Insight Editions · No. 240
Gödel Meets Einstein
Time Travel in the Gödel Universe
Palle Yourgrau
Yourgrau’s case that Gödel’s rotating-universe papers hold a serious, neglected argument for the ideality of time: where the past can be revisited, it never passed.
A study guide to Gödel Meets Einstein 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's rotating universe (1949) permits time travel; for Gödel that proves there is no objective lapse of time, and, by a modal argument, none in our world either.
A philosopher's study of Gödel's writings on Einstein: relativity, McTaggart's A- and B-series, Kant, tense logic, indexicals, being and death, Brouwer and Zeno.
1931 incompleteness; 1949 Gödel universe and its closed timelike curves; 1991 first edition; 1992 Hawking's chronology protection conjecture.
Readers who want the argument structure behind Gödel's case against time, with the logic stated precisely and the critics given their due.
The Book on One Page
Palle Yourgrau argues that Kurt Gödel's papers on Einstein contain a serious, unjustly ignored argument for the ideality of time. In 1949 Gödel found solutions of Einstein's field equations, the rotating or Gödel universes, with closed timelike curves: a rocket can reach its own past. Commentators remember the time travel; Gödel's point was the reverse. If you can revisit the past, it never passed out of existence; so where there is "time travel" there is no time, and because the Gödel universe obeys our laws, time is ideal here too.
- Limit cases. Gödel's method throughout is to build limit cases: formal constructions that restrict what intuitive content a formalism can carry. Incompleteness showed formal proof cannot capture arithmetic Truth (big T); the Gödel universe shows relativity's t cannot capture intuitive time (little t). Yourgrau asks why Gödel blamed the formalism in one case and intuition in the other.
- What time demands. Intuitive time is McTaggart's A-series: a privileged, moving now; a lapse from possible to actual; only the present exists; a fixed past and open future; absolute simultaneity. The "relative A-series" (every time is now for itself) is just the B-series.
- Four steps. Special relativity relativises the now, and existence cannot be relativised; general relativity restores a cosmic time in lucky universes; the Gödel universe abolishes it and adds CTCs; the modal step moves from a possible world with our laws to ours.
- Becoming cannot be saved by relativising. Čapek's many rivers of time, Stein's light-cone becoming and Sklar's relativised reality either keep only the relative A-series or relativise existence. Gödel: a lapse "which is not a lapse in some definite way" is "as absurd as a coloured object which has no definite colours".
- Against Earman. The 1999 chapter answers John Earman's critique: the modal premise is that lapse is essential to time (□(T → L)), not that it holds in all worlds; in a Gödel world even motion is unreal; the verificationism is Einstein's, not Gödel's. Gödel "verifies" Kant by realigning him: realist about spacetime, idealist about time.
- Formal models see only traces. Thomason's branching-time logic and the Kaplan–Perry semantics of "now" capture at best a spacelike trace of A-time. Privileged positions (the now, the actual, the true) are formally diaphanous.
- The price of idealism. Without A-time we lose potential infinity, the direction of series, incomplete totalities, and the being/existence distinction that makes sense of birth and death. Brouwer's intuitionism is the mathematics of A-time; the appendices press the point through Brouwer and a contrarian reading of Zeno.
| Ch | Title | One-line payload |
|---|---|---|
| P | Prefaces (1999, 1991) | Gödel's four-step argument; Hawking's 1992 "anti-Gödel law" fulfils the book's prediction. |
| 1 | The Reception of Gödel's Results | Commentators kept the time travel and lost the idealism; Gödel's results are limit cases. |
| 2 | Gödel's Idealism | Parmenides, Plato, McTaggart: the A-series is what Gödel demanded of time. |
| 3 | Time Travel and the Gödel Universe | Closed time is spacelike; paradoxes are consistency constraints; the modal argument. |
| 4 | Not Everything Can Be Relativized | Čapek, Stein, Rietdijk, Sklar: becoming cannot be relativised without relativising existence. |
| 5 | From Kant to Star Trek | Cosmic time, the Gödel metric, Earman's critique, time circles vs cycles, Kant realigned. |
| 6 | Formalization and Representation | Branching-time logic and indexical semantics fail to capture the open future and the now. |
| 7 | Being and Time | The cost of idealism: potential infinity, direction, being vs existence, Brouwer. |
| A | Brouwer and the 'Revolution' | Brouwer's time is Husserl's and Wittgenstein's: disjoint from physics. |
| B | Zeno's Revenge | Limits evade Zeno; a contrarian case that the rationals are only potentially infinite. |
The New Expanded Edition (1999)
Yourgrau sets out the book's whole programme: Gödel's 1949 rotating universes made time travel a rigorous possibility, and Gödel used them in a four-step argument that time is ideal. The book's thesis is that this argument is far stronger than physicists or philosophers have admitted, and that it raises the problem of the existence of time to a new level. The 1999 edition adds a long chapter on time travel and two appendices on potential versus actual infinity.
Two friends and a child of the imagination
At mid-century Kurt Gödel and Albert Einstein were colleagues at the Institute for Advanced Study in Princeton and the closest of friends. When Gödel turned a mathematical eye on Einstein's cosmology, the result was a family of new solutions to the field equations of general relativity: the rotating universes, now called Gödel universes, in which there provably exist closed timelike world lines. In 1949, then, Gödel became the first thinker to show rigorously that time travel is consistent with the established laws of nature.
Gödel was already, in one colleague's words, "the greatest logician since Aristotle", and that colleague added that even this undervalued him. When Harvard gave him an honorary doctorate in 1952, W. V. Quine's citation called him the discoverer of "the most significant mathematical result of the century". Gödel told his mother that "most significant" meant only "of the greatest general interest outside mathematics", not that he was the century's greatest mathematician.
Three limits, three ontological leaps
Yourgrau groups incompleteness with Heisenberg's uncertainty principle and Einstein's relativity. Each is a high-water mark of a rigorous science; each is proved by formal means; and each sets a limit in principle on what the formal science can do.
| Thinker | Formal result | The limit it sets |
|---|---|---|
| Heisenberg | Uncertainty principle | No simultaneous exact knowledge of a particle's position and momentum |
| Einstein | Relativity | No signal, no information, faster than light |
| Gödel | Incompleteness (1931) | No formal axiomatic system, no computer program, captures all arithmetic truth |
A further shared trait: all three draw ontological conclusions (about what is real) from epistemic premises (about what can be known or measured). The book's interest is that Gödel did this twice, in logic and in cosmology, and that the second case has been ignored.
Gödel's argument in four stages
- Special relativity. The relativity of the intuitive now to an inertial frame (and no frame is privileged) implies the relativity of existence, since an objective lapse of time means that only the present really exists. But existence cannot be relativised "without destroying its meaning completely". So if special relativity is true, intuitive time disappears.
- General relativity. Matter curves spacetime, and some frames are privileged: those that follow the mean motion of matter. In our expanding universe these can be stitched together into a single objective cosmic time. Time reappears.
- The Gödel universe. (a) Gödel found models of general relativity in which cosmic time provably cannot be defined. (b) In certain non-expanding models there are closed timelike curves (CTCs), permitting time travel; and if you can revisit the past, it never really passed out of existence. Time disappears again.
- The modal step. The Gödel universe is merely possible. But it differs from ours only in the global distribution of matter and motion; both obey the same laws and could offer observers the same experience of time. A lapse of time ought to be grounded in the laws of nature, not in the accidental layout of matter. So if time is an illusion there, it is an illusion here. Time disappears even in the actual world.
The disappearance of the argument
Cosmologists were intrigued in 1949 and went on to find other models with temporal and causal anomalies. Gödel at least woke general relativists up to the fact that Einstein's theory does not exclude such worlds. But interest in his universe narrowed to two questions: is our universe a Gödel universe? (it is not) and is the solution physically realistic or a mere mathematical construct? Gödel himself never gave up the observational question; his Nachlass holds two notebooks of calculations on the angular orientation of galaxies.
In the first edition (1991) Yourgrau predicted that some general relativist would propose "an anti-Gödel law" to rule out the rotating universe being lawlike compossible with ours. One year later, in 1992, Stephen Hawking proposed the chronology protection conjecture, a new law designed to rule out precisely such causally anomalous models. Matt Visser (1994) went further and suggested it be taken as an axiom rather than proved.
Philosophers did little better. Popular accounts appeared (Hawking, Paul Davies, John Barrow), and some philosophers accepted Gödel's case for the possibility of time travel, but almost nobody took seriously his argument for temporal idealism. Yourgrau claims his 1991 book was, and in 1999 remained, the only full-length sympathetic study of any part of Gödel's philosophy. John Dawson's biography (1995) gives the cosmology a few paragraphs and reads Gödel's Platonism psychoanalytically, as "arrested development" culminating in paranoia. Hao Wang's posthumous book of conversations (1996) gives the philosophy of time a few pages, mainly on Husserl; Wang admitted he had avoided the topic, even cutting Gödel short. Lawrence Sklar confessed that, like most, he sticks "to the easier stuff on closed timelike loops". John Earman (1995) advises us to "brush aside" the link between general relativity and idealistic philosophy.
The preface is combative. Yourgrau accuses Giora Hon of reproducing his ideas and words without adequate acknowledgement, and answers reviewers (Nerlich, Schlesinger, Oaklander) point by point; only Adriano Palma is said to have caught the book's spirit. Readers should weigh the "only serious study" claim as a 1999 snapshot: since then Savitt, Dorato, Belot, Manchak and Yourgrau's own A World Without Time (2005) have widened the discussion considerably.
The book's central theses
- Incompleteness and cosmology are internally related. The incompleteness theorem proved, by formally acceptable means, that Principia Mathematica cannot define the intuitive concept of arithmetic Truth (big T). The Gödel universe proved, by relativistically acceptable means, that relativity cannot capture the intuitive concept of time (little t).
- The asymmetry question. Yourgrau asks what no one else had: why did Gödel conclude for T that the limitation lies in formal mathematics, but for t that intuitive time itself is an illusion, rather than that relativity is limited?
- Two methodological signatures. Gödel (1) probes mathematically the limits of formal methods in capturing intuitive concepts, and (2) tends to move from the possible to the actual, as in his formalised ontological argument (after Leibniz: if a being with all perfections, including necessary existence, is possible, it is actual) and in the modal step about time.
- Limit cases. Gödel builds ingenious limit cases: extreme formal constructions that restrict which intuitive contents a formalism can carry. The Gödel universe is a limit case for Einstein's geometrisation of time (the reduction of all essential properties of time to geometric ones). Properties of time that vanish in that extreme geometry are thereby shown not essential to time as relativity conceives it.
- No formal science has analysed time. Neither relativity, nor tense logic, nor the semantics of indexicals like "now" has succeeded. The book criticises Richmond Thomason's indeterminist tense logic for the open future, and the "New Theory of Reference" accounts of "now" by David Kaplan and John Perry.
| Perry's "essential indexical" | Yourgrau's reply |
|---|---|
| "now" uttered at t refers to t, just as "here" uttered at p refers to p | No place is really here; on Perry's account no time is really now either |
| Context enters through a semantic rule | Then context reflects nothing in the objective order: call it "the inessential indexical" |
Answers to critics, in brief
- Not a brief for Gödel. Yourgrau does not "take sides" with Gödel or Einstein, against Schlesinger's reading that the book defends the unreality of time. He defends neither McTaggart's A-series (the shifting now; past, present, future) nor his B-series (fixed before/after relations between dated events).
- The relative A-series. His actual claim: in special relativity every time is merely present-relative-to-itself, and such a relative A-series is equivalent to the B-series. Those who claim to reconcile the A-series with relativity (Howard Stein and others) conflate the genuine with the relative A-series.
- Policing a formalism. Against Nerlich: Thomason's indices include points and complete branches, yet he defines necessity without invoking the branches, which begs the question of how to have truth about the future without necessity.
- The dead are persons. Against Nerlich again: death affects whether, not what, we are. Persons are essentially living, so when we die we cease to exist, but the dead remain dead persons, not concepts or memories.
- Possible worlds are worlds. Against Robert Stalnaker, merely possible worlds are real worlds, as possible and past persons are persons: an expectant couple expects a baby, not the concept of a baby.
What is new in 1999
- Chapter 5, "From Kant to Star Trek": a deeper study of time travel, of the modal argument, of Earman's critique, of Gödel's analogy with Kant, and of his interest in Husserl's phenomenology of internal time-consciousness. (Steven Savitt had, by contrast, acknowledged the force of the argument as Yourgrau reconstructs it.)
- Appendix A: Brouwer's intuitionism, which admits only the potential infinite, and the intuitive time it needs, compared with relativity, Husserl, Wittgenstein and Gödel.
- Appendix B: Zeno's paradoxes and the tension between the Cauchy–Weierstrass limit (seemingly potential infinity) and Cantorian actually infinite addition. Yourgrau's radical conclusion: the received view that the rationals, let alone the reals, form an actually infinite collection may need re-examining.
The link back to Princeton is Gödel himself: he proved the consistency of Cantor's continuum hypothesis (that the continuum's cardinality 2^ℵ₀ = ℵ₁) with Zermelo–Fraenkel set theory, Paul Cohen later proved its negation consistent too, and Gödel defended Cantor's Platonism about actually infinite sets. He also accepted Plato's doctrine that all genuine reality is timeless. Yourgrau's ideal, borrowed from Gödel via Wang, is to be precise without being technical.
The concept of existence cannot be relativized without destroying its meaning completely.Kurt Gödel, 1949
The First Edition (1991)
Of Gödel's three philosophical essays published in his lifetime, only one is about time: "A Remark about the Relationship between Relativity Theory and Idealistic Philosophy" (1949). Its claim is that Einstein has not so much explained time as explained it away. The first edition sets out to show that this claim belongs to a unified Gödelian philosophy, not to a logician's idiosyncratic aside.
The 1949 essay both announced new solutions of Einstein's field equations with closed timelike world lines and argued that they support the view that time is ideal: a subjective appearance or, more strongly, an illusion. Yourgrau asks whether these are isolated remarks or part of a comprehensive philosophy, and whether the logician had one at all.
- New sources after 1978. Gödel's Nachlass revealed the longer unpublished drafts, "Some Observations about the Relationship between Theory of Relativity and Kantian Philosophy". Studies by Georg Kreisel (1980), Solomon Feferman ("Conviction and Caution", 1984) and above all Hao Wang (Reflections on Kurt Gödel, 1987) began to show Gödel as philosopher.
- Prior work on time. Howard Stein ("On the Paradoxical Time-Structures of Kurt Gödel", 1970, and his introductory note in the Collected Works, 1990) and Paul Horwich ("Time Travel", 1981) addressed Gödel's time directly. The Collected Works (1986, 1990, 1995) gathered the published papers.
- Scope. No attempt is made to evaluate the technical side (see David Malament and Lawrence Sklar, both 1984). The book addresses the metaphysics: Parmenides, Kant, McTaggart, reality, idealism, becoming. Yourgrau offers no apology, since Gödel himself raised these issues, and shying away would mean not taking him seriously as a philosopher.
- A recurring structure. Yourgrau had already argued (1989) that Gödel's thoughts on mathematics, time and theology share one structure: the possible has strong implications for the actual, and that his Platonism is close to Frege's.
- Caution on the drafts. The quotations from Gödel's unpublished drafts, assembled from Princeton's Firestone Library, may contain material Gödel meant to cut from the final version; Wang assured Yourgrau that including it would not violate Gödel's wishes.
The Reception of Gödel's Results
Gödel's 1949 essay for Einstein's Festschrift used new solutions of general relativity to argue that time is ideal, yet commentators remember only that it made time travel possible. Yourgrau argues that this inverts Gödel's purpose and that the inversion hides a unity in Gödel's work: in logic and in cosmology alike, Gödel built "limit cases", formal constructions that constrain what intuitive content a formalism can carry. Einstein had geometrised time; Gödel built the limit case for that geometrisation.
The occasion and the two texts
In the 1940s Gödel, already famous for his foundational results, turned to the universe. The occasion was an invitation to honour Einstein in the Library of Living Philosophers volume. The result was "A Remark about the Relationship between Relativity Theory and Idealistic Philosophy" (1949a); a longer and more suggestive draft, "Some Observations about the Relationship between Theory of Relativity and Kantian Philosophy", survives unpublished in the Nachlass. Gödel calls time "that mysterious and seemingly self-contradictory being" which nevertheless seems to form the basis of the world's existence and our own. His new solutions, he claims, support Parmenides, Kant and McTaggart, all sceptics about the ultimate significance of time.
A divided reception
- Einstein (1949): the first commentator; brushed aside the idealism half of the title ("Entirely aside from the relation of the theory of relativity to idealistic philosophy..."). Privately he joked that Kant's view of time was the emperor's new clothes, with the form of intuition in place of the clothes.
- Milič Čapek (1961) and Karl Popper (1982): chide Gödel for the senseless idea of real time travel; Čapek blames excessive spatialisation of time.
- Howard Stein (1970), David Malament (1984), Lawrence Sklar (1984), Paul Horwich (1987): take the results seriously and study them carefully.
- Malament: "good evidence" that the Gödel model does not describe our universe, but it cannot be ruled out a priori, nor can any model just because it allows time travel.
In the rotating universes (the R-universes, in Gödel's unpublished drafts) it is possible "to travel into any region of the past, present, and future, and back again, exactly as it is possible in other worlds to travel to distant parts of space". What troubles Yourgrau is that the primary point has been lost: for Gödel, deducing the possibility of time travel from general relativity shows that time is ideal, a mere appearance imposed on reality by the "optical refraction" of the mind's lens.
One cannot have it both ways. Horwich (1987) presents himself as defending "Gödel's claim" that time travel is physically possible. But Gödel took the possibility of time travel to show that t, the time variable, does not denote genuine successive time here. If there is no genuine time, there is no genuine time travel.
Stein reads Gödel more subtly: Gödel's "chief aim is to put in question our ordinary notions about time". Stein himself is less impressed by the paradoxes, and denies that the "actual feasibility" of time travel would justify rejecting such structures a priori. He grants that in such a universe, given free actions and enough energy, a person "can decide, at any point in his career, to visit the space-time vicinity of any epoch of his own past".
The Berkeley comparison, and two differences
The situation resembles Berkeley, who insisted that tables and chairs really exist while reinterpreting all talk of them as talk of ideas. Gödel likewise says the R-universe "permits time travel" while reading "time" as a relativistic formal simulacrum of the real thing. Two differences matter:
- Not mental. Gödel says time is an appearance or illusion, not that it is mental. If the mental domain had genuine succession, he would have concluded only that time's sphere is limited.
- Only time. Gödel's idealism covers time, not space or spacetime. Any explanation of the 1949 paper must explain this asymmetry.
Why the philosophy went missing: caution
Gödel was sparing with his overall philosophy, which he knew ran against the temper of the times. Wang reported (1974) that Gödel suppressed the heuristic behind incompleteness: "the highly transfinite concept of 'objective mathematical truth' as opposed to that of demonstrability", while the printed 1930–31 text used only consistency and ω-consistency. Under Hilbert's shadow, where truth and the transfinite were suspect, a prudent man kept his formal results free of philosophical "contamination".
- Feferman, "Conviction and Caution" (1984): notes "the tight rein" Gödel kept on his true thoughts; he never said publicly, until Wang in 1974, what he thought of the Vienna Circle, though he was already a Platonist when he met it.
- Missed research. Feferman speculates that caution kept Gödel from pursuing computable functions (as Turing did) and truth (as Tarski did); Gödel had anticipated Tarski's undefinability of truth.
- Wang (1987) shows the quarrel with the zeitgeist was larger still. Gödel, shaped by Plato and Leibniz (later Husserl), had his own ontological argument ("theistic, not pantheistic, following Leibniz not Spinoza"), disliked the Tractatus because it claimed philosophy is impossible, and held "there is mind separate from matter". Given that, calling his caution "excessive" seems unfair.
Formalism in the broad sense
Formalism, broadly, is the tendency to recognise nothing in a domain beyond what a chosen mathematical formalism captures. Incompleteness gave formal evidence against its most precise version, the Hilbert programme, which identified (or replaced) mathematical truth with provability in a formal system; the notion of a formal system became fully precise for Gödel only with Turing's work. Gödel told Wang that formalists took formal demonstrability as an analysis of truth and so "were of course not in a position to distinguish the two". His philosophical essays (1944, 1964) hold that formalisms do not speak for themselves; they need something informal, intuitive, for their interpretation.
Putnam and the Skolem paradox. Hilary Putnam (1985) used the Löwenheim–Skolem theorem to argue that a formalised version of Cantor's proof that non-denumerable sets exist can be reinterpreted in a countable model, so it proves at most a denumerable totality. The effect comes from refusing to set the formal system in a wider interpretive context, forcing the formalism to speak for itself. Paul Benacerraf (1985) answered Putnam in an explicitly Gödelian spirit. Putnam's move combines the two hallmarks of modern rigour: axiomatisation and first-order formalisation.
The broader target is the mathematisation of nature: from Descartes' geometry (matter reduced to extension) and arithmetisation of geometry, to the geometrisation of time in Einstein's special and general theories. The historical Descartes was less pure: for him, says David Lachterman, geometric constructions were "outward counterparts" of the mind's inward activity. Not so for Hilbert (1902), and not so for Einstein. G. J. Whitrow sums Einstein's programme up as "the geometrization of physics", time "completely absorbed into the geometry of a hyper-space".
Time as an ontological suspect
Wang reports that Gödel traced his interest in time to his early work on Kant and Leibniz. Yourgrau offers a rational reconstruction (with Lakatos: if such reconstructions caricature history, history may equally caricature them):
- Platonism. Logic and mathematics, "just as physics", rest on axioms "with a real content which cannot be explained away" (Gödel 1944). Like Frege, Gödel separates epistemology from ontology, thought from object, proof from truth; Hilbert, Brouwer and Wittgenstein blur these.
- The timeless and the temporal. With Plato, Gödel confines pure mathematics to the eternal and time to the sensible world; Plato rigorously excluded time from pure geometry.
- Saving the phenomena. Like Plato, Gödel tried to integrate the two domains, but his result was far more negative, because he faced the task under Einstein's extreme mathematisation of time.
| Aristotle, Newton, Kant | Einstein–Minkowski | |
|---|---|---|
| Mathematics of time | Applied: time is "the number of motion with respect to before and after"; calculus measures continuous motion | Constitutive: time is a coordinate of a four-dimensional geometry |
| The flux | Represented only by analogy: a line being drawn, a point generating a line by "fluxion" | Gone: the world line is given whole, "in block" |
| Status of "geometry" | A useful, potentially deceptive metaphor | Literally a geometry, so non-standard geometries of time become thinkable |
| Future | Past fixed but supplemented; future partly open | Everything given together |
Kant is explicit: time is "nothing but the form of inner sense", and because inner intuition "yields no shape, we endeavour to make up for this want by analogies"; we represent time "under the image of a line, which we draw". For Aristotle, says Michael White, the operative simile is a moving point generating its trajectory, leaving "the past fixed but continuously supplemented, and the future at least partially indeterminate". Mary Tiles: philosophy of time must go via intuitions and non-verbal representations, not only analysis.
Einstein, by contrast, found it "more natural to think of physical reality as a four-dimensional existence, instead of... The evolution of a three-dimensional existence", a rigid four-dimensional analogue of Lorentz's aether. Louis de Broglie: in spacetime past, present and future are "given in block", and a particle's existence is a world line.
What it is to spatialise time
The essence of a space, in the most general sense, is not extension but the ontological neutrality of positions: where a thing is located has no effect on whether it exists. The temporal mode of being, pre-relativistically, is the opposite: to exist is to exist at a privileged location, now. To spatialise time is to collapse the temporal mode of being into the spatial one.
This definition also explains Gödel's description of sets as "quasi-spatial": a set's members do not come into being successively. Some deny that relativity forces spatialisation; Gödel did not, and he believed his limit case made the implication inescapable.
Limit cases: the unity of Gödel's results
John Myhill (1951) distinguished a formal question, which asks us to supply only the content of the answer, from an informal question, which asks for both form and content; we cannot know whether an informal question is objective or subjective until it is answered. Gödel's genius, on Yourgrau's reading, was to turn informal questions into formal ones, then settle the question of content by building constructions that limit the contentful interpretations they admit, by analogy with limitation results such as Löwenheim–Skolem.
- Frege: the reverse of a limit. The completeness theorem shows that for first-order predicate logic with identity, first presented in Frege's Begriffsschrift, the syntactic property provable is coextensive with the semantic property valid (true under all interpretations). Here proof can simulate truth, and no limit can be imposed on either reading.
- Hilbert: a genuine limit. Hilbert wanted every branch of mathematics completely formalised, with truth replaced by provability in a system itself proved consistent by finitary means. Incompleteness shows proof cannot simulate truth: no predicate T can consistently bear the interpretation "is true" in such systems. Form cannot replace non-finitary content. (Yourgrau notes Michael Detlefsen's challenges to this orthodox reading.)
- Einstein: a limit for time. Exploiting Einstein's geometrisation of time, itself a kind of formalisation, Gödel built a model whose closed timelike world lines make it impermissible to read t as genuine successive time, the time that carries objective becoming. A further modal argument extends this to the actual world.
Geometry versus geometrisation
The distinction is decisive. Many domains occupy a logical space with a geometric signature. Erwin Schrödinger gave colour space a geometry without trying to reduce the qualities of colours to its structure: a geometry, not a geometrisation. The Einstein–Minkowski geometry is offered as a geometrisation: it replaces the qualitative, intuitive features of time with the structural. Hermann Weyl (1949) made the parallel with Hilbert explicit: intuitive space and time must be relinquished as building material for physics and replaced by an abstract four-dimensional arithmetical continuum, leaving "a symbolic construction of exactly the same kind as that which Hilbert carries through in mathematics".
Yourgrau flags the question the rest of the book pursues: why did Gödel blame the formalism in Hilbert's case but intuitive time in Einstein's? Incompleteness shows formal proof falls short of intuitive truth; by parity, the Gödel universe could show relativity falls short of intuitive time. Gödel instead concluded that the intuitive content itself vanishes, absolutely. Yourgrau does not settle this here, and readers should note that the parity argument is a live objection to Gödel rather than a mere puzzle.
Formalisation in Gödel's hands is neither a tool to eliminate pure form (as, roughly, in Brouwer) nor content (as in Hilbert), but to illuminate each by the other: Wang's "dialectic of the formal and the intuitive".
Limit concepts and the danger of "all"
A limit case in mathematics echoes a limit concept in philosophy: delimiting a whole domain in thought so it can itself be manipulated. Frege delimited all quantificational validities; this let Hilbert create metalogic, treating logical systems as mathematical objects. The move from finite numbers to the single completed series of all natural numbers is Gödel's "first infinite jump". Cantor then delimited the infinite with actual infinity and built a transfinite arithmetic.
Limit concepts carry a powerful "all", and "all" breeds paradox:
- Aristotle applied excluded middle to "all future times" and ran into the sea-battle problem.
- Kant's Dialectic: the limits of all time, all space, all causal chains lead reason, used uncritically, into antinomy, putting cosmology itself in question.
- Frege's Axiom V treated the extension of a concept as a completed object; if two extensions are identical the concepts agree on all arguments, including those extensions, which yields Russell's paradox.
- Cantor: there is no set of all things.
- Reality itself is "the limit of all limit concepts", hence the most liable to antinomy, as Parmenides shows.
Gödel's constructions are two-faced. As theorems they are as indisputable as mathematics. But they admit alternative interpretations, inside and outside mathematics, that force the "philosophically minded" to make hard choices that would be far less visible under less extreme constructions.
A ruler that cannot measure itself. Picture a ruler so well made that physicists decide length simply is "what the ruler reads". Someone then uses only the ruler's own markings to build a rod that plainly has a definite length, yet the ruler can never read it off; or a closed track along which "further along" eventually returns to the start. Either the ruler fails to capture length, or length was never what we thought. Gödel faced exactly this fork twice: with Hilbert he blamed the ruler; with Einstein he blamed length.
Gödel's Idealism
To understand why Gödel drew an idealist moral from relativity, Yourgrau goes back to Parmenides: a robust concept of reality as "all that is the case" seems incompatible with temporal becoming. Through Plato, Aristotle and McTaggart's A- and B-series, he isolates what Gödel demanded of time: an objective, absolute, ontologically loaded now. Relativity delivers only B-theoretic structure; for Gödel the rotating universe is the limit case that makes this visible, and a time without the A-series "can hardly be called time".
Relativity as Gödel found it
- Special relativity (STR). The relativity of distant simultaneity and the absence of a privileged frame mean there is no world-wide instant. Einstein: the four-dimensional continuum "is now no longer resolvable objectively into sections... 'now' loses for the spatially extended world its objective meaning". Simultaneity, once a two-term transitive relation, becomes a three-term relation: simultaneous at a frame.
- The invariant interval. Frames agree on ds² = dx² + dy² + dz² − c²dt². For timelike separated events (ds² < 0) the temporal order is the same in every frame; since such events are causally connectable, STR preserves causal time order.
- General relativity (GTR). Matter curves spacetime, so some frames may be privileged: those that "follow in their motion the mean motion of matter". In every cosmology before Gödel's, these co-moving frames fit into one consistent cosmic time. Newtonian, "equably flowing" time is gone, but with luck in the distribution of matter, causal time and cosmic time survive.
- Gödel's unlucky universe. In the R-universe the "compass of inertia" rotates everywhere relative to matter, co-moving frames cannot be fitted into a cosmic time, and there are nearly closed and closed future-directed timelike curves. Always heading into your causal future, you arrive in your past.
Gödel's two interpretive claims: (1) in the R-universe the objective lapse of time is an illusion; (2) although whether ours is an R-universe is empirical, the collapse of objective succession in a universe that differs from ours only in non-lawlike features of the distribution of matter and motion shows that in our universe too time is merely ideal.
Is time compatible with reality?
Reality in the strong sense used by Parmenides, Plato and Gödel is approached by stages of generalisation:
The widening concept of reality
- The earth: the primitive "world".
- The universe: the totality of physical things. (Cosmologists still speak loosely of the "origin of the universe" in the Big Bang, which strictly is the origin of the rest of it.)
- The facts: "The world is all that is the case... The totality of facts, not of things" (Wittgenstein). The universe is only a large object; reality includes what is the case with it.
- Modal space: facts might have been otherwise, so reality seems to include all possible worlds, with ours, α, privileged as actual.
- The regress. That α is actual is not a fact internal to α; a "superfact" that α is actual could itself have failed to obtain. A vicious regress: reality as "all that is the case" eludes us.
The modal problem has a temporal twin, and that is where Parmenides enters: "Nor was it ever, nor will it be, since it is now all at once... Thus becoming is extinguished and perishing is not to be heard of."
The Parmenidean dilemma, reconstructed
- Reality is the totality of all there is, "all that is the case".
- If time is real, then yesterday's "all there is" left something out: what was not yet but came to be.
- We cannot repair this by including past and future, since if time is real only the present exists (Gödel: an objective lapse of time is one "whose essence is that only the present really exists").
- So if the "is" of "all there is" is tensed, reality grows by accretion (Richard Jeffrey); but growth would draw on what is now nothing, and "it is not possible for 'nothing' to be".
- If the "is" is tenseless, time has already been eliminated as real.
- Hence either reality is incomplete (an illusion) or time is. Since ontology cannot treat reality as an illusion, time is ideal.
"An expanding universe is one thing, an 'accumulation of Reality' (and thus of truth) another." Gödel himself writes that an objective lapse of time means "reality consists of an infinity of layers of 'the now' which come into existence successively", which implies reality itself comes into existence successively. Reality is the limit of all limit concepts, and Kant's warning about antinomy applies.
Turning Kant on his head
An unpublished remark of Gödel's says his cosmological results only confirmed his suspicion of change and tensed existence, since tensed existence "entails that we can, at different times, rightly assert about the same thing that it exists and that it does not exist". Kant had said the opposite: "only in time can two contradictorily opposed predicates meet in one and the same object, namely one after the other."
The obvious reply is that this is unfair: tensed existence says only that something exists at one time and not at another. Gödel would ask whether both times really exist. If not, reality is incomplete. If so, this "exists" is tenseless, position in time carries no ontological weight, and time has been spatialised, which for Gödel is to make it ideal. The force of the "all" in "all that exists" drives the argument.
This is why Gödel names McTaggart as a precursor. There is something always unfinished about the temporal, so it can never be completely described. Michael Dummett: McTaggart's argument shows that if time is real "we must abandon our prejudice that there must be a complete description of reality". Čapek, from a dynamic relativistic view, asks whether the "totality of the universe" stays meaningful once absolute simultaneity goes.
From Plato to McTaggart
- Plato (Sophist, Parmenides, Timaeus) separates Being (eternity, the world of forms and mathematics) from Becoming (appearances). Time is "a moving image of eternity". Eternity, says Harold Cherniss, "is not a measure but a mode of being"; an image always falls short of its original. J. M. E. Moravcsik: "temporal being is not complete being; what is temporally both is and is not."
- Aristotle separates actual from potential being and ties potentiality to matter, expressed in motion (locomotion and alteration). Time separates the potential from its always partial actualisation (Kosman). Full actuality means being outside time; hence the unmoved mover of Metaphysics XII.
- McTaggart names what the classical tradition left implicit: the A-series and the B-series.
| A-series | B-series | Relative A-series | |
|---|---|---|---|
| Terms | Past, present, future; now | Before, after; dates | Now-relative-to-t |
| Fixed? | No: positions shift with the nunc fluens | Yes, once and for all | Yes |
| Example | "It is now 1991" (false in 50 years) | "Yourgrau writes this in 1991" (always true) | Writing is now-for-1991; McTaggart is now-for-1908 |
| Privileged time | One: the now | None | None: every time is now for itself |
| Formalisable | Elusive | Easily (physics) | Easily, because it is the B-series |
The relative A-series is the B-series in disguise. "Now-with-respect-to-t" means no more than "simultaneous-with-t". Anyone who claims to have saved the A-series by making every time present to itself has kept only the B-series. Keeping the genuine and the relative A-series apart is the hinge of the whole book.
McTaggart argued that time requires both series, that the A-series is contradictory, and therefore that time is unreal; Dummett (1960) defended a version. Yourgrau uses the distinction, not the argument. The A-series's elusiveness explains both why Plato, Aristotle and Kant needed analogies for time and why the temporal seems ontologically anomalous: mildly for Plato and Aristotle, extremely for Parmenides, McTaggart, Gödel (and in a way Kant).
The five concomitants of the A-series
If the A-series were realised:
- One time is privileged, the now, and the privilege passes on to a succession of times.
- Time lapses irreversibly in one direction: from the possible to the actual, from future to present to past.
- Positions in time are not ontologically neutral: only the present confers existence.
- The past is closed and determinate, hence "necessary" in the sense of unavoidable; the future is open.
- Simultaneity is absolute, a non-relative two-term relation.
Aristotle's picture shows why these belong together. The A-series has ontological weight because it is the coming into actuality of the potential; it is successive and incomplete because matter is always realising itself; it is branchwise asymmetric because the past is unique and linear while the future branches into open possibilities. And the "flow" must be analogical: a literal flow, like a river's, happens in time at a rate.
A bias toward the spatial
The nunc fluens must be primitive, sui generis: time is like a river, but where the analogy fails, intuition must step in. Whitehead concluded that analysis of time must appeal ultimately to direct awareness. But how does a conceptual scheme biased toward the visual, whose logos is geometry, recover the temporal? Hearing is the better modality.
Score and performance. A musical score is graspable at a glance, like a geometry; a performance unfolds. A chord has the non-successive unity of a set; a melody has the successive unity of a sequence. Dipert and Whelden define a chord as a set of tones and a piece as an ordered n-tuple of chords; but defining ordered sets via unordered ("quasi-spatial") sets sacrifices the temporality of sequence. Penrose asks how Bach or Mozart could conceive a whole fugue "all conjured up together" when temporality is one of music's essential ingredients. The question is whether time, like music, can be captured conceptually (a score) or needs a temporal model (a performance). Brouwer asks the same about mathematics: can a tenseless formal system represent mathematical activity, or must we construct "in real time"?
Robert Tragesser observes that the pages of Frege's Begriffsschrift present "a rather beautiful geometry of signs". Classical mathematics turns points into instants and lines into world lines: tools fitted to the B-series. This underlies Zeno's arrow, which at no instant is in a state of motion, discussed by Plato in the Parmenides and Aristotle in the Physics.
Time and classical logic
If the A-series is to be mathematised at all, it needs a non-classical systematisation: something like Brouwer's intuitionistic continuum, not the pointwise decomposable continuum of classical analysis. Čapek: mathematical continuity "means infinite divisibility, that is, discontinuity infinitely repeated, the very opposite of continuity in the dynamical sense". The now is not the knife-edge of a point. Aristotle stumbled here himself: generating time by a moving point clashes with his view that time consists of lengths bounded by instants (White 1989).
Aristotle as proto-intuitionist (with Brouwer):
- Numbers generated in time (Categories VI).
- Only potential infinity, never actual (Physics III).
- Doubt about excluded middle for future contingents (De Interpretatione IX, the sea battle).
Čapek also warns that "direction" is borrowed from geometry and kinematics. The direction of time, from potential to actual, is not spatial, so reversibility is a non-starter for the A-theorist: nothing can be "de-actualised". A literal reading would even let us ask whether the future approaches the present or vice versa, since all spatial motion is frame-relative.
What Gödel demanded of time
Gödel held that, if time is not ideal, it must include the A-series as well as the B-series, with all five concomitants. His unpublished Kant paper:
- "An objective lapse of time, such as is contained in the intuitive concept of time, is impossible in the R-worlds."
- Relativity "has proved only the relativity of time not its subjectivity", one might say. But "absoluteness (or uniqueness) forms itself an essential characteristic of intuitive time", and a relative passing "would be something radically different... For existence by its nature is something absolute".
- A structure lacking the character of "passing by" "can hardly be called time".
| Mainstream philosophy of time | Gödel | |
|---|---|---|
| A-series | Ignored (van Fraassen, Newton-Smith) or dismissed (Mellor, Horwich) | Essential to time |
| Flow | Earman (1967): time does not "go", "unroll" or "flow"; literal readings fail | Agrees no literal flow exists in physics |
| Conclusion | B-time alone is real time (Mellor's Real Time) | Then time is ideal |
Gödel blocks the escape: one may say an objective lapse of time is meaningless, "but this is no way out of the dilemma", for that opinion is itself the idealist view of change. Yourgrau adds that A-theorists all agree flow needs analogy; Earman's demand for a "literal-minded" account rules analogy out without argument.
Objectivation and secondarisation
Gödel's drafts describe a history of objectivation: level after level, features of intuitive time are stripped from the scientific picture as merely subjective, so the class of intuitive features with physical significance keeps shrinking. Galileo's primary/secondary distinction began this; Sklar calls it "secondarisation" of phenomena. We accept heat = mean kinetic energy only by separating heat from the sensation of heat.
The disanalogy for time. Heat as a process is independent of the sensation of heat. But the experience of succession is not independent of genuine succession: even a succession of experiences is an objective succession, so it implies time. Kripke makes the parallel point about pain = C-fibre stimulation: the sensation is the essence of pain, so it cannot be secondarised.
- Kant's level. Gödel: Kant's world of appearances is already one level "in which a great many subjective elements... Have already been eliminated". Margaret Wilson: Kant is a phenomenalist only because the scientist's reductive explanation is set within ideal forms of intuition. Gödel thinks transcendental idealism exaggerates a fruitful distinction "into a boundless subjectivism, whereby its effect is annulled", as with the unknowable things in themselves.
- Kant's slipperiness. Time is the form of inner sense, yet in the Second Analogy subjective time-consciousness is grounded in lawlike causal order: "the subjective succession by itself is altogether arbitrary". But is an arbitrary succession a succession? Modern causal theories of time, inspired by relativity's causal order, pull these strands apart and run into trouble.
- Special relativity's level. Time becomes far more spacelike: Gödel, like Einstein, reads it as a "rigid four-dimensional space" replacing an evolving three-dimensional one. Čapek and Stein prefer a dynamic reading. Whitrow hedges: passage "corresponds to the continual advance of the light cones", yet relativity lets us consider "either the set of all these light cones or the continual transition".
The last objective remnant of time
In relativity there is a spacelike trace of becoming: the objective past-to-future ordering of the light-cone structure. But no objective becoming corresponds to our experience of it. Time is a fourth dimension, distinguished from space only geometrically, not, as in Kant, as a distinct form of intuition.
| Newton-Smith (1980) | Gödel |
|---|---|
| The opposite sign of the time coordinate shows an "absolute distinction between space and time" | The one negative square is "the last objective remnant of time", which a further level of objectivation might remove |
| So the puzzles about past, present and future are "alleged" | The sign is a B-theoretic fact; it says nothing about the A-series |
Penrose (1989) sides with Gödel: physics treats time much as space; time "does not really 'flow' at all"; and it is hard to believe that what makes time flow is merely the asymmetry between 3 space and 1 time dimension.
The limit case arrives
Because the negative sign remains and GTR still allows a cosmic time in suitable universes, objectivation is incomplete: relativity alone is not yet a limit case, so its counterintuitive consequences stay hidden. The R-universe completes it. No cosmic time is definable, and by always travelling toward the causal future one reaches the local past, cancelling the lapse of time.
| Cyclical time | Circular (closed) time |
|---|---|
| A process repeats n times, like a second hand going round a dial | One returns to the very same past |
| "How many times round?" makes sense | The question has no meaning |
| Compatible with succession | Makes nonsense of a succession in which only the present exists |
If one can travel at will to any part of time, it is because all parts of time already exist to be travelled to, exactly as all parts of space do. The question for the next chapter is whether Gödel's philosophical conclusions can be sustained.
Existence by its nature is something absolute.Kurt Gödel, unpublished draft
Time Travel and the Gödel Universe
Closed time cannot be successive time: if stage 5 precedes stage 1, every stage is already complete at every stage, and all five concomitants of the A-series vanish. Yourgrau shows why time travel in the Gödel universe changes nothing, why its paradoxes are really constraints on consistency, and how Gödel's modal argument carries the result from a possible world to ours. Because there is "time travel" in such worlds, there is no passage of time, and so no genuine time travel at all.
Succession versus the circle
In picture (A), time unfolds stepwise; when an earlier step has occurred the later ones have not (they are not yet actual); and at every step there is a definite, unique answer to "which stage has now been reached?". None of this applies to (B). Stage 5 no more waits on stages 1–4 than they wait on it. If stage 5 were not already complete at stage 1, stage 1 could never occur, since it comes after stage 5. In circular time, temporal position cannot carry ontological weight: closed time is spacelike, which is exactly Gödel's point.
Gödel: "the decisive point is this, that for every possible definition of a world time one could travel into regions of the universe which are passed according to that definition. This again shows that to assume an objective lapse of time would lose every justification in these worlds."
All five concomitants fail
The A-series in closed time:
- No privileged now. Nothing singles out one moment.
- No lapse, no A-direction. A geometric, B-theoretic direction can be specified, but not the direction from possible to actual: travel to the past is travel to the already actual, which cannot be re-actualised. The very distinction between actual and possible times breaks down.
- Ontological neutrality. Every position is equally real.
- Closed future. The future is as determinate as the past; if the past is "necessary", so is the future.
- No absolute simultaneity. The model is relativistic.
Richard Sorabji ("Closed Space and Closed Time", 1986) sees "no reason why there should not be a direction for the flow of time" in closed time, even though each event lies equally in past and future. Yourgrau's verdict: only a conflation of A- and B-series permits this. A B-direction (an orientation of the curve) is not a flow.
The two faces of the result
Gödel mentions paradoxes that would follow if one could travel "into the near past of those places where he has himself lived", then argues such trips are not practically feasible: the required velocities and fuel are prohibitive. "Therefore it cannot be excluded a priori, on the ground of the argument given, that the space-time structure of the real world is of the type described."
- Face one (mathematics). Nearly closed and closed timelike world lines exist in the model. If t keeps its standard interpretation, this is time travel.
- Face two (philosophy). Gödel rejects that interpretation, because nothing corresponding to the passage of time remains. The very features that make the curves closed prevent them from being temporal. There is not even travel, in the intuitive sense.
Nobody remakes the past
What remains are B-theoretic questions. The traveller does not remake history: she can visit the past precisely because all parts of B-time are equally real. If she will visit it, she already has. A circle is not a cycle, so she is not revisiting. Nothing can be undone, since what has happened is a fact and nothing alters facts.
The main B-theoretic consequence of time travel in a Gödel universe is the symmetry of past, present and future with respect to facticity. With no open future, nothing can be changed, in the future or in the past.
So the grandfather (or parent) paradox is not a contradiction waiting to happen. The traveller's existence guarantees she does not kill her parents: either the trip is infeasible (Gödel's suggestion) or she always fails. Closed time imposes strict consistency constraints: the "beginning" presupposes the "end", and what is possible now depends on what was actual "before". There are no genuine possibilities, hence no free will in the sense of the ability to do otherwise. In a world of pure fact the only requirement is logical consistency, which reality, unlike a formal system, cannot fail to satisfy.
- Ad hoc failures. The gun jams; a bolt of lightning startles her.
- Horwich (1987): such a universe will have "the sort of initial conditions that will give rise to certain types of coincidence".
- Sklar (1984): if there is a lawlike constraint on initial conditions, so that some world-states are "lawlike incompossible with themselves", what solid distinction remains between laws of nature and mere initial conditions?
The paradox of memory
Gödel's epistemic worry is subtler. A traveller to his local past could meet his earlier self, "now he could do something to this person which by his memory, he knows has not happened to him". Gödel says knows, not "thinks he knows". The issue is not fallible memory, which we have already; it is that a present decision could disconfirm even a confident first-person memory.
The memory test. You clearly remember that, on your tenth birthday, no stranger gave you a blue coin. You now board a Gödel rocket, arrive at your tenth birthday, and resolve to hand your younger self a blue coin. Either something stops you, every time, or your memory was wrong all along, and it was your intention that exposed it. Ordinarily, as Dummett stresses in "Bringing About the Past" (1978), only judgments about the future must take our intentions into account; memory is immune. Time travel would erase that mark of the past–future difference.
Gödel does not say whether every class of memory report could be undermined this way, or most of them. Such cases produce no explicit antinomy, but they threaten the epistemic status of memory, which is tied closely to time-consciousness. One may reply that in a time-anomalous world we should expect memory to change. Then the question becomes whether what remains deserves the name "time". Earman (1967) asks how far the concepts of time, causality and identity in a coherent time-travel story would resemble ours. Gödel's reason is different and deeper: no sense can be made of objective passage in such a world, a point lost on those, like Earman, who make no sense of flow even in ours.
Gödel's unpublished draft: "That at least that passing of time which is directly experienced has no objective meaning in the R-worlds also follows from the fact that it is possible in these worlds to travel arbitrarily far into the past and future and back again." He keeps the phrase "time travel" because he trusts readers to remember that all temporal phenomena there are ideal.
The fruits of geometrisation
Gödel generalised Einstein's geometrisation of time to conclusions Einstein resisted. Einstein always denied that one could "send wireless signals into the past"; in ordinary relativity this is impossible, but because of the structure of Minkowski spacetime, not, as in Kant, because space and time are distinct forms of intuition. A purely geometric approach left room for Gödel's counterintuitive geometries.
Einstein's reply (1949). The problem "disturbed me already at the time of the building up of the general theory of relativity, without my having succeeded in clarifying it... Does it make any sense to provide the world-line with an arrow, and to assert B is before P, A after P?" He added: "It will be interesting to weigh whether these are not to be excluded on physical grounds." Yourgrau: this does not touch the modal force of Gödel's argument.
Kreisel reports Penrose's view that the direct physical interest of Gödel's papers is limited, in line with Einstein's comment, though they served as a cross-check on conjectures in global relativity. Stein (1970): "only one cosmos is given us", so which others are physically possible "is moot". Yourgrau agrees the question is moot but resists Stein's reason: only one cosmos is given in sense experience, yet theorists routinely confront the actual world with an ensemble of possible universes, and experimentalists decide which is actual.
Modus ponens or modus tollens?
The Gödel universe seems, to Gödel at least, not to violate any law that holds in ours. Yourgrau's 1991 prediction follows: someone might propose "a new law of nature, an anti-Gödel law" to rule out the R-universe being lawlike compossible with ours. But what would motivate it, except that time travel is counterintuitive? And bringing out relativity's counterintuitive implications is exactly what Gödel was doing.
| Sceptics: modus tollens | Gödel: modus ponens |
|---|---|
| If Gödel's extrapolation from Einstein were physically significant, time would be ideal | If relativity is true and the R-universe is lawlike compossible with ours, time is ideal |
| Time is not ideal | Relativity is true; the R-universe is compossible |
| So the model is a mathematical curiosity, a cross-check | So time is ideal: a consequence hidden under less extreme geometries |
Which way to run the argument depends on Gödel's philosophical reading of relativity, just as his incompleteness results needed the distinction between truth and proof that the formalists could not draw. The troubling features of the Gödel universe (cancelling passage by visiting the past, exposing the unreality of the future by visiting it, undermining time's direction) all concern A-time, which on Gödel's reading had already gone in special relativity.
Possibility implies actuality: the modal argument
Gödel's modal argument, reconstructed
- In a Gödel universe there is no objective lapse of time (the A-series fails).
- The Gödel universe is lawlike compossible with ours: same laws, differing only in the distribution of matter and motion.
- So observers there could have the same direct experience of time as we do.
- So our direct experience of a lapse of time is compatible with its non-existence.
- If even direct experience cannot establish an objective lapse of time, nothing can.
- Therefore there are, and can be, no grounds for asserting an objective lapse of time in our world. "If the experience of the lapse of time can exist without an objective lapse of time, no reason can be given why an objective lapse of time should be assumed at all."
Gödel was used to letting possibility imply actuality: his ontological argument has the form "if God is possible, God is actual", and for a mathematical Platonist the possibility of a structure implies its mathematical existence. Here the pattern is more elusive, because the cosmological model must be granted physical significance for anything to transfer.
Shades of Kant: scepticism to idealism
The argument mirrors Kant's Transcendental Deduction and Refutation of Idealism: if our possession of the categories is compatible with all our experience of an external world being illusory, nothing further can show it is not. Both thinkers move from scepticism (p and not-p fit all possible experience) to idealism, a move that, as Strawson notes, rests on a verificationist bridge principle (Kant's "principle of significance").
Gödel's draft makes the principle explicit for special relativity: "in perfect conformity with Kant", the observations do not force us to give up Newtonian time and space, only the observations plus principles such as "two states of affairs which cannot be distinguished by observations are also objectively equal". J. S. Bell (1989) made the same point: Einstein declared "really resting" meaningless because it cannot be detected; Lorentz kept a real rest frame defined by the aether even though the laws conspire to hide it. "The facts of physics do not oblige us to accept the one philosophy rather than the other."
Yourgrau's reconstruction exposes the argument's soft spot: step 5 needs a verificationist premise, which most philosophers (and arguably the Platonist Gödel, elsewhere) reject. If one allows that an objective lapse could exist undetectably, as Lorentz allowed an undetectable rest frame, the inference from the Gödel universe to ours fails. Later critics, notably Earman and Savitt, press exactly here, and the issue returns in Chapter 5.
The core idea. In our world, what is "timelike" in relativity is only accidentally correlated with the intuitive features that make time time. By projecting pure Einsteinian time into a universe where it survives but the correlations vanish, Gödel believed he had shown that even here Einsteinian time is only a formal simulacrum of successive time.
Kant and relativity "go in the same direction"
Gödel's long draft is a commentary on the "insufficiently appreciated" similarities between relativity and Kant, although he says "I am not an adherent of Kantian philosophy in general". "Kant and Relativity Theory go in the same direction, but Relativity Theory goes only one step."
| Kant | Relativity | Gödel universe | |
|---|---|---|---|
| Time relative to | Universal human sensibility | Each inertial frame or observer | No consistent world time at all |
| Higher objective time? | None: other minds might not experience time at all | Yes: cosmic time (if matter cooperates) and invariant timelike order | None |
| Scope of ideality | Space and time | Simultaneity only | Time only, not space or spacetime |
| Basis | A priori philosophy | Empirical physics | Empirical physics plus the modal step |
In both Kant and relativity time is ideal in that it is imposed by a subject or observer, not intrinsic: it "does not inhere in the objects, but merely in the subject which intuits them". Relativity "goes only one step" because of cosmic time and the invariance of timelike order. The Gödel universe removes both, bringing Gödel's position close to Kant's, but grounded in Einstein's physics and confined to time. The next chapter tests Gödel's reading of special relativity against those who reconcile relativity with becoming.
Not Everything Can Be Relativized
Howard Stein objects that "to be relative is not to be illusory". Yourgrau answers that for Gödel it is: the passage of time is a change in what exists, and "the concept of existence cannot be relativized without destroying its meaning completely". Testing the best attempts to put becoming back into relativity (Čapek, Stein, Sklar's relativised reality), he finds each either keeps only the relative A-series or relativises existence. The A-series is incompatible with certain cosmologies, which is why Gödel is an idealist about time but not about space or spacetime.
Stein's "apparent discrepancy"
In his introductory note to Gödel's "Remark" (1990), Stein sees a tension: the unpublished "Some Observations" stresses that temporal relations are relative to observers but still objective, while the "Remark" calls time ideal. The resolution is that, for Gödel, for time to be relative is for it to be an illusion. The objective but relative relations of the draft belong to the B-theoretic structure, which "can hardly be called time" and should be called space or spacetime.
Gödel anticipated the objection in footnote 5 of the "Remark": a relative lapse of time, "if any meaning at all can be given to this phrase, would certainly be something entirely different from the lapse of time in the ordinary sense, which means a change in the existing". In a word: some things cannot survive relativisation. If time were real, the world-in-time would by now have advanced so far and no farther, as in the open succession of Chapter 3; that requires absolute world-wide cuts, the "infinity of layers of the now which come into existence successively", which relativity denies.
Not subjective time either
If a merely relative succession is ruled out as genuinely temporal, so is a merely subjective one. One might suggest that time exists at least subjectively, especially for Kant, whose inner representations are always successive. Kant forbids it: "I can indeed say that my representations follow one another; but this is only to say that we are conscious of them as in a time-sequence... Time is not, therefore, something in itself." The appearance of succession does not imply the succession of appearances.
I really do appear to myself temporally; but I do not really temporally appear to myself.P. F. Strawson on Kant
Frege advised staying away from objective truths about the subjective altogether, though Hacker argues he did not escape trouble himself.
Gödel's gap: what replaces Kant's mind?
Gödel shares Kant's ideality of time but rejects the "boundless subjectivism" of universal transcendental idealism and its unknowable things in themselves. Kant reached universal idealism because the mind has a form of its own that it imposes, so it knows only that form a priori. Gödel never says what picture of mind he puts in its place.
| Kant | Aristotle (De Anima) |
|---|---|
| Mind imposes its own forms (space, time) | Mind has no form of its own |
| Knows objects only as they appear | A "form of forms" that can know things as they are |
| Leads to universal idealism | A universal "intellectual space" of forms, like the receptacle of Plato's Timaeus |
Yourgrau suggests Gödel, steeped in Greek philosophy, may have leaned toward the Aristotelian picture, and hopes the philosophical notebooks in the Nachlass will show it. Gödel also called the positivist interpretation of quantum mechanics, assumed to be the theory's final stage, another overgeneralisation of Kant's Copernican turn.
Two contrasts remain. Seen as rival accounts of the physical world, Kant and relativity are incompatible, and Gödel sides with relativity. And although Gödel agrees there are a priori constraints a form of being must meet to count as time, he derives the ideality of time from an empirical theory. How two thinkers reach the same result from categorically different sources is taken up in Chapter 5.
Čapek: Heraclitus to Gödel's Parmenides
Milič Čapek's The Philosophical Impact of Contemporary Physics (1961) tries to integrate becoming into relativity. He blames Gödel's time travel on excessive spatialisation; but Gödel credits the spacelike view to Einstein, and his models prove rather than presuppose it. Unlike Sorabji, Čapek does not try to put becoming into the Gödel universe; he denies such models physical significance a priori, because they allow time travel.
Čapek and Gödel agree that passage has ontological force, which is where Čapek's trouble starts. "On each world-line, my present particular 'now' separates my past from my future." Which now?
- Not the absolute now: special relativity has none.
- Not "my frame's" now: even within my frame every t is merely now-relative-to-t, the relative A-series, which discriminates nothing.
- Not all frames' nows together: if every observer's now is genuine, there is no room for the nunc fluens, the transition of privilege from one time to the next, since all nows are already privileged.
- Not "only presently observable observers are real": Čapek dismisses future observers in whose past light cone my now lies, as not presently observable. That holds only from his frame; relativity forbids privileging any frame.
So Čapek must posit a separate river of time for each frame: "this plurality of contemporaneous or co-becoming temporal series", or Whitrow's "complex of advancing light-cones". Gödel's objection is not Earman's (no literal sense for flow) but that he finds senseless the idea that "time lapses in different ways for different observers". On the many-rivers picture there is no non-arbitrary answer to how much time has flowed so far.
A lapse of time which is not a lapse in some definite way seems to me as absurd as a coloured object which has no definite colours.Kurt Gödel
Three challenges for becoming in special relativity
- Non-arbitrariness. Can STR recognise my set of nows as non-arbitrary, or even say what that would mean?
- A universal measure. Do my nows fix a valid standard of how much time has lapsed to date?
- Conventionality. Does the world-wide instant reflect an irreducibly conventional element in fixing distant simultaneity?
Can we save becoming? The modal analogy
Sklar ("Time, Reality, and Relativity", 1981) argues the reality of the present is not decisively refuted, but only by reading STR as resting on verificationism, which then slides toward point solipsism. Gödel presupposes STR's epistemology, so this does not engage him.
| Modal space | Time in special relativity | |
|---|---|---|
| Privileged item | The actual world | The now |
| Relativised view | David Lewis (1973): "actual" is indexical; every world is actual-relative-to-itself | Relative A-series: every time is now-relative-to-itself |
| Chauvinist option | Insist ours is absolutely actual (consistent, if brute) | Insist one inertial frame is privileged: ruled out by STR |
Modal chauvinism is available; inertial chauvinism is not. Against Lewis one can dig in and declare our world absolutely actual. Against relativity one cannot, even by brute force, declare one's own frame the universal standard of becoming, without leaving special relativity behind.
The one loophole is Lorentz's reading of STR, with a hidden absolute rest frame. Popper (1984) argued that if Aspect's experiments on Bell's theorem were interpreted as establishing action at a distance with infinite velocity, they would be the first experiments to decide between Lorentz's and Einstein's interpretations, since infinite velocity entails absolute simultaneity and absolute space, even if, as Newton foresaw, we could never identify the frame at rest.
Bell tests since 1999. Loophole-free Bell tests (Delft, Vienna and NIST, all 2015) confirmed the violation of Bell inequalities, and Aspect, Clauser and Zeilinger shared the 2022 Nobel Prize in Physics for this line of work. But the no-signalling theorem guarantees that entanglement cannot transmit information, so no experiment can single out a preferred frame. Neo-Lorentzian "preferred foliation" views survive in Bohmian mechanics and some collapse theories, and Tim Maudlin argues that non-locality strains relativity's spirit; most physicists still treat the rest frame as empirically idle. Popper's decisive experiment has not happened.
Stein: becoming from the light cones
Čapek holds that the formal details of STR confirm becoming; Stein (1968) gives the argument. His principle: "At a space-time point a there can be cognizance of, or information or influence propagated from, only such events as occur at points in the past of a." Becoming is therefore relative to a point, not a time: "at a given time" is not invariant, so definiteness of truth-value must mean "definiteness at a given space-time point (or event)... To be vivid, 'definiteness for me now'". Stein sees this as analogous to pre-relativistic relativisation to "now".
Yourgrau doubts the analogy. That a's past light cone is frame-invariant is a geometrical fact holding at every point. How do geometrical facts introduce transition or becoming? Stein needs only "what-is-past-for-a, for all a": the relative A-series again. And "for me now", said by any point a about itself, has lost the privileged now altogether.
Geometry cannot supply becoming
Two problems for Stein and Čapek
- The geometric appeal. They claim the light-cone geometry justifies "process" and "becoming". But a rigid four-dimensional block satisfies every geometric constraint of relativity. So (a) how can geometry justify going beyond the block, and (b) what must be added to the formalism to make becoming visible?
- Becoming without a unique now. They hold becoming is objective, a change in the existing, yet sacrifice the unique now and absolute simultaneity. Then we need a new picture of existence in which many rivers all flow and all add to what exists.
On (1), Stein seems to share Earman's literal-minded representation of time (nothing is more literal than mathematics) while drawing the opposite conclusion from the same equations. Čapek is less consistent: abstract constructs are "the only way... To represent the structure of the transphenomenal plane", yet "all geometrical analogies are intrinsically inadequate for expressing the nature of time", and only when "succession and change are incorporated into the very foundations of geometry" is there hope. Einstein himself said the Now "means something special for man" but "this important difference does not and cannot occur within physics", echoing Plato's exclusion of time from pure geometry.
On (2), Čapek even doubts the "totality of the universe" once absolute simultaneity goes. Then: (a) what is relativity a theory of, if not the whole universe? (b) how can there be a relativistic cosmology without that concept? (c) how can genuine coming-into-being proceed in parallel along different frames, as a "co-becoming or co-fluidity of world-tubes"? Without a theory of how formalisms represent, Gödel and Einstein on one side and Stein and Čapek on the other draw opposite conclusions from the same mathematics; that is the subject of Chapter 6.
Rietdijk's proof of determinism
Rietdijk (1966), "A Rigorous Proof of Determinism Derived from the Special Theory of Relativity" (also Putnam 1967)
- Observer α (frame i_α) and observer β (frame i_β, moving relative to α) are coincident, so β is real for α.
- Event e lies in α's future light cone.
- STR allows that in β's frame, e is simultaneous with β, so real for β.
- Transitivity of reality: what is real for someone real for me is real for me.
- So e is real for α: the future is as real as the present.
- If future facts are fully determinate, the future is not open: determinism.
Escape 1. Relativise reality. Sklar (1974): since STR gave up the transitivity of simultaneity, give up the transitivity of reality; the presentist can "relativize his notions of reality... At the same time that he relativizes his concepts of simultaneity", a consequence "surprising, no doubt, but by no means inconsistent". Yourgrau grants consistency but asks (1) how the original motive survives if "the" future is a frame-relative, near-subjective matter, and (2) whether it makes sense to relativise reality at all.
Escape 2. Symmetrise irreality. Sklar (1981): if distant simultaneity is conventional, there is no fact of the matter about which distant events are real-for-me, just as for the A-theorist there is no fact about the future. Spatially remote events become as unreal as temporally remote ones. Sklar sees no basis for STR other than the verificationism present from the start. Gödel half agrees: Wang reports him saying positivism was fruitful in STR only because of "a very exceptional circumstance", that simultaneity is directly observable, unlike most basic entities. (For a different view of conventionality, see Michael Friedman, 1983.)
Being, not knowing
Yourgrau turns Sklar around. The future is not unreal because it is epistemically remote; there is an epistemic distance because the future is not really there. The foundations of the A-theory are ontological, which is why the past, though determinate, is also unreal for the A-series.
Distant sufferings. You learn of famine in a far country you will never reach, and of the grief of a family in fourteenth-century Florence. You do not treat the first as unreal because it is out of reach; you do treat the second as over. Yeats: "Swift has sailed into his rest; savage indignation there cannot lacerate his breast." Swift has not sailed to a remote place but into a different mode of being: non-existence. This "natural ontological attitude" (Yourgrau borrows Arthur Fine's phrase), not the finite speed of information, grounds the asymmetry between temporal and spatial remoteness.
The twofold challenge of STR to the open future: either (1) relativise reality to a frame and threaten the integrity of existence, or (2) symmetrise irreality to the spatially remote, trading on a suspect analogy between epistemic remoteness (behind STR, on some readings) and existential distance (behind the A-theory).
Can the lapse of time be merely local?
Michael J. White suggested a softer response: treat present–future asymmetry as local, like flatness on a globally curved surface. Yourgrau rejects it. A local–global distinction suits the spacelike B-series; but a closed curve in A-time is an antinomy, as the two figures of Chapter 3 showed. Gödel, who rejected a merely relative lapse, could not accept a merely local lapse; to apply the local–global distinction to A-time would itself spatialise time.
Why idealism about time but not space. A-theoretic time is incompatible with certain cosmologies, and the soundness, or for Gödel even the possibility, of such cosmologies counts against the A-theory. Nothing analogous threatens space or spacetime. This is the key to Gödel's selective idealism, and to the Kantian title of his unpublished essay.
From Kant to Star Trek: A Philosopher's Guide to Time Travel
The longest chapter, new in 1999, rebuilds Gödel's case in full: special relativity removes the absolute now; general relativity restores a cosmic time in our universe; the Gödel universe abolishes it and adds closed timelike curves; and a modal argument carries the verdict home. Yourgrau defends the argument against John Earman, separates time circles from time cycles, and shows how Gödel could claim that "Einstein verifies Kant" while splitting Kant's doctrines of time and space. Anyone who is not shocked by Gödel's results about time has not yet understood them.
Einstein "verified" Kant?
Gödel wrote that relativity furnished "a very striking illustration, in some sense even a verification, of Kantian doctrines". That is unusual: most thinkers stress what relativity refuted in Kant (a priori knowledge of space and time, Euclidean space, absolute simultaneity), and after Minkowski's fusion of space and time it seems perverse to hold Kant right about time but not space. Relativity had also become orthodoxy, eclipsed by quantum mechanics as the philosophical puzzle of the day.
| Putnam (1979) | Bohr, and Gödel |
|---|---|
| "I do not believe that there are any longer any philosophical problems about Time; there is only the physical problem of determining the exact physical geometry of the four-dimensional continuum." | Anyone not shocked by quantum mechanics has not understood it. Gödel told Wang that time is the philosophical problem, "that mysterious and seemingly self-contradictory being". |
(Einstein in Prague in 1911, pointing at an asylum below his study: "Here you see that portion of lunatics who do not concern themselves with quantum theory.")
An ironic division of labour
| Time travel is possible | So time is ideal | |
|---|---|---|
| Gödel | Yes | Yes |
| Čapek (1976), Popper (1982) | No | Yes, if there were time travel (so they reject it) |
| Horwich (1987) | Yes | No: time travel is unproblematic |
Nobody accepts both of Gödel's propositions. Čapek and Popper see the clash with experience and so reject time travel; Horwich accepts time travel and ignores the idealism. Horwich's naturalism ("the promise of metaphysics was an intellectually dangerous illusion") is the opposite of Gödel's, who "wished to do for metaphysics what Newton did for physics" and called his outlook "rationalistic, idealistic, theological, and optimistic".
Gödel's work on Einstein no more expresses a natural-scientific worldview than incompleteness expresses Hilbertian formalism. A metamathematical parallel: Gödel admitted that the completeness theorem "is indeed an almost trivial consequence of Skolem 1922", but others missed it because of "a widespread lack... Of the required epistemological attitude toward metamathematics and toward non-finitary reasoning". His freedom from that prejudice made incompleteness possible. By analogy, those who treat the temporal component of spacetime as an analysis of intuitive time "are of course not in a position to distinguish the two".
Einstein's ambivalence
- Against spatialisation. In his reply to Émile Meyerson, Einstein disliked reading time as a mere fourth spatial dimension.
- For the block. Elsewhere: more natural to think of reality "as a four-dimensional existence", the "rigid four-dimensional space of special relativity".
- Popper, 1950. Popper urged Einstein to give up the view that the world was "a four-dimensional Parmenidean block universe in which change was a human illusion"; Einstein agreed it had been his view, and Popper called him "Parmenides".
- Besso, 1955. On the death of his friend Michele Besso: "For those of us who believe in physics, this separation between past, present, and future is only an illusion, however tenacious."
- Gödel's comment. Robert Goldman phoned Gödel on 14 August 1977, months before his death, about the Besso letter. Gödel: "I think he meant it as some sort of joke." Einstein, he added, believed science could never explain consciousness and that space and time were human constructions; "reality was something deeper".
- Carnap. Einstein once said the experience of the Now "means something special for man... But that this important difference does not and cannot occur within physics". Carnap thought Einstein failed to distinguish experience from knowledge.
What Gödel means by "time"
Gödel draws a sharp line between intuitive time and t, the temporal coordinate of spacetime, which is for him a simulacrum. Collected from his texts, intuitive time is:
- Also called "Kantian", "pre-relativistic" time, "what everyone understood by time before relativity".
- "A one-dimensional manifold that provides a complete linear ordering of all events in nature".
- Involving "the objective lapse of time", which is "directly experienced" and "involves a change in the existing".
- Time "whose essence is that only the present really exists".
- A lapse that "means (or is equivalent to the fact) that reality consists of an infinity of layers of 'now' which come into existence successively".
- And: "something without these properties can hardly be called time".
So Gödel's intuitive time is tied to McTaggart's A-series; relativity's t (and, Yourgrau will argue, current tense logic) to the B-series. With McTaggart, Gödel holds that real time needs both, and that this fails. A-time is not mental time: "it is snowing now" reports snow, not a mood. Whether anything, including a subject's states, is located in the A-series is an objective question. Kant even makes time more basic than space: all representations belong to the mind's inner state, so "time is an a priori condition of all appearance whatsoever".
Special relativity: no objective lapse
The argument from special relativity
- Relativity of simultaneity: simultaneity holds only relative to an inertial frame.
- Observational equivalence: no experiment distinguishes inertial frames.
- Verificationist principle: "two states of affairs which cannot be distinguished by observations are also objectively equal".
- So there is no unique, frame-independent now.
- Intuitive time requires such a now (only the present exists).
- So in the world of special relativity there is no objective lapse of time.
What survives is the invariant interval ds² = dx² + dy² + dz² − c²dt²: timelike-separated events (ds² < 0) are causally connectable, giving causal time. Gödel's conclusion can be put: causal, relativistic time cannot be identified with intuitive time. At most it defines a B-series, and not even one complete linear B-series; one can build a family of B-series relative to timelike world lines (Abner Shimony). Relativising the B-series is harmless; relativising the A-series is not.
Shimony's point P. Shimony (1993) concedes the Minkowski structure "is neutral with respect to the status of transiency", which already undercuts Stein, whom he praises. He then argues the now is non-relative: two observers whose world lines cross at P both call P "now", so "A-determinations must be ascribed to the point P itself". But observers with diverging world lines will in general disagree about distant nows, and Shimony admits A-determinations cannot reach outside a point's light cone. So each observer has his own non-relative now, all "co-becoming": exactly Gödel's "each observer has his own set of 'nows', and none of these various systems of layers can claim the prerogative of representing the objective lapse of time".
Čapek's generalisation objection (1976): Gödel and "modern neo-Eleatics" never consider that Newtonian time may be a special case of time in general, as Euclidean space is of space. Yourgrau replies that this treats time as a space in the general sense (ontologically neutral positions), which is Gödel's point. Gödel happily grants that Minkowski spacetime generalises space; intuitive time cannot be generalised without generalising away what distinguishes it from space. And Čapek rejects the further generalisation Gödel makes, to closed relativistic time, though it is achieved inside the letter of relativity.
General relativity: the return of cosmic time
In general relativity, matter "largely destroys the equivalence of different observers and distinguishes some of them conspicuously from the rest, namely, those which follow in their motion the mean motion of matter". In every cosmology before 1949 their local times fit together into a cosmic time. James Jeans (1936) drew the moral: once relativity covers astronomy, "the symmetry between space and time... Must be discarded", and "we are free to believe that time is real".
Gödel's notes: all observers remain equivalent under the laws, but the actual arrangement of matter "may offer quite different aspects to different observers", more "natural" to some. The "true mean motion" is reached by averaging over regions so large that enlarging them no longer changes the value (in our world, many galactic systems), and "of course a true mean motion in this sense need not necessarily exist".
Few people who know the big bang story notice that "the current state of the universe", "the age of the universe", "the origin of the universe" are notions ruled out by special relativity. They live only in general relativity, which describes the actual universe (special relativity being its matter-free idealisation), and only if privileged fundamental particles (galaxies, clusters) exist.
Cosmic time, defined (after Gödel, Malament and Earman)
- Temporal orientability: the light cones can be consistently divided into past and future halves (here a relativistic, not an A-theoretic, notion).
- Chronological precedence: x ≪ y iff there is a smooth, future-directed timelike curve from x to y.
- Global time function: a map t: M → ℝ with t(x) < t(y) whenever x ≪ y.
- Construction: take the congruence of world lines of the fundamental particles. If it is twist-free, there is a one-parameter family of spacelike hypersurfaces orthogonal to every world line; these slices, globally synchronised, give cosmic time.
- Malament's rope: think of the world lines as fibres of a rope. If the rope is twisted, no cut is orthogonal to every fibre. "Orthogonal sliceability is equivalent to fiber untwistedness", which is equivalent to global non-rotation.
Our universe appears non-rotating, so cosmic time can be defined, and with Jeans one may claim a single non-arbitrary world time "objectively lapses" here.
The Gödel universe
Gödel's three replies to Jeans: (1) in the rotating universes the matter congruence is not twist-free, so no cosmic time is definable; indeed in the Gödel universe p ≪ p for every point p: every event chronologically precedes itself; (2) there are future-directed closed and nearly closed timelike curves; (3) the modal argument. Rotation is relative to "the axis of a completely free gyroscope" inside the universe, since nothing lies outside it.
The Gödel metric (Gödel 1949, units c = 1)
- ds² = a²[ dt² − dx² + ½e^(2x) dy² − dz² + 2e^x dt dy ], a constant a > 0.
- Source: pressure-free dust of uniform density ρ with 8πGρa² = 1 and a negative cosmological constant Λ = −1/(2a²).
- Homogeneous (every point is like every other) and rotating: the matter rotates relative to local gyroscopes with angular velocity ω = 2√(πGρ).
- In cylindrical coordinates (Hawking and Ellis), circles about any axis are spacelike for small radius, null at the critical coordinate radius r = ln(1 + √2) (where sinh r = 1), and timelike beyond it: closed timelike curves through every event.
- The smallest timelike circles are "of the same order of magnitude as the world radius in Einstein's static universe"; a rocket must reach at least c/√2, and Gödel estimated the fuel needed as prohibitive.
Does our universe rotate? Gödel kept checking; his Nachlass holds two notebooks on the orientation of galaxies, and he later found rotating expanding models (1952). Modern limits are severe. Analyses of COBE, WMAP and Planck cosmic microwave background data, notably Saadeh et al. (2016), found no sign of global vorticity or shear and disfavoured anisotropic expansion at odds of over 100,000 to 1; bounds put the present rotation rate many orders of magnitude below the Hubble expansion rate, the exact figure depending on the model. A 2025 proposal (Szigeti, Szapudi and colleagues) that a very slow rotation, about one turn per 500 billion years, might ease the Hubble tension prompted renewed CMB and large-scale-structure fits; follow-up analyses leave little room for a rotation large enough to matter. Our universe is not a Gödel universe; Gödel's argument never claimed it was.
Earman's critique and the "benign neglect"
Gödel's final argument, from the mere compatibility of rotating worlds with the laws, is the most controversial: if someone says absolute time lapses in our world, "he accepts as a consequence that whether or not an objective lapse of time exists... Depends on the particular way in which matter and its motion are arranged in the world. This is not a straightforward contradiction; nevertheless a philosophical view leading to such consequences can hardly be considered as satisfactory."
Steven Savitt ("The Replacement of Time", 1994) found Yourgrau's reconstruction forceful. John Earman (Bangs, Crunches, Whimpers, and Shrieks, 1995) did not. Yourgrau's complaints about the tone:
- "The neglect has been benign." Earman notes the "relative neglect" of Gödel's moral, while discussing the one book devoted to ending it, and then calls the neglect deserved. Why discuss it? "A deeply held conviction of someone of Gödel's stature deserves serious consideration": attention to the man, not the philosophy.
- Burying Caesar. He ends by endorsing Einstein's brushing aside of idealism as "the correct response to Gödel". Earman does concede that Gödelian time travel "implies that there is no time in the usual sense in which to 'go back'", a complaint with "a good deal of justice", which is Yourgrau's own point from Chapter 1.
- "A pretty piece of ordinary language philosophizing." Earman's verdict on "existence cannot be relativised". But Gödel was no friend of ordinary language philosophy; of Kripke he said he was "though not a positivist, still doing linguistic philosophy".
Gödel on conceptual intuition (from Wang)
- "We perceive objects and understand concepts. Understanding is a different kind of perception." (Compare Church: a concept may be "understood or grasped" as a body is seen; and Frege: "one apprehends or thinks a thought".)
- "The analysis of concepts is central to philosophy. Science only combines concepts and does not analyze them. Einstein's theory is itself not an analysis of concepts... Physical theories are not the correct metaphysics."
- Against Kant: intuition tells us not only that 7 + 5 = 12 but that there are infinitely many primes and that arithmetic is consistent.
- "Intuition is not proof; it is the opposite of proof." "The positivists attempt to prove everything from nothing."
- "The notion of existence is one of the primitive concepts with which we must begin as given. It is the clearest concept we have. Even 'all'... Is less clear."
His thesis adviser Hans Hahn wrote "The Crisis in Intuition" (1933), ending that intuition is "a force of habit rooted in psychological inertia". Gödel's sympathies lay with Husserl's intuition of essences; he urged colleagues to read Husserl.
Is positivism dead?
Warren Goldfarb says Gödel "took positivism to be a more widespread and longer-lived orthodoxy than it was". Yourgrau disagrees: the spirit of positivism (anti-metaphysics, scepticism about conceptual intuition and Platonism) lives on. Hawking: "I... Am a positivist." Nancy Cartwright: "positivist that I am, [I] am opposed at once both to metaphysics and to transcendental argument". Horwich, quoted above. Michael Friedman locates positivism's failure narrowly in Gödel's refutation of Carnap's conventionalism; Yourgrau calls that "a positivistic conception of positivism".
| Gödel's "left" | Gödel's "right" |
|---|---|
| Scepticism, materialism, positivism | Spiritualism, idealism, theology (metaphysics) |
| Empiricism | A priorism |
| Hilbert's formalism (mathematics drifting left) | Mathematics "by its nature" |
"The spirit of the time always goes to positivism and materialism; for instance, Plato was followed by Aristotle." Yourgrau's test for Earman: does he seriously dispute that existence cannot be relativised? Would he wait for the Hubble Telescope to settle a clear question about a primitive concept? Sklar (1974) and Quentin Smith (1995) do relativise reality to an observer, apparently unaware of Gödel's objection.
Earman's missing premise
Earman supplies the premise he thinks Gödel needs: "The existence of an objective lapse is not a property that time can possess contingently." Formally, with L = "time lapses", N = necessity, P = possibility:
Earman's reconstruction
- L → N(L) (if time lapses, it lapses necessarily)
- P(¬L), established by the Gödel universe, a physically possible world where ¬L; so ¬N(L)
- Therefore ¬L (by modus tollens on 1)
Yourgrau's disambiguation of premise 1
- (A) If time lapses in any world, it lapses in all physically possible worlds. Gödel shows no sympathy for this.
- (B) If time lapses in any world, it lapses in every world in which time exists: □(T → L), where T = "time exists". Lapse is essential to time.
Under (B) the question becomes: why is it unsatisfactory that time exists, and lapses, here but not in the Gödel universe? Because the worlds share the same laws (Einstein's field equations) and, presumably, the same phenomenology of time; they differ only in the global distribution of matter and motion. Gödel: a lapse of time "would have to be founded, one should think, in the laws of nature".
The pain test. Suppose physicists found a possible world with our laws, inhabited by beings with exactly our nervous systems and our behaviour, in which pains exist but do not hurt, while in our world they do, the difference resting on the distant distribution of galaxies. You would conclude something had gone wrong: pain is not the kind of thing whose existence the testimony of direct experience plus the laws can leave open. Gödel's intuition is that time is like pain in this respect. And whether a river flows is not settled globally whatever happens locally: a river cannot flow locally but not globally.
Earman: "Gödel gives us no guidelines for judging superiority of conceptual insight", and others "claim to perceive the nonexistence of CTCs as essential to the concept of time", so ruling out the Gödel model a priori. Yourgrau: Gödel agrees that no CTCs is essential to intuitive time; it is not essential to relativistic spacetime, the simulacrum. Gödel is not founding a theory of intuition, only invoking one intuition to test the completeness of one formal system, as he invoked an intuitively true but unprovable sentence against Principia Mathematica.
The anti-Gödel law arrives. On the basis of the Gödel universe's lawlike compossibility with ours, Yourgrau's 1991 edition predicted an "anti-Gödel law". In 1992 Hawking proposed the chronology protection conjecture: the laws of physics prevent the appearance of closed timelike curves. Yourgrau asks how one can accept the geometric basis of general relativity and then append an ad hoc condition to rule out its solutions. "Those who live by geometry must be prepared to die by it."
Chronology protection in 2026. Hawking's argument rested on the divergence of quantum vacuum fluctuations (the renormalised stress-energy) at the Cauchy horizon where CTCs would first form. Kay, Radzikowski and Wald (1997) proved that the quantum stress-energy is ill defined on such horizons, which supports protection but also marks where semiclassical gravity stops being reliable (Visser's "reliability horizon"). Counter-examples (Li and Gott's self-consistent vacuum, 1998) showed the divergence is not universal. The consensus remains that chronology protection is unproved: a conjecture that only a full theory of quantum gravity could settle. Researchers still publish "dynamical chronology protection" mechanisms in modified gravity (2025–26), a sign that no accepted law yet forbids Gödel's solution. Note too that the Gödel universe has CTCs through every point and no horizon at which they "form", so Hawking's mechanism does not touch it directly; it is excluded, if at all, by global energy and causality conditions or by fiat.
The epistemology of the Gödel universe
Observations of non-rotation and expansion seem to rule out that ours is a Gödel universe, but these observations are epistemically remote: subtle telescopes and long theoretical chains. Our everyday temporal experience is entirely consistent with our world turning out to be one.
The telegram from the observatory. At the end of a long life of anticipations and regrets, you learn from Hubble Telescope data on the most distant galaxies that the universe rotates: time does not flow, so your whole life has been an illusion. Next day a telegram says the data were misread; time exists after all, and your life was not an illusion. Gödel's point is that it is absurd for the reality of time, which we live through directly, to hang on such evidence.
Malament (reviewing Sklar) made the same move about the direction of time: "Is it at all plausible that our only means of determining whether the dessert comes after the soup is by first determining patterns of weak interaction in particle physics? Surely not." Sklar: there is "something implausible" in an analysis making "an obviously present feature of the world hinge upon" features that may not exist and are observable only in "the most unusual of situations". Substitute lapse for direction and cosmology for particle physics, and you have Gödel.
Earman asks what non-geometrical feature time in our universe lacks. The answer: the lapse itself, the successive coming into existence of layers of now, which Earman finds hard to make "non-psychologistic sense" of. Like the formalists who took proof as an analysis of truth, those who take the relativistic t as an analysis of time cannot distinguish the two.
- Earman's concession and reply. Grant that observers in a Gödel universe are under an illusion. Still, "apart from our experiences of time lapse we have all sorts of other experiences" supporting the inference that we inhabit a universe meeting all the geometrical conditions for a lapse.
- Savitt's burden shift. Geometry gives only necessary conditions. If Gödelians can have experience like ours, "why should our experience be taken as any indication that in our world time does objectively lapse?" To the blunt challenger: "How do you know that?" The other condition, successive coming into existence, "is precisely what is at issue".
- Earman's possible rejoinder. A stick in water looks bent though straight, so seeming-bent does not entail bent; but in favourable conditions (no refraction) seeing is enough. Likewise, when large-scale geometry is favourable, experience of passage might suffice.
The true meaning of the Gödel universe
Yourgrau's answer exposes the paradox Earman's picture hides. In our world, matter and motion are arranged so that cosmic time exists and time lapses. In a world with the same laws but another pattern, there is no time, and so no motion, since motion is change of position over time. When the "motion" of matter assumes the pattern of the Gödel universe, it ceases to be motion at all. The Gödel universe is not a frozen world; it is defined by a pattern of motion, universal rotation. How can the mere pattern of something's movements decide whether it really moves?
The ballet. There are possible ballets in which the dancers stand frozen on stage. Are there possible ballets, as lively as Swan Lake, where the pattern of the steps alone guarantees that no dancer is really moving? The Gödel universe is such a ballet: the choreography of matter ensures nothing moves.
Second, epistemic intimacy. For Gödel, time is like pain: if it is a genuinely possible mode of being, then if we so much as seem to experience it, we do. Unlike pain, it is also a fundamental feature of empirical reality: hence "the basis of the world's and our own existence". Eddington: "we have direct insight into 'becoming' which sweeps aside all symbolic knowledge as on an inferior plane." So if even direct experience fails to establish the lapse in the Gödel universe, then, since time is the kind of thing for which direct experience must be decisive if it is possible at all, time is essentially ideal, even here.
Earman finally says Gödel "would seem to need an additional premise asserting something like a verifiability theory of meaning... Few people will be attracted by such a premise". Yourgrau: Gödel's verificationism is Einstein's, the one that defines time "by direct measuring with clocks". Gödel told Wang positivism was fruitful in special relativity only because local simultaneity is directly observable. Earman's complaint should be aimed at Einstein, and J. S. Bell questioned exactly that principle.
Local versus global
Could the local properties of time stay as perceived, objectively lapsing, whatever the global structure, as the round earth is locally flat? No: Gödel would object to a "local lapse" as he did to a relative one. If it is even possible, consistently with our experience, that a large closed timelike line runs through our spacetime, and if Minkowski time is the time we experience, then the experienced lapse is actually and locally an illusion: its ontological effect is annulled by the mere possibility of return, however large the radius. To deny this is to drop "a change in the existing" and treat spacetime as a space, conceding the ideality of time. (Malament's remark that past and future "do not break down locally" in the Gödel model concerns the relativistic causal future, not the A-theoretic one.)
Analogies with incompleteness
Sklar (1984): Gödel did not need the cosmology to refute Jeans; conceptual reflection shows cosmic time is no restoration of pre-relativistic absolute time. Yourgrau compares Zermelo and Brouwer, who said the arithmetisation of metamathematics was not needed to refute Hilbert, since reflection shows formal proof is no substitute for truth.
- Same strategy. In both cases Gödel used his opponents' own methods (formal systems against Hilbert; cosmic time against Jeans) to prove within their framework what he wanted to emphasise. Hilbert could ignore Gödel's intuitions, not his theorems.
- Same gap. Formal results alone never prove the philosophical conclusion; the last step is intuitive and can be resisted. Gödel was guarded about Hilbert's programme, far less guarded about time. Hilbert himself resisted at first; Bernays recalled he was "only angry and frustrated" before trying a "semi-infinitary" rule (1931), which Gödel privately told Carnap would compromise the programme.
- Opposite morals. Incompleteness supported realism about mathematical truth; the cosmology supported idealism about time. Kreisel reports Gödel treated realism and idealism as branches of philosophy: the first about the things, the second about how we come to know them. Then incompleteness uses idealist methods (definitions, proofs) for realist conclusions, while in physics Einstein's idealism (time defined by clocks; indistinguishable means equal) leads to Kant's.
| Incompleteness (1931) | Gödel universe (1949) | |
|---|---|---|
| Formal system | Principia Mathematica and kin | General relativity |
| Intuitive concept | Arithmetic Truth (big T) | Successive time (little t) |
| Limit-case result | No predicate can mean "is true"; a true sentence is unprovable | t cannot mean intuitive time: no cosmic time, CTCs |
| Background context | Informal classical number theory, where the Gödel sentence is seen to be true | No developed theory in which an objective lapse is visible |
| Gödel's verdict | The formalism is limited | The intuitive content is illusory |
The unanswered question. Why did Gödel not conclude that relativity is incomplete with respect to time? Yourgrau, who says no one else had even posed the question, admits "we don't really know enough" to answer it. His best suggestions: (1) arithmetic has an uncontroversial informal setting where the unprovable sentence is seen true; time has none, and special relativity even conflicts with an objective now; (2) Gödel may have hoped Husserl's phenomenology would supply that setting, but Husserl forbids ontological conclusions from phenomenology; (3) Gödel told Wang physics should be rethought around "real time", something like the causal structure of the world, with "cause" in the sense of an axiom causing its theorems: Aristotle's formal cause. The question remains open, and it is a fair objection to Gödel's confidence.
Aristotle's sea battle offers another parallel. Truth about the future seems to bring necessity. Aristotle keeps excluded middle as a theorem while rejecting bivalence: ⊢ □(p ∨ ¬p) but neither ⊢ □p nor ⊢ □¬p, reading □ as "determinately (now necessarily) true". "It is necessary for there to be or not to be a sea battle tomorrow, but it is not necessary for a sea battle to take place tomorrow, nor for one not to take place." Read □ as "provable" and p as the Gödel sentence, and this is exactly the form of incompleteness (Myhill noticed this too). Stein and Horwich dismiss Aristotle's dilemma; Aristotle, Aquinas and tense logicians like Thomason do not (Chapter 6).
Gödel's attachment to Plato surfaces in Wang's records: his favourite passage was Republic 6, where geometers use visible figures but think of the Ideas; and "if there is nothing precise to begin with, it is unintelligible to say that we somehow arrive at a precise concept... We begin with vague perceptions of a concept, as we see an animal from far away". Gödel's attitude to Einstein's clock-based methodology also stays ambiguous: he prefaces it with "if time is defined by direct measuring with clocks", and Wang says he was very interested in Lorentz versus Einstein. (Heisenberg recalled Einstein dismissing the positivism he had once used for special relativity: "Perhaps I did use such philosophy earlier... But it is nonsense all the same.") Gödel also blamed quantum mechanics' positivism for the absence of substantial progress in basic laws for about forty years.
Time circles, not time cycles
Take a closed timelike curve to be a time circle. A cycle repeats a process n times, and one can always ask how often it has occurred; of a circle the question makes no sense. A time circle is a completed circle, not one being inscribed: no point is privileged as now, every point is fully actual, nothing is merely potential. Since "the lapse of time in the ordinary sense... Means a change in the existing", a Gödel universe with time circles has no genuine succession, and therefore, strictly, no travel and no "time travel". The picture of a rocket racing at more than c/√2 misleads; the right philosophy of such a rocket trip, as of ours, is Kant's.
The cycle fallacy in print and on screen.
- Star Trek: the crew, supposedly in a time circle, repeat a time cycle n times; and they are ordered not to change the past, as if the grandfather paradox were a live risk. The past is a fact; nothing changes it.
- Alan Lightman, Einstein's Dreams: "Suppose time is a circle... The world repeats itself, precisely, endlessly."
- John Wheeler: in the Gödel universe "one could in principle live one's life over and over again".
- Popper: a closed world line "must be infinitely and absolutely repetitive", from which he derives a contradiction.
- Čapek: phases of the traveller's time "co-exist simultaneously". But simultaneity is sameness of time; in Gödel's world there is no intuitive time at all. And it is not true that the time machine alone "retained its moving, its succession" (Čapek on H. G. Wells).
A Minkowski diagram represents the journey as well as the path. One needs no special effects to film the Gödel universe: "Simply look around yourself and you will, if Gödel is correct, already be witnessing a universe no less 'timeless' than the GU."
Čapek also claims the time traveller, able to interact with past events, would make envious "even the medieval God" whom Aquinas denied the power to undo the past. Wrong: closed time means symmetry of facticity. Everything is like our past: closed, complete, in Aristotle's sense "necessary". The traveller is as much already in the past as returning to it, and any felt freedom to undo it is an illusion; no one is free in the sense incompatible with determinism.
| Horwich (1987) | Yourgrau |
|---|---|
| Charles may return to 1066 and could attend the battle, "compatible with the supposition that he did not attend it"; not doing something does not show one was not free to do it | In a world with complete symmetry of facticity, there is no earlier moment when the event was not already determinate. If the facts rule out an event, in what sense is one free to bring it about? |
Quantum time travel, 1991–2026. David Deutsch (1991) showed that quantum mechanics on a CTC always admits a self-consistent fixed-point state (D-CTCs), dissolving grandfather paradoxes as mixtures: the would-be killer emerges from the loop in a state that makes the story consistent. Lloyd, Maccone, García-Patrón, Giovannetti and colleagues (2011) proposed post-selected CTCs (P-CTCs), modelled on teleportation, and simulated one photonically; in P-CTCs paradoxical histories simply get zero probability, enforcing Novikov-style consistency. Aaronson and Watrous (2009) showed D-CTC computers could solve PSPACE problems, a sign of how much power closed time would confer. Ringbauer et al. (2014) simulated D-CTC dynamics with photons, and Tobar and Costa (2020) argued that classical reversible dynamics on CTCs can allow non-trivial freedom of choice with consistency. None of this shows that CTCs exist; it shows that, if they did, physics can accommodate them consistently, which confirms Yourgrau's point that the paradoxes are consistency constraints, not contradictions.
How Gödel splits Kant
Kant's noumenal freedom survives because he makes space and time both ideal; the Gödel universe yields full determinateness because spacetime is real, a "space". (Allen Wood reads Kant's moral progress in time as "moving images of our eternal moral attitude", echoing the Timaeus.) So how does Gödel enlist Kant while splitting time from space?
- Not a universal idealist. Popper wrongly lists Gödel with Schrödinger as an idealistic opponent of realism. Gödel says Kant "should be modified": scientific knowledge can, "at least partially and step by step", go beyond the appearances and approach things in themselves. Relativity's equations describe the world as it is; Gödel is a realist about spacetime. Einstein agreed: "pure thought can grasp reality, as the ancients dreamed."
- Why time and not space. Gödel gives two reasons: "the passing of time has no analogue for space"; and with matter present the symmetry breaks, since matter supplies "a 3-parametric system of physically distinguished one-dimensional time-like subspaces (the world lines of matter), whereas there exists in general no one-parametric system of physically distinguished 3-dimensional space-like subspaces". So no possible world departs from intuitive space as far as the rotating worlds depart from intuitive time.
| Kant | Special relativity (Gödel's alignment) | |
|---|---|---|
| Time exists only | In appearances, relative to universal human sensibility | Relative to an observer or inertial frame |
| Things in themselves | Noumena, unknowable to theoretical reason | World lines and spacetime, knowable, with their relations to us |
| Phenomenal world | Essentially in time | The "appearances" to each observer are as if temporal |
| Knowledge of forms | A priori | A posteriori, from experiment |
"Kant and relativity go in the same direction, but relativity goes only one step." Gödel's comparison is a deliberate realignment, not a literal match; as he wrote, many of Kant's assertions, "literally understood... Are false but in a broader sense contain deep truths". Gödel uses "world of appearances" in three senses:
- The observer's world: the world as it appears relative to a frame, temporal in something like Kant's sense, but many worlds for many frames.
- The noumenal world of relativity: spacetime as a whole, not in time, "in perfect accord with Kant". It is noumenal also because modern physics, unlike Newton's, splits laboratory language from theory.
- Appearance in the widest sense: still the world as cognised by beings with human sensibility, however abstractly. So, against Kant, the world of appearances in this sense is not in time either.
The ironies: the a posteriori basis of relativity, which grounds Gödel's temporal idealism, undercuts Kant's, which rested on a priori knowledge; and "doubtless Kant would least of all have held an overstepping of the world of appearances to be possible, on the basis of conclusions drawn from experiments". Gödel's epistemological optimism (Goldfarb's phrase) follows: if the world's form is not imposed by the knower yet is known, the gap between thing-in-itself and appearance closes.
Does the appearance of time prove time?
Kant himself wrote: "Alterations are real, this being proved by change of our own representations... Alterations are possible only in time, and time is therefore something real." Many follow:
The argument from experienced succession
- Popper: if we experience genuine succession, it is real; it cannot be explained away by time-slices "rising into consciousness", since that rising is itself a succession.
- Peter Geach: our varied states "must really be, as it appears to be, a change", since they are combinable only in succession.
- William Lane Craig: no B-theorist has answered the charge that even mind-dependent becoming makes the succession of conscious contents objective.
- Shimony: "Without this minimal amount of objectivity there cannot even be an illusion of transiency." Nowness "must either be more than an illusion or less than an illusion".
Yourgrau's replies: (1) these thinkers owe an account of how one subject's inner times relate, by simultaneity, to others' and to the relativistic outer world; (2) Husserl, the phenomenologist Gödel turned to, demands "the complete exclusion of every assumption, stipulation, and conviction with respect to objective time"; the order found in experiences "cannot be met with in the empirical, objective order, and do[es] not fit into it". Gödel told Wang "time is the only natural frame of reference for the mind", meaning Husserl's immanent time, not clock time. (3) Kant answered his own words at once: my representations follow one another only in that "we are conscious of them as in a time-sequence... Time is not, therefore, something in itself". For Gödel, too, "the psychic realities in us are for our self-consciousness also a kind of outer world which we know only as an appearance".
The appearance of a succession is not a succession of appearances. Kant's own idealism cannot be turned against Gödel. (Wayne Waxman reads Kant the same way; Izchak Miller makes the converse phenomenological point, that a succession of presentations is not a presentation of succession, but wrongly derives awareness of succession from simultaneous structural features. Structure is geometry, and geometry cannot account for succession.)
Yourgrau does not deny the force of Shimony and the others. His claim is narrower: an objective correlate of experienced flow is inconsistent with relativity, yet our intuition does not refute relativity; it may signal relativity's incompleteness, as Einstein–Podolsky–Rosen signalled an incompleteness in Bohr's quantum mechanics that led to Bell's theorem. Gödel: "As we present time to ourselves, it simply does not agree with fact. To call time subjective is just a euphemism for this failure."
Gödel's cosmology is Janus-faced, like his arithmetisation of metamathematics: contributing mathematically to the "left" while pointing philosophically to the "right". Kant built on Newton and Euclid; Gödel built on Einstein.
Where philosophers stand on Gödel's argument in 2026. Most philosophers of physics follow Earman and Savitt in rejecting the modal step, usually by denying that the lapse of time must be nomically necessary (it may depend on contingent structure such as a cosmic time) or by rejecting the verificationist bridge. Others, such as Mauro Dorato and Gordon Belot, have scrutinised the argument's physics and argued that becoming can be anchored in contingent structure (a cosmic time, or individual world lines), so its absence in a Gödel universe proves nothing about ours. Growing-block and local-becoming views (Arthur, Savitt, Dieks) accept frame-independent becoming along world lines, which is the Stein strategy Yourgrau criticises. Kant scholars have revisited the comparison (for example Silvia De Bianchi, 2024, on Kant's absolute time and the rotating universes), and the argument is still taught as the sharpest relativistic case against the A-theory. Presentism remains a minority view, held mainly with a preferred foliation. Yourgrau's own A World Without Time (2005) brought the argument to a wide audience; the scholarly verdict remains that Gödel proved the possibility of CTCs beyond doubt, while his idealism stays contested.
As we present time to ourselves, it simply does not agree with fact. To call time subjective is just a euphemism for this failure.Kurt Gödel to Hao Wang
Formalization and Representation
Gödel turned an intuitive question into a formal one and settled it; but a formalisation is only as good as its translation, and the languages of mathematics and logic were built for the tenseless. Yourgrau tests two formal treatments of A-time: Thomason's branching-time logic of the open future, and the Kaplan–Perry semantics of "now". Both, he argues, capture at best a spacelike trace of time. Privileged positions (the now, the actual world, the true) are "formally diaphanous": a formal model can label them but cannot say what makes them privileged.
The dialectic of the formal and the intuitive
Gödel exploited relativity's formalism against the spirit of Einstein's enterprise, as he had exploited Hilbert's formal number theory (via Frege, Dedekind, Peano) against Hilbert's programme. Relativity's resistance to the A-theory reflects a double wish: to view time from within and sub specie aeternitatis, from Plato's eternity. Wang calls Gödel's characteristic move a "transmutation" of an intuitive problem into a "cleaner" formal one. Einstein had already formalised time, but left room for rival readings; Gödel's limit case removed that room.
- Incompleteness turned "is there, even for the strongest consistent axiom system for number theory, an undecidable proposition?" into a precise question, answered yes.
- Wittgenstein's complaint: "a piece of mathematics cannot solve problems of the sort that trouble us." Yourgrau: Gödel's piece of mathematics about relativity does speak to the problem of the ideality of time.
From formalisation to mechanisation
The lineage runs from Frege, who formalised logical consequence "to prevent anything intuitive from penetrating here unnoticed", to Turing and the mechanisation of thought. Hegel's "cunning of reason": Frege despised mechanical symbol manipulation ("just as it is possible to speak like a parrot") yet built a notation that "does the thinking for us, so to speak". Gödel defined formal systems as those in which reasoning "can be completely replaced by mechanical devices", and Judson Webb (1980) argues the arch-anti-mechanist's results support mechanism. Unlike Turing, Gödel never left intuition out.
| Hilbert and the formal school | Frege and Gödel |
|---|---|
| Axioms are implicit definitions, true by stipulation | Axioms have real, extra-systemic content |
| They apply to any system that satisfies them | If true, they describe a genuine reality |
| Formalisation replaces intuition | Formalisation translates an intuitive problem, lending "wings and armour" (Myhill) |
Formalisation "can penetrate where intuition falters", but its results are only as plausible as the original translation. For time the problem is acute, since the very language of the formalism was designed for the tenseless.
Aristotle's sea battle
Aristotle (De Interpretatione IX) derives a result that seems to contradict the open future. From excluded middle (p ∨ ¬p) and the necessity of the past, consider past predictions about the then-future: one of two opposed predictions was true, and being past is now necessary; so the future is as necessary as the past. "There is nothing to prevent someone's having said ten thousand years beforehand that this would be the case." Unlike Gödel, Aristotle treats the result as a reductio (modus tollens) and revises the formal side.
Aristotle's principles on the open future (F p = "p will obtain")
- What is, when it is, is necessary. "What is, necessarily is, when it is."
- What is now true about the future is now necessary. "What anyone has truly said would be the case cannot not happen."
- Excluded middle holds necessarily: (a) Nec(p ∨ ¬p), (b) Nec(Fp ∨ ¬Fp), but not (c) Nec(Fp) ∨ Nec(¬Fp).
- Bivalence fails for some, especially future, propositions: "of the affirmation and the negation, neither is true rather than the other".
- Correspondence is tensed: (a) truth comes from correspondence with present facts; (b) the contingent future is not now a fact. Hence (4a′) reject the principle that if p comes to obtain, it was already true earlier that p would obtain.
These principles reflect Aristotle's picture of time generated by a moving point: a fixed but growing past, a partly indeterminate future.
Thomason's branching time
Richmond Thomason, "Indeterminist Time and Truth-Value Gaps" (1970), with related work by John Burgess (1978), Richard Jeffrey (1980) and Michael J. White (1980), formalises this picture.
Thomason's model
- Frame. A set K of times with a relation < that is transitive and backward-linear: if β < α and γ < α with β ≠ γ, then β < γ or γ < β. Many futures, one past.
- Branch (history). A maximal linear subset h of K. H_α = the set of branches through α.
- Two valuations. V_α evaluates relative to a time α; V_α^h relative to a branch h through α. Tense-free formulas get T or F from V_α (bivalent there).
- Branch-relative clauses. (a) V_α^h(A) = V_α(A) for tense-free A; (b) V_α^h(FA) = T iff V_β^h(A) = T for some β ∈ h with α < β, else F; (c) V_α^h(PA) = T iff V_β^h(A) = T for some β ∈ h with β < α, else F. These are bivalent.
- Time-relative supervaluation (after van Fraassen). V_α(FA) = T iff FA is true on every branch through α; F iff on none; otherwise undefined. Future contingents get truth-value gaps.
- Validity is the same on both kinds of valuation.
- Necessity. V_α^h(LA) = T iff V_α^g(A) = T for all g ∈ H_α (L is an S5 operator: inevitability).
The Ockhamist results (A. N. Prior's term): past and present are now necessary, but only facts with no trace of the future.
- ⊨ p ⊃ Lp for tense-free p: too late to change what is already the case.
- ⊭ Fp ⊃ LFp: a branch may make Fp true while another branch through α makes it false.
- ⊭ PFp ⊃ PLFp: that it was going to be does not mean it was inevitable.
Thomason claims to have put the alternative futures "into the ontological structure of time" and given rigorous form to the view that future contingents are neither true nor false. The structure is not ad hoc; it is a plausible picture of time.
Future truth without future necessity?
Yourgrau's objection targets clause 3. Thomason uses the branch h as an index to define truth about the future, but then ignores that index when defining necessity: V_α^h(LA) quantifies over all branches through α, as if evaluated from time α alone. So he gets future truth without necessity only by changing the perspective in mid-definition, contradicting Aristotle's principle 1.
White's defence and Yourgrau's reply. Michael J. White objects: (a) with several branches "to the right" of α, the model distinguishes what will happen from what must; (b) "fixed", "determined", "already a fact" can mean mere actuality, not necessity; the model draws that distinction for the future but not the past. Yourgrau on (a): the other branches are futures that, by hypothesis, will not become actual. For Aristotle, what is possible now is what can still yield to deliberation; if Fp is true, ¬Fp is no longer subject to my control. Two senses of "possible" are run together: open to deliberation, and merely might-have-been. On (b): once we assume the future is already a fact, the future branches collapse into one history, and there are no more alternative futures, in the deliberative sense, than alternative pasts. In the might-have-been sense, nothing in the model explains why there are many possible futures but only one possible past. (Could I be myself with a different past? That needs separate arguments, such as Kripke's necessity of origin.)
Aristotle undefeated
Principle 1 stands. If the past is necessary because it is one big unalterable fact to which present truths correspond, then if there are truths about the whole future, the future is also one big unalterable fact. Yourgrau finds no modal fallacy here. Thomason's model also fails to secure the open future in three ways:
- A privileged branch is not excluded. The model's indices include complete world histories, and nothing stops one from being "the" history. Blocking it is like blocking a super-liar ("This sentence is false or neither true nor false") in formal theories of truth.
- Only the present exists? Nothing in the model says so; each time is its own point of reference, which gives at most a relative A-series.
- Gaps are not enough. An open future requires more than that a time's perspective omits the future: we must be unable to widen our perspective to include it.
A completely static world could satisfy Thomason's structure, as it could Stein's light cones. White concedes that to capture Aristotle's "developmental" time "we must envision the model as being viewed from a (dense and continuous) succession of nows along one linear (but not predetermined) branch". Then the model itself is a spacelike trace, and representation must go beyond formalisation to analogy or intuition. (Brouwer says the same of mathematics: even intuitionistic logic cannot represent mathematical activity.)
The correspondence problem
The deepest gap concerns principle 4. The model reflects it: V_β(p ⊃ PFp) = T for all β (White's "Cicero's principle"), yet if contingent p obtains at β, for α < β, V_α(Fp) is undefined. So if yesterday you predicted rain today and it rains, you did not speak truly yesterday. This is a glaring exception to the correspondence theory, which Aristotle himself introduced, and the model gives no reason for it.
For an A-theorist reality "grows by accretion of facts" (Jeffrey): the future is not there to correspond to. Truth states how things are, in the present tense. (Facts about the past concern things that no longer exist, such as the dead: the subject of Chapter 7.)
| Truth-value link (Dummett, White) | Tensed correspondence (Aristotle) |
|---|---|
| What I now say will obtain is true now if it comes to obtain | A future contingent is neither true nor false now |
| White: presupposes the B-theory, so begs the question | The future is not now a fact to correspond to |
| Yourgrau: rests on semantic symmetry: claims about New York are made true by New York, about the past by the past, about other worlds by those worlds, so about the future by the future | Reality grows; truth is confined to what exists now |
A genuine antinomy. The question forces us to choose between a perspective within time and one outside it, though each is natural and unavoidable. We have the future tense, which seems to lift thought out of its confinement to the present; yet on the A-theory time is an incomplete totality, as in some foundations of set theory where correspondence is likewise modified (Charles Parsons). This is why Gödel distrusted tensed existence and chose Kant's resolution of antinomies: idealism.
Myhill again: "metamathematics must be in the final analysis informal; for the process of discussing formalisms by means of another formalism must either terminate in a formalism which is not discussed, or be informal." The formal must be read against the intuitive, not the reverse.
What a formalism cannot see
Three "formally diaphanous" privileges
- The true. In Kleene's strong three-valued table, t and f are "designated", but within the model that label is a mere mark; all values are formal objects on a level. Łukasiewicz's three-valued logic began as an attempt to formalise Aristotle's escape from determinism. Even two-valued tables inherit the problem from Frege's treatment of truth values as objects; Dummett: "it is part of the concept of truth that we aim at making true statements", which Frege's theory leaves out. Church: within formal logic the special significance of the intended interpretation can only be "postulated as different".
- The actual. In possible-worlds semantics, Kripke: "K is the set of all 'possible worlds'; G is the 'real world'". G is merely labelled. Nor can the model express that α might not have been actual; treating that metafact symmetrically starts a regress of superworlds.
- The now. A Minkowski diagram puts "now" at the origin only relative to the diagram's frame.
| ∨ (Kleene strong) | R = t | R = f | R = u |
|---|---|---|---|
| Q = t | t | t | t |
| Q = f | t | f | u |
| Q = u | t | u | u |
Stalnaker (1976) equates "the actual world" with "Reality". Doubly misleading: Reality must include the intersystemic fact that α alone obtains, which α cannot express (Wittgenstein: "It is not how things are in the world that is mystical, but that it exists"), and neither α nor the whole model can express that α might not have been actual.
| Lewis (1973) | Stalnaker (1976) | Yourgrau |
|---|---|---|
| "Actual" is indexical; every world is actual for itself | Only ours is a world; the others are concepts or properties | Distinguish being from existence |
| Modal analogue of the relative A-series | Modal chauvinism | Possible worlds, past and future people, other times are genuine beings; they differ in existential status (existent/non-existent, actual/possible, present/past or future) |
Kripke, the name most tied to the formal model, has said little beyond rejecting Lewis's realism and warning against taking the models too literally. How literally, then? The formal model is silent.
The ontology behind spacetime diagrams
In a Minkowski diagram a particle's history is a world line; by Cartesian coordinates a set of ordered quadruples of reals; by the Frege–Russell definition of number and the Wiener–Kuratowski definition of ordered pairs, a vast unordered set of sets. Minkowski meant the world line not as a path a particle moves along but as a definition of motion. Russell: "When a body moves, all that can be said is that it is in one place at one time and in another at another" (tenseless "is").
If for Plato time is a moving image of eternity, a world line in a Minkowski diagram is a frozen image of time. The flux of becoming and the privilege of the now are not represented, and given the relativity of simultaneity it is unclear they can even be added by intuition.
The branching model fares better: its tree captures part of the open future, and, bracketing relativity, one can read a moving now into it (Yourgrau's Figure 8: a solid line marking how far time has advanced). But then (a) the diagram must be redrawn every moment, or set in motion, needing analogical supplement; (b) progress is still pictured linearly, a partial line always unfolding; (c) the model still cannot answer the three questions: does future truth entail fatalism? Is a future-tensed proposition now true if it corresponds to a future fact? Can a privileged complete branch be ruled out? Gödel's models are the exception: incompleteness shows truth cannot collapse into proof, preserving Church's syntax–semantics line, and the Gödel universe rules out reading t as lapsing time.
Frege and the decontextualisation of thought
"Now" belongs with I, here, this and, more controversially, actual: demonstratives or indexicals, strongly context-sensitive. They trouble Frege's semantics.
Frege, a mathematician by training, sought the foundations of arithmetic, "the most certain part of the most certain science". Unlike Wittgenstein ("A proposition of mathematics does not express a thought"), and like Gödel, he held that mathematics expresses thoughts and yields knowledge. His Begriffsschrift ("a formula language, modelled on that of arithmetic, for pure thought") replaced subject and predicate with function and argument, making truth values the values of sentential functions (concepts). The truth bearer, the thought, must be true in itself, not relative to context; Aristotle, by contrast, held "somebody is sitting" true and later false. Frege: a thought "true today but false in six months"? "No, for it is not the same thought at all."
Frege's five foundational tasks, and their fate
| Task | Frege's solution | Outcome |
|---|---|---|
| 1. Define the natural numbers (settle whether 0 = the Moon) | Numbers as logical objects, via extensions (Grundlagen, Grundgesetze) | Benacerraf (1965): such definitions are arbitrary |
| 2. Explain access to numbers as objects without Kantian intuition of succession | The logical source of knowledge | Frege says almost nothing about it; treats epistemology as psychology |
| 3. Make deductive inference explicit | Quantification theory (Begriffsschrift); induction reduced to logic | First-order part proved complete by Gödel (1930) |
| 4. Derive arithmetic's basic laws from logic | Grundgesetze | Gödel (1931): any consistent formal system for arithmetic is incomplete |
| 5. Account for extensions | Axiom V | Russell's paradox (1902): inconsistent |
Kronecker: "God created the integers, and man has made the rest." René Thom retorts that this reveals more about the banker who grew rich speculating than about philosophy; "the geometric continuum is the primordial entity", and "perhaps, even in mathematics, quality subsists". Russell's paradox arose, in part, from wanting a perspective both within and outside a totality: a quantifier ranging over the extension of a predicate within its own scope.
Sense and reference
"Identity gives rise to challenging questions which are not altogether easy to answer." Wittgenstein: to say two things are identical "is nonsense", and to say one thing is identical with itself "is to say nothing at all". Frege's early view (identity is about signs) changes the subject ("Mark Twain is Samuel Clemens" is about the man) and makes identity arbitrary; the opposite view makes all identities trivial and a priori, yet Hesperus = Phosphorus was a discovery. Since any statement can be recast as an identity (Fa ≡ a = ιx(x = a ∧ Fx), which Gödel used in reformulating Frege's argument that a sentence designates its truth value), everything would go without saying.
Frege's solution splits meaning into sense (mode of presentation, cognitive value) and reference (truth and designation). "Whereof there are two in an identity is material for cognition; whereof there is one concerns truth and reference." The sense of a sentence is a thought.
Magritte's pipe. In The Use of Words I, a lifelike pipe is painted above the words "Ceci n'est pas une pipe". The picture points by resemblance to a pipe; the painted sentence, itself only another painting, denies it. Confined to images, nothing settles which wins: a mere picture always needs an interpretation. Frege concludes that unless something is true in itself, saying what it means, there is no truth at all. That truth bearer, the thought, cannot be physical or mental: it belongs to a third domain that "must be recognized". (Quine's refusal to follow led to the indeterminacy of translation and the abandonment of analyticity.)
Thought, thinking and time
Thoughts are timeless: "It is of the essence of a thought to be non-temporal and non-spatial." Yet words do not always suffice: "If a time indication is needed by the present tense one must know when the sentence was uttered... Therefore the time of utterance is part of the expression of the thought." Yourgrau attributes to Frege a principle of exportation: the pragmatic contribution of context can always be exported into semantic thought content. Hence: "If someone wants to say the same today as he expressed yesterday using the word 'today', he must replace this word with 'yesterday'."
| Gareth Evans (1985): dynamic Fregean thoughts | Yourgrau |
|---|---|
| "Today is fine" expresses a tensed thought, grasped tomorrow as "yesterday was fine" | There are no dynamic Fregean thoughts: thoughts are non-temporal and encountered "as the same thought" |
| Grasp requires an ability to keep track of a day over time | Nor are Evans's thoughts dynamic: for Aristotle "today is fine" may become false; for Evans it stays true, re-expressed |
Frege separates thought content from the process of thinking. Is thinking a natural process like digestion? "My digestion cannot contradict yours." The categories of thought (truth, implication, validity) have no internal relation to nature's (motion, contiguity, resemblance, causation); hence the clash between Frege and the causal theory of reference of Kripke, Donnellan, Kaplan and Putnam. Frege offers no synthesis: grasping a law "is a process which takes place on the very confines of the mental", a mysterious spatial metaphor.
- Thought in clothing. Frege presents "a thought, in itself immaterial, dressed in sensible linguistic form". Better: language embodies thought, as body embodies soul.
- No reciprocal action. Handing over a hammer changes it under pressure; communicating a thought does not. "How does a thought act? By being apprehended and taken to be true." Like the unmoved mover or a Platonic Form, a thought moves by being apprehended.
- Plato's alternative. The Sophist: if knowing acts on what is known, reality "must, in so far as it is known, be changed" (Yourgrau notes an echo of Heisenberg); and "are we really to be so easily convinced that change, life, soul, understanding have no place in that which is perfectly real?" Frege, preoccupied with "reality at rest", neglects reality in motion: time.
A tension in Frege, and a repair
Frege names three modes of cognition, each suited to a domain: we have ideas, sense physical objects, grasp thoughts. Yet officially all reference goes through sense. Then the experience of red is invisible to thought, which reaches only a sense that determines it. Worse: if a sense can be introduced into thought only via another sense, sense₁ needs sense₂ and so on, a vicious regress.
The cure is the other half of Frege's own theory: we can grasp senses directly, and, by the same token, introduce the deliverances of experience into thought simply by having them. For experiences, the mode of presentation is essential (Sklar on secondarisation, Kripke on pain), so a designating sense would change the subject. Having, sensing and grasping are legitimate modes of presentation alongside designation by sense: cognitive relations akin to Russell's acquaintance, not causal ones.
Why Fregean thought sees only the tenseless. If "Column A" (named to echo the A-series) is a tensed access to a tensed reality, then reality in motion reaches us through direct, non-conceptual channels, not through the timeless hierarchy of senses. Plato in the Sophist, says Iris Murdoch, "conjures up a moving knower to follow a moving known". Russell: "introspection is necessary in order to understand the meaning of 'past'... 'past', like 'present', is a notion derived from psychology." This does not prove A-facts exist; it gives them semantic breathing room, should they survive Gödel and Einstein.
Kaplan, Perry and the "inessential indexical"
David Kaplan's non-Fregean theory of demonstratives inspired John Perry's "Frege on Demonstratives": Frege cannot explain both the shifting reference of "now" and the constancy of meaning beneath it.
Perry's account
- Content: "Today is fine" said on day t expresses the ordered pair ⟨t, being fine⟩; content carries information and fixes truth.
- Role: a semantic rule: "today" used on day t refers to t. Roles explain psychological states and action.
- Example: "Today is fine" (on t) and "Yesterday was fine" (on t + 1) share content ⟨t, being fine⟩ but differ in role: only the first makes you put the umbrella away.
D. H. Mellor (Real Time, 1981) applied this to "now": all times are now relative to themselves, so only the relative A-series, the B-series, is real time, and the A-series is a semantic hallucination. A strict Kaplan–Perry approach spatialises its domain: Lewis's indexical "actual" is its modal twin.
Yourgrau's refutation of Perry. A role is a universal rule: "for all times t and speakers s, if s uses 'now' correctly at t, s refers to t". To use it, you must instantiate the quantifier, which requires already having a particular t in mind. By a dated description ("31 July 1990, 8 p.m.")? That is a Fregean sense, which Perry rejects. By taking t to be now? Circular. Perry has mistaken a constraint on any mode of presenting a time with "now" (used at t it must determine t) for the mode of presentation itself. So Mellor's semantic case against the A-series fails too. (Kaplan's defenders reply that indexicals need no prior identification of t: the context supplies it, and the agent need only be in the context. Yourgrau's objection holds only if grasping a role must be a cognitive identification.)
"Thank goodness that's over"
A. N. Prior argued that relief that a pain is over is hard to explain without tense. Mellor's B-theoretic answer: what matters is the date of the pain relative to the token utterance: "thank goodness that's over" is said just after the pain ends; "I wish this pain would end", simultaneously with it.
The temporal fetish. On the B-theory every pain of mine, past, present and future, is equally real, each merely located in time as a pain in my foot is located in space. Someone could have a foot fetish and care specially about pains located in the foot; why should all humans have a temporal fetish, caring specially about pains simultaneous with certain sentences they utter? Mellor's answer misses what Prior pointed to: the human significance of the present moment, which Einstein too admitted "means something special for man".
So the challenges from tense logic and semantics are not conclusive, but the challenge from physics remains strong. Chapter 7 asks the price of following Gödel: if it is too high, we may prefer modus tollens against him to a costly modus ponens.
Being and Time
What do we lose if, with Gödel, we give up successive time? Yourgrau counts the cost in mathematics and in human life. Without A-time we lose potential infinity, the direction (beyond the order) of series, and incomplete totalities; and we lose the distinction between being and existence that makes sense of birth and death. The best model of a mathematics fit for A-time is Brouwer's intuitionism, and time "seems at once to demand and to resist a reconciliation" between its intuitive face and its place in our best objective models.
Time and infinity
All finite cardinal numbers are distinct and simultaneously present in God's intellect.Georg Cantor
Since Aristotle, mathematicians distinguished actual (completed) from potential (always incomplete) infinity, leaving the actual to theologians (Augustine: "every infinity... Is made finite to God"). As late as 1831 Gauss protested "against the use of an infinite quantity as a completed one, which in mathematics is never allowed". A quantity is potentially infinite if it grows without limit or goes on forever, and there A-time enters.
- Kant: "the infinity of a series consists in the fact that it can never be completed through successive synthesis. It thus follows that it is impossible for an infinite world series to have passed away."
- Cantor: the potential infinite appears in a variable finite quantity that increases beyond all limits, such as "that so-called time which is counted from a definite initial moment".
- Mary Tiles: potential infinity "is never fully actual but always in the process of becoming. It is an inescapably temporal notion."
Cantor, by making actual infinity primary, broke the old tie between time and infinity, as Plato had separated time from geometry. Cantorian infinity is opposed to A-time specifically: if time goes on forever, could it have gone on longer? That makes no sense; yet Cantor's infinities are always extendable: 2^α > α for every cardinal α.
A Gödelian disjunction. Nothing essentially incomplete survives in the B-series. So: either time is not ideal, or the only permissible mathematical infinity is the actual one. If time, the form of inner sense, is ideal, then neither nature nor even the mathematical imagination can supply potential infinity as something objective. That threatens intuitionists like Dummett, for whom we reach infinity only "by considering a process of generation or construction which will never be completed". (Gödel's idealism spares relativistic spacetime and Cantor's sets.)
Can anything witness potential infinity?
Jonathan Lear ("Aristotelian Infinity", 1980) proposes that an actual process of division, which stops after finitely many steps, is all a witness to the potential infinite could be: "while such a process is occurring, one can say that the infinite is actually coming to be". He glosses Metaphysics 1048b: the line is potentially infinite "precisely because there are possible divisions that will remain unactualized".
Yourgrau doubts a finite process can witness the infinite. We need a firmer basis for Gödel's "first infinite jump", and access to the modal fact that there will always be more possible divisions than we will ever make. Frege, reviewing Cantor: the potential infinite presupposes the actual, since "the 'receding limit' must have an infinite path if it really is to recede ever and ever further". The same question can be put to branching time: does the shrinking open future presuppose an actually infinite branch? If so, potentialists face the very problem of access to actual infinity they were avoiding.
Gödel is untouched. He grounds Cantor's sets in a noetic, Platonic intuition, an analogue of perception, not in Kantian–Hilbertian space or time intuition (compare Charles Parsons's more Kantian account). If Gödel is right about time, and if we are not to deny every mathematical infinity, his little-followed theory of mathematical intuition gains surprising new support.
Direction versus order
A potentially infinite series has a definite origin and an irreversible direction. A series running into the past from the present would be actually infinite, since all its members must already have elapsed. Hence Kant's First Antinomy: time can be infinite only toward the future. This is the long-running "Kant versus Cantor" debate over an infinite past (Huby, Whitrow, Craig).
Yourgrau's two theses on the infinite past
- Mathematically, potential and actual infinities can coexist, provided each has a sound epistemic basis: A-temporal intuition for the first, intellectual intuition (if Gödel is right) for the second. Cosmology bears on them only through those bases.
- Cosmologically, if time is A-theoretic, the past cannot be infinite (the future can). That is irrelevant to Cantor's sets, but highly relevant to how relativity is interpreted.
Following Russell (1903): a relation R orders a series iff R is connected (for x ≠ y, xRy or yRx), asymmetric (hence irreflexive) and transitive. Its direction runs from x to y when xRy. The naturals are ordered by "less than" and by "greater than"; unless something (such as the A-series) privileges one, there is no the direction. Reichenbach noted that special relativity needs only order; entropic analyses of "the direction of time" also deliver order, not direction.
| Intuitive notion | Set-theoretic reduction | What is lost |
|---|---|---|
| Ordered pair ⟨x, y⟩: x comes first | Wiener–Kuratowski: ⟨x, y⟩ = {{x}, {x, y}}; criterion ⟨x, y⟩ = ⟨w, z⟩ iff x = w and y = z | Direction: {{y}, {y, x}} would serve equally well; and which set the pair "is" is arbitrary (Benacerraf-style; Dipert, Kitcher) |
| Successor of n (Aristotle, Kant, Brouwer: generated in time) | Frege's purely logical successor; the series as actually infinite | Direction; the potentialist could not ignore it, since a potential series' direction is irreversible |
| Iteration in time (Aristotle, Kant) | Polyadic quantification (Frege; Friedman) | The temporal element of "and so on" |
Quine calls Wiener–Kuratowski a "philosophical paradigm": the question "what is an ordered pair?" is dissolved by "clearer notions". Yourgrau notes that Cantor himself took the ordinal, an ordered sequence, as basic: "intimations of the temporal in the very heart of the 'space' of Cantor's sets".
Complete and incomplete totalities
Cantor: a multiplicity whose elements can be thought "as 'being together'" without contradiction is a consistent multiplicity or set; one whose unification leads to contradiction is an "absolute infinite or inconsistent collection". The universe of all sets is not a set: if its cardinal were α, its power set would have 2^α > α members. So the metatheory of set theory needs collections that are not sets, incomplete totalities, called since Gödel proper classes. Can they be grounded intuitively without the A-temporal ideas behind potential infinity and the open future?
Gödel favoured the iterative conception: a set is "attainable from the integers... By iterated application of the operation 'set of'", which "has never led to any antinomy whatsoever". A set presupposes its members' "prior" existence, which suggests time and process. Parsons ("What is the Iterative Conception of Set?") thinks Wang comes too close to actual temporal notions, and urges us to "replace the language of time and activity by the more bloodless language of potentiality and actuality".
Parsons's modal picture
- Objects that can "exist together" (in some possible world) form a set: in any world, {z : Fz} is the set of existent z with Fz there.
- No world is closed under set formation: for any world i there is possibly a set of all objects in i that are not members of themselves, which cannot exist in i.
- The universe of sets is a proper class: essentially potential, "predicative", akin to an intension "constructed by the mind", not to an extension. Talk of all sets is "systematically ambiguous".
Yourgrau's objections. (1) How can set theory be "the very paradigm of a platonistic theory" if the totality of sets is merely potential and mind-constructed? Parsons has leaned toward the idealism he warned against. (2) Is a "potential universe" a universe? (3) Cantor held absolute infinity to be actual, not potential. (4) How can a multiplicity be merely potential without successively coming to completion, which is time again? (5) "Exist together" sounds spatio-temporal; Dedekind already told Cantor "I don't know what you mean either by the 'being together of all elements of a multiplicity' or its opposite."
Yourgrau's alternative follows Wang: the universe of sets "is a multitude (many) and not a unity (one)". In Russell's terms, a set is a class as one, the universe a class as many, a manifold. A manifold of sheep is not one thing collecting them (the flock is in the meadow); it is the sheep. Plato's Parmenides: "when a thing becomes 'the same' as something, it does not become 'one'... If it becomes the same as the many, it must become many." Members of a manifold can enjoy ontological parity, like positions in a space, matching Gödel's "quasi-spatial" sets, and avoiding both mere potentiality and the obscure "existing together".
The modal account of potential infinity
Parsons models potential infinity in an S4 modal logic where every world is finite but each has an accessible world with one more member; converting potential into actual infinity means collecting the worlds' finite segments of ω into one world.
- Two sizes of potentiality. The potential infinity of the finite segments of ω seems "smaller", preceding Cantor's hierarchy; the potential "universe" of sets transcends it entirely. How are two merely potential infinities compared?
- Already actual? An infinite sequence of possible worlds representing all finite segments of ω looks like an actual infinity before any conversion. Denying it requires a merely potential totality of possible worlds: a doubling of modality ("possible possible worlds"), which is not the same as iterated operators like ◇◇p, which Kripke semantics handles via accessible worlds.
- Only figuratively temporal. Parsons likens his logic to a tense logic where the past is necessary, but "realisation" is figurative: "nothing is really happening in his model: all is static, geometric, relational". As with Thomason, the model is indeterminist only if we confine our view to one world at a time.
Being versus existence
If all the possible worlds form a modal manifold with parity, their shared status must be possibility, not existence. Then what are we quantifying over? Yourgrau's answer: distinguish being from existing. Read ∃x not, with Quine, as existence, but as being: being something that, if contingent, can come to exist and cease to exist.
This is not Meinongian. Kripke (1963) "got into trouble" saying Sherlock Holmes does not exist but might have; that invokes a naive semantics on which all names, fictional ones included, refer. Yourgrau's thesis is "Parmenidean", about objects, not names. Kripke's Addendum separated the two: he could no longer say Holmes might have existed, but "some of the entities which actually exist might not have existed, and there might have been entities other than those which actually exist".
Birth and death
Like a running grave, Time tracks you down.Dylan Thomas
Death is the unavoidable deliverance of intuitive time, and Reichenbach observed that "the fear of death is thus transformed into a fear of time". Any serious philosophy must save this phenomenon. When we die, our survivors still grieve for us, not for nothing; yet the dead have not "moved into the past" like changing residence (Harry Silverstein's spatialised reading). Aristotle: "for living things, it is living that is existing."
Yourgrau's four "alarming" propositions
- The dead don't exist. (From Aristotle in one step; Kaplan agrees: "past individuals are also, in my view, nonexistent.")
- The dead cannot be buried. What does not exist is not there to be shovelled under; you bury something, but not me. Socrates to Crito: "Any way you like, if you can catch me."
- We don't really know where we (as babies) came from. That babies come from the womb is a fact about bodies.
- Being ≠ existence. Birth and death are changes in one subject, Socrates, from non-existence to existence and back; change needs a persisting substratum. So the being of Socrates is not the existence of Socrates.
The logic of being and existence
- Let the quantifier ∃x express being ("there is such a person").
- Add a primitive existence predicate E!.
- "Socrates is (now) dead" implies, by existential generalisation, ∃x(¬E!x): there is someone who does not exist.
- Kaplan ("Afterthoughts", 1989) proposes "a narrow existence predicate to distinguish the robust being of true local existents... From the more attenuated being of the nonexistents".
- Plato, Parmenides 161e: a one that does not exist "must in some sense even possess being... Otherwise, we should not be speaking the truth in saying that the one does not exist."
Once the dead are non-existent objects, other non-existents follow: the unborn, the never-to-be-born, merely possible worlds, past and future times. Hence an ontological symmetry of categories: future people are people (not concepts or names), past people are people (not memories), possible worlds are worlds (not sets of sentences, against Stalnaker). Kant: "A hundred real dollars do not contain the least coin more than a hundred possible dollars." Yet this is not spatialisation, since mode of being (existence vs non-existence) still distinguishes them: a course between the Scylla of Lewis and the Charybdis of Stalnaker.
Physics is not ontology. The logic of human beings is not the logic of bodies. We are not bodies but are embodied. (As the sentence embodies the thought in reconstructed Frege.) What we need is a logic of embodiment (first steps: Strawson's Individuals; Montgomery Furth; Cherniss for Platonism). Modal logicians rarely take individuals seriously: Kripke defends transworld identity against Lewis but declines to say how literally to take worlds; when charged with metaphysics, logicians answer "mathematics!". Kaplan goes further: individuals are not purely qualitative, so the world "might be in a state qualitatively exactly as it is, but with a permutation of individuals": the sign of serious metaphysics.
Birth precedes death
Being and existence still leave out direction. Living takes place in A-time, and birth precedes death; that is why we fear death and not prenatal non-existence, a fact invisible to spatialisers like Silverstein and Mellor. On B-theoretic order alone, pre- and post-natal non-existence are symmetric; the A-direction of a life pointed toward its death, like that of a potentially infinite series, is irreversible. So, as Plato and Nietzsche insist, how we die matters; the end outweighs the beginning, and in the Phaedo philosophy is the "practice of dying".
Gödel's theological worldview
Gödel's four long letters to his mother (1961), not meant for publication, spell out his "theological worldview": "the world and everything in it has meaning and reason, and in particular a good and indubitable meaning". Hence:
- This life is a means. "Our worldly existence, since it has in itself at most a very dubious meaning, can only be means to the end of another existence."
- Birth as precedent. "Since we came into existence one day in this world without knowing how and whence, the same can happen again in the same way in another world." This echoes the Phaedo: "the living have come from the dead no less than the dead from the living."
- Wasted potential. "What would be the point of bringing forth an essence (the human being) that has so wide a range of possible developments... But is never allowed to realize one thousandth of them?"
- Mind keeps developing. Mind "is not static but constantly developing... There is no reason why the number of mind's states should not converge to infinity."
- Learning hereafter. "The greater part of learning will take place in the next world", and we "could very well be born with latent memories".
So Gödel's papers on Einstein give only half his philosophy: the theological perspective complements the spacetime one. "We do not know what we are (namely, in essence and seen eternally)." Plato's subject, says Cherniss, is "a mean between" ideal being and phenomenal becoming; no one from Plato and Aristotle (the mind's form is formlessness) to Kant (phenomenal and transcendental ego) and Wittgenstein ("the subject does not belong to the world") has dissolved this intermediate self.
The meaning of eternity
For Plato and Gödel, eternity is genuine being, not long-lasting life. So Wittgenstein's protest ("is some riddle solved by my surviving forever?") misses: it is death that closes off development and introduces the riddle, and what matters is development toward perfection and transfiguration. Humans are born with the greatest gap between what they are potentially and actually; a perfect child is one on the way to adulthood, not a Peter Pan, and a perfect adult points beyond. Heraclitus: "Man is called a baby by God, even as a child by a man."
| Gödel (and Plato's "threshold" view) | Wittgenstein (and Plato's "indwelling" view) |
|---|---|
| Transcend time by crossing death's barrier | Escape time within time: "a man lives eternally if he lives in the present" |
| Life a preparation; philosophy the practice of dying | Each moment self-contained, "an end in itself", like a work of art seen sub specie aeternitatis |
| E. N. Lee: nous as release from wandering into the company of Forms, "the exile's view" | Lee: nous as a sphere rotating about a fixed centre, "the real as perspectivally neutral" (Timaeus, Laws X; Parmenides' sphere) |
| Stresses the partiality of eternity within time | Aristotle's energeia (complete activity, like Glenn Gould contemplating the Goldberg score) versus kinesis (hunting for the score) |
The two perspectives are complementary: there is no point crossing a threshold unless you are ready to dwell beyond it, and good reason to make our present dwelling like the next. Pascal: "We never keep to the present... Thus we never actually live, but hope to live."
An unreconciled tension. How can Gödel combine a Platonic picture of gradual this-worldly preparation over time with a doctrine that time is ideal? For Plato time has a lesser, reflected being; Gödel cites Parmenides, for whom it has none. Kant's transcendental self might mediate, but Gödel disliked the Tractatus, which echoes it. Yourgrau suggests the answer may lie in Leibniz, the philosopher Gödel admired most and wrote least about. It remains open.
Decontextualisation and its limits
The fundamental Platonic problem is reconciling the ideal with the real. Lived temporal reality points to ideals that make sense of it but would be paradoxical if realised. Plato separated the Ideas, yet wrote only dialogues: we see the Ideas only as seen by us. Cherniss: Parmenides, transcending human discourse, "speaks the language not of mortal men but of gods"; for Plato, dialogue is "an externalization of the process of thought".
Four cases of the ideal and the real
- (a) Geometry. Pure forms are the true subject and even the fabric of matter (Timaeus), yet invisible: if only the smile of the Cheshire cat remains, the cat vanishes.
- (b) Human life. Our lives make inadequate sense if forever incomplete; as we "see" that the numbers never end, or that slightly converging lines must meet, Gödel "sees" lives completing themselves from the theological viewpoint. Yet perfection would make us kin to lifeless Ideas, killing the erotic drive (Symposium): "true philosophers make dying their profession".
- (c) Language. Sentences are bodies of thoughts; but how could we ever compare a material sentence with an immaterial thought to learn their link?
- (d) Mathematics. Platonism separates mathematical Forms to make sense of practice; at the cutting edge they elude us: the continuum hypothesis stays undecided after Gödel (1938) and Cohen (1963), despite Gödel's search for new axioms. Brouwer's contextualised mathematics enriches the dialectic.
The continuum hypothesis in 2026. CH remains independent of ZFC. Programmes for new axioms continue: Woodin's Ω-logic case against CH (later qualified) and his Ultimate-L programme, which would favour CH; and Asperó and Schindler's 2021 proof that the forcing axiom Martin's Maximum++ implies Woodin's axiom (), which gives 2^ℵ₀ = ℵ₂. Opinion is split between those seeking a decisive new axiom and pluralists* (Hamkins's multiverse) who think CH has no determinate answer. Gödel's hope for a compelling new axiom has not yet been met.
Temporal mathematics
Since Kant, two currents have run in opposite directions. In pure mathematics, time was purged: Hilbert formalised geometry, Frege rebuilt arithmetic in tenseless logic (the tense-free successor), Cantor founded a hierarchy of completed infinities. In applied mathematics, this purified instrument was imported into physics, culminating in Einstein's geometrisation of time. The physicists of time have been using an instrument designed by the mathematicians expressly to be insensitive to time.
Desiderata for a mathematics of A-time
- Incomplete totalities in its domain: the open future, potentially infinite series, proper classes.
- No unrestricted universal quantifier: no Parmenidean or Wittgensteinian "all that is the case", no Fregean absolute "all".
- Direction beyond order, as in the Aristotelian–Kantian succession of numbers.
- Modified classical logic: restricted bivalence, tensed correspondence (as Aristotle, and Parsons for proper classes).
- Complete formalisation abandoned as an ideal, and limit cases ruled out in principle.
Brouwer: putting time back
Brouwer's intuitionism meets all five most strikingly. The First Act: mathematics is "an essentially languageless activity of the mind having its origin in the perception of a move of time, i.e. Of the falling apart of a life moment into two distinct things, one of which gives way to the other, but is retained by memory". The empty form of this "two-ity", stripped of quality, is "the basic intuition of mathematics".
- A-theoretic, not B-theoretic. Only potential infinity (B-structures are complete); genuine succession (none in the B-series); irreversible direction from a closed past to a free future. That Brouwer never says "now" is beside the point.
- The Second Act: generating "infinitely proceeding sequences p₁, p₂, ... Whose terms are chosen more or less freely", with restrictions that "may be made to depend on possible future mathematical experiences of the creating subject". A mathematical entity "in its state of free growth, at some time acquire[s] a property which it did not possess before".
- No truth before experience. "Expected experiences... Are true only as anticipations and hypotheses; in their contents there is no truth." Against Dummett's suggestion that a constructivist can hold proved statements "true all along", this was not Brouwer's view.
- No decontextualisation. Unlike Frege, Brouwer will not separate thought content from the act of thinking: "in mathematics no truths could be recognized which had not been experienced."
Yourgrau's analogy. As Brouwer's intuitionism stands to its formalisations (Heyting's intuitionistic logic), so intuitive A-time stands to its spacelike projections: branching tense logic, or Minkowski world lines. W. W. Tait doubts Brouwer's "move of time" has "any real content"; Yourgrau's concern is intuitionism as Brouwer conceived it.
Complementarity
For a mathematical space like Cantor's universe of sets, or Einstein's "rigid four-dimensional space", a Platonistic philosophy fits. Perhaps the two views are complementary. Paul Bernays (1946), on Gödel: as the observer is essential in the new physics, "the role of mental activity cannot be fully abstracted from" in mathematics, and the "objective, theoretical (existential) standpoint" and the "intuitionistic (constructive)" one "could be regarded as 'complementary' aspects".
That still leaves time unreconciled: "that mysterious and seemingly self-contradictory being". Time "seems at once to demand and to resist a reconciliation between its subjective, intuitive manifestations and its appearance in the best models that have been offered to date." By building the Gödel universe, Gödel raised an informal question to a new level of clarity, waking us from our dogmatic (relativistic) slumbers.
The first prerequisite for learning anything is thus utterly lacking: I mean, the knowledge that we do not know.Frege, on number; Yourgrau, on time
Brouwer and the 'Revolution': The Philosophical Background of 'Temporal Mathematics'
Potential infinity is an essentially temporal notion, and the time it presupposes is McTaggart's A-series; so Brouwer's rejection of Cantor's actual infinity presupposes A-time. Yourgrau asks what kind of time Brouwer meant, and finds it close to Husserl's internal time-consciousness, Wittgenstein's "form of time" and Gödel's intuitive time: a time disjoint from the time of physics. None of these idealists ever located their intuitive time relative to relativistic spacetime; Gödel seems to have wanted to, but death intervened.
Und Brouwer, das ist die Revolution!Hermann Weyl, 1921
Absolute infinity: actual or potential?
Having proved Cantor's theorem, Cantor could not call the whole hierarchy of sets a universal set; it is an "inconsistent multiplicity", today a proper class, whose cardinal is "absolutely infinite". Absolute infinity is no member of the hierarchy of transfinite cardinals; for Cantor it is theological, not mathematical. Is it a kind of actual infinity?
- Yourgrau: yes; how can sets be fully actual Platonic objects if their totality is not? Those who call proper classes potential owe a non-temporal account of that potentiality (non-temporal, since actually infinite collections cannot be generated in time).
- Shaughan Lavine (Understanding the Infinite, 1994) agrees that Parsons's modal reading fails: "Without the temporal notion of constructing some sets after others the iterative conception loses much of its appeal... It is not yet, however, clear how to interpret the necessary notion of possibility without relying on metaphors of time and construction."
- Ignacio Jané (1995) argues Cantor changed his mind: the absolute, once "a kind of the actual infinite", became incompletable, existing "only potentially". Yet Jané admits "it is difficult to make sense of the potential infinite... Without appealing to activities or processes taking place in some time-like medium", and that Cantor "remained silent about how to make sense of irreducible potentiality".
Popper on Brouwer's Kantianism
Popper ("Epistemology Without a Knowing Subject") notes that Brouwer followed Kant on mathematics but dropped Kant's space intuition, refuted by non-Euclidean geometry, and, via the arithmetisation of geometry, founded everything on Kant's time intuition. Popper objects, as this book does: "If we say that Kant's theory of space is destroyed by non-Euclidean geometry, then we are bound to say that his theory of time is destroyed by special relativity." Brouwer might never have developed intuitionism had he known the analogy; but he "could have retained his own theory of a personal time", unaffected by relativity.
| If "personal time" is introspective time | If it is Brouwer's "primordial temporal awareness" |
|---|---|
| Subject to empirical psychology | Brouwer distinguishes intuitive time from "scientific time", which is "definitely a posteriori" |
| Which cannot challenge relativity | "The Intuition of Time... Is a measureless one-dimensional continuum" |
| Popper's escape works trivially | Shared with Bergson: science imposes a spatial frame, "a spatialization of time"; real time is "pure fluidity", grasped by intuition |
So Brouwer's time is disjoint from the time of physics: it lies in an aspect of reality with no internal relation to the empirical world studied by relativity or psychology.
Husserl, Wittgenstein, Gödel: four phenomenologies of time
- Husserl: phenomenology requires "the complete exclusion of every assumption, stipulation and conviction with respect to objective time"; "one cannot discover the least thing about objective time through phenomenological analysis"; the order of experiences "cannot be met with in the empirical, objective order". His analogy: it makes no sense to say a point of the visual field is one metre from the corner of the table; "the house appearance is not next to the house".
- Wittgenstein (Philosophical Remarks): "possibility in mathematics is precisely the same as it is in the case of time". Time is "not an infinite reality, but infinite in intension"; "infinity is an internal quality of the form of time"; "the infinity is only in the rule". Not a prediction that we could always add another element, but part of the grammar of time that it never admits completion. And: "Perhaps this whole difficulty stems from taking the time coordinate from time in physics and applying it to the course of immediate experience." If memory is no seeing into the past, it "contradicts every concept of physical time that I should have perception into the past". He told Drury his work could be called "phenomenology".
- Brouwer, per van Stigt: the primordial intuition is "the simultaneous presence in consciousness of two sensations, one recognized as past, the other as present".
- Gödel: "It is the role time plays in the description of the outer world which is similar in Kant and in relativity theory, while its relation to inner sensibility does not concern physics at all." "There is no such natural coordinate system for the mind; time is the only natural frame of reference." "Time is no specific character of being... I do not believe in the objectivity of time. The concept of Now never occurs in science itself." An entity corresponding to intuitive time "in relativity theory... Exists only in our imagination".
Potential infinity is not weakened actual infinity. For those who reject the actual infinite, an infinite series is understood intensionally, as a rule, not extensionally as a stock of objects to which one keeps adding. That is why Wittgenstein's "form of time" is not the temporal dimension of the physical world.
Brouwer the idealist
Van Stigt: Brouwer's philosophy of the physical world "can with some reservations be described as idealism", sharing features with Platonism, Berkeleyan solipsism and Kant. Mathematics is "an 'idealized' reality... A reaching for the perfect reality", but "a reality whose exclusive seat is the individual human mind". "Unlike Plato he held that perfection can be achieved without abandoning time." "No objectivity is claimed for mathematics by Brouwer."
| Plato and Gödel | Brouwer | |
|---|---|---|
| Mathematical world | A higher level of existence | A higher level of existence |
| Its location | Timeless, "itself by itself" | The individual, temporal mathematical mind |
| Objectivity | Yes | None claimed |
| Relation to time | Perfection lies outside time | Perfection achieved in time |
Brouwer's time has the A-series features. Van Stigt: the subject "is imprisoned in time", surveying only past constructions, lacking "the divine power of overseeing the whole of time from the outside". "What any formalization must fail to grasp is the dynamic aspect of the present and the special indeterminacy of the future." "The present is the only real moment... The present is the dividing point between determinacy and freedom." The free future's "dimension transcends the denumerable". Like music, mathematics "only exists in time... It is distinct from the musical score".
The unfinished task. Neither Brouwer nor any idealist of his kind (Husserl notwithstanding) ever systematically located intuitive time relative to the time of the empirical world, the temporal component of relativistic spacetime. Gödel's own idealism is double: Kantian about time, realist about spacetime, if he accepts relativity's perspective; but insofar as he adopts Leibniz's monadology, he rejects empirical science's perspective altogether. Gödel seems to have wanted to complete the task late in life; we do not have his answer.
Zeno's Revenge: On the Mathematical Dogma of Infinite Divisibility
The received view is that Cauchy and Weierstrass refuted Zeno: 1/2 + 1/4 + 1/8 + ... = 1. Yourgrau argues that the limit concept evades Zeno rather than answering him, and that if the rationals in [0, 1] form an actually infinite totality, an "actually infinite sum" of Zeno's series would, by Cantorian reasoning, be ℵ₀, not 1. His radical conclusion: either drop the actual infinity of the rationals (siding with Aristotle and Brouwer) or separate the line from its points, as Gödel suggested. The argument is provocative and, on standard mathematics, flawed; both sides are set out below.
Where the nonsense starts is with our habit of thinking of a large number as closer to infinity than a small one.Ludwig Wittgenstein
The dialectic of the formal and the intuitive, writ large
Yourgrau extends Wang's phrase to Western culture. The formal side is usually the quantifiable.
| Domain | Formal | Intuitive |
|---|---|---|
| Ethics (Euthyphro) | What is loved, chosen; the letter of the law | The good itself; the spirit of the law |
| Politics (Republic) | Majority rule, counting votes | Rule of the good in the Kallipolis |
| War | Orders | Their justification |
| Economics (young Marx) | Market price | Human value |
| Relativity | Clock time | Internal time-consciousness |
| Mathematics | Proof, algorithm, number, discrete, finite, actual infinite | Truth, insight, line, continuous, infinite, potential infinite |
Formalism is the leitmotif of the twentieth century: the computer (Gödel: "a formal system made concrete"), structuralism (Chomsky, Lévi-Strauss, Frye), cubism, Schoenberg's twelve-tone method, Glenn Gould's structural Bach. Gödel's theorem is the central formal result about formalism; the Gödel universe, Yourgrau adds, is a second. The Euthyphro question is a litmus test for all realisms: Wittgenstein says 2 + 2 = 4 because we say so; Gödel and Frege, that we say so because it is so. (Gould and Gödel were both hermits seen bundled up in winter clothes in summer, and Gould scored the time-travelling film Slaughterhouse-Five.)
Gödel counted himself a logicist with Frege, taking set theory as logic and its axioms as analytic in the sense of true by the content of concepts, grasped by mathematical intuition. Church's thesis, which reduces the intuitive notion of algorithm to recursive functions, is another attempted reduction across the dialectic; Gödel was ambivalent about it (Martin Davis, "Why Gödel Didn't Have Church's Thesis").
The orthodox refutation of Zeno
Zeno of Elea, Parmenides' follower, argued that the sum of infinitely many finite parts must be infinite. The orthodoxy holds that the ε–δ limit and Cantor's transfinite numbers dispose of him.
Yourgrau's four theses
- (a) Modern foundations have not refuted Zeno; they have evaded him.
- (b) If the infinite totality of rationals actually exists, Zeno was right: Σ 1/2ⁿ = ℵ₀, not 1.
- (c) So the intuitionists' scepticism, Aristotle to Brouwer, about the actual infinity of the rationals gains unexpected support.
- (d) The "size" of the first transfinite number ℵ₀ has been seriously underestimated, as Paul Cohen stressed that Cantor underestimated 2^ℵ₀, which he hypothesised to be ℵ₁.
Gregory Vlastos (Encyclopedia of Philosophy, 1967) says Zeno assumed (c1) an infinitely divisible existent contains a complete infinity of parts and (c2) if each member of an infinite set has some size greater than zero, the aggregate is infinite. Zeno lacked "the conception that the sum of an infinite series could be defined as the limit on which its successive partial sums converge". Tobias Dantzig (Number): once convergence and limits are accepted, Zeno's argument "loses its force".
The limit is not a sum
Σ 1/2ⁿ = 1 means: for every ε > 0 there is an N such that for all m > N, |1 − Sₘ| < ε, where Sₘ is the sum of the first m terms. Zeno asked how all the infinitely many parts could be added without yielding something infinite; the answer "we refuse to add them all" avoids the question.
- Wittgenstein: "the sum 1 + 1/2 + 1/4 ... Is of course a limit of a series of sums, and not a sum in the sense in which, e.g., 1 + 1/2 + 1/4 is a sum."
- Konrad Knopp (Theory and Application of Infinite Series): the value "is not a sum in any sense previously in use, but only the limit of an infinite sequence of sums"; better to speak of the value of the series.
- Abraham Robinson: the ε–δ method did not force its users into a position on actual versus potential infinity. Joseph Dauben: it "re-established analysis on acceptably finite terms".
Cauchy and Weierstrass laid to rest Newton's infinitesimals, Berkeley's "ghosts of departed quantities", but not "the ghost of Zeno". If Newton equated the indefinitely small with zero, Dantzig and company conflate an arbitrarily large finite sum with an actually infinite addition.
Knopp's shillings. Knopp gives someone 1 shilling, then 1/2, 1/4, 1/8... And, if he "never comes to an end", asks whether the fortune grows without bound. No: every partial sum sₙ = 2 − 1/2ⁿ⁻¹ < 2. Yourgrau: if he never comes to an end, he gives all infinitely many gifts, an actually infinite sum the limit definition does not touch. One cannot have it both ways.
Decomposition implies reconstitution
The current position is unstable: all the rationals between 0 and 1 actually exist, yet we cannot add them all. Aristotle was coherent: rationals (ratios) and divisions exist only potentially; a line potentially consists of finitely many sub-segments; division can always go further but is never completed. Vlastos even grants Aristotle would have accepted (c2). Mary Tiles: when reals are defined as limits of rational sequences with the intended geometric reading, "the actual infinite must be accepted" if every point on the axes is indexed by a number.
Thesis 1: what can be actually infinitely decomposed can be actually infinitely reconstituted. If [0, 1] actually falls apart into 1/2, 1/4, 1/8, ..., the parts constitute the whole, so it must be possible to glue them all back. The limit theory declares that actually infinite addition senseless. The coffee cup: if a cup falls into infinitely many pieces, every piece must literally be reglued; from the fact that the limit of "partial gluings" is the whole cup, nothing follows about gluing all the pieces at once.
Why ℵ₀? The array argument
Mark Sainsbury (Paradoxes, 1988) answers Zeno differently: "it is essential to the idea of infinite divisibility that the parts get smaller without limit". Yourgrau finds it implausible that Zeno missed this; Vlastos even attributes to Zeno (c3) "an infinite sequence of terms of decreasing size must have a smallest term", which entails (c4) a last term, contradicting Zeno's own text.
Sainsbury's reply implies that series of constant terms, Z₁ = 1/2 + 1/2 + ... or Z₂ = 1 + 1 + ..., have infinite actual sums. Cantor's arithmetic agrees: Fraenkel: "1 + 2 + 3 + ... = ℵ₀... The proper addition of infinitely many numbers, as opposed to the improper addition (by means of the limit concept)"; ℵ₀ · n = ℵ₀.
Every row (a term of Zeno's series added to itself infinitely often) is infinite. The first column is Zeno's series. Why treat rows and columns asymmetrically?
- The cardinal defence (the terms become vanishingly small): every term of Z is finite, not infinitesimal, and for each there is a row of copies summing to ℵ₀. Why should combining different finite terms give less? "How could the mere smallness of the addenda speak to the size of their infinite sum?"
- The ordinal defence (partial sums converge to 1): in finite sums, more terms are offset by proportionally smaller terms. In an actually infinite sum the number of terms becomes infinite but no term shrinks to the infinitesimal, so the offsetting proportion disappears. Both claims hold: the limit of Z's partial sums is 1, and its actually infinite sum, if it exists, is infinite.
- No approach from below. Knopp says Z₂'s partial sums "approach infinity"; Yourgrau, with Wittgenstein, denies it: every finite partial sum is as far from infinity as every other. "We can only grasp the totality of numbers at one stroke." Gödel's "first infinite jump" to the totality of naturals dwarfs later jumps such as impredicative definitions. Against Lavine's "immortal mathematician" who counts forever: stepwise knowledge of infinity is no knowledge of it. Gödel: "We recognize possibilities in our minds in the same way we see objects with our senses."
Critical lens: the ℵ₀ argument does not survive standard mathematics. (1) Cardinal arithmetic adds cardinals, not lengths. ℵ₀ · n = ℵ₀ concerns how many elements; 1/2 is not a cardinal, and "1/2 + 1/2 + ... = ℵ₀" mixes how many with how much. In the extended reals the constant series diverges to +∞, which is not ℵ₀. (2) Measure theory answers Zeno directly. Since Lebesgue (1902), length is countably additive: if [0, 1) is the disjoint union of [0, 1/2), [1/2, 3/4), ..., its length is exactly the series' value, 1. The infinite sum here is not a finite approximation but a theorem about the whole, and Adolf Grünbaum (Modern Science and Zeno's Paradoxes, 1967) made this the standard reply. (3) The array is consistent. By Tonelli's theorem for non-negative terms, summing the array by rows or by columns gives the same total, +∞ either way: each column converges to 1, and there are infinitely many columns. No asymmetry, no contradiction. (4) Yourgrau's point that a limit is not a finite sum is correct and well taken (Knopp says the same), and so is the observation that the arithmetisation of the continuum carries philosophical commitments. But the claim that the rationals cannot be actually infinite does not follow. Few mathematicians or philosophers of mathematics accept theses (b)–(d).
What has to give
If the rationals in a finite interval form an actual infinity reached by pointwise decomposition, and the limit theory forbids reconstitution by actually infinite addition, Yourgrau sees an outright contradiction once the actual sum is ℵ₀. Options:
- Brouwer: reject the completed, pointwise-decomposable continuum for a "fluid", free-choice, never-completed one. Van Stigt: "Brouwer's intuitive continuum is a medium in which points can be inserted, a potential for 'cutting'"; "the scale is not the continuum itself"; it "shares the dynamic character of the 'time continuum'".
- Gödel: "In space intuition, a point is not a part of the continuum but a limit between two parts... All the points do not add up to the line but only make up a scaffold (Gerüst) or a collection of points of view." An Aristotelian idea.
- Not acceptable: simply identifying, or putting in one-one correspondence, the actually infinite rationals (or reals) with the points of the line.
Paul Bernays to Wang: one "should not overestimate the philosophical relevance of the possibility of embodying classical mathematics in set theory"; for the intuitive sources we "keep to the old dualism of arithmetic and geometry"; "even the concept of the number series is geometrically motivated. From the strictly arithmetical point of view the progress of numbers is only a progressus in indefinitum."
Yourgrau stresses that limits remain fine as "mathematical engineering", a calculating device; what he questions is limits as a metaphysics of the calculus. He answers a colleague who said he had made "most mathematics" unintelligible: he has not doubted infinite sums (Cantor makes sense of them), nor limits, nor decimal expansions, only the joint acceptance of Cantor's arithmetic and of limits as the way to reconstitute a decomposed continuum. (He does say the familiar proof that 0.999... = 1 fails if 0.999... Is not read as a limit, which few would grant.)
Recapitulation
| Convergent series | Divergent series | |
|---|---|---|
| Example | 1/2 + 1/4 + 1/8 + ... | 1/2 + 1/2 + 1/2 + ... |
| Cantor/Dedekind treat its "sum" as | Improper: a limit, lim(1/2, 3/4, 7/8, ...) = 1 | Proper: ℵ₀ · 1 = 1 + 1 + ... = (1/2 + 1/2) + ... = ℵ₀ |
| Yourgrau's charge | Cantor mixes finitary limits (defining each real) with infinitary actual sets (the reals as a whole) | Cantor rejects the infinitely small (no infinitesimals as reciprocals of transfinites), yet wants the line actually decomposed |
The three theses of the recapitulation
- Decomposition implies reconstitution: if [0, 1] falls apart into all its rationals, it can be reassembled by an actually infinite addition.
- The proper sum of Zeno's series is ℵ₀ (its limit is 1). The illusion of 1 comes from extrapolating the finite inverse relation (more terms, smaller increments) to the infinite case.
- So the rationals are at most potentially infinite, the rational line and the real continuum cannot be actually infinitely decomposed, Brouwer's continuum gains support from the "safe" rationals, and Zeno's thesis is confirmed. (Robinson's non-standard infinitesimals, built on Gödel's compactness theorem, are an entirely different notion.)
Since potential infinity is tied to time, and time, even after Einstein and Husserl, remains an enigma, perhaps for Gödel the enigma, "the infinite is far from having yielded to us its fondest secrets". Plato first drove time out of mathematics in the Republic; Cauchy, Weierstrass, Cantor, Dedekind and Frege seemed to finish the job. Yourgrau thinks Frege and Gödel came close to fulfilling "Plato's Program" of an unhypothetical foundation for mathematics, and that Zeno's ghost still walks.
Yourgrau's exercise for the reader. If "the nonsense starts with our habit of thinking of a large number as closer to infinity than a small one", how is it that children do learn infinity by considering large numbers, and that mathematics can often simulate the infinite with very large finite numbers (Lavine's "theory of zillions")? One answer: the large numbers are not stepping stones toward ℵ₀ but occasions for grasping, at one stroke, the rule that there is always one more.
Master Takeaways
Yourgrau's book is a sustained defence of Gödel as a philosopher: his relativistic cosmology is a limit case that forces a choice between intuitive time and Einstein's geometrised time. The decisive idea is that time travel and time cannot coexist: closed timelike curves make every moment equally real, and a time in which nothing comes into being is not time.
- Where there is time travel, there is no time. In a universe with closed timelike curves every moment is already complete; nothing comes into existence successively, so there is no objective lapse, and strictly no "travel" either.
- Existence cannot be relativised. If the passage of time is a change in what exists, a frame-relative now makes existence frame-relative, which Gödel calls destroying the concept. Hence special relativity already leaves no objective lapse of time.
- The relative A-series is the B-series. "Now-relative-to-t" means "simultaneous with t". Every attempt to save becoming by making each time present to itself keeps only before and after.
- The Gödel universe is a limit case. It is a genuine solution of Einstein's equations in which no cosmic time exists and CTCs pass through every event; it makes visible what milder geometries hide.
- The modal step carries the weight. Same laws, same experience, different distribution of matter: if lapse is essential to time (□(T → L)) and must be grounded in laws, not in the layout of galaxies, the Gödel universe shows time is ideal here too. It relies on a verificationist bridge (indistinguishable states are objectively equal), which Gödel takes from Einstein.
- Big T versus little t. Incompleteness showed proof cannot capture Truth and Gödel blamed the formalism; the Gödel universe showed relativity cannot capture time and Gödel blamed intuition. Why the asymmetry is the book's sharpest open question.
- Paradoxes are consistency constraints. In closed time past, present and future are equally factual: nothing can be changed, attempts to kill one's earlier self always fail, and the line between laws and initial conditions blurs (Sklar).
- Circles are not cycles. A time circle has no "how many times round?"; Star Trek, Lightman, Wheeler and Popper picture repetition and so miss Gödel's point.
- Spatialising time means ontological neutrality. The essence of a space is that location does not affect existence; to spatialise time is to deny the present any existential privilege.
- Geometry cannot supply becoming. A rigid four-dimensional block satisfies every constraint of relativity; light-cone structure gives order, not flow (Stein, Čapek, Shimony fail to add it).
- Hawking's chronology protection (1992) is the "anti-Gödel law" Yourgrau predicted in 1991; it remains a conjecture, and the Gödel universe is ruled out by observation and energy conditions, not by any accepted law.
- Our universe does not rotate measurably, so it is not a Gödel universe; Gödel's argument never needed it to be.
- Formal models of the open future cheat. Thomason's branching logic gets future truth without necessity only by dropping its branch index when defining necessity; Aristotle's principle that present truth about the future brings necessity stands.
- Privileged positions are formally diaphanous. The true, the actual and the now can only be labelled in a model, never shown to be privileged (Church, Kripke, Minkowski diagrams).
- "Now" is not explained by a rule. Perry's role ("now" used at t refers to t) can be applied only by someone who already has t in mind; Mellor's B-theoretic semantics inherits the flaw, and Prior's "thank goodness that's over" survives.
- Potential infinity is temporal. It is "never fully actual but always in the process of becoming": abandon A-time and only actual infinity remains.
- Order is not direction. Set-theoretic reductions (Wiener–Kuratowski pairs, Frege's successor) preserve order but lose the direction that potential series, and lives, possess.
- Being is not existence. The dead do not exist, yet there is someone who is dead: ∃x(¬E!x). Birth and death are changes of a persisting subject.
- Direction explains the fear of death. We fear our future non-existence but not our prenatal non-existence, an asymmetry invisible to the B-theory.
- Brouwer's intuitionism is the mathematics of A-time: potential infinity, free-choice sequences, no truth before experience, and a continuum "in which points can be inserted".
- Limits are not sums. The ε–δ definition of Σ1/2ⁿ = 1 sidesteps Zeno's question about adding all the parts; but measure theory's countable additivity answers it, and Yourgrau's claim that the "proper" sum is ℵ₀ fails.
Frameworks & mental models
| Name | What it says | Use it when |
|---|---|---|
| Limit case | Build an extreme formal model that rules out an intended interpretation of the formalism | Testing whether a formal model really captures an intuitive concept |
| A-series / B-series / relative A-series | Past–present–future with a moving now; fixed before–after; "now-for-t" (= B-series) | Diagnosing whether a theory of time includes passage or only order |
| Five concomitants of the A-series | Privileged now; irreversible lapse; non-neutral positions; open future; absolute simultaneity | Checking what a model of time leaves out |
| Gödel's four steps | STR removes the now; GTR restores cosmic time; Gödel universe removes it; modal step | Reconstructing the argument for temporal idealism |
| Modus ponens vs modus tollens | The same conditional can refute the antecedent or prove the consequent | Judging whether an odd consequence refutes a theory or reveals it |
| Circle vs cycle | Closed time returns to the same moments; cyclical time repeats processes | Reading time-travel stories and arguments |
| Geometry vs geometrisation | Giving a domain a geometry (Schrödinger's colours) vs reducing it to geometry (Einstein's time) | Asking whether a model represents or replaces a phenomenon |
| Objectivation | Each level of science strips more subjective features from a concept | Tracing what physics has removed from intuitive time |
| Epistemic vs existential distance | The future is unknown because it is not there, not unreal because unknown | Answering conventionalist and verificationist readings of relativity |
| Being vs existence | Quantifier for being, predicate E! For existence | Talking coherently about the dead, the unborn, possible worlds |
| Order vs direction | A connected, asymmetric, transitive relation vs a privileged sense | Evaluating reductions of sequences, entropy arrows, B-theories |
| Formally diaphanous | A model can label the actual, the true, the now but cannot show their privilege | Interpreting possible-worlds and tense-logic models metaphysically |
Critical lens on the whole book. Yourgrau is a partisan advocate who disclaims taking sides; his strongest work is the reconstruction of Gödel's argument and the analysis of the A-series, both still standard references. Weaknesses: (1) the modal step depends on a verificationist premise and on treating lapse as nomically grounded, both widely rejected, and Yourgrau's defence leans on analogies (pain, rivers, ballet) rather than argument; (2) he says little about local becoming along world lines (Stein, Dieks, Savitt), which has become the main relativistic A-theory since 1999; (3) the polemic against Earman, Dawson and Hon is personal and sometimes ungenerous; (4) Appendix B's mathematics is unsound by the standards of measure theory and cardinal arithmetic; (5) he largely ignores quantum mechanics and thermodynamic arrows, where much later work on time lives; (6) claims of priority ("the only serious study") date quickly. Read the book for the shape of Gödel's argument and the stakes, not for its verdicts.
Who's who
| Person | Role in the book |
|---|---|
| Kurt Gödel (1906–1978) | Logician; incompleteness (1931), rotating universes (1949), temporal idealist and mathematical Platonist |
| Albert Einstein (1879–1955) | Gödel's friend; geometrised time; brushed aside the idealism; "the Now... Cannot occur within physics" |
| Immanuel Kant | Time as form of inner sense; Gödel claims relativity "verifies" his idealism about time |
| J. M. E. McTaggart | A-series and B-series; argued time is unreal (1908) |
| Parmenides, Plato, Aristotle | Being vs becoming; time as moving image of eternity; potential infinity and the sea battle |
| Hao Wang | Gödel's interlocutor; source of his philosophical remarks |
| Howard Stein | "To be relative is not to be illusory"; becoming relative to spacetime points |
| Milič Čapek | Dynamic reading of relativity; many co-becoming rivers of time |
| John Earman | Main critic of Gödel's and Yourgrau's argument (1995) |
| Steven Savitt | Accepts the epistemic force of the reconstructed argument (1994) |
| Lawrence Sklar, David Malament | Philosophers of spacetime; relativised reality; cosmic time and "fiber untwistedness" |
| Paul Horwich | Defends time travel while ignoring Gödel's idealism |
| C. W. Rietdijk | Relativity of simultaneity implies determinism (1966) |
| James Jeans | Cosmic time lets us believe "time is real" |
| Stephen Hawking | Chronology protection conjecture (1992) |
| Richmond Thomason | Branching-time tense logic of the open future (1970) |
| Gottlob Frege | Sense and reference; timeless thoughts in a "third domain" |
| David Kaplan, John Perry, D. H. Mellor | Indexical semantics of "now"; B-theory of real time |
| A. N. Prior | Tense logic; "Thank goodness that's over" |
| Charles Parsons | Modal account of sets and potential infinity |
| L. E. J. Brouwer | Intuitionism grounded in the intuition of time |
| Edmund Husserl | Phenomenology of internal time-consciousness |
| Zeno of Elea | Paradoxes of infinite divisibility |
Connect it
- Models versus the thing modelled. Gödel's lesson that a formalism can be internally perfect yet miss its intended concept applies directly to enterprise data and AI models: a metric, schema or model can satisfy every formal constraint and still fail to capture what the business means by "customer" or "risk". Building deliberate limit cases (adversarial tests, edge scenarios) is how you find out.
- Bitemporal data. Banking systems already distinguish valid time (when a fact held) from transaction time (when it was recorded), a working B-series with an "as-of" now. Yourgrau's point that a relative now is not a privileged now explains why "current state" must always be pinned to a reference time in audit and regulatory reporting.
- Truth-value gaps in forecasting. Aristotle's sea battle and Thomason's supervaluations map onto probabilistic forecasts: a credit model's prediction about default is not true or false today, and treating scenario branches as already settled is a classic model-risk error.
About this book
Gödel Meets Einstein: Time Travel in the Gödel Universe by Palle Yourgrau, 1999. 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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