David Hilbert · The twenty-three problems
Can mathematics prove its own consistency?
That time, that place
The International Congress of Mathematicians met in Paris in 1900, the year a century turned.
Hilbert presented a list of problems for mathematics to solve. He spoke about ten of them; the printed version had twenty-three.
The list's power lay in the judgement behind the choices. For the next hundred years, solving one of them was the highest honour available to a mathematician.
Why this question
The second problem was the special one: prove that the axioms of arithmetic do not contradict one another.
It can sound odd. How could arithmetic, where one and one make two, contain a contradiction? But paradoxes had been surfacing in set theory through the 1890s, and ground that had looked solid was turning out not to be.
Hilbert's plan ran: rebuild all of mathematics as a system derived from finitely many axioms by mechanical rules alone, with no appeal to meaning. Then, inside that system, prove that the system yields no contradiction.
Mathematics would be safe for good.
What was found
In the autumn of 1930, at Königsberg, he gave a retirement address ending: we must know, we shall know. It was broadcast, and the recording survives.
The day before, at a conference in the same city, a twenty-four-year-old named Gödel had given a short talk. In any sufficiently strong system there are statements that are true but unprovable within it — and such a system cannot prove its own consistency.
The second problem got the answer that there is no answer.
Yet something came out of the wreck. Trying to make precise what Hilbert meant by mechanical rules alone, Turing imagined a machine that rewrites symbols by rule. The idea of the computer came from there.
One of the twenty-three is still open: the eighth, the Riemann hypothesis.
Mathematics is complete in principle, and every truth can be proved
None — this was not a discovery but a list, and the list set a century's direction
Gödel answered the second problem with impossibility, and the attempt to state it precisely produced the idea of the computing machine