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.
David Hilbert. Courtesy the University of Göttingen
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.
Does one unpack ‘add 7 and 5’ to find ‘12’? Kant thought not. Rather, one constructs 12 from 7 and 5 in the mind. More than a century later, a young Dutch mathematician called L E J Brouwer, following Kant’s thought, argued that, if the mathematician’s work consists in such mental constructions, then one cannot assume any statement is either true or false. That was controversial: to strip the mathematician of that assumption, Hilbert retorted, would be like ‘proscribing … to the boxer the use of his fists’.
Gottlob Frege, working in the late 19th century, found Kant’s account of mathematics intolerable for different reasons: arithmetic, he claimed, is not psychological. The numbers exist whether anyone counts them or not. He set about reducing arithmetic to pure logic, grounded in the mind-independent categories of True and False. The project consumed two decades of his life and produced some of the most consequential advances in the philosophy of language and mathematics in history. It is widely regarded as having sparked the modern discipline of logic. And yet, as the second volume of his masterwork, the Grundgesetze der Arithmetik, sat at the printer in 1902, the foundations cracked. The British philosopher Bertrand Russell sent Frege a letter pointing out an inconsistency at the heart of the system. A melancholy note in his diary later summed up his defeat: ‘My efforts to become clear about what is meant by number have resulted in failure.’
Russell had been wrestling with the foundations of mathematics for some years before his letter. It was only after meeting the Italian mathematician Giuseppe Peano at the International Congress of Philosophy in 1900 that he acquired the tools to attempt the problem. In place of Frege’s edifice, Russell set out to build his own, together with his colleague Alfred North Whitehead. Principia Mathematica appeared in three massive volumes between 1910 and 1913: all of mathematics, grounded in pure logic.
Mathematics carries what’s already there without adding anything, as plumbing moves water without supplying any
Russell’s student Ludwig Wittgenstein spent the war in the Austro-Hungarian artillery, volunteering again and again for the most exposed posts he could find – an observation position under fire on the Eastern Front, from which he was decorated for bravery at least three times. He carried notebooks. In them, between bombardments, he worked the Frege-Russell philosophy into a complete account of how language touches the world: the world is made of facts, a sentence is a picture of a fact, and whatever can be said at all can be said clearly. He finished the manuscript in an Italian prisoner-of-war camp in 1918 and sent it to Russell from the camp. Wittgenstein took himself to have solved the problems of philosophy, gave away his inherited fortune, and left to teach primary school in the Austrian mountains.
Ludwig Wittgenstein in 1929. Courtesy Wikipedia
The Tractatus became the manifesto of a group of philosophers in Vienna including Moritz Schlick, Rudolf Carnap and Otto Neurath, who called their position logical positivism. A statement is meaningful only if there is some way of settling whether it is true. Analytic statements are settled by unfolding the definitions of the words involved: for example, ‘All bachelors are unmarried’ is true because a bachelor is, by definition, an unmarried man. Synthetic statements are claims about the world, and are settled by – in principle – going out and looking: ‘It rained in Vienna on 3 June 1929’ is false because one could have observed that the sky over the Ringstrasse stayed clear that day. Anything else is grammar without content.
The Vienna Circle, as they became known, claimed that mathematics is analytic and science is synthetic. The function of mathematics is to extract information from scientific claims: it carries what is already there without adding anything, the way plumbing moves water without supplying any. If the Vienna Circle were right, then Kant’s and Brouwer’s restrictions on scientific and mathematical knowledge were entirely unnecessary. If.
By 1929, after almost three decades of labours by Hilbert, Russell, Frege, Wittgenstein, the Vienna Circle and Brouwer, among others, the foundations of mathematics looked very nearly finished. One beam remained that would lock the whole frame together. In the plenary addresses for the International Congress of Mathematicians in Paris in 1900, Hilbert had set out 23 problems that would define the next century of the subject. The second problem was this: prove that arithmetic is consistent – that is, no chain of reasoning from the axioms, however long, would yield a contradiction. The beam would not go in.
Kurt Gödel in 1950. Courtesy the IAS, Princeton
While studying physics at the University of Vienna in the 1920s, Gödel had attended Schlick’s seminar on Russell, which drew him into the problems of logic and, by way of them, into Hilbert’s programme. His PhD in 1929 showed that the standard system of first-order logic is complete – every logically valid statement can, in principle, be derived from the axioms of logic. It was the result Hilbert wanted, though for pure logic rather than mathematics. A year later, he turned to the natural next step: the consistency of arithmetic, Hilbert’s second problem.
What he announced at the Königsberg meeting the day before Hilbert’s radio address, no one had expected. Taking Principia Mathematica as the articulation of mathematics within logic, Gödel showed that there were true mathematical statements that cannot be proved using the resources of Principia Mathematica alone. Among them is the statement ‘Principia Mathematica is consistent.’ The trick at the heart of the proof is to encode statements about the system within the system itself – a move with the same vertiginous quality as Russell’s letter to Frege. But where Russell’s letter had killed a system, Gödel’s theorem killed a programme.
The one-time prophet of the Vienna Circle left nothing of its philosophy standing
His argument was so general that it applied to any reasonable system for arithmetic. Hilbert’s dream of an axiomatic system capturing all of mathematics – or even just arithmetic – was over.
The response in Vienna was disbelief, then dismay. The Circle had built its philosophy on the assumption that mathematics was analytic; Gödel had shown it wasn’t as simple as that. Carnap struggled for years to absorb the result. Wittgenstein dismissed it, but was already repudiating the Tractatus himself. Meaning, he came to argue, lay not in matching picture to reality but simply in how people discourse in public. The one-time prophet of the Circle left nothing of its philosophy standing.
Hilbert never publicly conceded that Gödel’s result doomed his programme. He died in 1943; his colleagues had by then been murdered or exiled by the Nazis. His tombstone in Göttingen bears the words he had spoken into the Königsberg microphone: Wir müssen wissen. Wir werden wissen. We must know. We shall know.
And Gödel? After the Anschluss, he escaped Vienna for Princeton, where he continued to make extraordinary contributions to logic while taking long walks with his close friend Albert Einstein. At his American citizenship exam in 1947, asked by the judge whether a regime like the Nazis’ could arise in the United States, he began to explain a logical loophole he had found in the Constitution that would permit exactly that. The judge cut him off and moved the proceedings swiftly to their conclusion. (We still don’t know what loophole he found.)
The orthodox response was to keep the truth and give up the proof. There is a realm of mathematical fact, standing independent of us, and axiomatic systems reach only part of it; incompleteness measures the distance between what is so and what we can show. Its most committed advocate was Gödel himself. However, two problems come with the realm. First, with logical positivism in ruins, Kant’s problem returns: if mathematical truth is independent of the mind, how can it structure sciences that exist within the mind? Second, the realm is posited chiefly to spare us two unwelcome alternatives – psychologism on one side, Brouwer’s restrictions on the other. Avoiding two conclusions is not an argument for a third.
In 1973, the British philosopher Michael Dummett was invited to deliver the annual philosophical lecture at the British Academy. He took as his subject ‘The Justification of Deduction’ – how to justify the ordinary practices of mathematicians without helping oneself to the assumption that their subject matter exists in a realm of its own. The problem, he told his audience, was perhaps ‘one of the most fundamental and intractable problems … in all philosophy’.
Dummett was the great Frege scholar of the 20th century; his Frege: Philosophy of Language (1973) brought an otherwise obscure figure to the forefront of the discipline. The book had been years in the writing, but in 1964, with the manuscript all but finished, Dummett put down his pen. He conceived it his duty, he explains in the preface, to involve himself ‘actively in opposition to the racism which was becoming more and more manifest in English life.’ When he returned to the manuscript several years later, a previously suppressed fragment of Frege’s Nachlass (ie, his literary estate) had surfaced, revealing the philosopher as a ‘virulent racist’ with ‘extreme Right-wing political opinions’. Once his book was finally published, Dummett recorded the bitter irony, and reflected that he had learned something dark about how a mind he revered could harbour views he despised.
Our linguistic practices – the ways in which we are prepared to assert, justify and infer – determine logic
What Dummett took from Frege’s work on logic was that the problems of mathematics are, at heart, problems of language. Where things had gone wrong, he thought, was that Frege had grounded his account in a mind-independent notion of truth. ‘In this respect (and in this respect alone),’ Dummett wrote, ‘Frege’s new approach to logic was retrograde.’ For Frege, logic answered to truth: the rules were good because they preserved something already fixed. Dummett turned this around.
Following Wittgenstein’s later thinking, Dummett argued ‘a theory of meaning is a theory of understanding.’ And understanding, he insisted, must be manifestable: whatever a competent speaker knows in knowing what a sentence means must be something she can, in principle, display in her use of language. In The Logical Basis of Metaphysics (1991), Dummett gave this account of language, logic and reality its most sustained form. Our linguistic practices – the ways in which we are prepared to assert, justify and infer – determine logic. Logic, in turn, determines what counts as a fact, what counts as truth, and ultimately what we are entitled to regard as real.
Consider how a child learns the word ‘red’. We do not begin by telling her about light of a certain wavelength, or the metaphysics of colour properties, or the conditions that would make ‘this is red’ true. We point at things. We correct her when she calls a ripe tomato green; we approve when she calls a fire engine red. Over time, she comes to use the word as we do – and at that point, she understands it. There is no further fact, hidden behind the practice, in which her understanding consists. The understanding is the practice, competently exercised.
Dummett died in 2011, having been knighted in 1999 for ‘services to philosophy and to racial justice’. He is remembered as one of the most significant British philosophers of the previous century. Yet for all its power, his programme remained a prospectus: he set out what a theory of language, logic and reality must achieve, but he never completed one.
In 2005, Tor Sandqvist, a doctoral student at Uppsala University, submitted a thesis that took up Dummett’s challenge: to justify the ordinary practices of mathematicians without any appeal to truth. Where Dummett had given the conditions a solution must satisfy, Sandqvist gave a solution. He produced a precise mathematical justification of deduction as mathematicians actually use it, grounded entirely in practices of assertion. The boxer’s fists were untied.
It was rediscovered a decade later in the Department of Computer Science at University College London, where a group of logicians recognised what Sandqvist had actually built, and developed it into a full framework on a par with the orthodox truth-conditional view. The framework now stands equal to the truth-based account, analogous to the situation after Gödel’s PhD thesis. What about mathematics? The UCL team have recently shown that the Dummett-Sandqvist framework gives a way to understand what Gödel’s Incompleteness Theorems are about, and it is not the limits of knowledge.
If meaning comes from practices of assertion, then our concepts are settled only as far as our practices have settled them. A system of definitions is a kind of constitution that determines something. The US Constitution, for example, is meant to enshrine a form of democracy. But what a constitution defines answers to how it is amended.
An amendment can betray what a constitution defines. But it can also unfold it
Principia Mathematica is a constitution for arithmetic. To read it as a constitution is to assume it is consistent; otherwise, it would be nonsense. So, when we amend it by asserting that ‘Principia Mathematica is consistent,’ we are not changing the subject: the new axiom says aloud only what the practice already did. Gödel’s theorems apply afresh to the enlarged constitution: the amended system has its own unprovable truth, its own consistency it cannot certify. No amendment is the last; each faithful extension opens room for the next. To grasp a concept like arithmetic is not to survey a finished structure; it is to hold a principle for going on: a rule for laying the next stretch of line, not a map of where the line ends. Dummett called this openness indefinite extensibility.
Faithful is the word doing the work, because not every amendment is. Gödel claimed the US Constitution could deliver a dictatorship. That does not make dictatorship a form of democracy. Whatever loophole he had in mind, it would have contradicted the practices the Constitution already enshrined. An amendment can betray what a constitution defines. But it can also unfold it, drawing out what the founding commitments demanded all along. Only the second kind is what Dummett has in mind.
The constitution of our language is our current practice for public discourse. As it is extensible in this way, what we claim as ‘true’ is not a fixed structure with a ceiling we are forever failing to reach, but something that can change. Gödel’s verdict never applied to mathematics, but only to some edition of the constitution. Within the Sandqvist system, we can attend to the extension systematically. Hilbert’s words outlive his programme: Wir werden wissen. We shall know. Not because the answers are waiting at the end of a ladder, but because truth goes exactly as far as we have built the rungs.