Having rejected the postulate of the necessity of causal links, Hume replaces it with the condition of regularity. Paraphrasing his words, if x is a cause of y, each event that is similar to x is followed by an event similar to y. Thus, the complete definition of causation can look like this: x is a cause of y iff x is spatiotemporally contiguous with y, x temporally precedes y, and for all x’, if x’ is similar to x, then there is a y’ such that y’ is similar to y and y’ follows x’. One of the main difficulties with this definition is the notorious vagueness of the notion of similarity between events. If we interpret similarity as being identical in some respect, then it is plausible that every object is similar to every other object, and as a consequence the condition of regularity can never be satisfied. On the other hand, similarity conceived as identity in all respects (qualitative identity) collapses into numerical identity (given the PII), and therefore the condition of regularity reduces to the previous two conditions (contiguity and temporal precedence). The main challenge to Hume’s analysis is to characterize the notion of a relevant aspect with respect to which the similarity between events is interpreted. But even if this can be done, Hume’s analysis is open to serious criticism. Critics point out that there are cases of regular successions of events without a causal link. Days regularly follow nights, and yet there is no causal relation between the two. Similarly, if according to the timetable after the departure of train A train B regularly arrives at the station, this does not indicate that one causes the other. Note that typically the regular but not causal correlations can be explained with the help of a common cause (the succession of day and night is explained by the rotation of the earth, and the timetable acts as the common cause of both the departure of train A and the arrival of train B). So it can be concluded that Hume’s regularity approach has difficulties with distinguishing between direct causal regularities and regularities arising from a common cause.
Another group of counterexamples to the Humean analysis contains cases of causal links where no regularity is present. It is sometimes argued that causal links between unique events (for instance the Big Bang) don’t satisfy the requirement of regularity. Actually, they satisfy it but trivially, making all spatiotemporally conjoined unique events causally connected with each other. But it is unquestionable that in real life we make numerous causal claims where there is no underlying regularity. We say that the failure of brakes was a cause of the car crash, although not every failure of that sort leads to a crash. This clearly shows that Hume’s analysis is in need of serious corrections.
Neo-Humean approaches to causation try to eliminate some of its weak points. One of them is the nomological conception. According to it, an event x of type A is a cause of an event y of type B iff x and y occur in conditions C and there is a law of nature according to which if an event of type A happens in conditions C, an event of type B occurs. There are two main differences between this approach and Hume’s regularity account: the presence of background conditions C and the reference to laws of nature. The role of laws of nature is to eliminate accidental regularities, while the presence of conditions C should take care of the problem of the apparent lack of regularity of some causal links (the failure of brakes leads to an accident only in certain specific circumstances). But these additions to the regularity approach bring new problems. What are laws? Hume himself insisted that laws are nothing over and above mere regularities, but if that’s the case the addition of laws to the definition of causation does not constitute an improvement with respect to the old, Humean version. The introduction of conditions C creates a different problem. By a crafty selection of appropriate “conditions” we can make virtually any succession of events a causal one. It may be claimed, for instance, that the snapping of my fingers causes the lights to go out if we include in the appropriate conditions that someone turns off the switch at the same moment. Yet another objection is that there are laws which are not causal. Pascal’s law states that a gas exerts equal pressure in all directions, but it is not correct to say that the fact that the pressure in one direction equals p is caused by the fact that the pressure in the opposite direction is also p. Finally, most laws of physics are symmetric in time, but from this it does not follow that backward causation is a common fact.
One of the most sophisticated versions of the regularity approach is the conception of causation proposed by J.L. Mackie. Mackie notes that the notion of a cause is closely related to sufficient and necessary conditions, but a cause of x cannot be simply defined as a necessary or a sufficient condition of x. Let us consider Mackie’s example with a fire of a house being caused by a short circuit. The short circuit by itself is not sufficient for the house to burn down; other conditions have to be present, such as the presence of inflammable materials, of oxygen, the absence of automatic sprinklers and smoke detectors, etc. Generally speaking the short circuit is not necessary for the fire either, for fires can start in many different ways, for instance after a strike of a lightning bolt. But in the actual conditions the short circuit was necessary, because without it the conditions themselves would not have created the fire. (Mackie speaks in this case about a necessary condition post factum.) The short circuit is an INUS condition for the occurrence of the fire, where the acronym INUS stands for an Insufficient but Necessary part of an Unnecessary but Sufficient condition. A more precise definition of an INUS condition is as follows: A is an INUS condition for B iff there are conditions X and Y such that (AX or Y) is a necessary and sufficient condition of B, but neither A nor X is a sufficient condition of B. In our example A is the short circuit, B is the fire, X refers to all the conditions which together with A were sufficient for the fire to occur, and Y stands for a disjunction of all alternatives ways of starting a fire. In most cases a cause of an occurrence B is an INUS condition of B such that it occurred, and no alternative conditions Y were present (however, Mackie admits the possibility that a cause itself may be sufficient for its effect, or even sufficient and necessary – he refers to all such cases jointly as “at least” INUS conditions of B).
It has to be noted that some further restrictions on the causal condition A have to be introduced, otherwise Mackie’s analysis will lead to obviously incorrect conclusions. To see this, let us use the letter C to abbreviate the complete sufficient condition of B that was actually present, and let us select any fact S irrelevant to the occurrence of B that happened simultaneously with C (for instance the fact that when the fire started somebody walked past the house whistling “Ode to joy”). The formula [(S or C) and (not-S or C)] is logically equivalent to C, and hence if there is a condition Y such that (C or Y) is a necessary and sufficient condition of B, then {[(S or C) and (not-S or C)] or Y} is a necessary and sufficient condition of B too. But now observe that the condition (S or C) satisfies the requirement for an INUS condition of B. (S or C) is not sufficient for B, nor is (not-S or C), and yet their conjunction is sufficient (as it is equivalent to C). But it is highly unintuitive to pick the disjunction S or C as a cause of B. To eliminate cases like this it may be suggested that causes should not have the form of disjunctions of simple events.
An interesting element of Mackie’s conception is that he admits that causal claims are always made in a context. In order to account for this fact, he introduces the notion of a causal field. Let us consider as an example the case of a person going down with flu. The answer to the question “What caused this man to contract the flu?” depends on the context. If we consider as the causal field the set of all moments in his life, and ask why he contracted the disease at this moment rather than any other, then the correct answer may be that he was infected by influenza viruses. But we can also select as the causal field the set of all people who came into contact with influenza viruses, and we may be interested in selecting the factor which is responsible for the fact that some of them contracted the disease, while the others did not. Mackie introduces the causal field to his definition of a cause, assuming that the conditions characterizing this field are present when the cause is present.
Reading:
M.J. Loux, Chapter 6 "Causation", pp. 187-203, Metaphysics: A Contemporary Introduction.
Wednesday, April 28, 2010
Causation and necessity
We can distinguish two types of causal statements: general causal statements and singular ones. General statements relate types of phenomena (for instance: smoking causes cancer), whereas singular causal statements connect individual occurrences (for example: the cause of the sinking of the Titanic was that it collided with an iceberg). While the two categories of causal claims are undoubtedly related, their relation is not straightforward. It may seem that general causal statements of the form “Phenomenon A causes phenomenon B” can be reduced to the following singular claim: “For all x, if x is of type A, then x causes some y of type B”. But this won’t work. From the fact that smoking causes cancer it does not follow that every smoker will suffer from cancer. General causal claims are very often statistical only, and their truth is typically hedged by the ceteris paribus condition. On the other hand, if we wanted to define singular causal claims of the type “x causes y” with the help of the general formula “The type of phenomena A to which x belongs causes the type of phenomena B containing y”, we would encounter an immediate problem connected with the fact that each individual event can be classified into many distinct types. In the following we will restrict our analysis to singular claims only, and therefore we will interpret causation as a relation between individual objects.
It may be useful to start an analysis of causation from the following questions:
(1) What are the relata of the causal relation?
(2) What are formal properties of the causal relation?
(3) What is the temporal relation between a cause and its effect?
(1) Typically three categories of objects are regarded as being capable of standing in the causal relation: things, events and facts. One natural way of speaking about causal links seems to identify causes as things. For instance, we can say that John smashed a window glass with a stone, and a car hit a pedestrian. This suggests that causes are things (John, car) while effects are events (shattering the window, hitting the pedestrian). But clearly this is an oversimplified way of speaking. If John is busy talking on the phone, there is no shattering, although the purported cause (John) is still present. If the car is parked in a garage, no pedestrian is in danger of being hit by it. Strictly speaking, it is not John but his throwing the stone that causes the breaking, and it is not the car but its particular movement that causes the hitting of the pedestrian. This observation leads to the most commonly accepted conception of causation, according to which both causes and effects are events (throwing the stone – shattering the glass, movement of the car – hitting the pedestrian).
However, some philosophers insist that this account is too restrictive, as it does not make room for cases of negative causation. Sometimes it seems natural to single out absences of events rather than events themselves as causal factors contributing to a given effect. We say that the lack of attention of the driver was a cause of the crash, and that the absence of sprinklers contributed to the fire. But there are no negative events (in Kim’s conception, events are property attributions, but it is customary not to admit negative properties). In order to admit negative causation (sometimes also called causation by omission) it is proposed that causes and effects be facts, not events. Facts are just ontological counterparts of true statements, so there is no problem with the assumption that there are negative facts corresponding to negative statements. But critics point out that negative causation is really not necessary, and moreover that admitting it opens the door to many unintuitive cases of spurious causation. It may be claimed that underlying every case of apparent negative causation there is an instance of positive causation (for instance the driver’s lack of attention could have been actually his talking on the phone). And we tend to dismiss statements of the sort “The fact that I had not been struck by lightning caused me to survive” if there was no reason to expect that the lightning was imminent.
(2) It should be clear that the causal relation is not reflexive (there are events that don’t cause themselves). But is it irreflexive (no event is a cause of itself)? That depends. If we admit the possibility of causal loops (as in time travel), and we agree that causality is transitive, then there may be cases of self-causation (x causes y and y causes x, therefore x causes x). Similarly, causality is not symmetric, but it is open to debate whether it is asymmetric (if there are causal loops, clearly it is not asymmetric). The case for transitivity looks plausible enough, but recently this feature of causality came under attack. Some philosophers point out that there are cases which seem to violate the transitivity requirement, such as the following one. A bomb had been planted at the door of a politician’s house, but luckily it was spotted by the security and defused. It is natural to assume that the placing of the bomb was a cause of its defusing (if there hadn’t been a bomb, there wouldn’t have been the act of defusing), and the defusing of the bomb causes the politician to survive. But it is unnatural to say that the placing of the bomb was a cause of the politician’s survival (clearly the counterfactual “If the bomb had not been planted, the politician would not have survived” is false).
(3) It is typically assumed that a cause happens earlier (or, at least, not later) than its effect. But, again, if we want to admit that it is conceptually possible to have backward causation, we have to reject this requirement.
The main question we have to ask now is “What is causation?”. Answers to this question can be given in the form of a reductive analysis, explicating the causal relation in terms of some more fundamental concepts. We will start with the most famous reductive analysis of causation given by David Hume. Hume observes that it is an uncontroversial fact that causation displays the following two properties: the cause and the effect are contiguous in space and time (they “touch” each other), and the cause temporally precedes the effect. Actually, both claims can be questioned. The issue of temporal precedence has been already mentioned in point (3). As for the contiguity, at best it can be applied to direct causes only. Clearly there is a temporal and spatial gap between my act of hurling the stone and the smashing of the window. But it can be claimed that there has to be a chain of events contiguous in space and time leading from the throwing to the breaking. Still, this does not seem to be conceptually necessary. There is nothing inconsistent in considering causal links acting at a distance with no intermediate stages. Actually, this is how gravitational interaction between massive bodies can be assumed to work in Newtonian mechanics. So it looks like the two conditions proposed by Hume are not necessary for causation to occur. But we have to agree with Hume that they are not sufficient either, for there are plenty of events following one another which are not causally connected.
Hume then asks, what should be added in order to have a sufficient condition for the presence of a causal link. One typical response is that the cause has to be necessarily linked with its effect, or in other words, that if the cause occurs, the effect must occur. But Hume famously questions this. Firstly, he notices that the purported necessity cannot be of the logical kind, for no contradiction arises from the supposition that a given event does not produce its expected effect. I can imagine without contradiction the stone magically passing through the glass, or bouncing off it. But perhaps the necessity connecting causes and effects is of a different kind (nomological, or physical). Hume’s response is that no such necessity is given to us in sensory experience. We never perceive two events as connected, only as conjoined.
Clearly, Hume’s criticism of the necessary character of causation has its roots in his version of empiricism. Hume insists that every meaningful concept should be traced back to some sensory experience (‘impression’). But this requirement may be seen as overly restrictive. Hume’s radical empiricism does not square well with modern science which commonly postulates the existence of unobservable objects and properties. According to Hume’s criterion, along with the notion of necessity we should abandon such concepts as that of atoms, electrons, electromagnetic field, etc., as they cannot be supported by any direct sensory data either. On the other hand, more moderate versions of empiricism can in principle accommodate the notion of a necessary causal link, if it is treated as a theoretical concept used to make empirical predictions and explain observable facts.
Reading:
B. Garrett, "Causation", pp. 53-66, What is this thing called metaphysics?
It may be useful to start an analysis of causation from the following questions:
(1) What are the relata of the causal relation?
(2) What are formal properties of the causal relation?
(3) What is the temporal relation between a cause and its effect?
(1) Typically three categories of objects are regarded as being capable of standing in the causal relation: things, events and facts. One natural way of speaking about causal links seems to identify causes as things. For instance, we can say that John smashed a window glass with a stone, and a car hit a pedestrian. This suggests that causes are things (John, car) while effects are events (shattering the window, hitting the pedestrian). But clearly this is an oversimplified way of speaking. If John is busy talking on the phone, there is no shattering, although the purported cause (John) is still present. If the car is parked in a garage, no pedestrian is in danger of being hit by it. Strictly speaking, it is not John but his throwing the stone that causes the breaking, and it is not the car but its particular movement that causes the hitting of the pedestrian. This observation leads to the most commonly accepted conception of causation, according to which both causes and effects are events (throwing the stone – shattering the glass, movement of the car – hitting the pedestrian).
However, some philosophers insist that this account is too restrictive, as it does not make room for cases of negative causation. Sometimes it seems natural to single out absences of events rather than events themselves as causal factors contributing to a given effect. We say that the lack of attention of the driver was a cause of the crash, and that the absence of sprinklers contributed to the fire. But there are no negative events (in Kim’s conception, events are property attributions, but it is customary not to admit negative properties). In order to admit negative causation (sometimes also called causation by omission) it is proposed that causes and effects be facts, not events. Facts are just ontological counterparts of true statements, so there is no problem with the assumption that there are negative facts corresponding to negative statements. But critics point out that negative causation is really not necessary, and moreover that admitting it opens the door to many unintuitive cases of spurious causation. It may be claimed that underlying every case of apparent negative causation there is an instance of positive causation (for instance the driver’s lack of attention could have been actually his talking on the phone). And we tend to dismiss statements of the sort “The fact that I had not been struck by lightning caused me to survive” if there was no reason to expect that the lightning was imminent.
(2) It should be clear that the causal relation is not reflexive (there are events that don’t cause themselves). But is it irreflexive (no event is a cause of itself)? That depends. If we admit the possibility of causal loops (as in time travel), and we agree that causality is transitive, then there may be cases of self-causation (x causes y and y causes x, therefore x causes x). Similarly, causality is not symmetric, but it is open to debate whether it is asymmetric (if there are causal loops, clearly it is not asymmetric). The case for transitivity looks plausible enough, but recently this feature of causality came under attack. Some philosophers point out that there are cases which seem to violate the transitivity requirement, such as the following one. A bomb had been planted at the door of a politician’s house, but luckily it was spotted by the security and defused. It is natural to assume that the placing of the bomb was a cause of its defusing (if there hadn’t been a bomb, there wouldn’t have been the act of defusing), and the defusing of the bomb causes the politician to survive. But it is unnatural to say that the placing of the bomb was a cause of the politician’s survival (clearly the counterfactual “If the bomb had not been planted, the politician would not have survived” is false).
(3) It is typically assumed that a cause happens earlier (or, at least, not later) than its effect. But, again, if we want to admit that it is conceptually possible to have backward causation, we have to reject this requirement.
The main question we have to ask now is “What is causation?”. Answers to this question can be given in the form of a reductive analysis, explicating the causal relation in terms of some more fundamental concepts. We will start with the most famous reductive analysis of causation given by David Hume. Hume observes that it is an uncontroversial fact that causation displays the following two properties: the cause and the effect are contiguous in space and time (they “touch” each other), and the cause temporally precedes the effect. Actually, both claims can be questioned. The issue of temporal precedence has been already mentioned in point (3). As for the contiguity, at best it can be applied to direct causes only. Clearly there is a temporal and spatial gap between my act of hurling the stone and the smashing of the window. But it can be claimed that there has to be a chain of events contiguous in space and time leading from the throwing to the breaking. Still, this does not seem to be conceptually necessary. There is nothing inconsistent in considering causal links acting at a distance with no intermediate stages. Actually, this is how gravitational interaction between massive bodies can be assumed to work in Newtonian mechanics. So it looks like the two conditions proposed by Hume are not necessary for causation to occur. But we have to agree with Hume that they are not sufficient either, for there are plenty of events following one another which are not causally connected.
Hume then asks, what should be added in order to have a sufficient condition for the presence of a causal link. One typical response is that the cause has to be necessarily linked with its effect, or in other words, that if the cause occurs, the effect must occur. But Hume famously questions this. Firstly, he notices that the purported necessity cannot be of the logical kind, for no contradiction arises from the supposition that a given event does not produce its expected effect. I can imagine without contradiction the stone magically passing through the glass, or bouncing off it. But perhaps the necessity connecting causes and effects is of a different kind (nomological, or physical). Hume’s response is that no such necessity is given to us in sensory experience. We never perceive two events as connected, only as conjoined.
Clearly, Hume’s criticism of the necessary character of causation has its roots in his version of empiricism. Hume insists that every meaningful concept should be traced back to some sensory experience (‘impression’). But this requirement may be seen as overly restrictive. Hume’s radical empiricism does not square well with modern science which commonly postulates the existence of unobservable objects and properties. According to Hume’s criterion, along with the notion of necessity we should abandon such concepts as that of atoms, electrons, electromagnetic field, etc., as they cannot be supported by any direct sensory data either. On the other hand, more moderate versions of empiricism can in principle accommodate the notion of a necessary causal link, if it is treated as a theoretical concept used to make empirical predictions and explain observable facts.
Reading:
B. Garrett, "Causation", pp. 53-66, What is this thing called metaphysics?
Tuesday, April 27, 2010
How things persist
Things exist in time. More specifically, they persist. A thing, such as a tree, remains the same object throughout its existence, although it changes some of its properties, loses some of its parts and acquires new ones. The relation that holds between the same object at different times is called ‘diachronic identity’. But the question is: What is this new type of identity? How does it relate to numerical identity? Is diachronic identity reducible to numerical identity, or perhaps to qualitative identity? These questions are answered differently by two major conceptions of how things persist in time: endurantism and perdurantism. Endurantism can be characterised broadly as the position according to which things persist in time by being wholly and completely present at every single moment of their existence. On the other hand, perdurantism assumes that at a given moment only a small part of a thing is present. The whole thing is an object which extends in time as well as in space.
Let us look more closely at both views. Endurantism seems to be in agreement with our pre-philosophical intuitions regarding persistence. When I look at a table in front of me, I believe that no part of it is missing from my view. Things have only spatial parts and spatial dimensions, but no temporal ones. According to endurantists, the expressions “Napoleon at the time of the battle of Austerlitz” and “Napoleon at the time of the battle of Waterloo” refer to one and the same individual: Napoleon. Hence diachronic identity reduces to numerical identity. Things are three-dimensional objects taking up various spatial regions at different times.
Perdurantism, on the other hand, claims that things are four-dimensional objects taking up regions of space-time. The expressions “Napoleon at the time of the battle of Austerlitz” and “Napoleon at the time of the battle of Waterloo” refer to numerically different objects: temporal parts of the four-dimensional entity that we call “Napoleon” and whose temporal dimension stretches from the moment of Napoleon’s birth to his death. Consequently, the relation of diachronic identity is not defined as numerical identity, but instead can be explicated as the relation that holds between any two temporal parts of the same thing: x is diachronically identical with y iff x and y are different temporal parts of individual z. (An alternative interpretation of diachronic identity under perdurantism is that it is reducible to numerical identity after all, when we say that “Napoleon at t1 is identical with Napoleon at t2” means “The four-dimensional object whose temporal part at t1 is Napoleon at t1 is numerically the same as the four-dimensional object whose temporal part at t2 is Napoleon at t2”.) Temporal parts of an object can be divided into stages and slices. A stage of a thing x is a part of x that occupies a non-zero interval (it has a non-zero temporal “width”), whereas a slice of x is a part of x taken in a zero-length moment of time, and thus it has no temporal dimension. Slices are three-dimensional, and they represent what we would usually refer to as objects of our perception. It is noteworthy that the way things persist according to the perdurantist is analogous to the temporal existence of events. An event, such as the battle of Waterloo, is never fully present in an interval that is shorter than its entire duration.
The main motivation for perdurantism comes from the problem of change which threatens the endurantist approach. Things change their properties in time: for some moments t1 and t2 and a property P it is the case that x has P at t1 and x does not have P at t2. But according to endurantism x at t1 is numerically identical with x at t2. If we apply Leibniz’s law, which states that if x = y and Px, then Py, then we have to conclude that an object has P and doesn’t have P, which is obviously a contradiction. Perdurantism avoids this problem by assuming that the properties P and not-P are attributed to numerically different individuals: different temporal slices. But the endurantist is not without options with respect to the problem of property change. One solution is to relativise properties to time. Let us consider a poker which is cold at t1 and hot at t2. We may say that the poker possesses the property of being-cold-at-t1 and being-hot-at-t2, and these properties are not mutually exclusive as long as t1 is different from t2. But an objection can be raised that this approach treats properties as if they were relations between objects and moments, and consequently no property can be intrinsic. And, besides, isn’t it legitimate to speak about properties simpliciter, without any temporal relativisation? Another solution, available to the endurantist, is to relativise the relation of possession between the object and its properties. Objects don’t just possess properties, but they always possess them at certain moments. This position is known as adverbialism, as it amounts to the adverbial modification of the verb “be” (the poker is-at-t1 cold and the poker is-at-t2 hot). One consequence of this approach is that there is no single relation of exemplification between particulars and universals, but an infinite (even uncountable) number of different relations of exemplification. Finally, let us notice that all the above endurantist solutions seem to assume that moments exist independently, and therefore commit themselves to the substantivalist view.
We will now consider an argument against endurantists which employs the notion of change of parts. The argument is due to Peter van Inwagen, with some corrections added by Mark Heller. Suppose that a person X underwent an amputation of his left hand. Let t1 denote a moment before, and t2 after the amputation. Let us also denote by ‘X-minus’ the whole consisting of X’s body without the left hand (regardless of whether the hand is attached to it or not). The endurantist should accept the following identity statements:
(1) X at t1 = X at t2
(2) X-minus at t1 = X-minus at t2
(3) X-minus at t2 = X at t2
But from these three premises we can derive, using the assumption of the transitivity of identity, that
(4) X at t1 = X-minus at t1
This conclusion is clearly unacceptable. My body is not identical at this moment with my body minus my left hand. Now we will have to look closely at the justification of all the premises (1)-(3), to see which one should be rejected. Premise (1) follows from endurantism and the assumption that an object can lose its part without losing its identity. Premise (2) is a simple consequence of endurantism. Premise (3) is implied by the principle according to which two numerically distinct objects cannot occupy the same spatial region at the same time. Now it should be clear that rejecting endurantism and accepting perdurantism solves the problem. If we agree that the expressions “X (X-minus) at t1 (t2)” refer to temporal slices of appropriate four-dimensional objects, then premises (1) and (2) are evidently false, although (3) is unquestionably true. Another possible interpretation of (1)-(3) under perdurantism is that actually these identities are between appropriate four-dimensional objects, identified by their three-dimensional slices. In that case (1) and (2) are true, but (3) becomes false (two distinct four-dimensional objects can nevertheless share their three-dimensional slices).
But the endurantist has some viable strategies of defence. Firstly, he can claim that a thing cannot lose any of its parts without losing its numerical identity. But this is a highly unintuitive supposition, and if it’s true, then with each passing second our bodies are turned into new things, because they are constantly losing old parts and acquiring new ones. Secondly, the existence of X-minus can be called into question. For instance Van Inwagen rejects the doctrine which he calls “the doctrine of arbitrary and undetached parts”. X-minus before the amputation is not a separate, autonomous object, but an undetached part of X, and its existence is questionable. Thirdly, the assumption that two distinct things cannot occupy the same space at the same time can be rejected. It is argued that a sculpture, for instance “The Thinker” by Rodin, is numerically distinct from the lump of material it is made of (bronze) and yet throughout some period of time the two things occupy the same space. Finally, some authors question the transitivity of identity. According to Peter T. Geach, identity is a relative and contextual notion. We can say that X at t1 is the same person as X at t2, and that X-minus at t1 is the same body as X-minus at t2, but from this it doesn’t follow that X at t1 is the same body as X-minus at t1, nor that X at t1 is the same person as X-minus at t2.
One problem for perdurantism is that it does not offer clear criterions of how to distinguish four-dimensional wholes which are genuine things from arbitrary regions filled with matter. A given temporary slice of a four-dimensional object has an infinite numbers of future continuations. Which one is selected as the right one, and why? Yet another difficulty was noticed by van Inwagen. Four dimensional objects are often presented as collections of stages (slices). But a collection of objects possesses its elements necessarily. From this it follows that Napoleon could not have different stages from the ones he really had (for instance, he could not have been born earlier or later).
Endurantism is typically associated with presentism, and perdurantism with eternalism. But other combinations are also possible. Perdurantism can logically coexist with the theory of the growing (shrinking) universe. In such a case things would be four-dimensional wholes that grow or shrink as time passes. It is also possible to have both endurantism and eternalism. It seems that the only combination which is logically impossible is that of perdurantism and presentism (although some authors disagree with that). Perdurantism assumes that things have different temporal parts, so it is essential to admit that moments other than the present one exist. Also, presentism implies that the universe is three-dimensional (as time is not a dimension, because it is reduced to a point). But perdurantism identifies things with four-dimensional objects, and four-dimensional objects cannot exist in a three-dimensional world.
Readings:
E.J. Lowe, Chapter 3 "Qualitative change and the doctrine of temporal parts", pp. 41-58, A Survey of Metaphysics.
M.J. Loux, Chapter 8 "Concrete particulars II: persistence through time", pp. 230-256, Metaphysics: A Contemporary Introduction.
Let us look more closely at both views. Endurantism seems to be in agreement with our pre-philosophical intuitions regarding persistence. When I look at a table in front of me, I believe that no part of it is missing from my view. Things have only spatial parts and spatial dimensions, but no temporal ones. According to endurantists, the expressions “Napoleon at the time of the battle of Austerlitz” and “Napoleon at the time of the battle of Waterloo” refer to one and the same individual: Napoleon. Hence diachronic identity reduces to numerical identity. Things are three-dimensional objects taking up various spatial regions at different times.
Perdurantism, on the other hand, claims that things are four-dimensional objects taking up regions of space-time. The expressions “Napoleon at the time of the battle of Austerlitz” and “Napoleon at the time of the battle of Waterloo” refer to numerically different objects: temporal parts of the four-dimensional entity that we call “Napoleon” and whose temporal dimension stretches from the moment of Napoleon’s birth to his death. Consequently, the relation of diachronic identity is not defined as numerical identity, but instead can be explicated as the relation that holds between any two temporal parts of the same thing: x is diachronically identical with y iff x and y are different temporal parts of individual z. (An alternative interpretation of diachronic identity under perdurantism is that it is reducible to numerical identity after all, when we say that “Napoleon at t1 is identical with Napoleon at t2” means “The four-dimensional object whose temporal part at t1 is Napoleon at t1 is numerically the same as the four-dimensional object whose temporal part at t2 is Napoleon at t2”.) Temporal parts of an object can be divided into stages and slices. A stage of a thing x is a part of x that occupies a non-zero interval (it has a non-zero temporal “width”), whereas a slice of x is a part of x taken in a zero-length moment of time, and thus it has no temporal dimension. Slices are three-dimensional, and they represent what we would usually refer to as objects of our perception. It is noteworthy that the way things persist according to the perdurantist is analogous to the temporal existence of events. An event, such as the battle of Waterloo, is never fully present in an interval that is shorter than its entire duration.
The main motivation for perdurantism comes from the problem of change which threatens the endurantist approach. Things change their properties in time: for some moments t1 and t2 and a property P it is the case that x has P at t1 and x does not have P at t2. But according to endurantism x at t1 is numerically identical with x at t2. If we apply Leibniz’s law, which states that if x = y and Px, then Py, then we have to conclude that an object has P and doesn’t have P, which is obviously a contradiction. Perdurantism avoids this problem by assuming that the properties P and not-P are attributed to numerically different individuals: different temporal slices. But the endurantist is not without options with respect to the problem of property change. One solution is to relativise properties to time. Let us consider a poker which is cold at t1 and hot at t2. We may say that the poker possesses the property of being-cold-at-t1 and being-hot-at-t2, and these properties are not mutually exclusive as long as t1 is different from t2. But an objection can be raised that this approach treats properties as if they were relations between objects and moments, and consequently no property can be intrinsic. And, besides, isn’t it legitimate to speak about properties simpliciter, without any temporal relativisation? Another solution, available to the endurantist, is to relativise the relation of possession between the object and its properties. Objects don’t just possess properties, but they always possess them at certain moments. This position is known as adverbialism, as it amounts to the adverbial modification of the verb “be” (the poker is-at-t1 cold and the poker is-at-t2 hot). One consequence of this approach is that there is no single relation of exemplification between particulars and universals, but an infinite (even uncountable) number of different relations of exemplification. Finally, let us notice that all the above endurantist solutions seem to assume that moments exist independently, and therefore commit themselves to the substantivalist view.
We will now consider an argument against endurantists which employs the notion of change of parts. The argument is due to Peter van Inwagen, with some corrections added by Mark Heller. Suppose that a person X underwent an amputation of his left hand. Let t1 denote a moment before, and t2 after the amputation. Let us also denote by ‘X-minus’ the whole consisting of X’s body without the left hand (regardless of whether the hand is attached to it or not). The endurantist should accept the following identity statements:
(1) X at t1 = X at t2
(2) X-minus at t1 = X-minus at t2
(3) X-minus at t2 = X at t2
But from these three premises we can derive, using the assumption of the transitivity of identity, that
(4) X at t1 = X-minus at t1
This conclusion is clearly unacceptable. My body is not identical at this moment with my body minus my left hand. Now we will have to look closely at the justification of all the premises (1)-(3), to see which one should be rejected. Premise (1) follows from endurantism and the assumption that an object can lose its part without losing its identity. Premise (2) is a simple consequence of endurantism. Premise (3) is implied by the principle according to which two numerically distinct objects cannot occupy the same spatial region at the same time. Now it should be clear that rejecting endurantism and accepting perdurantism solves the problem. If we agree that the expressions “X (X-minus) at t1 (t2)” refer to temporal slices of appropriate four-dimensional objects, then premises (1) and (2) are evidently false, although (3) is unquestionably true. Another possible interpretation of (1)-(3) under perdurantism is that actually these identities are between appropriate four-dimensional objects, identified by their three-dimensional slices. In that case (1) and (2) are true, but (3) becomes false (two distinct four-dimensional objects can nevertheless share their three-dimensional slices).
But the endurantist has some viable strategies of defence. Firstly, he can claim that a thing cannot lose any of its parts without losing its numerical identity. But this is a highly unintuitive supposition, and if it’s true, then with each passing second our bodies are turned into new things, because they are constantly losing old parts and acquiring new ones. Secondly, the existence of X-minus can be called into question. For instance Van Inwagen rejects the doctrine which he calls “the doctrine of arbitrary and undetached parts”. X-minus before the amputation is not a separate, autonomous object, but an undetached part of X, and its existence is questionable. Thirdly, the assumption that two distinct things cannot occupy the same space at the same time can be rejected. It is argued that a sculpture, for instance “The Thinker” by Rodin, is numerically distinct from the lump of material it is made of (bronze) and yet throughout some period of time the two things occupy the same space. Finally, some authors question the transitivity of identity. According to Peter T. Geach, identity is a relative and contextual notion. We can say that X at t1 is the same person as X at t2, and that X-minus at t1 is the same body as X-minus at t2, but from this it doesn’t follow that X at t1 is the same body as X-minus at t1, nor that X at t1 is the same person as X-minus at t2.
One problem for perdurantism is that it does not offer clear criterions of how to distinguish four-dimensional wholes which are genuine things from arbitrary regions filled with matter. A given temporary slice of a four-dimensional object has an infinite numbers of future continuations. Which one is selected as the right one, and why? Yet another difficulty was noticed by van Inwagen. Four dimensional objects are often presented as collections of stages (slices). But a collection of objects possesses its elements necessarily. From this it follows that Napoleon could not have different stages from the ones he really had (for instance, he could not have been born earlier or later).
Endurantism is typically associated with presentism, and perdurantism with eternalism. But other combinations are also possible. Perdurantism can logically coexist with the theory of the growing (shrinking) universe. In such a case things would be four-dimensional wholes that grow or shrink as time passes. It is also possible to have both endurantism and eternalism. It seems that the only combination which is logically impossible is that of perdurantism and presentism (although some authors disagree with that). Perdurantism assumes that things have different temporal parts, so it is essential to admit that moments other than the present one exist. Also, presentism implies that the universe is three-dimensional (as time is not a dimension, because it is reduced to a point). But perdurantism identifies things with four-dimensional objects, and four-dimensional objects cannot exist in a three-dimensional world.
Readings:
E.J. Lowe, Chapter 3 "Qualitative change and the doctrine of temporal parts", pp. 41-58, A Survey of Metaphysics.
M.J. Loux, Chapter 8 "Concrete particulars II: persistence through time", pp. 230-256, Metaphysics: A Contemporary Introduction.
Saturday, March 27, 2010
Time's arrow
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.
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.
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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