Mathematical Problems (The 23 Problems)
One man hands the new century a to-do list — and bets there is no problem we cannot, in principle, solve.
At the dawn of a new century, one mathematician stood up and handed his whole field a homework list — 23 problems — and told it where to go for the next hundred years.
The idea, unpacked
In August 1900, David Hilbert — then the most influential mathematician alive — gave a lecture in Paris that did something unusual. Instead of presenting a finished result, he laid out a list of important problems nobody had solved yet, and argued that these were the questions that would shape mathematics in the century ahead.
Underneath the list ran a bold belief. Hilbert insisted that any clearly stated mathematical question must, in the end, have a clear fate: either someone solves it, or someone proves it can't be solved. There is no third option where the answer is hidden from us forever. “In mathematics,” he said, “there is no ignorabimus” — no “we shall never know.”
Where it came from
Hilbert spoke at the International Congress of Mathematicians at the Sorbonne. He read out only about ten of the problems that day; all 23 appeared when the lecture was printed. He chose them with care — hard enough to be worth a career, but not so hopeless that no one could make progress. Some, like the consistency of arithmetic, reached down into the very foundations of mathematics; others were sharp, specific puzzles about numbers and shapes.
The list landed with enormous authority because of who said it. To solve a “Hilbert problem” became one of the surest ways to make your name. Over the following century, mathematicians picked them off one by one — and a few of the answers turned out to be far stranger than Hilbert could have guessed.
Why it mattered
The list gave a sprawling field a shared sense of direction — a rare thing in any science. But its deepest lesson was about the limits of knowledge itself. Hilbert's optimism — that everything provable is within reach — was tested hardest by his own first two problems. Thirty years later, Kurt Gödel proved that any system rich enough to do arithmetic must contain true statements it can never prove. The dream of settling everything turned out to be impossible, in a precise and permanent way. Mathematics learned, from the inside, where its own walls are.
A way to picture it
Think of a great expedition leader pinning a map to the wall and circling 23 unclimbed peaks: “These are the summits that matter — go.” For decades, climbers reached them one after another. But for two of the circled peaks, the explorers discovered something humbling: not that the climb was merely hard, but that, with the tools everyone agreed to use, the summit could never be reached at all — and they could prove it. Knowing exactly which mountains are unclimbable is itself a kind of summit.
Where it sits in the story
Hilbert's faith ran straight into the work of Gödel (whose incompleteness theorems are elsewhere in this Library) and Turing, who together mapped the limits of proof and computation. A century later, the spirit of the list lives on: the Clay Institute's seven Millennium Prize Problems (2000) are its direct heirs, and one of them — the Riemann hypothesis — was Hilbert's Problem 8, still waiting to be solved.
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?
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.
A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street.
Every system of infinitely many real numbers, i. e., every assemblage of numbers (or points), is either equivalent to the assemblage of natural integers, 1, 2, 3, … or to the assemblage of all real numbers and therefore to the continuum …
To prove that they are not contradictory, that is, that a definite number of logical steps based upon them can never lead to contradictory results.