I want to talk about a basic fallacy that I think infects much of the way a lot of people think about time.

This is a fallacy that arises from combining two intellectual operations that are each, in themselves, perfectly legitimate.  It might be compared to the way language allows grammatical constructions that can lead to fallacious reasoning.  For instance, the phrase “five-sided square” is linguistically permissible, but not logically permissible.  That English allows us to speak of a “five-sided square” should not lead us into meaningless ponderings about the nature of a five-sided square.  Or again, “nothing” is a noun in English, which leads to the logically fallacious idea that it can take predicates—that we can speak of the properties of nothing—whereas it is only logically correct to use it as a quantifier, to say, for instance, that “nothing can prevent the inevitable.”  In a similar way, it is my contention that by combining two mental operations that are each valid and useful on their own, people are led into fallacious and nonsensical ideas about time.  These “mental operations”, as I call them, are so basic and so familiar that unless we are quite careful about it, we never even notice that we’ve done anything questionable in combining them.

The plan for this essay is that first we’ll discuss the “mental operations” I have in mind and show the inappropriateness of combining them.  Then we’ll look at how this leads to misunderstandings, first in science fiction that deals with time and then in philosophical approaches to time.

Let’s introduce these mental operations.  The first is simply picturing a changing image¹.  This is so fundamental to the way our minds work that we hardly even notice that it’s a tool in our mental toolbox.  Of course, we have direct experience of visual images changing over time, and we can reproduce something similar purely in our minds, as visual imagination.  But this faculty is not restricted to visualizations of images we might really experience; it can be used more abstractly.  Imagine, for instance, a time-lapse of a person’s face as they age, or a visualization of an object in space moving along a trajectory under the influence of gravity. 

Still more abstractly, we can imagine a representation of something that is not inherently visual changing over time.  For instance, a weather map, colour-coded to indicate the temperature at each point on the map.  One can imagine such a map changing, say, over the course of a day, as temperatures rise and then fall again, or over the course of a week as a cold front moves in from the west.  Here, the x- and y-dimensions of our image are still “real” spatial dimensions, but the content of the image (in this case, the colour) represents something non-visual.

At yet another stage of abstraction, we can take an image where the x- and y-dimensions no longer represent real space.  For instance, consider a frequency analysis of an audio waveform.  The x-axis represents frequency in Hertz and the y-axis represents amplitude.  Thus, for instance, a plot showing a single spike at x = 440 Hz indicates a pure sine wave at that pitch, while a plot showing a horizontal line of constant height indicates white noise, with an equal contribution from all frequencies.  Again, we can imagine this changing over time, for instance, as different notes are played, raising spikes on the plot at different frequencies.

In all of these examples, we are using, in our imagination, one or more spatial dimensions to represent something—in some cases, they represent the actual dimensions of space while in others (as in the example of frequency analysis of a waveform), they represent something else.  But in all of these cases, we are using time to represent time.  This is true whether we are imagining things in “real time”, at a rate of one second in our imagination per one second of time in the thing we are representing, or in a kind of “time lapse” or “slow motion”.  We don’t necessarily imagine these things “to scale” in time, either; for instance, in picturing the process of a star going supernova, one would imagine the slow build-up of heavy elements in the star’s core (which, in its final stages, still takes thousands of years) in fast motion, and then the core collapse (which takes less than a second) in slow motion.  But in all these cases, the time over which our “image” is changing represents the time over which the thing that our image represents is changing.  For clarity, let us call the former representational time and the latter physical time.

The second “mental operation” I have in mind is the use of a dimension of space to represent time.  Again, this is a very familiar idea, even fundamental to the way we think about time.  We often use the same words to talk about time that we do to talk about space; we can look “forward” to something, we can “move” the date of a meeting, we can do something “throughout” a whole day, and so on.

A calendar is a ready example of the use of space to represent time; we say that this square over here represents a certain day, and that square over there represents a different one.  The sequence by which one day follows the next in time is represented as a sequence of squares or symbols laid out in space and viewable all at the same time.  Similarly, when we talk about a “timeline” (again, using a spatial word, “line”), we picture a process or a sequence of events laid out before us spatially, perhaps with arrows pointing to specific points on it, telling us that this event is “here”, that event “there”.

As simple as this trick is, it should be emphasized that it is, in a manner of speaking, a trick.  We’re using one thing (position in space) to represent another thing (position in time).  Time and space are different things (even if, in a theory like General Relativity, they are related things), and we should not forget that when we speak of an event happening here or there on a timeline, the “here” and “there” aren’t literal.

This idea is used in a more formal and precise way when we make graphs that plot a quantity as a function of time.  We might, for example, plot the air temperature at some location as a function of time over a 24-hour period, and get a kind of hump shape, lower at the ends and higher in the middle, to be read as saying that it was cool at the beginning of the day, warmed up, and then cooled down again in the evening.  This is hardly groundbreaking stuff, but we have employed the trick of representing time using one dimension of space here to allow us to “see” and, thus, to think about the temperature over the course of a whole day all at once.

Those, then, are the two mental operations at issue here: first, picturing something changing over time and, second, using a dimension of space to represent time.

A small digression.  I have largely used the language of sight in talking about these mental operations; I have said that we picture a changing image, for instance.  One might then raise the question of whether, and how, this applies to those who are sightless or aphantasic.  But I don’t think that the visual aspect of this process is fundamental or intrinsic.  Rather, for me and, I would guess, for the majority of people, vision occupies the prime place among the senses, and to conceptualize or imagine a thing invariably involves some degree of visualizing it in the mind’s eye.  However, it seems patently obvious that someone who has been blind from birth can nonetheless imagine a physical system changing over time, or imagine a “timeline” that represents time as a dimension of space, even if the manner of imagining them is quite different from mine.  I would provisionally conjecture that the two mental operations I have described, though I have at times used visual language in expressing them, are universal to humans.

My contention is that it’s in combining these two operations that many fallacies about time have their origin.  Just as there’s no grammatical reason that the adjective “five-sided” can’t be applied to the noun “square”, there’s no reason that we can’t take a picture that we’ve generated using the second operation (representing time as a dimension of space) and then apply the first operation to that picture (imagining it changing over time).  But, just as “five-sided square” is logically incoherent concept, so too is the result of this combination.

That this combination of mental operations does not generate meaningful results should be clear, but let’s look at an example.  Consider a ball bouncing up and down.  We can mentally represent this using the first operation: picture the ball (or some representation of the ball) at some height above the floor, then picture it, in sequence, falling, accelerating, hitting the floor, rising again, slowing, reaching a maximum height just a little below where it started, falling again, and so on.  We’re using time to represent time.

Or we could mentally represent it using the second operation.  We plot height on the y-axis and time on the x-axis.  Then we get something like this:

Here, we’ve used the horizontal dimension of space to represent time, so that we can consider the whole process at once.

But what if we picture this plot changing?  Suppose we picture the width of each bounce on this plot slowly contracting along the x-axis, and the height of each peak slowly decreasing.  That we can imagine such a thing, like a movie of the graph changing shape, is obvious.  But does it follow that this means anything, as far as the situation that we are representing (a ball bouncing on the floor) is concerned?  Each instantaneous moment of this changing picture is a representation of an imagined sequence of events over time.  Each one represents a different imagined scenario involving a bouncing ball.  But the evolution of the graph over time in our minds does not, cannot, represent any kind of evolution over time in the physical scenario we are considering.  Physical time is already represented by the x-axis of the graph; it therefore cannot be the case that the representational time over which we imagine the graph changing also represents physical time.

To put it another way, since there is only one physical dimension of time in human experience, we cannot let two representational dimensions both simultaneously represent time.  We can represent physical time either with a dimension of space or with (representational) time, but not both at once.

To be sure, there are cases where it may make sense to think of two kinds of time, making it sensible to represent one by a dimension of space and the other by (representational) time; but these cases are restricted and very specific.  As an example, we might think of a person’s plans changing over time as meetings and appointments are scheduled or cancelled.  We might well picture this as a calendar—a representation of time laid out in space—filled with blocks representing scheduled activities, which move around, appear, or disappear over representational time.  Thus, we have used representational space to represent planned time—the sequence of activities the person foresees at some particular moment—and representational time to represent real time, over which that person’s plans are changing.

To give another such case, imagine a writer constructing a history for a fictional world.  She creates a list of dates and events, and a narrative telling of the rise and fall of nations.  But over time, as she develops the work, she makes changes to the chronology; she shifts dates, alters the details of events, and so on.  We might speak here of an “internal” dimension of time—time within the writer’s fictional world—and “external” time—real time over which the writer develops and changes that world’s chronology.  This is in many ways analogous to the case of the changing calendar; again, we have real, physical time, and orthogonal to it we have an imagined timeline, much like the planned time of the calendar.

Perhaps all this seems very obvious and not worth saying.  Indeed, I hope it does, because if you are with me thus far, then my hopes are high of convincing you of the consequent of my argument: that the improper combination of these two mental operations results in a great deal of confusion in the way many people think about time—confusion prominently seen in both science fiction and philosophy.

Let us begin with science fiction.  A common sort of time-travel² story involves a character travelling into the past, effecting some change there, and thereby altering the subsequent course of events from that which (from their point of view) had already occurred.  “Changing the timeline” is a convenient and usual way to refer to this.  Often, the central problem of the story is that of “fixing” the timeline by correcting the change, or by effecting other changes that counteract it.  Prominent examples of this sort of story include the movie Back to the Future and the Star Trek episode “The City on the Edge of Forever”.

The problem here should, by now, be evident, but let’s rehearse it.  The idea behind this sort of story arises, it seems to me, like this.  The writer imagines a timeline laid out abstractly before them—that is, using our second operation, representing time as a dimension of space.  They then observe that an event here, at this point (a spatial word) in time, is the cause of these other events later on.  They imagine that if the first event had occurred differently, the later events would also have been different.  And then they imagine changing that first event, and thereby changing history: one timeline vanishing and another appearing.  This is an instance of our first operation, of imagining an image changing over time.  Thus, they have performed the double operation that, as we have seen, is inadmissible.  They have imagined the timeline changing in time, forgetting that time was already included in it in the first place.

“Changing the timeline”, then, is a nonsensical phrase.  A “change” means that the state of affairs at a certain time is different from the state of affairs at a prior time.  If we are thinking of a timeline—of time represented as a dimension of space—then a “change” can only mean the state of affairs at this spatial point in the representation is different from that at this other spatial point.  We cannot speak of a change in a timeline, because that timeline, in its static form, must already encapsulate all change.  (The exception, as discussed above, is when we are talking about changes in real time to an imagined timeline, like alterations to the outline of a story).

I should perhaps clarify that this is not intended as artistic criticism.  To the extent that one can ignore the incoherence of the idea, this sort of tale can be quite effective.  The two examples cited above are, in fact, particular favourites of mine.  But that does not render their treatment of time travel any less nonsensical.

Another concept that comes up often in science fiction is the “time loop”.  Typically, this involves a character, or characters, living through the same events over and over.  In almost all cases, those characters retain some or all of their knowledge through each repetition.  A distinct, but related, idea is that of a “causal loop”, in which event A causes event B, and through some form of time travel, event B causes event A.  As we shall see, the former “time loop” concept is incoherent (unless reinterpreted as having nothing to do with time travel per se), while the “causal loop” idea is logically coherent, but sometimes misunderstood.

A body in a time loop might be represented graphically like so:

And a causal loop could be represented thus:

Here, we might imagine, for instance, a person who receives a signal (event A), which causes her to travel backward in time (event B) and then send a signal (event C), which is in fact the very signal she received at A.

Obvious though it may be, let us look at these images and observe that there is only one loop in each of them.  If we ask, “how many times did event A happen?” the answer is—can only be—once.  Where people go astray in thinking about this is, again, in combining our two mental operations: they imagine this graphical representation and then imagine a point moving along the loop.  Then thereby create multiple iterations through the loop, saying, “We have moved through the loop once… now we have moved through the loop twice… now thrice…”  But, as in the other examples, this is an error resulting from their having represented time twice, first as a spatial dimension on the loop diagram, and then as the time over which a point moves through that loop.  Again, the temporal dimension has been doubled.

The retention of the subject’s memory through each iteration of the time loop is, therefore, incoherent—because there is only one iteration.  The scenario can be logically consistent only if the subject is returned to the exact physical state (including that of the brain and all its memories) as at the beginning of the loop.  Moreover, a person in a true time loop like this would have to have had no existence prior to the loop; otherwise we end up with a physically nonsensical vertex where the original and the looped copy must merge.  Of course, since the subject of the loop is then disconnected from the human family tree (the person would be motherless and fatherless), the existence of such a scenario would be incredibly unlikely even if allowed by physics.  A cleverly expanded version of such an improbable loop was presented by Heinlein in the classic story “All You Zombies—”.

It may be noted that sometimes, as in the Star Trek: The Next Generation episode “Cause and Effect”, a time loop is presented that seems to be not so much a case of actual time travel as one of the physical configuration of a system being repeatedly “reset”.  When at length the Enterprise breaks out of the loop, the crew finds that time has indeed been passing normally on the “outside”.  What happened to them, then, seems to be more like a system undergoing cyclic evolution—like a gigantically complicated version of a pendulum swinging back and forth—than of any real time travel or causal reversal.  Whatever doubts one might have about the physical possibility of such a situation, it doesn’t present any logical inconsistencies, if we assume that no actual time travel is happening here.

The “causal loop” can be thought of as a generalized version of the “time loop” discussed above, in which it is not necessarily a person, or even an object, that executes the loop, but a chain of causality.  We might say that, starting from a certain point A in spacetime, there is a path along which the arrow of time is consistently pointing forward for which, if we follow that path in the forward direction, we eventually come back to point A.  There needn’t be any physical object executes a closed path along this trajectory, but the existence of such a path allows a causal chain beginning at A to return to A, in other words, for an event to be one of its own causes.

For an example of such a loop, treated in a logically coherent way, we can turn again to Star Trek: The Next Generation, this time to the two-part episode “Time’s Arrow”.  In this story, the android Data’s disembodied head is found in a mine near San Francisco.  Later, as a result of an investigation into this strange discovery, Data travels back in time to the 19th century.  After several adventures, he finds himself at the mine, where his head is severed.  His body is brought back forward in time to the 24th century, while his head remains in the mine until its discovery, with which the episode began.  After the return of Data’s body from the past, chief engineer Geordi LaForge succeeds in reattaching the 500-year-old head to his body.  (I have omitted various complexities from this summary, for the sequence of events involving Data is sufficient to study the causal loop.) 

If we accept the time travel here, the logical coherence of this story, and others like it, is unimpeachable.  Nonetheless, people are still sometimes led astray in the way they think about such stories.  If, for example, someone imagines a “first version” of the timeline in which Data’s head is not buried near San Francisco, which is then changed “when Data travels back in time”, this someone is committing the fallacy we have been considering, imagining a timeline that changes.  Similarly, one might follow the causal loop in one’s mind—Data travelling back in time, leading to the loss of his head, leading to the discovery of the head, leading to an investigation, leading to Data travelling back in time, leading to the loss of his head, etc.—and ask, “What if this time through the loop, Data makes a different decision?”  This is also an erroneous way of thinking, for again one has first, however abstractly, represented time using space and then represented it again using time to “move through” the loop.  The loop only happens once.  Each event along its path through space and time happens once.

From science fiction, let us now move to philosophy (I leave it to the reader to judge whether we move from the frivolous to the serious or vice versa).  I do not propose to look into the way time has been treated, or mistreated, in all the manifold philosophical schools of thought that have been propounded since the days of Thales and Confucius and Siddhartha (for fallacies of the sort considered here are at least that old).  I propose, instead, to restrict myself to the tradition of analytical philosophy that I feel to be my philosophical home.  This will suffice to illustrate the ways in which the fallacy infects philosophical thought.

There is a debate of long standing in the philosophy of science between what we might call the “flow-of-time presentism” and “block-time eternalism”.  I will briefly rehearse the claims usually made by each, before arguing that the origin of this debate comes down to our now-familiar fallacy.

Flow-of-time presentism asserts, first, that only the present is real, and that what the present is—and therefore, what is real—changes as time passes.  It is, therefore, also a core claim of this view that time does pass.  Proponents of this view hold that time is dynamic rather than static, that time flows toward the future, or that we move through time.  The special and general theories of relativity raise some physical problems for this view, as they muddy the notion of a present moment, but defences or modifications of the view have been raised to meet this objection (see, e.g., Hawley (2006)).  The former typically adduce purported reasons to accept the metaphysics of presentism rather than a metaphysics suggested by relativity; the latter retain the “flow-of-time” part of the claim while modifying the presentism—Tim Maudlin’s view, for instance, might be characterized as “flow-of-time eternalism”.  All of these approaches agree in asserting that a) there is an objective present moment (albeit, in light of relativity, perhaps a local one), b) time has an intrinsic direction, and, crucially, c) time is dynamic; it flows or passes or is moved through.

In contrast, adherents of block-time eternalism say that all times are equally real and that other times should be thought of more or less the same way as other locations in space.  Time does not pass or flow, and the universe should be thought of as a static four-dimensional “block”, of which one dimension is temporal and the other three spatial.  The past still exists and the future already exists.  Proponents of this view will often speak of time being an illusion, or say that the universe is “fixed” and non-dynamic.  This view has sometimes been linked to determinism, and claims have been made that the non-deterministic features of quantum physics present a challenge to this view.  It has also been challenged as being counterintuitive or even patently, experientially false (as for instance by Avshalom Elitzur, who insisted, “I don’t think that next Thursday has the same footing as this Thursday. The future does not exist. It does not!”).

Between these two extremes, there are other positions that have been held.  We have already mentioned “flow-of-time eternalism”.  There is also a “growing block” view, which holds that the present and past are real, but the future is not.  It is not my purpose here to taxonomize all of these views, and I believe the points that follow concerning the two extremes apply equally to all modifications and compromises between them.

The supposed distinctions between these views could be attacked from several angles.  An anti-realist (as, indeed, I am), for instance, could say that the reality or non-reality of the past and future is a meaningless question, because “reality” in this sense is meaningless.  However, I think that this debate can be rejected as nonsensical on grounds that even a realist could agree with.  I don’t claim that the arguments that follow are novel; if anything is novel here, it is my conjecture as to the origin of the belief that flow-of-time presentism and block-time eternalism contain meaningfully distinct claims.

Let’s start with the “flow of time”.  What could it possibly mean to say that time flows, or that time passes, or that we move through time?  To move, to flow, to pass, means to occupy one position in space at some time t1 and to occupy a different position at a later time t2.  The motion of something, in other words, is a change in the spatial position of that something over time.  What it means to say that I have moved through a room is that at one time I was at this place in the room and at a later time I was at that place (and at intervening times, I was at other places). 

We can employ “movement” or “motion” more abstractly and use it to describe changes in other things over time.  I might say that I’m moving quickly through a book, meaning that at time t1, I was on a certain page, and at some other time t2 (not very long after t1), I was on a much higher-numbered page, having “passed through” (i.e. been reading) the intervening pages in the interval.  Or we might say that the air temperature has fallen over the course of a day, meaning that in the morning air had some temperature and in the evening it had a lower temperature. 

But what about “I moved through time”?  It can’t mean “I moved (through space) over time”, for the flow-of-time proponents would say that we move through time even when we are stationary.  If we try to make sense of it in the same way we make sense of moving through space or through a book, we are bound to say that it means: “At some time t1, I was at time t1, and at a later time t2, I was at a different time t2.”  But this is, of course, a tautology, and is something that block-time eternalists will readily assent to; therefore, it cannot be the claim that the flow-of-time presentists are making and that the block-time eternalists deny.  Nonsense, or tautology, also emerges when we try to analyze claims like “time passes”.  Nonsense: “There is a change in the spatial position of time over time.”  Tautology: “Time has value t1 at time t1 and a different value t2 at a later time t2.”  (For a similar refutation of flow-of-time, see Price (2009)).

What interpretation is left?  I think that, however they may be dressed up in sophisticated language and made to sound philosophically presentable, what lies behind these claims is actually a very natural—and equally erroneous—thought process.  What has happened is that one has imagined some kind of timeline, representing time using a dimension of space (our second mental operation), and one has then imagined a point, representing oneself, or representing the present moment, moving along that timeline (our first mental operation).  Once again, time has been erroneously doubled, represented twice. 

To be sure, there are situations in which it is useful and meaningful to say things like, “A few days passed.”  What is meant by this, of course, is that some event (to which we have presumably just referred) happened at time t1; another event (which presumably we are about to relate) happened at time t2; and the interval between t1 and t2 was a few days.  But this use of “pass”, understood in this way, is of course unobjectionable to the block-time eternalist as well.

Nor are the adherents of block-time eternalism immune to this erroneous way of thinking.  To say that the universe is a “static”, “unchanging”, “fixed”, or “non-dynamical” four-dimensional object is also to reduplicate time.  What, after all, does it mean for a system to be static?  It means that at time t1, at a later time t2, and at all intervening times t1 < t < t2, the state of the system is the same.  But time is already represented as part of the four-dimensional “block”; it is nonsensical to say that the block is static.  As for time being an “illusion”, it is difficult to even puzzle out what could be intended by such a claim.  The block-time advocate will certainly agree that time is one of the dimensions making up the 4D “block” that is the universe; surely that means it is in no sense an illusion. 

It is similarly nonsensical to say that “the past still exists” or “the future already exists”.  To say that a thing “already exists”—indeed, to say that it “exists” (a present tense verb)—is to say that it exists now.  If I state at time t1 that a thing (still, already) exists, I am saying that it exists at time t1.  But “the future” means everything at time t > t1.  To say that the future already exists is as nonsensical as it would be to say that everything north of the equator is at the equator.  One could say that the past did exist and that the future will exist, and these statements I would readily accept as true.  True—but empty, for they are tautologies.  They are true purely by virtue of the meanings of the words “past” and “future” and of the verb tenses of English.

What lies behind the block-time eternalists’ statements is, it seems to me, the same erroneous doubling of time that gave rise to the claims of the flow-of-time presentist.  Clearly what has happened is that the eternalist has imagined the whole history of the universe laid out before them, using a dimension of space to represent time.  This is the “block” that they talk about, its height and width representing space (a third dimension of space being suppressed in the image) and its length representing time.  So far, so good.  But then, forgetting that the one and only physical dimension of time is already represented in the image before them, they imagine that block sitting there, unchanging, as time passes.  They say that the whole block is there, that it exists, now, forgetting that “now” can only refer to a slice of the block at some point along its length.  Like the presentist, they have doubled time.

Both the presentist and the eternalist might object that I am merely playing games with language, that there is sophistry in my parsing of the meanings of “flow” and “pass” and “exists”.  In particular, it might be objected that “exists”, though grammatically present tense, is not semantically restricted to the present.  It might be said that the eternalist’s claim about the future, for instance, should be taken as, “The future always exists.”  But this does not evade the problem.  A thing “always exists” if it existed, exists, and will exist, at all times.  Then the claim is “the future did exist in the past, does exist in the present, and will exist in the future.”  The first two elements of this conjunction can be dismissed as nonsensical, just as above, while the third is a tautology.  Perhaps the eternalist will say instead that “exists” is meant in an atemporal sense; that the past and future exist as part of the four-dimensional manifold that is our best description of the universe.  But now we have weakened the claim to the point where the flow-of-time presentist will surely not disagree with it. 

Far from this analysis being linguistic sophistry, I would say, on the contrary, that the claims I have analyzed and rejected as meaningless are (to adapt Wittgenstein’s phrasing), not merely a disease of language but a disease of thought.  For just as the grammar of natural language allows meaningless locutions like, “five-sided square” or “the future already exists”, so too does the grammar of thought allow us to erroneously represent time twice, once as a dimension of space and again as time.  One might venture to psychologize further here and say that the real disagreement between the two schools is simply which of our two ways of representing time their adherents prefer, with flow-of-time advocates preferring the “time as time” representation and block-time supporters preferring the “time as space” view.

I will stress once again that I don’t think my criticisms of changing timelines, flow-of-time presentism, or block-time eternalism, are novel.  If anything is novel here, it is my conjecture as to the origin of these fallacies: the double representation of time.  That fallacies like these so readily sprout in the garden of human thought I ascribe to the very basic nature of the two mental operations which, in combination, give rise to them. 

---

1. I use visual language here, but see below for a consideration of blindness and aphantasia as they relate to this. ↩︎
2. In light of the argument I have been making, the very term “time travel” may cause some disquiet.  I use it because it is the conventional phrase.  What it means, really, is something like a local reversal of the “arrow of time”.  It is doubtful that any such thing could actually occur, though there may be loopholes in General Relativity or Quantum Field Theory that allow it, but it must at least be admitted that such a thing is logically possible. ↩︎