Insight Editions · No. 253
What Makes Time Special?
Craig Callender
Callender starts from physical time and explains why evolved, embedded agents like us model time as a flowing present, which is neither in physics nor a mere illusion.
A study guide to What Makes Time Special? by Craig Callender. Shared free for personal reading; not for sale and not endorsed by the authors or publishers. Please read and buy the original.
Physics contains no now, no flow and no open future, yet time is genuinely special: it is the direction in which the laws tell their strongest stories. Manifest time is what creatures like us, stuck in time, with fast signals, asymmetric memories, enduring selves and evolved emotions, would naturally build.
Relativity, Cauchy surfaces and cosmic time; quantum non-locality and collapse; causal sets and canonical quantum gravity; the Best System account of laws and well-posed Cauchy problems; the psychology of synchrony, recalibration and the self; Prior's "thank goodness"; the A/B/C debate.
Einstein and Carnap on the Now; McTaggart; Putnam, Stein, Malament; Popper; Sorkin; Kiefer; Lewis; Geroch; Hartle's IGUS; Butterfield; patient PH; Velleman; Parfit; Prior; Caruso.
Readers who want the hard physics of time and the cognitive science of the "now" in one argument, and anyone tired of choosing between "time is an illusion" and "physics is incomplete".
The Book on One Page
Craig Callender takes Einstein's worry seriously: the experienced Now "does not and cannot occur within physics". Rather than resign himself, or patch physics with metaphysical extras, he follows Carnap: start from physical time and explain why beings like us model time as manifest time. The flowing now is neither in physics nor a mere illusion; it is the natural model for an embedded, evolved agent in a world whose time really is special.
- The two-times problem. Manifest time has an objective, global now, a flow, and a fixed past facing an open future; physical time has none of them, and unlike Eddington's "two tables" we have no bridge between the images. Callender goes "from there to here": from physics to the mind.
- Physics will not supply a now (Chapters 2–5). An informal dilemma governs relativity: the better a structure represents the manifest present, the less relativistic it is, and vice versa (Putnam versus Stein, Malament's theorem, lightcone, puny, donut, dome and diamond presents). Cauchy, FLRW and CMC times are gauge choices or idealised truncations. Quantum non-locality's preferred frame would be undetectable; collapse gives only "drips and splashes" of actualisation. Causal sets reproduce the dilemma; canonical gravity's semiclassical time derives time from timelessness but may smuggle it in.
- But time is special (Chapters 6–8). Physics does not "spatialise" time: the Lorentzian signature, one-dimensionality, mobility asymmetry, arrows and natural kinds all distinguish it. What glues these fragments? On the Best System account, time is "the great informer": the direction in which theories are strongest. Rigorously, only hyperbolic equations with data on spacelike surfaces have well-posed Cauchy problems, which yields one-dimensional, Lorentzian time without closed timelike curves. Sideways and null "pages" exist but tell worse stories.
- The present is not perceived (Chapter 9). The "presence of experience" is the Temporal Knowledge Argument, which overgeneralises to "here" and "I". Psychology shows synchrony itself fragments: ventriloquism, recalibration, uncorrelated measures, task-dependent binding.
- Why the now seems common and objective (Chapter 10). Fast signals and slow macroscopic change mean no time stamps are needed; evolution tuned shared integration windows; and because we cannot turn around or travel in time, the disagreement that exposes "here" as egocentric never shows for "now".
- Why time seems to flow (Chapter 11). An IGUS with change detectors, a specious present, clocks and asymmetric memory still lacks the "whoosh"; the enduring, narrated self supplies it, and we slide from ego-moving to time-moving schemas. Agency makes the future feel open.
- Why the past matters less (Chapter 12). Prior's "thank goodness that's over" proves no tensed metaphysics; causal and knowledge asymmetries drive an affect asymmetry that mediates the value asymmetry and many other time biases.
- Beyond the ABCs (Chapters 13–14). Metaphysics that rests on naked existence explains nothing, like the old life-force doctrine; the real explanation is physical hooks + environment + embedded subject + evolution.
| Ch | Title | One-line payload |
|---|---|---|
| 1 | The Problem of Time | Manifest time (now, flow, past/future asymmetry) is near-universal but absent from physics; explain it from physics, Carnap-style |
| 2 | Lost Time: Relativity Theory | Classical time hides an animatable foliation; relativity shatters the master clock, and every candidate present is either unrelativistic or useless |
| 3 | Tearing Spacetime Asunder | Cauchy, FLRW and CMC times are gauge choices or truncations no good theory would make law |
| 4 | Quantum Becoming? | A quantum preferred frame would be undetectable by experience; collapse yields no manifest openness |
| 5 | Intimations of Quantum Gravitational Time | Causal-set becoming repeats the dilemma and turns ontologically vague; semiclassical time emerges from timelessness |
| 6 | The Differences Between Time and Space | Signature, dimension, mobility, arrows and kinds distinguish time; what glues the fragments? |
| 7 | Laws, Systems, and Time | On the Best System account, time is the direction of greatest informative strength |
| 8 | Looking at the World Sideways | Only hyperbolic equations with spacelike data are well-posed; sideways and null stories are worse |
| 9 | Do We Experience the Present? | No sensed tensed presentness; even perceived synchrony falls apart under study |
| 10 | Stuck in the Common Now | Fast signals, evolved windows and being stuck in time make the now seem objective, unlike the here |
| 11 | The Flow of Time | An IGUS with memory and an enduring, narrated self models time as flowing |
| 12 | Explaining the Temporal Value Asymmetry | Causal, knowledge and affect asymmetries explain "thank goodness that's over" and other time biases |
| 13 | Moving Past the ABCs of Time | Existence-based metaphysics of time explains nothing; it faces the life-force doctrine's fate |
| 14 | Putting It All Together | One schema explains every feature: hooks + environment + subject + evolution |
The Problem of Time
Callender sets up the book's problem: we all build a model of time, manifest time, with an objectively special present, a flow, and a fixed past facing an open future, yet physical time contains none of these and offers no obvious counterpart. Rather than dismiss the now as an illusion or patch physics with metaphysical extras, he proposes to start with time as physics describes it and explain why creatures like us would come to model it as manifest time.
1.1 Manifest Time
Children build their representation of time in fits and starts. Young children stumble on serial order recall; their "yesterday" covers all past events. By around four they know a birthday is near and can place the start of school in a sequence. Gradually they abstract from particular events to calendar frameworks (days, weeks, years), separate the history of the world from the history of the self, and learn to decenter: Lily grasps that when Ewan lost his first tooth, Isla's birth lay in his future. The endpoint is a detached, unified model of time itself.
Etched into that proto-theory are three features that make a human life recognisably human. Unlike any spatial perspective, we treat our current temporal perspective as objective: the present is not one viewpoint among many but the moment that is really happening. This slice divides the world in two, and we invest the halves with different properties: the past is fixed, the future open. Past headaches mean little; future ones matter. As the Epicureans noticed, our pre-natal non-existence troubles nobody, while future death causes great angst. And the tripartite division keeps updating, so time seems to flow.
Callender calls this regimented common-sense conception manifest time, echoing Wilfrid Sellars' "manifest image". It is not time as directly experienced; it is a theory, informed by experience, that psychology suggests we reach in late childhood. D. H. Mellor calls it "the time of our lives". The trouble is that our best science suggests it is "more or less rubbish".
Science corrects folk beliefs all the time (Euclidean space, the Sun circling the Earth). What makes time different is double: what we are wrong about matters enormously to us, and we have no story of why we are wrong. For space we can say curvature is locally tiny; for falling bodies and the Sun we have the Earth's motion. For time, "we're wrong and we don't know why."
Einstein's worry and Carnap's reply
Rudolf Carnap reports that Einstein was "seriously" worried by the problem of the Now: the experience of the Now is something essentially different from past and future, yet this difference "does not and cannot occur within physics", a matter of "painful but inevitable resignation". Carnap objected that everything objective can be described in science: physics describes the temporal sequence of events, and psychology can describe and in principle explain the peculiarities of human experience of time, including our attitudes to past, present and future. Einstein held that something essential about the Now lies outside science.
The choice looks bleak. If physics is right, the now and flow seem to be an elaborate, unexplained illusion; if physics is wrong, someone must show where and build a better theory. Einstein accepted the first horn. Callender rejects the resignation: there is an explanatory gap between time in life and time in science, like the one in the consciousness debate, and time is tied to too much (causation, freedom, identity) to leave mysterious. Carnap never carried out his project; this book attempts it. Reconciling the two images is, for Callender, the principal goal of philosophy of time.
Getting timing and duration roughly right
Watching a gull over a lagoon, Callender sees continuous flapping and hears its cry as the beak opens, a smooth, gapless, ordered experience. Underneath is chaos: light from the gull arrives much faster than the sound (slower still on a humid night), parts are in shadow, a child runs, a dog barks. The brain nonetheless builds a representation of temporal order that roughly matches the objective order. In fact we get it "wrong" to get it "right": the perceptual system adjusts signals so that properties of one event are experienced together (Chapter 9).
Duration works similarly. Pan's Labyrinth may feel endless to one viewer and fleeting to another; subjective duration depends on wakefulness, interest, mood, caffeine and much more. Yet presented with evidence, nobody disputes that the film objectively took 112 minutes, during which the Earth turned about 7.8% of its daily rotation, or that time did not really "freeze" for people at home during a car accident. We separate subjective from objective duration, and for salient macroscopic events our estimates track the objective ones fairly well. Without temporal cues an objective hour is judged on average to be 1.12 hours (Campbell, 1990), and even at the millisecond scale we tend to stay within 10% of objective duration (Mauk and Buonomano, 2004). Combined with memory, these abilities give us a rough map or timeline of order and duration.
The egocentric/allocentric distinction in time
Manifest time has dozens of features: it is ordered, metrical, flowing, global, one-dimensional, shared, continuous, absolute, productive. Callender narrows to the temporal counterparts of what spatial psychologists call egocentric (perspectival) representation.
| Frame | Space | Time |
|---|---|---|
| Egocentric | Fork on the right, car in front: changes when you spin | A-series (McTaggart), "deictic time", tensed: Socrates' death is past |
| Allocentric | Kansas City lies between San Diego and Providence | B-series, "sequence time", tenseless: Socrates drank hemlock before he died |
| Truth-value stability | Allocentric claims survive turning around | "March 1968 is later than December 1967" holds wherever the now is; "Socrates will drink hemlock" does not |
| Key structure | A here anchors egocentric frames | A now anchors past/present/future; strip it and only before/after remains |
J. M. E. McTaggart (1908) noted that we arrange events in an A-series moving from future through present to past, defined relative to a distinguished now. Linguists call the same thing deictic time. The B-series runs from earlier to later with no deictic centre.
Temporal decentering is the ability to adopt a temporal frame at a time other than one's own, the analogue of seeing that the fork on my left is on your left. Without it, the idea of past/present/future updating itself makes little sense. In probably the first study, Cromer showed children a picture strip of a child climbing a ladder and sliding down too hard, and asked them to point to the card where the child might say "I will go down the slide". Children succeed at around ages four to five, which Cromer read as mastery of decentering. Spatial decentering is prompted by simply turning one's head; temporal decentering seems to need memory of the self in past times (McCormack and Hoerl). Unless you can recall that what is now past was once present, little in experience supports it.
Cromer's task leans heavily on language comprehension, as Callender concedes, so it may test grammar as much as decentering. He leans on convergence with other studies (Friedman, Weist) that also place a transition around ages four to five. The developmental timeline is suggestive, not settled.
The adult result is a fully decentered conception in which the past/present/future division slides up and down a unified timeline. And here is the twist: although {past, present, future} is an egocentric representation, manifest time treats it as objective. That is the key asymmetry between spatial and temporal representation, and the book's three targets are the now, the flow, and the past/future asymmetry: central to life, more or less special to time, and apparently in conflict with physics.
The figure above compresses manifest time: a global now splits the world, the slice keeps moving futureward (flow), and the two sides differ in kind (past/future asymmetry).
1.1.1 Now
T-shirts, Japanese-character tattoos and self-help guides exhort us to "live in the now"; Pooh's favourite day is "today". Taken literally the advice is redundant: we have no choice but to live in the present, because that is when all conscious action and decision seems to happen. Events do not seem merely strewn along a four-dimensional manifold; we think there is a further fact about when they are happening, one set of events being "lit up".
The psychological present has been contested for over a century. We process time on many scales handled by different mechanisms:
- Microseconds. Sound localisation needs discrimination of intervals as short as 600 µs, the time sound takes to travel from one ear to the other, done by axonal delay lines.
- Tens to hundreds of milliseconds. Motion detection, speech recognition, motor coordination, multisensory integration, music.
- Around a second. Predicting events, like a skateboarder darting across your path.
- A day. Circadian rhythm, driven by a protein transcription–translation feedback loop.
None of these mechanisms is necessary for the others, so what counts as "the present"? Experience seems to include succession and duration (the second hand looks like it moves), which E. Robert Kelly (writing as "E. R. Clay") named the specious present in the late nineteenth century. Estimates of its length range from milliseconds to many seconds, partly because researchers conflate the vehicle/content distinction: how long the percept lasts versus how long a stretch it represents. There is no clear experiential boundary to the present. Michon (1978) defines it operationally as an interval in which sensory information, internal processing and behaviour are integrated in one span of attention. Callender suspects "present" may not be a natural kind of psychology at all, but keeps the phrase "subjective present" for immediate experience at these short scales.
Inspired by these brief experiential presents, we attribute a distinguished present to the world itself. It marks our language: English forces the choice between am, was and will be happy, and Alfred Gell reports that all known languages mark presentness somehow. Now-talk can drop clock reference entirely ("Look out!" means look out now; those who answer "When?" get hurt). Like other indexicals, the now motivates action: learning that it is now 11:45 sends you rushing to a noon meeting in a way that learning "11:45 is 15 minutes before noon" does not.
We also take the present to be global and shared: snap your fingers and you assume the snap is simultaneous with events on an extrasolar planet. That indefinite extension is what lets the now slice the world into unambiguous past and future. Contrast the spatial here: we never objectify front/back or left/right, which we know hang on a reference frame. Treating the present as objectively special while treating no spatial counterpart that way is "a critical difference between manifest space and time".
1.1.2 Flow
Poets, philosophers and musicians compare time to a river; English, Japanese, Chinese and the Niger-Congo language Wolof all ascribe motion to time. Some flow-talk is admittedly subjective ("time flies when you're having fun"), but some is aimed at time itself.
- Primary sense of flow. The tripartite structure {past, present, future} continually updates itself in a temporally directed manner: future events, not past ones, become present.
- Dependence on the now. So defined, flow presupposes an objective now, though the two are best seen as two aspects of one package; we would hardly treat the subjective now as objective if its content never changed.
- Phenomenology contested. Experiences change, but whether they reflect time itself flowing is disputed, and the question is tied to whether presentness is phenomenal.
Callender separates two notions of change:
| Change | Souped-up change | |
|---|---|---|
| Definition | Having different properties at different times | Alteration of properties with respect to the present |
| Example | The ball is at x at t₁ and y at t₂ | The ball is now at y |
| Conflict with physics? | None: physics has handled it for centuries | Yes: it needs a now that physics lacks |
| Space/time difference | Leaves changing colour vs a plaid shirt's colour pattern look alike | Only the now makes autumn leaves "real change" and the shirt mere variation |
We update the spatial here as we walk, but because we do not treat it as objective we never think space flows.
1.1.3 Past/Future asymmetry
Choosing between flourless chocolate cake and peach cobbler, you weigh futures that feel genuinely open; once the cake is eaten, its fate is sealed. The story holds two asymmetries: the present moved one way, and at any moment the future is open and the past fixed. Callender reserves past/future asymmetry for the second. It is bound up with the sense of agency: futures seem to branch and we prune them to one.
Philosophers of time have focused on the preference asymmetry, which comes in two forms:
- Past vs future. We dread future pains and feel relief when they are over, but feel no dread for past pains. Had you ordered the cobbler you would regret it now, not before ordering.
- Near vs far future. We prefer distant pain to proximal pain; we procrastinate. Credit-card behaviour and society's response to climate change show the costs.
Spinoza held that a rational agent would direct attitudes equally toward past and future; even if retraining were possible, it would be a radical departure from normal human life. Both asymmetries rely on the tripartite division. The second does too, despite appearances: we prefer pain in 2027 rather than 2017, but when 2027 comes we again want it pushed back, rather than saying "good, the pain is arriving just when I wanted".
Do not conflate the past/future asymmetry with B-series asymmetries. The causal asymmetry (your decision caused the cake to arrive) holds from every temporal perspective; it attaches to earlier-than. The openness asymmetry attaches to past/future. Entropy increase and cosmic expansion likely underlie the past/future asymmetry, but they are not it.
Physics posits no open future, and manifest space has no counterpart: nobody thinks the "east" side of a plane through their here is fixed and the "west" side open.
1.1.4 Is manifest time universal?
Callender himself conceives time as a set of directions on a spacetime with a semi-Riemannian metric; Shakespeare did not. What they share is not high-level theory, which may even contradict manifest time (cyclic cosmological myths, physicists' closed time models). What stays roughly invariant are the core aspects: the specialness of the present, flow, and past/future asymmetry, which are developmentally prior to cosmological beliefs and grounded in bodily experience. Believing in a closed cosmology gives nobody agency over the past, and Lady Macbeth's "what is done, is done" reaches across centuries to a shared past/future asymmetry.
Callender admits rational reconstruction: consistency demands a global present (otherwise some regions are neither past nor future), though few people ever ask whether the now reaches Andromeda. If manifest time commits us to less than his clean version, so much the better: it is easier to recover less.
He then answers four relativist challenges:
- Measurement and pace of life. Clocks, church bells, punch-clocks, railway timetables and imperial time standardisation transformed societies; Levine (1998) showed the "pace of life" varies with city size, temperature and culture (Sicily is slower than New York). None of this shows anyone lacks a special present or a settled past.
- Durkheim against Kant. Émile Durkheim (1912) argued time cannot be a universal category because it comes in socially specific units. Callender diagnoses a conflation of units with facts: a gallon to me and about 3.8 litres to you measure the same milk, and Mayan and Aztec calendars differ without any evidence they disagree that there is a past. Gell concludes there is "no fairyland" where people lack past, present and future or where time swings like a pendulum.
- Sapir–Whorf. Whorf (1956) claimed the Hopi have no notion of time flowing from future through present into past because their language lacks tense markers. Malotki (1983) showed Hopi has ample tense, temporal units and a word for yesterday. Even without the words, the inference from lexicon to concept is the "greater sin". Japanese Ao covering blue and green never stopped Japanese speakers distinguishing the colours. Weak Whorfian effects are real (grammatical gender of "bridge" shifts adjective associations, per Boroditsky), but there is "no reason to be afraid of the big bad Whorf" (Casasanto).
- Spatial construals of time. The Aymara of the Andes put the future behind and the past in front; the Australian Pormpuraaw place the past east and the future west (geocentric, not egocentric); the Papua New Guinean Yupno (Núñez et al., 2012) gesture downhill toward the river mouth for the past and uphill to its source for the future. A New Scientist piece concluded Yupno time "flows uphill and is not even linear". Callender's reply: a thumbs-up does not tie your concept of "positive" to gravity, and the Yupno still map uphill/downhill onto future/past and the here onto the now. The evidence offered for variation is "some of our best evidence that the core features don't vary".
If the core of manifest time is universal, that calls for explanation: it is the natural by-product of solving problems common to our species: communicating, navigating, organising confusing sensory input. Other animals facing the same problems may share parts of it; experiments on blue jays' discounting of delayed rewards imply an indexing to a present, hence something A-series-like.
1.2 Physical Time
There is no single physical conception of time; physics changes, and quantum theory and relativity clash over time itself (relativity makes time dynamical, altered by matter-energy; quantum theory standardly treats it as non-dynamical). But physical time has always clashed with manifest time.
Physics models a system by a state in a state space of all physically possible states, plus a dynamics giving the transition (probabilities) from a state at one time to another. A ball on a table needs positions, velocities and a law. Building such a theory forces many decisions about time:
Choices physics must make about time
- Instants. Are there instants at all, and are they ordered partially or totally?
- Topology. Continuous, dense or discrete; open or closed; one-dimensional or not; beginning with no end, beginning and end, or branching.
- Metric. Durations between instants; also needed for velocities and accelerations, defined as limits over temporal intervals with topological neighbourhoods.
- Markovian dynamics. Does the law need only the present state, or also information from three hours ago?
- Time-reversal invariance. Does the law care about the direction of time?
- Time-translation invariance. Does the evolution depend on what time it is, all else equal?
1.2.1 No manifest time
The striking fact is how little physics needs. No physical theory has ever required an A-series: Aristotelian, Newtonian, Einsteinian and quantum physics all manage without a special now, and hence without flow or past/future asymmetry, both of which depend on a now. "Now" appears in no physics index.
The controversial question is whether physics at least supplies structure fit for animation: a present you could pick and move along a time function. Callender previews his answers:
- Relativity not only lacks a present; it barely has structure worth calling a present. His loose dilemma: to the extent a structure corresponds to manifest time's present it is not relativistically invariant, and vice versa. Relativity ignores the now "with extreme prejudice" (Chapters 2–3).
- Quantum mechanics. Karl Popper and others claim it frees manifest time from relativity's constraints. Chapter 4 asks whether that means bending science against its virtues.
- Quantum gravity. Programmes pull in opposite directions (Chapter 5). Canonical quantum gravity threatens to eliminate time altogether; causal set theory advertises a rescue of genuine flow. Neither restoration nor elimination comes easily.
The historical trend is a demotion of time: Boltzmann made time's direction a feature of local material asymmetries, not of time; Einstein made time an aspect of spacetime; canonical quantum gravity threatens to finish the job.
1.2.2 What makes time special?
The positive project asks an old question: how does time differ from space? Physics is often said to "spatialize" time; Callender denies it. The kernel of truth is that physics treats the spatial counterparts of the three manifest notions the same way: it ignores them too. There is no physical here, no flowing here, and no ontological east/west split. Physics neither spatialises time (Bergson's charge) nor temporalises space (Milič Čapek's reply); it sheds the same manifest properties from both. Those absences pass unnoticed for space only because they were never part of manifest space.
Inside relativity, a deep distinction survives between timelike and spacelike directions, plus the direction of time, the natural kind asymmetry, dimensionality and mobility. Callender's new question: why do these logically detachable "temporal" features hang together? Why, for instance, is our mobility restricted in exactly the directions the metric distinguishes? His answer links laws of nature and time:
- Chapter 7 (informal, from the Best System theory of laws): time is that set of directions in which our physical theory can tell the best stories.
- Chapter 8 (formal): laws are productive: data on a 3D slice can be marched forward to produce the next slice. Tilt the world to light-like surfaces, or on its side so data marches "east to west", and ask whether laws stay productive.
These differences matter because organisms evolving in such a world are shaped by them: they are "hooks" that may explain why we model time as manifest time.
1.3 The "Two Times" Problem
Most physicists dismiss flow as a grand illusion. Perhaps loosely correct, but the dismissal hands the explanatory job to psychologists, who don't know it's on their desk, so manifest time stays unexplained.
The model is Arthur Eddington's "two tables" from his Gifford Lectures (published 1928). Table No. 1 is the familiar one: extended, permanent, coloured, substantial. Table No. 2, the scientific one, is "mostly emptiness" with scattered electric charges whose combined bulk is "less than a billionth of the bulk of the table". Eddington granted that the two are "ultimately to be identified after some fashion".
Callender's resolution: there is one table, modelled two ways for two purposes. The harder residual puzzle is that both models succeed, so something must be right about attributing and withholding solidity. We know roughly how to bridge them: start from the physical table and show approximations of manifest features emerging at our scales (inter-particle forces, shielding; solid-state physics' tight-binding model explains why wavefunctions are "repulsed" from the same atomic state). Colour and texture need a perceiver embedded in the world.
| Two tables | Two times | |
|---|---|---|
| Bridge known? | Roughly: tight-binding model, perception science | No: "explorers deep in the mist" |
| Physics supplies counterparts? | Yes, swarms of charges resisting each other | No: physics works without and offers no obvious counterparts |
| Stakes | Low: few care whether tables are solid | High: manifest time has a central place in our lives |
| Status | A kind of détente | Einstein's worry is genuine |
1.4 From There to Here
R.E.M. Sang that you "can't get there from here"; Callender goes from there (physical time) to here (manifest time). Drop Iggy, an IGUS (information gathering and using system), into a relativistic spacetime. Using the differences between time and space, facts about Iggy and her interaction with her environment, can we explain why Iggy models time as manifest time? That is the method of the book's second half.
Why start from physics? Because it is the only science that takes time itself as a target and tests its theory severely. The Hafele–Keating experiment (flown 1971, published 1972) carried synchronised caesium clocks on commercial flights to confirm time dilation; atomic clocks are now accurate to within 100 quadrillionths of a second; every smartphone using satellite positioning corrects for relativistic effects.
The reverse path, from "here" to "there", was taken by Henri Bergson, who in his doctoral thesis (1889, before relativity) contrasted lived time (durée réelle) with science's "mechanistic" time and judged the latter a distortion, and by phenomenologists who try to derive physical time from experience.
Three motives for starting from physical time
- Parsimony. Physics might get time essentially right. If so, Occam's Razor says posit no more temporal structure than physics needs. With brains and macroscopic environments added, can we avoid extra physical or metaphysical temporal structure?
- Better methods. Physics reaches its temporal features by inference to the best explanation under rival criticism and severe testing; manifest time is a vague set of theses formed in development to thrive among cycles of day, night, feeding and sleeping. Fitness-conducive beliefs need not be true; nobody would keep folk physics on such grounds.
- Tensers have failed. Analytic metaphysicians ("tensers") have dressed physical time in primitive flows, tensed, transient, ersatz, Meinongian, existent, priority, thick and skipping presents, moving spotlights, becoming, and half a dozen kinds of branching. They explain manifest time by fiat and leave a mystery why physics misses their accessories (Chapter 13).
Callender adds candidly that he is by nature both reductionist and scientistic, but says these motives, not temperament, drive the choice. The critics he has in mind include Arthur Prior ("the theory of relativity isn't about real space and time") and Lee Smolin (in the real world "it is always some particular moment", which mathematics can never capture).
He will not try to close the gap between experience and physics, which would mean solving the mind–body problem; "natural science has never explained a single mental experience directly", and deducing temporal experience from physics is "a fool's errand". Two methodological lessons follow from the tables analogy:
- Use the science of time. A- and B-properties and logical analysis are useful but blunt: you cannot see the second law of thermodynamics, the relativistic signature, or its link with laws through them.
- Use more than physics. Carnap switched from "physics" to "science". Explaining the now needs psychology, biology, and other natural and social sciences alongside philosophy.
Iggy thrown into spacetime. Picture a relativistic block: worldlines, light cones, no now. Put a small information-gathering robot in it, with sensors of different speeds, a memory, a body and a need to act and to talk to others like it. Without adding any objective present, ask what representation of time such a robot would be forced to build. If the answer is "one with a common now, a flow, and an open future", the two-times gap starts to close, without any physics being changed.
The book promises a man who hears speech before seeing lips move, children's sense of self, a theory linking time to laws, determinism from "east" to "west", no-go theorems for relativistic presents, time bias in judgments of responsibility, quantum-gravity time and the evolution of the psychological present: an "all-out interdisciplinary attack" in which philosophy contributes conceptual tools and freedom from any single discipline.
Lost Time: Relativity Theory
Callender makes good on the claim that physics has no now, flow or direction, and goes further: relativity contains no good structure to "animate" into a flowing present. Classical spacetime at least hides a unique time and space that a tenser could animate; relativity shatters the master clock into one clock per worldline. His informal dilemma: the better a structure represents the manifest present, the less relativistic it is; the more relativistic it is, the worse it represents the manifest present.
Callender is explicit that explaining manifest time will require embedding a subject like us in a world like ours, not finding a structure in physics that plays the manifest-time role.
2.1 Classical Physics
Classical physics is not one theory: Newton, Hamilton, Euler, Lagrange and Jacobi formulated it in different, sometimes inequivalent ways, but all share the same temporal structure. For comparison with relativity, Callender uses the coordinate-free four-dimensional formulation of Élie Cartan (1923, 1924) and Kurt Friedrichs (1927).
- Events and manifold. Take events as primitive (birthday parties, King Philip's War, a firecracker). Represent all events by a smooth, connected 4D manifold M, which allows local coordinate systems that "play nicely" together; classically M is usually topologically Euclidean.
- Metric. F = ma tells bodies how far to go in how long. When physics predicts that a falling body reaches a terminal velocity of roughly 32 ft/s, the laws in effect say how long a second is. Topology cannot see such facts; metric structure is needed.
- Two metrics. Galilean or neo-Newtonian spacetime drops Newton's absolute rest and carries a temporal metric t_ab and a spatial metric h_ab. Both are degenerate tensor fields, with signatures (1,0,0,0) and (0,1,1,1), and they are orthogonal: h^ab t_bc = 0.
- Affine connection ∇. Lets us compare vectors at different points by parallel transport, defining "straightness"; assumed flat.
- Matter. Mass density ρ.
A Newton–Cartan world: ⟨M, t_ab, h_ab, ∇, ρ⟩
- t_ab gives every vector a temporal length ≥ 0: positive means timelike, zero means spacelike. It sorts all vectors exhaustively and, with ∇, assigns temporal lengths to curves (your path between birthdays: one year).
- h_ab assigns lengths only to spacelike vectors; elsewhere it is undefined.
- Consequence: spatial distance ≠ 0 implies temporal distance = 0; temporal distance ≠ 0 implies spatial distance is undefined. Non-simultaneous events have no spatial distance, which expresses the absence of absolute rest (otherwise we could say how far apart two finger-snaps at different times were).
2.1.1 Recovering space and time
Nothing labelled "space" or "time" was needed: two metric fields and a connection do all the work. Yet a unique time and space can be recovered:
- Spaces. A hypersurface Σ is spacelike if all its tangent vectors have zero temporal length.
- Time functions. Assuming M is simply connected, t_ab yields a family of global time functions t: M → ℝ that increase along every future-directed timelike curve. There are infinitely many, but they agree on durations and simultaneity; a curve γ from p₁ to p₂ has duration t(γ(p₂)) − t(γ(p₁)), which depends only on the endpoints. The numerical origin is meaningless (t′ = t + b describes the same facts).
- Classical clock hypothesis. Ideal clocks record the t_ab-length of worldlines.
- Linking them (Malament). A hypersurface is spacelike iff the global time function is constant on it. So each spacelike slice is "all of space at a moment", and simultaneity foliates M into a unique sequence of spaces: M = Σ × ℝ.
Equivalently, t_ab defines an earlier-than relation that totally orders classes of simultaneous events. "Neither precedes the other" is an equivalence relation, simultaneity, which partitions M into cells, each a subset of 3D Euclidean space thanks to h_ab. Time is the quotient set M/sim: its points are instants, with the structure of a 1D real affine space (the Euclidean line). For the mathematically inclined, the theory is a fibre bundle with time ℝ as the base space and a copy of ℝ³ (space) as the fibre over each instant. Time is distinguished: one cannot swap base and fibre. "In a sense, time is all of space collapsed to a point."
2.1.2 Classical ideal clocks and manifest time
An ideal clock is a point-sized test object that perfectly records time along any allowed worldline. No real body is one, but it has the status of the laws themselves: measurements confirming the laws confirm the ideal clocks they posit. Since classical duration is path-independent, if Jack stays home and Jill goes to the gym between leaving work at noon and meeting for dinner, the same time elapses for both. One master clock serves the whole universe.
Features of the classical ideal clock
- Absolute. Its rate is independent of the existence and arrangement of matter; physical clocks (Newton's "sensible measures") only approximate it.
- Time translation invariant. Laws are unchanged under t → t + k: an experiment at 2 p.m. turns out as one at 3 p.m. The zero of the number line could never show up physically, so it cannot ground a privileged moment.
- Time reversal invariant. t → −t is a symmetry. Classical physics needs only a betweenness relation, B(<; a, b, c) ⇔ a < b < c or c < b < a: a line capable of holding a direction, not a directed line.
Callender notes, against the "causation-obsessed" axiomatisations of Minkowski geometry (Robb's 1914 primitive after; Reichenbach 1924, Grünbaum 1973, van Fraassen 1985), that Mundy (1986) and John Schutz (1997) axiomatise it with only events, paths and betweenness, no direction of time required.
Score against manifest time. At first glance good: global equivalence classes of simultaneous events, a gapless continuum, uniqueness and absoluteness. But the three core ingredients are missing: no distinguished now, no flow, no past/future asymmetry. Time-translation invariance resists any physically motivated special moment; time-reversal invariance resists any physically motivated past/future difference. Newton wrote in the Scholium that time "flows", but if he meant manifest flow, he posited more than his physics needs. Classical physics is called "friendly" to tense, but "if all it takes to be friends is to be ignored, that is not so impressive a partnership." Its simultaneity classes are good candidates for animation, nothing more.
2.1.3 Trautman–Cartan theory
Classical physics may even be actively hostile to manifest time. Standard Newtonian gravity uses the Poisson equation ∇_a∇^a φ = 4πGρ on a flat background. Andrzej Trautman (1965) reformulated Cartan's theory with a curved connection, absorbing gravity into geometry: free-falling bodies follow geodesics, space stays flat, but ∇ is curved along timelike directions (a vector carried round a 4D loop returns rotated). The field equation becomes R_ab = 4πρ t_ab.
In Trautman's version the contingent matter distribution fixes which paths are geodesics, and so the rate at which even ideal clocks tick depends on matter, "presumably imperceptibly, by where I left my keys". Since common sense (and a premise of Gödel's argument for the ideality of time) holds that duration cannot depend on matter, and since Trautman's is arguably the most natural formulation given what came next, classical physics already harboured dormant threats to manifest time.
2.2 Relativity
Einstein said reconceiving time is what let him discover special relativity in 1905. Its radical consequences were initially masked by similarity to Lorentz's theory; after general relativity (1915) and its vindicated predictions, the clash with manifest time was plain by the 1920s, and some Kantians and phenomenologists opposed relativity for that reason.
Terminology: "relativity" means general relativity; Minkowski spacetime is one of its solutions. Callender thinks philosophy of time has been harmed by an unwarranted concentration on special relativity.
Relativity replaces the two classical metrics with a single spacetime metric g_ab. Distances are neither spatial nor temporal but spatiotemporal: the metric gives a "distance" between the supernova SN 1987A in the Large Magellanic Cloud and its sighting from the southern hemisphere on 23 February 1987, or between it and Benjamin Franklin's death. Using the line element ds² = g_ab dx^a dx^b (Euclidean case: ds² = dx² + dy² + dz²), two differences stand out.
| Classical metrics | Relativistic metric | |
|---|---|---|
| Type | Symmetric, positive semidefinite | Symmetric pseudo-Riemannian, Lorentzian signature (+++−) |
| Sign of ds² | Never negative | Can be negative (indefinite) |
| Vector classes | Timelike / spacelike (spacelike = zero temporal length) | Timelike (ds² < 0), null (ds² = 0), spacelike (ds² > 0) |
| Cones | Future and past lobes meet along a 3D hypersurface | Lobes meet only at the point p: the lightcone |
| Variability | Fixed once and for all | Varies point by point with matter |
| Duration | Path-independent | Path-dependent |
The signature change is not a coordinate trick. You can relabel dt² as "−dt²" just as you can call cats "dogs" and then say cats bark; the signature is a coordinate-invariant feature. Lightcones live in each point's tangent space; massive bodies follow everywhere-timelike curves, light everywhere-null curves, and tachyons (always spacelike) are permitted but not posited.
The second difference comes from the Einstein field equations, G_ab(g_ab) = 8πT_ab: Einstein tensor (curvature) against the stress–energy tensor (energy density, pressure). Compact on paper, they stand, Callender says, for 256 coupled partial differential equations until symmetry is invoked, so they are rarely solved explicitly, and they do not uniquely fix geometry from matter: many distinct vacuum spacetimes share T_ab = 0. The equation is local, so relativity is very permissive: any triple ⟨M, g, T⟩ satisfying it at every point is a model, with curvature positive here, negative there, flat elsewhere. Many physicists want extra conditions to "censor" unphysical solutions; Callender sees this as a sign the theory is still under construction.
Proper time and the shattered master clock
The proper time along a path P of a timelike worldline is τ_P = ∫_P √(−g_ab dx^a dx^b), an invariant all observers agree on. The Clock Hypothesis says ideal clocks measure it. (A classical proper time exists too, with g_ab = −t_ab, as does a proper length along spacelike curves.)
Relativistic proper time is path-dependent, as spatial distance is (Interstate 5 versus Interstate 15 from San Diego to Los Angeles), for two separate reasons:
- One Lorentzian metric. Spatial components enter the integral. In flat Minkowski space, ds² = −dt² + dx² + dy² + dz²; at rest only dt contributes, so any moving path between the same events differs.
- Point-dependent metric. Paths through different events sum different local contributions (near-Minkowskian at r, near-Schwarzschild at r′). Only this kind is available to the Trautman classical theorist.
The twin "paradox" is simply this path-dependence: the travelling twin is younger (fewer grey hairs, more optimistic, higher vertical leap) because her path has shorter proper time, a "hard fact of life", as David Malament says.
- Clocks a short height apart show measurable gravitational time dilation (Chou et al., 2010, optical clocks): the clock at your waist ticks slower than the one at your head.
- GPS devices, phones, tablets and car navigators correct for relativistic dilation, so they "test" relativity constantly.
- ds² < 0, = 0, > 0: the three relativistic categories (timelike, null, spacelike) under the (−+++) convention.
Two numerical slips. Callender's "256 coupled partial differential equations" overstates the count: G_ab and T_ab are symmetric 4 × 4 tensors, so the field equations are 16 component equations of which 10 are independent (and the Bianchi identities leave 6 that fix the geometry). He also describes the Chou et al. Clocks as "only a meter higher". The 2010 NIST experiment actually resolved dilation for a height difference of about 33 cm; later optical-lattice work (JILA, 2022) resolved it across about a millimetre within a single atomic sample. The direction of his point only gets stronger.
Classically, proper times along all worldlines mesh into one global master clock that matter "listens to". In relativity one can often define a global time, but matter does not listen to it: the master clock has shattered into a plurality of miniature ideal clocks, one for every timelike path.
2.3 Where's Time?
Debates over finding manifest time in relativity never converge because they require two decisions neither logic nor physics settles: (1) what counts as manifest time, a concept rough at the edges, and (2) which physical structure to identify with it. Some seek candidates for tensed animation; others just want something to call "the present". Callender's aim is fixed by his project: to show that physical time gives would-be vindicators of manifest time no natural candidate for animation. Since the now founds flow and past/future asymmetry, he focuses on the present.
Seven features associated with the manifest present
- Equivalence relation. "Present with" is reflexive, symmetric and transitive (the present is usually identified with the real for relation).
- Global and bisecting. It divides the universe cleanly into past and future.
- Achronal. No present event is in the causal past of another (relaxable for specious-present fans, provided Lincoln's whole life is not present).
- Simultaneous. Its events are simultaneous with one another.
- Radar-synchronous. An operational version of 4 (see Box).
- Productive. Present events influence events on the "next" present.
- Common. Different observers can share it.
These are at most necessary conditions, and none of the tensed aspects are included. Callender asks only that some reasonable constellation be kept, so as not to trivialise the question. His foil is McEvent, the relativistically impeccable event of the first McDonald's opening: crown it "the now" and you have reconciled nothing, since it lacks almost all seven features (no common settled past, for one).
Box: Einstein–Poincaré "radar" synchronisation. Both Einstein and Poincaré gave operational definitions of simultaneity. Operations do not define concepts, but agreement between a structure and natural operations is a reason to think it plays the "simultaneity role". Observer A sends light at a₁ to B's mirror at event b; it returns at a₂. Assuming constant light speed, A sets t_b = a₁ + ½(a₂ − a₁) = ½(a₁ + a₂). Enough observers and mirrors yield a whole hyperplane. Within an inertial frame the result is reflexive, symmetric and transitive, and synchronised clocks stay synchronised, which lets us define a global time function. But radar synchrony is an equivalence relation (for special observers) only when the metric is static (it admits a timelike Killing vector with an orthogonal hypersurface, roughly time translation plus time reversal invariance). Most spacetimes are not static, so radar synchrony fails in most of them.
Like a Where's Waldo? picture, classical spacetime eventually yields a "time" with all seven features. In relativity "Waldo doesn't exist".
2.4 Minkowski Spacetime
Minkowski spacetime is flat, empty and unrepresentatively symmetric, so arguments relying on its symmetries are "parochial" (Kit Fine's proposal to relativise existence to a global inertial frame works only where such frames exist, which our universe lacks). Still, the famous Rietdijk–Putnam–Penrose (RPP) argument (Rietdijk 1966, Putnam 1967, Penrose 1989) and Howard Stein's reply (1968) illustrate the dilemma perfectly.
Putnam's decisions. (1) The "man-in-the-street's" conception of time, which wants ontological differences grounded in the now, so the now must be a non-trivial equivalence class. (2) Its implementation: the constant-t hyperplanes of an observer's natural (inertial) coordinates, i.e. Planes of simultaneity.
The figure shows Alice and Bob passing at event o. Their simultaneity planes through o tilt oppositely, so on a distant planet (Callender uses Gliese 15 Ab, in the constellation Andromeda) Alice's present contains a and Bob's contains b, days apart. At o, event c between them is real for Alice and unreal for Bob. Unless reality is person-relative ("nonsense, or at the very least, desperate"), either the metaphysics is wrong or some inertial frame must be arbitrarily privileged. The Lorentz transformation t′ = (t − vx/c²)/√(1 − v²/c²) makes the disagreement immediate, and neither walker notices it, since signals from the planet take years to arrive.
Putnam's reductio
- Your here-now (o) is real; your future is not real.
- Whatever is simultaneous with a real event, for some observer, is real ("no one is special").
- Event b on the distant planet is simultaneous with o for you (Bob), so b is real.
- For observers at b moving suitably, b is simultaneous with event d on your worldline, which lies in your future.
- So d is real. Let d be your future death: it is as real as your here-now, contradicting premise 1.
The argument is neither coordinate-bound nor loose. Alfred Robb (1914) defined orthogonality from invariant causal structure; for any point on an inertial worldline, the Robb-orthogonal points are exactly the Einstein–Poincaré simultaneous ones. Dropping the restriction to one worldline, the advocate of manifest time meets a theorem (Malament, 1977, Proposition 3):
Theorem 1 (Malament's theorem). If a binary relation R on Minkowski spacetime is (i) implicitly definable from causal connectibility κ, (ii) an equivalence relation, and (iii) non-trivial (Rpq for some p ≠ q), then R is the universal relation. Take R as the person-in-the-street's "present with": if R colours events, it colours them all.
Malament's paper is famous for the conventionality-of-simultaneity debate, but read alongside RPP its Proposition 1 shows Putnam's now is geometrically definable, Proposition 2 that it is unique, and Proposition 3 that it is not generalisable across inertial observers. Within one frame, Putnam's now satisfies features 1–6, better than most rivals.
Stein's becoming and its collapse
Stein proved that a unique "definite for" relation R can be defined from Minkowski geometry: R is reflexive, transitive, antisymmetric, and holds whenever the vector from x to y is past-pointing. His theorem: if R holds for any pair, it holds iff the vector xy is past-pointing. So the region definite for any event is exactly its past lightcone, and "temporal becoming" looks compatible with special relativity.
The price is the present. Defined as co-determinate events, antisymmetry (Rxy ∧ Ryx → x = y) shrinks the present to a single point, one per event. It satisfies none of the seven features, and animating it is "more or less absurd". Callender also doubts the intelligibility of an event that exists yet is indeterminate (does future-me have a freckle?), and notes that definite for cannot be real for, since nobody thinks reality is antisymmetric.
Theorem 2 (Callender's no-go theorem, from Callender 2000). Add the weakest possible globality, Non-uniqueness: ∃x∃y (Rxy ∧ Ryx ∧ x ≠ y) (at least one event shares its present with another). If R on time-oriented Minkowski spacetime is (i) definable from time-oriented metric relations and obtains at least once, (ii) transitive, (iii) such that y ∈ J⁻(x) implies Rxy, and (iv) non-unique, then R is universal. It is "a trivial modus ponens of Stein's modus tollens".
| Putnam's now | Stein's now | |
|---|---|---|
| Congenial to manifest time? | Yes: global, equivalence class, features 1–6 in a frame | No: a single point |
| Relativistically invariant? | No (Malament) | Yes (Stein's theorem) |
| Verdict | Dilemma horn 1 | Dilemma horn 2 |
2.5 Lorentzian Time
H. A. Lorentz and Henri Poincaré found the Lorentz transformations before Einstein, working in a classical "3 + 1" setting before Minkowski's geometric picture (1908). So a classical time is compatible with the relativistic data. Heaviside, Larmor and FitzGerald had found that electromagnetism hides the ether frame; Einstein in 1905 objected that the classical theory "leads to asymmetries which do not appear to be inherent in the phenomena", but that was a theoretical defect, not an empirical failure.
Lorentz kept a traditional time on purpose (1927): "My notion of time is so definite that I clearly distinguish in my picture what is simultaneous and what is not." Classical electrodynamics predicts that moving charged systems physically contract, mimicking Einstein's kinematic Lorentz–FitzGerald contraction; John Bell (1987) showed Maxwell's equations ensure we can never detect the true state of motion. On this view the background is neo-Newtonian, absolute simultaneity returns, and one can picture simultaneity planes "flashing in and out of existence", though we can never tell which frame flows.
Today this looks conspiratorial (why are matter fields Lorentz-symmetric but spacetime not?), though Callender grants that conspiracy is relative to background assumptions. Two reasons still tell against "going Lorentzian" to save manifest time:
- Beyond electromagnetism. Atoms held only by electromagnetic forces are unstable, so Lorentzian theories of the strong and weak forces are needed. Weak objection: the standard approach needed relativistic extensions too, so this may just define a research programme (Harvey Brown).
- General relativity. The real choice is Lorentz versus full relativity, our best theory of gravity, and there is no Lorentzian interpretation of general relativity worth the name. Lorentz (1900) and Poincaré both tried and failed to fold gravity in; modern attempts (Broekaert) "pale in comparison to the real thing". One programme thrives, the other does not.
Historically, preferring Lorentz to Minkowski bent no scientific norms. Preferring Lorentz to full relativity now, solely to save manifest time, means bending science "in ways that don't optimize its virtues".
2.6 Outside Minkowski: Domes, Donuts, and Diamonds
Tooley, Swinburne, Crisp, Lucas and George Ellis find general relativity friendlier to manifest time. One bad reason: there is no general RPP result outside Minkowski. But that is often because general spacetimes are so hostile that the question cannot be posed. Gödel spacetime contains no everywhere-spacelike global hypersurface at all; the RPP-style argument there is "so short that no one bothers to make it".
The classical recipe for instants used two conditions that coincide classically and come apart relativistically: (i) zero temporal distance (ds² = 0) and (ii) orthogonality to time. Relativistically, ds² = 0 picks out null surfaces; orthogonality g(u, v) = 0 to a timelike v picks out spacelike surfaces with ds² > 0. You must choose. Callender surveys the candidates:
| Candidate present | Definition | Relativistic? | Manifest failures |
|---|---|---|---|
| Lightcone present | Surface of p's (past) lightcone, ds² = 0 | Yes, fully | Not an equivalence relation, not spacelike, not synchronous, not shared; may include the Big Bang (Savitt, 2000) |
| Puny present | Infinitesimal subspace orthogonal to observer O's 4-velocity at p | Yes | Divides not even a finite region into past and future |
| Donut present | All events spacelike to p, a 4D volume | Yes | Not achronal (whole distant lives fit inside); with sharing, Weingard's (1972) colouring argument makes everything present |
| Private space (dome) | Hypersurface of inextendible geodesics orthogonal to O at p (Milne, Wüthrich 2013) | Unique for O at p | Can turn non-spacelike far away; may hit the Big Bang in FLRW; may miss some worldlines; two observers' domes clash: "too private" |
| Public space | Hypersurface orthogonal to a family of timelike geodesic observers from p | Unique if non-rotating | Good only for geodesic observers, and you are not one |
| Diamond present | Alexandroff interval I⁺(p) ∩ I⁻(q) for p ≪ q on a worldline | Yes; globally hyperbolic under conditions (Ribeiro, 2010) | Local, worldline-dependent, does not bisect the world, typically not shared |
The Weingard argument for donuts runs: colour red (real) all events spacelike to p; pick a red q ≠ p and colour everything spacelike to q; some red events now lie inside p's lightcone; repeat, and all of Minkowski spacetime turns red. Variants work in many other spacetimes.
Savitt's diamonds
Steven Savitt (2009, 2014) wants, like Callender, to explain why humans come up with manifest time, not to vindicate it. Limit the proper time between p and q to a psychologically realistic one second, and the diamond is about 300,000 km across at its widest. Two people walking past each other at 4 km/h have diamonds that coincide to within one half of one millionth of one percent, so Alice and Bob agree on every noticeable terrestrial event. Savitt: it is no more surprising that we took this "brief, fat structure" to be unbounded than that we took the Earth to be flat.
Callender's objection: what does the explaining? Diamonds try to serve "two masters", relativity and psychology. Relativity permits diamonds big enough to contain most of the universe and small enough to share nothing; the one-second length comes from psychology. Once psychology is admitted, "I don't see why the diamonds should matter." Carl Hoefer adds that a realistic psychological present may need more of the past lightcone than the diamond includes: why should q's future-bounded cone fix how much of p's past counts? And for the metaphysician, diamonds are worldline-dependent, so "saving" animated time this way makes reality observer-dependent.
The Subway foot-long. "Being a foot long" is not Lorentz invariant, yet nobody needs an invariant structure for Subway sandwiches to explain why customers and staff speak successfully of foot-longs: everyone in the shop moves at tiny relative velocities. Likewise, creatures in a relativistic world may call something "the present" without that thing being an invariant structure. The explanation comes from the creatures' situation, not from a privileged geometric object.
2.7 Conclusion
Physics alone neither rules in nor rules out ontology: it mentions neither basketballs (which exist) nor ghosts (which do not). Ghosts are irrational to believe in because no good theory needs them; basketballs are rational because good macroscopic theory does. So the real question is whether good theory as a whole needs a now. All this chapter has shown is that relativity does not employ a special now, and adding one does violence to the theory. Chapter 3 tries slightly modified relativity.
Tearing Spacetime Asunder
Physicists, despite Minkowski's warning that space and time would "fade away into mere shadows", split spacetime into space and time all the time in "3 + 1" formulations. Callender asks whether the time functions this produces (Cauchy time, cosmological FLRW time, CMC "York" time) can serve as a preferred, animatable now. Each either depends on an arbitrary gauge choice, so an Alice-and-Bob argument returns, or is unique only in a drastically truncated, idealised class of solutions that no good physical theory would adopt as law.
3 + 1 formulations, invented by Georges Darmois (1927), slice spacetime into 3D spatial hypersurfaces ordered by a 1D time parameter. They are used to solve the Cauchy problem (Chapter 8), in quantum gravity (Chapter 5), and in numerical relativity, where simulations evolve data on an initial "spatial" surface. Outside Minkowski there are often geometric and material reasons to prefer some time functions; the question is how to understand those "preferences".
3.1 Cauchy Time
Callender's daughter's kindergarten clipped each child's behaviour to green, yellow or red (biting being the ultimate crime). Relativists grade spacetimes similarly, with respect for causality as the goal. The hierarchy is the Causal Ladder; for those who fear causality violation it is a Jacob's Ladder, and the higher a spacetime climbs, the closer to heaven.
Box: Spacetime behaviour chart (the causal ladder, bottom to top)
Notation: p ≪ q if a future-directed timelike curve runs from p to q (chronological precedence); p < q if a causal curve does. I⁻(p) = {q : q ≪ p} (chronological past); J⁻(p) = {q : q < p} (causal past); futures likewise.
- Chronologically vicious: ∀p, p ≪ p. A closed timelike curve through every point.
- Non-totally vicious: some p with p ≪̸ p.
- Chronological: no p with p ≪ p. No closed timelike curves.
- Causal: p < q and q < p imply p = q. No closed causal curves.
- Past (future) distinguishing: I⁻(p) = I⁻(q) implies p = q.
- Strongly causal: every p has a neighbourhood no timelike curve enters more than once. No almost-closed causal curves.
- Stably causal: no arbitrarily small perturbation of the metric creates a closed timelike curve.
- Globally hyperbolic: strongly causal, and J⁺(p) ∩ J⁻(q) is compact or empty for all p ≪ q.
Globally hyperbolic spacetimes are called "predictable". Global hyperbolicity is equivalent to each of:
- A Cauchy surface. There is a closed, achronal, connected Σ that every inextendible timelike curve crosses exactly once. Any p traces back to Σ, so full data on Σ plus deterministic equations predict p.
- Total domain of dependence. There is such a Σ with D(Σ) = M, where D(S) is the set of points every inextendible timelike or null curve through which meets S. No region is causally isolated from Σ: "Like Big Brother, Σ sees all."
A centrepiece of relativity, due to Choquet-Bruhat and Geroch (1969), is that the Cauchy problem is solved for globally hyperbolic spacetimes.
Geroch's splitting theorem (1970) adds: there is a continuous onto map t: M → ℝ such that (i) each level set Σ_a = t⁻¹(a) is a Cauchy hypersurface and (ii) t strictly increases along every causal curve. (ii) makes t a cosmic time function (Hawking, 1969, proved any stably causal spacetime has one); (i) makes it what Callender calls a Cauchy time function. And M is homeomorphic to Σ × ℝ, with Σ of arbitrary topology. Two bonuses: all slices are homeomorphic, so no spatial topology change, and no closed timelike curves.
The figure shows a Cauchy foliation: every timelike worldline (green) crosses each slice once; the lapse sets how much proper time separates neighbouring slices.
Splitting the metric and the laws
The split so far is purely topological. With a Cauchy time function the metric splits too: g = −N²dt² + g_t, where g_t is a Riemannian (+++) metric on each slice and N(xⁱ, t) is the lapse function, the proper time per unit t normal to Σ (the shift vector is set to zero). A clock moving spatial distance dℓ records dτ² = N²dt² − dℓ²: path-dependence survives, and even a stationary clock is affected by the local lapse, which acts as a "disturbing field". Fixing N fixes the foliation.
In the Arnowitt–Deser–Misner (ADM) formalism, relativity becomes a constrained Hamiltonian system: the 3-metric plays "position", the extrinsic curvature "momentum". Einstein's equations become 4 constraint equations on each slice plus 12 evolution equations that push the data forward or backward, together with 4 arbitrary conditions on lapse and shift (see Gourgoulhon, 2012, for the variants). The evolution laws are N-dependent, so they demand a choice of foliation.
| Wish-list feature | Level surfaces of Cauchy time |
|---|---|
| Bisect the universe | Yes: disjoint global slices, each met once by every inextendible timelike observer |
| Achronal | Yes |
| Productive | Yes: D(Σ_t) = M |
| Equivalence relation | Yes |
| Radar-synchronous | Only locally, for observers orthogonal to Σ_t |
| Unique / invariant | No: the foliation is a gauge choice |
Changing the presentation cannot conjure an invariant now. General covariance survives the 3 + 1 move as freedom to rebuild M from infinitely many initial slicings. So the Alice-and-Bob story returns: let Alice use geodesic slicing (N = 1) and Bob maximal slicing (zero mean curvature); in all but contrived spacetimes some events are present for one and not the other. Some foliations misbehave (geodesic slicing breaks down when its fiducial observers collide in a non-uniform field), so a general analogue of Malament's theorem is out of reach, but a reductio on "the future is unreal" still works.
3.2 "Unique" Time Functions
Add restrictions beyond global hyperbolicity and some subspecies of Cauchy time become unique or physically preferred. Relativity is "replete with times": cosmic, Cauchy, Misner, Maupertuis, WKB, Kodama, Chitre–Misner, Kijowski–Tulczyjew "material time", and the Andersson–Galloway–Howard cosmological time function. Nearly all exist only in a proper subset of solutions.
FLRW time
Squint at the sky and matter looks isotropic: radio sources, X-rays, γ-rays and the 3 K thermal background radiation are evenly spread. Assume we are not special and isotropy holds everywhere, so the universe is homogeneous. These two assumptions form the Cosmological Principle. Add Weyl's Principle: galaxy worldlines form a non-intersecting bundle of timelike geodesics orthogonal to the homogeneous slices. In comoving coordinates (galaxies at rest, g₀ᵢ = 0, t = galaxies' proper time), the metric takes the Friedmann–Lemaître–Robertson–Walker (FLRW) form:
ds² = dt² − a(t)²dℓ²
Here dℓ² is a homogeneous, isotropic 3-geometry (three-sphere S³ or elliptic space P³, Euclidean ℝ³, or hyperboloid H³), and the scale factor a(t) is loosely the "radius of the universe". Matter becomes a uniform fluid of density ρ(t), and the 256 coupled equations of Einstein (Callender's count again) collapse into 2 simple relations between scale factor and density, the Friedmann equations, the workhorses of cosmology.
Why FLRW time looks like classical time restored
- Unique slices. The hypersurfaces of constant spatial curvature are unique.
- Physics listens. For fundamental (isotropic) observers, coordinate time is proper time.
- New synchronisation. Moments are picked out not by radar but by density ρ(t): every fundamental observer at a given t reads the same "hand" on the density clock.
- CMC (York) time generalises it: leaves are Cauchy surfaces of constant mean curvature (mean of the principal curvatures, the trace of extrinsic curvature). Crisp and Monton propose it for tensed animation; where it exists (closed space, energy conditions) it is sometimes provably unique (Rendall, 1997; Andersson et al., 2012). FLRW ⊂ CMC ⊂ Cauchy.
Swinburne, Crisp and Monton see the tenser's dream here. Callender (with Bourne, 2004; Callender, 2010) gives four objections:
- Averaging. FLRW and CMC ("M" for mean) rest on elaborate averaging. Your office is not isotropic; neither is the CMB at fine scales. Averaging loses information: why should a metaphysically fundamental time hang on information loss?
- The actual world is neither. We are not fundamental observers: Earth moves, the Milky Way heads toward Virgo, and we move at a few hundred km/s relative to the cosmic rest frame, producing the CMB dipole anisotropy. Galaxies collide, breaking Weyl's non-intersection assumption. A CMC spacetime containing all matter has trivial local geometry and no interesting local physics. The "good approximation" reply misses the point: the question is whether the realistic model keeps the unique time, and Callender suspects the more realistic the model, the less likely uniqueness becomes.
- Residual arbitrariness. Relativity does not rank isotropic observers above others, nor the CMC foliation above one on which the number of McDonald's increases monotonically. "Fundamental observers aren't really fundamental."
- No link to manifest time. Why would the present be the events sharing a ρ-value? FLRW time gives intuitive durations (the universe is 13.6 billion years old, not the 0 years of a photon's proper time from the Big Bang). That is pragmatic usefulness, not metaphysical preference.
Callender's 13.6 billion years is slightly dated: the Planck CMB analysis (2013, refined 2018) puts the age of the universe at 13.8 billion years (13.787 ± 0.020). Nothing in the argument depends on the digit.
Objection 2 has gained force since 2017. Tests of the Cosmological Principle now report tensions: the quasar and radio-galaxy number-count dipole (Secrest and colleagues, from 2021) came out roughly twice the amplitude expected from our motion relative to the CMB frame, a claim still being reassessed in 2025–2026 analyses, and the Hubble tension (local H₀ ≈ 73 versus CMB-inferred ≈ 67 km/s/Mpc) persists. DESI baryon-acoustic results (2024–2025) also hint at evolving dark energy. None of this overturns FLRW cosmology, but it makes "the unique FLRW slicing" an even more idealised object to promote into a metaphysical now.
3.3 Time, Stuff, and Laws
Every special time function restricts the solutions of Einstein's equations: Hawking's cosmic time to stably causal spacetimes, Cauchy time to globally hyperbolic ones, CMC time to spatially compact globally hyperbolic ones, FLRW time to constant-curvature ones. Call the full solution set 𝒮 and the restricted set 𝒮′. The basic question: is the restriction lawlike or factlike? Is the tenser proposing a new theory with only 𝒮′, or observing that the actual world happens to allow their favourite time?
Tensers want the ambitious version: time is metaphysically, hence physically, necessarily tensed, so tense must hold in all physically possible worlds, which means restricting relativity to 𝒮′, effectively a new physics. (John Earman, 1995, discussing Gödel, considers contingent tense, comparable to whether space is open or closed, but no tenser holds it.) As far back as 1936, Sir James Jeans came close: extending relativity to astronomy forced dropping the space/time symmetry, so "time regained a real objective existence, although only on the astronomical scale", and "we are free to believe that time is real". Callender hears in the qualifier an emergent time, which most tensers cannot accept. (Combine Jeans with Trautman and the relativistic revolution merely moved curvature: classically time is curved and space flat; for Jeans, space curved and time not.)
The fair test: is a system of laws corresponding to 𝒮′ on balance simpler and stronger than one for 𝒮? Theories trade simplicity against informational strength (possible worlds ruled out). Relativity is not finished; physics already imposes unobserved restrictions (non-degenerate, continuous metrics; smooth hypersurfaces; no signature change), and early-1970s quark models were weeded out for being too weak even when empirically adequate (Massimi, 2007). Einstein had a more or less complete theory by 1917, but that does not end the balancing act.
FLRW restriction: clearly no. Gravitational science needs Schwarzschild, de Sitter, FLRW perturbations and much more; truncating relativity to the Friedmann equations would wipe out all its local physics.
Global hyperbolicity: tempting but risky. Relativity permits "ill-behaved" solutions with closed timelike curves, spatial topology change, supertasks and naked singularities. Roger Penrose's Cosmic Censorship programme aims to expunge naked singularities; his original (1979) proposal is equivalent to requiring global hyperbolicity, hence a Cauchy time function. Two arguments favour making it law:
- Strength for little simplicity. It rules out naked singularities, naked points at infinity, topology change, multiply connected topologies, CTCs and gravitationally indeterministic regions, arguably surplus structure.
- The conspiracy of good behaviour. Einstein's equations are local, so the metric could at any point tip its lightcones and ruin global hyperbolicity, yet "so far as we're aware that never happens". "The parent in me asks: surely there is a reason?"
Against it, Hawking (with Penrose, 1996): assuming global hyperbolicity "may be ruling out something that gravity is trying to tell us"; it should be deduced from physically reasonable assumptions, ideally by a theorem that unreasonable physics does not occur in generic spacetimes with reasonable matter. Criminal sweeps catch the good with the bad. We do not know our spacetime is globally hyperbolic: arXiv's gr-qc archive lists scores of non-globally-hyperbolic models of our world.
Malament's indistinguishable twins. Take Minkowski spacetime M (globally hyperbolic) and a spacetime M′ made by gluing anti-de Sitter spacetime to Minkowski (not globally hyperbolic). Malament (1977) shows no observer in either could ever tell which one they inhabit. So the existence of a Cauchy time, and therefore of the tenser's candidate now, may be empirically undecidable in principle.
So few endorse the restriction as more than pragmatic.
3.4 Conclusion
Are there relativistic structures that (a) correspond to manifest time and (b) are invariant? Without a general proof, no. Invariant structures abound (points, infinitesimal slices, causal diamonds) but none is animatable. The theory can mimic a classical now (Cauchy level surfaces, FLRW instants, Milne's public space), but these are either not unique across observers or unique only in a serious truncation. Idealise enough and a distinguished time appears, but unless you change relativity's laws, the selection is pragmatic. That was RPP's complaint in Minkowski spacetime, and its echo runs through all of relativity: without a clear physical need for a preferred time, relativity doesn't posit one.
Quantum Becoming?
Quantum theory is often hailed as the escape from relativity's assault on the now: non-locality supposedly demands a preferred frame, and wavefunction collapse supposedly gives an open future and a fixed past. Callender grants the strongest version, Bohmian mechanics with a real preferred foliation, and shows the tenser still loses: either the quantum frame and the becoming frame coincide, and then no physical experience could ever reveal it, or they differ, and the tensed theory conflicts with physics. Collapse-based openness, meanwhile, is "more a series of partial drips and splashes than a worldwide surge".
Callender sets aside energy–time uncertainty, quantum clocks and the quantum arrow of time. For the earlier questions quantum theory "will merely force us to repeat ourselves": non-relativistic QM lives on Galilean spacetime, special-relativistic QFT on Minkowski, and QFT on curved spacetime on globally hyperbolic spacetimes (Wald, 1994, 2009). Time in quantum theory is not discrete, ill-defined or non-classical; that impression comes from confusing quantum clocks with time, like confusing spatial coordinates with particle positions (Hilgevoord and Atkinson, 2011).
4.1 Quantum Mechanics
Interpretation (above all the measurement problem) leaves the theory's "stuff" open, but the predictive core is simple. A system is a complex wavefunction ψ(q, t) on configuration space ℝ³ᴺ, evolving by the Schrödinger equation iℏ ∂ψ/∂t = Hψ, with H the Hamiltonian. Predictions come from Born's Rule: probability density ρ(q, t) = |ψ(q, t)|² (generally, P(a = λᵢ | ψ) = |cᵢ|² for ψ = Σcᵢeᵢ). Schrödinger evolution is linear, unitary and deterministic; getting definite outcomes seems to need an instantaneous, nonlinear, non-unitary, probabilistic jump. Their uneasy marriage is one framing of the measurement problem. Quantum field theory keeps this outline, attached to relativistic rather than classical time.
"Time is that which makes contradictions possible": a die cannot show 5 and 3 at once, only at different times. A fair die gives each face a probability of 16.67%, and for that to be a probability the faces must sum to one. Probability axioms (Kolmogorov) do not distinguish time from space, but quantum theory does: probabilities of exclusive outcomes must sum to one at a time, ∫ dP(r, t) = 1 over all space. That forces unitary evolution, U†U = UU† = I, which preserves inner products and so conserves probability. Quantum mechanics requires a global ideal clock with at least the structure of a Cauchy time function; both its dynamics and its probability apparatus "listen" to it. (Complex wavefunctions serve this time-slice probability structure, per Barbour; Oeckl's general boundary formulation tries to replace hypersurfaces with spacetime regions.)
4.2 Popper's Experimentum Crucis
Karl Popper (1982), after experiments following Bell's work: "If there is action at a distance, then there is something like absolute space", and quantum reasons for absolute simultaneity would send us back to Lorentz. For him, the sixty-year underdetermination between Lorentz and Einstein met its decisive experiment in Alain Aspect's 1981 demonstration of the violation of Bell's inequality. Unlike Chapter 3's cosmological preferences, this looks like a lawlike reason for a foliation, from a fundamental theory.
Callender works with non-relativistic QM: relativistic QFT predicts the same non-local correlations (Smith and Weingard, 1987) and has the same measurement problem (Maudlin, 1994).
From EPR to Bell
- Two spin-½ electrons leave a source in the singlet state ψ = (1/√2)(|↑x⟩₁|↓x⟩₂ − |↓x⟩₁|↑x⟩₂), which is not an eigenstate of x-spin.
- Measured along any common axis, the results always disagree (spin conservation, spherical symmetry), so measuring particle 1 tells you particle 2's result with certainty.
- EPR (1935): assuming locality, particle 2 must already have had a definite spin. Dilemma: quantum mechanics is non-local or incomplete.
- Bell (1964): any local theory obeys an inequality that quantum mechanics violates. Aspect (1981) and successors confirmed the violation; loopholes have been closed or shrunk.
- Consensus: correlations between spacelike events not explicable by any local hidden variables. Complete or not, quantum mechanics seems non-local.
No experimental conflict. Outcomes are random and uncontrollable, so the correlations cannot be used to signal or detect a privileged foliation (the no-signalling theorem). If Popper thought Aspect's data empirically picked out a frame, "he badly misinterpreted them". The charitable reading: the best theory behind the data requires a hidden distinguished frame. Here Popper has company in Bell, who saw "an apparent incompatibility, at the deepest level" between any sharp formulation of quantum theory and fundamental relativity, possibly requiring "radical conceptual renewal".
Loophole-free Bell tests (Delft, Vienna, NIST, 2015) closed the detection and locality loopholes together; the 2022 Nobel Prize in Physics went to Aspect, Clauser and Zeilinger for these experiments. Non-locality is now as experimentally secure as anything in physics, and the no-signalling point still holds: none of these tests reveals a preferred frame.
Collapse in Minkowski spacetime
The measurement problem: Schrödinger evolution entangles the superposition with the device D and observer O, giving (1/√2)(|sees "up"⟩_O|"up"⟩_D|↑x⟩ + |sees "down"⟩_O|"down"⟩_D|↓x⟩), which by the eigenstate rule represents no definite seeing. As Bell put it: "Either the wavefunction as given by the Schrödinger equation is not everything or it is not right." The Copenhagen answer: it is not always right; it collapses to one branch with Born probabilities.
Put real collapses into Minkowski spacetime (Aharonov and Albert, 1981). Particle 1 is measured for x-spin at L, particle 2 for z-spin at R, spacelike to L. Bob's foliation puts L first; Alice's puts R first.
| Stage | Bob's story (L first) | Alice's story (R first) |
|---|---|---|
| t = 1 | Singlet ψ | Singlet ψ |
| t = 3 | L finds x-up: ** | ↑x⟩₁ |
| t = 4 | ** | ↑x⟩₁ |
| Disagreement | Which event collapsed the singlet, which outcomes were chancy, what states existed at t = 3 |
If the wavefunction is real and really collapses, one story is right, which picks out a preferred set of inertial observers, contrary to relativity. Whether that means Galilean spacetime or a preferred foliation of Minkowski is debatable (Maudlin, 1996); either way "Bell's worry isn't idle".
4.2.1 Quantum preferred frames and time
RPP assumed nothing in the world fails to commute with Minkowski symmetries; quantum theory seems to violate that. J. R. Lucas (1998) savours the irony: realistically read, QM distinguishes a probabilistic future of superposed eigenstates from a definite past, with the present as the moment of collapse, "a unique hyperplane advancing throughout the whole universe of collapse into eigen-ness". Two attractions: a unique global hyperplane and a genuinely open future.
4.2.2 Caveats and alternatives
Bell's worry deserves more attention in physics. It is ignored because (a) the no-signalling theorem guarantees empirical "peaceful co-existence", and (b) one only sees it when interpreting the theory. Burying one's head is harmful when we seek to unify the two theories. But Copenhagen is a poor model (Albert, Bell), and other sharp formulations may not clash with relativity; Fleming's (1996) hyperplane-dependent states may even count as relativistic. Callender sets that aside, because his point needs the opposite: grant an interpretation that does privilege a frame, namely Bohmian mechanics, "the best worked out 'realistic' no-collapse interpretation", whose relativistic extensions (Bohm and Hiley, 1993; Dürr et al., 1999, for Dirac; Berndl et al., 1996, and Nikolić, 2005, for Klein–Gordon) need a preferred foliation. Things can get no better for the tenser.
4.2.3 The coordination problem
In Bohmian mechanics there are the wavefunction plus beables (Bell's term), here particles with configuration Q = (Q₁…Q_N), moved by the guidance equation:
dQ_k/dt = (ℏ/m_k) Im(ψ̄∇_kψ)/(ψ̄ψ) (Q₁, …, Q_N)
that is, probability current over probability density. Schrödinger's equation and the guidance equation are the fundamental laws; add the distribution postulate (initial particle density = |ψ₀|², which some Bohmians, including Callender elsewhere, treat as a law) and Bohmian statistics match standard QM exactly.
Bohmian EPR. Spin is not fundamental; it is contextual. A Stern–Gerlach magnet splits the wavepacket into upper and lower parts; a particle starting in the upper half ends "spin-up", and since the dynamics is first-order and deterministic, trajectories cannot cross in configuration space. Put the two-particle configuration in the upper-left of the singlet packet. If A measures first, A gets x-up and B must get x-down; if B measures first, B gets x-up and A x-down. The outcome depends on who measured first, which is not invariant for spacelike events, so one foliation must be physically preferred. Bohm's theory is not refuted because the distribution postulate makes the configuration unknowable in advance, so no-signalling holds and relativistic symmetry holds at the statistical level; only the sub-quantum level picks a frame.
Now the question: is the tenser's metaphysically preferred foliation the same as the Bohmian physically preferred one? If not, the tenser merely trades one conflict with physics for another. So they need a linking principle. Dean Zimmerman (2011) offers a generalisation from experience: e₂ causally depends on e₁ only if, when e₂ was happening, e₁ was happening or had already happened. Raised to a principle of presentism (only the present exists), it ties the becoming frame to the Bohm frame: the state (ψ₀, Q₀) on one slice causes (ψₜ, Qₜ) on a later one. Simplicity alone might also suggest identifying them.
Callender's dilemma for the quantum tenser
- Assume physicalism and a complete Bohmian world: you are a Bohmian system, and every experience or introspection is a Bohmian "measurement".
- If the distribution postulate is a law, no Bohmian measurement can reveal the preferred frame.
- So no feeling, impression or introspection can reliably indicate which frame is the becoming frame. Zimmerman cannot have "generalised from experience".
- Brains could "smell the ether wind" only by acting like a Michelson–Morley device with a non-null result, which requires suspending or completing Bohm's laws.
- Horn A: frames coincide, so experience is no guide to becoming, undercutting the tenser's chief evidence. Horn B: frames differ, so the tensed theory conflicts with physics over the order of events.
Two escape routes, both costly. Dualism: a non-physical mind might sense the becoming frame; Callender concedes the possibility but says there is then too little shared ground to argue. Monton (2006) replies that experience of flow shows that there is a preferred frame though not which, as you can feel wind without knowing its speed. Callender: the disanalogy is that the Bohm frame is not felt at all, so long as feeling is a physical interaction. He also notes Antony Valentini's (2008) proposal to adapt cosmic time to the Bohmian frame is more principled than CMC time, because it rests on something fundamental rather than averaged.
4.3 Quantum Becoming via Collapses?
Lucas, Popper, Abner Shimony, Henry Stapp, Michael Tooley and G. J. Whitrow take inspiration from objective collapse: a "worldwide tide of actualization", a preferred foliation sweeping through the universe, distinguishing "an alterable future and an unalterable past". The street runs both ways: Joy Christian (2001) pursues Penrose's gravity-induced collapse precisely because it treats "the incessant fading away of the specious present into the indubitable past" as an objective attribute of the world.
Does open/fixed map onto superposition/eigenstate? Callender says no:
- Basis egalitarianism. Hilbert space lets ψ be written in any basis. X-spin-down is a superposition of z-up and z-down: collapsing to fixity in x buys openness in z.
- Already-definite systems. Measuring an x-eigenstate for x-spin: is the outcome open because future, or fixed because eigenstate? If open, QM has little to do with openness; if fixed, the "fixed" claims are contingent ones, unlike "tomorrow 2 + 2 = 4".
- Preferred basis reply. Collapse in a privileged basis, e.g. Position, as in GRW (Ghirardi, Rimini, Weber, 1986). But GRW collapses only to near-eigenstates, and Schrödinger evolution re-spreads the wavefunction at once. The "spread of actualization" is "more a series of partial drips and splashes than a worldwide surge".
Collapse models have since been squeezed experimentally. An underground experiment at Gran Sasso (Donadi and colleagues, 2020) searched for the spontaneous radiation predicted by the Diósi–Penrose gravity-related collapse model and ruled out its simplest parameter-free version; continuing X-ray and cantilever experiments keep narrowing CSL/GRW parameter space. Collapse remains a live option, but a smaller one.
Chance and the open future
Shimony and Popper find openness, flow and even freedom in genuinely probabilistic processes. The intuition: if a radium atom has a 0.5 chance at t of decaying tomorrow, there must not "already" be a determinate future containing a decayed atom, or the chance would be 1. A tenseless block has such a future, so non-trivial chances require tensed time (Shanks, 1991). Callender's two worries:
- It depends on the theory of chance. It works most naturally for a Popperian propensity view. On David Lewis's (1994) account, chance is a theoretical quantity that improves the best systematisation of nature; facts about whether the atom decays after t are inadmissible at t and do not affect the chance at t (see Loewer, 2001; Hoefer, 2011). Non-trivial chance fits a tenseless world.
- It mirrors the fatalism argument. "The sea battle tomorrow spoils freedom today just as the radium atom's decay tomorrow spoils non-trivial values of chance today." If the fatalist argument is flawed (Callender cites Sobel's 1998 critique), so is this one: the actual world may contain a decayed atom tomorrow while today it has a one-half chance of decaying, as frequentism and many other views allow.
4.4 Conclusion
Relief from relativity is not easily found in quantum theory. Whether non-locality demands a preferred frame is unclear; if it does, the frame is probably no help to a tenser whose evidence is experiential. If there are collapses, the future they leave open is quite unlike manifest time's, and the argument to it is "fraught with obstacles and alternative paths".
Intimations of Quantum Gravitational Time
Quantum gravity is a set of rival programmes, and each makes stark choices about time. Callender examines the best case for restoring time, causal set theory's "asynchronous becoming", and the best case for eliminating it, semiclassical time in canonical quantum gravity. Causal sets reproduce the old relativistic dilemma and can deliver global becoming only as a strange, ontologically vague growth; semiclassical time is the first rigorous derivation of time from timelessness, though it may smuggle time into its approximations. The minimal time of relativity is both harder to kill and harder to improve upon than is commonly thought.
G. J. Whitrow (1961) framed natural philosophy as a contest between eliminating time and treating it as fundamental. Quantum gravity revives the contest. The programmes (superstring theory, loop quantum gravity, canonical quantum gravity, causal sets, causal dynamical triangulation, non-commutative geometry) each pursue a core "clue". Quantum theory, even without a preferred foliation, brings two temporal commitments that clash with relativity:
- Global inertial frames. Quantum states are tied to them; relativity generally has none.
- Absolute time. Quantum time is non-dynamical; matter does not affect the ticking quantum states obey.
From the quantum side, relativity offers no time fit to push states forward: bodies ignore coordinate time, and proper times are too local. From the relativistic side, "the problem is not so much that there isn't any time in quantum gravity (as is often said), but that there is one in quantum mechanics". These tensions are the problem of time. Start from quantum theory (string theory) and you begin with full-blooded time; start from relativity (loop quantum gravity) and time is already demoted. Because the field is split, "one can find confirmation of almost anything one thinks about time."
5.1 The Best of Times: "Asynchronous Becoming" in Causal Sets
(This section draws on joint work with Christian Wüthrich.) A causal set (causet) is a discrete set of events partially ordered by causality, which "grows" as new events are born to the future of old ones, in a "generally covariant" way. Founder Rafael Sorkin (2006) claims the classical sequential growth dynamics refutes the idea that general covariance forces us to abandon becoming: "things really happen", and the "Now" might be restored "without paying the price of a return to the absolute simultaneity of pre-relativistic days." Callender's verdict, borrowing from Aerosmith: largely "the same old song and dance".
5.1.1 The basic kinematics of CST
Inspiration: Malament's (1977) theorem that the causal structure of a well-behaved relativistic spacetime fixes its geometry up to a conformal factor, i.e. Everything except scale. In CST, counting discrete events supplies the scale and causal relations supply the rest: "order + number = geometry".
Causal set kinematics: a causet is ⟨C, ⪯⟩ with
- Partial order. ⪯ is reflexive, antisymmetric and transitive.
- Local finiteness. For all x, z: card{y : x ⪯ y ⪯ z} < ∞. The structure is discrete.
Faithful approximation. A spacetime ⟨M, g⟩ approximates ⟨C, ⪯⟩ if an injection C → M (i) preserves causal relations, (ii) places on average one element per Planck-sized volume, and (iii) leaves no spacetime structure below the mean spacing. Build one by a random Poisson sprinkling into any globally hyperbolic, bounded-curvature spacetime. The spacetime is not fundamental, just an approximate way of talking about the causet. Sprinkling must be random, not a lattice, or a coarse-grained lattice could be told apart from its boosted twin, breaking Lorentz symmetry (Dowker, 2004).
Two open issues: the novel form of emergence involved (Samuel Fletcher on the Hauptvermutung), and the inverse problem or "entropy crisis": most kinematically allowed causets look nothing like low-dimensional spacetimes (Smolin, 2006), which dynamics is hoped to fix.
Kinematic presents. The CST counterpart of a spacelike slice is a maximal antichain: a maximal set of pairwise causally incomparable events. Like simultaneity slices, these are not unique for a given here-now and not invariant under causet automorphisms: Putnam's problem returns. Wüthrich and Callender (2015) found one new twist: using non-Hegelian subsets (distinct events with exactly the same relational profile to every other event, never causally related to each other), one can define, invariantly, a co-presentness relation whose distinct members are always spacelike, unlike Stein's point-presents. But such sets need extraordinarily unlikely coincidences of births: they are small and rare; and when large, they form the regular lattice patterns that break Lorentz invariance. (David Meyer notes the expected number of non-Hegelian pairs from N samples sprinkled into a 1 + 1 Minkowski Alexandrov neighbourhood is 1 as N → ∞; larger sets contain more pairs, so their expected number should quickly tend to zero.) Non-Hegelian sets have cardinality between 1 and n for a causet of n events.
In CST the informal dilemma becomes strictly true: the bigger the set of co-present events, the less relativistic it is, and vice versa.
5.1.2 Taking growth seriously
Becoming must come from dynamics. A law of sequential growth gives stochastic transition probabilities from each n-element causet to (n + 1)-element ones. Rideout and Sorkin (Rideout, 2000) proved that a few natural conditions force the dynamics into the class of generalised percolation. The simplest member is transitive percolation:
- Label births with integers 0, 1, 2, … consistently with causal order (x ⪯ y implies label(x) < label(y); not conversely, since a newborn may be spacelike).
- Start from the singleton "big bang", element 1.
- When element 2 is born it links causally to 1 with probability p (1 ≺ 2) and fails to with 1 − p; element 3 links to 1 and to 2 each with probability p, and so on.
- Enforce transitive closure: 1 ≺ 2 and 2 ≺ 3 imply 1 ≺ 3.
Rideout: the passage of time "is taken to be a manifestation of this continuing growth"; the process does not happen "in time" but constitutes time.
But a block reading is available. Mathematically a stochastic process is a triple (sample space, σ-algebra, probability measure); transition probabilities are only building materials. Here the sample space is Ω(∞), labelled causets run to infinity (Brightwell et al., 2003), so the theory can be read as tenseless histories with a probability measure.
| Grade of becoming | Content | Callender's assessment |
|---|---|---|
| Local, observer-dependent | Sorkin: our now is approximately local; asking whether a distant spacelike event "has happened yet" lacks sense; an "asynchronous multiplicity of nows", no "super observer" | Equally available in Minkowski: Stein's past-lightcone becoming, Clifton and Hogarth's (1995) worldline becoming |
| Dowker's "occurrence" (2014) | Only CST with dynamics gives events that occur; GR's events do not | 3 + 1 formulations make GR "dynamical" too; if occurrence is a label, GR can label; if it means existence, spacelike events not existing for each other is a radical fragmentation of reality (Pooley 2013: a non-standard A-theory, after Fine 2005) |
| Global, objective | Spacetime itself grows | Faces Smart's objection, the gauge problem, and ends in ontological vagueness |
Smart's objection. J. J. C. Smart (1966) asked "how fast does time fly?": if time changes and change is having different properties at different times, a moving present needs two times. Here: an external growth time and an internal metric time inherited from embedding. How fast are elements born? C. D. Broad (1938) answered that temporal becoming is sui generis, not qualitative change; Bradford Skow (2012) treats the second time as a metaphor for primitive tense operators. Others find this physically opaque. Callender brackets the worry to give CST its best chance.
The gauge problem. CST advocates call label time "fictitious": choosing a labelling is like choosing a coordinate x⁰, and any dynamics favouring one would be non-relativistic. So they impose discrete general covariance (label invariance): the product of transition probabilities along any two paths to the same causet must be equal.
Alice and Bob's birthdays. From the singleton, path α births Alice's birthday a (timelike to the singleton) at label 1, then Bob's birthday b (spacelike to both) at label 2. Path β births b first, then a. Discrete covariance gives both paths the same probability, and the labels are pure gauge: there is no fact about which birthday came into being "first". So in what sense is growth a process in time? John Earman (2008) suggests adding a hidden "actuality" that takes one path, but Arageorgis (2012) rightly notes this contradicts reading labels as gauge.
Gauge-invariant growth. The number of elements is gauge invariant, so the world does go from C₁ to C₂ to C₃. But at C₂ there is no fact, not mere ignorance, about whether it contains a or b, only that it contains one. "Determinately" cannot penetrate the disjunction: the hallmark of ontological vagueness, whose possibility is fiercely disputed (Ted Sider, 2003, argues existence cannot be vague, and that accepting this commits one to four-dimensionalism). CST so read offers a model of a world that is ontologically vague yet grows element by element.
Posts. Will anything ever become determinate in a future-infinite causet? Generally yes. Transitive percolation produces many posts: events comparable to every other event. Rideout and Sorkin (1999) read these as a cosmology cycling "through phases of expansion, stasis, and contraction... Back down to a single element". Suppose N events precede post p. At stage N − 1, all ancestors except p's three immediate predecessors are determinate; two of those three exist, but which two is indeterminate. At stage N all ancestors determinately exist; at N + 1, p determinately comes to be; at N + 2, one of p's two successors exists, indeterminately which. (Objection: at stage N it is not yet settled that p will be a post. Even so, cardinality grows determinately, a weak objective becoming; and the mathematics is well-defined only in the infinite limit.)
Strange features of causal set becoming
- The past can be indeterminate. Aristotelian views make only the future vague (tomorrow's coin is determinately heads-or-tails); here, at C₃ it is not determinate which way C₂ was.
- "The past isn't what it used to be." Growth can give a spacelike event a new timelike link, making it determinate after the fact.
- Two-way becoming. Transitive percolation is time-reversal invariant; relax "births only to the future" and the world can grow both futureward and pastward from a here-now, or be future-finite and grow only into the past.
Causal set theory's most famous result remains Sorkin's heuristic prediction (late 1980s, before the 1998 discovery) of a cosmological constant of the observed order of magnitude, from number fluctuations of order √N. Work since 2017 has developed this "everpresent Λ" into fluctuating dark-energy models fitted to supernova, BAO and CMB data (e.g. Das, Nasiri, Yazdi and colleagues, 2023), and studied causal-set non-locality and "swerves" of particle trajectories as possible observational signatures. Quantum (decoherence-functional) sequential growth dynamics remains under development, and the inverse problem is not solved. The metaphysical gauge question Callender and Wüthrich raise is still open.
5.2 The Worst of Times: Disappearing Time in Canonical Quantum Gravity
Prophets of doom and philosophers of timelessness (Zeno, McTaggart, Gödel) share a flaw: the philosophers eliminate time only by raising the standard for what counts as time impossibly high. Zeno rejects modest "Russellian" accounts of motion; McTaggart demands an A-series (with a C-series) and spurns the B-series; Gödel wants an A-theoretic lapse. The natural response is to "make do with less": they knock time over only by "adorning time with so much heavy jewelry".
Semiclassical time is different. Its standard for time could not be sparer: time is just the parameter with respect to which quantum matter fields evolve, the functional role of t in the Schrödinger equation. "No high falutin' properties necessary." And still, fundamentally, there is none.
Canonical quantization. Put relativity in ADM 3 + 1 Hamiltonian form and quantize it as one quantizes electromagnetism to get QED: replace 3D particles with 3D spatial geometries. The result includes the Wheeler–DeWitt equation:
ĤΨ = 0
where Ĥ is the super-Hamiltonian and Ψ a wavefunction over 3-geometries. Ĥ was meant to stitch 3-geometries into a quantized 4D world, but the side where time should appear is zero: no evolution. This is the problem of time.
That alone does not mean a still universe. Peter Bergmann, a former Einstein postdoc, made clear as early as 1959 (published 1961) that time "disappears" even in classical Hamiltonian GR, a consequence of general covariance (Belot and Earman, 2001). Yet GR handles Mercury's perihelion, light bending and gravitational redshift, a very busy universe. If the Hamiltonian form is equivalent, time is in it, hidden; constrained systems often look timeless until a change of variables reveals time.
Three strategies for the problem of time
- Tempus ante quantum (hidden time). Identify a fundamental time before quantizing: a function of the scale factor, York time, or a matter clock or reference fluid.
- Tempus nihil est (no time). Do physics without time: Carlo Rovelli (2009) says "forget time"; Julian Barbour (1999) calls time an illusion. Challenge: explain why the world seems temporal.
- Tempus post quantum (emergent time). No time fundamentally, but time, change and prediction arise in certain regimes after quantizing. Example: semiclassical time (Banks, 1985; Kiefer, 2012; after DeWitt, 1967).
5.2.1 Semiclassical time
Kiefer traces the idea to Neville Mott's treatment of an alpha particle colliding with an atom (1931). Mott used the time-independent Schrödinger equation for the whole system, then approximated the fast alpha particle as a plane wave and used it as a clock. Write Ψ(r, R) = ψ(r, R)e^(ik·R) (r for the atom, R for the alpha particle) and identify
i∂/∂t ∝ ik·∇_R
With that identification, the atom obeys the time-dependent Schrödinger equation. The identification is justified by functional role: the atom "listens" to ik·∇_R exactly as it would to i∂/∂t. And it works only where the plane-wave approximation holds; where correlations do not allow the derivation, the directional derivative "relinquishes its right to be dubbed time". Time emerges only when subsystems have the right correlations with the whole.
Applied to the universe. Ψ is a functional of 3-metrics h and a matter field φ. The total energy is zero, but subsystems can have energy, "and where there is energy, there is the potential for time development". Assume a semiclassical regime: gravity shows no quantum effects at large scales; quantum degrees of freedom change fast relative to gravitational ones; spacetime is roughly flat at large scales; relevant masses far exceed the Planck mass. A Born–Oppenheimer-style approximation gives
Ψ ≈ exp(iS₀[h]/ℏ) ψ[h, φ]
where S₀ obeys the Einstein–Hamilton–Jacobi equation (a classical spacetime) and ψ describes the quantum field. Then ψ approximately satisfies iℏ∇S₀·∇ψ ≈ H_M ψ (H_M the matter Hamiltonian, ∇ functional derivatives in h). Identifying ∂/∂t ∝ ∇S₀·∇ yields a Schrödinger equation for the matter field: "like a rabbit from a hat", a time-dependent equation from a time-independent one, with heavy gravitational degrees of freedom as the clock for light matter fields.
Consequences:
- Time is contingent. If the universe's initial quantum state were, say, a Gaussian, the procedure fails: no time (unless initial conditions are themselves lawlike).
- The theory is incomplete. Not every regime splits into a near-classical clock and quantum matter, and we want quantum gravity precisely for non-semiclassical domains. Outside the approximation, no time, so no unitarity, so no probability, so no predictions. "What is the approximate solution an approximation to?" (Unruh and Wald, 1989, attempted answers.)
5.2.2 Justifying the approximations
Even granting technical success, does time truly disappear? The picture is impressionistic: there are no "objects" or "subsystems", only correlations in the universal wavefunctional, and the measurement problem and origin of probabilities remain (Callender and Weingard, 1996; Everettians might let subsystems emerge).
The transcendentalist worry: assuming time to get time. Edward Anderson (2007) counts over 20 approximations the programme needs. None formally contains a t; no smoking gun. But are they justified only temporally? Mott's plane wave is justified by the alpha particle's high velocity, and velocity is change with time; the semiclassical regime was motivated by quantum variables changing fast relative to gravitational ones. Callender likens this to David Baker's (2007) charge against Everett: decoherence needs probability to make the branches, while probability needs the branches. Since decoherence and the semiclassical approximations are deeply related, the two chicken-and-egg problems are one. The "package deal" reply, assuming everything at once, gives him the queasiness of Wittgenstein's ladder: here no ladder is thrown away, but two must be climbed at the same time. He leaves the verdict open.
Still, semiclassical time deserves "pride of place in the pantheon": unlike Zeno, McTaggart and Gödel, it relies on no contentious notion of time. It is, to Callender's knowledge, the first rigorous and physically motivated explanation of how time can arise from timelessness. Loop quantum gravity, a descendant of canonical gravity, has suggestive results (Bojowald et al.) but no equivalent derivation, which may be why Rovelli turned to his thermal time hypothesis; string theory starts from a background spacetime and has no comparable result.
The related Page–Wootters mechanism (1983), in which a globally stationary entangled state yields time evolution for a subsystem relative to a "clock" subsystem, received a tabletop experimental illustration with entangled photons at INRIM Turin (Moreva and colleagues, 2014; multi-time extensions 2017). Since then, quantum reference frames and quantum clocks (Giacomini, Castro-Ruiz and Brukner from 2019; Höhn, Smith and Lock's "trinity" of relational quantum dynamics, 2021) have shown that Page–Wootters, relational Dirac observables and clock-neutral formulations are equivalent under conditions, sharpening exactly the "time from correlations" picture of semiclassical time. The problem of time in full quantum gravity is still unsolved.
5.3 Conclusion
Two rigorous but divergent paths: in one, time disappears; in the other, it promises a return to pre-relativistic glory. Quantum gravity may take either or a third. The lesson so far: relativity's minimal time is resilient.
The Differences Between Time and Space
Physics does not support manifest time, but that does not mean it "spatializes" time. Callender revives the old question of how time differs from space, with scientific rather than linguistic data and without assuming a single essence. He finds five physical differences: the metric signature, one-dimensionality, the mobility asymmetry, the direction of time and the natural kind asymmetry. The metric "minus sign" is no formality but the deepest of them, and since the five are logically detachable, a new puzzle arises: the Fragmentation Problem, what glues the features of time together.
Lee Smolin (2006) locates "the scene of the crime" in Descartes and Galileo graphing motion with time as an axis, freezing change into a static picture; Bergson before him, and after relativity Einstein, Meyerson, Čapek and others, debated whether relativity made time another dimension of space. Callender's reply: spatial representation may tempt us psychologically (Boroditsky, 2000, shows we recruit spatial metaphor for time), but we also graph happiness without spatialising it. Until it causes a clear mistake in physics it is no worry. And physics draws sharp distinctions between timelike and spacelike directions. Missing them throws away "pegs" that creatures like us may "hook" onto, making the two-times gap harder to close. This chapter and the next two argue: (1) features thought inessential, above all the metric, are central to time; (2) time is the "great informer", the direction in which our best theories tell the strongest stories (after Misner, Thorne and Wheeler's 1973 "great simplifier").
6.1 The Project Reconceived
Research on "what makes time different from space" (Taylor 1955–1983, Thomson 1965, Webb 1977, Romney 1977, Schlesinger 1975–1991, Huggett 1960, Mayo 1961–1976, Dretske 1962, Meiland 1966, Shorter 1981; Skow, 2007, a recent exception) dwindled over the last three decades. Two blows did it:
- Methodological. The old method hunted for sentences whose truth or meaning changed when temporal and spatial terms were swapped, then consulted intuition to pick the essential difference. That assumes ordinary language holds the key and that there is an essence. Asking what separates protons from neutrons, you would consult a physics textbook, not intuitions, and it would have been idle (before quarks) to decide whether charge or mass "constitutes" the difference.
- Scientific. Minkowski reduced space and time to "shadows", and relativity makes properties once intrinsic to time (open or closed) depend on the contingent matter distribution.
Neither blow is fatal. Relativity's block still separates timelike from spacelike directions independently of matter, matter fields vary with time differently from space, and "few equations are invariant under a transformation of spatial and temporal variables, and no important ones". The fix: use scientific data, and drop the essence assumption. There may be no the difference between space and time. The new question is whether the many differences are connected, or merely coincide in our world.
6.2 Time in Physics
Ignore features shared with space (Hausdorff, connected) and those fixed by initial conditions (time closed, space open). Five candidate differences remain.
6.2.1 The metric
The metric fixes distances between all events, angles between vectors, and so causal structure: "If I believed in the difference between space and time, the metrical distinction would have my vote." Hans Reichenbach (1957): Minkowski's world "expresses the peculiarity of the time dimension mathematically by prefixing a minus sign". Skow (2007) finds the minus sign "too formal and abstract" and too unlike classical time. Callender thinks its importance "badly underestimated".
Signature, not sign. A coordinate change can hide the minus (string theorists' lightcone coordinates x± = (1/√2)(x⁰ ± x¹) spread time over two null coordinates), but the signature is invariant. Take an orthonormal basis at p with g(v_μ, v_ν) = 0 for μ ≠ ν and ±1 for μ = ν; p vectors give +1, q give −1: signature (p, q). For a non-degenerate continuous metric on a connected M, it is constant. The "West Coast" (1, 3) versus "East Coast" (3, 1) choice is conventional, but Stein (1968) gives an intrinsic version: of any four mutually orthogonal vectors, three have self-inner-products of one sign and the fourth the opposite; a vector is timelike iff its self-product is opposite in sign to any non-zero vector orthogonal to it. Or define signature as |#positive − #negative eigenvalues| = 2 for Lorentzian 4D spacetime.
A Lorentzian metric is a Riemannian metric plus a time direction field. Any paracompact manifold admits a positive-definite metric; it admits a Lorentzian metric iff it admits a non-vanishing, unordered direction field (n, −n). Given a Riemannian h with h(n, n) = 1, g_ab = h_ab − 2(h_ac n^c)(h_bd n^d) is Lorentzian, and n, −n are the eigenvectors of the negative eigenvalue: the direction field runs in the timelike directions. Relativity banishes "time" and "space" yet is committed at bottom to a difference between timelike and spacelike.
Classical physics too. Add Newton's preferred rest frame to Galilean spacetime as a timelike field λ^a with ∇_aλ^b = 0; then λ^aλ^b − h^ab = η^ab, a Minkowski metric (Penrose, 1968; Thomas Barrett, 2015). Newton's spacetime already contains Minkowski structure, and a single Lorentzian metric can split in a classical limit into Cartan's two degenerate metrics (Malament, 1986). "Classical and relativistic physics have always singled out time from space metrically."
Physical consequences of the signature
- No continuous flip of time. A timelike vector pointing to t cannot be continuously rotated or boosted to −t; spatial directions can be rotated into their opposites.
- Asymmetric connectibility. Events connectable by a timelike curve are also connectable by a spacelike one; not conversely.
- Closed curves. Chronological spacetimes have no closed timelike curves but may have closed spacelike ones (Friedmann spacetimes do).
- Topology from time. In past–future distinguishing spacetimes, continuous timelike curves determine the topology (Malament, 1977). No spacelike analogue exists, because lightcones bifurcate the timelike but not the spacelike: timelike curves "see" all of M, spacelike curves can be trapped locally.
- Downstream. Much of what is right in the causal theory of time, causal set theory (Rideout and Sorkin, 1999) and Maudlin's directed timelike lines (2010) flows from this.
Speculative physics can demote the signature: signature change by allowing degenerate or discontinuous metrics (Dray et al., 1997), e.g. ds² = t dt² + a(t)²dx², Lorentzian for t < 0 and Riemannian for t > 0; or signature as the output of spontaneous symmetry breaking (Greensite, 1993). In mainstream physics, the signature is "as central to spacetime as it gets".
6.2.2 Dimensionality
Every successful fundamental theory makes time one-dimensional. Classically, the instants form a continuum whose order topology has small inductive dimension 1: time is ℝ. Relativistically there is no set of instants, but two senses survive:
- Local. Decompose any vector at p relative to a 4-velocity: the parallel components form a 1D subspace, the orthogonal ones a 3D subspace (linear-algebra dimension).
- Global. Where a splitting theorem holds, ⟨M, g⟩ = ⟨ℝ, −dt²⟩ × ⟨Σ, h⟩, and the ℝ factor is 1D (not unique, but always 1D).
Both senses depend on the signature, which again shows the minus sign is deep.
Zoe, Adrian and the mesons. MacBeath (1993, like Thomson, 1965) imagines Zoe's group and Adrian's group: Zoe lives to 80 years, Adrian to 40, but Adrian's friends pack into one year what takes Zoe's two. Measured in Adrian's unit, "yonks", Adrian lives 80 yonks and Zoe 40, so Zoe packs in twice as much life per yonk. Wouldn't it be rational to posit two times? Callender: this already happens. Atmospheric muons ("mesons") survive more than ten times longer than their half-life allows by terrestrial clocks, and little meson-scientists would see our processes stretched just as symmetrically. Faced with nearly MacBeath's facts, physics did not posit two times; it posited many reference frames related by symmetry transformations (King, 2004, makes the same point). Extra time is conceivable, but science does not reach for it.
Multi-time physics assumes temporality rides on q ≥ 2 in signature (p, q): (4, 2) brane worlds, (3, 3) Kaluza–Klein theories and (11, 2) F-theories exist in the literature, but none has caught on:
| Problem | What goes wrong with a second time |
|---|---|
| Stability | Dorling (1970): the mechanism that forbids certain decays vanishes |
| Causality | Closed timelike curves appear even in flat, Euclidean-topology spacetimes, since a loop need never turn spacelike |
| Unitarity | Ynduráin (1991): probability is not conserved; proton-decay limits rule out a compact extra time even if rolled up to 1/10th of the Planck radius |
| Tachyons | Even a scalar field in a two-time "Minkowski" spacetime has faster-than-light modes |
| Ghosts | Ghost modes in the propagator for massless vector particles (Foster and Müller, 2010) |
The signature labels are loose. Vafa's F-theory is usually given 12 dimensions with signature (10, 2); the (11, 2) signature belongs to Itzhak Bars's thirteen-dimensional "S-theory" and two-time physics. The point stands either way: two-time models exist on paper and none has become mainstream.
Craig and Weinstein (2009) show the free wave equation is well-posed with extra time directions, but adding interactions spells trouble: "the exception proves the rule". Working models need ad hoc fixes (shrink the extra time, let gravitons explore only one time, thermalise one dimension).
Why such trouble? Physical dynamics is tuned to evolving in the "minus" direction, and conservation laws (particle number, probability current) derive from that dynamics. Demand them along two directions unrelated by symmetry and particles can "sneak" into one time's past: four particles at Time₁ = 1, two at Time₁ = 2. Either forbid sneaking or balance comings and goings by elaborate restrictions. "It's very hard to tell a physical narrative in two metric directions when all the worldlines, dynamics, and conservation principles are tuned to just one." Callender suspects the same reason underlies the trouble with interacting tachyonic and ordinary matter.
6.2.3 Mobility asymmetry
If Callender never hikes Machu Picchu, the reasons are mundane (time, money, will); his future lightcone just happens not to include it. The Jurassic is off his bucket list for a deeper reason: probably no future-timelike path connects him to his past lightcone. Lightcones in our world are not "tipped over", so we move back and forth spatially but not temporally; a Tralfamadorian (Vonnegut) looking at the 4D block would see worldlines wander in three directions but never the fourth.
But Einstein's equations permit time travel (Taub-NUT, Gödel; see Earman, 1995; where one can reach any event along a future timelike path; the spacetime is chronologically vicious). If the field equations define the possible, the Jurassic is barred only as Machu Picchu is, by contingent matter-energy (too much fuel), and the asymmetry is factlike. Yet the pastward bar is total, unlike walls that block only some northward trips, which makes some suspect time travel should be outlawed, as with cosmic censorship. An intermediate option: laws permit time travel but "stack the deck" against it, as gravity curtails upward travel and air rarely gathers in a corner. Saying "most" GR solutions forbid time travel needs a natural measure on the solution space, which we lack (McCoy, 2016). Status: lawlike, factlike or deck-stacked is open.
6.2.4 Direction of time
Temporal directions carry arrows: thermodynamic entropy increase, retarded radiation, time-asymmetric K-meson decay, black holes without white holes. Space has local asymmetries (moss on the north side of trees in the northern hemisphere) and parity-violating particles, but nothing comparable in scope. The heat equation ∂T/∂t = k(∂²T/∂x² + ∂²T/∂y² + ∂²T/∂z²) is invariant under x → −x but not t → −t.
- Standard view. Arrows come from asymmetric boundary conditions; only K-meson decay is nomic (inferred from the CPT theorem plus CP violation, though Arntzenius and Greaves, 2009, contest even that).
- Alternatives. Ilya Prigogine wanted time-asymmetric fundamental laws; others an anisotropy of time itself. Callender is sceptical of both.
- Boltzmannian. Time-symmetric laws plus a low-entropy initial constraint and no matching final one. Since we project the arrow (heat will flow hot to cold next week), the constraint is arguably lawlike: "the laws stack the deck in one direction of time".
- Could be local. Huw Price (1996) notes a Gold universe (after Thomas Gold) with low entropy at both ends, entropy rising away from each.
6.2.5 Natural kind asymmetry
Physics threads events into objects: genidentity (Reichenbach, 1956), events belonging to one object's history. Your birth and your reading this sentence are so threaded; a photon emitted and later trapped can be one photon; a neutrino event and a later electron event are not one object.
One ball or two? A ball bounces off a wall. Slice the world into space-at-a-time frames and flip them: one ball approaches, hits, rebounds, a dull movie that the laws generate frame by frame. Now slice along a spatial direction, each frame part space, part time. Frame 1 shows the incoming and outgoing ball side by side; frame 2 shows them closer; frame 3 one ball at the wall; frame 4 nothing beyond the wall. Callender naturally calls the two images "versions of one and the same ball", not "two balls becoming one and then none": he has chosen threads of identity. The laws of nature make that choice too, privileging the first slicing.
What counts as a worldline? Only curves with v = ds/dt < c everywhere (massive) or = c (massless); tachyonic curves (v > c) and curves that change type are not worldlines. That is no mere word: massive systems are always timelike, massless always null, and spacelike systems would need imaginary mass. Laws take as input properties of things that persist through time (strangeness, charm, charge, mass), never "gruesome" properties. You may call the edge of your desk "a worldline", but no law evolves one part of the edge into another.
6.3 The Fragmentation of Time
The five features are a priori logically detachable. Callender's temporally fragmented worlds:
| World | Signature | Mobility asymmetry | Thermodynamic arrow | Comment |
|---|---|---|---|---|
| 1 | (+++−) | None (time travel) | Along the minus direction | Our world with time travel; the minus direction still special |
| 2 | (++−−) | Along x⁴ | Along x⁴ | Two "times", one with time travel and no thermodynamics |
| 3 | (++++) Riemannian | None | Along x⁴, for no metrical reason | Cf. Greg Egan's Orthogonal trilogy |
| 4 | (++−+) | Along x² | Along x¹ | Maximally fragmented; metric and worldlines pick x³ |
Such worlds suggest the intuition-based search for "the" temporal direction is bankrupt. Better: ask whether the features are physically, not just logically, connected. Our world is not fragmented. Coincidence, or glue? What unifies time? This is the Fragmentation Problem.
The model is the classical equality of inertial and gravitational mass. Students rarely wonder why they are equal because both are called "mass", yet classically they could vary independently. General relativity yoked them together and dissolved the mystery. Finding glue for time's fragments would be a comparable advance. There may be no answer, or several.
One partial glue already exists: the thermodynamic arrow implies nearly monotonic entropy increase, hard (not impossible) to square with travel round a closed timelike curve, so if the arrow is lawlike, macroscopic creatures should find time travel difficult. (A cylinder spacetime wrapped along a spacelike axis, with radius tuned to recurrence times, lets entropy rise and fall consistently around a CTC; silly and unstable, but it shows the Second Law does not strictly forbid CTCs.)
6.4 Conclusion
There may be no single difference between space and time, and intuitions about fragmented worlds are unreliable. But the metric signature is perhaps the deepest division, singling out time for special treatment with ramifications through all of physics. Chapters 7 and 8 propose a new difference, informative strength, and use it as the glue.
Laws, Systems, and Time
Callender proposes a new difference between time and space that also glues time's fragments together. Using the Best System (Humean) account of laws, he argues that simplicity alone cannot single out time (space is a "great simplifier" too), but strength can: time is the direction on the manifold of events in which our best theory tells the strongest, most informative stories, the direction in which it gets the most determinism. That idea explains why time is one-dimensional, why closed timelike curves are shunned, why the direction of time is nomic, and why natural kinds thread through time; in its radical version, the lightcone geometry itself falls out of the laws.
Tensed, flowing, becoming or branching time would distinguish time from space, but the book assumes none of these. Among block-friendly options, an old idea links physical modality to time:
- Causal theory of time. From Leibniz, revived after relativity by Reichenbach, Grünbaum, van Fraassen (with Reichenbach's "mark method"): temporal relations, not spatial ones, reduce to causal relations. It founders on detail (it does not work for relativity as a whole) and motivation (genidentity is no clearer than time) (Earman, 1972). Mellor (1998, tenseless) and Tooley (1997, tensed) take causation as primitive, which buys immunity at the cost of clarity: "explaining the obscure in terms of the obscurer".
- Laws govern in time. Bradford Skow (2007): the direction in which laws govern defines the temporal direction (also Maudlin, 2010; Sider, 2001, and Loewer, 2012, in passing). Callender finds the tie too strong. Real laws act across space: the ten vacuum Einstein equations split into six evolution equations G_ij = 0 and four constraint equations G₀₀ = 0, G₀ᵢ = 0, which hold across a spacelike slice and cannot be derived from the evolution equations. Denying them lawhood is unacceptable; Skow seems to define laws by the time/space split rather than find the split in the laws.
7.1 System Laws and Time
Views of laws range from no laws, to laws that govern in a rich sense, to laws as the best summary of the facts. Humeans explain laws from the facts, not facts from laws. The most attractive Humean theory is the Best System Account (David Lewis, 1973; Cohen and Callender, 2009; Barry Loewer, 2004).
The Best System Account of laws
- Consider deductive systems that make only true claims about what exists.
- Score them on theoretical virtues, chiefly simplicity (measured in a language with a primitive predicate for each fundamental property) and strength (informativeness about particular matters of fact). Others can be added: unification, probabilistic fit, Akaike simplicity.
- Virtues trade off: a simple theory may say little; a huge list is strong but not simple.
- The laws are the axioms common to all the best systems: the best balance of virtues.
- Motivation: physical laws aim to describe as much of the world as possible as compactly as possible.
Time as "great simplifier". On Poincaré's (1913) view, we choose the measure of duration that makes physics simple and powerful. Define time by a pendulum on a boat and physics must explain why the Sun speeds up and slows down as the boat nears or leaves the poles (gravity, humidity, etc. Alter the period). The BSA implies this, but nothing about it is special to time: meter sticks shrinking at the poles would complicate physics just as much, so space is a great simplifier too, and simplicity alone cannot separate timelike from spacelike.
Strength is the other half. Balancing simplicity with strength, a best system will not catalogue events; it will contain algorithms that generate parts of the event-domain from other parts, the shorter the better. Loewer suggested the distribution of basic properties may make one set of directions special for this. Cosmology shows how: the universe is homogeneous and isotropic in three of its four directions (the "perfect" cosmological principle, holding in all four, is false), which lets the system write simple, strong laws, the Friedmann equations.
Algorithms recover infinite information from finite input: one cannot list all members of a denumerable set, but one can give explicit instructions for the nth member (Boolos and Jeffrey, 1989). With a few inputs (initial scale, matter density) the Friedmann equations return the universe's properties at every time step. Algorithms have nothing essentially to do with time; the distribution of physical properties picks out time, and the system exploits that asymmetry by marching information forward in one set of directions. "The system picks the algorithm that picks the time."
Without simplicity we might accept "sideways" algorithms or a patchwork of algorithms running different ways in different regions; neither packs as much inferential punch per line of code. "Simplicity has already had its day; it is time for strength to shine."
7.2 Time is the Great Informer
Some algorithms are especially general and powerful. Of Newton's three laws, the first and third act as constraints across a hypersurface; the Second Law is the algorithm that in principle recovers all events on M. To generate strength, laws seize on a "time" parameter.
Callender's proposal. Given a manifold M_d of elementary events with metric g (used only to make sense of "directions", not presupposing which are temporal): a temporal direction at p is that direction (n, −n), an unordered pair of nowhere-vanishing vectors, in which our best theory tells the strongest, i.e. Most informative, "story". It is local and arrowless, like the direction field of Chapter 6; if the world permits, a global time parameter may also exist.
Determinism as the paradigm of strength. Let a history H map ℝ to tuples of fundamental properties. A theory is deterministic iff for any two lawful histories H₁, H₂, if H₁(t) = H₂(t) at one t, then H₁(t) = H₂(t) for all t (Earman, 2006). That definition presupposes a global time function; reverse it: a smooth t: M → ℝ is a global temporal function if lawful histories agreeing at one value of t agree at all values. (Bad foliations can ruin determinism; and some CTC spacetimes are deterministic despite having no global time function, while Gödel spacetime has no global spacelike foliation yet may have local directions of strength.)
The proposal does not assume determinism. Indeterministic theories can be strong: a Markovian process q(t) predicts q(t + dt) from q(t) alone, so a probabilistic analogue says time is the dimension in which the laws are Markovian. But determinism and Markovianity are only marks of strength; strength is how much of the actual world the theory implies.
- Newtonian mechanics is indeterministic unless force functions are restricted or boundary conditions "enable" it; then finite-time existence and uniqueness hold. For almost all initial conditions it is "almost deterministic" (Saari, 1977, proved global existence for n = 4 particles; it is open for n > 4).
- Quantum mechanics: deterministic if the Hamiltonian is essentially self-adjoint, whose unique extension generates a one-parameter group of translations; read them as time translations and you get determinism; read them as spatial and you get a false theory.
- Relativity: Einstein's equations are nonlinear PDEs with gauge freedom, so only "a smattering of results" on determinism exist (Chapter 8).
7.3 Binding Time
Calling the strength direction "time" is earned only if it connects with time's fragments.
7.3.1 One-dimensionality
A deterministic best system is already maximally strong; adding determinism in a second set of directions buys no strength and costs simplicity, and in a complex world two such directions are hard even to imagine. The same logic holds for indeterministic evolution. Relativity has many global time functions (Adrian's and Zoe's), but because physical properties look the same from each, they are related by a symmetry: the laws use a set of directions, which via the metric is one-dimensional.
Nature need not be kind: in a compact universe the best theory might put data on the 4D boundary and solve for the interior; patches might need unstitchable algorithms or times at right angles. "Nature must be kind. So far, it has been."
7.3.2 Closed timelike curves
The wormhole billiard ball. Identify two timelike-related regions R₁ and R₂ (the mouths of a wormhole). A particle entering R₂ emerges from R₁ with the same velocity. Initial data on a surface S before the mouths: the particle heads in, misses R₁, enters R₂. In one solution it loops back in time several times and finally exits rightward. In another, midway it is struck by a later version of itself, brought to rest, enters R₂, emerges from R₁, knocks its earlier self rightward, and exits just as before. Same inputs, same outputs, wildly different goings-on inside, and infinitely many such solutions (made famous by Kip Thorne, 1994).
- Blow to strength. Arntzenius and Maudlin (2002, 2009) conjecture this is generic in CTC spacetimes: the time-travel region is "constrained but not determined" by S, and the dynamics supplies no probabilities. Knowing everything on S, you learn almost nothing about a later region. "When searching for strength, looking for the directions in which worldlines don't close is a good start."
- Blow to simplicity. Time travel needs a pre-established harmony of initial conditions: try to kill your infant self and a last-minute neuron change, police or "a few well-placed banana peels" must stop you. Such consistency constraints (i) are not local, (ii) depend on global structure, (iii) depend on where the surface lies, and (iv) "appear not to be simply statable". (i)–(iii) hurt strength, (iv) simplicity.
So a strength-and-simplicity-maximising system avoids CTCs, either by exploiting an existing mobility asymmetry or, more radically, by drawing worldline identity so that worldlines do not bend back.
7.3.3 The direction of time
Most agree temporal arrows come from statistics plus asymmetric boundary conditions, not fundamental laws. But thermodynamics carries "a staggeringly large amount" of information, so a best system will add what is needed to recover it: on a Boltzmannian account, a low-entropy initial state (the Past Hypothesis) and a uniform probability distribution over compatible microstates. Since we project thermodynamics, these become laws, and both are imposed across spacelike hypersurfaces: we never normalise probabilities over spatial directions or posit an "East Hypothesis" on a (++−) surface. These laws distinguish time from space as well as past from future, again for the sake of strength. Within time there is a further asymmetry: they add more retrodictive than predictive strength.
7.3.4 Natural kind asymmetry
For Lewis, laws systematise perfectly natural properties (intrinsic, non-modal, monadic); on Cohen and Callender's and Loewer's variants, laws and kinds are chosen as a package. Either way the kinds, with their time/space split, are bound to the laws, enshrining the epistemic point that we pick kinds that support good explanation and prediction. Persistence is tied in too: Sider (2001), following Reichenbach and Carnap, says a sequence of temporal parts is a continuant only if it falls under a causal law. Stitchings are relative to laws and laws to stitchings, so a law-level time/space split carries over into worldlines.
7.4 Metaphysical Variations
What does the best system systematise?
| Conservative version | Radical version | |
|---|---|---|
| Input | Events on a given Lorentzian ⟨M, g⟩ plus fields | Events on bare M plus fields |
| Output | Laws L that privilege some time | Metric g and laws L together (and perhaps the natural kinds) |
| Status of "time" | An honorific for the directions of the main algorithms | Even the timelike/spacelike metric split comes from systematising |
| Surprises | No algorithms, no time; algorithms along non-timelike directions make timelike directions non-temporal; many algorithms, many times | Lightcones fall on the "system" side, not the "systematized" side; still objective, since BSA laws are objective |
On Lewis's view the system is the source of all physical modality, so chance and causation hang on it too: chances are normalised at times, and causal asymmetry runs along timelike directions. "An awful lot of metaphysics can hang on" the honorific.
Callender finds the radical version most natural. In relativity spacetime is dynamical, so systematising matter on a fixed background is "wildly inappropriate" and could never yield Einstein's equation. Even for fixed backgrounds, we moved from Newtonian to Galilean spacetime (dynamics did not need absolute rest) and from Galilean to Minkowski (no need for absolute simultaneity), matching spacetime symmetries to dynamical symmetries: geometry and laws come as a package. ("Centre of the universe" was an Aristotelian kind, "absolute rest" a Newtonian one; kinds belong in the package too.) This also helps Huggett's (2006) best-system relationism against Belot's (2011) objection that there is no simplicity metric for geometry: look for the simplest package of kinds, laws and geometry, not geometry alone.
Geroch's causal cones. Suppose the world is a bare 4D manifold M plus fields φ^α obeying, at every point, a first-order quasi-linear system k^{Aa}_α(φ) ∇_a φ^α = j^A. Maxwell, Klein–Gordon and Einstein fields can all be written this way. Such systems admit a hyperbolization: a tensor h_{Aβ} such that, for some covector n_a, n_a h_{Aβ} k^{Aa}_α is symmetric and positive-definite. That defines the causal cone at p: tangent vectors ξ^a with ξ^a n_a > 0 for every such n_a, a non-empty, open, convex cone. Robert Geroch (2011) shows these systems have well-posed initial value problems, so field values along curves inside the cones depend on values at p. Geroch's interest: cones could be wider than light cones and differ field to field. Callender's: in our world all fields' causal cones coincide. Creatures whose rods, clocks and flashlights are themselves field configurations would attribute these robust, agreeing cones to spacetime geometry, although only fields and a manifold exist. Informative strength plus the laws define the metrical geometry; the principal time/space difference lives in k^{Aa}_α.
7.5 Questions and Connections
Objective, really? Aristotle asked whether time, as the number of motion, would exist without a soul to count; Thomist commentators replied that a world with motion is countable whether or not anyone counts. Likewise, given (shifting) standards for a strike, a 1970s pitch by Mark "The Bird" Fidrych objectively was or was not a strike; given standards of strength, a direction objectively is or is not temporal.
Relativism? The proposal is "knee deep in relativism, but this is a virtue": time is relative to standards and to a choice of natural kinds. If Martian scientists carved nature differently and their strongest direction were orthogonal to ours, is one side wrong? That reduces to whether there are objective natural kinds. Callender prefers an abundant conception of properties, so Martian time counts as time for them if it plays the time role in their vocabulary.
Ties? If two orthogonal systems were equally good, there could be two times or none. Not our world, where the choice is lopsided. But E. A. Milne (1948) once proposed a cosmology with two times: the ideal clock of atomic phenomena related logarithmically to that of gravitational phenomena, no single system marching both to one time. There one might speak of a particle-physics time and a gravitational time.
Causal theory of time. Causal theories define "later than" via physically possible causal chains (light signals). From Alfred Robb (1914) onward they derived the Lorentz transformations from modest foundations, but stumbled on explicating possibility, genidentity and causation empirically (actual signals do not suffice), and on relativistic solutions with CTCs and no global slices; enthusiasm waned from the 1970s. Two aids: Malament (1977) and Hawking, King and McCarthy (1976) proved that in past–future distinguishing spacetimes, causal structure fixes topology and the metric up to a conformal factor, so ⟨M, ≪⟩ nearly yields ⟨M, g⟩ (and the restriction may be justified on systems grounds); and the best system can supply the modality in ≪, defining one event as later than another iff the best system says a physically possible causal chain links them.
Governing. The view is a "systematized" version of the Tooley–Skow–Maudlin governing idea (1997, 2007, 2007), with three advantages: (1) a Humean explanation of why laws pick out time, grounded in the distribution of properties rather than a primitive; (2) no ban on axioms acting across spacelike surfaces; (3) above all, connections binding the fragments, which even non-Humeans, for whom good explanation is a symptom of modality rather than constitutive of it, can borrow.
The proposal leans on the Best System Account, itself contested: anti-Humeans (Armstrong, Maudlin) reject laws as summaries, and the BSA's notions of simplicity, strength and "balance" are notoriously vague and possibly language-relative. Callender's embrace of relativism about time via abundant natural kinds is a bold bullet to bite. The geometry-from-fields picture also sits uneasily between relationism and substantivalism, since the fields still live on a background manifold, as he notes. (Small slip: the baseball pitcher is Mark Fidrych, not "Fidrich".)
Since 2017, the Best System debate has moved toward Callender's emphasis on usable strength. Pragmatic or "best predictive system" versions (Michael Townsen Hicks's "dynamic Humeanism", 2018; Chris Dorst, 2019; Jaag and Loew, 2020) argue that what counts as a good system reflects the needs of limited, embedded agents, which fits his claim that time is where the system's predictive algorithms run. Meanwhile Eddy Keming Chen and Sheldon Goldstein's minimal primitivism (2022) offers a non-Humean governing view of laws that does not need a fundamental direction of time, a rival answer to the same question of how laws can single out temporal directions. Work on "action at a temporal distance" within the BSA explores laws that are non-Markovian, testing the link between Markovianity and time.
7.6 Conclusion
Even in relativity's block, sharp differences separate timelike and spacelike directions, above all the signature. Callender's indirect answer to what makes them line up: the laws single out time, loosely "the direction in which physics tells its best stories", and this picture reveals informal connections among the other temporal features, justifying calling the asymmetry the system seizes on temporal. Chapter 8 makes the connection rigorous.
Looking at the World Sideways
Physics writes its story on spatial pages and flips them in the temporal direction; Maudlin (2007) calls this time's "production", perhaps the last remnant of manifest time in physics. Callender asks whether we could write on tilted pages and read "from east to west". Making Chapter 7 rigorous, he defines strength as possession of a well-posed Cauchy problem, and proves that for the broad class of second-order linear equations only hyperbolic equations with data on spacelike surfaces are well-posed, so strength picks out a one-dimensional, Lorentzian time, without closed timelike curves, aligned with production. The loopholes (data on light-like pages, or on partly timelike pages with constraints) exist but tell worse stories.
Either answer would matter. If basic physics runs sideways, the last vestige of manifest time is gone and seekers of an arrow of production lose. If not, the laws are tuned to temporal directions, vindicating Chapter 7. The verdict: "a qualified No. One cannot tell the stories of physics in the non-temporal directions; or rather, one can, but the stories aren't as good." Rigour costs scope: fewer features are bound, but more tightly.
8.1 Strength and Well-posed Cauchy Problems
Not every notion of strength splits time from space, so the choice matters.
The dew drop. Ralph Waldo Emerson wrote in 1836 that "each particle is a microcosm" that "faithfully renders the likeness of the world". Take it literally: given the exact state of one dew drop plus the laws, could you deduce where your lost socks are, or the next supernova? In a classical gravitational universe, with a few extra assumptions (total particle number; bounded speeds and accelerations; no sourceless gravitational radiation; smooth trajectories; no collisions), Schmidt (1997, 1998) proves that an accurate enough specification of any piece of one worldline fixes that whole worldline and then all the others. A water droplet in Concord, Massachusetts in 1731 "contains (with some help) enough information to deduce your whole life". Feed in a timelike segment and strength flows spatially. On Schmidt's notion, every part determines the whole, so strength cannot distinguish directions at all.
Callender wants strength for creatures like us, Earman's (1984) "strength in intended applications" rather than Lewis's strength in itself. Schmidt's theorems need inaccuracy going to zero (Bishop, 2003), information no finite creature could store, and false assumptions (no collisions). "A Laplacian demon may need no more than a water droplet... But there is a reason why actual scientists ponder more than droplets."
At the other extreme, Manchak (2008, building on Geroch, 1977) proves prediction is impossible in relativity: to predict p from q, all past causal curves through p must meet q's past lightcone, so part of what q sees must be a Cauchy surface whose future domain of dependence includes p. That fails in many spacetimes, Minkowski included; and if knowing the future requires knowing the surface is Cauchy, prediction fails everywhere.
A Goldilocks problem. Grant too much (Schmidt) and a droplet predicts everything; grant too little (Manchak) and nothing can be predicted. Numerical relativists sit in between: they assume Cauchy surfaces, put reasonable data on them, and evolve, granting more than the scrupulous allow but far less than the generous (say, the 10,104th derivative of the solution at a point). Callender stipulates the middle notion: the most widely used sense of determinism in physics, true of many fundamental laws, and a goal of much work.
Determinism in mathematical terms: a Cauchy problem with a unique solution, a one-to-one correspondence between solutions and Cauchy data on an (n − 1)-dimensional hypersurface S. For a second-order equation, data are u(x, 0) and u_t = ∂_t u(x, 0). Data on t = 0 only gives an initial value formulation; solutions near S give a local problem, everywhere a global one (for linear equations local usually implies global; not for nonlinear).
Hadamard well-posedness (un problème bien posé, Hadamard 1902)
- Existence. A solution u exists for any data d in an admissible set X.
- Uniqueness. u is uniquely determined, within a set Y, by d.
- Continuous dependence. u depends continuously on d in a suitable topology. With a norm: ‖u(x)‖ ≤ F(x)‖u(0)‖ for a non-decreasing non-negative F.
Why (3) matters to us: then u(t) = G(u(0), u_t(0)) with G continuous, so to locate an asteroid at time t within ε, there are d₁, d₂ > 0 such that initial errors below them keep the prediction within ε. Without continuity, rounding and noise yield wildly different solutions, and a theory needing infinitely precise data "is worthless to us".
Hadamard said ill-posed problems lack physical meaning; Callender does not go that far (many textbook problems are ill-posed yet significant), but well-posedness is the ideal physicists and simulators pursue. Between full well-posedness and mere unique existence lie many grades: Lyapunov versus Hölder stability (the backward heat equation fails the first but meets the second), and "almost uniqueness" (the Monge–Ampère equation has exactly two solutions; see Ames and Straughan, 1997).
Boundary data types (two variables x, y):
- Dirichlet: u(x, y) = f(x, y) on the boundary.
- Neumann: du/dy (x, y) = g(x, y), the normal derivative.
- Cauchy (mixed): u = φ(x) and du/dy = ψ(x) together. Usually "mixed" if the surface is meant to be timelike and "Cauchy" if spacelike; Callender says "Cauchy" for both to avoid begging the question.
"Spatial" Cauchy problems exist, e.g. The sideways heat equation (x and t swapped: find the temperature inside a rod from temperature and heat flux at one end), and it is ill-posed. Nothing in Cauchy data itself presupposes a time/space split; data could march "east". (McCabe, 2005, objects that choosing Cauchy rather than Dirichlet or Neumann problems stacks the deck toward Lorentzian signature; Callender concedes the argument would not run for those, but notes most fundamental equations are hyperbolic, so the choice is not idiosyncratic.)
8.2 The Worlds
The possible worlds are the solutions of second-order linear partial differential equations on ℝᵈ:
( Σ a_ij ∂²/∂x_i∂x_j + Σ b_i ∂/∂x_i + c ) u = 0
with a symmetric coefficient matrix a_ij, vector b_i and scalar c, smooth in the coordinates. This covers the wave, heat, Schrödinger, Klein–Gordon, Euler and Poisson equations and parts of Dirac. Generality:
- Higher order reduces to second order by auxiliary fields (though not every first-order system becomes second-order, which matters because Ásgeirsson's theorem needs second order).
- Nonlinear equations are locally approximated by linear ones; nonlinear terms cannot make an ill-posed linear problem well-posed (at best they spoil well-posed ones), and Leray (1953) proved quasi-linear second-order hyperbolic equations are well-posed with data on spacelike surfaces.
- Effective equations inherit the split from fundamental ones. Geroch (1996): perhaps all fundamental PDEs are hyperbolic, elliptic and parabolic ones being approximations. Poisson is a truncation of hyperbolic Maxwell; Schrödinger the non-relativistic limit of relativistic field theory; Newtonian mechanics its classical limit; Navier–Stokes holds only macroscopically.
(Callender notes Albert Lautman (1946) as the only earlier philosopher to connect PDE structure with time.)
8.3 Proposal
The rigorous proposal. For worlds described by the second-order linear equation, a temporal direction at p on ⟨M_d, g⟩ is that direction (n, −n) in which our laws allow a well-posed Cauchy problem, i.e. The direction normal to a (d − 1)-dimensional hypersurface through p on which Cauchy data yield a well-posed problem. A PDE determines invariant structures (e.g. A field of Monge cones) that fix which hypersurfaces work; the proposal turns this around to implicitly define the timelike.
Picture a manifold of events compactly described by such an equation, with no axis labelled "time". Turn ⟨M, g⟩ this way and that, looking for a direction with a well-posed Cauchy problem. Does this single out directions, and do they match what we call time?
8.4 The Argument
PDE type depends on the eigenvalues of a_ij: let P = number of positive and Z = number of zero eigenvalues.
| Type | Eigenvalue condition | Example | Well-posed Cauchy problem? |
|---|---|---|---|
| Hyperbolic | Z = 0; P = 1 or P = d − 1 | Wave, Klein–Gordon | Yes (with data on spacelike surfaces) |
| Parabolic | Z > 0 | Heat, diffusion | No: too many solutions (non-unique) |
| Elliptic | Z = 0; P = 0 or P = d | Laplace, Poisson | No: non-existence, non-uniqueness or discontinuity |
| Ultrahyperbolic | Z = 0; 1 < P < d − 1 | Two-time wave equation | No: ruled out via Ásgeirsson's theorem |
Type is coordinate-independent: a_ij is symmetric, so its eigenvalues are real, and their signs are invariant under smooth non-singular coordinate changes.
Callender's argument that strength finds time
- Strength (for us) = possession of a well-posed Cauchy problem, with data on a non-closed hypersurface.
- Among second-order linear equations, only hyperbolic ones have well-posed Cauchy problems. All linear diagonal second-order hyperbolic systems on a globally hyperbolic region with data on a smooth spacelike Cauchy surface Σ are well-posed (theorem, see Wald, 1984). Flip a sign in ∂²φ/∂t² = ∇²φ − m²φ to make it Laplacian, parabolic or ultrahyperbolic and well-posedness is lost.
- Define characteristic surfaces y = c by Σ a_ij (∂y/∂x_i)(∂y/∂x_j) = 0; a hyperplane is spacelike when the strict inequality holds for its normal. For hyperbolic equations the characteristic conoids form two sheets from each point, splitting space into three regions: surface elements between both sheets are spacelike; directions into a single sheet's interior are timelike (Garabedian, 1964). These are lightcones in PDE dress, and the Lorentz group is exactly the group preserving the wave equation.
- By Ásgeirsson's mean-value theorem (1937), data on a mixed (part timelike) surface over an arbitrarily small region determine the data on a much larger region of the same surface. So data cannot be freely chosen; arbitrary data typically yield no solution. Hadamard saw why: points on such a surface lie in each other's domains of dependence, an "infinite series of relations"; a pulse from one part is carried by characteristics to another.
- So arbitrary Cauchy data give a well-posed problem only on surfaces carefully bisecting the two lobes of the characteristic cones: spacelike surfaces.
- Therefore the direction of strength is timelike, and the desire for well-posedness alone, not smuggled-in relativity, does the work. (Courant and Hilbert, 1962, give the details.)
8.5 Illustration
A toy wave equation. Take u_tt − u_xx = f(x, t), but instead of putting data on the x-axis, put u and its normal derivative on an arbitrary curve C. For a point P off C, the two characteristic lines through P hit C at Q and R; u(P), if it exists, is fixed by the data on arc QR and by f inside triangle PQR. Define C as spacelike if, as P approaches a point of C, Q and R approach that same point. No prior time/space split is assumed. Yet u(P) is consistent with arbitrary data on C only if C is spacelike in this sense. Take the temporal direction to be locally orthogonal and strength has "found" time. (Callender saw the example in a PDE textbook in a library and could never trace it again; he apologises to the author.)
8.6 Is It Time?
Three facts weld the strength direction to time's other features:
- Dimension fact. A well-posed Cauchy problem needs data on a (d − 1)-dimensional submanifold; in four dimensions, on 3D hypersurfaces. So time is one-dimensional.
- Signature fact. The metric signature is tied to the signature of the free field equations: Klein–Gordon, Dirac and Weyl equations represent the irreducible unitary representations of the local spacetime symmetry group, so changing the signature changes those equations (McCabe, 2005, shows flipping the metric sign flips Klein–Gordon's). For covariant field equations a_ij has the same eigenvalues as the metric. With the argument above, data must sit on the (+++) submanifold, not a mixed (++−) one: time marches in the "−" direction, and the spacetime signature is Lorentzian. Production runs with strength and signature; productivity in two directions would break the link between the laws' signature and the metric's.
- CTC fact. Spacetimes with closed timelike curves have no Cauchy surfaces, hence no well-posed Cauchy problems: no maximum strength where worldlines bend back on themselves.
Max Tegmark (1997) used the same facts for an anthropic argument: predicting observers find themselves in Lorentzian spacetimes, as fish caught in nets with one-foot holes tend to be over a foot long. Callender borrows Tegmark's observations but declines the further claim that this kind of prediction is necessary for observers. There is also a small literature deriving Lorentzian signature from spinor structure or electromagnetism (van Dam and Ng, 2001; Borštnik and Nielsen, 2002; Itin and Hehl, 2004), which could add indirect links between strength and signature. And note the argument's scope: second-order linear equations, Cauchy (not Dirichlet or Neumann) data, and Hadamard's notion of stability; other notions of strength may behave differently, as Callender admits. A weaker route, unique existence for Geroch's quasi-linear symmetric first-order systems, again requires a hyperbolization, so the time/space split enters as a sufficient condition there too. (Name check: the relativist Callender calls "Jeff Wincour" is Jeffrey Winicour.)
So, for worlds of this form and strength as well-posedness, one can rigorously show that strength distinguishes time from space and connects it with the topological (one-dimensional), metrical (Lorentzian), productive and mobility (no CTC) features.
8.7 Turning Pages in Non-temporal Directions
Relativity already lets us tilt the "now" anywhere short of null. Can we tilt further?
8.7.1 Pages of light
Characteristic initial value problems put data on null surfaces. Introduced by Bondi and colleagues (1962) for gravitational waves, with Sachs (1962), Penrose (1963), Dautcourt (1963) and others, they are now used in numerical relativity, the holographic principle and radiation problems. Setups vary: a past or future lightcone with its vertex; two intersecting null wedges meeting on a spacelike 2-surface; past and future cones meeting on a 2-sphere; a null surface plus a timelike worldtube. Data are the solution on the surfaces plus data on the intersection, not normal derivatives: boundary information replaces Cauchy derivatives.
Advantages: closer to how observation comes in; above all, data are unconstrained or constrained only by ordinary differential equations, escaping the Cauchy approach's "nasty elliptic constraints". Rendall (1990) proved well-posedness for intersecting null surfaces; Choquet-Bruhat and colleagues (2011) proved existence for the lightcone case, and Luk (2011) continued the work.
A flashlight's whole history as "initial" data. Switch on a flashlight; photons race up the future lightcone. Intuitively it "takes time" for light to get from one event to another, and we usually march data in that very direction. In the characteristic problem we can instead take the entire lifetime of the photons as the initial "moment" and push it "forward" to the next slice. From the standpoint of manifest time this is bizarre: data placed on a "moment" that itself takes time. Here the arrow of production and the arrow of strength part company with the intuitive time/space split.
Tegmark dismisses the case: you would have to "live on the lightcone". Too quick, says Callender: most cosmic evidence arrives as light, so our data do live on the past lightcone (Komar, 1965). Still, the threat may be illusory. To his knowledge the null problem has never been solved in characteristic coordinates; Rendall's proofs reduce it to a Cauchy problem; Choquet-Bruhat et al. Use normal coordinates; and (per Jeff Winicour) physics' characteristic problems have enough hyperbolicity to guarantee a well-posed Cauchy problem anyway. No realistic system is known where the characteristic problem is well-posed but the Cauchy problem ill-posed. For now the characteristic approach looks parasitic on the Cauchy approach.
8.7.2 Pages of time
Now tilt all the way: put Cauchy data on a mixed (++−) surface and evolve in the remaining + direction. Courant and Hilbert's argument is valid, but an ill-posed system can become well-posed when constraints are imposed on the data. Classical electromagnetism is the precedent: Maxwell's equations with arbitrary fields at a time are not well-posed, but are once Gauss's laws constrain the initial fields.
- Geroch's suggestion. For Klein–Gordon on a partly timelike hypersurface, add the Constraint: every point of the surface has a neighbourhood with a Klein–Gordon solution. Local uniqueness plus Constraint gives global uniqueness near the whole surface by patching overlapping local solutions. Given the similarity of Klein–Gordon and Einstein Cauchy problems, timelike well-posed problems plausibly exist in GR too.
- Craig and Weinstein (2009) rigorously show a well-posed Cauchy problem for data on a mixed (++−) surface, but with a highly non-local constraint.
Both "cheat" by referring off the surface. Geroch's Constraint is local, so the loss in predictive power might be unnoticeable to us; but simplicity, in the background so far, returns: Occam's razor should prune such examples for their ugly constraints, especially Craig and Weinstein's non-local one, and especially if strength means usable strength for creatures like us.
8.8 Conclusion
The reconceived project bore fruit. Rather than chase "the" difference, Callender found connections among seven aspects of time in physics:
Seven aspects of physical time
- (a) One-dimensionality.
- (b) Metrical distinction (signature).
- (c) Mobility asymmetry.
- (d) Direction of time.
- (e) Asymmetry of strength.
- (f) Natural kind asymmetry.
- (g) Production.
Chapter 7 linked all seven informally; this chapter links (a), (b) and (c) to (e) rigorously, and perhaps (f).
The physics shows that "one can get robust counterfactual dependencies in all directions, null, timelike, and spacelike". The world does not "unfold" along time as the tenser hopes. Yet the tenser's intuition holds a central insight: there really is a difference between space and time, embodied in its connection to the laws; forget simplicity and strength and it tends to vanish, but seek the simplest and strongest system and it stands out, an objective difference that survives modern physics' onslaught against time.
Do We Experience the Present?
Before explaining manifest time, Callender must remove a rival explanation: that we simply perceive an objective, tensed present that physics misses. He argues that no experience confirms tensed metaphysics; the "problem of the presence of experience" is really the Temporal Knowledge Argument, which trades on general features of indexicals and overgeneralises to "here" and "I". He then turns to psychology and asks a rarer question: do we even directly experience synchrony? Temporal ventriloquism, recalibration, divergent measures and illusions suggest the more we probe the subjective "now", the more it falls apart.
Brian Greene (2004) admits that by day he thinks the mind imposes the Now on time, but "by nightfall... It's hard to maintain full resistance" to the idea that science misses a fundamental quality of time. Callender grants that resisting manifest time is almost psychologically futile and aims to explain why. This chapter and the next have a "Humpty Dumpty" structure: break the now apart here, reassemble what is left in Chapter 10.
9.1 Metaphysics of Time
| Tensed theories (now is metaphysically special) | Core claim |
|---|---|
| Presentism | Only the present exists |
| Growing block | Past and present exist; the present is the cusp of being |
| Moving spotlight | Eternal block plus an irreducible now crawling up worldlines |
| Branching | Futures diverge from a settled trunk; the now is where branches drop off |
| Tenseless (detensers) | Time has only the properties of allocentric science; the now carves no joint (Grünbaum 1973, Smart 1966, Williams 1951) |
Tensers include Broad (1923), Dummett (1960), Geach (1979), Lucas (1973), McCall (1969) and Prior (1967). Callender doubts the tenser/detenser framing (Chapter 13), but notes the strategic situation: detensers pursued a "nuclear" strategy, impossibility proofs (McTaggart's paradox, special relativity), failed to win outright, and so left the tenseless theory hollow on experience. In a "conventional" dispute over which theory best explains the phenomena, the conventional wisdom (conceded by detensers like Balashov, 2005, and Falk, 2003) is that tensed theories fit experience better. Callender disagrees: it would be "a minor miracle" if experience could discriminate between two metaphysical hypotheses. Philosophers gesture at "feeling time pass" while the people who study experience in detail (phenomenologists, cognitive scientists) stay silent on metaphysics. "It's time detensers stand up for themselves."
9.2 The Problem of the Presence of Experience
The label covers several distinct problems, few about experience and none damaging to tenselessness. Experience troubles the detenser only if one reads the theory into the data: William Lane Craig (2000) says we are "appeared to presently"; George Schlesinger (1991) says the present is "palpably real". Callender allows we may experience subjective simultaneity but doubts we experience presentness. "Do I have an experience of presentness or am I simply at present having an experience?" He senses his computer's hum and his back pain, but no "stamp of presentness".
Four reasons experience does not reveal objective presentness
- Objects look the same whatever their age (Mellor, 1998). The point of light from Jupiter shows the planet an hour in the past; other points may be roughly simultaneous or a million years old, yet all look alike. We are not sensing Jupiter's presentness. Craig replies that observed events usually fall in a brief present interval and that our basic belief survives "extraordinary" exceptions. Callender: the claim was about observing tense, not believing in it, and stars and thunder after lightning are anything but extraordinary.
- "My experiences are present" is a tautology. The spotlight-on-a-wall picture begs the question; on the block view your first school day is an existing, past experience. The non-question-begging version, that in the present all one's experiences are present, every detenser accepts.
- Presentness is an odd phenomenal property. Phenomenal properties normally distinguish experiences (loud from quiet, red from blue), but every experience is present (Hestevold, 1990). There may be indexical phenomenal properties ("the way this feels"), but no route from them to a global objective present.
- Experience cannot reveal logical form. Tensers make presentness monadic; detensers make it relational. Sydney Shoemaker (1996): "to the right of" is at least triadic but not experienced as such; "heavy" feels monadic though it is relative to one's build. Because we are not among the objects of our perception, relations to ourselves are represented pre-reflectively as monadic. Simon Prosser (2006) argues temporal experience makes this one-place substitution especially natural.
Spatial and weight cases come with disagreement data (the fork is on my right and your left; the weight is heavy for me, not for a bodybuilder), and nobody is wrong. Physics and psychology also supply disagreement data about the present, but "for a host of fascinating reasons" we don't notice it and so represent the present as monadic. Chapter 10 explains why.
9.3 The Temporal Knowledge Argument
David Mozersky (2006), a detenser, summarises the stock challenge: we may know all week that the movie and Friday, 1:00 p.m. are simultaneous, yet learning that Friday, 1:00 p.m. Is present seems to teach a new fact, so tenseless relations cannot be all there is. Note the experiential gloss on what is really an argument about knowledge. John Perry (2001) calls it the Temporal Knowledge Argument and says it lies beneath almost every argument for tense; Callender agrees and calls its continued use "something of a scandal".
It is the temporal twin of Frank Jackson's Knowledge Argument (1986) (colour-deprived Mary learns a "new fact" on seeing red), which few endorse, Jackson included. (Even its defenders need not accept the temporal version: Chalmers (1996) stresses Mary's knowledge is not merely indexical, and indexical gaps vanish for an omniscient observer.)
Why the Temporal Knowledge Argument fails
- Perry's case (1979): I knew all along the meeting is at 5 p.m.; only the thought "now is 5 p.m." explains my getting up. Indexicals are essential: substituting non-indexical descriptions does not preserve cognitive significance.
- The same holds for "I": the shopper sees a trail of flour from someone's leaky bag and acts only on learning "that's me".
- And for "here": "I knew the meeting was in Room 215, but didn't know that Room 215 is here."
- The shopper case does not prove a me over and above the person with the cart; the spatial case does not prove a here over and above Room 215.
- So the temporal case does not prove a now over and above the network of temporal relations. The argument overgeneralises. (A few bravely accept the personal analogue: Caspar Hare, 2007, and Craig, 1996; Kit Fine, 2005, discusses "first-person realism".)
- A relevant disanalogy would need a further, currently non-existent argument, not phenomenology.
The standard semantics. Following David Kaplan (1979) and Perry (1979), indexicals have a content (the person, time or place referred to) and a character (the rule linking utterance to content: "I" refers to the speaker, "now" to the time of utterance). Semantic evaluation needs more than what is encoded: "the coffee is on the right" leaves out the frame, an unarticulated constituent for Perry, contextual enrichment for Bach (1994) or Recanati (1993). Applied to time, the truth condition for "e is now" is that e is (roughly) simultaneous with the judgment: the token-reflexive account (Smart; from Castañeda, 1966, 1967), with the date theory (Le Poidevin, 2003) as a rival.
Critics say the token-reflexive account misses some phenomenology. Balashov says some experiences are "more radically" present, that they "simply occur", that they are "known to be occurring simpliciter", that the experience is "sui generis": "more Latin... But we never get more lumen". Falk says presentness is "an inextricable part of all sensory awareness".
The critics have a point, misdirected. The token-reflexive account correlates judgments with experiences, and character correlates words with contents; neither ever claimed to describe a psychological experience. Psychology needs modes of presentation (Recanati: I think of myself "as myself, not as the utterer of such and such a token"), an internal now-symbol, as recognised by Frege (1956), Gareth Evans (1982), Jenann Ismael (2007) and others: descriptively thin concepts that let us act without knowing the date, place or description of ourselves. Complaints about missing phenomenology point at most to a mental token for the now. "How one could see through this extra layer all the way to the metaphysics of time is beyond me." (Christoph Hoerl, 2009, even denies we have tensed contents.)
Even a genuine phenomenal present would not favour tense automatically: hallucinations exist, but pink elephants do not.
9.4 From Metaphysics to Psychology: Perceived Synchrony
What is uncontroversial is that we weld sensory input into coherent wholes and judge simultaneity: the doorbell rang as Odie the dog got up (subjective simultaneity or perceived synchrony); Odie's bark goes with his moving mouth (audiovisual integration). Ernst Pöppel (1988) grounds the experienced present in such integration: an integrative mechanism shapes sequences into unitary forms, and "that which is integrated... Seems to us present". Callender doubts we experience anything as present but takes integration to be an essential part of the story.
The challenge. All signals come from the brain's past lightcone, so we never experience physical simultaneity. An explosion sends light, heat, sound, pressure and chemicals at very different speeds.
- Light: ~300,000,000 m/s; sound: ~330 m/s in normal atmospheric conditions, and up to about five times faster underwater; smell depends on the wind.
- A touch on the nose reaches the cortex roughly 30 ms before a touch on the toes.
- Inner-ear mechanical transduction is many times faster than retinal phototransduction, and neural transmission from visual cortex onward takes about 30 to 50 ms longer than from auditory cortex.
- These offsets cancel the speed difference at roughly 10 m from the observer, where the horizons of simultaneity for light and sound intersect; if anything they overcompensate, since people report synchrony when the audio lags slightly.
- In speech, the sound can lag the lips by as much as 250 ms before people notice (Dixon and Spitz, 1980); at most distances lips and voice never look like a badly dubbed Godzilla film.
The brain "gets it wrong... To get it right", since such signals usually do share a source. The limits are familiar: thunder after lightning ("hearing it is a good thing: it means you survived the lightning"), a car door slamming far up the street; up close, we may react to sound before vision arrives.
Brain time versus event time (Holcombe, 2013). On brain time theories, experienced order mirrors the order in which signals reach the brain's consciousness-relevant regions (Köhler, 1947); asynchrony tolerance is just a wide window of temporal integration. On event time theories, experienced order also uses features carried in the signals. Callender's analogy: "dumb" keyboards versus "smart" keyboards that change with the app (showing "@" in email). Dumb compensation: asynchronous signals seem synchronous because we fail to notice the gap (still evolutionarily clever). Smart compensation: the brain adapts the window, or bends a signal stream, based on the information carried, e.g. Using a shared spatial origin as a hint to bind.
9.4.1 Temporal ventriloquism
Spatial ventriloquism: the puppet's moving mouth captures the voice's location. The modality appropriateness hypothesis (Welch et al., 1986) says the most accurate modality dominates: vision for space, hearing for time. So does sound pull vision in time? Yes. In Fendrich and Corballis (2001), subjects reported a flash's timing by the clock-position of a rotating marker: a click just before made the flash seem earlier, a click just after made it seem later. Replicated many times (Scheier et al., 1999; Morein-Zamir et al., 2003; Spence and Squire, 2003): if auditory and visual stimuli fall within 200 ms of each other, the visual event is attracted toward the sound; touch does the same to vision. The effect is robust, reduces cross-modal lag, and plausibly optimises matching of signals from one event (Alais and Burr, 2004). It shows a wide window is not the whole story: one stream is bent toward another.
9.4.2 Temporal recalibration
Recalibration changes the windows themselves after repeated exposure. Measure a subject's point of subjective simultaneity (PSS), expose them to a fixed lag for a few minutes, re-measure: the PSS shifts toward the adapted lag (Fujisaki et al., 2004; Vroomen and de Gelder, 2004), to a degree that depends on the lag size and sometimes by more than 50 ms, in every pairing of audition, vision and touch, and even within vision (colour change versus motion reversal, Arnold and Yarrow, 2011). Fujisaki et al. Showed the shift alters the timing of a sound-induced visual illusion, so it is sensory, not merely cognitive: the brain "attempts to adjust subjective simultaneity... By detecting and reducing time lags between inputs that likely arise from the same physical events".
- In the wild. The brain can recalibrate different audiovisual pairs from different events in opposite directions (Roseboom and Arnold, 2011), and recalibration of a pair can depend on a third stimulus (Roseboom et al., 2009).
- Action and effect. Cunningham et al. (2001) introduced a lag between mouse movement and an on-screen spot; subjects adapted until action and effect felt simultaneous. When the lag was removed, they saw the effect before they moved the mouse. Haggard et al. (2002) confirmed intentions alter experienced simultaneity (see also Eagleman and Holcombe, 2002; Stetson et al., 2006).
- Distance. Sugita and Suzuki (2003): 10 ms white-noise bursts over headphones paired with flashes from an array of 5 green LEDs at 1 m to 50 m (brightness equalised at the eye), subjects imagining the LED as the source and judging which came first. Subjective simultaneity shifted by about 3 ms per metre up to about 40 m, matching sound's speed of about 1/3 m per ms at sea level: "the brain probably takes sound velocity into account". Replications are mixed; real distance computation is costly, so the brain likely uses proxies. Zampini et al. (2005) show stimuli from the same location are more often judged simultaneous.
9.4.3 Comments
These findings challenge pure brain-time theory. Kopinska and Harris (2004) propose simultaneity constancy, like seeing a penny as round from every angle or an object's colour as constant under shading. Harris et al. (2010) suggest three stages: fit stimuli into windows; delay unfamiliar stimuli by fixed rules (say, a 40 ms delay for sound to be bound with light); fine-tune delays for familiar pairings.
But the brain is not always chasing correct order. These processes work at the millisecond scale, not minutes or years (Holcombe, 2013). Over a century ago Vittorio Benussi (1913) found temporal order illusions: low tone, noise burst, high tone is heard as low tone, high tone, noise, plausibly because the two tones form a gestalt. And all this is largely unconscious; how it reaches consciousness, "I wish that I knew". The claim is only that this work underlies any conscious present.
9.5 Interlude: Measuring Subjective Simultaneity
The schematic shows the two canonical measures: a bell curve for simultaneity judgments and an S-curve for temporal order judgments. In practice the two PSS values often do not coincide.
How psychologists measure subjective simultaneity
- Simultaneity judgment (SJ). Present two stimuli at various stimulus onset asynchronies and ask "synchronous or not?". The plot is roughly Gaussian. PSS = the peak (e.g. Beep and flash judged simultaneous most often when the beep follows the flash by 80 ms, not at objective simultaneity). Just noticeable difference (JND) = the average interval still called "synchronous" at least 75% of the time (e.g. Gaps up to 110 ms), often read as the window of simultaneity.
- Temporal order judgment (TOJ). Ask "which came first?". The plot is S-shaped. PSS = the 50% crossover (as many "light first" as "sound first"); JND = half the distance between the 25% and 75% points.
- Integration tasks. Ask whether stimuli fuse: 1 ms clicks to the two ears 2 ms apart are heard as one tone; at 3 or 4 ms apart, as two (Hirsh and Sherrick, 1961; Fraisse, 1984; Pöppel, 1988). Illusions also measure binding: the Stream–Bounce illusion (two discs crossing seem to bounce if a click sounds at the crossing, to stream through each other otherwise) and the McGurk effect (lips saying /ga-ga/ dubbed with /ba-ba/ are heard as /da-da/; close your eyes and it vanishes).
| Threshold | Auditory | Tactile | Visual |
|---|---|---|---|
| Fusion (two stimuli heard/felt/seen as one) | ~2–3 ms (most discriminating) | ~10 ms | ~20 ms |
| Order (which came first is discernible) | ~30 ms | ~30 ms | ~30 ms |
Between the fusion and order thresholds lies a striking zone: two events, but no perceivable order (flashes 25 ms apart may be seen as two without any sense of which came first). One might think two events are individuated by their temporal order; not so. Order thresholds vary little across modalities, so this zone is largest for hearing. Cross-modal windows are much wider than within-modal ones: for maximum audiovisual integration, sound may lag vision by as much as 100 ms.
9.6 Exploding the Now
Reporting stimuli as simultaneous does not show we directly experience synchrony; separating sensory from cognitive is notoriously hard. Introspection presents a tidy, unified percept, but the spatial binding problem shows colour, shape, orientation and motion can come apart: dots coloured black when moving right and white when moving left become unbound at some rates (Moutoussis and Zeki, 1997). "Our experienced 'here' is not as unified and tidy as we think. Nor is our experienced now."
Introspection predicts that SJ-PSS and TOJ-PSS should agree, and that maximum Stream–Bounce or McGurk illusion should occur near the PSS. Neither holds at these short scales:
- No correlation. PSS from SJ tasks and PSS from TOJ tasks are uncorrelated (e.g. van Eijk et al., 2008). You may reliably detect that audio came first while calling the pair synchronous, or be unable to tell order while calling them asynchronous.
- Negative correlation. Elliot Freeman et al. (2013) found the point of maximum bounce is significantly negatively correlated with TOJ-PSS; likewise maximum McGurk. Click and crossing seemed simultaneous (TOJ) when the crossing led, but bounce peaked when it lagged. Freeman et al. Abandon the unity hypothesis that the brain enforces a unified percept. (Chapter 10's patient PH shows a gap of a whopping 450 ms between his point of maximum integration and his PSS, longer than a baseball's flight from pitcher to catcher.)
The SJ/TOJ gap might reflect question-driven response biases ("simultaneous?" invites yes; "which first?" invites no) and the fusion-to-order zone, so it may not show disunity. The negative correlation with illusions is harder to explain away: you either see a bounce or not.
Add modality-specific fusion thresholds, different multimodal windows, and saccadic suppression: the eyes dart about three times a second, and visual processing is suppressed during each saccade (the "stopped clock" effect: a second hand seems to pause just as your gaze lands on it). In a realistic scene (talk, collisions, finger snaps, radio), "it's not so clear that our conscious experience has a single temporal resolution".
9.7 Does Synchrony Pop Out?
Cognitive scientists ask whether a detection is automatic (fast, attention-light, low-level) or not (slow, serial, attention-hungry), using visual search "pop-out" tests (Beck, 1966; Anne Treisman, 1982): tilted lines pop out among vertical ones; words do not pop out of letter jumbles.
"Pip and pop". Some studies suggest audiovisual synchrony detection is slow and serial (Fujisaki et al., 2006), but Keetels and Vroomen (2012) note those stimuli were fast or lacked sharp transitions. Erik Van der Burg and colleagues (2008) showed subjects arrays of variously oriented lines changing colour (red/green) at random times; the task was to find the vertical or horizontal target. A brief pip synchronised with the target's colour change dramatically sped search and was barely affected by adding distractors, even though the pip gave no spatial hint and could come at or just after the change (ruling out mere warning). A 2009 follow-up found the same with touch. If correctly interpreted, these are the first demonstrations that synchrony, like orientation, can pop out.
Research since 2017 has reinforced the "no single now" picture. Rapid recalibration (the PSS shifting after a single preceding audiovisual trial) is now well established and has been tracked across the lifespan; studies report it is particularly strong for audiovisual pairs and for speech. Work on Bayesian causal inference models of multisensory integration treats binding as the brain inferring whether signals share a common cause, a formal version of Callender's "smart compensation". Neuroimaging and modelling continue to find distinct, task-dependent timing mechanisms rather than one master clock, consistent with Durgin and Sternberg's dictum.
9.8 Conclusion
On a beach, bubbles drift seaward on an offshore breeze; two pop before reaching the water, the big one first. Callender can compare worldlines, but did their popping at the same time pop out, or was it inferred? Perhaps, for two salient bubbles; but it is very unlikely that synchrony popped out across everything at once (the child yelling, the surfer ducking a wave, sand on his toes).
The search for a sensed tensed presentness found only linguistic features of temporal indexicals, shared by all indexicals. The modest search for experienced synchrony found lab-scale pop-out but no evidence of a general, ecologically valid one. "No single underlying representation is responsible for all judgments of time" (Durgin and Sternberg, 2002), and there may be no task-independent representation of synchrony at all. The metaphysician's present is undermined by physics, and the experience on which it was supposed to rest falls apart under scrutiny.
Stuck in the Common Now
Why do we treat the now as objective and global but never the here? Callender names this the Representational Asymmetry. Psychology makes the puzzle sharper: like relativity, it supplies disagreement data (patient PH, varying PSS values, the Greenwich "personal equation"). His answer combines Butterfield's "no time stamps needed" argument, an evolutionary decision model of integration windows, and a basic physical fact: we can turn around and walk back in space, but physics forbids turning around or travelling in time, so the disagreement that exposes the here as egocentric never shows up for the now; the reification of the now is a natural response to the fact that we really are objectively stuck in time.
Manifest time's now is objective and global, flows, and is the moment the world is "on". Flow and "being on" wait for Chapter 11; here the target is objectivity and globality.
Egocentric and allocentric, in space and time
Gareth Evans (1982): in "egocentric space" the subject takes himself as the origin, with axes up/down, left/right, in front/behind. Allocentric representations need no viewpoint.
| Egocentric | Allocentric | |
|---|---|---|
| Space | "The coffee cart is here"; "the fork is on the left" | San Diego and Providence are 2366 miles apart; Kansas City lies between them |
| Time | Halley's comet visited in the past; is not visiting now; will visit in 75 years | The 1986 and 2061 visits are 75 years apart; 2061 is after 1986; my class graduated simultaneously with a visit |
| Agreement | Spatial: we routinely disagree and translate ("your left") | Shared; science builds on it |
Science prizes intersubjective agreement, so it builds allocentric models: the jelly is inside the donut, Jupiter at closest approach is about 365 million miles away, wherever you are; no objective Here, Left or Front. Manifest space agrees: nobody thinks their here is objective, since the dinner table refutes it nightly.
A telling linguistic asymmetry: spatial egocentric talk smuggles in time (the food truck is here, not was or will be: it picks out the now, whereas "Chubby's Burger is at the corner of 101 and J Street on August 3, 2012" is true from no viewpoint), but temporal egocentric talk does not invoke space ("Napoleon's death is past" needs no location). Most or all languages build egocentric temporal indexing into sentences (tense, aspect) with no spatial counterpart. (Alex Holcombe notes "anywhere" and "everywhere" thrived while "anywhen" and "everywhen" died out.)
Representational Asymmetry: we think, speak and act in ways that treat egocentric temporal categories as objective, but not egocentric spatial ones.
The Doctor Who line ("Dragonfire", 1987), "You're going. You've gone for ages, you've already gone, you're still here, just arrived, haven't even met you yet... Strange business, time", treats the now like a here and strikes us as odd. Students are shocked by the relativity of simultaneity but barely notice the relativity of co-location. Since past and future are defined relative to the now, treating them as objective commits us to an objective now. So: manifest space and physical space agree that egocentric spatial categories are not objective; manifest and physical time disagree about the temporal ones. Why?
10.1 Disagreement and the Case of PH
Relativity's threat to the common now is famous; psychology's is not.
PH: the man who hears you before he sees you. In the summer of 2013 news broke of PH, a 67-year-old retired pilot living in England, high-functioning on standard tests, whose brain lesions after an operation left vision consciously lagging hearing. He noticed while watching television at his daughter's house and told her both her televisions needed fixing: everything looked badly dubbed. Elliot Freeman and colleagues (2013) confirmed it: to achieve subjective synchrony, voices had to be delayed by 210 ms relative to lip movements, so his TOJ PSS is 210 ms, against mostly lower values in controls; yet to get the McGurk illusion, the visual signal had to lag the sound by 240 ms. His point of maximal integration is negatively correlated with his PSS, the two differing by a whopping 450 ms. PH does with speech what we do with thunder and lightning, faster and reversed. He has a subjective now, but it binds different events than yours.
What struck researchers equally was that PH was not so unusual. His very high PSS and the negative correlation also showed up in controls; 10 of 37 healthy subjects consistently reported auditory or visual lead for objectively simultaneous stimuli. They report no problems because they only notice in the lab. "Many of us are binding different events together into our subjective presents."
- PSS varies between people. Stone et al. (2001) found an individual's SJ PSS stable (as motor control requires), but interpersonal differences astonishingly large: one observer needed sounds to precede flashes by 21 ms, another needed flashes to precede sounds by 150 ms.
- Fusion thresholds vary with the person (auditory fusion from about 2 ms to 5 ms), with age (older people fuse more) and with training (up to a minimum that cannot be shrunk).
- The personal equation. In 1796 at the Greenwich observatory, Astronomer Royal Nevil Maskelyne fired his assistant David Kinnebrook for systematically deviating from him by 0.8 s in timing stellar transits (Mollon and Perkins, 1996). The task (noting the clock's audible beats as a star crossed a hanging wire) relied on multisensory integration. Astronomers later learned even experts' senses of simultaneity could differ by up to a second; Kinnebrook may have been blameless (though he had also turned down a marriage match Maskelyne proposed, another "personal equation").
If an instructor snaps her fingers and declares that moment real, PH might ask: "Which moment?" And we are not so different from him. This psychological disagreement is in a way worse than relativity's: relativistic disagreement about simultaneity is calculated, not observed, since simultaneity is not an invariant; psychological disagreement can in principle be observed. A psychological Putnam argument follows: you experienced e₁ and e₂ as now, your friend experienced e₂ after e₁; if neither is special, your now is not distinguished.
10.2 Manufacturing the Now: Signals, Speed, and Stamps
Physics explains why simultaneity seems absolute: our relative velocities are tiny compared to light's, so Lorentz transformations approximate Galilean ones. In terrestrial units the lightcone is effectively instantaneous; the spacelike region around an event, huge in a Minkowski diagram, is tiny in human affairs.
Snap both hands. Stretch your arms wide and snap both hands "at the same time". For the two snaps to be spacelike separated, they must fall within a few billionths of a second of each other (light crosses your arm span in about 6 ns). Possible, but unlikely on any given try, and by the time you notice them both lie in your past lightcone. Relativistic disagreement needs events in those regions, so it never matters in daily life. Can a similar argument explain agreement about the experienced now?
10.2.1 Time stamps not needed
Callender builds on Jeremy Butterfield's "Seeing the Present" (1984, one of his favourite papers). Microscopic properties change ferociously: the vast clump of electrons, protons and neutrons (about 1.2 × 10 to the high twenties of each) making up Odie the dog move at up to almost two million metres per second, colliding at a fantastic rate, with spins rotating and particles decaying, yet Odie "has not moved since breakfast". Quantum mechanics, chemistry, solid-state physics and statistical mechanics conspire to make the macroworld stable: electrically neutral, chemically stable, slow to fluctuate. Even chasing a cat, Odie tops out at 25 mph.
Butterfield's argument, as Callender reconstructs it
- Light from Odie a few feet away takes about 3 nanoseconds to reach the eye; transduction and neural processing take far longer, roughly 0.5 s in total, from the object's state at t to the belief at t′.
- Macroscopic properties change slowly relative to t′ − t, so the lag rarely falsifies the belief: at t′ Odie is still dog-shaped, furry and brown.
- Communicating the belief adds my signing, your perception: still about a second, still harmless. The belief is true at utterance and at reception, essential for coordinating action.
- The same holds for sound (slower in air, faster to transduce) and touch.
- So in typical environments we can form and share reliable beliefs about macroscopic objects without time stamps (calendar or clock dates): "the bird is in the tree", not "...at 12.58.03 GMT".
- The disagreement data are masked, so we treat the now as monadic and omit its relata, conflating egocentric with allocentric representation.
- A now patch is a spatiotemporal region over which typical observers do not need a time stamp to get around reliably. Sewing patches together and extrapolating gives a global common now, reified because everyone agrees.
This also explains why most of history accepted Max Jammer's (2006) visual simultaneity hypothesis, that everything seen is simultaneous with the seeing (Aristotle, Descartes, Kepler), with Empedocles, Avicenna, Alhazen, Roger Bacon and Francis Bacon a dissenting minority.
Two observations:
- Contingency on channel. With smell or the post the story fails: burnt-toast smell lingers all day without dating the burning; an undated bank letter may be worthless. Our main channels are fast relative to macroscopic change.
- The mind is missing. Like every "mind-dependent theory of the present" (Grünbaum, 1968), the time-stamp story is "behaviorist": it explains reliable belief and communication, but not why we experience events together (a bark with a red coat). Those experiences are the patches to be sewn together, and their construction can be explained by evolution.
10.2.2 The common now
Differences in PSS and integration points rarely matter in ecologically valid settings because everyday events are coarse-grained: a finger snap's sound attaches to the whole motion unless you attend closely. "If I snap my fingers and say 'right now!' this picks out the same coarse-grained moment for everyone in the room." Integration windows as wide as 250 ms, fast light, fast hearing and slow macroscopic change hide our differences.
Evidence. Boenke et al. (2009) ran TOJ tasks varying stimulus duration and intensity. With stimuli lasting 9 ms, individual PSSs scattered (some negative, sound-first; some positive, light-first). Lengthening the stimuli from 9 to 40 ms and then from 40 to 500 ms made PSSs converge: negative ones shifted positive, positive ones negative, toward an attractor value, by amounts depending on the duration change and initial offset. No one sees the ten o'clock sitcom while hearing the eleven o'clock news, but convergence appears even at short durations. Individual differences surface only in special cases (Greenwich, offside calls in football, PH). In normal interactions with events of a second or more, "we'll agree on enough of it to not notice the difference", and agreement survives communication, as Butterfield says.
Why we agree: shared evolutionary history. Evolution shaped our windows of simultaneity, and successful communication was essential to survival.
The integration decision (after Colonius and Diederich, 2010)
Two stimuli (visual V, auditory A) either share a source (C = 1) or not (C = 2); the creature integrates (I = 1) or not (I = 0).
| Integrate (I = 1) | Don't integrate (I = 0) | |
|---|---|---|
| Common source (C = 1) | U₁₁: correct binding | U₁₀: missed binding (type II) |
| Separate sources (C = 2) | U₂₁: false binding (type I) | U₂₀: correct separation |
With window t = |V − A|, expected utility is maximised by the rule: integrate iff P(C = 1 | t) / P(C = 2 | t) > (U₂₀ − U₂₁) / (U₁₁ − U₁₀).
Getting time exactly right is not always best: bind the low rumble with the tiger's sight, but don't waste time integrating a roar right next to your ear. An organism that withheld lightning from its percept until the thunder arrived would sit "in a kind of fugue state" and get eaten in storms. In the toy model:
- Higher prior of a common cause widens the optimal window.
- With the costs U₂₁ and U₁₀ fixed, a higher payoff U₁₁ for correct binding widens it.
- With those costs fixed, a higher payoff U₂₀ for correctly keeping separate signals apart narrows it (the threshold ratio rises).
Payoffs depend on predators, prey, weather and more, so subjective nows are environment-dependent. Treat the payoffs as fitness, populate a toy world with creatures of varying t, run replicator dynamics, and watch the dominant window evolve. Groups already test whether humans are Bayesian optimisers of multisensory integration (review by Marc Ernst, 2005); Callender takes no stand, since evolution is no perfect optimiser and the real payoff matrix is unimaginably complex. The model's value is to show that binding is a cost-benefit balance.
A zoology of nows. If windows are tuned to environments, they should differ across species. Sound travels nearly five times faster in seawater than in air: do ocean dwellers navigating by sight and sound have different nows? Do animals in cold, dry, high places differ from those in hot, wet lowlands? Prey of snail-like predators can afford wider windows than prey of cheetahs. And size matters: a giraffe's foot tendon is almost 3 m from its brain, giving long sensorimotor delays and coarse resolution (More et al., 2013). Does the giraffe have a "long now" to go with its long neck? The contingency of the subjective now makes it empirically testable.
Patches or dispositions? A bird tweets as it lands on a branch 20 m away: is the song in your now? A now only 1 ms long allows sound localisation but not music recognition; a 1 s now can hold birdsong; whether call and landing are bound may even depend on how vivid the bird is. With no single representation of synchrony, presents are better pictured not as spatiotemporal patches but as each person's probabilistic disposition to bind stimuli in an abstract stimulus space. The probability that you bind what I bind is very high at our usual scales, so we never notice our patches are not global; "everywhere we go we meet people who agree with us on the present". (Sewn-together thin patches would anyway fall short of simultaneity, which should be an equivalence relation; the thinnest patches may not even be transitive, per Hansteen, 1968, 1971.)
10.3 Wiggling in Time vs Wiggling in Space
The story so far explains why we might confuse egocentric with allocentric temporal discriminations, but a parallel story could be told for the spatial here (integration of inputs from one location arriving at two eyes or ears is the true spatial analogue of simultaneity constancy, not "location constancy"). It does not yet explain the asymmetry. Psychology could in principle be wired otherwise (science-fiction creatures conscious of past and future), but given our wiring, the answer bottoms out in physics: unlike Billy Pilgrim in Slaughterhouse-Five, we are "stuck" in time as we are not in space.
The physical roots of the Representational Asymmetry
- Mobility asymmetry (central). No time travel, or at least none exploitable. If we can walk west we can usually walk back east; nothing we count as an object has ever done the temporal equivalent.
- No turning around. One-dimensional timelike directions mean a past- or future-pointing arrow cannot be continuously rotated to point the other way; an east-pointing arrow can be rotated to point west. "One can't spin around and see other nows as one can other heres."
- Worldlines. We and everything we interact with follow continuous non-spacelike worldlines.
- Upshot. The two commonest ways of generating spatial egocentric disagreement, rotating and moving back and forth, have no temporal counterparts.
Spatially, the evidence against a privileged here is everywhere: walk 30 feet and it changes; on the phone, "here in California it's warm", "here in New England it's dark and cold". We say "wish you were here", never "wish you were now". Temporally, when a dog and a cat run past we agree now both are happening, and physics forbids any motion that would generate disagreement. Everyone you meet thinks it is now, not just this century but this very second. Hence massive agreement, hence reification: "The conflation is not so much a mistake or illusion so much as it is a natural response to the fact that we really are objectively stuck in time."
10.4 Conclusion
Philosophy of time has long asked whether presentness, pastness and futurity are monadic or relational, tensers saying some monadic properties are irreducible. Experience cannot settle it (Chapter 9).
The island and Gold's Gym. Someone raised alone on an island takes "heavy to lift" as monadic, splitting the world's objects into heavy and not. One day he walks into Gold's Gym and meets disagreement data: weights heavy for him are light for others. Why did he think it monadic? He never represented himself in the content of his experience (Shoemaker), and on the island he never met the disagreement. Manifest time portrays the present as monadic; then we meet PH, individual PSSs and varied integration: we leave the island. We had thought it monadic because of contingent facts: few PHs, coarse-grained events, signals fast relative to macroscopic change. This is the psychological counterpart of the physical explanation of why simultaneity seems absolute.
Callender concedes he has not shown the scientific explanations can be purged of tensed concepts, so a tenser could adopt them wholesale. His reply: the tenser's supposed "best case", experience, is now empty; the theory predicts and explains manipulations of the perceived present, and tenses do nothing to help. The roles are reversed: the tenser relies on a "half-hearted hope" that cognitive science needs tense. Critics may still say he explains the belief in an objective now and our agreement, not the felt vividness of presence; and the evolutionary story is a plausible sketch rather than a tested account, as he admits for the payoff matrices.
When Einstein lamented the missing now, Carnap said psychology could explain it; dismissing the now as "mere illusion" is wrong, because there is a real phenomenon with an explanation. The trick was to clear away the indexical distractions, focus on the Representational Asymmetry, and go interdisciplinary: physics, evolution, psychophysics, cognitive science, linguistics and philosophy. "Physics by itself can't explain the now, but science can." Next: "let's get this present moving."
The Flow of Time: Stitching the World Together
Time's flow is the most entrenched part of manifest time, and many metaphysicians posit an objective passage to honour it. Callender instead builds, following James Hartle, an IGUS (information gathering and using system) and asks what it needs to model time as flowing. Perception of change, a specious present, felt duration and asymmetric memory get most of the way, but not the "whoosh". The missing ingredient is the self: an enduring, narrated self that "crawls" up the worldline, which we effortlessly reframe as time itself moving; add a sense of agency and the future looks open. All of it uses physical time plus independently known mechanisms.
Ursula Le Guin: perhaps it is we who move, as a reader moves through a book "all there, all at once, between its covers". Broad (1923) pictured presentness as "the spot of light from a policeman's bull's-eye traversing the fronts of the houses"; Santayana (1942): time "runs like fire along the fuse of time". Bergson (1922), Broad (1923), Čapek (1991), Ellis (2006), Maudlin (2002), Prigogine (1980), Shimony (1993), Smolin (2013), Whitehead (1929) and Zimmerman (2007) all make room for objective flow.
Flow faces problems. Physics has none. Smart's (1966) "how fast does time flow?" questions whether "1 sec per 1 sec" is coherent; branching and moving-spotlight models raise "at what time" questions. And coherence is a low bar: metaphysical models must also earn their explanatory keep (Chapter 13). Physicists call flow an "illusion", which moves the problem onto psychologists' desks without telling them, so it falls between fields, though free will, causation and the self all connect to it.
11.1 Sharpening Focus
Callender brackets non-experiential arguments (e.g. Maudlin's (2007) claim that passage explains the thermodynamic arrow) and targets the argument from experience, an inference to the best explanation:
The argument from experience for passage
- We have experiences as of passage (not "of passage", which reads the theory into the data; Schuster (1986) calls passage "as indubitable an aspect of our perception... As the sights and sounds").
- If we have such experiences, an objective, fundamental flow is their best explanation.
- So objective, fundamental passage exists.
Weakness: advocates rarely describe the experience; the richest descriptions come from debunkers. Callender also targets the deep intuitions of manifest time, experiential or not, and aims to give the passage theory "some healthy competition", since against an impoverished field a poor explanation easily looks best.
| Cluster | What people point to | Callender's take |
|---|---|---|
| Perceived change and duration | Le Poidevin (2007): we "see time passing" in a second hand or hourglass; Norton (2010): even in a silent waiting room we perceive breathing, heartbeat, thought; "time flies when you're having fun" | Genuinely phenomenal, but explicable within physical time; felt duration is never cited as evidence for objective passage |
| Novelty / real change | Broad (1959): "continual supersession" of the latest phase is "the rock-bottom peculiarity of time"; Fiocco (2007): the phenomenological novelty of each moment | Non-question-begging version: the tripartite division updates; a fact of manifest time, not sensed |
| Growing nearer | Night approaches, dawn recedes; something persists and moves through time | Manifest time, arguably not phenomenology |
| Asymmetric power | Torre (2011): "power over the future, yet lack power over the past"; ice-cream options feel open | Tied to agency and the past/future asymmetry |
A footnoted surprise: Wearden (2005) had some subjects watch nine minutes of Armageddon and others sit nine minutes in a boring waiting room. The film group said time flew, yet judged the interval 10% longer, plausibly because they had more salient events to remember. Manifest time is the product of messy psychology and social life, "not the product of the refined world of PhD dissertations", so fuzziness is expected.
11.2 Meet IGUS
Thomas Gold (1967): before eliminating the flow idea, "we ought to understand how it arises", the rules that give "a beast this kind of phony picture of time". When Adolf Grünbaum told a Cornell workshop in 1963 that flow was an illusion of consciousness and no concern of physicists, Richard Feynman ("Mr X" in the transcript) objected that minds obey physical law too; in principle "one should be able to create a feeling of nowness in a sufficiently elaborate machine" (as told by James Gleick, 2011). Neither pursued it; James Hartle's "The physics of now" (2005) did, using Murray Gell-Mann's (1994) notion of an IGUS (humans, robots, many organisms).
IGUS's architecture
- Perception. Visual (easily generalised). A new snapshot every interval τ goes into register P₀; older images shift down P₁ … Pₙ, the oldest erased. So IGUS holds a coarse-grained record over (n + 1)τ.
- Process U distils the registers into a schema: a model of the world with locations of food, mates, enemies and inferred generalisations.
- Process C takes only P₀ plus the schema, computes the most likely future, and decides by cost–benefit: given the schema and a hungry tiger in P₀, "run!"
IGUS is a poor imitation of us (our memories are reconstructive and not linear registers; unconscious and conscious processing interpenetrate; our visual systems are not cameras), as Hartle admits. But it forms and updates percepts at intervals, builds models from memory, and acts on new information: by construction it is "conscious" only of the present, which spans an interval, while being aware of a changing schema. Its simplicity lets us isolate what makes it model time as flowing. Callender avoids phenomenological talk except for perceived change, and adds components before granting that its schema represents flow.
11.3 Getting IGUS Stuck in Time
IGUS has genidentity over time and is massive, so it inherits the constraints of always-timelike worldlines: the mobility asymmetry and one-dimensional temporal movement, plus the benefit of an invariant temporal order for its "experiences".
An overlooked asymmetry (Mellor, 1981): at a given place IGUS has different mental states at different times (C's contents change), but at a given time it does not have different mental states in different spatial parts: there is one C box. Your head does not believe hoppy beers are best while your right leg prefers wheat beers (split brains aside). So there is temporal variation of temporally perspectival beliefs at a place, but no spatial variation of spatially perspectival beliefs at a time; even blinking or turning round takes time.
Populate IGUS's world with macroscopically stable objects, fast signals and other IGUSs; let them emit signals, move and integrate intra- and inter-modally; restrict them to low relative velocities, signals and processing fast relative to macroscopic change, and macroscopic observation scales. Then, by Chapter 10, IGUS will model a global shared now. For Hartle, IGUS's present is the stretch of its worldline over which P₀ holds one content. For a distant IGUS′, an indefinite stretch of its life may be "simultaneous" with P₀; for a nearby, slow IGUS′, only a short stretch, too short for its content to change. In natural units, lightcones flatten out over hundreds of kilometres for a one-second interval: a common, if local, present, worth representing in the schema. IGUS divides the world into past, present and future, and because P₀ refreshes rapidly (humans can refresh percepts many times a second; see Holcombe, 2009), these nows update.
John Norton's challenge. Norton argues that the delivery of moments is "perfect": no "dislocations of serial order", no experience of next year thrown in with today; if passage is not objective, why can't an inventive researcher induce dislocations of a day or a year? Callender: on short scales dislocations are routinely induced (Stetson et al.'s, 2006, subjects saw the light jump before they moved the mouse; cross-modal lags reverse order; shifting attention reverses order within a modality, per Stelmach and Herdman, 1991). On long scales the absence of dislocation is no mystery: stimuli arrive in succession and can only be welded once they have arrived, so "no stimuli from Thursday can make it into any Tuesday window". Short-scale dislocation and long-scale order are both predicted by neural circuitry and physics; objective passage predicts neither. The evidence favours the mind-dependence of flow.
11.4 Outfitting IGUS
Like a Lego Mindstorms robot (bought "for the children") with snap-on gyro, ultrasonic, touch and colour sensors, IGUS can be fitted with modules for temporal experience.
11.4.1 Sensing motion and change
Broad (1923): "we do not merely notice that something has moved...; we also often see something moving". Humans are change detectors, and the mechanism is partly known: Reichardt detectors (Reichardt, 1969; modelled on fly neurons; Adelson and Bergen, 1985; van Santen and Sperling, 1985). Two edge detectors O₁ and O₂; O₁'s signal passes through a delay filter; if the edge reaches O₂ at the right speed and direction, both signals arrive together and the coincidence unit fires "motion!", encoding speed. Such detectors already exist on chips; IGUS can have them.
Motion and motion qualia come apart. In the waterfall illusion, fatigued retinal cells make a static scene seem to drift upward. In colour phi, a blue dot on the left followed quickly by a red dot on the right is seen as one dot moving right and changing colour mid-way. Le Poidevin (2007) and L. A. Paul (2010) think colour phi is key: if the brain can create illusory motion from static inputs, it can create illusory flow from the static block. Callender disagrees: colour phi shows you cannot infer motion from motion qualia, nothing more. In the block, balls really roll downhill while phi dots do not; perception makes type I and type II errors; "to think it's all like color phi obscures the difference between the illusory and non-illusory cases" (also Deng, 2013; Hoerl, 2014). The second hand's "whoosh" versus the hour hand's stillness reflects different mechanisms: we literally see the second hand move; we merely notice the hour hand has moved (Hoerl).
11.4.2 Specious present
The burst-mode basketball. Hold down your camera's burst button on a bouncing basketball. Each photo shows the ball somewhere; none shows motion. Flip them fast and you see animation, but then the flipper, outside any snapshot, supplies the motion. If experiences are snapshots like IGUS's P₀, how do we experience succession, duration or motion at a moment?
Five responses to the specious present puzzle
- Content is not stills. Snapshot vehicles may carry contents of motion, succession and duration; "brain time needn't be experienced time".
- Vehicles are not snapshots. Experiences are temporally extended, like short films, whose own temporal structure conveys temporal content (Ian Phillips, 2013; criticised by Watzl, 2012).
- Neither assumption. Extended vehicles and non-still contents.
- Retention by recent memory. A snapshot experienced alongside the recent memory of others (back to Whewell); add an arrow from IGUS's recent registers to P₀ or C.
- Scepticism. We do not directly experience succession at all.
Models: Rick Grush's trajectory estimation model (2007; anticipate what comes next, then correct) and Rao et al.'s (2001) fixed-lag smoother (wait a fixed interval, say 80 ms, before committing to a percept) implement option 1, inspired by representational momentum and the flash-lag effect. All are temporally asymmetric: memory models inherit memory's asymmetry; smoothers wait then decide; trajectory estimation shows the last non-overwritten estimate.
Whatever the answer, change perception needs nothing more than physical time (Dorato, 2015); each option can be snapped onto IGUS.
11.4.3 Felt duration
Psychology often means felt duration by "flow". Philosophers do not use it for passage, presumably because it varies so much. Researchers distinguish sub-second, "automatic" timing from longer, "cognitive" timing, likely using different mechanisms (Rammsayer, 1999): "how long were you on the phone with mom?" differs from "was the left flash longer?".
- Factors shifting duration judgments include stimulus intensity, attention, colour, gender, age, cognitive load, task demands, modality, predictability, mood, task description, caffeine, and elicitation method.
- Prospective vs retrospective judgment: knowing in advance you will be asked changes the estimate dramatically.
- The oddball effect (Tse et al., 2004): in a stream of identical stimuli, a surprise stimulus of the same duration is judged longer.
- Clock models: Hoagland's (1933) chemical clock, Treisman's (1963) internal (pacemaker–accumulator) clock, Thomas and Weaver's (1975) attentional model, Ornstein's (1969) memory-storage model (remembered duration = storage used by encoded information); periodic (pendulum-like) and accumulator (hourglass-like) designs.
The real story is "not how badly these clocks perform but how well". Physics supplies the true invariant duration of timelike paths; organisms estimate it under cost–benefit pressure. Nobody builds a caesium-133 hyperfine clock, and few fail to tell a second from a day; with practice we are "pretty darn good" (Montare, 1988). IGUS could measure duration Ornstein-style by storage used in P₀–Pₙ. If flow is felt duration, IGUS can feel it.
Duration research since 2017 has strengthened the "no special clock" view. Roseboom and colleagues (2019) showed that a model which simply accumulates salient changes in a visual-classification neural network reproduces human duration estimates, including their biases with busy versus quiet scenes, without any pacemaker. Work on episodic memory and event segmentation continues to explain why eventful periods feel longer in retrospect (the Wearden pattern). Both fit Callender's claim that felt duration is an estimate built from perceptual and memory resources, not a window onto passage.
11.5 Memories and Flow
IGUS is bathed in asymmetry: heat flows hot to cold; waves diverge from sources and never converge on sinks; signals arrive only from the past lightcone; entropy rises everywhere, inside IGUS too. This plausibly yields the knowledge asymmetry (yesterday's Dow is in the paper; tomorrow's would make this book unnecessary) and its instance, the memory asymmetry.
The link to thermodynamics is not sealed. The outline: a record must be counterfactually stable, still produced under slight perturbations of the microstate (Mlodinow and Brun, 2014); thermodynamics prevents memories of the future, because perturbed future microstates evolved backward give wildly different presents (see David Albert, 2000). Reichenbach (1956) tried via "quasi-entropy" and the fork asymmetry; Albert via a constraint on initial conditions; critics include Callender and Frisch; Landauer's Principle (erasure dissipates energy) is contested (Earman and Norton, 1998, 1999; Norton, 2005; Maroney, 2005, 2010). And Norton's (2000) toy model shows memory needs neither time-irreversible laws nor asymmetric primitive causation, nor any asymmetry of time itself.
IGUS's three memory asymmetries
- Input. Each register holds information only from the past lightcone of its interval's latest point; correlations are screened off by past, not future, interactions.
- Flow. Images move P₀ → P₁ → … → Pₙ, never the reverse.
- Erasure. Only the oldest is erased.
The schema can store a "layered cake" of temporally organised memories (a → ba → cba → dcba…). If the memory arrows define "earlier", then with Hartle IGUS "experiences" its conscious register updating from earlier to later.
Human memory is richer. Procedural memory (skills, like navigating the kitchen at night) versus declarative; declarative splits (Endel Tulving, 1972) into semantic (facts without context: that Bill Walton came off the bench for the 1986 Boston Celtics) and episodic (Callender catching a striped bass at 14: the forearm pressure, the boat dragged around, the buzzing line), remembered "from the inside" (Shoemaker, 1963); autobiographical memory is the self-involving kind. Memories of memories (recalled each time he sees the photo at his father's house) form a multi-layered cake, arguably essential for conceiving a real, linear past (John Campbell).
Barry Dainton (2001) shows why memory feeds flow: we know only our earlier life, so "it is easy to believe that our lives don't extend beyond the present"; continuation seems an addition to a completed past; remembering and anticipating such beliefs gives "the impression of living at the outermost limit of an ongoing future-directed process". Epistemic openness slides into ontological openness, and since everyone shares the asymmetry, disconfirmation never comes. Broad's growing block, "fresh slices of existence... Added to the total history of the world", faces a question: how could he tell fresh slices of existence from fresh memories being added? The phenomenology would be the same.
Callender agrees with Dainton and Mellor that memory is indispensable, but finds two gaps:
- No movement yet. Nothing seems to "crawl up" IGUS's worldline: no whoosh. Nothing literally does, but we conceive of something doing so.
- Thin openness. Memory yields only epistemic openness, not the deep sense of power over the future.
11.6 The Enduring Witness
Maurice Merleau-Ponty (1945): the river metaphor has persisted from Heraclitus because we surreptitiously put "into the river a witness of its course". The extra ingredient is the self, found in Eastern philosophy, continental and analytic philosophy, science fiction and developmental psychology. It supplies the something that moves, the "you are here now", and an intimate sense that the future is open.
Why selves? Daniel Dennett (1989): organisms need a self/not-self boundary to know what to defend and plan for; "the lobster has to know not to eat itself". Even amoebae encode something minimal. Humans need far more: planning (meet to hunt mammoths at dawn; a conference in Grindavík next summer; pistachios in a minute), much of it social, so the self must be one diachronic object distinct from others.
The two features of the self Callender needs (epistemology, not metaphysics)
- Endurance. In the manifest image we are endurantists: the self is wholly present at each time ("You're all there!"), at your first kiss and your fiftieth birthday, not a temporal part of a 4D worm (perdurantism). Callender regards the two as ways of carving one world; only the conceptual distinction matters.
- Narrative identification. The enduring self we posit is the subject of our narration (Ricoeur, 1986; Dennett, 1992; Schechtman, 2011). Dennett makes the self a "center of narrative gravity", a useful fiction; Velleman posits non-illusory selves for life episodes; Galen Strawson (2004) contests narrativity. Callender needs only a minimal version: at least two events in a non-logical, temporal relation (Lamarque, 2004); an unfolding, structured sequence from the individual's point of view (Goldie, 2000); related teleologically, giving reasons from the first person (Schechtman, 2007): I boiled water for tea, because I was chilly, to stay sharp grading, to be a good teacher.
Fit endurance to narration and you get what flow needed. David Velleman (2006), articulating an idea from Buddhism: whatever the future draws nearer to must exist at different times with its identity intact, and the enduring self is such a thing (or its illusion). I do not merely extend from a 9:00 pm stage to a 10:00 pm stage closer to midnight, as my head is closer to the ceiling than my feet; "I exist in my entirety within the stroke of 9:00, and I exist again within the stroke of 10:00... From my perspective, then, midnight draws nearer."
Slogan: the illusion of the enduring self is responsible for the illusion of the flow of time. What crawls up the worldline is not Weyl's moving spotlight but "the character in a kind of story". At each moment the protagonist is rebuilt from the resources then available; "you are always the leading edge of the story". (E. J. Lowe, 2009, argues passage metaphysically depends on enduring persons; Callender's is the deflated version.)
Memory and self connect: episodic memory is needed for a sense of personal identity over time (Atance and O'Neill, 2005; Klein and Nichols, 2012), and Ismael (2007) stresses reflexive autobiographical memory in narration. The patient D.B. knew his past traits (semantic competence) but, unable to form episodic memories, seemed unaware he had a past. (Even Hume granted memory is "the source of personal identity".) We order the jumble of memories using photos, procedural–declarative correlations and testimony (I know some Russian; I remember a college class; a friend saw me there), building a narrative, accurate or not, of the self that survives through time. Asymmetric memories are the raw material, so the self they build crawls one way.
11.7 From Flowing Selves to Animated Time
Getting from a moving self to moving time is easy because we slide between them constantly.
| Ego-moving perspective | Time-moving perspective | |
|---|---|---|
| Picture | The ego moves through a static temporal terrain | The ego is still; times stream past |
| Talk | "We're coming up on Christmas"; "approaching the deadline" | "Christmas is coming up on us"; "the deadline is approaching" |
| "Next Wednesday's meeting has been moved forward two days" | Friday | Monday |
Spatial and linguistic priming push people to one reading; Mandarin and English speakers differ (Fuhrman et al., 2011). Switching schemas mid-conversation costs a little time (Gentner, 2001: just off a London–O'Hare flight, a time-zone question that changes schema mid-way is harder), but we manage. Núñez et al. (2012) suggest the real divide is present-moving versus time-moving relative to the present; either serves, since the self is wholly present.
McTaggart (1908) himself wavered on whether time moves from past to future or future to past: if events move past a fixed present, the movement is future-to-past; if presentness moves along events, past-to-future. "Thus we say that events come out of the future, but we say that we ourselves move towards the future": metaphysical versions of the two schemas, like scenery past a train window.
The argument. Reality is a static worldline; the pressure for a self that endures yields a self-conception in which the "I" is wholly present at each moment and moves (ego-moving schema); because we slide effortlessly to the time-moving schema, this becomes a picture of dynamic time. To make IGUS believe time whooshes by, have it represent itself as an enduring self in its schema.
11.8 Temporal Decentering and the Self
The theory predicts a non-accidental link between the development of the self and of temporal representation.
Children of 2 or 3 first attach times to particular events, then build a rough order, then a causal order; by approximately age 4 or 5 they temporally decenter.
Adults decenter easily: when Socrates drank hemlock his death was future; now it is past. McTaggart's A-series relies on an ordering (his C-series) along which a moving now creates past and future. Spatial decentering comes free: turn your head, compare your two eyes' views. Temporal decentering is hard because the mobility asymmetry forbids turning round in time. Enter selves: decentering is "associated with a grasp of the fact that there were previous points in time at which one's temporal perspective on events was quite different but one's own nonetheless" (McCormack and Hoerl, 1999).
Delayed self-recognition (Povinelli et al., 1996; Povinelli and Simon, 1998). Daniel Povinelli videotaped 2-, 3- and 4-year-olds playing a game involving head pats, during which an experimenter secretly stuck a bright sticker on the child's head. Minutes later the child watched the video, sticker placement included. Adults would reach up at once ("me three minutes ago is me now"). Few 2- and 3-year-olds reached for the sticker; most 4-year-olds did (e.g. 25% of 3-year-olds versus 75% of 4-year-olds), so the capacity develops between ages 3 and 4. Younger children recognised themselves but dissociated from their video selves: the experimenter put a sticker on "his/her" head, not mine. Asked who had the sticker, a 3-year-old named Jennifer said: "It's Jennifer... But why is she wearing my shirt?" (Povinelli, 2001). (Some, like Zelazo and Sommerville (2001), think the deficit is in reasoning about the self and date the self-conception to as early as 2; all agree that by age 5 children typically have the bare bones of the mature enduring self.)
Callender's predicted picture: memory develops → a self forms (Fivush, 2001: autobiographical memory is essential) → temporal decentering → an A-series conception of flowing time. He predicts correlations among autobiographical memory, self-conception and decentering (all change around age four), testable by comparing DSR scores with memory and decentering tasks; he does not claim which comes first, and a common cause (handling conflicting non-spatial representations) is possible. Decentering gives an A-series; the narrating self gives the preferred perspective (the leading edge of the story); Chapter 10 makes the now look objective.
11.9 The Acting Self
One feature remains: the unwhooshed direction seems open. Two massive asymmetries matter: the knowledge asymmetry and the causal–counterfactual asymmetry (causes precede effects; the future depends counterfactually on the present, the past does not). Together they make creatures uncertain about later events on their worldlines yet able to control them. David Lewis (1979): the "mysterious asymmetry between open future and fixed past is nothing else than the asymmetry of counterfactual dependence"; the forking futures are those that would come about under counterfactual suppositions; the fixed past is the one that would stay actual.
IGUS anticipates by building probabilistic models of possible futures, never pasts, since its deliberations are productive only futureward. Already modelling a present and a moving self, it will read these allocentric asymmetries egocentrically: the future as a whole open, the past as a whole fixed.
Self-fulfilling predictions. For Velleman, the sense of freedom comes from the fact that, within limits, whatever we decide to do, we do: decide on chocolate when the waiter arrives and chocolate is what you get, and until the moment of decision you could "bilk" anyone's prediction (Ismael, 2011). At the car dealership, at the altar about to say "I do", resisting an itch, all our evidence says the decision is the trigger. "We feel like minor gods." Terence Horgan (2007): you experience your arm's motion "as generated by yourself". Decisions are not self-fulfilling about the past (deciding on strawberry yesterday does nothing; any probabilistic influence on the past is undetectable for macroscopic creatures, per Kutach, 2011). So as the self evolves up its worldline it feels part of "a worldwide surge of 'actualization'": passage and the open future are bound together. Fit IGUS with a sense of agency whenever it is positioned to make self-fulfilling predictions, phenomenal or merely in its schema (Bayne, 2008, on the controversy).
11.10 The Explanation of Passage
The complete recipe for a passage-modelling IGUS
Gadgets: - Detectors of movement, velocity, succession, duration and other objective temporal properties. - Signalling, movement, intra- and inter-modal integration, fast relative to macroscopic change. - A self-conception built from patterns in memory registers. - A sense of agency when making self-fulfilling predictions about the next time step. - Sensory modalities tuned to macroscopic scales.
World: - A sharp timelike/spacelike difference, with all it implies. - Low relative velocities, macroscopic observation scales. - Large physical asymmetries grounding or accompanying the knowledge and causal asymmetries.
Output: a schema in which time flows, "a natural reaction to the situation in which it finds itself". Details will differ (IGUS lacks our saccades, so it won't see the paused second hand), but we share the core.
Split-screen and always-behind IGUSs. With Hartle, imagine an IGUS whose C box reads both P₀ and (say) P₇: it sees the present and a past moment at once, two "presents". Or an "always behind" IGUS wired to P₇ instead of P₀: it experiences as present what is encoded in P₇, remembers P₈–Pₙ, and has reliable premonitions of P₀–P₆, which it knows but has not yet "experienced". Is that stretch open or fixed for it? Physics allows both wirings; biology punishes them. In a largely Markovian world, catching a fly needs only its position and speed from P₀; P₇ adds nothing but energy costs; the P₇-wired IGUS catches far fewer flies, and a "dumb" one crunching all its registers catches none. Passage is contingent, and selection explains why we have ours.
11.11 Conclusion
The theory needs elaboration, and may prove false or underdeveloped. But compared with "simply positing a primitive metaphysical flow and crossing one's fingers hoping that somehow we sense it", it advances independently suggested mechanisms, makes specific claims, unifies phenomena and suggests experiments: by any reasonable standard of theory choice, a better theory of passage than any currently on offer in metaphysics.
The account leans on contested psychology: the narrative self is disputed (Strawson's "episodic" people deny living narratively), the ego-moving/time-moving literature is debated, and the developmental correlations are predictions, not findings. Critics of deflationism (e.g. Defenders of phenomenal passage) will say Callender explains why we believe time flows, and why we represent it so, but not a felt quality of passage, which he denies exists. The "always behind" and "split screen" cases also show the explanation is evolutionary-functional, not a derivation from physics, as he intends.
Explaining the Temporal Value Asymmetry
We prefer pain to be past rather than future, even when the past pain was greater. Arthur Prior made this the basis of his famous "Thank goodness that's over" argument for tensed time. Callender (drawing on joint work with Christopher Suhler, 2012) dismantles the argument's three readings and replaces it with a scientific explanation modelled on how economists and psychologists explain discounting of the distant future: the physical causal and knowledge asymmetries generate an affect asymmetry, and that affect, shaped by evolution, mediates the past/future value asymmetry and a whole family of newly discovered temporal biases. Tensed metaphysics, by contrast, explains by stipulated primitives, much as a posited life force once "explained" life.
Hume already noted that future distance weighs more on the will than the same distance in the past, "as none of our actions can alter the past" (Treatise 2.3.7.6).
Parfit's hospital. You wake in a hospital bed, unsure who you are. A nurse finds two charts: you are either Patient A, about to undergo a special operation requiring you to be conscious and without pain relief for an hour, followed by an amnesia pill; or Patient B, who has just undergone the same operation, which by complications took five hours of agony, a hospital record, also followed by amnesia. Which would you rather be? Five exceeds one, yet most people strongly prefer to be B: past pain is discounted so steeply that some would prefer any amount of forgotten past pain to almost any future pain. "It's almost as if that past pain happened to someone else." (Derek Parfit, Reasons and Persons, 1984, §64.)
| Past/Future value asymmetry (PF) | Proximal/Distant discounting asymmetry (PD) | |
|---|---|---|
| Definition | All else equal, we prefer past pain (future pleasure) to future pain (past pleasure) | All else equal, we prefer distant future pain (proximal future pleasure) to proximal future pain (distant future pleasure) |
| Compares | Past vs future events | Near vs far future events |
| Examples | Relief that a headache is over; eagerness for holidays to come | Skipping a flu shot for a coffee; under-saving for retirement |
| Who studies it | Mostly philosophers (Sidgwick 1907, Parfit 1984, Brink 2011 on ethics) | Economists, psychologists, evolutionary theorists, in rich detail |
| Explanations offered | Prior's (1959) tensed metaphysics | Psychological, economic and evolutionary mechanisms |
The figure plots care against time from one present: the familiar hyperbolic decline toward the future (PD) and the less studied collapse of concern once an event becomes past (PF). The only existing explanation of PF comes from metaphysics: Prior's argument, one of the two or three central arguments in philosophy of time.
12.1 "Thank Goodness That's Over"
Prior (1959), on his Australian colleague J. J. C. Smart's tenseless, four-dimensional view: when I say "Thank goodness that's over", do I mean "Thank goodness the latest part of that is earlier than this utterance"? "I certainly do not... It's the overness, the now-endedness, the pastness of the thing that I'm thankful for." Read naively (eternalism makes all events "of equal significance", per Cockburn, 1998), the argument is a non-starter: believing all times exist no more makes them equally significant than physicalism makes your mother as significant as your desk. Callender finds three serious readings (one framed as a knowledge argument by Kiernan-Lewis, 1991):
Three readings of Prior's argument and Callender's replies
- Indispensability of tense. What fact is the eternalist thankful for? Not that the headache is earlier than the utterance. Prior is right, but this is just the Temporal Knowledge Argument (Chapter 9): attitudes create non-extensional contexts. One can thank goodness Superman arrived without thanking goodness Clark Kent did (Dainton, 2001). The same holds for space and persons ("thank goodness the fire is not here"; "thank goodness the person with the pie in the face is not me"). The answer is an indexical mode of presentation, not new ontologies of places and persons, so by parity not of times. (The temporal asymmetry is inessential: "the headache is now" raises the same point.)
- "Over". "Over" signals a privileged perspective: the seat of experience has passed the headache and won't again. Mellor (1998) replied that thanking goodness for "earlier" explains as well, but misses the sense of the thanker's spot as the cut between open future and dead past: the flow. Callender's Chapter 11 answers this: before the headache, the self cannot narrate it "from the inside"; afterwards the experiences are built into the story, so later selves won't go "through them" again. Over and done, not merely earlier.
- Explanatory challenge. Dean Zimmerman (2007): relief when pain becomes past and disappointment when pleasure does; a theory of time making this "utterly mysterious" is suspect (also Cockburn, 1998; Carroll and Markosian, 2010). What is the tensed explanation? Craig (2000): "past pain is non-existent pain, and so no pain at all!"; Zimmerman: past headaches "have no properties whatsoever, including being painful".
Why the tensed explanation is no explanation. (a) On presentism future pains do not exist either, yet we dread them; so non-existence cannot be what explains indifference. Craig's real answer is that future pain will be present: non-existence is a red herring. (b) Then Mellor's tu quoque applies: a primitive link between attitudes and what will be is no better than stipulating that relief fits pains earlier than one's experience. (c) A primitive directionality is compatible with eternalism, so at best the argument supports adding a direction, not presentism. Callender's verdict: "theory read into the data, primitives galore, and explanation by stipulation again and again", "all the hallmarks of a pseudo-scientific explanation". (A growing-block or becoming theory does no better, Chapter 13.)
Still, reading 1 reminds us a naive treatment of tense fails; reading 2 forces a theory of apparent flow; reading 3 rightly asks why our attitudes are so asymmetric.
12.2 The Proximal/Distant Asymmetry
Humans and other vertebrates show an immediacy bias: with lending and borrowing at 5%, many prefer $120 this year and $70 next to $100 this year and $105 next, despite the lower lifetime value (Read, 2008, is a good introduction). Documented features:
- Magnitude effect. Small amounts are discounted more than large ones.
- Sign effect. Losses are discounted less than gains.
- Direction effect. Delaying an outcome is discounted more than expediting it.
- Hyperbolic, not exponential. An exponential discount factor D(d) = (1 + r)^−d is stationary; a hyperbolic one, D(d) = (1 + kd)^−1, changes with delay. Hence preference reversals: people prefer $100 now to $200 next year, yet $200 in six years to $100 in five. Wanting to be richer in 2025 than 2020 is not inconsistent; the updating is: in 2025 the year 0–1 that was "worth the wait" from 2020 becomes the year not worth it, and the person bound by 2020 preferences is frustrated. Planning a dinner next week, I value fitness and resolve to skip dessert; after dinner, I want dessert.
Explanations exist at several levels, and discounting is sometimes rational (given certain hazard distributions), sometimes not:
- Psychology. Future anhedonia: we expect distant hedonic states to be less intense (Kassam et al., 2008).
- Knowledge asymmetry. Delay correlates with uncertainty (imperfectly: Halley's return is more certain than a coin toss in a minute). Keren and Roelofsma and Sozou proposed that immediacy bias stems from implicit uncertainty; Weber and Chapman confirmed that adding uncertainty eliminates the immediacy effect, and adding delay eliminates the certainty effect.
- Evolution. Maclaurin and Dyke (2002): for most of our history the costs of predicting and influencing the distant future outweighed the benefits; "we care more about proximate future pain than distant future pain for the same reason that herbivores care more about proximate predators". Robson and Samuelson link discounting models to fitness.
An overlooked point: PD is tensed too. It is not one value-versus-time curve but a family of curves, one per present; what matters is delay from now, not calendar date (Strotz, Rasmusen). So PD and PF both depend on the tripartite division, and PF should not be read as a preference for minimising later pain at the expense of earlier pain.
12.3 The Humean Solution
Look first at psychology, environment and evolutionary history: "causal accounts". Hume supplied the causal asymmetry, Parfit mentioned natural selection, Paul Horwich (1987) combined them.
- Causal asymmetry. Our actions sometimes bring about future goals, rarely or never past ones (Kutach, 2011). Since Reichenbach, people have tried to ground it in the thermodynamic arrow (critics Loewer, 2007; Frisch, 2010); Callender finds it "almost impossible to believe" it is unconnected to the universe's physical asymmetries, another physical "hook".
- Effective desires. Horwich: desiring that future desires be satisfied raises their chances; desiring that past ones were satisfied does not. Wanting to have played basketball yesterday changes nothing; wanting to play tomorrow does.
- Selection. Satisfying desires generally aids survival and reproduction, so caring about future desires was selected for; caring about past ones was not, since no effort after the fact helps.
This assumes, coarsely, that we maximise expected utility and that expected utility tracks fitness (Okasha, 2011). Craig (1999) scoffs at a "'just so' story of sociobiological mythology"; Maclaurin and Dyke's defence, tying particular emotions to fitness, blunts the jibe. But a richer story is available.
12.4 The Knowledge Asymmetry
The Humean solution ignores an equally important asymmetry. "We know more about the past" is loose; David Albert's (2000) version: take the macroscopic information available now and evolve it with the laws forward and backward. All our knowledge of the future comes from this procedure, but much of our knowledge of the past does not. It predicts Halley's comet in 2061, but not tomorrow's Dow Jones; yesterday's Dow is in the newspaper. (Albert traces it to a special initial condition; Callender needs only that it exists.) Its relation to causation is open: a causal theory of knowledge (Goldman, 1967), knowledge grounding causation (Horwich), a common origin (Albert), or independence.
Together the two asymmetries define a basic predicament: we are uncertain about many of the events we can potentially control. "When we look into the future we face a yawning darkness" (literally: no light from future events reaches us). A storm could ruin the crops, your partner could hate the present, a meteor could hit you. "And that makes us nervous."
12.5 The Affect Asymmetry
The causal and knowledge asymmetries leave a trail, notably the affect asymmetry: emotions are more intense for anticipated than retrospected events (Van Boven and Ashworth, 2007). Told that Coca-Cola will test a vending machine that raises prices in hot weather, people are angrier than when told it was tested months ago (Caruso).
- Knowledge → affect. Uncertainty intensifies emotion: haunted houses frighten because we don't know what's next; we like surprise gifts. Wilson et al. (2005) gave students on campus an unexpected dollar coin with a card; when the reason was unclear, the good mood lasted longer. Bar-Anan et al. (2009) had viewers repeat uncertainty phrases while watching film clips; their reactions, positive and negative, were more intense. Other routes: focalism (Wilson and Gilbert, 2005; overweighting one event, likelier for an unknown future than a richly known past) and more extensive mental simulation of future events, which mediated affect differences in Van Boven and Ashworth's study.
- Causation → affect. Not knowing what you are responsible for changing is especially nerve-racking. Emotions evolved to organise and motivate action (Frijda, 1986, 1988; Griffiths, 1997; Nesse, 1990; fear's physiology readies fight or flight, LeDoux, Panksepp, 1998). Future-directed affect helps bring about good outcomes (food, mates) and avoid bad ones (starvation, predators); past-directed affect changes nothing, given the causal asymmetry, and wastes time and energy, so it would be selected against. (Antonio Damasio's work, and Bechara et al., 1994, 1996, on ventromedial prefrontal patients shows how decision-making collapses without appropriate affect.)
Box: mental mechanisms and evolution. Callender explicitly avoids "explanatorily impotent" evolutionary psychology that posits a dedicated module for each function (as in Cosmides and Tooby, 1997; Haidt and Joseph, 2007), criticised by Buller and Hardcastle (2007) and by Suhler and Churchland (2011). His starting point: valuation and choice mechanisms exist in nearly all animals, applied to proximate options. As hominin cognition learned to represent distant options (barren trees that will bear fruit in months) and counterfactuals, the old machinery was applied to them. Given the evolutionary function of emotion plus the causal and knowledge asymmetries, selection would favour weighting future costs and benefits over past ones. No new dedicated mechanism is claimed.
12.6 Explaining the PF Asymmetry
The decisive evidence is Eugene Caruso, Daniel Gilbert and Timothy Wilson's (2008) five experiments, the first in psychology aimed squarely at PF (stories about equidistant past and future events, order counterbalanced).
Caruso, Gilbert and Wilson. - Experiment 1. Five hours of boring data entry one month ahead versus one month ago: fair pay for the future job averaged $125.04, for the past job $62.20, twice as much, a highly significant difference, with equal (seven-point) ratings of difficulty and qualification. - Experiment 3. Helping a neighbour move one week ahead versus one week ago, rewarded with a wine coupon: subjects chose wine 38% more expensive for future help ($162.24 versus $117.96, from a list priced $10 to $400), with equal ratings of difficulty, length and satisfaction. On a seven-point scale, future helpers felt more "tired, stressed, and dreadful", and this index of negative affect fully mediated the effect of temporal location on valuation. - Experiment 4. Data entry by oneself versus by a random community member: the asymmetry appeared only for oneself; stress ratings showed the same pattern, and affect again fully mediated the interaction.
So the affect asymmetry, itself plausibly rooted in the causal and knowledge asymmetries, explains PF. Caveats: one study, relying on self-report affect. In its favour: consilience with extensive evidence that affect is central to valuation and decision generally (Damasio, 1994; Panksepp, 1998; Thagard, 2006), e.g. Reactivated negative affect steers people from options that went badly before (Bechara et al., 1996) and positive affect sustains cooperation (King-Casas et al., 2005).
Caspar Hare (2009) objects that whatever his "swampy ancestors" could do, he can focus on controllable future things without future bias. Callender: introspective confidence cannot rebut an explanation of unconscious biases, and an evolutionary account explains a general tendency, not an inevitability.
12.7 Other Temporal Biases
Almost every emotion is temporally biased, and so are many judgments. (You may not feel these in yourself, since reading both conditions together prompts consistency; real studies test one condition per group.)
| The future seems... | Study | Finding |
|---|---|---|
| More intentional | Burns et al. (2012) | Elderly Gertrude kills her husband with the wrong pills (poor eyesight and tiny labels; but a rocky marriage and a big life-insurance policy). Told it is a month ahead, mock jurors judge it more intentional, are more upset, want harsher punishment |
| More potently willed | Plaks (2012); Helzer and Gilovich (2012) | Peter does 60 push-ups, 15 more than his personal best: willpower is judged more decisive if the feat lies a year ahead than a year ago; not more willpower, but more potent willpower |
| More likely | Rothbart and Snyder (1970); Brun and Teigen (1990); Hanko (2007), Hanko and Gilovich (2008) | People bet more on an unrolled die than a rolled-but-unseen one, and feel guesses about a baby's sex are better before birth |
| More rational, more prototypical | O'Brien (2013); Kane et al. (2012) | Past selves rated more emotional, future selves more rational; prospection is grounded in "scripts, schemas, stereotypes" |
| More blameworthy and praiseworthy | Caruso (2010) | The hot-weather pricing machine provokes more outrage if upcoming; future transgressions are worse, future donations more praiseworthy |
| Closer, brighter, lighter | Caruso et al. (2013); Helzer et al. (2011) | The temporal Doppler effect: Valentine's Day a week ahead feels nearer than a week past; the future is drawn bright and light, the past dark and heavy |
Confounds are hard to escape in a world this asymmetric (the past soda trial evidently did not spread such machines; the future one might), and the same worry applies to philosophers' thought experiments like Parfit's. But the picture is of a large, deep suite of biases in emotion, desire, intention, rationality and probability, with real stakes: juries might award more for future suffering than equal past suffering; organisations might do controversial things first and announce later, to draw less condemnation.
Zimmerman wanted an explanation that vindicates the bias as rational. Psychology rarely does that, and some of these biases are outlandish: should a tensed metaphysics make it apt to punish future crimes more than past ones, or to think future dice likelier? Rationality will be contextual, as with PD (Bayesian agents facing certain hazard distributions should discount; in other environments the tendency misfires). Compare hunger: a superb shortcut that offloads calorie accounting, yet we do not always feel hungry when we need food, "a peek at my waistline proves this false".
12.8 Explaining Other Time Biases
- Compensation and praise. Since the future intensifies affect, future suffering merits more compensation and future generosity more praise. If each chapter of this book gives you a headache, Callender owes you more for the one still to come than for all the past ones. Affect largely mediates these biases (Caruso, 2010).
- Will, prototypicality, predictability, likelihood, premeditation. Fischhoff and Beyth's (1975) hindsight research found "creeping determinism": past events seem inevitable. Callender blames the knowledge asymmetry: looking back at last night's impulsive candy bar, he sees every push (hunger, a too-healthy dinner ahead, the closed bakery, the shifty character outside the better shop) and every consequence (the sugar crash, the empty milk carton); it is locked in like a puzzle piece. The future lacks that detail, and the causal asymmetry adds that nothing can be done about the past anyway.
- The will. Helzer and Gilovich (2012): looking back, people struggle to see the will's role; looking forward, they see it as "a powerful force", forgetting the snags that will recur, and become overconfident. Background knowledge dilutes the will in the past; in the empty future "the will stands out", amplified by agency. "That's why those past New Year's diets didn't work but this next one will." The future seems prototypical and predictable because we extrapolate from the present for want of detail.
- Dice. We do not really think the future roll likelier; we think our guesses about it are better, as if by a "magic power" over the die, an exaggerated will that the causal asymmetry rules out for the past.
- Gertrude. Burns et al. (2012) found that adding an index of negative affect made the effect of temporal perspective nonsignificant, and intention ratings likewise mediated affect: a bidirectional correlation. We are inveterate mind readers who need to predict others, who act only futureward; and future wills look more potent.
Not every case reduces directly to causation, knowledge or affect: we are nearly certain of Halley's next visit yet do not think it fixed, and we know almost nothing of individual Etruscans yet do not think their lives open. Given the sheer number of past/future differences, organisms would naturally adopt a heuristic that paints the whole future one way and the whole past the other. Caruso: reactions to future events are "overlearned responses to the natural environment", an "overgeneralized response to future situations even when these situations are not actually under one's control" (similarly Parfit; Van Boven and Ashworth). Hence the yellow future and the brown past.
12.9 Conclusion
Since 2017 future bias has become an active empirical topic in experimental philosophy. Preston Greene, Andrew Latham, Kristie Miller and James Norton have run large surveys separating hedonic from non-hedonic future bias, finding that bias toward the future is robust for pains and pleasures, weaker or absent in some non-hedonic cases, and that it correlates only loosely with whether people endorse a "dynamical" (flowing) view of time. Related studies by Latham, Miller and Norton probe whether the folk conception of time is dynamical at all, with mixed results. This supports Callender's separation of the value asymmetry from metaphysics, while showing the asymmetry is more fragmented than a single affect story may suggest.
Tensed models "explain" PF by positing a primitive link between a metaphysical arrow and our preferences, which opponents can posit equally, which says nothing about why our attitudes took this shape, and which predicts nothing about other biases: Like a special life force posited to explain life, "these arrows name a problem more than they provide a solution." The Humean account has "morphed into a scientific research program" that is fruitful, unifying and predictive. Testable conjectures: PF vanished for strangers; will it reappear for emotionally close others (do I treat my children's past labour like my own)? Affect's role can be probed with skin conductance and fMRI. "When evaluated along typical dimensions of theory choice in science, there is simply no contest between the two." Which biases are rational is left for future work.
The practical stakes are large. Temporal biases run from personal finance to social policy, so understanding how the causal and knowledge asymmetries leave a trail of biological and psychological asymmetries in their wake is, for Callender, of paramount importance; the next step is to sort the rational biases from the misfiring ones.
Moving Past the ABCs of Time
Why has the book ignored A- and B-predicates, token-reflexive analyses and the presentism–eternalism debate? Callender gives a deliberately "biased" history: analytic philosophy of time drew metaphysics from the semantics of tense, then retreated into debates about naked existence, which buys immunity from science at the price of explanatory power. Existence explains everything and nothing; by making the tensed present undetectable to defend it, philosophers leave it unable to do the job it was posited for: "Time's ontic voltage can't light the smallest bulb." His approach is a naturalised modification of the old programme, mapping the A-series onto manifest time and the B/C-series onto physical time.
13.1 Analytic Philosophy of Time: A Potted and Biased History
- 1908 McTaggart argues A-properties are essential to time but contradictory, so time is unreal; the same year Minkowski declares space and time "doomed to fade away into mere shadows": a "rough year for Time" (Savitt, 1994).
- Mid-century In the era of ordinary-language analysis, the A- versus B-theory debate becomes a question of whether A-predicates are semantically reducible to B-predicates.
- 1968 Richard Gale's canonical text defines the A-theory as a bundle of four doctrines.
- 1980s The "new B-theory": B-theorists may "win" by giving tenseless truth conditions, not synonyms; the debate shifts to whether A-facts or B-facts are ontologically basic.
- Today Debate centres on existence: presentism, possibilism (growing block), eternalism, and many variants.
A-theory: A-properties (past, present, future) fundamental, B-relations derivative. B-theory: the reverse. Once philosophers asked whether B-predicates could replace A-predicates without change of meaning, and inferred tensed reality if not, B-theorists "obligingly played the stooge" with ever cleverer translations, doomed because indexical language does not reduce to non-indexical language.
Gale's four-part A-theory (a "real mongrel")
- Semantic irreducibility of A-predicates to B-predicates.
- Temporal becoming: the future is unreal but becoming real; future events acquire now-ness.
- Ontological past/future differences: past fixed, future open.
- Change requires an A-series (Fitzgerald, 1985).
These are logically independent (on a natural reading), yet (1), a purely linguistic thesis, carried almost the whole argumentative burden, as if securing (1) delivered (2)–(4). "Now we scorn arguments whose premises are entirely about language and whose conclusions are metaphysical." (Heather Dyke, 2003, chronicles recent cases.)
The 1980s refashioning changed little on the ground: battles moved to token-reflexive versus date analyses, whose pitfalls all have exact twins for "here" and "I", so why the issue concerns time was never clear. The Routledge Philosophy of Time two-volume set still devotes seventeen chapters to analyses of tensed statements (imagine seventeen chapters on "here" in a philosophy-of-space collection: "there would be a riot"). Replace (1) with the metaphysical (1′), that A- or B-properties are basic, and (2)–(4) still do not follow. Even if A-properties exist, all we learn is that truth-values of tensed statements change, a "paltry thing" that says as much about truth as about time: nothing about whether past or future are real, whether time branches, is discrete or continuous, or why we know more of the past. "It's like being told that a particle accelerator revealed the existence of something represented by a noun." We know quarks are more basic than protons because a tested theory says how quarks compose protons; nothing comparable exists for A-facts and B-facts.
Two libraries. For Newton, Helmholtz, Poincaré, evidence about time was like evidence about space: scientific theory choice (the Newton–Leibniz dispute turned on which physics explained the phenomena best). After the linguistic turn, time was studied through language, space through science. At UC San Diego's Geisel Library (named for "Dr. Seuss"), philosophy-of-time books sat in Humanities and Social Sciences on the left, philosophy-of-space(time) books in Science and Engineering on the right, and the contents justified the split: space books draw on Berkeley, Kant, Helmholtz, Newton, Leibniz, Maxwell, Poincaré, Einstein; time books on Augustine and semantics, written almost entirely by non-scientists. Evidence standards drifted: to find Bigfoot you search the woods; to infer dark matter you weigh the best theory of countless gravitational events; to learn time's nature, philosophers ask whether the date analysis "leaves something out".
Today's debate claims to be ontological:
| View | What exists |
|---|---|
| Presentism | Only present events |
| Possibilism / growing block (becoming) | Past and present events |
| Eternalism | Past, present and future events, "equally" |
Arguably orthogonal to A/B (Josh Parsons, 2002). Variants multiply: thick and thin presentism, priority presentism, existence presentism, moving spotlights, Meinongian presentism, fundamentally tensed presentism, many branching models, absolute becoming (forward and reverse), with stock worries like the truth-maker objection. The Australasian Journal of Philosophy once asked for more referees because presentism submissions exceeded its capacity.
13.2 The Explanatory Challenge
Callender is not sure one model is right, nor that they are mere notational variants (as he, 2000, Meyer, 2013, Savitt, 2006, and Dorato, 2006, have suggested they might be). His complaint: the models do little explanatory work, and evidence standards ignore philosophy of time's goal. What would count as evidence for thick skipping presentism (an extended present that jumps)? A stop-motion world? No: we don't experience the missing moments. For the future's non-existence? A sense of free will? No: laws still constrain us, we cannot sense the gap between physical and metaphysical necessity, and block-compatible models of agency abound.
| Dark matter models | Metaphysical models of time | |
|---|---|---|
| Variety | WIMPs, WILPs, MACHOs, RAMBOs, axions... | Presentisms, growing blocks, spotlights, branchings... |
| Observations | Some eliminated; all aim at possible observations | No observational consequences sought |
| Independent checks | "Warm" dark matter (particles of roughly 300 eV to 3000 eV) suffers if particle physics has no independent need for it | No effort at independent checks, unification or consequences |
Diagnosis. Science threatened metaphysics of time by indifference: physics and psychology succeeded without a now, flow or skipping presents, and any addition that mattered (a preferred frame) lacked scientific reasons. Philosophy of time found "the perfect response": make itself about the one property guaranteed never to conflict with science, naked existence. What distinguishes the present? It alone exists (presentism) or is the edge of existence (possibilism). Flow? The changing extent of what exists. Fixed past, open future? Past exists, future does not. (Callender cites C. D. Broad (1933), via John Earman, praising McTaggart's ignorance of science as an advantage, since physics had "gone to the heads" of eminent physicists.)
Why existence cannot explain
- Work is done by natures, not existence. Jean Perrin's case for atoms turned on transmission speeds and specific heats, the atoms' specific causal powers. "Without specific essentia, their esse explained nothing." Much explanation is causal, and naked existence causes nothing.
- Existence is no first-order property (Hume, Kant, Frege–Russell; McGinn, 2000). A hundred real dollars and a hundred imagined dollars have the same amount, colour and weight. For Frege existence is captured by the quantifier, a predicate of predicates: for the future to exist is for "a is future" to be true of some event a. It changes no object "one jot".
- Even as a property it adds nothing. Donald Williams (1962): any doctrine that does not impute too much to existence imputes much too little; spell out its "ontic voltage" (Santayana's "strain and rumble of the universal flux", the neo-Thomist Perfection, the Existentialists' absurdity) and you say too much.
- It is unobservable. Williams respects anyone who stares at a doorknob and thinks he sees it existing; "what one observes is not the existing of the knob but just the knob". Phillip Bricker (2004): how do you know you exist? Nothing about your parents, house, shirt or mental state differs, and you are not incomplete like Sherlock Holmes, who neither has nor lacks a freckle on his left shoulder. That the question makes sense shows existence does not affect experience.
Growing block or shrinking block? (adapted from Dainton, 2001). In Broad's growing block, t_j is later than t_i iff reality's sum total at t_j is greater: the block-arrow. Now define the world-arrow by memories, causation, counterfactual dependence, knowledge and attitudes: t_j is later than t_i iff memories of t_i events exist at t_j and not conversely. Nothing in the theory forces the arrows to align, so consider a world where they point opposite ways: by the world-arrow it is a shrinking block, where your memories of childhood are of events that have not yet come to exist and your anticipations are of what already existed. Assume memory partly underlies the attitude asymmetry and sense of freedom, and supervenes on matter configurations. Then the block-arrow explains nothing: it is explanatorily inert. Tie the arrows together by fiat ("I can remember only what already exists") and you have the dormitive virtue of opium. Telling someone ghosts don't exist is useful, but to explain the creaking at night "one needs to roll up one's sleeves, climb into the attic, and find the open shutter or noisy critters".
(Presentism and possibilism may add a primitive direction, but then the direction, not existence, does the work, and eternalists such as Weingard (1977), Christensen (1993) and Maudlin (2002) posit one too.)
The irony. By owning existence, philosophy of time made itself immune to experiment (existence can't be seen) and theory (it has no causal powers), and bought disciplinary autonomy amid the rise of metametaphysics. But the models were posited to explain temporal phenomenology, the fixed past and open future. Defended as undetectable ("don't worry about that preferred frame... We're doing metaphysics, not physics"), they are then asked to explain experience, which would require us to detect what no physical detector can. Kristie Miller (2013): making the privileged present empirically undetectable undermines the very motivation for positing it (also Dieks, 2016). The effects of "tensed existence" are the phenomena used to infer it: pseudo-explanation that labels our ignorance (paraphrasing Keating, 2002, on the life-force doctrine).
13.3 The ABCs of Physics
Callender's approach naturalises earlier work (notably Mellor, 1998). Compare McTaggart's series with his two images:
| Series | Content | Example relations | Relation to Callender's images |
|---|---|---|---|
| C-series | Order only: a betweenness relation among triples (M, N, O, P in that order, not M, O, N, P) | N is between M and O | Probably closest to physical time, which is fundamentally time-reversal invariant |
| B-series | C-series plus direction; two-place, asymmetric, irreflexive, transitive; also metrical | x is earlier than y; M is two days before N; simultaneity defined | Kernel of physical time: no egocentric past/present/future; with updating, earlier/later picks out causal connectibility, from which Robb, Reichenbach, Grünbaum and Winnie recovered Minkowski structure, and Malament (1977) recovered past–future distinguishing spacetimes up to conformal factor |
| A-series | Events move future → present → past; one-place, also metrical | is present; is past; "the year 2000 is ten years past" | Shares manifest time's key feature: the objectification of {past, present, future}; but manifest time adds openness, absoluteness and more |
Two "picky" disanalogies: physical time lacks fundamental direction (so the C-series fits better), and relativity has spatiotemporal, not temporal, relations. The causal programme failed for full relativity, whose time is "a lot richer" than any web of earlier/later relations; even where it works, packaging physics in B-vocabulary would obscure one's resources, as nobody would do with Eddington's two tables. McTaggart's great insight, the objectified egocentric division, also proves useful in developmental psychology.
13.4 Eliminating Tense?
Callender has not "purged" his sciences of tense, and uses causal and knowledge arrows without sanitised versions. Is he begging the question? Not unreasonably:
- Tenseless arrows exist. Albert (2000), Loewer (2012) and Kutach (2011) offer causal and knowledge arrows compatible with a tenseless metaphysics. On Albert and Loewer's "imperialistic" statistical mechanics (the Mentaculus), all special sciences are probabilistic corollaries of a tenseless probability posit over a special initial condition, fitness included: a kind of transcendental argument that the sciences used rest on a tenseless base.
- Clock time suffices. Otherwise, case by case: most of the sciences used rely on ordinary clock time, easily shown compatible with physical time.
He concedes he has not shown the cleansing can be done, and refutes no metaphysical model. What he does is give them less to do: no more going straight from tensed attitudes to presentism; one must go via psychology, and tie presentism to the self, and point out hidden presentist assumptions in each scientific success. Nobody is doing that.
13.5 Conclusion
The old doctrine that life needs a special life force was never refuted; it was made unemployed, redundant, and the synthesis of organic compounds from inorganic matter only made it more contrived. Callender's hunch is that presentism and its kin will share this fate as explanations of the present, flow and past/future division grow richer and testable, and recognising this frees us to study time with many more tools.
This is the book's most polemical chapter, and Callender calls his history "potted and biased". Metaphysicians reply that (a) they never claimed tensed models are empirical hypotheses; (b) explanation is not the only theoretical virtue (fit with intuition, truth-making, accounts of change and persistence); and (c) existence-based views do make contentful claims, e.g. About truth-makers for past-tense statements. His life-force analogy assumes the scientific explanations of Chapters 9–12 will mature; they are, as he admits, sketches. And some presentists (e.g. Zimmerman, Crisp) have engaged physics seriously, as Chapters 3–4 themselves show.
Putting It All Together
Callender assembles the account. Evolved binding windows, shared because of hard physics and soft environment, make us think the present is objective; an enduring, narrated self at the leading edge of asymmetric memories makes time seem to flow; causal and knowledge asymmetries plus evolved affect make the future open and the past settled. Every feature of manifest time is explained by the same schema: physical hooks + environment + an embedded subject + evolution; trying to explain the flowing now with physics or metaphysics alone "is like trying to bake a cake with only flour."
The theory in three moves
- The present. Signals arrive at different speeds through different senses; binding thunder with lightning would cost too much time, failing to bind lips with speech would cripple communication. Evolution shaped the windows; physical constraints and environmental factors make them widely shared though variable at the edges. Intersubjective agreement "tricks us" into taking the present as a global feature of the universe.
- Flow. Presents must update. Organisms need a self (primitive ones must not eat their own arms); long-range, social planning demands an enduring self, constantly narrated with goals and history. Built from temporally asymmetric memories, the self takes itself to be the leading edge of an extended agent; because the story keeps going, it thinks it grows through time, "tantamount to conceiving of time as flowing".
- Past/future asymmetry. A natural response to the physical, causal and knowledge asymmetries: evolution will not build creatures that waste effort trying to change the past or care about it as much, so it installs an asymmetric affect structure; with agency, the future looks open and the past "dead and settled".
Williams's pure intelligence. In "The Myth of Passage" (1951), Donald C. Williams asked: suppose a pure intelligence "bred outside of time", taught the nature of the manifold and the design of the human space-time worm with its "mnemic organization", were incarnated among us; "what could he have expected the temporal experience to be like except just about what he actually discovers it to be?" Implement that intelligence as IGUS, embed it in our world, and ask how it will represent time. Callender's answer: a "resounding Yes", it would divide the world into past, present and future, update the division, invest it with deep significance, and believe time flows.
14.1 Common Structure
All three explanations rest on the physical differences between time and space from the book's first half, the physical hooks:
- The present needs time's one-dimensionality, our always-timelike worldlines, the mobility asymmetry, signal speeds, and higher-level facts about the physical arrow.
- Flow and the past/future asymmetry inherit all of these, since they depend on the present.
- Decentering needs a self precisely because we cannot decenter by moving in time as we do in space: the same features that make us "stuck in the now" make an enduring self the only route to temporal decentering.
- The past/future asymmetry depends on the causal and knowledge asymmetries, which Callender believes are tied to the thermodynamic and radiation arrows.
Hooks are far from sufficient. Add initial and boundary conditions (the environment), still not enough; add an embedded subject interacting with others and the world; add evolution, which shapes simultaneity windows, the self's boundaries and our emotions. Only then do we approach an explanation, which is why one was so long coming and why Einstein's worry can be answered in Carnap's spirit.
14.2 A Unified Flowing Now
Decomposing manifest time into present, flow and direction did violence to its unity: we would hardly believe in a privileged present if it didn't update, and fixity and openness need the now to slice the world. As windows of simultaneity evolved, so did past/future discounting, memory, self and agency; "it's biologically naive to think of these conceptions as arising independently". Some connections:
- The self and headaches. Beyond the indexical reply to Prior: the self crawling up your worldline has built the past headache into its story; the future one is uncertain (triggering asymmetric emotion) and genuinely not yet experienced. "The past one is part of what you now are, whereas the future one isn't", so it is over as well as fixed.
- Narrative and openness. Through agency and felt freedom, the narrative self helps make the future feel open; and, as Genevieve Lloyd (2008) notes, the autobiographer constantly transforms the past into something necessary to fit the developing story, "impos[ing] a pattern of necessity on the fragments", which supports the past's fixity.
Intentional binding. Patrick Haggard and colleagues (2002) had participants press a key that produced a tone after a fixed lag of 250 ms; using a Libet clock, they reported the time of either the press or the tone. Some presses were voluntary, others involuntary, induced by transcranial magnetic stimulation. For voluntary actions, the perceived press and the perceived tone shifted toward each other: perceived authorship compresses the interval between action and outcome. "We use temporal contiguity to determine causation, as Hume thought, but we also use causation and agency to determine temporal contiguity." The effect is robust across modalities and is used as a measure of felt authorship.
What intentional binding connects
- Agency and the present. How we organise phenomena in time depends partly on our sense of agency: the self that exists wholly in the moment and feels authorship links to the temporal organisation of Chapters 9–10.
- Freedom and the future. Barlas and Obhi (2013): binding is strongest when participants choose among many keys ("greater freedom"), weakest with no choice. Agency, time and freedom appear linked, how and why still open.
- Valence and time bias. Emotion and the sense of agency both evolved to prepare and motivate action. With binding, we pull good outcomes we caused closer in time and push bad ones away (Yoshie and Haggard, 2013), echoing Chapter 12.
Intentional binding has been heavily re-examined since 2017. Marc Buehner and others had already found comparable "causal binding" for mechanical causes without voluntary action, and later reviews and meta-analyses (e.g. Tanaka and colleagues, 2019) conclude that binding reflects causal and predictive relations at least as much as intention, so it is a contested proxy for the sense of agency. This complicates, without undercutting, Callender's point: what binds action and effect in time is the creature's causal model of its world, again a hook-plus-subject explanation rather than a window onto metaphysical time.
14.3 Animals
Animals face the same pressures and asymmetries, so many probably model time similarly. Temporal discounting studies on primates and jays (do cotton-top tamarins show an immediacy bias, and of what mathematical form? See Santos and Rosati, 2015) matter because discounting is tensed, measured from the now; if so, some animals carve the world into past, present and future. Few will have our rich self or sense of freedom ("perhaps time doesn't flow in any interesting sense to the lobster"), and presents should vary widely: sea creatures live where sound is about five times faster; a giraffe's foot nerves run about 3 m to its brain, perhaps a "long now" to go with the long neck (More et al., 2013). Expect, as ever in zoology, "a huge spectrum of surprising differences and commonalities".
14.4 An Illusion?
Deniers of an objective now are often said to call it an illusion, and some mind-dependence theorists embrace the label. Callender declines it. An illusion is a perceptual experience in which an object appears to have properties it lacks, as in the Müller–Lyer lines.
| Aspect of the flowing now | Status | Illusion? |
|---|---|---|
| Sensed motion, synchrony, duration | Genuinely perceptual | No: they generally track real motion, synchrony and duration; they misfire no more than perception generally |
| Objective global now, flow, open future | Parts of our implicit model of the world, not perceptual experiences | No: not misperceptions at all, though false as fundamental physics |
"Strictly speaking, nothing we counted as an experience of the now or of passage turned out to be illusory." In a loose sense calling manifest time an illusion does little harm, but for many aspects it is "a bit like calling our impression of a shape an illusion".
14.5 Conclusion
Callender's view of philosophy follows Michael Friedman: philosophy supplies metascientific perspectives that open new possibilities for science, as reflection on geometry from Kant through Helmholtz and Poincaré made Einstein possible (Clark Glymour adds Putnam on computational learning theory, Lewis on convention; Fodor and Chomsky behind the shift from behaviourism). This book aspires to that: explaining manifest time falls between physics and psychology, and only a philosopher would ask physicists about sideways initial value problems or psychologists to apply theories of the self to passage.
Research programme and testable claims the book generates
- Time biases. The Hume–Parfit causal answer to Prior has already become a psychology research programme; Callender predicts emotional attachment will make the first-person PF asymmetry appear for third persons.
- Development. Correlations among independent measures of self, autobiographical memory and temporal decentering in childhood.
- Mathematical physics. Work on "looking at the world sideways" (mixed and null initial value problems) and time's role in laws.
- The subjective present. Game-theoretic and evolutionary models of simultaneity windows, and comparisons across species and environments.
- Quantum gravity. Challenges for programmes that would eliminate time (hidden time in the approximations) or rescue it (causal set becoming).
He has not answered the "big" questions, e.g. Whether future events "already" exist, and doubts their answers would explain much. Scott Aaronson describes philosophical progress as a "bait-and-switch": replace an unanswerable riddle Q with a scientific question Q′ that captures part of what we wanted, and solve Q′. So here: from "why does only the present seem real?" to the physical, biological and psychological facts that leave a creature stuck in a moment; from "why dread future but not past headaches?" to how emotions tie to causal and knowledge asymmetries; from "how does time differ from space?" to whether laws can be productive in non-temporal directions. Answering Q′ can make Q look less interesting, as biochemistry did for the life-force doctrine, leaving "are viruses alive?" a terminological question. "The eternalism debate may itself be eternal. While we wait for the verdict, I suggest we tackle some of the low hanging fruit dangling before us."
Master Takeaways
The whole book in its strongest claims, the frameworks worth keeping, and where the argument is most exposed.
- Physics has never needed a now. Aristotelian, Newtonian, relativistic and quantum physics all work without a distinguished present, so without flow or an open future; the question is only whether they contain structure fit to animate.
- Relativity obeys an informal dilemma. The better a structure represents the manifest present, the less relativistic it is, and vice versa: Putnam's planes are global but frame-relative (Malament's theorem), Stein's becoming is invariant but shrinks the present to a point.
- Time is special, not "spatialized". Physics treats the spatial counterparts of now, flow and openness the same way (it ignores them), but the Lorentzian signature, one-dimensionality, mobility asymmetry, arrows and natural kinds sharply distinguish time.
- Time is the great informer. On the Best System account, time is the direction in which our best theory tells the strongest stories; rigorously, only hyperbolic equations with data on spacelike surfaces are well-posed, binding together dimension, signature, production and the absence of closed timelike curves.
- We do not perceive the present; we build a common one. Fast signals, slow macroscopic change and evolved integration windows let us drop time stamps, and because we cannot turn around or travel in time, nothing exposes the now as egocentric the way dinner tables expose the here.
- Flow is the enduring self, reframed. An agent with asymmetric memories narrates an enduring self at the leading edge of its story; sliding from ego-moving to time-moving schemas turns "I move through time" into "time flows".
- Classical physics already threatened common sense. In Trautman's geometrised Newtonian gravity, matter distribution sets the rate of ideal clocks.
- Relativity shatters the master clock into one ideal clock per timelike path; proper time is path-dependent, tested down to height differences of centimetres and below.
- Preferred cosmic times are idealisations. FLRW and CMC time rest on averaging, fail in the actual world, and would require truncating relativity; global hyperbolicity may even be empirically undecidable.
- A quantum preferred frame would not help tensers. If Bohmian mechanics' frame were the becoming frame, no experience could reveal it; if not, tense conflicts with physics.
- Quantum gravity cuts both ways. Causal-set "asynchronous becoming" either matches relativistic options or becomes an ontologically vague, gauge-dependent growth; semiclassical time derives time from timelessness but may assume time in its approximations.
- Two times would wreck physics. Extra time dimensions bring instability, closed timelike curves, non-unitarity, tachyons and ghosts, because dynamics and conservation laws are tuned to one "minus" direction.
- The Temporal Knowledge Argument overgeneralises. If "now is 5 p.m." proved an objective now, "Room 215 is here" and "that's me" would prove an objective here and me.
- Perceived synchrony is not unified. Ventriloquism, recalibration, uncorrelated SJ and TOJ measures and negatively correlated illusions show no single representation of "now".
- Disagreement data exist for the now too. Patient PH, individual PSS values and the Greenwich "personal equation" show our nows differ; coarse events and shared evolution hide the differences.
- Memory is necessary but not sufficient for flow. Asymmetric memory yields epistemic openness and a sense of growth, but not the "whoosh" of something moving.
- Agency makes the future open. Self-fulfilling decisions and counterfactual dependence make us "minor gods" toward the future and powerless toward the past.
- "Thank goodness that's over" proves no metaphysics. Its readings reduce to indexicality, to flow (explained by the self), or to an explanatory challenge that tense answers only by stipulation.
- Affect mediates the value asymmetry. Causal and knowledge asymmetries make anticipation more emotional than retrospection; in Caruso's studies, negative affect fully mediates paying twice as much for future work.
- The future seems more intentional, willed, likely and blameworthy. These biases have real legal and policy stakes, and their rationality is contextual, like hunger's.
- Existence explains nothing. Presentism and its kin buy immunity from science by resting on naked existence, which has no causal powers; they risk the fate of the life-force doctrine: never refuted, made redundant.
- The flowing now is not an illusion. Its perceptual parts mostly track real motion, synchrony and duration; its other parts are an implicit model, false as fundamental physics but sensible for creatures like us.
Frameworks & mental models
| Name | What it says | Use it when |
|---|---|---|
| Manifest vs physical time | A regimented folk theory (now, flow, open future) versus the time that works in physics | Diagnosing why a "time" claim feels obvious yet finds no physical counterpart |
| Two-times problem | Like Eddington's two tables, but with no known bridge | Framing any reconciliation of lived and scientific time |
| Informal relativistic dilemma | Manifest-friendly presents are not invariant; invariant ones are not manifest | Evaluating any proposed "relativistic now" |
| Seven features of the present | Equivalence, global, achronal, simultaneous, radar-synchronous, productive, common | Testing whether a candidate structure deserves to be called "the present" |
| Fragmentation Problem | Time's physical features are logically detachable; what glues them? | Asking why several properties co-occur, like inertial and gravitational mass |
| Time as the great informer | Time is the direction of maximal informative strength in the best system | Explaining why laws seem to "govern" in time |
| Well-posedness test | Existence, uniqueness, continuous dependence; only hyperbolic + spacelike data pass | Asking whether a story can be told "sideways" |
| Representational Asymmetry | We objectify egocentric temporal categories, not spatial ones | Explaining belief in a global objective now |
| Now patches / no time stamps | Fast signals and stable macro-objects make dates unnecessary locally | Explaining agreement on "now" without a physical now |
| Integration payoff matrix | Bind or not, common source or not; integrate iff likelihood ratio exceeds cost ratio | Modelling simultaneity windows as cost–benefit trade-offs |
| IGUS | A minimal information gathering and utilising system | Asking what an agent needs to model time like us |
| Enduring narrative self | The self as protagonist at the leading edge of its story | Explaining the felt "whoosh" of flow |
| PF vs PD asymmetries | Past/future value asymmetry versus proximal/distant discounting | Separating "glad it's over" from procrastination |
| Causal–knowledge–affect chain | Physical asymmetries drive emotional asymmetry, which drives valuation | Explaining time biases in judgment and choice |
| Hooks + environment + subject + evolution | The explanatory schema for every feature of manifest time | Building any naturalistic explanation of a lived category |
The book as a whole. Callender's first half is rigorous; his second half is, by his own account, a sketch. Four exposures stand out. (1) The positive theory of time rests on the Best System Account and on relativism about natural kinds, both contested, and the well-posedness result covers second-order linear equations with Cauchy data. (2) The psychological story leans on findings that have since become more contested (intentional binding as a measure of agency, some recalibration results) and on a narrative self that many reject. (3) He explains why we believe and represent time as flowing, not any felt quality of passage, which he denies; phenomenal-passage theorists will say he changes the subject. (4) His treatment of analytic metaphysics is openly "potted and biased": metaphysicians do not regard tensed theories as empirical hypotheses and value virtues beyond explanation. Factual slips are minor (a pitcher's misspelt name, a relativist's misspelt name, a clock-height figure later bettered, an inflated count of 256 Einstein field equations, a slightly dated 13.6-billion-year cosmic age). Where the book is strongest is in showing that "physics leaves the now out" and "the now is an illusion" are both too quick.
Connect it
- Distributed systems and data architecture. Relativity's lack of a global now has an exact engineering twin: distributed systems have no global clock, so architects use event time versus processing time, logical clocks (Lamport, vector clocks) and watermarks. Butterfield's "no time stamps needed" holds only where signals are fast relative to change; in event streaming and payments, they are not, so time stamps and ordering guarantees are mandatory.
- AI agents as IGUSs. An LLM-based agent with a context window, memory store and planner is a literal IGUS. Callender's analysis suggests why agents need an explicit "now" anchor, asymmetric memory and a persistent self-model to plan coherently, and why multimodal models must solve the binding-window problem when fusing audio, video and sensor streams with different latencies.
- Banking behaviour and conduct. The PD asymmetry (hyperbolic discounting) underlies credit-card revolving, under-saving and BNPL; the PF asymmetry and the "future is more blameworthy" bias mean customers and regulators judge an announced fee or policy change more harshly than one discovered after the fact, a reason for consistent, early disclosure rather than a licence to act first.
- Risk and model governance. The knowledge asymmetry ("all knowledge of the future comes from evolving the present macrostate") is the logic of forecasting models; creeping determinism in hindsight is the bias that makes post-incident reviews overrate how predictable past losses were.
About this book
What Makes Time Special? by Craig Callender, 2017. 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.
Insight Editions · rajeshrmahapatra.com/insight-editions