Not only vicious causal loops can create a problem for time travel. Suppose that we have a situation in which a later event A causes an earlier event B, whereas B causes A. This does not seem to lead to any logical contradiction, yet it gives rise to a serious conceptual problem. Suppose that in the year 2011 you are visited by yourself from the year 2020, and your older self hands you in a complete blueprint for a time machine. You then build it, and in 2020 you enter your machine to visit yourself in 2011 and deliver the plans. The question is: where did the blueprint come from, and who invented the time machine? Yet another paradox goes as follows. The traveller from the future hands you in an empty notebook and instructs you to enter the time machine, go to the future and write an entry about your journey in the notebook. Then you are supposed to give the notebook to the traveller in the future, whereupon he enters the time machine, goes back in time and gives you in the past the instruction to write another entry in the notebook while travelling to the future. This creates an immediate problem: how many entries are there in the notebook? It looks like the number of entries should correspond to the number of ‘loops’ made, but actually there is only one loop, not many repeated ones.
Apparently this last example is akin to the grandfather paradox, as the traveller attempts to do something in the past (i.e. convince you to enter the time machine and write something in the notebook) which would cause some future event (the traveller’s entering the time machine with an empty notebook) to disappear. So perhaps the solution would be, again, that something should prevent you from writing an entry in the notebook, and thus the situation in the future will be exactly as it should: the time traveller, despite his attempts, will enter the time machine in the future with an empty notebook. On the other hand, if the notebook you receive in the past already contains an entry, the problem looks more like the blueprint paradox we discussed above. You try to write another entry, but you fail, so you hand in the notebook with the old entry to the future traveller. But the question remains: who wrote the entry?
Even if the notion of time travel can escape some of the most threatening conceptual difficulties, the question remains whether it is nomically possible. We will not attempt to answer this question generally, but only with respect to one physical theory, namely the special theory of relativity. First let us notice that the conceptual framework of STR seems to leave room for a realisation of a journey to the past. Let us consider three events A, B and C such that C is in the absolute past of A, whereas B is space-like separated from both A and C. We know that there is a frame of reference f in which event B looks later than event A, so at least in principle it seems possible to send somebody from A to B along the ordinary direction of time from the past to the future. But there is also a different frame of reference f’ in which C is later than B, hence another trip from B to C seems feasible. Combining these two together, we can see that the journey took us from A to its absolute past. But this scenario is not physically realizable, due to the fact that in order to reach B from A (and C from B) the traveller would have to accelerate beyond the speed of light, and this is prohibited by the laws of relativistic dynamics. Let us note in passing that STR does not exclude the possibility of the existence of superluminal particles (called tachyons) – particles that travel faster than light. But if tachyons exist, they cannot cross the light barrier by slowing down to a subluminal speed.
How can we explain the fact that time has an objective direction, in contrast with space? And what does it mean precisely that time has a direction? The directionality (or asymmetry) of time can be defined as the property of being ordered by the asymmetric relation “being earlier than”. But points on a spatial line can be similarly ordered in one direction. To distinguish between the two cases let us observe that the temporal relation of being earlier than is an intrinsic relation of two moments which does not depend on anything extrinsic from these moments. On the other hand, for a spatial line to be ordered in a similar fashion we have to choose some external “point of reference”. For instance, we can order points from left to right, but this ordering depends on the position of the external observer. Analogously, it is possible to order points on a meridian, but again we have to choose the South Pole and the North Pole to do that.
Is the directionality a primitive and irreducible property of time, or can it be grounded in some more fundamental property? There are three standard ways of grounding time’s arrow: in the psychological arrow, causal arrow and thermodynamic arrow. The psychological arrow is based on the observation that perceptions always precede memories. My perception of a given occurrence is designated as being earlier than my memory of the same occurrence. But clearly this criterion can be applied to a very narrow category of events only: the mental ones. How, then, can we use the criterion of such a limited scope to order all the events? The answer is that in some circumstances it may be sufficient to define a direction for only two points in time, and the rest will be ordered accordingly. Let us suppose that the set of events is equipped with a symmetric structure defined by the three-place relation of being between. This means that we can say that an event x lies between events y and z (in short, B(x, y, z)) without deciding whether y is earlier than z or vice versa. Now the question is: What has to be added to the structure defined by the relation B in order to have a full linear order? It is easy to notice that if we decide for a pair of selected events a and b which of them is earlier and which later, the rest of the events will be ordered thanks to the relation B. For instance, let’s suppose that a is earlier than b (E(a, b)), and consider two events x, y lying between a and b (i.e. such that B(x, a, b) and B(y, a, b)). In this case, E(x, y) iff B(x, a, y) (or, equivalently, B(y, x, b)). This definition can be easily extended for all possible distributions of x and y with respect to a and b.
From this it follows that in principle we need only one instance of a perception and its memory to give time its direction (under the condition that the relation of betweenness is already given). But in practice we obviously need more such instances. The psychological arrow suffers from obvious shortcomings. It is strongly anthropocentric, as it requires the existence of humans (sentient beings) in order for time to have a direction. But surely there would be earlier and later events even if there were no humans. Another difficulty is that this criterion excludes a priori the possibility of clairvoyance (which may be impossible physically, but does not look like a contradictory concept). Knowing the future requires that a ‘memory’ of a given event is earlier than its perception (by ‘perception’ we mean the ordinary way of seeing things, and not the one claimed by the clairvoyants), but this contradicts the criterion.
The causal arrow assumes that if x is a cause of y, x is earlier than y. This reduction of the directionality of time encounters the following two objections. First, it excludes the possibility of backward causation (we considered this possibility when analysing time travel). Second, grounding the direction of time in the causal relation requires that we define causality without referring to temporal precedence. And yet some conceptions of causality, famously including Hume’s analysis, rely on just that. If we insist that causality grounds time’s arrow, we have to find an alternative way of making sure that the causal relation will be asymmetric.
The most commonly accepted of all three arrows is the thermodynamic arrow. It relies on the second law of thermodynamics which states that for isolated systems their entropy (a measure of disorder) never decreases. The second law explains why we observe so many irreversible processes (heat transfer, dissolution, etc.). Thus the thermodynamic criterion states that if x and y are two macrostates of a given isolated system, and the entropy of x is smaller than the entropy of y, then x is earlier than y. But there is a well-known foundational problem associated with this account of time’s arrow. The theory which describes the interactions that underlie all thermodynamic processes is just classical mechanics of many particles, and this theory is known to be time-reversible (if a process is admissible by the laws of classical mechanics, so is its reversed version). The standard attempt to explain the observed thermodynamic asymmetry is due to L. Boltzmann, and it is based on the fact that one macrostate of a system can be realized by a great number of various microstates (defined by the positions and velocities of all individual molecules of the system). Boltzmann proved that for a given macrostate the vast majority of corresponding microstates are such that their dynamic evolution leads to the increase of entropy. Thus the second law can be seen as probabilistic only, but the probability that a system will actually violate it is extremely small.
However, one of the main problems with Boltzmann’s argument is that it is essentially symmetric, i.e. it can be used in support of the claim that the system at a given moment evolved from a state of higher entropy. One way of avoiding this difficulty is to assume that the initial state of the universe had extremely low entropy, and hence the overwhelming tendency of the systems is to increase their entropy. This means that the thermodynamic arrow has to be grounded ultimately in a singular fact about the origin of the universe, together with the probabilistic laws of statistical mechanics. One interesting consequence of this fact is that it is theoretically possible that when the universe reaches the state of maximal entropy, the tendency can be reversed and the majority of systems will follow the entropy-decreasing evolution. It is not clear whether this would mean that the direction of time got reversed.
Reading:
E.J. Lowe, Chapter 18 "Causation and the direction of time", pp. 325-344, A Survey of Metaphysics.
Saturday, March 27, 2010
Time in special relativity and time travel
We will now discuss the changes in the notion of time and space brought about by the development of the special theory of relativity. Let us start with a brief characteristic of the classical account of space and time as incorporated in the Galilean-invariant version of Newtonian mechanics (i.e. the version that dispenses with the concept of absolute motion and absolute position). The main assumption is that no inertial frame of reference is privileged, and uniform motion is relative. However, the notion of simultaneity remains absolute, i.e. frame-independent. For each moment of time the set of events occurring at that moment defines a three-dimensional space with the usual Euclidean metric (distance) attached to it. Hence, Galilean space-time foliates naturally into separate spaces defined at different times. However, there is no absolute connection between points in spaces at different times (no absolute co-location). The question of which point of space at t2 is a continuation of a point at t1 does not receive a frame-independent answer. If we think that spatial points (places) are objects which retain their identity over time, then no such objects are present in the Galilean version of classical mechanics.
A step towards special relativity is the realisation that the relation of simultaneity has no obvious empirical content, due to the fact that all signals (including light) travel at finite speeds. What we observe as our ‘present’ is actually already in the past (the farther, the more distant the observed event is). The standard, operationally defined notion of simultaneity is given as follows: two events x and y are simultaneous iff light signals sent from x and y meet exactly half way between x and y. But this definition is obviously not frame-independent. Suppose that the definition of simultaneity is satisfied in a frame f, and let us consider another frame f’ which moves with respect to f in the direction of the event y. The spatiotemporal point where the two beams of light meet will not be located in the middle of the distance between their locations x’ and y’, but rather closer to x', so from the perspective of f’ y happened earlier than x. We have to add that the signal definition of simultaneity presupposes that the speed of light is constant in all frames of reference.
In special relativity neither simultaneity nor co-location are invariant notions (independent of the frame of reference). Thus space-time cannot be absolutely divided into space and time. However, there is a relation between events (spatiotemporal points) which stays the same in all frames of reference. This relation is defined by the so-called spatiotemporal interval: cdt^2 – dx^2 – dy^2 – dz^2., where c – the speed of light, and dt, dx, dy and dz are temporal and spatial intervals between the two events. Two events for which the spatiotemporal interval is positive are called ‘time-like separated’. Such events can be connected by a signal travelling slower than light. If the interval equals zero, the events can be only connected by a beam of light. Events separated by a negative interval are called space-like separated. Such events cannot directly communicate by way of sending signals.
The basic structure of relativistic space-time (so-called Minkowski space-time) can be given with the help of light cones. For a given event x, its forward light cone consists of all events reachable from x by beams of light. Similarly, x’s backward light cone contains all events which can reach x using beams of light. The area within x’s backward light cone is called its absolute past, and within the forward light cone its absolute future. The events outside of both light cones are neither past nor future with respect to x, but they cannot be interpreted as being simultaneous with x. Their temporal relation with x is frame-dependent: for every event y space-like separated from x there is a frame of reference f in which x and y are simultaneous. But if we choose a different event y’ also space-like separated from x, the frame of reference in which y’ is simultaneous with x will be generally different from f.
Now we will discuss some issues related to the asymmetry of time. Let us start with the problem of time travel. Is time travel conceptually possible, or does it involve logical contradiction? First we have to decide what process can be called time travel. For a given object we can say that it travels in time if there is a difference between its own time and the external time of the world. If the interval measured with the object’s time is shorter than the external interval, we can speak about travel into the future. Such travel is not only possible but actually happens, according to the special theory of relativity. Due to the effect known as time dilation, if an object moves, its own time measures shorter intervals than the external time. If a traveller embarks on a journey and then comes back, his clock will show that his journey was shorter than when measured by the external clocks (this is the basis of the so-called twin paradox).
The most radical type of time travel is when the traveller goes into the past, i.e. the duration of his journey measured according to the external time is actually negative. Some philosophers claim that travel into the past involves contradiction, because a time traveller could change the past, and this is impossible. More specifically, the concept of changing the past is applied to states of affairs. In order to change a given past state of affairs – for instance, by scratching an inscription on a rock a thousand years ago – this state of affairs (the rock being unscratched) has to be both real (‘before’ the change) and unreal (‘after’ the change). But of course the time of the occurrence of these two contradictory states of affairs is the same, so the contradiction seems unavoidable. But it can be observed that the same problem applies to the apparently uncontroversial case of a change in the future. To literally change a state of affairs at a future time t requires that this state of affairs exist before but not after my action. But, again, this leads to logical contradiction. This difficulty can be avoided, though, when we apply the notion of change to things, not states of affairs. I certainly changed the past rock: before my intervention it wasn’t scratched, and afterwards it bore an inscription.
The most celebrated grandfather paradox exploits the possibility of vicious causal loops that seems to be opened by admitting time travel. The time traveller goes back to the times of his grandfather’s youth and kills him in the past. If his grandfather dies before he can have any children, the traveller will not be born in the future, and a contradiction ensues: the traveller both exists in the future (because he came from there to kill the grandfather) and does not exist (because his grandfather dies childless). This paradox has the following general form: there are two events A (the beginning of the journey to the past) and B (the killing of the grandfather) such that A is later than B, A causes B and B causes non-A. In order to avoid the problem, serious restrictions have to be imposed on the possible interactions of the time traveller with the past. Speaking loosely, each time the traveller tries to kill his grandfather something must get in the way to prevent him from accomplishing this task. Note that this extends to any action of the traveller (even seemingly innocent, such as leaving footprints on the grass) which might lead to the consequences threatening the entire future travel to the past as it precisely occurred. One general solution may be to assume that causal links directed from the future to the past do not ‘couple’ with the causal links leading in the ordinary temporal direction. But this would effectively imply that the traveller could only observe the past and not interact with it the normal way.
Reading:
B. Garrett, "Time travel", pp. 94-99, What is this thing called metaphysics?
A step towards special relativity is the realisation that the relation of simultaneity has no obvious empirical content, due to the fact that all signals (including light) travel at finite speeds. What we observe as our ‘present’ is actually already in the past (the farther, the more distant the observed event is). The standard, operationally defined notion of simultaneity is given as follows: two events x and y are simultaneous iff light signals sent from x and y meet exactly half way between x and y. But this definition is obviously not frame-independent. Suppose that the definition of simultaneity is satisfied in a frame f, and let us consider another frame f’ which moves with respect to f in the direction of the event y. The spatiotemporal point where the two beams of light meet will not be located in the middle of the distance between their locations x’ and y’, but rather closer to x', so from the perspective of f’ y happened earlier than x. We have to add that the signal definition of simultaneity presupposes that the speed of light is constant in all frames of reference.
In special relativity neither simultaneity nor co-location are invariant notions (independent of the frame of reference). Thus space-time cannot be absolutely divided into space and time. However, there is a relation between events (spatiotemporal points) which stays the same in all frames of reference. This relation is defined by the so-called spatiotemporal interval: cdt^2 – dx^2 – dy^2 – dz^2., where c – the speed of light, and dt, dx, dy and dz are temporal and spatial intervals between the two events. Two events for which the spatiotemporal interval is positive are called ‘time-like separated’. Such events can be connected by a signal travelling slower than light. If the interval equals zero, the events can be only connected by a beam of light. Events separated by a negative interval are called space-like separated. Such events cannot directly communicate by way of sending signals.
The basic structure of relativistic space-time (so-called Minkowski space-time) can be given with the help of light cones. For a given event x, its forward light cone consists of all events reachable from x by beams of light. Similarly, x’s backward light cone contains all events which can reach x using beams of light. The area within x’s backward light cone is called its absolute past, and within the forward light cone its absolute future. The events outside of both light cones are neither past nor future with respect to x, but they cannot be interpreted as being simultaneous with x. Their temporal relation with x is frame-dependent: for every event y space-like separated from x there is a frame of reference f in which x and y are simultaneous. But if we choose a different event y’ also space-like separated from x, the frame of reference in which y’ is simultaneous with x will be generally different from f.
Now we will discuss some issues related to the asymmetry of time. Let us start with the problem of time travel. Is time travel conceptually possible, or does it involve logical contradiction? First we have to decide what process can be called time travel. For a given object we can say that it travels in time if there is a difference between its own time and the external time of the world. If the interval measured with the object’s time is shorter than the external interval, we can speak about travel into the future. Such travel is not only possible but actually happens, according to the special theory of relativity. Due to the effect known as time dilation, if an object moves, its own time measures shorter intervals than the external time. If a traveller embarks on a journey and then comes back, his clock will show that his journey was shorter than when measured by the external clocks (this is the basis of the so-called twin paradox).
The most radical type of time travel is when the traveller goes into the past, i.e. the duration of his journey measured according to the external time is actually negative. Some philosophers claim that travel into the past involves contradiction, because a time traveller could change the past, and this is impossible. More specifically, the concept of changing the past is applied to states of affairs. In order to change a given past state of affairs – for instance, by scratching an inscription on a rock a thousand years ago – this state of affairs (the rock being unscratched) has to be both real (‘before’ the change) and unreal (‘after’ the change). But of course the time of the occurrence of these two contradictory states of affairs is the same, so the contradiction seems unavoidable. But it can be observed that the same problem applies to the apparently uncontroversial case of a change in the future. To literally change a state of affairs at a future time t requires that this state of affairs exist before but not after my action. But, again, this leads to logical contradiction. This difficulty can be avoided, though, when we apply the notion of change to things, not states of affairs. I certainly changed the past rock: before my intervention it wasn’t scratched, and afterwards it bore an inscription.
The most celebrated grandfather paradox exploits the possibility of vicious causal loops that seems to be opened by admitting time travel. The time traveller goes back to the times of his grandfather’s youth and kills him in the past. If his grandfather dies before he can have any children, the traveller will not be born in the future, and a contradiction ensues: the traveller both exists in the future (because he came from there to kill the grandfather) and does not exist (because his grandfather dies childless). This paradox has the following general form: there are two events A (the beginning of the journey to the past) and B (the killing of the grandfather) such that A is later than B, A causes B and B causes non-A. In order to avoid the problem, serious restrictions have to be imposed on the possible interactions of the time traveller with the past. Speaking loosely, each time the traveller tries to kill his grandfather something must get in the way to prevent him from accomplishing this task. Note that this extends to any action of the traveller (even seemingly innocent, such as leaving footprints on the grass) which might lead to the consequences threatening the entire future travel to the past as it precisely occurred. One general solution may be to assume that causal links directed from the future to the past do not ‘couple’ with the causal links leading in the ordinary temporal direction. But this would effectively imply that the traveller could only observe the past and not interact with it the normal way.
Reading:
B. Garrett, "Time travel", pp. 94-99, What is this thing called metaphysics?
Sunday, March 14, 2010
Absolutism and relationism
Now we will consider the question of the ontological status of time itself, and its relation to the material world. The problem can be stated as follows: is time a fundamental substance, capable of independent existence, or is it ontically dependent on things/events? One particular way of cashing out this general question is to ask whether it is possible for time to exist without any change. Of course the possibility in question has to be considered as metaphysical (stronger than logical but weaker than physical). The situation in which time exists but there is no change can be described as follows: there is a non-zero interval (t, t’) such that for any two moments t1 and t2 from this interval, all objects have exactly the same properties at t1 and at t2 (we can call the world in the interval a “frozen universe”). But now it can be argued that because all situations within the interval are indistinguishable, the statement that the length of the interval is non-zero has no empirical meaning. We could also appeal to the Principle of the Identity of Indiscernibles to argue that the moments t and t’ should be identified. However, Sydney Shoemaker has proposed an argument showing that under certain circumstances the hypothesis of the frozen universe can offer some advantages even to an empiricist. Suppose that the universe consists of three parts A, B, and C, and that the data gathered shows that region A goes through the period of a freeze every three years, region B freezes every four years, and region C freezes every five years. From these, empirically confirmed hypotheses it follows that the entire universe will freeze every 60 years, but of course this consequence can never be empirically confirmed or disconfirmed. Facing the choice between two empirically equivalent hypotheses we should choose the simpler one, and this is the one which assumes that there are no gaps in the regular patterns of freezing for regions A, B, C. So methodological postulates accepted by empiricists favour the hypothesis that the entire universe can freeze.
The issue of the dependence between time and the material world can be considered in an even more radical way. We may ask whether it is possible for time to exist without any events taking place. Can there be a period of time consisting of “empty” moments? Notice that this would be a case of time without change, but not all cases of time without change are cases of empty time. The view that it is fundamentally, metaphysically possible for such a situation to occur is known as absolutism, or substantivalism, with respect to time. On the other hand, those who believe that moments cannot exist without participating events (whether they are changes or not) are called relationists. Relationists believe that only spatiotemporal objects (things, events) and their temporal and spatial relations exist in the fundamental sense. Temporal objects (moments) are derived from the fundamental temporal relations. We should notice that absolutism and relationism can be formulated with respect to space as well as time. Relationists with respect to space believe that spatial points and the relations between them are mere reflections of events and their spatial relations (in particular, the relation of co-location).
Leibniz gave a strong argument against absolutism and in favour of relationism. Suppose that absolutism with respect to space is right and that space and spatial points exist independently of the material objects occupying them. Then shifting the entire world 5 metres in one direction would produce a distinct state of affairs (different points would be occupied by different objects) which nevertheless is indiscernible from the original one. Leibniz points out that such a possibility violates the principle of the identity of indiscernibles, and the principle of sufficient reason. We may also add that this argument shows that absolutism violates the principle of ontological parsimony, because it postulates the existence of objects (spatial points) which are not necessary to explain observable phenomena. In addition to this argument, known as the static shift argument, Leibniz also produced another one, based on a dynamic shift. The absolutist should consider the following two states of affairs as distinct: one in which the entire universe is stationary, and the other, in which it moves at a constant speed in a particular direction. But again, there is no observable effect that could distinguish between the two.
Newton was a proponent of absolutism. In support of his view, he pointed out that it is possible to distinguish between being at rest and moving, but this possibility applies only to a certain category of motions, namely accelerated motions. One example of such motions is rotation, which produces observable effects in the form of the centrifugal forces. Newton used this fact in his famous bucket argument. Consider a bucket full of water, suspended on a rope. The rope is twisted, and then released. In the first stage the bucket will start rotating, but the water will for some time remain stationary. The surface of the water will be flat. In the second stage the water being dragged by the sides of the bucket begins its rotational motion. This stage is characterised by a concave surface of the water, due to the centrifugal forces. Finally, the bucket is stopped, but the water inside it will continue spinning for some time. The concavity of the surface is still visible. Newton compared the first and the third stages, arguing that they are perfectly symmetrical with respect to the relative motions of the bucket and water. And yet only in the third stage we observe the concave surface. This can be only explained by postulating the existence of absolute space, against which the water rotates in the third, but not the first stage. But later critics, including Ernest Mach, pointed out that the situation is not exactly symmetrical. In the first case the water is stationary with respect to the rest of the universe, whereas in the second the water moves with respect to the fixed stars. Mach observed that Newton’s argument would be valid if we somehow managed to make the entire universe spin around the water inside the bucket. But this is impossible to achieve. Mach’s position is sometimes interpreted as suggesting that the inertial effects (e.g. centrifugal forces) are a result of the influence of all the masses in the universe on a given system. But Mach himself was sceptical of such a hypothesis, due to its apparent unverifiability.
We can distinguish several variants of the relationist account of space and time. The most radical version of relationism claims that space and time do not exist – the only entities are events and things which enter spatial and temporal relations. More moderate version of relationism accepts that time and space can be defined by abstractions from events, using relations of simultaneity and co-location. Moderate relationism rejects the existence of empty points of time and space, but this fact gives rise to a conceptual problem. Suppose that the universe consists entirely of three equidistant things A, B and C. In spite of the fact that no physical thing exists between A and B, or A and C, we would like to be able to say that there are points of space on line joining A, B and C. One solution could be to assume that a point exists if it is possible for a physical object (an event) to exist at this point. Such a position can be called “modal relationism”. But we have to note that modal relationism comes dangerously close to absolutism. The key is of course the notion of possibility, which has to be defined in a way that saves the distinction between relationism and absolutism.
The historical development of the debate between absolutism and relationism in the context of physical theories has followed a rather twisted path. Newtonian mechanics was originally founded on the idea of absolute space and time, whose mere reflections are temporal and spatial relations given to us in experience. But soon it became clear that Newtonian mechanics can be formulated in a relationist-friendly way, in the so-called Galilean invariant form. In this formulation, spatial and temporal coordinates of objects are defined not as positions in absolute space and time, but rather relatively to some frame of reference of a particular type, known as an inertial frame. All laws of classical mechanics have the same form in all inertial frames of reference, so no particular frame is privileged. The only types of frames which are physically distinguishable from the inertial frames are the ones that accelerate. In the special theory of relativity the absolute character of acceleration is retained, therefore the theory is not fully relationistic. Einstein attempted to include the principle of relationism in his general theory of relativity. Thanks to the principle of equivalence, accelerating frames of reference are locally indistinguishable from inertial frames of reference in a gravitational field. Although the general theory of relativity has certain mathematical features that make it well suited for a physical expression of relationism (mostly its covariant character), it turns out that some consequences of its fundamental equations seem to support substantivalism. In particular, there are solutions to Einstein’s field equations which describe an empty space-time (with no distribution of masses and energy). Another possible solution describes a solitary and yet rotating object, which goes against the idea that all motions are relative.
Readings:
B. Garrett, "Time: Three Puzzles", pp. 87-94, What is this thing caled metaphysics?
E.J. Lowe, Chapter 14 "Absolutism versus relationism", pp. 253-270, A survey of metaphyics.
The issue of the dependence between time and the material world can be considered in an even more radical way. We may ask whether it is possible for time to exist without any events taking place. Can there be a period of time consisting of “empty” moments? Notice that this would be a case of time without change, but not all cases of time without change are cases of empty time. The view that it is fundamentally, metaphysically possible for such a situation to occur is known as absolutism, or substantivalism, with respect to time. On the other hand, those who believe that moments cannot exist without participating events (whether they are changes or not) are called relationists. Relationists believe that only spatiotemporal objects (things, events) and their temporal and spatial relations exist in the fundamental sense. Temporal objects (moments) are derived from the fundamental temporal relations. We should notice that absolutism and relationism can be formulated with respect to space as well as time. Relationists with respect to space believe that spatial points and the relations between them are mere reflections of events and their spatial relations (in particular, the relation of co-location).
Leibniz gave a strong argument against absolutism and in favour of relationism. Suppose that absolutism with respect to space is right and that space and spatial points exist independently of the material objects occupying them. Then shifting the entire world 5 metres in one direction would produce a distinct state of affairs (different points would be occupied by different objects) which nevertheless is indiscernible from the original one. Leibniz points out that such a possibility violates the principle of the identity of indiscernibles, and the principle of sufficient reason. We may also add that this argument shows that absolutism violates the principle of ontological parsimony, because it postulates the existence of objects (spatial points) which are not necessary to explain observable phenomena. In addition to this argument, known as the static shift argument, Leibniz also produced another one, based on a dynamic shift. The absolutist should consider the following two states of affairs as distinct: one in which the entire universe is stationary, and the other, in which it moves at a constant speed in a particular direction. But again, there is no observable effect that could distinguish between the two.
Newton was a proponent of absolutism. In support of his view, he pointed out that it is possible to distinguish between being at rest and moving, but this possibility applies only to a certain category of motions, namely accelerated motions. One example of such motions is rotation, which produces observable effects in the form of the centrifugal forces. Newton used this fact in his famous bucket argument. Consider a bucket full of water, suspended on a rope. The rope is twisted, and then released. In the first stage the bucket will start rotating, but the water will for some time remain stationary. The surface of the water will be flat. In the second stage the water being dragged by the sides of the bucket begins its rotational motion. This stage is characterised by a concave surface of the water, due to the centrifugal forces. Finally, the bucket is stopped, but the water inside it will continue spinning for some time. The concavity of the surface is still visible. Newton compared the first and the third stages, arguing that they are perfectly symmetrical with respect to the relative motions of the bucket and water. And yet only in the third stage we observe the concave surface. This can be only explained by postulating the existence of absolute space, against which the water rotates in the third, but not the first stage. But later critics, including Ernest Mach, pointed out that the situation is not exactly symmetrical. In the first case the water is stationary with respect to the rest of the universe, whereas in the second the water moves with respect to the fixed stars. Mach observed that Newton’s argument would be valid if we somehow managed to make the entire universe spin around the water inside the bucket. But this is impossible to achieve. Mach’s position is sometimes interpreted as suggesting that the inertial effects (e.g. centrifugal forces) are a result of the influence of all the masses in the universe on a given system. But Mach himself was sceptical of such a hypothesis, due to its apparent unverifiability.
We can distinguish several variants of the relationist account of space and time. The most radical version of relationism claims that space and time do not exist – the only entities are events and things which enter spatial and temporal relations. More moderate version of relationism accepts that time and space can be defined by abstractions from events, using relations of simultaneity and co-location. Moderate relationism rejects the existence of empty points of time and space, but this fact gives rise to a conceptual problem. Suppose that the universe consists entirely of three equidistant things A, B and C. In spite of the fact that no physical thing exists between A and B, or A and C, we would like to be able to say that there are points of space on line joining A, B and C. One solution could be to assume that a point exists if it is possible for a physical object (an event) to exist at this point. Such a position can be called “modal relationism”. But we have to note that modal relationism comes dangerously close to absolutism. The key is of course the notion of possibility, which has to be defined in a way that saves the distinction between relationism and absolutism.
The historical development of the debate between absolutism and relationism in the context of physical theories has followed a rather twisted path. Newtonian mechanics was originally founded on the idea of absolute space and time, whose mere reflections are temporal and spatial relations given to us in experience. But soon it became clear that Newtonian mechanics can be formulated in a relationist-friendly way, in the so-called Galilean invariant form. In this formulation, spatial and temporal coordinates of objects are defined not as positions in absolute space and time, but rather relatively to some frame of reference of a particular type, known as an inertial frame. All laws of classical mechanics have the same form in all inertial frames of reference, so no particular frame is privileged. The only types of frames which are physically distinguishable from the inertial frames are the ones that accelerate. In the special theory of relativity the absolute character of acceleration is retained, therefore the theory is not fully relationistic. Einstein attempted to include the principle of relationism in his general theory of relativity. Thanks to the principle of equivalence, accelerating frames of reference are locally indistinguishable from inertial frames of reference in a gravitational field. Although the general theory of relativity has certain mathematical features that make it well suited for a physical expression of relationism (mostly its covariant character), it turns out that some consequences of its fundamental equations seem to support substantivalism. In particular, there are solutions to Einstein’s field equations which describe an empty space-time (with no distribution of masses and energy). Another possible solution describes a solitary and yet rotating object, which goes against the idea that all motions are relative.
Readings:
B. Garrett, "Time: Three Puzzles", pp. 87-94, What is this thing caled metaphysics?
E.J. Lowe, Chapter 14 "Absolutism versus relationism", pp. 253-270, A survey of metaphyics.
Tuesday, March 9, 2010
The A and B theories of time
McTaggart’s argument for the thesis that the A-series involves a contradiction is more complicated, and we will be only able to give its rough outline here. The argument starts with the unquestionable assumption that the three spheres constituting the A-series – the past, present and the future – are mutually exclusive. And yet McTaggart claims that the existence of the A-series requires that each event be past, present and future; thus a contradiction ensues. A typical response to this claim is that events are past, present and future not simultaneously, but in succession. The battle of Waterloo is past, but was present, and had been future. My current lecture is present, but was future and will be past. But McTaggart demands that we explain precisely what we mean by the words “was”, “had been” or “will”. One possible explication is as follows: an event x was present means that x is present at some past moment. Similarly, an event x will be past if and only if x is past at some future moment. But now we can observe that we have applied the distinction among the past, present and the future to moments, and again it can be claimed that the A-series requires that every moment is past, present and future. To avoid this conclusion we can only repeat the same procedure: we can distinguish between the situation in which a moment m was present (past, future), is present (past, future) and will be present (past, future). In order to explain the use of grammatical tenses, we have to appeal yet again to the second-order past, present and future moments at which the first-order moments can be classified as present, past and future without a contradiction, so it should be clear now that an infinite regress looms large.
It is not clear whether the above regress is vicious. Some commentators claim that the regress can be avoided without falling victim to a logical contradiction. For instance J. Lowe insists that the A-series can be expressed in a language that employs temporal adverbial modifiers “presently”, “pastly”, and “futurely”. Lowe notes that each event x has to satisfy three disjunctions: (1) x is either pastly past, or presently past, or futurely past; (2) x is either pastly present, or presently present or futurely present; and (3) x is either pastly future, or presently future or futurely future (note that the disjunctions (1) – (3) are not necessarily exclusive). Lowe’s point is that we don’t need to explain the adverbial modifiers in a way that leads to a regress, and he also thinks that the A-theorist should be content with such a characteristic of the A-series. This last claim can be questioned, though. The partition of events into Lowe’s nine temporal spheres falls short of making time move. The truth of (1)-(3) is logically compatible with a completely stationary time, in which a given event x always belongs to the same spheres. McTaggart can repeat his main point: in order for the A-series to exist, every event has to be pastly present, presently present and futurely present, pastly past, presently past and futurely past, and pastly future, presently future and futurely future.
McTaggart’s distinction gives rise to two theories of time: the A-theory and the B-theory. The main difference between them lies in their approach to the idea of the passage of time. The A-theory accepts the existence of the objective passage of time, while the B-theory rejects it. The B-theorists invoke two arguments against the passage of time. Firstly, if the passage of time existed, it would make sense to ask how fast time flows. The rate of time’s flow would have to be measured in seconds per second, which is a dimensionless quantity. Secondly, the movement of time requires some stationary background against which it can happen (similarly to ordinary motion, for which the background is precisely time itself). But this implies the existence of a second-order time, which presumably requires yet another, higher-order time and so on. B-theorists insist that we can translate our ordinary way of speaking about time into the language of the B-theory, based on the fundamental relation “earlier than”. The main challenge for the B-theory is how to express grammatical tenses in the tenseless B-language. A typical suggestion goes along the following lines. The temporal expressions, such as “past”, “present”, “now”, “yesterday”, “tomorrow”, ten days ago”, belong to the category of the so-called indexicals, i.e. expressions whose meaning depends on the context of utterance. Other words in this category are “here”, “there” “I”, etc. When I utter the word “here” while standing in Trafalgar square, this word refers to a different place than when I utter it under the Eiffel tower. Similarly, when I say “It is cold now”, I mean something like “It is cold at the moment of my utterance”. The expression “Napoleon was defeated at the battle of Waterloo” can be interpreted as “Napoleon’s defeat at the battle of Waterloo is earlier than the moment of utterance”.
The proponents of the A-theory of time do not give up easily. They point out that the experience of the passage of time is too fundamental to dismiss it as some sort of illusion. They accuse the B-theorists of interpreting time as an extra spatial dimension (so-called spatialisation of time). The defenders of the passage of time claim that it is possible to meet the B-theorists objections. The passage of time does not have to be literally interpreted as a kind of motion, to which the ordinary notion of velocity would apply. Rather, it consists in the fundamental fact that events come into being successively. Responding to the argument from the rate of flow Tim Maudlin claims that it misses the point. He points out that there is nothing fundamentally wrong with dimensionless quantities, quoting the example of an exchange rate of one currency for itself (dollars for dollars). But we may note that even if Maudlin is right and the notion of the velocity at which time passes is not meaningless, still it is quite unsettling that in his approach this velocity can assume only one value (one second per second) as a matter of conceptual necessity. In other words, time cannot speed up, nor can it slow down. A different solution to this problem has been proposed by Peter Forrest. According to his approach, time passes by adding new layers of spacetime of positive thickness to the already existing universe. The thickness of the successive new layers is the measure of the rate of the flow of time. This picture of the passage of time dispenses with the stationary background against which the passage is supposed to occur.
The A and B theories of time are naturally associated with particular positions regarding the reality of temporal spheres of events. The B-theory is typically connected with the view known as eternalism (or the block universe view). According to eternalism, all events, past, present and future, enjoy the same fundamental status of reality. The battle of Waterloo did not vanish – it exists but in a different part of spacetime than the region occupied by us. Past events are analogous to events that occur in spatially remote regions of the universe: they happen elsewhere, but are not less real because of that. The unintuitiveness of this position is best exposed in Arthur Prior’s “Thanks goodness it’s over” argument. He points out that the eternalist cannot satisfactory explain why we feel relieved when something bad comes to an end. For instance, when my teeth stop aching after taking a pain killer, I feel relieved, but why should I, given that my past pain did not cease to be real? You may reply that the pain belongs to the past now. But according to the B-theory, this means that my (real) pain is earlier than the moment of utterance. Why should I be happy about this?
The A-theory of time is compatible with more than one ontological position regarding the reality of past and future. The most radical is the view known as presentism, which claims that only present events exist. Both past and future events are not real (the former are no longer real, the latter not yet real). The universe consists just of one three-dimensional layer of events which moves as time passes. Presentism is threatened by two main arguments: one from science, and the other from semantics. It is commonly accepted that presentism is incompatible with the special theory of relativity. According to special relativity, the relation of simultaneity is relative with respect to the frame of reference (we will talk about this later). Consequently, the set of events simultaneous with my current “present” depends on the selected frame of reference. But presentism requires that only one set of mutually simultaneous events be real, hence it privileges one particular frame of reference, and this fact violates the principle of relativity. The argument from semantics turns on the fact that some statements about past events (and future events too) are true. But what is the true sentence “Napoleon lost the battle of Waterloo” about, if neither Napoleon, not the battle exists? What is its truthmaker?
Two alternative views compatible with the A-theory are: the growing block theory and the shrinking block theory. The first assumes that past and present events exist, but not future events. The second accepts the opposite: present and future events exist, but past events do not. Both theories are susceptible to similar objections as presentism, although the argument from semantics is now limited to the case of future statements for the growing block theory, and the case of past statements for the shrinking block theory.
Readings:
E.J. Lowe, Chapter 17 "Tense and the reality of time", pp. 307-324, A Survey of Metaphysics.
M.J. Loux, Chapter 7 "The nature of time", pp. 205-228, Metaphysics: A Contemporary Introduction.
It is not clear whether the above regress is vicious. Some commentators claim that the regress can be avoided without falling victim to a logical contradiction. For instance J. Lowe insists that the A-series can be expressed in a language that employs temporal adverbial modifiers “presently”, “pastly”, and “futurely”. Lowe notes that each event x has to satisfy three disjunctions: (1) x is either pastly past, or presently past, or futurely past; (2) x is either pastly present, or presently present or futurely present; and (3) x is either pastly future, or presently future or futurely future (note that the disjunctions (1) – (3) are not necessarily exclusive). Lowe’s point is that we don’t need to explain the adverbial modifiers in a way that leads to a regress, and he also thinks that the A-theorist should be content with such a characteristic of the A-series. This last claim can be questioned, though. The partition of events into Lowe’s nine temporal spheres falls short of making time move. The truth of (1)-(3) is logically compatible with a completely stationary time, in which a given event x always belongs to the same spheres. McTaggart can repeat his main point: in order for the A-series to exist, every event has to be pastly present, presently present and futurely present, pastly past, presently past and futurely past, and pastly future, presently future and futurely future.
McTaggart’s distinction gives rise to two theories of time: the A-theory and the B-theory. The main difference between them lies in their approach to the idea of the passage of time. The A-theory accepts the existence of the objective passage of time, while the B-theory rejects it. The B-theorists invoke two arguments against the passage of time. Firstly, if the passage of time existed, it would make sense to ask how fast time flows. The rate of time’s flow would have to be measured in seconds per second, which is a dimensionless quantity. Secondly, the movement of time requires some stationary background against which it can happen (similarly to ordinary motion, for which the background is precisely time itself). But this implies the existence of a second-order time, which presumably requires yet another, higher-order time and so on. B-theorists insist that we can translate our ordinary way of speaking about time into the language of the B-theory, based on the fundamental relation “earlier than”. The main challenge for the B-theory is how to express grammatical tenses in the tenseless B-language. A typical suggestion goes along the following lines. The temporal expressions, such as “past”, “present”, “now”, “yesterday”, “tomorrow”, ten days ago”, belong to the category of the so-called indexicals, i.e. expressions whose meaning depends on the context of utterance. Other words in this category are “here”, “there” “I”, etc. When I utter the word “here” while standing in Trafalgar square, this word refers to a different place than when I utter it under the Eiffel tower. Similarly, when I say “It is cold now”, I mean something like “It is cold at the moment of my utterance”. The expression “Napoleon was defeated at the battle of Waterloo” can be interpreted as “Napoleon’s defeat at the battle of Waterloo is earlier than the moment of utterance”.
The proponents of the A-theory of time do not give up easily. They point out that the experience of the passage of time is too fundamental to dismiss it as some sort of illusion. They accuse the B-theorists of interpreting time as an extra spatial dimension (so-called spatialisation of time). The defenders of the passage of time claim that it is possible to meet the B-theorists objections. The passage of time does not have to be literally interpreted as a kind of motion, to which the ordinary notion of velocity would apply. Rather, it consists in the fundamental fact that events come into being successively. Responding to the argument from the rate of flow Tim Maudlin claims that it misses the point. He points out that there is nothing fundamentally wrong with dimensionless quantities, quoting the example of an exchange rate of one currency for itself (dollars for dollars). But we may note that even if Maudlin is right and the notion of the velocity at which time passes is not meaningless, still it is quite unsettling that in his approach this velocity can assume only one value (one second per second) as a matter of conceptual necessity. In other words, time cannot speed up, nor can it slow down. A different solution to this problem has been proposed by Peter Forrest. According to his approach, time passes by adding new layers of spacetime of positive thickness to the already existing universe. The thickness of the successive new layers is the measure of the rate of the flow of time. This picture of the passage of time dispenses with the stationary background against which the passage is supposed to occur.
The A and B theories of time are naturally associated with particular positions regarding the reality of temporal spheres of events. The B-theory is typically connected with the view known as eternalism (or the block universe view). According to eternalism, all events, past, present and future, enjoy the same fundamental status of reality. The battle of Waterloo did not vanish – it exists but in a different part of spacetime than the region occupied by us. Past events are analogous to events that occur in spatially remote regions of the universe: they happen elsewhere, but are not less real because of that. The unintuitiveness of this position is best exposed in Arthur Prior’s “Thanks goodness it’s over” argument. He points out that the eternalist cannot satisfactory explain why we feel relieved when something bad comes to an end. For instance, when my teeth stop aching after taking a pain killer, I feel relieved, but why should I, given that my past pain did not cease to be real? You may reply that the pain belongs to the past now. But according to the B-theory, this means that my (real) pain is earlier than the moment of utterance. Why should I be happy about this?
The A-theory of time is compatible with more than one ontological position regarding the reality of past and future. The most radical is the view known as presentism, which claims that only present events exist. Both past and future events are not real (the former are no longer real, the latter not yet real). The universe consists just of one three-dimensional layer of events which moves as time passes. Presentism is threatened by two main arguments: one from science, and the other from semantics. It is commonly accepted that presentism is incompatible with the special theory of relativity. According to special relativity, the relation of simultaneity is relative with respect to the frame of reference (we will talk about this later). Consequently, the set of events simultaneous with my current “present” depends on the selected frame of reference. But presentism requires that only one set of mutually simultaneous events be real, hence it privileges one particular frame of reference, and this fact violates the principle of relativity. The argument from semantics turns on the fact that some statements about past events (and future events too) are true. But what is the true sentence “Napoleon lost the battle of Waterloo” about, if neither Napoleon, not the battle exists? What is its truthmaker?
Two alternative views compatible with the A-theory are: the growing block theory and the shrinking block theory. The first assumes that past and present events exist, but not future events. The second accepts the opposite: present and future events exist, but past events do not. Both theories are susceptible to similar objections as presentism, although the argument from semantics is now limited to the case of future statements for the growing block theory, and the case of past statements for the shrinking block theory.
Readings:
E.J. Lowe, Chapter 17 "Tense and the reality of time", pp. 307-324, A Survey of Metaphysics.
M.J. Loux, Chapter 7 "The nature of time", pp. 205-228, Metaphysics: A Contemporary Introduction.
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