# Four-dimensionalism and the psychology of time

July 29, 2019. Modern physics suggests a view of time called four-dimensionalism: just as all places exist simultaneously, all times should exist simultaneously. I examine some consequences for the psychology of time, and contrast with the (more intuitive) philosophy of time called presentism.

## Introduction

If physics is to be believed, there is nothing special about the present. “Right now” is similar to the statement “right here”, but it refers to when the speaker is. But what is privileged about the moment of utterance? Is it any more privileged than the bath tub or the park next to the subway where the speaker happens to declare they are located? In physical terms, the “present” is just a projection function:

$P(t, x, y, z) \mapsto t.$

On the other hand, time feels different from space. Unlike space, where I can travel at will between the park and the bath tub, I cannot choose to return to last Tuesday. We seem to go in one direction through time, at a fixed speed. Of course, physics does not contradict this. Relativity teaches us that time is, indeed, different from space, and closely bound up with causation. Since experience is a causal affair, with earlier brain states influencing later ones and not the other way round, this is as it should be.

But there is more to the peculiar psychology of the present. While the direction of our movement through time is easily explained, the sensation of being in a particular moment takes more work. Our qualia seem to be attached to now; folk Cartesianism which takes this as incorrigible proof that the present is special and perhaps the only “time” which really exists: “I can only tell that right now exists, and since I cannot experience any other times, they must not exist”. This view is called presentism.

As I will argue below, despite its folk appeal, I don’t think presentism is a viable metaphysics of time. Nevertheless, I think there is an explanatory gap between common psychological experience and the physics of time. I will argue that the gap can be filled by carefully considering what a person is, and how this is linked to time and causation. Ultimately, we will need to resolve a variant of Zeno’s arrow paradox: what property of an arrow’s time slice explains its motion? Similarly, what properties of a person’s time slice explains their sensation of being trapped in a moving window of experience?

## No time like the present

I’ll recap some properties of time, before defining (and arguing against) presentism. This will leave us with some explaining to do!

#### Simultaneity and causal order

Einstein taught us that simultaneity is relative. Any time two observers move relative to each other, they will disagree about which events occur at the same time. However, whatever your reference frame, you will get the same answer for the following quantity:

$s^2\big(P(t_1,\vec{x}_1), Q(t_2,\vec{x}_2)\big) = - (t_1-t_2)^2+|\vec{x}_1-\vec{x}_2|^2.$

Here, $P$ and $Q$ are points in spacetime, and $(t, \vec{x})$ are spacetime coordinates in an inertial (non-accelerating) reference frame. The point is that, by virtue of the principle of relativity, whatever reference frame we use to define the coordinates, the function $s^2$, called the invariant spacetime interval, is the same.

The spacetime interval has the follow physical interpretation:

• If $s^2(P, Q) = 0$, and $t_1 < t_2$ ($t_2 < t_1$), then $P$ and $Q$ are null-separated, and we can send a light ray from $P$ to $Q$ ($Q$ to $P$).
• If $s^2(P, Q) < 0$, and $t_1 < t_2$ ($t_2 < t_1$), then $P$ and $Q$ are timelike-separated and a massive observer, or message, can travel from $P$ to $Q$ ($Q$ to $P$).
• If $s^2(P, Q) > 0$, then $P$ and $Q$ are spacelike-separated, and cannot influence each other causally.

Note that the time-ordering in the first two cases does not change, whatever the reference frame. For instance, if $t_1 < t_2$, then $t_1’ < t_2’$ in any other inertial reference frame. The last requirement, that spacelike separated observers cannot influence each other, is not only an empirical observation, but needed to ensure that cause and effect are ordered. This is due to the relativity of simultaneity. If I could send messages faster than light, it is easy to set up situations where, bouncing signals off a moving mirror, I could send communicate with myself in the past. We then run into all the paradoxes of time travel. To me, this suggests that the ordering of cause and effect is a basic consistency requirement for the universe.

If you want to be a little more precise about cause and effect, you can use the spacetime interval to define a partial order $\prec$ on events. If $P$ can influence $Q$ (they are either null- or timelike-separated), we write $P \prec Q$, and note that this relation is

• reflexive, $P \prec P$;
• antisymmetric, $P \prec Q$ and $Q \prec P$ implies $P = Q$; and
• transitive, $P \prec Q$ and $Q \prec R$ implies $P \prec R$.

Two points are not comparable, or causally ordered, just in case they are spacelike separated.

#### Presentism defined

What has this got to do with the present? If I understand the notion properly, presentism is the view that only some time slice of the universe really exists. By “time slice”, I mean some collection of events $\Sigma$ which are mutually spacelike separated, or equivalently, mutually incomparable according to the causal partial order. I used to think that the relativity of simulaneity was knock-down argument against presentism. Since being simultaneous is not an equivalence relation, and the notion of being co-present (existing together) is, surely presentism is wrong? No: all this shows is that being co-present is not the same as being simultanous according to each other’s clocks. If we define the “present” as a slice $\Sigma$, which will have a transitive notion of co-presence after all.

One issue with $\Sigma$ is that it is not unique. That is, any event $P$ is contained in an infinite number of spacelike slices $\Sigma$. Which is the “real” one? At large scales, our universe has a preferred cosmological time $t$, and perhaps (the presentist could argue) this naturally singles out a slice $\Sigma(t)$. This solution is somewhat ad-hoc, but more importantly, doesn’t explain how to “pick out” the right slice locally, for instance in the presence of a black hole. A friend falls into the event horizon, and I wonder “Which slice is real?” I don’t think the presentist can help, since the natural operational definitions (e.g. I can send a message to the slice) are causal, and therefore take me out of the set of timelike slices $\Sigma$. (Here, I am ignoring the issue of global hyperbolicity, but it is not really relevant since the same issues arise near a star.)

One suspects that, in asking these questions about the person falling into a black hole, language has “gone on holiday” (Wittgenstein). But the presentist can simply assert that the slice exists, and leave the problem of selecting the correct $\Sigma$ to the axiom of choice, or God, or some comparable higher power. What can we say then?

#### Demiurges and the reality of change

A subtler problem is change. Presentism claims that what is real, i.e. the present slice $\Sigma$, is changing. But changing with respect to what? If I label or define slices with respect to some type of “universal proper time” $\lambda$ (e.g. cosmological time), the answer seems to be: with respect to $\lambda$, at a rate of 1 unit of $\lambda$ per unit of $\lambda$. As people often joke, we are moving into the future at 1 second per second.

The answer is of course tautological. But it is worse than that for the presentist. To see why, let’s discretise time into steps $t_n$. Presentism suggests that, ontologically, a slice $\Sigma(t_0) = \Sigma_0$ is brought into being, then destroyed and replaced by $\Sigma_1$. We repeat the process with $\Sigma_2$, and so forth. How long do these slices exist? It seems like there is no way to measure this, since $t_n$ is attached to the slices themselves. But to ask what exists “now” presupposes some sort of time in our creation and destruction process!

We have the following infinite regress problem. Suppose a demiurge $D_1$ is running this “program” of creating and destroying the slices $\Sigma_n$. To even ask which slice is currently loaded, the demiurge also needs to have a time $t^{(1)}$ (once again discrete for simplicity). But of course, we then need another demiurge $D_2$ with time $t^{(2)}$ to program the slices for the demiurge $D_1$. And so on. I think this a bad regress since no demiurge explains what “present” actually means. Perhaps, like the choice of spacelike slices, the presentist can appeal to God, in this case as a sort of limit or union of demiurges, $D = \lim_{n\to\infty} \cup_n D_n$.

I don’t think this helps, but let us suppose for the sake of argument that it is possible to fix the regress. But what if our demiurges make errors, as demiurges are wont to do? Perhaps they slices $\Sigma_4$ and $\Sigma_5$ actually coexist for at some time $t^{(1)}$, or different slices of the demiurge themself coexist at some time due to a lapse of attention in one of the higher demiurges! Even worse, demiurges could get the order wrong, making $\Sigma_9$ before $\Sigma_3$. Can observers living in the slice $\Sigma$ tell how long in demiurge time it is running for? Or somehow feel “present” in both $\Sigma_4$ and $\Sigma_5$ when they are run at the same time? Or “sense” the reversal of causal order when $\Sigma_3$ comes before $\Sigma_9$?

Of course not. Observers living in $\Sigma$ have no access to any of this data, since this would give them super-causal powers. In fact, I think this means we cannot tell what any demiurge is doing, including $D_1$. The ability to tell which slice is present is unreasonably powerful for a creature causally bound to the slices. In other words, even if presentism is a correct picture of the ontology of time, there is every reason to think that we would be unable to distinguish this state of affairs from the more conventional, four-dimensional picture where all the slices $\Sigma_n$ exist at the same time.

(These faulty demiurges are partially inspired by Greg Egan’s “dust theory”, featuring in his novel Permutation City. The use of demiurges has a similar flavour to Augustine’s argument against duration in the Confessions.)

## Mental states and processes

Perhaps, as I have argued, presentism is not a sensible picture of time. But the claims of our qualia are not lightly dismissed. An infinite tower of demiurges creating and destroying one another may be absurd, but the related convictions that I am experiencing right now, and that change is real, are not absurd. How can we explain these in the framework of four-dimensionalism?

#### Mental states and “nowness”

Conscious experience is famously difficult explain from the physicalist viewpoint. But whatever qualia are, I assume they are the result of physical processes in the brain, and supervene on brain states. Run the process, or set up the state (a subtle distinction we return to later), and the associated conscious experience should appear.

It is then straightforward to explain why we feel like we are always “in the present moment” from the four-dimensionalist perspective. If the time slice of the brain $B(t_n) = B_n$ always exists, it is always in a particular state; it is therefore always producing the sensations $S_n$ associated with that state. Of course, those sensations are attached to a specific time and place, and the sensation of being “experiencing things right now” is no different, fundamentally, from “experiencing things right here”. The novelty or strangeness of these time indexicals is that we do not feel like we control our motion in time, so there is perhaps something surprising about being “at this present moment”. How did we get there? It is not due to some deep ontology that our brains are somehow enabled to track. It is simply a result of causal relations between brain states.

Before I elaborate on this last statement, let me be clear. I think that brain states $B_n$ always exist; this is what four-dimensionalism dictates. If they always exist, they are always producing the associated qualia $S_n$ and so a brain state leads to a permanent state of feeling. You may ask how it is possible to always be “feeling” $S_n$ when the sensations feel temporary. I will discuss this at length in the next section, but the point of the demiurge example is that the ontology of time slices has nothing to do with conscious experience. This is why presentism is tempting but wrong. It conflates properties of conscious experience with properties of time.

#### The arrow of Zeno

It is not clear that brain states that are permanently feeling have anything to do with our everyday experience. Sensations last a moment, and are quickly succeeded by new ones, almost as if they are being created and destroyed by particularly reliable demiurge. The question, then, is whether the content of the sensation $S_n$ can account both for the feeling of duration and succession.

There is a helpful pre-Socratic analogue to the problem we are considering here. Zeno’s third paradox of motion involved an arrow in flight. We can state the paradox as follows:

• The arrow, at any moment of time, is stationary.
• Time is composed of moments.
• Stationary objects remain stationary.

Combining these three, we seem forced to conclude that the arrow can never move. Similarly, one might imagine that a brain state $B_n$, always producing the qualia $S_n$, can never give rise to the “flow” of conscious experience. (A similar argument was put forward independently by Le Poidevin, building on Broad’s Scientific Thought.)

The answer to the paradox is simple. There is a difference between a stationary arrow and a moving arrow examined at an instant of time. The moving arrow has properties the stationary arrow lacks, which not only explain the difference in subsequent motion but have measurable effects on the slice itself.

Similarly, intrinsic properties of the brain state $B_n$ must generate the feelings of duration and succession. Let’s first consider the simpler of the two, succession. At a physical level, “succession” reflects the fact that earlier brain states $B_{m<n}$ are part of the causal input to the brain state $B_n$. The previous states $B_{m<n}$ are in fact stored and represented in $B_n$ via memory, but at an abstract level, succession is just the statement that causal relations are mirrored in cognitive ones.

Duration is trickier, but not so different from the velocity of an arrow. The velocity is the rate at which the arrow acquires distance, but both the distance and time must be given units, i.e. a physical basis for measurement. Similarly, the duration of an experience cannot make any sense without units. Luckily, our brains are naturally equipped with physiological process (such as circulation) which can be used to measure time, and input channels (with thresholds, capacities, and so forth) which can be used to measure intensity. It seems plausible that the subjective “rate” of experience is some function of the strength, nature and speed of sensory input, relative to these physiologically defined units of measurement, along with how efficiently they can stored in sensory and short-term memory.

## Conclusion

To reconcile our experience of time with four-dimensionalism, we seem to be driven to some strange conclusions. Every brain state $B_n$ in the history of a thinking being always exists, by four-dimensionalism. If we are physicalists about sensation, then those brain states are always accompanied by sensations. The brain states always feel. Four-dimensionalism therefore suggests an extreme version of Nietzsche’s doctrine of eternal recurrence: we are, and will always, live all moments of our life simultaneously, but with the impression of “nowness” and temporal succession by virtue of the physical properties brain states.

But “always feeling” does not mean “feeling always”. We should not confuse the properties of the time slices with properties of brain states, or even worse, the sensations they give rise to. This is the fundamental (but understandable) error of the presentist. Still, some work is required to see where the duration and succession of our experience of time comes from. I have sketched some arguments, and hope to fill out the details in the future.