Insight Editions · No. 233
A World Without Time
The Forgotten Legacy of Gödel and Einstein
Palle Yourgrau
The friendship of Gödel and Einstein at Princeton and the rotating universe it produced, read as Gödel’s argument that time as we intuit it does not exist.
A study guide to A World Without Time by Palle Yourgrau. Shared free for personal reading; not for sale and not endorsed by the authors or publishers. Please read and buy the original.
Gödel turned Einstein's relativity against time itself: in a rotating universe with time travel, time does not pass, so time is ideal, in our world too.
Intellectual history from Vienna 1924 to Princeton 1978: logic, positivism, relativity, exile, friendship and the philosophy of time.
1931 incompleteness; 1942–55 the friendship; 1949 the Gödel universe; 1978 Gödel dies at 65 lb.
Readers who want the logic of incompleteness and the case against time, with a partisan narrator to argue with.
The Book on One Page
Palle Yourgrau tells the story of Kurt Gödel and Albert Einstein, best friends at Princeton's Institute for Advanced Study from about 1942 until Einstein's death in 1955, and of the discovery their friendship produced: the Gödel universe (1949), a solution of Einstein's equations in which a rocket can reach its own past. His thesis is that Gödel used this world to prove that time, as we intuitively understand it, does not exist, and that physicists and philosophers have ignored or evaded the result.
- A century of limits. Einstein (relativity), Heisenberg (uncertainty) and Gödel (incompleteness) each drew conclusions about reality from limits on knowledge. Gödel and Einstein, realists both, resisted the positivist use of that move.
- Vienna and the positivists. Gödel grew up inside the Vienna Circle, whose heroes were Wittgenstein and a misread Einstein, and whose plan (via Hilbert's formalism) was to dissolve mathematics into a game of signs after Cantor's infinities and Russell's paradox caused a crisis.
- The spy. Gödel used formalism itself to show truth outruns proof (1931) and that no consistent system proves its own consistency. His lifelong programme: prove, by formal means, the limits of formal methods in capturing intuitive concepts.
- Exile and friendship. After the "Vienna syndrome" kept him in Austria until 1940, Gödel joined Einstein in Princeton: two isolated philosopher-scientists, sceptical of Copenhagen, unfashionable, walking home together every day.
- Big T and little t. Applied to relativity, the programme asks whether t, the time coordinate, is time. In four steps (no world-wide now in special relativity; cosmic time in general relativity; no cosmic time and closed timelike curves in the Gödel universe; a modal step from that possible world to ours) Gödel concludes that time is ideal, as Kant and the idealists held.
- Silence and decline. Physicists falsely charged him with error, then asked only whether our universe rotates; Hawking's chronology protection conjecture (1992) became, for Yourgrau, an "anti-Gödel amendment". Both men spent their last years on unripe problems; Gödel starved himself to death in 1978.
- The verdict. Against Goldfarb, Dreben and Earman, Yourgrau argues that Gödel was a philosopher posing as a logician, and that his treatment of time is a model of how philosophy should be done.
| Ch | Title | One-line payload |
|---|---|---|
| 1 | A Conspiracy of Silence | Two exiles walk home; Gödel's universe kills time; the world looks away. |
| 2 | A German Bias for Metaphysics | Opposites in everything except Kant, realism and mathematics as a window on reality. |
| 3 | Vienna: Logical Circles | Mach, Boltzmann, Schlick's Circle, Cantor, Russell and Hilbert set the stage for incompleteness. |
| 4 | A Spy in the House of Logic | The incompleteness theorems, step by step, and their slow, hostile reception. |
| 5 | It's Hard to Leave Vienna | The "Vienna syndrome" and Gödel's late escape via Siberia in 1940. |
| 6 | Amid the Demigods | Princeton outsiders; realism against positivism; the question "is t time?" |
| 7 | The Scandal of Big "T" and Little "t" | The four-step argument from relativity to the ideality of time, and its neglect. |
| 8 | Twilight of the Gods | Lost causes, the recipe for discovery, and Gödel's death by starvation. |
| 9 | In What Sense Is Gödel (or Anyone Else) a Philosopher? | Yourgrau's defence of Gödel as philosopher against his academic critics. |
A Conspiracy of Silence
Yourgrau opens with two refugees walking home through Princeton: Kurt Gödel, the greatest logician since Aristotle, and Albert Einstein, the most famous scientist of all. Out of their walks came the Gödel universe (1949), a solution of Einstein's own equations in which a traveller can reach the past, and with it an argument that time, as we ordinarily understand it, does not exist. Yourgrau's charge is that this discovery, and the friendship that produced it, were met by a "conspiracy of silence" that keeps the world from seeing what relativity really implies.
The stranger on the shore at Blue Hill
In the summer of 1942, with German U-boats hunting off the coast of Maine, the people of Blue Hill grew alarmed at a hunched figure with a thick German accent pacing the shore at midnight. Rumour made him a spy signalling to submarines. He was Gödel, on holiday at the Blue Hill Inn with his wife Adele, and his quarry was not shipping but Cantor's continuum hypothesis, the question of how many points lie on a line. The couple appeared at dinner but were never seen eating. Gödel's parting word to the town was a letter accusing the innkeeper of stealing the key to his trunk.
Back in Princeton he no longer had to walk alone. At the Institute for Advanced Study (IAS), an exclusive club whose members had one duty, to think, his companion was Einstein. Late in life Einstein said his own work no longer meant much and that he went to the office "just to have the privilege of walking home with Kurt Gödel." Yourgrau adds an irony: it was the smiling pacifist, not the scowling logician, who had once helped the German navy, by improving gyroscopes during World War I. Gödel's own gyroscopes would spin at the centre of the universe.
Gödel was the only one of our colleagues who walked and talked on equal terms with Einstein.Freeman Dyson
Three limits that defined a century
Yourgrau groups Einstein, Heisenberg and Gödel as authors of the three most fundamental scientific results of the twentieth century, and notes that each result is a limit. Each also drew an ontological conclusion about reality from an epistemic principle about knowledge, which he calls the signature move of the age, the "dance of limit and limitlessness".
| Thinker | Result | Limit it set | Conclusion drawn about reality |
|---|---|---|---|
| Einstein | Relativity (1905, 1915) | No signal faster than light; time defined by clocks | Time is relative to a frame, not absolute |
| Heisenberg | Uncertainty principle (1927) | Δx·Δp ≥ ħ/2: no joint sharp knowledge of position and momentum | For Heisenberg, a limit on reality itself |
| Gödel | Incompleteness (1931) | No finite or recursive formal axiom list captures all mathematical truth | No machine exhausts mathematics; if minds grasp it, minds are not machines |
The incompleteness theorem was soon hailed at a Harvard ceremony as "the most significant mathematical truth of the century." Gödel himself drew the anti-mechanist moral at once, to the annoyance of later artificial intelligence enthusiasts. Some heard in incompleteness an echo of the uncertainty principle; Gödel was not among them.
The shared method of the century was to infer what is from what can be known. Einstein used it as a heuristic in 1905; Heisenberg made it a creed; Gödel and Einstein ended up opposing the creed.
Against the Copenhagen tide
Heisenberg, who stayed loyal to Germany, joined the winning side in physics: the positivist Copenhagen interpretation of his mentor Niels Bohr. Einstein had inspired Heisenberg and had pioneered quantum theory himself, but he came to loathe its positivism. In Prague in 1911 he pointed out to Philipp Frank the asylum below his study: "that portion of lunatics who do not concern themselves with quantum theory." When Heisenberg defended his method as the very philosophy behind special relativity, Einstein replied that he may have used such philosophy earlier, "but it is nonsense all the same."
In Princeton, Einstein (pacifist, bohemian, socialist, Jew) was a man apart, and with Gödel he stayed isolated because both opposed the positivist worldview that dominated physics. Yourgrau calls them philosopher-scientists rather than "intellectual engineers". Their fame also made them unapproachable to everyone but each other.
| Einstein | Gödel | |
|---|---|---|
| Art | Violin; Mozart, Bach, Beethoven | Operettas; Snow White and the Seven Dwarfs |
| Mathematics | A "refugee" from it; Minkowski called him a "lazy dog" | Felt secure only once a problem was in symbols |
| Mood | Cheerful, tried to lift his friend's spirits | Gloomy, later paranoid |
| Direction of travel | Mathematics to physics | Back from logic towards physics |
Gödel liked fables because "only fables present the world as it should be and as if it had meaning." Einstein said he had left mathematics because he could not find in that garden of many paths the one leading to the fundamental; through Gödel he learned that mathematics too has such a path. Einstein's quip on calculation was crude and memorable: once you start, "you shit yourself up before you know it." Gödel's pessimism became paranoia; Oskar Morgenstern once found him hiding in the cellar behind the furnace.
Yourgrau's aside linking Alan Turing to Gödel and Snow White is speculation. Turing was at Princeton University (1936–38), not an IAS member, and Gödel was largely absent from Princeton in those years. Turing's codebreaking work at Bletchley Park was against German naval Enigma, and he was sentenced in 1952 to hormone treatment; the link from his poisoned-apple death (1954) to Disney is folklore, not evidence. The "lazy dog" anecdote is reported by Minkowski's circle second-hand.
The bomb in the lap of the philosophers
Time, which baffled Plato and Augustine, seemed to meet its match in Einstein, who turned it into the fourth dimension of space-time. Even David Hilbert conceded that every boy in Göttingen knew more four-dimensional geometry than Einstein, "yet Einstein did the work and not the mathematicians." On their walks Einstein tutored Gödel in general relativity, sowing it in a mind later described as Einstein crossed with Kafka.
Gödel then performed a stranger trick. He found new cosmological solutions of Einstein's field equations, the rotating or Gödel universes, where matter warps space-time so much that there are timelike, future-directed paths into any region of past, present or future. He even computed the speed and fuel a rocket would need. For the first time, time travel was shown to be consistent with the laws of physics.
Gödel's inference was immediate: if we can revisit the past, it never really passed, and a time that does not pass is no time at all. Time, "that mysterious and seemingly self-contradictory being," which seems to form the basis of our existence, would be the world's greatest illusion. If relativity is real, time is ideal. Einstein called the paper an important contribution to general relativity, and he saw that time had slipped his net again.
Then: nothing
After a brief flurry, in which a few physicists tried and failed to refute Gödel and then generalised his solutions, interest died. Yourgrau's comparison: a friendship as remarkable as one between Michelangelo and Leonardo has vanished, while biographers chase Lieserl, Einstein's "love child". Exhaustive Einstein biographies give the Gödel friendship a sentence or two at most.
The only recent recognition, he says, came from Stephen Hawking, whose chronology protection conjecture is in effect an anti-Gödel postulate: an apparently ad hoc amendment to the laws of nature that rules out time-travel worlds like Gödel's. For Yourgrau this is the highest compliment, proof of how dangerous Gödel's result is. He also reports that Hilbert answered incompleteness by proposing a new rule of logic to block it. Gödel the ironist, unlike the classicist Einstein, threw a bomb built from cosmology's own materials into the foundations of physics.
Two framings here are partisan. Hawking's conjecture (1992) was not aimed at Gödel; it was prompted by wormhole time machines (Morris, Thorne, Yurtsever 1988) and rests on calculations showing quantum vacuum fluctuations blow up near the onset of closed timelike curves. It applies to time machines built in an initially normal universe, so it leaves Gödel's eternal rotating cosmos largely untouched. Hilbert's ω-rule proposal (1931) was a response to the foundations crisis, not narrowly a device to "refute" Gödel, who in any case proved a theorem, not a hypothesis. And "conspiracy of silence" is rhetorical: physicists (Raychaudhuri, Stein, Malament, Earman, Hawking and Ellis) worked on Gödel's solution steadily from the 1950s.
- 1942: the Gödel–Einstein friendship begins, by Yourgrau's dating; it lasts until Einstein's death in 1955.
- 1949: Gödel publishes the rotating universe in Reviews of Modern Physics (the issue marking Einstein's 70th birthday) and the philosophical argument in Schilpp's Albert Einstein: Philosopher-Scientist.
- 3: fundamental "limit" results of the century (relativity, uncertainty, incompleteness).
A German Bias for Metaphysics
Gödel and Einstein were opposites in body, taste, temperament and faith, and they disagreed openly about physics. Yourgrau argues that what bound them was deeper: both were heirs of the Austro-German philosophical tradition, both cut their teeth on Kant, and both treated mathematics as a window onto reality rather than a calculating tool. Both also knew that time, not space, is the real challenge to science, because science can only grasp what it can geometrise, and Einstein had once more turned time into space.
Opposites in every visible way
| Gödel | Einstein | |
|---|---|---|
| Body | 5 ft 6, emaciated; weighed 65 lb at death | 5 ft 9, sturdy, later a "professorial paunch" |
| Food and health | Ate as little as possible; hypochondria after childhood rheumatic fever; distrust of doctors | Loved German cooking and his pipe; vowed to die "with the minimum of medical assistance" |
| Dress | Buttoned up, overcoat and scarf in summer, hair combed | No socks, tie or belt; combs verboten |
| Laugh | Soft, high chuckle; sentences trailed off as queries | Volcanic belly laugh |
| Music | Light classics and operettas; disliked modern art and comedies; loved Disney | Mozart above all; a fine sight-reading violinist |
| Religion | Baptised Lutheran; theist "following Leibniz, not Spinoza" | Jew; "deeply religious unbeliever"; hero Spinoza |
| Marriage | Married Adele, a Viennese café dancer | Undecided whether to marry Elsa or her daughter |
Gödel's anorexic conviction that the less one ate the better became a neurosis that killed him; during periods of stress he was confined to sanatoria. The Blue Hill innkeeper thought he lacked "the strength to make babies", yet in youth he chased women openly, according to his friend Olga Taussky-Todd, and Adele teased him about the pretty students at the IAS, which she called an Altersversorgungsheim, a home for elderly pensioners. His method for avoiding appointments was to fix a precise place and time, which gave him certainty about where not to be; at unavoidable institute teas, Paul Halmos observed him steering between guests to avoid physical contact. As a child of four he was Herr Warum, "Mr. Why".
Einstein's household was the reverse. Philipp Frank read his apparent cynicism as an urge "to make the serious things in the world tolerable by means of a playful guise," the same quality Frank heard in Mozart. The professional violinist Boris Schwartz judged his playing steady, in tune and pure in tone.
Small slips: Elsa's daughter was Ilse Löwenthal (not "Ilsa"), aged about 21 in 1918. The famous line about marrying "either Mama or me" comes from Ilse's letter to Georg Nicolai. Yourgrau's portraits lean heavily on anecdote from memoirs (Taussky-Todd, Halmos, Menger), which are vivid but uneven as evidence.
Faith, Jewishness and Russell's sneer
1879, Einstein's birth year, was also the birth year of Gödel's mother Marianne and of Frege's Begriffsschrift, the founding text of modern mathematical logic. Einstein, sceptical of his ancestral faith, rediscovered his "tribal companions" when Nazism rose and became a passionate, thorny Zionist, unbothered by his earlier attacks on nationalism. His Spinozist leaning showed in a career that made transcendent-seeming things (space, time, light) immanent and subject to physical causality.
Gödel's God was "more than a person", able to play the role of one, whereas Spinoza's was "less than a person". He noted that the founders of modern science were not atheists, scorned the "ninety per cent of philosophers" who try to knock religion out of people's heads, and used modern logic to reconstruct Leibniz's ontological argument for God. Nazis in Vienna took him for a Jew; Adele drove off a gang of rowdies with her umbrella.
The misreading recurred in Princeton. Around 1943–44 Gödel joined a tiny discussion group with Einstein, Wolfgang Pauli and Bertrand Russell. Russell found the talk too philosophical "in the old-fashioned sense" and wrote that the other three were "Jews and exiles... [with] a German bias for metaphysics." Gödel replied that he was not a Jew, "even though I don't think this question is of any importance," and admired how the Jewish people survived for millennia without a country, "merely by their faith." Yourgrau: "the failings of an entire century are crystallized in this fact."
- Olga Taussky-Todd: Vienna fellow student; source on Gödel's private life and symbolic habits.
- Paul Halmos: mathematician whose memoirs record Gödel at institute teas.
- Karl Menger: Vienna Circle mathematician; diagnosed Gödel's "vicarious persecution complex" on Leibniz's behalf.
- Paul Erdős: told Gödel he became a mathematician so that people should study him, not Leibniz.
- John Bahcall: young IAS astrophysicist told by Gödel he did not believe in natural science.
- Bertrand Russell, Wolfgang Pauli: the other members of the 1940s Princeton discussion group.
Exile and the start of a friendship
Both left comfortable posts, Einstein in Berlin, Gödel in Vienna, when the Nazis rose, and both were snapped up by the new IAS. Strangers in Europe, they became friends only around 1942, and the friendship lasted until Einstein's death in 1955, a loss Gödel never got over. Einstein, a German Jew "in a nest of WASPs", felt out of place; Adele suffered in the provincial town and pleaded for Harvard, but Gödel would not take a job that required teaching.
Disagreement as a bond
Gödel thought Einstein enjoyed his company precisely because he did not hide disagreement, in politics, philosophy and physics. He was sceptical of Einstein's search for a unified field theory and of natural science itself, useful as it was for building "TVs and bombs". Physics, he held, took the wrong turn when it followed the naturalist Newton instead of the idealist Leibniz. He even believed key Leibniz manuscripts had been suppressed by "those who do not want man to become more intelligent." When Menger asked why Voltaire's papers had survived, Gödel answered: "Who ever became more intelligent by reading Voltaire?"
They also split on mathematics versus intuition. Einstein resented Minkowski's four-dimensional rewriting of special relativity at first, though it led to general relativity, and stayed wary of any step he could reach "only mathematically". For Gödel, symbols came first, yet he also held, notoriously, that there is mathematical intuition. For Gödel the equations led to new insight against common sense; for Einstein common sense was the final touchstone for what the mathematicians offered.
What united them: Kant and the reality of mathematics
Russell's prejudice aside, his phrase hit a truth. Both men read Kant's Critique of Pure Reason at sixteen; Einstein took a Kant course at the Zurich Polytechnic; Gerald Holton records that their favourite topic in Princeton was Kant. Gödel described his own relativity work, to Einstein's dismay, as showing that relativity had "verified" Kant's idealism. Kant also helped free Einstein from Ernst Mach's over-reliance on immediate sense data, though Einstein mocked the Kant cult: a highway of milestones where "all the little dogs come and each deposits his bit."
From Kant they shared the doctrine that a science is rigorous to the degree it is mathematical, and from their own temperament the Socratic hunt for what things "really are". Einstein had told Kant what time really was (the fourth dimension), Newton what gravity really was (space-time curvature) and everyone what energy really was (E = mc²). Neither could accept the Bohr–Heisenberg view that mathematics only predicts measurements. A snapshot of the pair in white straw hats reminded one observer of two gentlemen farmers from a Faulkner novel.
Why time is the hard case
For Kant, space is the form of outer sense and time the form of inner sense, and because inner intuition "yields no shape" we represent time "under the image of a line." Science, built on Euclid's axiomatic geometry, has a "geometrical Midas touch": whatever it touches becomes geometry. Einstein generalised Euclid to non-Euclidean geometries and in 1905 folded time into the geometry of space-time; G. J. Whitrow called the project "the geometrization of physics." Einstein never lost his nostalgia for his boyhood "holy geometry booklet."
Yet the most striking feature of time is that it flows, and flowing is itself something that happens in time, the knot Augustine tied himself in. General relativity sharpened the problem. Special relativity made simultaneity local and relative, yet general relativity, to Einstein's displeasure, described a universe expanding over time, and Hubble showed it really expands. Expanding absolutely, in which time? Yourgrau's question for the next chapters is why the logician of the timeless world of sets and numbers was the right person to take up Einstein's unfinished hunt.
The flowing river. Picture time as a river carrying events from future to past. Now ask how fast it flows: one second per... Second? The rate needs a second time against which the first one flows, and that second time needs a third. The line picture captures order (earlier, later) perfectly and captures passage not at all. That gap is exactly what Gödel will exploit.
Vienna: Logical Circles
Gödel came of age in interwar Vienna, inside the Vienna Circle, whose creed of logical positivism he never shared. Yourgrau traces the creed's ancestry (Mach against Boltzmann), its misreading of Einstein as an ally, and its plan, carried out through Hilbert's formalism and Wittgenstein's Tractatus, to dissolve mathematics into a game of signs after Cantor's infinities and Russell's paradox threw it into crisis. The stage is set for incompleteness: a grenade aimed at Hilbert that landed in the laps of the positivists.
From Brno to Vienna
Kurt Gödel was born into the German-speaking minority of Brno (the city where Mendel founded genetics), with his brother Rudolf, four years older and later a radiologist. Kurt was a standout in everything, reputedly never made a mistake in Latin, and got his only less-than-perfect grade in mathematics. He reached the University of Vienna in 1924, began in physics, studied the history of philosophy with Heinrich Gomperz, and moved steadily towards greater rigour: physics → mathematics → mathematical logic → mathematical philosophy.
Vienna's "ratio of intellectual genius to square footage" was staggering: Freud, Mahler, Richard Strauss, Schoenberg, Klimt, Kokoschka, the architect Adolf Loos ("You are me!" he told Wittgenstein in 1914), Boltzmann, Mach, Popper, Wittgenstein. Yourgrau adds Hitler, who in 1904 attended the same Linz Realschule as Wittgenstein, two years behind him.
The decisive teacher was Philipp Furtwängler, cousin of the conductor, who lectured on number theory to hundreds, paralysed from the neck down, from a wheelchair, without notes, while an assistant wrote on the board. Gödel called them the best lectures he ever heard and credited them with his turn to mathematics. The disembodied scene suited the subject: the natural numbers seem to have an independent existence, which makes number theory a breeding ground for Platonists. Kronecker said "God made the natural numbers; all else is the work of man"; for Gödel all numbers were "the work of God."
The claim that a Linz class photograph shows Wittgenstein near Hitler comes from a speculative 1998 book (Kimberley Cornish) and is not accepted by historians; the two were in different years and there is no evidence they knew each other.
The new logic Gödel inherited
Four building blocks of modern logic, as Yourgrau sets them out
- Frege's Begriffsschrift (1879). Logic both axiomatised and formalised in an artificial symbolic language whose rules can be followed mechanically, without knowing what the symbols mean. For Frege it was a genuine science with its own content, and the foundation for arithmetic.
- Hilbert's metamathematics. The mathematical study of formal systems themselves. Hilbert took the idea of a metalanguage from L. E. J. Brouwer on a 1909 holiday at Scheveningen; Brouwer later became his nemesis. Frege, by contrast, believed in one logic only, with no "stepping outside", a view his admirer Wittgenstein inherited.
- Consistency and completeness. Hilbert's questions about a system: is it consistent (cannot prove a statement and its negation) and complete (proves every true statement)? Answering these questions produced incompleteness.
- Axiomatic set theory. Frege's concepts gave way to their extensions, i.e. Sets. After Russell found an inconsistency in Frege's theory of extensions, Zermelo and Fraenkel built set theory as its own axiomatic field, replacing both Frege and Cantor's "naive" set theory.
Gödel's adviser Hans Hahn ran a seminar on Russell and Whitehead's Principia Mathematica. Gödel skipped that one but attended Moritz Schlick's seminar on Russell's Introduction to Mathematical Philosophy (written in prison for Russell's anti-war protests) and Rudolf Carnap's on foundations. Carnap, Frege's student at Jena, would become one of the Circle's most influential members.
Life outside the Circle
The brothers shared a comfortable flat, often joined by their mother Marianne, toured as far as Marienbad in Rudolf's new Chrysler (Kurt drove fast and daydreamed at the wheel until Adele ended his driving), and took box seats at Max Reinhardt's Josefstadt Theatre. Karl Menger recalled that Gödel grasped problems instantly and answered "with a certain shyness and a charm". Shy was not timid: told by Carnap to write encyclopedia entries for recognition, he said he did not need such devices.
At 21 he fell for Adele Porkert, a dancer at the nightclub Der Nachtfalter (The Moth): six years older, married (briefly and unhappily), Catholic, her face marked by a port-wine stain. His parents objected. Only after his father's death in 1929 (aged 54) did marriage become possible, and it took nine more years, until 1938. Hao Wang thought the delay partly explains why they had no children, which weighed on Adele. Georg Kreisel credited her with "a real flair for le mot juste" and a habit of teasing Kurt about ghosts and demons. Gödel's friends from Circle cafés (the Reichsrat, Schottentor, Arkaden) included Carnap, Hahn, Herbert Feigl, Marcel Natkin and, closely, John von Neumann. He joined the Circle in 1926, introduced by Hahn, and edited the proceedings of Menger's mathematical colloquium.
The Vienna Circle
Meetings were by invitation, 10–20 members, in a dingy ground-floor room in the Boltzmanngasse building housing the mathematics and physics institutes; early arrivals cleared chairs from the blackboard, and one table was for smokers. Gödel rarely spoke, signalling assent or dissent with a slight tilt of the head. Popper was not invited because of his views on Wittgenstein; Wittgenstein declined. Gödel and Wittgenstein apparently never met, though Gödel saw him once at a lecture in Vienna by Brouwer (the 1928 lecture that is said to have pulled Wittgenstein back to philosophy).
Positivism, the Circle's creed, holds that physical science, based on sense experience, exhausts what can be known; philosophy's job is to police thought's pretensions and dissolve metaphysics as confusion about language. Kant had already called for a critique of reason, but he still built a system claiming knowledge beyond science; the positivists refused even that. The "logical" twist came from Frege, Russell and Hilbert: logic as a tool to formalise (and police) science and as a foundation for mathematics.
Their bible was Wittgenstein's Tractatus Logico-Philosophicus (title suggested by G. E. Moore, after Spinoza), finished while its author was a prisoner of war. Its preface claimed "the final solution of the problems," and its close, "what we cannot speak about we must pass over in silence," gave Menger his joke: after one session, he and Gödel had "out-Wittgensteined these Wittgensteinians: we kept silent." Olga Taussky-Todd testified that an argument could be settled by citing the Tractatus.
The Tractatus solved the positivists' worst problem. If science is rigorous to the degree it is mathematical, but mathematics is not an empirical science, positivism needs an account of mathematics. Wittgenstein supplied one: mathematics contains no truths and no knowledge, only conventional rules for manipulating signs. Carnap's slogan: mathematics is the "logical syntax of language." Gödel, Frege's true heir, spent his life fighting this doctrine.
Yourgrau notes one subtlety: Wittgenstein was not a positivist. For him, what must be passed over in silence was exactly what had value, whereas the Circle took it to be nonsense.
Einstein, the unofficial saint
Einstein was the Circle's second hero (Wittgenstein once said he was in a sense a follower of Einstein). In special relativity he had rejected Newton's untestable absolute space and time: time is physically real only so far as clocks measure it, and since not all clocks can be definitively synchronised, time is relative to a frame; with no test for absolute motion, a "rest frame" is chosen by convention. The positivists read this as their maxim, the meaning of a term is its method of verification, producing good physics.
For Einstein it was a claim about the physical world; for the Circle it was a religion: no science or philosophy could give a Weltbild, only calculation and prediction, and no faculty of intuition could reach the theorist's concepts or the mathematician's objects. Einstein and Gödel prized intuition. To John D. Rockefeller Jr.'s "I put my faith in organisation," Einstein replied, "I put my faith in intuition." Gödel's most (in)famous remark threw down the gauntlet:
We do have something like a perception of the objects of set theory... The axioms force themselves upon us as being true.Kurt Gödel
He saw no reason to trust mathematical intuition less than sense perception.
The war of the titans: Mach against Boltzmann
Schlick's chair (from 1922) had been held by Boltzmann, and before him by Ernst Mach (history and philosophy of the inductive sciences, 1895–1901). Mach, professor of mathematics at Graz (1864) and physics at Prague (1867), contributed to acoustics, optics, thermodynamics and more; in 1887 he founded the study of supersonics (the sonic boom; "Mach 2" is twice the speed of sound). He had a cult following, including the poet Hugo von Hofmannsthal, later Strauss's librettist.
| Mach | Boltzmann | |
|---|---|---|
| Basis of science | Sensations of the five senses | Mathematical theory and models |
| Atoms | "I don't believe that atoms exist!" (Imperial Academy, 1897) | Real, though invisible |
| Models | Opposed; science built by induction | Models describe physical reality |
| Abstract mathematics | Suspicious of it | Source of new insight |
| Legacy | Cleared ground for relativity; drifted towards idealism | Statistical mechanics; probability in physics |
Mach called traditional philosophy "pseudoproblems" (a word Carnap loved) and asked whether the sun "really" stands still: with no detector for absolute position, the question is meaningless. Einstein credited Mach as an influence on special relativity, but the influence was negative: "Mach's way cannot give birth to anything living; it can only exterminate harmful vermin," he told Michele Besso. Mach's other maxims did harm. He opposed models, believed science grows by induction, and resisted atoms; Einstein worked by Gedankenexperimente instead, and later blamed the anti-atomism of Mach and Ostwald on "philosophical prejudices." Einstein did share Mach's wariness of abstract mathematics for a while ("since the mathematicians pounced on the relativity theory I no longer understand it myself"), but came round to Minkowski, without whom general relativity was impossible.
Mach's empiricism also collapsed into idealism: sensations are private and subjective, so a science built on them loses the shared external world. Yourgrau's irony is that the Platonist Frege turns out the better empirical realist; in his essay "Thought" he warned that a scientist left only with ideas "finally undermines the foundations of his own construction." Lenin took time out from revolution to attack Mach in Materialism and Empirio-Criticism (1909).
Boltzmann (pianist, once a pupil of Bruckner until his mother found the composer's wet raincoat on the bed; "my sweet fat darling" to his fiancée) quit Vienna for Leipzig after Mach arrived in 1895, clashed there with Wilhelm Ostwald, attempted suicide, and returned in 1901 when Mach went to the Austrian upper house; his philosophy lectures overflowed the largest hall. He showed that the second law of thermodynamics (isolated systems move towards maximum entropy) holds under a statistical interpretation, even though the laws of mechanics are time-symmetric. That bears on the direction of time and on irreversible processes like breaking an egg, and paved the way for probability in quantum mechanics.
1905: Einstein uses both titans at once. In the same volume of Annalen der Physik he used Mach's verificationist critique of Newton to build relativity and Boltzmann's realism to explain Brownian motion by real atoms, with testable predictions. Einstein later described the scientist as an "unscrupulous opportunist": a realist when describing a world independent of perception, a positivist when justifying concepts by sensory relations. Rescue came too late: in 1906, on holiday at Duino near Trieste, Boltzmann hanged himself while his wife and daughters were swimming. Wittgenstein, who had hoped to study physics under him, went instead to aeronautics in Manchester, and from there to Frege and Russell.
Boltzmann left a mark on the Tractatus too. He located a physical system by coordinates (three of space, one of time, plus temperature, pressure and so on), giving an "ensemble of possible states". Wittgenstein generalised it into "logical space": "the facts in logical space are the world."
Corrections: Boltzmann's Leipzig rival was Wilhelm Ostwald (the source prints "Oswald"). Mach left his chair after a stroke (1898) and entered the Herrenhaus (upper house) in 1901. Yourgrau's two-sided portrait (Mach as ruthless polemicist, Boltzmann as martyr) is a common simplification: Boltzmann's depression had many causes, and Mach's critique of absolute space shaped general relativity positively too, through "Mach's principle."
Schlick, a murder, and the Circle's end
Moritz Schlick had studied under Max Planck, wrote widely on relativity, and is best known for General Theory of Knowledge. Quiet, liberal, a monotone lecturer in an elegant vest, he idolised only "figures of the first order": Einstein, then Russell, then most of all Wittgenstein. Asked by Menger if he was really from Berlin: "Sad, but true." Unlike Mach he accepted atoms, but with Carnap and Hempel he raised Einstein's methodological maxims into a formal postulate, the verifiability theory of meaning. Gödel later wrote that positivism's hegemony made the Circle mistake Einstein for an ally and underestimate the difficulty of reducing mathematics to sign manipulation.
The murder of Schlick, 1936. Hans Nelböck, an unstable former doctoral student, spurned by Sylvia Borowicka (who preferred Schlick) and unable to find work, hounded Schlick for months. Police were told; a bodyguard hired. On the university steps Nelböck fired four rounds at point-blank range: "Now you damned bastard, there you have it!" A newspaper hoped the murder would hasten "a truly satisfactory solution to the Jewish question", though Schlick was a German Protestant whose one Jewish assistant, Friedrich Waismann, had already been dismissed. Nelböck got a lenient ten years, became a folk hero after the Anschluss, was released on probation and spent the war as a geological technician for the Reich. The Circle's history ran from a suicide to a murder.
The shooting took place on 22 June 1936, not 21 June as the source has it.
What relativity meant
At the turn of the century Newton's mechanics (velocities add; measurements depend on the observer's motion) clashed with Maxwell's equations, which give light a fixed speed c, apparently relative to the ether at rest in absolute space. Measure light's speed in your frame and any deviation from c reveals your absolute motion. Yet the Michelson–Morley experiments found none, even though frames in relative motion cannot all be at rest in the ether. Bullets from a moving train gain the train's speed; light from the same train does not.
Hendrik Lorentz, Einstein's father figure, supplied the Lorentz transformations, making the laws hold equally in the ether's rest frame and any inertial frame; for him they marked distortions of measurement. Einstein drew the opposite moral: space and time themselves, Kant's forms of intuition, must be redefined, with the speed of light built into the definition of time. He combined the letter of positivism (no properties of space and time beyond what rods and clocks fix) with the spirit of German metaphysics (saying what space and time are). Heisenberg and others saw a standard-bearer of positivism; Yourgrau sees an opportunist, as Gödel would later be when he used Hilbert's formalism for Platonist ends. General relativity and Einstein's war on Copenhagen moved further still from the creed. In Einstein the positivists had acquired not a friend but an enemy.
Cantor's paradise and the third crisis
David Hilbert, inspired by Einstein, had published the field equations of general relativity five days before Einstein (Yourgrau's dating), causing uncomfortable moments. Positivists distrusted Cantor's transfinite numbers as Platonist; Hilbert defended them: "No one shall expel us from the paradise that Cantor has created."
Rigorous infinity was needed. Weierstrass, Cauchy and Cantor built the theory of limits, giving sense to "point" and "instantaneous velocity", replacing the ghostly infinitesimals of Newton and Leibniz; Cantor, Frege and Dedekind built the real numbers out of infinite sequences, essential because the continuum of space and time needs the reals.
Cantor's two discoveries
- Infinity is actual. Against the tradition from Aristotle to Gauss that infinity is only potential, Cantor argued there is a definite number, ℵ₀ (aleph-null), numbering the natural numbers. It cannot itself be a natural number: "the king cannot arise from the class of peasants."
- There is an infinity of infinities. Two sets are equal in size if their members can be put in one-to-one correspondence. Assume a pairing of naturals with reals; the diagonal argument builds a real number missing from the list. So there are more reals than naturals: 2^ℵ₀ > ℵ₀. Generalised: the power set of any set is larger than the set, so every infinite number has a bigger one.
The list that is never complete. Write every real number between 0 and 1 in an infinite list, one per row. Build a new number: its first digit differs from row 1's first digit, its second from row 2's second, and so on down the diagonal. The new number differs from every row in at least one place, so it is not on the list, however the list was made. Any "complete" list of reals is incomplete. Keep this move in mind: Gödel's theorem turns the same diagonal trick on proofs.
Paradoxes followed. Cantor himself proved there is no set of all sets: its number would exceed every transfinite number, yet every such number has a successor in size. Set theory could not even say how many sets there are. Then came Russell's paradox. If every property determines a set, consider the set of all sets that are not members of themselves: it is a member of itself if and only if it is not. So some properties determine no set, and nobody knew which sets exist. Russell found it while trying to refute Cantor's diagonal proof; the arrow missed Cantor and hit Frege. Russell's polite, lethal letter of 1902 ("There is just one point where I have encountered a difficulty...") wrecked Frege's life's work: "The sole possible foundations of arithmetic seem to vanish."
| Crisis | Trigger | Paradoxical new number |
|---|---|---|
| First (ancient Greece) | Pythagorean theorem | Irrationals, not a ratio of naturals |
| Second (17th century) | Newton–Leibniz calculus | Infinitesimals, non-zero yet negligible |
| Third (1900s) | Russell's paradox in set theory | Transfinite sets and the universe of sets |
Formalism: Hilbert's hermaphrodite
The third crisis lifted the positivists' star: mathematics had flown too high on infinity. Hilbert, a positivist in spirit, found a way to keep Cantor's paradise anyway. Formal proofs, strings you can write on a blackboard and perceive with the senses, would play the role of the physicist's instruments. Hilbert was the Moses leading mathematicians through the desert of positivism back to paradise, preserving Cantor's letter if not his spirit.
| Gödel's "leftward" worldview | Gödel's "rightward" worldview |
|---|---|
| Scepticism, materialism, positivism | Spiritualism, idealism, theology or metaphysics |
| Peak: the Copenhagen interpretation | Natural home of mathematics, an a priori science |
| Used the set-theoretic antinomies as "a pretext for a leftward upheaval" | Resisted, but "around the turn of the century its hour struck" |
Gödel's verdict on formalism: "that curious hermaphroditic thing," trying to satisfy both the Zeitgeist and the nature of mathematics. Truth gives way to a formula game: axioms are not true, inferences are "a mere game with symbols," and a point or number is by implicit definition anything satisfying the axioms. The positivists' refusal to consider a priori insight was, for Gödel, "an a priorism with the sign reversed." Schlick admitted that a system built by implicit definitions "does not at any point rest on the ground of reality," which makes Wigner's "unreasonable effectiveness of mathematics" even more unreasonable.
Yourgrau is openly partisan here. He casts the Circle as a dogmatic "religion" and Hilbert as a closet positivist; many historians see the Circle as diverse (Neurath's politics, Carnap's tolerance principle, internal fights over protocol sentences) and Hilbert's programme as a finitist epistemology, not a denial of mathematical truth. The Hilbert priority claim is also contested: the printer's proofs of Hilbert's paper (dated 6 December 1915) lack the explicit field equations, so most historians now credit Einstein's 25 November 1915 paper. Yourgrau's gloss on the reals ("rationals as sequences of naturals") is loose: rationals are ratios of integers, and reals are built from sequences or cuts of rationals.
Gödel's own adviser Hahn closed his essay "The Crisis in Intuition" by calling intuition "a force of habit rooted in psychological inertia." And the Tractatus admitted only empirical facts: "there exists no such thing as a mathematical fact," as Gödel summarised it. Between Hilbert and Wittgenstein, the ghost of mathematics seemed laid to rest. Gödel was about to show from the heart of Wittgenstein country that it was not. Vienna's circle could not, after all, be completed.
A Spy in the House of Logic
In 1930–31 Gödel took up Hilbert's programme to secure mathematics by formalising it, and proved it could not be done: any consistent formal system strong enough for arithmetic leaves some truths unprovable, and cannot prove its own consistency. Yourgrau walks through the proof (three languages, Gödel numbering, the formula that says "I cannot be proved"), its slow reception, and its real meaning. Gödel was a spy: he used the positivists' own weapon, formalism, to show that truth outruns proof and that syntax cannot replace meaning.
A paradox on every bedspread
After Russell's letter of 1902, fear spread. Gödel put it bluntly: Russell showed "that our logical intuitions... Are self-contradictory." Not every property determines a set; the notion of set, which seemed clear, led Frege straight to contradiction. Cantor's hierarchy was full of paradox, and even Euclidean geometry looked shaky once non-Euclidean geometries were shown consistent: if Euclid could be denied without contradiction, where did its truth lie? Everywhere, intuition was suspect. Hilbert declared: "Where else would reliability and truth be found if even mathematical thinking fails?" Clarifying the infinite was needed "for the honour of human understanding itself."
In Vienna, Gödel was becoming part of the problem without knowing it. At the very Königsberg conference where he announced incompleteness, his adviser Hahn admitted that one simple fact stood in the way of empiricism: "the existence of logic and mathematics." Gödel's formalisms were Janus-faced by design: irrefutable theorems on one side, anti-positivist philosophy on the other. Friends of formalism could not reject conclusions drawn from inescapable premises; its enemies had to admit that something philosophically deep could come from within the formal. This is the reason they shoot spies.
Formalism: the leitmotif of the century
Hilbert's programme aimed to replace intuitive mathematics in each domain with a system of axioms in a pure formula language, manipulated by mechanical rules of syntax (like a modern computer program), and to prove the system:
Hilbert's three requirements for a formal system
- Consistent. No two contradictory theorems can be derived from the axioms.
- Complete. Every true statement expressible in the system (under its intended interpretation) is derivable.
- Finitary consistency proof. The proof of consistency must use only finitary methods, surveyable in principle by sense experience, never appealing to a completed infinity of Cantor's kind, so that it is not circular.
Yourgrau widens formalism into the leitmotif of the twentieth century: form over content, syntax over semantics, proof over truth. Its machine is the computer, a pure syntax engine. The same spirit ran through the arts and sciences:
| Field | Formalist turn | Figures |
|---|---|---|
| Painting | Cézanne's realism hid a free play of geometric forms; cubism made it explicit | Cézanne, Gris, Braque, Picasso |
| Music | Brahms's hidden logic ("I can begin to hear the sound of machinery," said Wittgenstein); serialism | Brahms, Schoenberg, Glenn Gould (structural "x-ray" of Bach) |
| Physics | Kantian intuitions of space and time replaced by space-time, constrained by Lorentz invariance | Einstein |
| Linguistics | Syntax dominant over semantics, modelled on Frege | Chomsky |
| Anthropology | Structuralism | Lévi-Strauss |
| Geometry | Points could be "beer mugs" as long as they obey the axioms | Hilbert |
| Ethics | Authenticity replaces conscience; consistency over content ("Hitler too was authentic") | Sartre |
Frege rejected Hilbert's implicit definitions: they cannot settle whether Julius Caesar is a number, "whether that same familiar conqueror of Gaul is a number or not." Formalism promised creative freedom and also safety: since we made the rules, we can police them for contradictions, and consistency, not truth, becomes the goal. If formalism is the century's leitmotif and Hilbert's formalism is its purest case, then Gödel's refutation of it is, Yourgrau says, the most significant intellectual accomplishment of the twentieth century.
What Gödel actually proved
Gödel began by trying to fulfil Hilbert's programme for analysis, starting with its subsystem arithmetic. Frege, Dedekind and Peano had axiomatised arithmetic in the five Peano postulates, from which all number truths were believed derivable by Frege's logic, with rules of inference so mechanical they could be followed, as John Myhill said, "by an imbecile or a computer."
First theorem: any consistent formal system (even an infinite axiom list, so long as it is surveyable by a finite mind, i.e. recursive) strong enough for arithmetic is incomplete: some arithmetical truth is unprovable in it, and so is its negation. Second theorem: such a system, if consistent, cannot prove its own consistency. Truth is absolute; proof is always relative to a system. Mathematical truth is not reducible to formal proof.
Since a computer proves only what follows from axioms it is given and cannot create new ones (Gödel's emphasis), no computer or system of computers, even infinite, captures all arithmetic truth. "Continued appeals to mathematical intuition are necessary." Ironically, a computer can be programmed to prove Gödel's theorems themselves.
Be clear about what he did not prove. He did not find an absolutely unprovable proposition beyond every system; Gödel, like Hilbert, doubted such things exist. For each formal system there is a perfectly ordinary, if complex, arithmetical sentence that slips through its net; a stronger system proves it, and has its own unprovable sentence, and so on. There is no "supervirus", only a particular bug in each system.
The triple fugue: IA, FA, MFA
- IA, intuitive arithmetic. Textbook number theory in ordinary mathematical language; its sentences have content and are true or false.
- FA, formal arithmetic. In Gödel's proof, the system of Principia Mathematica (Russell, after Frege): axioms plus explicit rules; its formulas are neither true nor false, only provable (theorems) or not. It is built to represent IA one-to-one.
- MFA, the metatheory of FA. The owner's manual for the machine: it says which strings are well-formed and, above all, what counts as a proof in FA. Its sentences are true or false.
The insight was that FA can represent not only IA but MFA too, because MFA can be represented in IA (the arithmetisation of metamathematics) and IA in FA. So FA can represent its own metatheory. Gödel then built one formula with two simultaneous meanings, an arithmetical one in IA and a proof-theoretic one in MFA, and showed it provably unprovable in FA yet intuitively true. He sidestepped the liar paradox, replacing it with an unprovability "paradox" that is no paradox, and showed self-reference to be harmless. It looked, Yourgrau says, more like Kafka than mathematics. Paul Cohen, the Fields medallist who later proved the continuum hypothesis independent, first thought it more philosophy than mathematics; after talking with Stephen Kleene, "I was rather depressed when I realized Gödel was right."
Reusable numbers: arithmetising syntax
Gödel borrowed from Descartes, whose coordinates arithmetised geometry, turning facts about figures into facts about numbers. Gödel numbering gives every primitive symbol of FA a unique natural number, and uniquely codes sequences of symbols (formulas) and sequences of formulas (proofs), relying on the fundamental theorem of arithmetic: every positive integer factors uniquely into primes. A statement about FA's syntax becomes a statement about numbers.
He then showed that every key syntactic notion, including "is a well-formed formula" and "is a proof of", corresponds to a recursive function in IA, one that can be mechanically computed. Addition is the standard example of a primitive recursive function, defined from the successor s(x) = x + 1:
- (a) x + 0 = x
- (b) x + s(y) = s(x + y): x plus the successor of y equals the successor of x-plus-y.
Since every natural number is 0 or a successor, this "bootstraps" any sum mechanically. Recursion was used implicitly by Dedekind, Peano and Skolem; Gödel gave the first precise account, crediting Jacques Herbrand.
Herbrand's silence. Herbrand wrote to Gödel after hearing of incompleteness from von Neumann. Gödel replied at length, respectfully, proposing they each write in their own language in future. No answer came, only a letter from Herbrand's father: his son had fallen to his death climbing in the Alps, aged 23 (1931).
The proof relation Bew(x, y), "x is a proof of y" (German Beweis), coded into numbers, is recursive; and every recursive truth of IA is a theorem of FA. So FA, a system of meaningless signs, is "double stuffed" with meaning: its theorems represent arithmetic and, via Gödel numbers, FA's own proof theory. The natural numbers are reusable.
"I cannot be proved": the proof step by step
Gödel built the formula G, whose Gödel number is g, which on one reading is a statement about numbers and on the other says "the formula with number g is unprovable in FA". Since that formula is G itself, G says "I cannot be proved." Unlike the liar's "I am not true", the question about G concerns provability, which is clear-cut, and as a statement about numbers G is simply true or false.
Part 1: if FA is consistent, G is not provable
- Assume FA is consistent (otherwise it is useless and the game is over).
- Assume G is provable: some sequence S, with number s, is a proof of G, so Bew(s, g) holds.
- Bew is recursive, hence representable: the formula Bew(s, g) is a theorem of FA.
- By logic, Bew(s, g) implies "there exists an x with Bew(x, g)", i.e. It is not the case that no x proves g.
- That is equivalent to not-G, so not-G is a theorem too.
- Both G and not-G are theorems: FA is inconsistent, contradicting step 1.
- Therefore if FA is consistent, G is unprovable.
Part 2: if FA is ω-consistent, not-G is not provable
- ω-consistency: FA cannot prove F(0), F(1), F(2)... Individually while also proving "not for all x, F(x)". (ω names the natural numbers; if FA is to represent them, this is non-negotiable.)
- From Part 1, G is unprovable, so for each x, not-Bew(x, g) is true, and by representability provable in FA.
- If not-G were a theorem, FA would prove "not for all x, not-Bew(x, g)".
- That violates ω-consistency. So not-G is also unprovable: G is undecidable in FA.
Rosser's improvement (1936): an undecidable sentence can be built assuming plain consistency only.
The honest oracle. An oracle answers only with sentences it can prove, and it never lies. You hand it the card: "The oracle will never assert this card." If the oracle asserts it, the card is false and the oracle has lied. So it never asserts it, which makes the card true. You now know a truth the oracle can never state. Replace "oracle" with any consistent formal system and "assert" with "prove", and you have Gödel's first theorem, and the reason a mind that sees the card's truth seems to outrun the machine.
The second nail
The second theorem, proved quickly by similar techniques: if FA is consistent, FA cannot prove its own consistency. Only an inconsistent system can prove itself consistent. There is no magic shield against hidden contradiction for a mathematician treating theories as uninterpreted formula games. (Yourgrau's gloss: by the same token there can be no foolproof antivirus program guaranteed to catch every program trying to alter the one it protects.)
John von Neumann, "the quickest of the quick", derived the unprovability of consistency himself after hearing Gödel's first result and wrote offering details; Gödel replied that the manuscript was already with the editors. It was von Neumann, against a cautious Gödel, who saw at once that this left no wiggle room for Hilbert: without Russell's controversial axiom of reducibility, "one cannot obtain a foundation for classical mathematics via logical means." Gödel for several years left open whether some finitary proof might escape the theorem. Yourgrau suggests von Neumann, father of the computer and Los Alamos architect, never got over not being first. Hermann Weyl spoke of the Gödel "debacle" and "catastrophe": Euclid's two-thousand-year ideal of axiomatisation was shattered just as Frege and Hilbert perfected it, a case of Hegel's "cunning of history".
From recursion to computers and Church's thesis
The clarification of recursive functions by Gödel and Herbrand fed straight into the theory of computation. Alan Turing defined Turing-computable functions; Alonzo Church defined λ-definable functions; Church's students Kleene and Rosser (and Turing, a sometime student) showed the classes equivalent. Hence Church's thesis: the recursive functions are exactly the effectively calculable ones.
How Church's thesis strengthens Gödel
- Generality. A formal system just is one whose syntax is mechanically, hence recursively, specifiable; so incompleteness applies to every genuinely formal system.
- Finality for Hilbert. If Hilbert's "finitary" methods are the mechanically specifiable ones, the second theorem directly refutes a finitary consistency proof for arithmetic.
Yourgrau overstates two links. Turing's 1936 machine came from Hilbert's decision problem, not from Gödel–Herbrand recursion; the equivalence was shown afterwards. And the anti-mechanist moral is weaker than he suggests: in his 1951 Gibbs lecture Gödel himself concluded only a disjunction, either the mind infinitely surpasses any machine or there are absolutely undecidable arithmetical problems. The Lucas (1961) and Penrose (1989, 1994) arguments that minds are not machines are widely judged to fail, since we cannot know our own reasoning is consistent. The "antivirus" gloss is closer to Rice's theorem and the halting problem than to incompleteness itself.
The whisper heard round the world
Gödel first told Carnap, in the Café Reichsrat, days before both went to the foundations conference at Königsberg (today Kaliningrad, birthplace of Kant and Hilbert). Carnap, student of both Frege and Husserl (fathers of the analytic and continental traditions), promptly recommended consistency as the touchstone of formal theories at the conference, and a year later still found the results hard to understand. Only von Neumann grasped it at once and became a lifelong friend. Hans Reichenbach's conference report in Erkenntnis did not mention Gödel, though the editors soon asked him for a postscript. Usually a surprising proof of a believed result (like Fermat's last theorem) makes a sensation; here the result itself was unexpected, so the proof was explosive.
- Paul Bernays: Hilbert's assistant, later Gödel's close associate; puzzled at first and asked Hilbert's help.
- David Hilbert: answered with anger, then denial, then an ad hoc "anti-Gödel" rule allowing infinitely many premises (the ω-rule), which Gödel noted destroys the very idea of a formal system: the cure would kill the patient. He soon dropped it.
- Ernst Zermelo: founder of axiomatic set theory (Zermelo–Fraenkel axioms still standard); Gödel's chief mathematical opponent; claimed a "major gap"; Gödel replied in ten handwritten pages, then gave up. Carnap: Zermelo "completely misunderstood".
- Bertrand Russell: in 1963 still asked whether 2 + 2 is "not 4, but 4.001", missing the point; glad he no longer worked on logic.
- Ludwig Wittgenstein: chief philosophical detractor; Gödel called his posthumous reading "completely trivial and uninteresting", whereas Russell misread him "in a very interesting manner".
- Emil Post: in a 1941 paper (rejected), amazed that ten years on the only lesson drawn was the need for many formal systems instead of one, rather than "a return to meaning and truth."
Hilbert showed better grace elsewhere. Before WWI he took on Jakob Grommer, a Talmud-school student without a gymnasium diploma and with deformed hands and feet: if such students wrote such dissertations, "it will be necessary to make a law forbidding the taking of the examination for the diploma." He asked Max Born his weakest subject (ideal theory) and examined him on it: "I was just interested to find out what you know about things about which you know nothing." Zermelo, at Bad Elster in 1931, resisted lunch with Gödel because he "did not like Gödel's looks", the food was short and the climb too hard.
Gödel later diagnosed the confusion: formalists took formal demonstrability to be an analysis of mathematical truth, so they could not even draw the distinction. Most logicians eventually accepted the results; Hilbert's programme joined squaring the circle and proving the parallel postulate among abandoned projects, and Gödel's methods (recursion, arithmetisation, Gödel sentences) became routine. Yet formalisation as an enterprise went on regardless; even Brouwer's intuitionists built formal systems.
Gödel's programme: the formal limits of the formal
Hilbert wanted to replace the intuitive with the formal. Gödel's programme was to establish, by formal means, the limits of formal methods in capturing intuitive concepts. Incompleteness is the negative case (formal proof cannot capture truth); Church's thesis is the positive case (general recursiveness does capture effective calculability). Yourgrau's claim: the 1949 relativity work is a third instance, with the conclusion reversed.
Church himself said his thesis grew from Gödel's idea, inspired by Herbrand, of identifying the intuitive concept with a formal one. Gödel was astonished that it worked: "for the first time" an "absolute definition of an interesting epistemological notion", and "a kind of miracle" that the diagonal procedure does not lead outside the defined notion. Church's thesis cannot itself be formally proved, since effective calculability is intuitive. What made incompleteness so irresistible and so aggravating was that the negative result was proved formally, from inside the formalism. This method, practised in plain sight, has stayed invisible, and so has its reappearance in Gödel's cosmology. Gödel the logician was a spy; Gödel the philosopher would prove harder still to see.
Incompleteness is now machine-checked: Shankar (1986, Boyer–Moore prover), O'Connor (2005, Coq) and Paulson (2013–15, Isabelle, including the second theorem). The theorems remain central to debates on whether large language models or any formal reasoning system could exhaust mathematics; the standard answer is that incompleteness limits every fixed consistent system, humans included, if humans are formal, but does not by itself show that minds are not machines.
It's Hard to Leave Vienna
Fame brought Gödel to the new Institute for Advanced Study three times, yet he clung to Vienna until the door was almost shut. Yourgrau calls this the "Vienna syndrome": an attachment to the German-speaking world so strong that its luminaries (Schrödinger, the Wittgenstein sisters, Furtwängler, Heisenberg, even Einstein in Berlin) stayed far longer than reason advised. Gödel escaped only in 1940, by the trans-Siberian railway, to join Einstein in what Vonnegut called "das Reich der Zwei", an empire shrunk to two.
Flexner goes shopping
In 1929, just before the crash, Louis Bamberger and his sister Mrs. Felix Fuld sold their Newark department store to R. H. Macy and resolved to fund a centre of learning. The reformer Abraham Flexner proposed an institute of pure thought with no teaching or administration (publication still expected), located in Princeton to borrow the university's culture, and founded first on mathematics and mathematical physics, for their purity and agreed standards of objectivity.
- 1932: first permanent professors, Oswald Veblen (poached from Princeton University, the first of several thefts) and Albert Einstein.
- Within a year: Hermann Weyl, James Alexander, John von Neumann.
- 24: "workers" (later "temporary members") in the inaugural year 1933–34, Gödel among them.
- 3: approaches before Einstein said yes: Caltech (1931), Oxford (spring 1932), Caputh (summer 1932): "I am fire and flame for it."
Veblen heard Gödel at Menger's colloquium in 1933 (a paper relating classical and intuitionist logic) and invited him. The lures: von Neumann's quantum mechanics seminar (Gödel had a "lively interest"; he borrowed Schrödinger, Dirac, Planck, Mach, Born and Lorentz from the library), and Alonzo Church, Veblen's student, whose new system (including the λ-calculus) Church believed was immune to incompleteness. Church was wrong; worse, his students Kleene and Rosser proved his original system inconsistent, and he became a champion of incompleteness's generality. Gödel added a third reason: to improve his English.
The Vienna syndrome
| Case | Strain | What happened |
|---|---|---|
| Erwin Schrödinger | Austrian | Returned from Oxford to Graz in 1936 ("an unprecedented stupidity"); mocked Nazi race theory in a Vienna lecture; after the Anschluss published a grovelling "Confession to the Führer" (30 March 1938); dismissed months later anyway for "political unreliability" |
| Hermine and Helene Wittgenstein | Austrian | Refused to leave; bought reclassification as Mischlinge of the first degree with a huge transfer of family wealth to the Reichsbank |
| Wilhelm Furtwängler | German | Stayed with the Berlin Philharmonic, sacked Jewish players under orders, dropped Mendelssohn and Mahler, yet helped Jews constantly; protected only by Hitler's admiration |
| Werner Heisenberg | German | Despised the Nazis, admired Einstein's "Jewish physics", yet stayed; an outcast at home, a traitor abroad |
| Albert Einstein | Berlin | Moved to Berlin in 1913 and stayed through WWI and the early Nazi years |
| Kurt Gödel | Austrian | Bought a flat in Vienna in 1939; left only in January 1940 |
Schrödinger's lecture, "World Structure in the Large and in the Small", introduced by Hans Thirring (Gödel's physics teacher), ended with a jibe at the wish to place one nation "over or under another." The audience cheered; the Nazis took note. To Einstein he later apologised for his "certainly quite cowardly statement": he had not expected "the fortress would be surrendered without striking a blow."
Buying a grandfather. Under the Nuremberg laws, three Jewish grandparents made one Jewish, and the Wittgenstein sisters seemed to qualify. A Befreiung (reclassification) required proof of only two, and no practice of Judaism or Jewish spouse. In 1939 there were 2,100 petitions; Hitler approved 12, grumbling that party members knew "more decent Jews than there are Jews in the whole of the German Reich." WWI medals (Paul lost his right arm; Ludwig wrote the Tractatus in the army) did not move the Reich; gold did. With help from Ludwig and Paul and from Arthur Seyss-Inquart, a vast sum went to Berlin, and the ministry accepted that grandfather Hermann Christian was the illegitimate son of the princely house of Waldeck. The sisters stayed.
Furtwängler's cousin was Gödel's teacher Philipp Furtwängler; Paul Wittgenstein had played left-hand concertos with him. Goebbels complained: "Can you name me a Jew on whose behalf Furtwängler has not intervened?" A plan to replace him with Herbert von Karajan, who joined the Nazi party twice, was dismissed by Hitler. Heisenberg's rivalry with Schrödinger was mutual disgust: Schrödinger was "repelled" by Heisenberg's "transcendental algebra" and lack of visualisability; Heisenberg found Schrödinger's physical picture "disgusting... Trash". Von Neumann proved their matrix and wave mechanics equivalent in a strong sense.
Two corrections. Seyss-Inquart was not just "a Viennese lawyer": after the Anschluss he was Reich Governor of Austria, and was hanged at Nuremberg (1946) for his later role in the occupied Netherlands. Gödel's IAS visits were 1933–34, autumn 1935 (cut short after weeks, followed by a breakdown) and autumn 1938 (followed by spring 1939 at Notre Dame), not 1935–36 and 1936–37 as Yourgrau dates them.
Red roses from Dr. Einstein
Einstein had left Germany as a teenager for Switzerland, where he studied, spent seven years as a patent clerk becoming "the new Copernicus", and became a Zurich professor. Why go back, in 1913, to Berlin, the heart of Prussia? Max Planck and Walther Nernst gave him a day to decide; he would meet them at the station with white roses for no, red for yes. The roses were red. Biographer Albrecht Fölsing lists reasons (Planck first championed relativity; no teaching; cousin Elsa Löwenthal was there), none conclusive. Yourgrau's diagnosis: the Berlin strain of the syndrome, which kept a lone pacifist in Germany through WWI hysteria and into the Nazi years.
The writing was on the wall early. In 1920 an "anti-relativity" club, the "Study Group of German Natural Philosophers", paid people to denounce Einstein's "Jewish physics", and on 24 August 1920 staged a rally in Berlin's Philharmonic Hall. Einstein attended and published a reply, then regretted touching the tar baby. It took over a decade more to make him leave.
The eleventh hour plus one
Gödel hated controversy, unlike Einstein, who would have toasted Roosevelt's "I am an old campaigner and love a good fight." So he published only what he could prove beyond reasonable doubt, and his manuscripts are full of valuable crossed-out material. The strategy of safety failed anyway: he was always seen as a man out of step with his times on truth, language-centred philosophy, God and spirits.
- Losses on every side. Hahn died (1934), Schlick was murdered, Herbrand fell, Menger and Carnap left for America, and his mother, forced back to Brno by money troubles, was at risk for her loud anti-Nazi views.
- Silence about politics. Gödel said almost nothing about events, which cooled his friendship with Menger. The Nazis noted his tie to "the Jewish professor Hahn" and that he "traveled in liberal-Jewish circles." His cleaning lady signed her bill "Heil Hitler!"
- Marriage. In 1938 he married Adele in a civil ceremony, surprising Rudolf, who had only just met her.
- The push. Late in 1939 a gang took him for a Jew, knocked off his glasses, and was driven off by Adele's umbrella; and the frail 32-year-old was ordered to an army medical, which he passed.
- The exit. New IAS director Frank Aydelotte pleaded that Gödel was an Aryan and a world-famous mathematician; a visa came in December 1939. Fearing British arrest of German citizens on the Atlantic, the couple crossed Russia by the trans-Siberian railway, sailed from Japan to San Francisco (4 March 1940), and took the train to Princeton, travelling often by night in fear of being sent back.
What Gödel and Einstein had left behind they never found again. Isolated, speaking the language of their adopted country's enemy, they made a two-man empire. Their thoughts turned to the theme at the centre of Einstein's relativity and of Gödel's interest in Kant and Leibniz: time.
Amid the Demigods
In Princeton, Gödel and Einstein became inseparable outsiders: walking daily, sceptical of Copenhagen, far from the bomb, increasingly philosophical and unfashionable. Yourgrau shows how their shared realism (ontology must not be made the servant of epistemology) led Gödel, invited to honour Einstein's 70th birthday, to treat relativity as he had treated Hilbert's programme. He asked whether t, the time coordinate of relativistic space-time, really is time, and built a "limit case" universe so extreme that the answer had to be no.
Two giants among the pygmies on stilts
Princeton was small, provincial and inbred; Russell found it "as like Oxford as monkeys can make it," and Einstein called it "an exceedingly amusing ceremonial backwater of tiny spindle-shanked demigods." Gödel, fluent in English, found it "ten times more congenial" than Vienna and praised the "prompt functioning of government officials", "10 × 10 × ..." better. Adele, who struggled with English and missed Vienna's mix of professors and dancers, could hardly breathe.
Household of a fragile genius. Childless, Adele kept love birds, a dog (Penny) and a cat (neighbour Bobbie Brown talked her out of docking its tail to make it look Manx), and the couple sponsored a foster child abroad. Homes: 245 Nassau Street, 108 Stockton Street (neighbour George Brown, Quine's student, who had taken notes on Gödel's early IAS lectures; Gödel removed the window screens to "breathe properly", letting in dust and insects), 120 Alexander Street ("directly opposite the most elegant hotel in town"), and from August 1949 a cinder-block house at 129 (later 145) Linden Lane, bought with a mortgage for three-quarters and a salary advance from director J. Robert Oppenheimer for the rest, a "somewhat shaky" arrangement. Adele added oriental rugs, chandeliers, a sun porch and a screen of evergreens, and cooked heavy German food.
Gödel's minders passed down a line: Oswald Veblen, then Einstein, then after 1955 the economist Oskar Morgenstern. As early as 1941, director Frank Aydelotte asked Gödel's doctor whether his "malady" might turn violent; it would not. The symptoms: "bad air" from the refrigerator and heating, winter coat, hat and gloves in midsummer, imaginary illnesses treated as real, distrust of doctors, avoidance of touch, endless night-time phone calls (Yourgrau compares Glenn Gould), and a paranoia that may have become psychosis. Philipp Furtwängler had wondered whether the illness was a consequence of proving unprovability, "or is his illness necessary for such an occupation?" His death certificate would cite "inanition" caused by personality disturbance.
Einstein's eccentricities were milder: the ice cream on Nassau Street, no socks, a leather jacket worn because what doesn't wear out needn't be replaced, his son Hans turning dandy in protest. Peter Bucky called him "a classic male chauvinist"; his pipe survived even a capsized sailboat, the Tinef. A bohemian Jew in a town whose university kept Jewish quotas (which Einstein protested), he was an outsider. Richard Feynman, kept out of Columbia by its quota, went to MIT, where Philip Morse assured Princeton that Feynman's "physiognomy and manner" showed no trace of Jewishness. Feynman later became the third winner of the Einstein Award, after Gödel and Schwinger (who shared the first, 1951).
The friendship
- 1933 Gödel and Einstein first meet at the IAS, introduced by the émigré Paul Oppenheim.
- 1942 They become friends, after Gödel has joined the institute.
- 1946 Helen Dukas records a car hitting a tree when its driver recognised Einstein.
- Feb 1951 Gödel ill; Einstein visits him "many times both in the hospital and at home."
- 1952 Einstein: Gödel "has really gone completely crazy. He voted for Eisenhower."
- 1955 Einstein dies; Gödel declines invitations to speak about him.
Gödel told his mother the routine: he met Einstein at his house between 10 and 11 a.m., they walked half an hour to the institute, and walked back at 1 or 2 p.m. talking politics, philosophy and physics, about 30% of Einstein's working day. Both prized privacy: that people never connected him with Einstein was "very satisfactory to me." Gödel did not find his own fame burdensome; that starts only when "one is known to every child in the street." Einstein, though cheerful, was a pessimist about humanity, because those who tried to do good, "like Christ, Moses, Mohammed," died violently or used violence on their followers. Both lived modestly while von Neumann lived in a mansion; they exchanged housewarming gifts and argued in German on the way home.
Museum pieces in wartime
Einstein called them "museum pieces" and himself "the village idiot". A few doors from his Fine Hall office, Niels Bohr and John Wheeler worked out nuclear fission; Oppenheimer, visiting in 1935, judged "Einstein is completely cuckoo," and after the war became his nominal boss. In 1935 Einstein likened atomic bombs to "firing at birds in the dark, in a neighbourhood that has few birds," yet co-signed the letter to Roosevelt (1939), making him the "father of the father of the bomb", though the letter's causal effect was minimal and he spent his last years preaching against the weapon.
| Einstein | Gödel | |
|---|---|---|
| War work | Navy calculations on torpedo explosions; deemed a security risk for socialist sympathies | None direct; volunteered as a civil defence aircraft spotter after the war (told Atle Selberg) |
| Fund-raising | Rewrote the lost 1905 relativity paper, dictating to Helen Dukas ("Did I really say that?"); sold to an insurer for $6.5 million | Adele sewed a dress a day for Austrian relief, later rewarded with a bust of her father |
| Research focus | Unified field theory | The continuum hypothesis |
| Indirect legacy | Relativity as background to the bomb | Recursive functions, the soul of the computer, and so of military computing |
Yourgrau's pairing: Gödel and the computer, Einstein and the bomb. Neither built the technology or shared its ethos; each supplied essential background.
The citizenship hearing, 1947. Gödel told Morgenstern he had found an "inconsistency" in the US Constitution. Einstein, his witness with Morgenstern, told jokes all the way to distract him. Judge Phillip Forman, who had sworn in Einstein, asked whether a dictatorship like Germany's could arise in America; Gödel began explaining how the Constitution formally permitted one, and the judge cut him short. (The FBI had meanwhile been reading his letters to his mother.) Gödel's later legal analogue of incompleteness: any code relying wholly on its letter will leave some cases undecided, leaving the country defenceless against crises it could not foresee.
The continuum hypothesis and Gödel's realism
Cantor (1878) conjectured that the number of points on a line, 2^ℵ₀, is the very next infinity after ℵ₀: 2^ℵ₀ = ℵ₁. He died without proving it, after repeated stays in sanatoria. Gödel proved in the late 1930s (published 1938–40) that the hypothesis is consistent with Zermelo–Fraenkel set theory: it cannot be disproved. In 1963 Paul Cohen of Stanford proved it cannot be proved either: it is independent.
| Cohen and Church | Gödel | |
|---|---|---|
| Meaning of independence | Like Euclid's parallel postulate: a choice of convention | The comparison with Euclid is misguided |
| Truth value of CH | None, objectively | Determinate; the set universe is a complete reality |
| Way forward | Pick axioms to taste | Find new axioms that "force themselves upon us", guided by intuition and rationality |
Scientists as philosophers
In 1935, with Boris Podolsky and Nathan Rosen, Einstein published EPR, "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" It argued that Copenhagen implies "spooky action at a distance", and its premises were philosophical: a complete theory needs a counterpart for every "element of reality," and "no reasonable definition" of reality lets what is real in one system depend on measuring another. Bohr and Pauli were furious. Gödel saw the parallel to his own realism: "Cantor might also be classified with Einstein and me. Heisenberg and Bohr are on the other side."
Gödel's own manifesto came in 1947, disguised as a popular survey, "What is Cantor's Continuum Problem?", commissioned in 1946 by Lester Ford for the American Mathematical Monthly's "What is...?" series. It defended Platonism and mathematical intuition: there is no reason to trust it less than sense perception, and since the continuum is real, new axioms will settle CH. His 1944 essay in Schilpp's Library of Living Philosophers volume on Russell had prepared the ground, approving young Russell's line that "logic is concerned with the real world just as truly as zoology" and scolding him for abandoning it under Wittgenstein.
Russell's non-reply. Schilpp's invitation (November 1942) said Russell considered Gödel "the scholar par excellence" in the field. Gödel submitted late; Russell, having answered the others, "lacked the leisure" to reply, noting only that it was "eighteen years since" he had worked on logic. Behind this lay ambivalence. Russell claimed never to have hoped for a consistency proof, which blunts only the second theorem; the first struck at the formalist project whose summit was Principia Mathematica, the very system Gödel chose to prove incomplete. Russell was eclipsed three times: by Gödel in logic, by Wittgenstein in philosophy ("not an altogether pleasant experience" to be "regarded as antiquated"), and by Einstein in public fame. Late in life he said he would have become a physicist. Arriving at the IAS in spring 1943 with a chip on his shoulder (his membership application had been declined despite Einstein's support), he failed to recognise von Neumann at his own lecture, and dismissed Gödel as an "unadulterated Platonist."
Einstein's philosophy was immanent in his physics, like Spinoza's God in the world; Gödel kept part of his philosophy separate from his mathematics, like Leibniz's God apart from it. Gödel wanted to be a great philosopher in the line of Plato, Leibniz and Kant, but turned to it late.
From Gödel's fourteen theses, "My Philosophical Viewpoint" (c. 1960)
- Concepts have an objective existence.
- Materialism is false.
- The world is rational.
- Religions are, for the most part, bad, but religion is not.
- There are other worlds and rational beings of a different and higher kind, connected to us "by analogy, not by composition."
- Goal: an axiomatic (not purely formal) theory of the basic concepts, including reason, cause, substance, necessity, value, God, cognition, force, time, form, matter, life, truth, possibility, to "do for metaphysics as much as Newton did for physics" (to Hao Wang).
Ontology over epistemology
Both men had used positivism's weapons without its creed. Einstein's clock-based definition of time in 1905 was the right tool for the job, not a calling card; Minkowski's geometry and then general relativity went far beyond it. Gödel used formal proof, the mathematical positivist's favourite tool, to prove the limits of formal proof: a mechanical demonstration of the limits of mechanical methods, a jewel of formalism and a warning against it.
The positivist mentality derives ontology (what is) from epistemology (what can be known). Mach rejected atoms because no one will ever see one. Heisenberg moved from the impossibility of knowing position and momentum together to the non-existence of such a state, and further to abandoning quantum reality. For Einstein and Gödel, philosophy without ontology is an illusion, and physics without philosophy is engineering.
Yourgrau traces Mach's positivism to Kant's "Copernican revolution": the known must conform to the knower. Kant was a transcendental idealist, not of Berkeley's "shallow British strain", but idealism is idealism: science studies appearances, not things in themselves, a doctrine both Einstein and Gödel rejected. Gödel praised Kant's split of subjective and objective elements but warned it tends to grow into "boundless subjectivism". In an unpublished 1961 essay he said Kant's assertions, literally false, contain deeper truths: Kant's claim that proving geometrical theorems always needs new geometrical intuitions is provably false, but with "mathematical" in place of "geometrical" it is a consequence of incompleteness. The method for such fresh insight, he thought, was Husserl's transcendental phenomenology, "Kant's critique of pure reason transformed into an exact science" (draft letter to Gian-Carlo Rota), an idealism compatible with realism. Husserl agreed about Kantians: "German idealism has always made me want to throw up" (1915). Einstein, for his part, thought engineering no match for physics; of his son Hans's choice: "One cannot expect one's children to inherit a mind."
An offer that couldn't be refused
In 1949 Einstein gave Gödel "a wonderful flower vase" as a housewarming gift; Gödel, "after long searches", gave him an etching for his 70th birthday (Adele knitted a sweater but kept it). Schilpp invited Gödel to the Library of Living Philosophers volume Albert Einstein: Philosopher-Scientist, the only volume on a scientist. Gödel rejected Schilpp's title, "The Realistic Standpoint in Physics and Mathematics", and promised 3–5 pages on "The Theory of Relativity and Kant." (He later drafted a Carnap essay he never submitted and declined to write on Popper.)
The perversity: Einstein is usually said to have refuted Kant, who tried to prove a priori that Newton and Euclid were final, and who said logic could never go beyond Aristotle, all three claims overturned. Yourgrau's explanation for the choice of Kant over Gödel's hero Leibniz: the "host framework" strategy. Einstein's physics, though Gödel thought physics should have followed Leibniz, is Newtonian in lineage; as with incompleteness, Gödel derived radical consequences from within the host framework, making the formal result inescapable and the philosophical moral hard to resist. Inside it he grants that space-time is real, leaving open whether, in his own system, something deeper (like Leibniz's monads) underlies it. The question becomes: if we believe in the reality of relativistic space-time, are we forced to be idealists about time?
Plato had already called time "a moving image of eternity" (Timaeus): things in time never are, but are always coming to be. Time, Gödel told Wang, remains the philosophical question even after Einstein. Compare Hilary Putnam: "I do not believe that there are any longer any philosophical problems about Time." Gödel's project sat closer to Heidegger's Being and Time and Husserl's Phenomenology of Internal Time-Consciousness, but, unlike Heidegger, Bergson or McTaggart, he wanted formal results with philosophical consequences, not another fog of prose.
The two axes: being and knowing
| Axis | Question | Tradition | Perspective |
|---|---|---|---|
| Ontology | What is? | Plato, Aristotle | Nagel's "view from nowhere": largely spatial conditions of existence |
| Epistemology | What can be known? | Descartes onward; Kant | Wittgenstein's "the world as I found it": essentially temporal |
| Reconciliation | Do justice to both | Kant (failed on ontology, says Gödel); Husserl | Gödel's goal |
Special relativity was an epistemologically inspired theory: a priori, time is fixed by synchronised clocks; a posteriori, clocks in different frames disagree on simultaneity and no frame is privileged. Einstein had already drawn ontological conclusions from epistemological premises. Gödel would out-Einstein Einstein, pushing that reasoning to its end by asking the question Einstein preferred to sidestep: is t really time? Einstein's reply in the Schilpp volume admitted the problem "disturbed me at the time of the building up of the general theory": is the temporal connection between world-points asymmetrical, so that "A is before B" excludes "A is after B", or not? Put simply: is space-time in essence a space or a time?
Einstein's own remarks were nearly inconsistent. He agreed with Émile Meyerson that the time component is not just a fourth spatial dimension, yet spoke of the "rigid four-dimensional space" of special relativity. On Michele Besso's death (1955) he wrote that for "believing physicists" the distinction of past, present and future is "only a stubborn illusion." Yet he told Carnap that "the now means something special for man, something which physics cannot speak to."
An atom smasher for the mind
Just as formalists could not distinguish formal demonstrability from truth, relativists and analytic philosophers of science conflated t with intuitive time. So the Einstein essay was not an "excursion" but a continuation of the Gödel programme: testing the limits of formal methods in capturing intuitive concepts. The tool was the limit case, a construction so extreme it rules out the intuitive interpretation, much as an accelerator separates particles indistinguishable at low energy. The extremity here is geometrical, not dynamical.
Gödel listed what anything deserving the name of time must have, including Einstein's asymmetric order (if A is before B, it cannot be after B), then built a world model of general relativity with continuous timelike world lines connecting any two events: even if B is observed after A, a fast enough spaceship can reach B before A. In such a world space-time is a space, not a time, and t is another spatial dimension. Journeys along these closed timelike world lines are time travel, proved consistent with relativity. Enthusiasts got excited about time travel and missed the main point: if time travel is possible, time itself is not.
The atom smasher. Two particles look identical in every everyday experiment. Smash them at enormous energies and one decays, the other does not: they were different all along. Gödel does the same to "t" and "time". In every familiar solution of Einstein's equations, t behaves like time, so nobody tells them apart. In the Gödel universe the geometry is pushed until t still exists, but events are no longer ordered once and for all into before and after. If t can survive without the order, t was never the same thing as time.
Einstein's verdict and his track record
Einstein called the essay "an important contribution to the general theory of relativity, especially to the analysis of the concept of time," but added it would be "interesting to weigh whether these are not to be excluded on physical grounds." That set the pattern: most responses asked only whether the Gödel universe is physically realistic. Yourgrau reminds us of Einstein's record with solutions he disliked:
| Year | Solution | Einstein's reaction | Verdict of history |
|---|---|---|---|
| 1916 | Schwarzschild: collapse beyond an event horizon | A mathematical anomaly | Black holes are real |
| 1917 | De Sitter: a non-static cosmos | Unphysical | Expansion is real |
| 1922 | Friedmann: universe must expand or contract | Unphysical (at first) | Hubble, 1929 |
| 1917 | His own cosmological constant to keep the universe static | Later: "my greatest blunder" | Dark energy (1998) revives it |
| 1949 | Gödel: rotating universe, closed timelike lines | Perhaps excluded "on physical grounds" | Open |
Each time Einstein rejected a solution not for violating general relativity but for violating his intuitions about the shape of the universe, and each time reality refused to cooperate. So his hesitation about Gödel's model is not decisive. On what the discovery meant, there was a deafening silence; one philosopher of physics admitted decades later, "Like most others, I avoid the puzzling issue of what Gödel really thought he was showing about time and stick to the easier stuff on closed timelike loops."
Yourgrau garbles the history of Λ. Einstein introduced the cosmological constant in February 1917 to make his own static model possible, before de Sitter's model and five years before Friedmann; it was not a patch against them. De Sitter's 1917 model was first presented in static coordinates; its expanding reading came later (Lemaître, Weyl, Robertson). The "greatest blunder" line comes only via George Gamow and may be apocryphal. Supermassive black holes sit in the centres of most large galaxies, not "every galaxy". And Gödel's universe differs from the other cases in one decisive way: it rotates and does not expand, so unlike black holes or expansion, observation counts against it as a description of our universe; Gödel's argument needs it only as a possible world.
The Scandal of Big "T" and Little "t"
This is the book's core. Gödel's six-page essay, "A Remark About the Relationship Between Relativity Theory and Idealistic Philosophy" (1949), was met by physicists with a false charge of error and by philosophers with silence. Yourgrau reconstructs its four-step argument: special relativity abolishes a world-wide "now"; general relativity seems to restore cosmic time; the Gödel universe abolishes it again; and a modal step carries the result from that possible world to ours. Big "T" (mathematical truth) outruns formal proof; little "t" (relativistic time) is not time at all, and Gödel concluded, not that relativity is incomplete, but that intuitive time is an illusion.
A birthday present nobody opened
Albert Einstein: Philosopher-Scientist became the most influential volume in Schilpp's series, mainly for the Einstein–Bohr debate on Copenhagen. Gödel had cancelled his seaside holiday and given up letters to finish his essay, and was surely disappointed by the near silence.
The phantom error. S. Chandrasekhar, after hearing Gödel lecture in Princeton, published with J. P. Wright in the Proceedings of the National Academy of Sciences (1961) a claim that Gödel had erred: his time-travel paths were not geodesics (free-fall paths). The editors never consulted Gödel. The real error was theirs: Gödel never said the paths were geodesics. The philosopher Howard Stein showed that time travel in the Gödel universe requires great acceleration (a rocket), not free fall, yet, by Yourgrau's account, could not get the correction published until Gödel intervened. Meanwhile the claimed error gave philosophers a reason to ignore the philosophy.
Philosophers did ignore it, save for a few technical discussions. Yourgrau's diagnosis: Gödel was never a member of the club. After WWII Wittgenstein's later, still language-centred philosophy dominated America; Gödel seemed to have slept through two Wittgenstein revolutions, and W. V. O. Quine, the dominant American philosopher, was absent from his thinking. Gödel knew it: he let "What is Cantor's Continuum Problem?" be reprinted in Benacerraf and Putnam's anthology only after being assured the editors would not deride it.
Why the age of the universe is a puzzle
Cosmologists happily say the universe is billions of years old. But if it is n years old, its present state comes n years after the beginning, and in 1905 Einstein showed there is no such thing as "the present state of the universe." By the relativity of simultaneity, what is "now" in one inertial frame is not "now" in another moving relative to it. In Einstein's words: "'now' loses for the spatially extended world its objective meaning."
Gödel saw an inconsistency between this and the everyday belief that time passes. Two popular assumptions try to paper over it; both fail:
- "Time flows, but only locally." Flow is not a geometrical fact that can be local. A river may curve near us and run straight elsewhere, but what could it mean for it to flow only in our neighbourhood?
- "The block universe fits experience fine." Act on it and see: if yesterday's and tomorrow's breakfasts are as actual as today's, should I still be choosing yesterday's order, or cancel both because they have arrived? If I am still lying on last summer's beach, why identify only with the "I" shivering now? Do I have temporal parts? ("The future is now," says the logo of Hudsucker Industries; good film, bad metaphysics.)
General relativity offers hope. It is the theory of gravity and hence of cosmology; gravity is space-time curvature, and matter in motion determines that shape. So some frames may be privileged, those following "the mean motion of matter", and time relative to them is cosmic time, the time meant (or that should be meant) by "the age of the universe". The astronomer James Jeans, whom Gödel cites, concluded that time thereby "regained a real objective existence, although only on the astronomical scale," and that "we are free to believe that time is real." Gödel put exactly this to the test.
What Gödel means by time: the A-series and B-series
| B-series | A-series | |
|---|---|---|
| Built from | Fixed relations before / after | The moving now: past, present, future |
| Character | Structural, "geometric", like space | Fluid, dynamic, unlike space |
| Example | 1865 comes before 1965, now and forever; your dentist appointment is at 3 p.m. On 19 May | That the appointment is now: true only at this moment ("Thank Goodness That's Over") |
| Captured by | Calendars, history books, relativity's t | Nothing in relativity |
| Philosopher | J. M. E. McTaggart coined both terms |
Five marks of A-series (intuitive) time
- A privileged now, and the privilege passes from time to time: what is now will soon be then.
- Passage with a direction: future becomes present, then past.
- Ontological weight of position: to exist in New Jersey is to exist as much as in New York, but to exist "in the past" is not to exist at all. Socrates had his time; it passed. (So the A-series is objective, not mind-bound, and must be distinguished from Husserl's inner time.)
- Fixed past, open future.
- Absolute simultaneity: whether my friend in Paris is on the phone at the same time as I write is a fact about what exists, not settled by choosing a frame.
Relativity's t can represent the B-series. It cannot represent the A-series, because there is no objective world-wide now whose flux constitutes change in what exists. The minus sign that distinguishes t in the invariant space-time interval, s² = c²t² − x² − y² − z² (or the reverse convention), shows only that time has a different geometry from space, not that it is a different kind of being, something that flows. Einstein, helped by Minkowski, defined time by its contribution to four-dimensional geometry rather than its qualitative difference from space, as he defined gravity as curvature. Hence Gödel's title: "idealistic philosophy" means Parmenides, Plato and Kant, for whom time was always an ontological suspect.
Gödel's definition of intuitive ("Kantian", "prerelativistic") time: "a one-dimensional manifold that provides a complete linear ordering of all events," an "objective lapse of time" that is directly experienced and "involves a change in the existing." Its essence is "that only the present really exists", i.e. Reality consists of "an infinity of layers of 'now' which come into existence successively." Something without these properties "can hardly be called time."
The four steps: the forest
The argument from relativity to the ideality of time, premise by premise
Step 1 (special relativity)
- P1. Intuitive time requires the A-series: an objective, world-wide now whose succession is the change in what exists.
- P2. Special relativity admits no objective world-wide now (relativity of simultaneity).
- P3. Existence cannot be relativised to a frame "without destroying its meaning completely".
- P4. So no frame-relative surrogate will do: "each observer has his own set of 'nows'," and none can claim to be the objective lapse of time.
- P5. Therefore, if special relativity holds, intuitive time does not exist.
Step 2 (general relativity)
- P6. In general relativity, matter's distribution privileges frames following its mean motion; in our world these can be meshed into a cosmic time.
- P7. So a remnant of objective time reappears, at least on the astronomical scale (Jeans).
Step 3 (the Gödel universe)
- P8. General relativity has a solution, the Gödel universe, in which all matter rotates relative to the compass of inertia (free gyroscopes).
- P9. In rotating solutions no cosmic time can be defined.
- P10. In the non-expanding rotating solutions there are closed timelike curves: a rocket always heading into its local future reaches its own past.
- P11. If the past can be revisited, it never passed; a time that does not pass is not intuitive time.
- P12. So in the Gödel universe there is no intuitive time.
Step 4 (the modal step)
- P13. The Gödel universe obeys the same laws as ours and differs only in the large-scale arrangement of matter and motion.
- P14. Whether an objective lapse of time exists cannot reasonably depend on how matter happens to be arranged; such a view, though not contradictory, "can hardly be considered as satisfactory."
- P15. Therefore time does not exist in our world either: it is ideal.
Into the trees: step 1 and the absurdity of relative existence
Many still hold that relativity only ties the flow of time to a frame, so reality itself is frame-relative. Gödel's reply: the concept of existence "cannot be relativized without destroying its meaning completely." Science, he told Hao Wang, does not analyse concepts but applies them, and existence is "the clearest concept we have." You can relativise velocity, and "here" and "there", left and right. You cannot coherently speak of "my reality" and "your reality"; such talk means "my view of reality", and there is still one objective reality, which contains the fact that you hold that view. So a doctrine that relativises existence must go. We can have time or existence, not both, and Gödel chose existence: a world without time.
Two observers, one coffee cup. You and I pass each other at high speed at the moment I pour a coffee. In my frame, a supernova far away is happening now; in yours, it has not happened yet. If "now" is the edge of what exists, then for me the supernova exists and for you it does not. If both views are equally valid, the supernova both exists and does not exist at that moment. Either existence is frame-relative (which Gödel calls meaningless) or "now" is not the edge of existence. Either way the A-series loses.
Special relativity is therefore not merely incomplete about intuitive time; it is inconsistent with it. If intuitive time is real, Einstein is wrong. Hope remains only in general relativity, which Gödel then shuts down.
Steps 2 and 3: no time for time travel
Absent Newton's absolute space, what is the universe rotating relative to? Gödel's answer: "a certain inertial field" that "determines the behaviour of the axis of a completely free gyroscope"; matter rotates relative to the directions those gyroscopes fix. (Einstein once improved naval gyroscopes; Gödel's spin at the centre of the cosmos.) In these universes no single objective cosmic time can be introduced, on pain of contradiction.
Two stakes in the heart are better than one. In the subclass that is not expanding, the geometry is so warped that timelike curves close on themselves. Gödel worked out the journey's length and duration and the rocket's speed and fuel: a top speed that is a large fraction of light speed and enormous fuel; one later calculation puts it at about 10¹² grams of fuel for every 2 grams of payload, even with a perfect engine.
In the Gödel universe the future light cones tip over with distance from any chosen axis (every point is equivalent, so any observer can serve as the axis). Beyond a critical radius, a circle around the axis becomes a timelike path: a rocket can circle it, always moving into its local future, and return before it left. By Gödel's lights the very success of the journey shows time, and with it speed and motion, to be an illusion. If you can revisit the past, it still exists: you cannot revisit New Jersey if New Jersey is gone, and you cannot return to time t if t has dropped out of existence. Past and future become as ontologically neutral as distances in space. Even the "friends of Gödel" who defend time travel's coherence miss this: for Gödel, if there is time travel, there is no time. His aim was not to make room for Star Trek.
Step 4: from the possible to the actual
The Gödel universe is only possible. Can its timelessness transfer to the actual world? Gödel argues yes, by a mode of reasoning close to his heart. As a Platonist, he held that for necessary beings (numbers, sets), possibility implies actuality, since a necessary being exists in all possible worlds or none.
| Ontological argument | Gödel's modal step on time | |
|---|---|---|
| Possible case | God is possible | The Gödel universe is physically possible |
| Bridge principle | A necessary being, if possible, exists in every world | Time's existence cannot depend on the contingent arrangement of matter |
| Conclusion | God is actual | Time is absent in the actual world too |
| Consequence for opponents | The atheist must be a "superatheist", denying God is even possible | The temporal realist must accept that time's reality is an accident of cosmic layout |
Gödel built his own ontological argument for God in Leibniz's footsteps, then refused to publish it for fear of being taken for a theist in an atheist age.
Current consensus on the argument (2026). Step 1 is widely accepted as a problem for a global A-series, though defenders of frame-relative or local becoming (Stein 1968, Savitt, Dorato) and of a hidden preferred foliation (some presentists, and quantum-gravity programmes such as shape dynamics) dispute that it is decisive. Step 4 is the weakest link for most philosophers of physics (Earman, Savitt, Dorato, Belot). An A-theorist can reply that time's existence may well depend on the actual distribution of matter, since our universe, unlike Gödel's, does admit cosmic time; and "same laws" carries less weight if one adds a physically motivated condition (global hyperbolicity, chronology protection) that Gödel's model fails. Yourgrau's reading of Gödel as a temporal idealist via the modal step is itself one interpretation among several; few philosophers today accept the argument as sound, though many accept that it shows relativity and the intuitive "now" are in deep tension.
Protecting time from Gödel
Six pages sufficed. Once more Gödel built a limit case: a structure whose "geometry" constrains its interpretations. Before incompleteness, one could mistake proof for truth; after it, no reasonable person could. Before the Gödel universe, one could mistake t for time; in its extreme geometry, t cannot even serve as cosmic time, a mere simulacrum of the real thing. He succeeded because his philosophical eye isolated the essential properties of the intuitive concept first.
Yourgrau's parallel continues: as Hilbert invented a rule of inference to evade incompleteness, Stephen Hawking proposed the chronology protection conjecture, which Yourgrau calls the "anti-Gödel amendment": a change to the laws to rule out closed temporal loops, with what Russell called "all the advantages of theft over honest toil." He claims it has won few adherents.
Chronology protection and CTCs in 2026. Hawking's 1992 conjecture is not a change to general relativity; it says that quantum effects (vacuum polarisation diverging at a chronology horizon, back-reaction) prevent the creation of closed timelike curves from normal initial conditions. Support: Kay, Radzikowski and Wald (1997) showed quantum field stress-energy is ill-defined on compactly generated Cauchy horizons; string-theory studies (e.g. Dyson 2003) find similar protection. Against: counterexamples where the divergence is tamed, and the conclusion that the question needs a theory of quantum gravity, which still does not exist. CTCs remain an active field in quantum information (Deutsch 1991 and post-selected CTC models, with photonic simulations from 2011). Most important for Yourgrau, the conjecture concerns building time machines, so it does not by itself forbid an eternal Gödel cosmos; that is excluded by observation, not by chronology protection.
The puzzle of the two incompletenesses
| Big "T": mathematical truth (1931) | Little "t": relativistic time (1949) | |
|---|---|---|
| Formal stand-in | Provability in Principia-style systems | The coordinate t of space-time |
| Limit case | G, true but unprovable | Gödel universe, CTCs, no cosmic time |
| Finding | Formal fails to capture intuitive | Formal fails to capture intuitive |
| Gödel's verdict | Formalism too weak; truth is real (mathematical realism) | Relativity is fine; intuitive time is an illusion (temporal idealism) |
| Why trust the intuitive side? | Millennia of unformalised arithmetic and iterative set theory, never contradictory | Our experience of passage could be the same in a Gödel universe, where time provably lacks passage |
Nobody has asked why Gödel reacted oppositely to the two incompletenesses. Yourgrau's proposal: in mathematics, intuitive practice has earned trust. The iterative concept of set ("anything obtainable from the integers... By iterated application of the operation 'set of'") "has never led to any antinomy whatsoever"; the paradoxes arose from dividing the totality of all things into two categories, at mathematics' "outermost boundary toward philosophy." Formalisms were the problem, not practice. With time, the theories are the well-confirmed side and experience is unreliable: our experience of lapse might be indistinguishable from what an inhabitant of a Gödel universe would feel, where lapse is provably absent. A form of experience compatible with both a thesis and its antithesis cannot testify for either. So experience must yield. "As we present time to ourselves," said Gödel, "it simply does not agree with fact. To call time subjective is just a euphemism." That is where Kant's idealism and Einstein's physics meet.
Attention went instead to the inessential: first whether time travel is consistent with relativity, then whether our world is an expanding Gödel universe. The modal argument vanished. Yourgrau: "Rarely have so many understood so little about so much."
"Who's Gödel?"
General relativity is the philosophical cousin among physical theories, and time in general relativity its most elusive corner; Gödel's extreme cosmologies sat in a niche of that corner, so the ripple faded.
Wheeler, Misner and Thorne visit. One spring day John Wheeler brought Kip Thorne and Charles Misner, finishing their textbook Gravitation (1973), to see Gödel, who sat in his overcoat with the electric heater on. They asked about links between incompleteness and Heisenberg's uncertainty principle. For Gödel this was in bad taste: uncertainty was the finest flower of Copenhagen positivism, a rejection of Leibniz's principle of sufficient reason, "really the end of all theoretical science in the usual sense", whereas incompleteness refuted positivism. Then Gödel asked whether Gravitation would discuss his rotating universes. No. He was still checking whether our universe is an expanding rotating Gödel universe, and had gone through the Hubble atlas of galaxies with a ruler, measuring each galaxy's axis and compiling statistics. The results were negative.
Yourgrau grants Wheeler's group a point Gödel would not: incompleteness and uncertainty share a form. Each used positivist-acceptable methods (formal systems; direct observation) and drew ontological conclusions from epistemological premises establishing those methods' limits, the hallmark of positivism and of special relativity too. That Gödel concluded mathematical realism and Heisenberg physical irrealism does not change the parallel. Freeman Dyson: the two great conceptual revolutions of twentieth-century science are "the overturning of classical physics by Heisenberg and the overturning of the foundations of mathematics by Gödel."
Wheeler heard Gödel's lecture at Einstein's 70th birthday in 1949 yet summarised it as worlds where "one could, in principle, live one's life over and over again." Yourgrau's objection: that confuses a circle with a cycle. A closed timelike curve is not a journey repeated over time; it is the curve itself, and asking how many times the traveller goes round is like asking how often the points of a circle have gone round. Wheeler himself wrote that special relativity makes time relative and "static", yet his chapter "The End of Time" means only that time ends at the Big Crunch and inside black holes.
"Who's Gödel?" A year later, in James Peebles's office, student Dan Hawley announced a finished dissertation on preferred rotation among galaxies. Wheeler said Gödel would be pleased. "Who's Gödel?" "The greatest logician since Aristotle." Wheeler phoned him; Gödel's questions soon exhausted Wheeler, then Hawley, then Peebles, who concluded: "My, I wish we had talked to him before we started this work."
Is the universe rotating? Gödel's ruler survey was right in spirit. Modern limits from the cosmic microwave background (e.g. Saadeh et al. 2016, which disfavoured any anisotropic expansion by odds of about 121,000 to 1) put any global rotation many orders of magnitude below the Hubble rate. In 2025 a few papers revived a very slow cosmic spin (roughly one turn per hundreds of billions of years) as a possible way to ease the Hubble tension, and fitted Gödel-type metrics to supernova data; these remain speculative and nowhere near the rotation needed for closed timelike curves. Gödel's own 1949 model, which does not expand, is ruled out as a description of our world.
Twilight of the Gods
After 1949 both men spent their last decades on lost causes: Einstein on a geometrical unified field theory, Gödel on new axioms to settle the continuum hypothesis, a refutation of conventionalism, a Husserlian epistemology and an ontological proof of God. Yourgrau explains the failure by a two-part recipe for discovery, seeing which problem is ripe and having or inventing the craft: their craft never faded, their choice of problems outran their age. The chapter ends with Gödel starving himself to death in 1978, weighing sixty-five pounds, his passing disturbing "no other orbit".
Everything is something else
Geometry began it all. Plato barred the ignorant of geometry from his Academy and, in Republic Book 6 (a passage Gödel admired), said geometers reasoning about drawn figures "are in fact thinking not of these but of the ideals they resemble." Euclid captured those forms, imperfectly but well, in the paradigm axiom system. Einstein, in "Geometry and Experience", separated geometry as formal deduction from geometry as an empirical account of space, and dared use non-Euclidean geometry for the real world.
His method was geometrisation. The limit speed of light came not from shrinking rods but from the structure of space-time; gravity not from action at a distance through an ether but from curvature; time was "tamed" as a coordinate. Einstein, too, was an ironist: time is really space, gravity is really curvature, energy is really mass. Geometry was also the password to the Gödel universe, and the key Einstein kept turning on his unified field theory, which John Wheeler in 1980 called "Einstein's now abandoned dream of a geometrical unification of the forces of nature."
Honours and withdrawal
- 1949–50 Gödel suffers a bleeding duodenal ulcer soon after finishing the Schilpp essay.
- 14 March 1951 First Einstein Award, split by Oppenheimer's suggestion between Gödel and Julian Schwinger; von Neumann calls Gödel's work "a landmark which will remain visible far in space and time." Einstein to Schwinger: "You deserve it"; to Gödel: "You don't need it."
- Dec 1951 Gibbs Lecture to the American Mathematical Society, the only logician ever to give it, and his last talk to mathematicians; published only posthumously.
- 1953 Promotion to full professor, delayed by objections ("One crazy man on the faculty is enough," said C. L. Siegel, meaning himself).
- Later Honorary degrees from Yale and Harvard (Princeton's came too late), the National Medal of Science; no new essays published.
Gödel never liked lectures: "I have difficulty in following them, even if I am well acquainted with the subject matter." Yourgrau's line: his sun rose and set at the same moment.
The need for roots: Einstein's last years
Einstein spent his final years trying to undo forces he had helped unleash: Mach's positivism, the reality of Planck's quantum, probability in physics, the new quantum mechanics. Politically he made bookends: the letter to Roosevelt at one end, at the other his last public act, signing Bertrand Russell's manifesto for nuclear disarmament (1955). His white whale was unification plus a philosophical flaw in Copenhagen; like Ahab he was ready to go down with the ship. After reaching the IAS he never visited Europe, never drove, never flew.
- Ilse (Elsa's daughter, whom he once thought of marrying): died of cancer in Paris, 1934, aged 37; Einstein declined to go with Elsa.
- Mileva Marić: his first wife; died alone and unreconciled in Zurich (1948).
- Lieserl: their daughter, born before marriage; vanished from the record.
- Eduard: gifted younger son; schizophrenic, institutionalised for life, never visited by his father.
- Hans Albert: elder son; distant, stayed distant in America.
- Elsa: second wife; died 1936. "I never thought he loved me so much," she told Antonina Vallentin, "and that comforts me."
Einstein admitted his strongest personal ties had all failed. His deepest root became his "tribe": "My relationship to the Jewish people has become my strongest human bond." Yourgrau thinks Simone Weil's The Need for Roots explains this better than biographers, who struggle with the clash between his tribalism and his rational internationalism. In March 1955 he told Queen Elisabeth of Belgium that the "exaggerated esteem" for his work made him feel "an involuntary swindler." The day before he died, he asked for his latest unification calculations; he refused surgery: "It is tasteless to prolong life artificially... I have done my share, it is time to go."
The absence of the muses
Stanislaw Ulam claimed Gödel feared he had merely found "another paradox à la Burali-Forte or Russell." Gödel: "This is absolutely false. Ulam doesn't understand my result." Others confirm only that, like many IAS members, he sometimes wondered whether he had done enough. In isolation he pursued new axioms for CH and a refutation of conventionalism, strengthened by Cohen's result; it ran through the Gibbs Lecture and the Carnap essay.
Yourgrau's two-element recipe for discovery
- Ripeness: insight into which problems are now soluble. Fölsing on Einstein: in 1905 spectral lines could get no more than a phenomenological treatment, and his not trying showed "a master of the art of the soluble". Planck warned that rethinking gravity after 300 years of Newton was hubris; Einstein saw it was ripe.
- Craft: having or inventing the technique. General relativity strained Einstein's mathematics to breaking point; Gödel invented Gödel numbering, the arithmetisation of metamathematics, and new set-theoretic methods for the consistency of CH.
| Discovery | Ripeness seen | Craft supplied | Note |
|---|---|---|---|
| Relativity (Einstein) | Light speed and gravity open to rethinking | Minkowski geometry, tensor calculus | Planck warned of hubris |
| Incompleteness (Gödel) | Formal systems' limits testable | Gödel numbering, recursion | Ended 2,000 years of confidence in pure deduction |
| DNA (Watson, Crick) | Structure now solvable | Fitting scattered pieces | Pauling had the wrong idea; Rosalind Franklin's data, "desk drawer" |
| Fermat's last theorem (Wiles) | Saw it could now be proved | Years in the attic | First proof had a flaw, fixed with help |
| Unified field theory (Einstein) | Not ripe | Intact | Still no unification; string theory beyond his era |
| Continuum hypothesis (Gödel) | Not ripe | Intact | No convincing new axioms |
Both stayed mathematically nimble to the end; the trouble was the first element. They bit off more than anyone could chew. Feynman (1982) on EPR: "I cannot define the real problem, therefore I suspect there's no real problem, but I'm not sure there's no real problem." The Carnap essay, "Is Mathematics Syntax of Language?", published posthumously in several versions, is a struggle more than a consummation.
The continuum hypothesis in 2026. Still undecided by accepted axioms. Large cardinals, Gödel's favoured source of new axioms, are known (Lévy–Solovay, 1967) not to settle CH. Hugh Woodin argued around 2000 for not-CH (2^ℵ₀ = ℵ₂) and later shifted towards CH via his "Ultimate L" programme. In 2021 Asperó and Schindler proved that the forcing axiom MM⁺⁺ implies Woodin's axiom (*), both of which give 2^ℵ₀ = ℵ₂, making that value a serious candidate. Gödel's hope of axioms that "force themselves upon us" remains unmet; the multiverse view (Hamkins) treats CH as having no single answer, close to Cohen's conventionalism.
Sowing without reaping
Gödel was invited to the Carnap volume in 1953, two years after his Gibbs Lecture, "Some Basic Theorems on the Foundations of Mathematics, and Their Implications." Both aimed to use incompleteness as a superweapon for Platonism against conventionalism. It blasted no shortcut through mathematical ontology, as Wittgenstein, who had once compared Principia to music and later stressed the "motley" of mathematics, had concluded his own way. Gödel kept faith that philosophy could be as beautiful as mathematics, but his endless drafts were not. To Schilpp, 1959: "In view of widely held prejudices, it may do more harm than good to publish half done work."
Unlike Platonists such as Frege and Church, Gödel saw that his ontology needed an epistemology, and sought it in Husserl, studying the "recalcitrant texts" for years and urging surprised colleagues to do likewise; nothing definitive emerged. Nor did philosophers or physicists accept his breakthrough on time.
Suspicions of piety
Gödel aimed higher still: a modal-logic version of the Anselm–Descartes–Leibniz ontological argument. Given suitable modal axioms, the step from God's possibility to His actuality is easy; the hard part, as Leibniz knew, is proving God possible. He wrote a compressed formal proof and refused to publish it, claiming it was a mere formal exercise, fearing colleagues would think he believed in God. He was right about them: "Ninety per cent of contemporary philosophers," he told his mother in 1961, "see their principal task to be that of beating religion out of men's heads."
"Good grief, Parsons is pious." In 1955 first-year Harvard graduate student Charles Parsons, interviewing for the Society of Fellows, mentioned theology readings and an interest in Pascal, forgetting that examiner W. V. O. Quine was a firm opponent of religion. Quine was heard to say, "Good grief, Parsons is pious." He was not elected, though invited to reapply.
Nobody was fooled about Gödel's piety. When the proof surfaced posthumously, problems were found in the details; whether it can be repaired, and whether a repaired version would convince, are open.
Gödel's ontological proof, machine-checked. Jordan Howard Sobel (1987) showed Gödel's axioms imply modal collapse: every truth becomes necessary, so nothing could have been otherwise. In 2013–14 Christoph Benzmüller and Bruno Woltzenlogel Paleo verified the proof with automated higher-order theorem provers, confirming both its validity in modal logic S5 (indeed KB) and the collapse; repairs by Anderson and Fitting avoid collapse at the cost of new premises. Philosophers remain unconvinced that the key premise (positive properties are consistent) is any more evident than the conclusion.
Solomon Feferman blames Gödel's "special caution" for lost contributions. Having shown arithmetic truth indefinable within arithmetic, he left the formal theory of truth to Alfred Tarski; having laid the groundwork of effective computability, he left its definitive analysis to Turing, with Church, Kleene and Rosser. "One may wonder how logic might have been different had Gödel been bolder."
"The world tends to deteriorate"
The protectors died: Einstein 1955 (an abdominal aortic aneurysm, diagnosed in 1948 and kept secret), von Neumann 1957, Veblen 1960. Gödel was stunned: "We live in a world in which ninety-nine per cent of all beautiful things are destroyed in the bud."
Gödel's philosophy of history (after Hegel)
- Non-causal structural laws. "There are structural laws in the world which can't be explained causally."
- No progress. "The world tends to deteriorate. Good things appear from time to time in single persons and events... But the general development tends to be negative."
- Christianity "was best at the beginning. Saints slow down the downward movement." (Compare Simone Weil's "drops of purity".)
- Philosophy "tends to go down"; it is "at best, at the point where Babylonian mathematics was."
With Adele ill (hypertension, arthritis, bursitis, gall bladder disease), Gödel became her carer and Morgenstern his witness. By 1968 Morgenstern was alarmed by his gauntness. Gödel said his doctors lied, their medicines were mislabelled and their textbooks badly written, and feared, realistically, commitment. Morgenstern, who had three times failed to get Princeton to award an honorary degree, told him at least he was a true friend; Gödel answered that a real friend would have brought him cyanide. In 1974 an enlarged prostate blocked his urinary tract; he refused surgery, pulled out his catheter (it was forcibly reinserted) and stayed permanently catheterised, swallowing a cabinet of drugs (milk of magnesia, Metamucil, Keflex, Mandelamine, Macrodantin, Gantanol, Achromycin, Terramycin, Lanoxin, Quinidine and more), everything except food.
There were lucid, even charming, moments, above all with Hao Wang, who in 1971–72 met him every other Wednesday for two hours (or by phone) and later published Gödel's unwritten philosophy. How much was trial balloon? Of Gödel, as Richard Rorty said of Plato, we are still separating the straight lines from the jokes.
The end. In late December 1977, weighing 65 lb, Gödel entered Princeton Hospital; he died of "malnutrition and inanition" caused by "personality disturbance." He was buried in Princeton Cemetery; Adele died in 1981. Einstein, fearing his grave would become a shrine for "the bones of a saint," had been cremated and his ashes scattered. At the IAS memorial Wang and André Weil (brother of Simone) spoke. To philosophers, then as now, Gödel was simply a logician trying to pass as a philosopher.
Corrections: Gödel died on 14 January 1978, not 11 January. Yourgrau's claim that Turing used a syringe to put cyanide in an apple "imitating Gödel's favourite fairy tale" is unsupported: the half-eaten apple found by Turing's bed was never tested, and the inquest found cyanide poisoning; the Snow White link is anecdotal (and Turing's mother believed the death accidental). The Watson–Crick aside understates Franklin's role: her Photograph 51 and data were shown to them without her knowledge.
In What Sense Is Gödel (or Anyone Else) a Philosopher?
The closing chapter is Yourgrau's defence brief. The academy, schooled in Kant, Wittgenstein, Carnap and Quine, dismissed Gödel as "a logician trying to pass as a philosopher"; Yourgrau argues he was a philosopher posing as a logician. He rebuts Warren Goldfarb, Burton Dreben, Charles Chihara and John Earman, defends Gödel's Husserlian turn and his kinship with Frege, and restates the time argument as philosophy done right. Gödel brought the most powerful formal theory of time ever built to bear on philosophy's oldest question about it, and "if this is not the right way to do philosophy, what is?"
Who buried Kurt Gödel?
Gödel's "special caution" failed on every count: it cost him contributions, did not hide his belief, and did not stop people calling him "precritical" (before Kant's critique), a cosmological amateur, and a castaway from the Wittgenstein revolution who would not pay homage to Quine. Hao Wang's posthumous records of their talks failed to repair the image.
Boston University, February 1995: "Gödel's General Philosophical Significance." The speakers included editors of Gödel's Collected Works, the "guardians" of his legacy ("Who will guard the guardians?"). John Dawson noted that only one book, Yourgrau's The Disappearance of Time (1991), had examined Gödel's cosmology for the philosophy of time, but nobody took up the subject. Warren Goldfarb (Harvard) said Gödel had achieved nothing in philosophy; asked from the floor whether Gödel, though a great logician, was "a philosophical fool", he answered with a polite smile. Burton Dreben (Harvard) called out, "Wait until you hear my talk!", then preached that skill at formal thinking confers no right to do philosophy (fascists, he said, were sometimes good at science), condemned Gödel as a pre-Kantian throwback, and was especially irked that Gödel did rational theology "in this day and age". Yourgrau notes the irony: Dreben's own standing rested on formal logic. The celebration resembled a wake.
This chapter is advocacy, and Yourgrau is a party: he was in the audience, he cites his own books as the exceptions, and he reports unpublished talks by Goldfarb and Dreben from memory and from a separate transcript. Goldfarb was an editor of Gödel's Collected Works (including the philosophical volume III, 1995), hardly a sign of neglect. Earman's appendix is a substantive technical critique, not mere condescension, and its central objection (the modal step) is shared by most philosophers of physics. Readers should treat the portraits of opponents as one side of a live debate.
The philosopher on the train
Tell a fellow passenger you are a philosopher and you claim to be a sage; say "philosophy professor" and, if you are a real philosopher, you lie again. Leo Strauss: you are as likely to find a real philosopher in a philosophy department as a Picasso in a fine-arts department. Safer to say you are an architect. Gödel would not have called himself a philosopher on a train; being officially a logician was cover. Yet he asked the questions Parmenides and Plato asked: time, being, death, God, the forms.
He shared Socrates' and Wittgenstein's distrust of philosophy as a paid trade: "To do philosophy is a special vocation. We do see the truth, yet error would reign." And: "Philosophy is a persecuted science," risky for anyone questioning the ruling paradigm. On ethics he was shrewd: a strict ethics "would be easy... only the result would be unpleasant, and one does not want to see it."
Philosophers who doubt philosophy
The discipline's leading figures doubt its right to exist. Kant's dove, feeling the air's resistance, imagines flight would be easier in empty space, as Plato "ventured out beyond [the senses] on the wings of the Ideas." Yet every system that sets limits to knowledge risks breaching its own limits:
| Thinker | Self-limiting doctrine | Self-refutation | Critic |
|---|---|---|---|
| Kant | Knowledge confined to possible experience | Can experience alone teach what experience can and cannot do? | (Yourgrau) |
| Vienna Circle | Verifiability criterion: meaning = method of verification | The criterion is not verifiable, so by its own lights meaningless | Carl Hempel |
| Early Wittgenstein | Only science can be said; value can only be shown | The Tractatus itself is "nonsense"; "throw away the ladder" | Frank Ramsey, reviewing at about 20: "What we can't say we can't say, and we can't whistle it either." |
| Later Wittgenstein | Philosophy is a mental cramp, language "on holiday" | Philosophy as therapy, yet he dominated academic philosophy for decades | Karl Popper, "Are There Philosophical Problems?" (the poker incident, Cambridge, 1946) |
| Quine | No "first philosophy"; philosophy continuous with science |
The irony Yourgrau stresses: those who most doubted philosophy (Kant the proto-positivist, Carnap the positivist, Wittgenstein the patron saint, Quine the enemy of first philosophy) are the most influential in Anglo-American academe. Wittgenstein, unlike Socrates, Frege or Russell but like Marx and Freud, gathered disciples. Gödel, a disciple of none, sealed his fate. Paul Horwich (1990): after Wittgenstein and Quine, "it is now widely believed that the sciences exhaust what can be known and that the promise of metaphysics was an intellectually dangerous illusion."
"Precritical"? Yourgrau's rebuttal of Goldfarb
Yourgrau reconstructs Goldfarb's view from a related talk, "Gödel's Philosophy" (Association for Symbolic Logic, Helsinki, July 1990), point by point:
| Goldfarb's charge | Yourgrau's reply |
|---|---|
| For Gödel all content of primitive concepts shows in precise axiomatic relations | That is Hilbert's implicit definition, which Gödel and Frege rejected; primitives must be grasped by intuition, "a clarification of meaning that does not consist in defining" |
| No hint that metaphysical truths are hard to reach; so Gödel is "precritical" | Gödel rejected Kant's critique from deep knowledge; the long drafts, "Some Observations about the Relationship of Theory of Relativity to Kantian Philosophy", prove it; "perverse" to call him precritical |
| Gödel underrated the difficulty of finding concepts and axioms | Gödel: "the problem of time is important and difficult. For twenty-five years Husserl worked on just this one problem"; constitution of time and mathematical objects "is difficult" |
| Gödel's phenomenology is subjectivist; watching the "passing show" of consciousness yields no concepts | That is Hume's method, not Husserl's; phenomenology reconstructs our original acts of using concepts; for conceptual realists it explains how the objective is given |
| Frege's review of Husserl's Philosophie der Arithmetik refuted it | That review targets Husserl's early psychologism, not the later Platonist phenomenology Gödel studied |
| Kantian "intuition" concerns objects in space and time; Gödel misuses the term | The mind meets ideal entities as the eye meets objects: partially, in stages, from shifting perspectives |
Gödel described phenomenology as focusing more sharply on concepts "by directing our attention... onto our own acts in the use of these concepts," reaching "a new state of consciousness." He found Husserl's way "long and difficult" and wrote to his mother, "I love everything brief... The longer a work is, the less there is in it." Phenomenology, "entirely as intended by Kant," avoids both idealism's "death-defying leaps" and positivism's rejection of all metaphysics; Kant's philosophy itself "rests on the idea of phenomenology, albeit in a not entirely clear way."
We perceive objects and understand concepts. Understanding is a different kind of perception.Kurt Gödel
Two stars for one. Gödel: "We begin with vague perceptions of a concept, as we see an animal from far away or take two stars for one before using the telescope." Suppose "effectively computable" is the double star. Turing's analysis is the telescope: once it resolves the concept, new angles stop producing surprises. That is Gödel's criterion for an adequate intuition: not perfection, but reaching the point where the concept no longer surprises us. He thought Turing had done this for computability; nobody has yet done it for "set" (hence CH) or for "time".
From Frege to Gödel
Yourgrau turns Goldfarb's appeal to Frege around. How did Frege find his analysis of number in The Foundations of Arithmetic? A natural number answers "How many?" (a real number answers "How much?"). But how many what? Not the nine planets as objects (each is one, and the group is one), but the concept "is a planet": the number belongs to the concept, which has nine instances. His definition: the number of F is the extension of the concept "equinumerous with F", concepts being equinumerous when their extensions correspond one to one; in effect the number three is the set of all trios. Not a lucky guess, not a survey of Jena's citizens, not theorem-grinding: Frege attended, as nobody before, to what we are doing when we use numbers. That redirection of attention is just what Gödel, after Husserl, called phenomenology. Wittgenstein too once said his method could be called phenomenology; where he spoke of words, Gödel spoke of concepts. Frege: knowledge of a concept "in its pure form" may take "immense intellectual effort... Over centuries."
What Frege and Gödel shared
- Uncompromising Platonism, in a sea of ultra-empiricism ("they swam... But managed somehow not to get wet"). Frege: arithmetic's objects are "given directly to our reason... Utterly transparent to it"; compare Plato's Phaedo.
- Logicism: mathematics except geometry derivable from logic (taken to include set theory) plus correct definitions; analytic does not mean empty, for analytic truths are true by the meanings of concepts.
- Geometry apart, resting on a priori physical intuition, as Kant suggested.
- Predication as the basic logical and metaphysical relation: falling under a concept.
- Conceptual space: concepts exist objectively, "but not in any definite place" (Gödel); "in the external world... There are no concepts... No numbers" (Frege's "third" domain, beside the physical and the mental). Grasping them is the only escape from the private world of ideas, and our access to the physical world too.
- Where Gödel went further: a systematic epistemology for reaching conceptual space, via Husserl, repairing a hole in Frege, the very point where Goldfarb attacked.
Charles Chihara (Berkeley) ridiculed Gödel's mathematical epistemology in essays and in Constructibility and Mathematical Existence (1990); reviewer E. P. James objected that Chihara followed the line of Gödel as "a logician par excellence but a philosophical fool," attacking a caricature of Gödel's views on perception. Gödel: "At present mathematicians are prejudiced against intuition."
"The correct response to Gödel"
| Who | Time travel possible in Gödel's world? | Does that make time ideal? |
|---|---|---|
| Milič Čapek (and Karl Popper) | No, because if it were, time would be unreal | Yes, the implication holds |
| Paul Horwich, Asymmetries in Time (1987) | Yes, the argument is valid | Ignored the question |
| John Earman, Bangs, Crunches, Whimpers, and Shrieks (1995) | Yes | No: the modal step fails; neglect of the moral was "benign" |
| Gödel, and Yourgrau (The Disappearance of Time 1991; Gödel Meets Einstein 1999) | Yes | Yes |
Earman began well, noting "the relative neglect of the philosophical moral" and that Gödel's convictions deserve "serious consideration", then judged the moral worthless, called Gödel's point about existence "a pretty piece of ordinary language philosophizing," and endorsed Einstein's "brushing aside" of idealism as "the correct response to Gödel." Yourgrau's replies:
- Einstein is the wrong ally. He spoke as a physicist about a new result; privately he shared Gödel's appreciation of Kant and had speculated about relativity and experienced time, but never meant to engage the philosophical literature.
- Gödel is no ordinary-language philosopher. His intuitions come from rational attention to concepts, not the marketplace of usage; language "had nothing to do with it."
- The modal step survives. Earman's crux is that our world lacks Gödel's geometry. But (1) experience of lapse cuts both ways, since Gödel's inhabitants might share it; (2) our geometry permitting a flow gives only necessary, not sufficient, conditions; (3) the laws are the same in both worlds. So nothing seems able to decide for time's existence here, and our world is in principle indistinguishable from one where time is provably absent. Earman, like others, conflates the geometrical prerequisites of lapse with lapse itself.
- Denying lapse concedes the point. Gödel anticipated it: to call objective lapse meaningless "is no way out of the dilemma; for by this very opinion one would take the idealistic viewpoint."
Yourgrau's reply to Earman has a gap he does not address: an A-theorist can hold that cosmic time in our universe (absent in Gödel's) is exactly the extra condition that makes lapse real here, treating the arrangement of matter as relevant rather than absurd, and can add that physical possibility is not settled by the field equations alone (energy conditions, causality conditions, quantum effects). Many also deny Gödel's premise that "only the present really exists" is the essence of time, in which case the B-theorist calls the argument a reductio of presentism, not of time. As of 2026, the literature since Yourgrau (Belot 2005; Dorato; Savitt; Wüthrich; work in Synthese and Studies in History and Philosophy of Modern Physics) is far larger than "a cottage industry", and is generally sceptical of the full argument while crediting Gödel with posing the sharpest challenge to temporal becoming in relativity.
The case for Gödel as philosopher
Few explicit papers, but enough: the Russell essay (with a prescient comparison of Russell's denoting with Frege's sense and reference), the Cantor essay, the Einstein essay, and posthumously the Gibbs Lecture, the Carnap essay and the long Kant drafts. He mastered Frege, Russell, Carnap, knew the Tractatus, studied Husserl, ignored the analytic–continental split, studied Leibniz endlessly, understood Kant deeply, and assembled Hegel quotations that Georg Kreisel thought could produce "a minor revolution in philosophy." He used history against prejudice: "Two methods to transcend such prejudices are: (1) phenomenology; (2) going back to other ages."
The Gödel programme, investigating the limits of formal methods in capturing intuitive concepts, is a philosophical enterprise carried out with formal and philosophical tools. Gödel wanted to reclaim logic from the positivists: "One bad effect of logical positivism is its claim of being intimately associated with mathematical logic... Mathematical logic should be used more by nonpositivistic philosophers."
In print he stripped work to the bones; in notebooks and talks with Wang he flew "as high as pure thought could take him," with no fear of crashing because he never meant to land. There is rough coal and there are diamonds: non-causal laws of history, the limits of mechanical biology, a concept of "absolute" proof, the possibility of an afterlife. Einstein accepted his end, having fired his best shots; Gödel resented his. He had just begun to fight.
Gödel the philosopher, 2026. The Collected Works were completed in five volumes (1986–2003), volume III (1995) carrying the Gibbs Lecture, the Carnap and Kant drafts and the ontological proof. Since 2019 De Gruyter has been publishing Gödel's Philosophische Notizbücher / Philosophical Notebooks (the "Max Phil" series, ed. Eva-Maria Engelen), transcribed from his Gabelsberger shorthand, which has fuelled new work on his Leibnizian metaphysics and his reading of Husserl (van Atten, Kennedy, Tieszen). His standing as a philosopher of mathematics is now widely acknowledged; his temporal idealism remains a minority view.
An ugly body in beautiful clothing
After Einstein's death, Bruria Kaufman, his last collaborator, and Gödel sorted the papers in his office and found a blackboard of equations that led nowhere. But Einstein's earlier theories revealed the mathematical symmetries hidden in physical noise, and this impressed Gödel. Mathematics "is given to us in its entirety and does not change, unlike the Milky Way"; its applicability "suggests that the mathematical and empirical parts are in harmony and that the real world is also beautiful." Like Plato in the Timaeus, he believed the world was built on a divine model: "Otherwise mathematics would be just an ornament and the real world would be like an ugly body in beautiful clothing."
The last notebooks. Gödel's biographer John Dawson, opening the boxes left at the IAS, found two bound notebooks of calculations that had nothing to do with logic: angular orientations of galaxies. The purest of logicians had never stopped checking whether we live in a Gödel universe. Yourgrau's last line: the effort was unnecessary, for in a deep sense we all do live in Gödel's universe.
Engaging in philosophy is salutary, even when no positive results emerge... Reality appears more clearly as such.Kurt Gödel
Master Takeaways
Yourgrau's book is three things at once: a double biography of a friendship, a history of the formalist century and its positivist philosophy, and a brief for one argument, that relativity, pushed to its limit by Gödel, shows time to be ideal. The history and the logic hold up well; the argument and the "conspiracy" are where the reader must push back.
- Truth is not proof. Gödel's first incompleteness theorem shows that every consistent, recursively axiomatised system strong enough for arithmetic leaves some arithmetic truth unprovable; mathematical truth cannot be reduced to formal derivation.
- No system certifies itself. The second theorem: such a system, if consistent, cannot prove its own consistency, which ended Hilbert's programme of a finitary consistency proof.
- Gödel's programme is the key to his work. Across logic, computability and cosmology he used formal methods to find the limits of formal methods in capturing intuitive concepts: negative for truth, positive for computability (Church's thesis), negative again for time.
- Relativity has no world-wide now. Because simultaneity is frame-relative, special relativity admits no objective present, and if existence cannot be relativised, it admits no A-series, no objective lapse of time.
- The Gödel universe is real mathematics. A rotating solution of Einstein's equations (1949) with no cosmic time and with closed timelike curves, reachable by heavily accelerated rockets, not free fall; Einstein called it "an important contribution."
- The ideality argument turns on a modal step. From "time is absent in a possible world with our laws" to "time is absent here", resting on the claim that time's existence cannot depend on the arrangement of matter; this is the step most critics reject.
- Big T versus little t. Gödel concluded realism about mathematical truth but idealism about time, because in mathematics intuition has a clean record while in physics the theory is better confirmed than our experience of passage.
- Positivism derives ontology from epistemology. Mach's anti-atomism, the verifiability criterion and the Copenhagen reading of uncertainty share this move; Einstein and Gödel used positivist tools against positivist conclusions.
- Einstein was an opportunist, not a positivist. In 1905 he used Mach's critique for relativity and Boltzmann's realism for Brownian motion; the Vienna Circle mistook his method for his creed.
- Formalism was the century's leitmotif. Form over content, syntax over semantics, proof over truth ran from Hilbert's beer-mug geometry to cubism, serialism, Chomsky and the computer; incompleteness is its limiting theorem.
- Cantor made infinity actual and plural. The diagonal argument shows 2^ℵ₀ > ℵ₀, and the power set of any set is larger; the same diagonal move powers Gödel's proof.
- The continuum hypothesis is still open. Gödel showed it consistent (1938–40), Cohen independent (1963); Gödel's hope for self-evident new axioms remains unmet.
- Einstein's track record weakens his doubts. He wrongly dismissed black holes and expansion as unphysical; his hesitation about Gödel's model is not decisive, though observation does rule out rotation on Gödel's scale.
- Ripeness matters more than skill. Late Einstein and late Gödel kept their technique but chose unripe problems: unification and the continuum.
- Hilbert and Hawking reacted alike, says Yourgrau. Each proposed a new principle to block an unwelcome result; the parallel is rhetorically sharp but historically loose.
- Gödel was a philosopher posing as a logician. His Kant drafts, Russell essay, Husserl studies and notebooks show a systematic, unfashionable metaphysics, now taken far more seriously than in 1995.
- Caution cost Gödel dearly. By publishing only the uncontroversial, he left the theory of truth to Tarski and computability to Turing, and still was seen as eccentric.
- The Vienna syndrome was a real trap. Cultural attachment kept Schrödinger, the Wittgenstein sisters, Furtwängler, Heisenberg and Gödel in danger far too long.
- Existence is absolute. Gödel's sharpest single point: you can relativise velocity and "here", but not what exists; any view of time that relativises reality must go.
- A circle is not a cycle. A closed timelike curve is not a journey repeated "over and over"; it is a single history, so asking how many times the traveller goes round makes no sense.
Frameworks & mental models
| Name | What it says | Use it when |
|---|---|---|
| Gödel programme | Use formal tools to test whether a formal concept captures an intuitive one | Asking whether a metric, model or specification really captures the thing you care about |
| Limit case ("atom smasher") | Push a structure to extremes until two conflated concepts come apart | Two notions agree in every ordinary case and you suspect they differ |
| IA / FA / MFA | Intuitive theory, formal system, metatheory; Gödel numbering lets FA talk about itself | Analysing self-reference and system limits |
| A-series vs B-series | Moving now (past/present/future) versus fixed before/after | Clarifying any argument about time, tense or becoming |
| Ontology vs epistemology | What is versus what can be known; positivism derives the first from the second | Spotting an inference from "we can't measure it" to "it isn't there" |
| Left vs right worldviews | Gödel: scepticism, materialism, positivism versus idealism, theology, metaphysics | Placing a philosophy on Gödel's own map |
| Host-framework strategy | Derive radical results from inside the opponent's framework | Making a conclusion hard to reject |
| Two-element recipe for discovery | See which problem is ripe, then have or invent the craft | Choosing research or transformation problems |
| Vienna syndrome | Attachment to a culture outweighing evident danger | Judging when loyalty becomes denial |
| Modal step | If a necessary feature is absent in a possible world with the same laws, it is absent here | Testing whether a property depends on laws or on contingent facts |
On the whole book. The logic chapters and the biography are excellent; the polemic needs discounting. (1) The "conspiracy of silence" is overstated: Gödel's solution has been studied steadily by relativists (Raychaudhuri, Stein, Hawking and Ellis, Malament) and, after Yourgrau's own books, by many philosophers. (2) Yourgrau is partisan about the Vienna Circle, Wittgenstein, Quine and Gödel's Harvard critics, whose views appear mostly through his rebuttals. (3) Hawking's conjecture is misdescribed as an ad hoc amendment aimed at Gödel; it concerns quantum effects on attempts to create time machines and leaves an eternal Gödel cosmos untouched. (4) The modal step, the hinge of the argument, gets far less critical scrutiny than it needs: cosmic time exists in our universe, and many hold that this contingent fact can matter. (5) There are factual slips (Schlick's murder date, Gödel's death date, IAS visit dates, the cosmological constant's chronology, Ostwald's name, Turing details). (6) Yourgrau cites his own work as the sole exception to neglect; read Chapter 9 as advocacy.
Who's who
| Person | Dates | Role in the story |
|---|---|---|
| Kurt Gödel | 1906–1978 | Logician; incompleteness (1931), CH consistency, Gödel universe (1949); Platonist and temporal idealist |
| Albert Einstein | 1879–1955 | Relativity; Gödel's walking companion from c. 1942; realist opponent of Copenhagen |
| Adele Gödel (Porkert) | 1899–1981 | Viennese dancer; married Gödel 1938; his protector |
| David Hilbert | 1862–1943 | Formalist programme; proposed the ω-rule after incompleteness |
| Georg Cantor | 1845–1918 | Set theory, transfinite numbers, continuum hypothesis |
| Gottlob Frege | 1848–1925 | Begriffsschrift (1879); logicism; wrecked by Russell's paradox |
| Bertrand Russell | 1872–1970 | Paradox (1901–02); Principia; sneered at the "German bias for metaphysics" |
| Ludwig Wittgenstein | 1889–1951 | Tractatus, the Circle's bible; later Gödel's chief philosophical detractor |
| Moritz Schlick | 1882–1936 | Founded the Vienna Circle; murdered by Nelböck |
| Rudolf Carnap | 1891–1970 | Circle logician; first to hear of incompleteness; "logical syntax of language" |
| Hans Hahn | 1879–1934 | Gödel's adviser; positivist; "The Crisis in Intuition" |
| Karl Menger | 1902–1985 | Mathematician; colloquium host; Gödel's Vienna friend |
| Ernst Mach | 1838–1916 | Positivist physicist; denied atoms; negative influence on relativity |
| Ludwig Boltzmann | 1844–1906 | Statistical mechanics; atomist; suicide at Duino |
| John von Neumann | 1903–1957 | Grasped incompleteness first; derived the second theorem independently |
| Ernst Zermelo | 1871–1953 | Axiomatic set theory; Gödel's chief mathematical opponent |
| Alonzo Church | 1903–1995 | λ-calculus; Church's thesis |
| Alan Turing | 1912–1954 | Turing machines; analysed effective computability |
| Werner Heisenberg | 1901–1976 | Uncertainty principle; positivist foil to Gödel and Einstein |
| Erwin Schrödinger | 1887–1961 | Wave mechanics; case study in the Vienna syndrome |
| Abraham Flexner | 1866–1959 | Founded the IAS; recruited Einstein |
| Oswald Veblen | 1880–1960 | IAS mathematician; recruited Gödel; his first minder |
| Oskar Morgenstern | 1902–1977 | Economist; Gödel's last minder and witness to his decline |
| Paul Cohen | 1934–2007 | Proved CH independent (1963) |
| S. Chandrasekhar | 1910–1995 | Wrongly claimed an error in Gödel's time-travel paths (with Wright, 1961) |
| Howard Stein | b. 1929 | Philosopher who corrected Chandrasekhar and Wright |
| John Wheeler | 1911–2008 | Physicist; visited Gödel with Misner and Thorne; "greatest logician since Aristotle" |
| Stephen Hawking | 1942–2018 | Chronology protection conjecture (1992) |
| Hao Wang | 1921–1995 | Logician; recorded Gödel's late philosophy |
| Warren Goldfarb, Burton Dreben | b. 1949; 1927–1999 | Harvard logicians; dismissed Gödel's philosophy at Boston, 1995 |
| John Earman | b. 1942 | Philosopher of physics; rejected the ideality argument (1995) |
| Palle Yourgrau | living | Brandeis philosopher; author; Gödel's advocate |
Connect it
- Specification versus intent (AI). Gödel's programme is the right question for any AI evaluation: does the formal metric (benchmark score, reward model) capture the intuitive concept (competence, helpfulness, safety)? Limit cases, adversarial inputs pushed to extremes, are how the two come apart, exactly as G separates proof from truth.
- No self-certifying system (enterprise architecture, model risk). The second theorem is a clean reminder that a system cannot vouch for its own consistency from within; in banking model risk management this is the logic behind independent validation and separation of first and second lines of defence.
- Rules cannot pre-decide every case (regulation). Gödel's remark that a legal code relying wholly on its letter will leave cases undecided maps onto principles-based versus rules-based regulation: rule sets for credit or financial-crime decisions need a route for human judgement on cases the rules never foresaw.
- Epistemic limits are not ontological facts (data). The positivist slide from "we cannot observe it" to "it does not exist" reappears when a risk, customer or event is treated as absent because the data model cannot represent it.
About this book
A World Without Time: The Forgotten Legacy of Gödel and Einstein by Palle Yourgrau, 2005. This is a chapter-by-chapter study of the book, with summaries, maps, tables and diagrams; short quotations keep their page references. It is shared free for personal study, is not for sale, and is not endorsed by the author or publisher. Please read and buy the original.
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