonstrated that although both sets are infinite, there are more real numbers than whole numbers. The continuum hypothesis is the statement that there is no infinity larger than the infinity of the whole numbers and smaller than the infinity of the real numbers. From work in 1939–1940 by Kurt Gödel and in 1963–1964 by Paul Cohen, mathematicians know that the continuum hypothesis is independent of the usual axioms of set theory. But that may not be a fully satisfactory resolution to Hilbert’s question. Gödel believed that the continuum hypothesis was false and sought the “right” set of axioms that would supplement the standard axiomatic framework and allow mathematicians to prove that. Aspero and Schindler will present their work continuing that search.

Yu Deng will speak about the work that he and coauthors Zaher Hani and Xiao Ma have recently done on the sixth problem in a session about partial differential equations. Hilbert’s sixth question was about axiomatizing physics. In 1900, much of modern physics had not yet been developed. Relativity and quantum theory, for example, were still in the future. Hilbert left room for various directions, but he had his eye on two branches of physics in particular: probability and fluid mechanics. The former was settled in 1930 by Soviet mathematician Andrey Kolmogorov. The latter is the process of moving from Newton’s laws describing the atomic-level interactions within a fluid to their macroscopic manifestation. Deng and his coauthors have painstakingly completed that derivation from the laws of atomic interactions to fluid dynamics, from both the particle and the wave viewpoints.