On the evening of 8 September 1930, the German mathematician David Hilbert stepped up to a microphone in Königsberg and addressed his countrymen by radio. He was 68, recently retired, and at the height of his fame. The talk was a defence of pure reason, delivered in the city of Immanuel Kant, and it ended with two phrases that would later be carved on his tombstone: Wir müssen wissen. Wir werden wissen. We must know. We shall know.

Are certain questions closed to the human mind? Are there truths about the world that we can never know? Hilbert thought that, at least within mathematics, this could not be the case. For him, mathematics was a game played with symbols: one could fix the right axioms as rules of the game and every mathematical problem could be solved. He was wrong.

The day before his Königsberg address, at a small academic meeting in the same city, a 24-year-old Austrian named Kurt Gödel had quietly announced that no such system for arithmetic could exist. Whatever reasonable set of rules you fix, something true must lie beyond them and thus beyond our knowledge. We can’t know. We won’t know. Gödel’s results, known as the Incompleteness Theorems, are landmark achievements of 20th-century logic. Recent work in philosophy and mathematics has begun to illuminate the gap they opened, and with it something deeper about the foundations of knowledge itself. The story begins in the same city, but two centuries earlier.

When Kant remembered a book by David Hume he’d read years earlier, he was woken from what he called his ‘dogmatic slumber’. Kant had believed that pure reasoning could teach us every absolute truth about the world, but Hume had asked a question that that stumped him: what reason do we actually have to believe that the future will resemble the past? We may have seen the Sun rise a thousand times, but how do we know it will rise tomorrow? The belief that the future will resemble the past is, Hume argued, merely a kind of habit. Kant couldn’t answer Hume’s challenge and was silenced for a decade.

For Hume, proving a theorem was like unpacking a suitcase to find an outfit. Kant could not leave it there

When Kant finally broke his silence with the Critique of Pure Reason (1781), he offered a radical answer. We can know the world only as it appears to our minds, and that is where our science sits; the noumenal world – the Ding an sich (the ‘thing-in-itself’) – simply lies beyond our reach. Causation is not something we discover in the world, but something the mind supplies for experience to be coherent at all. When we watch a cat push a glass off a table, we don’t see a bare sequence of motions and then infer, by some metaphysical principle, that the push caused the fall. No: we see the push as causing the fall. The causal structure is not a hypothesis we add to the experience; it is part of how the experience comes to us in the first place. It is in our minds already.

So far, so good: Kant has met Hume’s challenge, and the workings of science are intelligible: we are structuring our experiences. But the work is not done, for there remains mathematics, the very language in which the sciences are framed. Hume was content that arithmetic and algebra are merely the unpacking of definitions; that proving a theorem is like unpacking a suitcase to find an outfit. Kant could not leave it there. If mathematics were only such unpacking, how could it underwrite the sciences? Either the sciences tell us nothing new about the world, or something has gone missing from the account...