{"id":136967,"date":"2026-06-18T14:00:20","date_gmt":"2026-06-18T18:00:20","guid":{"rendered":"https:\/\/www.simonsfoundation.org\/?p=136967"},"modified":"2026-06-18T16:12:38","modified_gmt":"2026-06-18T20:12:38","slug":"hilberts-23-problems-at-icm-2026-where-are-we-now","status":"publish","type":"post","link":"https:\/\/www.simonsfoundation.org\/2026\/06\/18\/hilberts-23-problems-at-icm-2026-where-are-we-now\/","title":{"rendered":"Hilbert\u2019s 23 Problems at ICM 2026: Where Are We Now?"},"content":{"rendered":"","protected":false},"excerpt":{"rendered":"","protected":false},"author":432,"featured_media":137371,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"_relevanssi_hide_post":"","_relevanssi_hide_content":"","_relevanssi_pin_for_all":"","_relevanssi_pin_keywords":"","_relevanssi_unpin_keywords":"","_relevanssi_related_keywords":"","_relevanssi_related_include_ids":"","_relevanssi_related_exclude_ids":"","_relevanssi_related_no_append":"","_relevanssi_related_not_related":"","_relevanssi_related_posts":"","_relevanssi_noindex_reason":"","footnotes":""},"categories":[181],"tags":[],"class_list":["post-136967","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-mathematics-and-physical-science","news_type-article"],"acf":{"block_editor":[{"acf_fc_layout":"image","type":"image_large","image":137365,"title":"","caption":"Sean McCabe for Simons Foundation; Portrait of David Hilbert by Alamy Art and History Collection","caption_full":"","less_margin":false},{"acf_fc_layout":"text","text":"\u201cWho 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?\u201d Mathematician David Hilbert asked this question on August 8, 1900, while addressing attendees of the second-ever International Congress of Mathematicians.\r\n\r\nIn 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. \u201cThey\u2019re the lifeblood of mathematics,\u201d says Jeremy Gray, a historian of mathematics at England\u2019s Open University and author of the 2000 book The Hilbert Challenge<\/em>, an account of the Hilbert Problems and their impact on the subsequent development of mathematics.\r\n\r\nHilbert\u2019s 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.\r\n\r\nNow, 126 years and dozens of ICMs later, Hilbert\u2019s problems are still relevant<\/a>. At this July\u2019s ICM in Philadelphia<\/a>, the first held in the United States since the 1980s, two lectures will provide updates on work probing two of Hilbert\u2019s 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\u2019s top mathematicians, and gratified to see that the problems he posed are still inspiring people in the field."},{"acf_fc_layout":"slideshow","slides":[{"image":137366,"caption":"The second International Congress of Mathematicians (ICM) was held in Paris from August 6 to 12, 1900. It ran alongside the massive Exposition Universelle (World Fair), which made it challenging to organize special excursions for attendees due to the numerous other exhibitions and events. France\u2019s president, \u00c9mile Loubet, can be seen under the Eiffel Tower among other members of the French government. Credit: Pictorial Press Ltd\/Alamy"},{"image":137367,"caption":"Pictured here is the gateway to the Universal Exhibition in Paris of 1900. In the center of the photo is the Eiffel Tower, which was built for the World Exhibition in 1889. Credit: Photos depoque en France\/Alamy"},{"image":137368,"caption":"The Palace of Electricity (behind) and the Water Castle (in front) at the Exposition Universelle, Paris. Credit: Charles Phelps Cushing\/Alamy"},{"image":137369,"caption":"Photo across the river Seine of the Globe C\u00e9leste beside the Eiffel Tower. The Globe C\u00e9leste was built to be a symbol of the Exposition Universelle of 1900 in Paris, much like the Eiffel Tower for the 1889 fair. Credit: Hirarchivum Press\/Alamy"},{"image":137370,"caption":"Lithograph of the panoramic view of the 1900 Universal Exhibition. Credit: Library of Congress; Lithograph by Lucien Baylac (1851\u20131913)"}]},{"acf_fc_layout":"text","text":"

Crafting the 23 Problems<\/h2>\r\nHilbert\u2019s 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. \u201cThe organic unity of mathematics is inherent in the nature of this science,\u201d Hilbert wrote, \u201cfor mathematics is the foundation of all exact knowledge of natural phenomena.\u201d\r\n\r\nHilbert\u2019s 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, \u201cIf I were Hilbert\u2019s lawyer,\u201d Gray says, \u201cI\u2019d say, no, no, no, they\u2019re vague for a purpose.\u201d\r\n\r\nThe vague, ambiguous and ill-posed questions on Hilbert\u2019s 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.\r\n\r\nHilbert\u2019s list could almost seem amateurish in comparison. That word, amateur, comes from the Latin for \u201clove.\u201d An amateur pursues a hobby or activity for enjoyment, not for financial gain, and perhaps Hilbert\u2019s 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, \u201cThe supply of problems in mathematics is inexhaustible, and as soon as one problem is solved, numerous others come forth in its place.\u201d\r\n\r\nReading Hilbert\u2019s address in the 2020s, one is struck by an almost disarming innocence and optimism. He took it as a given \u2014 in mathematical terms, as an axiom \u2014 that there could be no \u201cignorabimus,\u201d the Latin word for \u201cwe will not know.\u201d He believed that every mathematical problem could be solved. Carved on his gravestone are two German sentences: \u201cWir m\u00fcssen wissen. Wir werden wissen.\u201d Or, \u201cWe must know. We shall know.\u201d\r\n\r\nIn 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.\r\n\r\nThe 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 G\u00f6del\u2019s incompleteness theorems dealt a fatal blow to Hilbert\u2019s 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."},{"acf_fc_layout":"quote","text":"\u201cThe organic unity of mathematics is inherent in the nature of this science, for mathematics is the foundation of all exact knowledge of natural phenomena.\u201d","attribution":"David Hilbert"},{"acf_fc_layout":"text","text":"

The 1900 International Congress of Mathematicians<\/h2>\r\nThe 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<\/em>, notes that 1,000 attendees had been anticipated, but only 250 attended. The importance of Hilbert\u2019s address was not immediately obvious, with only \u201cdesultory\u201d discussion afterward, in Scott\u2019s account. But that changed in the years following the presentation, and today Hilbert\u2019s address is considered the meeting's most lasting contribution.\r\n\r\nHilbert\u2019s 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 G\u00f6ttingen University, which during the 19th century had grown into the world\u2019s leading mathematics department. \u201cG\u00f6ttingen was the<\/em> place,\u201d Gray says. \u201cSo, it\u2019s not just the problems, it\u2019s the place and the person.\u201d The luster of Hilbert and G\u00f6ttingen helped cement the problems in the general mathematical consciousness. To solve one of Hilbert\u2019s problems would promote a mathematician to the \u201chonors class,\u201d in the words of Hermann Weyl, a student of Hilbert who would become a leading mathematician of the first half of the 20th century.\r\n\r\nAs Hilbert hoped, his questions, both resolved and not, have spurred countless new investigations. In 1974, the American Mathematical Society held a symposium called \u201cMathematical developments arising from Hilbert problems.\u201d A hefty 600-page book collects its proceedings. Headlines continue to tout new progress or resolutions to Hilbert\u2019s problems, and a Hilbert problem solver\u2019s reputation is assured."},{"acf_fc_layout":"video","embed_code":"https:\/\/vimeo.com\/1040571695?share=copy#t=0","title":"Mathematician Yu Deng (University of Chicago) presenting on Hilbert\u2019s Sixth Problem at the 2024 Simons Collaboration on Wave Turbulence Annual Meeting.","caption":"Simons Foundation","video_autoplay":false,"video_loop":false,"video_width":1980,"video_height":1080,"video_container_size":"video_medium"},{"acf_fc_layout":"text","text":"

ICM 2026 and the Future of Hilbert\u2019s Problems<\/h2>\r\nMore than a century after he gave it, Hilbert\u2019s 1900 address still looms large at the ICM. At this year\u2019s meeting<\/a>, which will take place at the end of July in Philadelphia, two addresses will dive into specific Hilbert problems.\r\n\r\nDavid Aspero and Ralf Schindler will speak on \u201cHilbert\u2019s first problem, revisited,\u201d in a session focusing on logic. Hilbert\u2019s 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\u20131940 by Kurt G\u00f6del and in 1963\u20131964 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\u2019s question. G\u00f6del believed that the continuum hypothesis was false and sought the \u201cright\u201d 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.\r\n\r\nYu 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\u2019s 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\u2019s 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."},{"acf_fc_layout":"text","text":"To see where the problems stand in 2026, read our list breaking down the status of each of Hilbert\u2019s 23 problems here<\/a>.<\/strong>"}]},"_links":{"self":[{"href":"https:\/\/www.simonsfoundation.org\/wp-json\/wp\/v2\/posts\/136967"}],"collection":[{"href":"https:\/\/www.simonsfoundation.org\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.simonsfoundation.org\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.simonsfoundation.org\/wp-json\/wp\/v2\/users\/432"}],"replies":[{"embeddable":true,"href":"https:\/\/www.simonsfoundation.org\/wp-json\/wp\/v2\/comments?post=136967"}],"version-history":[{"count":13,"href":"https:\/\/www.simonsfoundation.org\/wp-json\/wp\/v2\/posts\/136967\/revisions"}],"predecessor-version":[{"id":137616,"href":"https:\/\/www.simonsfoundation.org\/wp-json\/wp\/v2\/posts\/136967\/revisions\/137616"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.simonsfoundation.org\/wp-json\/wp\/v2\/media\/137371"}],"wp:attachment":[{"href":"https:\/\/www.simonsfoundation.org\/wp-json\/wp\/v2\/media?parent=136967"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.simonsfoundation.org\/wp-json\/wp\/v2\/categories?post=136967"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.simonsfoundation.org\/wp-json\/wp\/v2\/tags?post=136967"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}