On this page
Most lectures are forgotten within a week. David Hilbert's talk in Paris in 1900 set the direction of mathematics for a century, because instead of reporting what he had done, he listed what nobody had done yet. This post covers who Hilbert was, why he chose problems over results, and what happened to six of them.
Who was David Hilbert?
According to his MacTutor biography, Hilbert was born on 23 January 1862 at Wehlau, near Königsberg in Prussia, and died in Göttingen on 14 February 1943. He did not shine at his first school, which stressed memorising Latin and Greek; he later called himself a "dull and silly" boy there. At his final school, where original thinking was encouraged, he received the top grade in mathematics.
He studied at the University of Königsberg, taught there until 1895, and then moved to Göttingen at Felix Klein's invitation, where he spent the rest of his career. His work ranged from number theory and the foundations of geometry to physics, and the idea of Hilbert space, used in quantum mechanics, grew from his work on integral equations.
What happened in Paris in 1900?
At the Second International Congress of Mathematicians in Paris in August 1900, Hilbert gave a lecture called Mathematical Problems. MacTutor's biographies of Max Dehn and Yuri Matiyasevich note that he discussed only ten problems in the talk, while the published version contains all 23.
The English translation by Mary Winston Newson appeared in the Bulletin of the American Mathematical Society in 1902, and a version of the text is hosted by Clark University. It opens: "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 future centuries?"
Why did Hilbert think problems mattered so much?
He said it directly in the lecture: "As long as a branch of science offers an abundance of problems, so long is it alive; a lack of problems foreshadows extinction or the cessation of independent development."
He also described what makes a problem good. It should be clear, and "a mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock at our efforts." And he stated his belief that every problem can be settled one way or the other: "in mathematics there is no ignorabimus", Latin for "we shall not know."
What happened to the problems?
Here are six, with what became of them.
Problem 1: Cantor's continuum hypothesis. Is there an infinity between the counting numbers and the real numbers? Kurt Gödel showed in 1938 that the standard axioms of set theory cannot disprove it, and Paul Cohen showed in 1963 that they cannot prove it either. The problem was answered, but not with a yes or a no.
Problem 2: the consistency of arithmetic. Hilbert asked for a proof that the axioms of arithmetic never lead to a contradiction. Gödel's 1931 paper on undecidable propositions showed, in the words of his MacTutor biography, that the consistency of such a system cannot be proved within the system itself, a severe blow to Hilbert's hopes.
Problem 3: cutting up solids. If two polyhedra have the same volume, can one always be cut into finitely many pieces and reassembled into the other? Hilbert noted that the matching problem for flat polygons had already been solved. Max Dehn showed that for solids the answer is no, and submitted his proof before the end of 1900. It was the first of the problems to be solved.
Problem 7: transcendental numbers. Hilbert asked whether a number like 2 to the power √2, his own example, is transcendental or at least irrational. MacTutor's biography of Aleksandr Gelfond records that in 1934 he proved that a to the power b is transcendental whenever a is algebraic and not 0 or 1, and b is an irrational algebraic number; Schneider proved it independently. So 2 to the power √2 is transcendental.
Problem 8: prime numbers, including the Riemann hypothesis. Still open. The Clay Mathematics Institute made the Riemann hypothesis one of its seven Millennium Prize Problems in 2000, with a prize of one million dollars, and notes that it also appears in Hilbert's list.
Problem 10: solving Diophantine equations. Hilbert asked for a process that decides, in finitely many steps, whether any polynomial equation with integer coefficients has integer solutions. In 1970 Yuri Matiyasevich, building on work by Martin Davis, Hilary Putnam and Julia Robinson, proved that no such process can exist.
What did Hilbert's optimism look like at the end?
When he retired in 1930, the city of Königsberg made him an honorary citizen. MacTutor records that his address ended with the words "Wir müssen wissen, wir werden wissen": we must know, we shall know. Problems 1 and 10 were settled, though not in the way he expected, and problem 2 changed what a proof of consistency could even mean.
What can a student take from Hilbert's way of working?
Collect problems, not just answers. Hilbert's power came from knowing exactly what was unknown. Keep a list of the questions you cannot yet solve.
Pick problems by his test. Hard enough to teach you something, not so hard you cannot begin.
A late start is not a verdict. The boy who struggled with memorising became one of the most influential mathematicians of his century.
An answer can be "it cannot be done". Dehn and Matiyasevich solved their problems by proving impossibility.
When you are building your own list of hard problems, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer. For the people behind two of these problems, read about Georg Cantor and Bernhard Riemann.
This week, write down three problems from your own syllabus that you cannot yet solve, and come back to one each day.
