Introduction and Definitions

One big question in the philosophy of science, at least among philosophers (I am not sure that scientists are so worried about it), concerns the nature of time. In particular, are the past and future equally real as the present, or is only the present real? And what do we even mean by "real" in this context?

I'm going to pick Edward Feser as a conversation partner here, and in particular his work Aristotle's revenge, chapter 4. Professor Feser is one of the few philosophers I have a great deal of respect for. While I had the broad details of my philosophy worked out before I encountered his work, I am still indebted to him. I found his work a help when struggling over the finer details of the philosophy, a challenge to make me think deeper about things even when we disagree, and someone I could reliably turn to if I got stuck on a particular argument, or needed clarification on some aspect of scholastic philosophy that my education in physics didn't prepare me for. That doesn't mean that I agree with him on every philosophical detail. Just most of them.

However, Professor Feser does have a notable weakness. Too often he has been found wanting in his understanding of contemporary physics. His examples tend to be drawn from more common experience, which are usually good, but sometimes can be misleading. He makes mistakes (although, to his credit, he does learn from them). That's not surprising: he is, after all, trained in philosophy, not physics, and he does do a better job with physics than almost all physicists would do when writing about philosophy. As a physicist who dabbles in the philosophy of science, I make no claims that my depth or breadth of knowledge of philosophy is equal to the professionals; likewise we cannot expect a philosopher who dabbles in the philosophy of nature to have the depth of knowledge of a professional physicists (unless, of course, he trained in both of them). This specialisation is deeply unfortunate, but, stuck within the limits of our brains and lifespan, there isn't much to be done about it.

But this weakness is particularly disadvantageous for an Aristotelian. The main argument against Aristotle's philosophy used today is that it has been disproved by contemporary physics. Now I don't believe that to be correct. Indeed, I believe that it is the opposite of the truth, with key Aristotelian concepts such as act and potency and form and finality captured at least partially by modern physicists, albeit called by different names. But my goal is not to be faithful to Aristotle, but to find a workable philosophy of quantum physics. I firmly believe that Aristotle's philosophy is a better starting point for that than the various enlightenment and post enlightenment philosophers.

I equally don't believe that we should end with Aristotle. I feel obliged to modify or replace aspects of the Aristotle/Aquinas framework where it is clearly contradicted by contemporary science. I am not (here) saying that the Aristotelians were either right or wrong on the subject of time, merely that we need to keep an open mind until the evidence is in. Just because the Stagarite proposed something doesn't mean that it was true. Nor does it mean that it was false. It means that we have to consider it, and test it.

I had better start by defining the scope of the debate. I will quote directly from Professor Feser:

Then we have the distinction between the A-theory (or tensed theory) and the B-theory (or tenseless theory) of time. According to the A-theory, the A- series [events ordered according to an absolute notion of time from the past to the present to the future] is both real and more fundamental to the nature of time than the B-series [events ordered according to a relative notion of time, i.e. earlier than and later than]. There is, on this view, an objective fact of the matter about whether some event is now or present, and an event's place in time is to be understood in terms of its relationship to what is now. The A-theory comes in three main varieties. The classical form taken by the A-theory is presentism, according to which only the present is real, with past events no longer existing and future events not yet existing. A second kind of A theory is known as the growing block theory, according to which the present is real and all events that were present and are now past also in some way still exist, but future events do not yet exist. The third kind of A- theory is the moving spotlight theory, which holds that past, present and future events all in some way exist, but only present events fall under the spotlight of the now, which moves along, and successively illuminates, this series of events. The B- theory, on the other hand, takes the B-series to be more fundamental to the nature of time, and the A series to be either reducible to the B-series or eliminable altogether. The B- theory takes an eternalist view according to which all events - whether past, present, or future - are all equally real, as different parts of a single unchanging Parmedian "block". Temporal passage and the now are, on this view, illusionary. An event is now or present only relative to our consciousness of it, and not as a matter of objective fact.

So the question I want to address in this post is which of these four views of time is correct, or is it something else? I should say that my thoughts here are still rough and undeveloped: they are a physicist's reaction to a philosophical problem, and as such there might well be subtleties that I have missed. Nonetheless, it is a subject which does overlap with physics, and I also fear that there are subtleties in the physics which philosophers might have missed. Feser follows Aristotle into advocating for presentism. An advocate of the A theory believes that a particular time can be objectively pointed to as distinct from all the others. An advocate of the B theory believes otherwise.

I find Professor Feser's use of the word illusionary here to be somewhat too restrictive. Things may be true absolutely (2+2=4), false absolutely (illusionary), or true and false conditionally (if I understood ancient Egyptian, then I would be able to read the writing on the Berlin Pedestal). I don't understand ancient Egyptian, but the statement that if I did I would be able to read hieroglyphics is nonetheless correct. Another conditional truth is That bus appears red. It might be red in its rest frame, but whether it appears red to me depends on my relative velocity with respect to the bus and whether the light is red shifted, blue shifted or neither. The statement should read That bus appears red to all those in the same inertial frame as it: the statement and the conditions under which it is true. Talking about things being either objective or illusionary is thus too limiting. To avoid a false dichotomy, I will define the A theory as any view where there is an objective present time, and the B theory as any view where one time being present is not objective, whether that entails that the concept is illusionary, or only subjective (a conditional truth), or something else.

Note also that in this debate, neither side makes any claim that spatial reality is localised, i.e. that only those things that are here in this room with me is objectively real, and everything else is an illusion. So I will assume that the definition of real encapsulates at least a spatial slice of the universe, and isn't localised to just one point.

Another point I need to mention before I begin is that there is one aspect of Aristotle's thought concerning space and time which is clearly wrong. The biggest division between Aristotle and Plato's other disciples was that Plato advocated the Pythagorean view that geometry was essential to understanding physics, and space and time in particular. Aristotle disagreed, and believed that causality was the key to physics. His physics (save for his discussion of the heavens) makes no mention of geometry. Of course, Aristotle's vision of physics was initially far more successful than Plato's. But Plato and Pythagoras have proved to be the victor here. Their weakness was that their mathematics was not sufficiently developed. They didn't know how to apply geometry to physics. That was worked out by the Merton Calculators of the fourteenth century, and further developed by scholars in Paris. Galileo took their ideas and ran with it, and physics hasn't looked back since. Every success of modern physics can be traced back to this innovation; unless the assumption that physical space time can be mapped to a geometrical space is true, the whole scientific project collapses. So in this respect, Aristotle was wrong. But does this mean that he was also wrong about presentism?

I'll start by discussing the physics and philosophy of space and time as I understand it. I'll then turn back to the definition of the four perspectives outlined above, and offer my criticisms of each of them. I'll then discuss Professor Feser's arguments against the B- theory, and finally his arguments in favour of the A-theory.

Space and time in modern physics

Two common arguments

Before proceeding with my thoughts, I ought to comment on two points which are commonly made. The first of these states that the microscopic laws of physics are time-reversible, from which it is concluded that things which we perceive as flowing from the past to the future could just as easily be seen as flowing from the future to the past, denying the direction in time. This is true for Newton's laws, special relativity, general relativity, quantum mechanics (with a caveat relating to the measurement problem), quantum electrodynamics and quantum chromodynamics. That's a pretty impressive catalogue of theories. The statement is that there is nothing in the microscopic theory of modern physics which displays evidence of the flow of time or the priority of the present.

The philosopher would, of course, respond that we should not assume that physics is complete (not just in the sense that we don't yet know the final unified theory): the process of mathematical abstraction could cut out features of reality that nonetheless play a key role in the philosophy of time. But my concern for now is over whether the statement is correct even within the context of physics. And I would argue that it isn't. Firstly, time-reversal symmetry isn't quite exact. In the context of the standard model, time-reversal symmetry is broken by the theory relating the eigenstates of the weak nuclear force and those of the strong nuclear force. It is often conjectured that there must be further breaking of time-reversibility in physics we don't yet know to account for the dominance of matter over anti-matter in the universe (time reversal symmetry is related in the equations to the symmetry connecting matter with anti-matter; if one is violated, then so is the other). Now, these effects are relatively small, and not likely to affect our conscious experience. However, there is another reason why we might think of the flow of time is recorded in physical theory, which I will come to later.

The second point is that there is one area of physics which is commonly cited as ensuring the direction of time. Thermodynamics, which is the study of the large scale properties, such as temperature and pressure, of ensembles of particles. In particular, the second law of thermodynamics, states (in an imprecise but common formulation) that entropy always increases over time. In other words, the universe is moving from a state which is very much out of statistical equilibrium to one where it will be in statistical equilibrium. It is here, it is said, that the flow of time enters physics.

I'm personally rather suspicious of this argument. If the passage of time is merely a matter of entropy increasing, then that would imply that it stops when a system reaches statistical equilibrium. This surely cannot be the case. While statistical equilibrium implies that observables remain static at the macroscopic level, at the microscopic level there is still dynamism and change, which implies a passage of time. And if the system reached an equilibrium state from a non-equilibrium state, then we can hardly change from time having a direction to becoming directionless. There must therefore, I think, be more to the question of the passage of time than just this.

Mappings and geometry

There is something out there called physical space time. To understand it (as with everything else), we need to make abstractions of that part of its nature that is open to abstraction. We perform a mapping to a geometrical space, which is an abstract object. So every point in physical space time corresponds to a point on the geometrical space, and vice versa. We have a great deal of freedom in choosing the geometrical abstraction, but not total freedom (we need the topology, metrical structure and so on of the geometry to match their equivalents in physical space time). We then perform a second mapping from the geometrical space time to a coordinate space, i.e. we assign a number to every point in the geometrical space. Again, this is a one to one mapping, and there is even more (but not total) freedom in how we do this. We can, for example, choose where to put the origin of the coordinate system, or what corresponds to one unit of distance, or how to orientate the axes. This freedom should not affect our predictions concerning physical space time. Wherever we place the origin of our coordinate system, we should make equivalent predictions. There are not many possible theories which pass this criteria, and one of the driving forces behind contemporary physics is that we are restricted to just those theories. Indeed, this is one of the key principles behind the theory of relativity.

The idea that physical space time could be mapped to a geometrical space found fruit in the fourteenth century with the Merton calculators, who first proposed a mapping between the three dimensions of space and a three dimensional Euclidean geometry (albeit that I am being anachronistic in the language I use to describe their work). Descartes then proposed the second mapping with his Cartesian geometry. The final step was the geometrisation of time, which I believe was first proposed by Isaac Barrow, Isaac Newton's mentor and predecessor in the Lucasian chair at Cambridge. This is where the first notion of space-time emerges. We now have space and time represented by an E³⊗ E¹ Euclidean geometry, with a corresponding coordinate representation. The symbol ⊗ means that the two sides act independently of each other; every transformation which involves both space and time coordinates can be factorised into a transformation acting only on the spatial coordinates and a transformation acting solely on the temporal coordinate.

This geometry held sway until the time of Einstein. In the nineteenth century, it was realised that Euclidean geometry was not the only possibility. The first alternative was known as hyperbolic geometry (first proposed by Gauss and worked out in detail by Bolyai and Lobachevsky). Later on others were developed as well, and Riemann's geometry is of particular interest to the physicist. Special relativity, as Minkowski recognised, can in part be formulated as the statement that the geometry that represents space-time is a four dimensional hyperbolic space (denoted SO(3,1)) rather than the Euclidean space coupled to an Euclidean time accepted by Newton. General relativity is the realisation that the space time is best represented by a Reimann geometry with a hyperbolic measure (don't worry too much about this language; I'll explain the important aspects of it as I go along). Special relativity is a bit simpler, and contains all I need for my discussion of the philosophy of time, so I will concentrate my discussion on this. The more complex Reimann geometry only gives the points I want to make even more force.

The most important difference (for my purposes here) between E³⊗ E¹ and SO(3,1) is that the older Euclidean geometry, time is entirely separable from space. There can be no transformation mixing time and space coordinates. In the SO(3,1) of special relativity, time and space are mixed together as a single entity. There are transformations of the coordinates which cannot be factorised into independent transformations over space alone and time alone. One cannot take time out of that vision of space time, and treat it independently of space, whereas one could with the older Euclidean theory.

As stated, there is no unique way of constructing a coordinate system. We are free to choose the origin (zero point), orientation of the axes, and separation of the ticks as we please. The principle of relativity states that each of these coordinate systems are equivalent to each other in the sense that they encode the same physics (obviously, once we introduce general relativity, the equations of motion might look different, so you might have a gravitational force in one frame and not in another; general relativity teaches us how to convert from one coordinate system to another while still describing the same events in the real world). So every physical object will appear at some point in each coordinate system; every relation between two objects has something corresponding to it in the coordinate system. The numbers used to describe it differ from one coordinate system to another, but the existence of the objects and their relationships is common to all of them.

In particular, the origin of a coordinate system need not be in the same point in space at every moment of time. It can move with some velocity compared to another coordinate system, so at one time the two coordinate systems have their origins at the same place, and at a later time the spatial origin of the two coordinate systems will be in different places. Special relativity is particularly concerned with how to translate between different coordinate systems of this sort, where the two coordinate systems are moving at a constant relative velocity. The rules to convert from one to another are known as the Lorentz transformations. General relativity expands this framework to allow for coordinate systems which are (for example) accelerating relative to each other.

The Lorentz transformations mix the spatial and time coordinates. Time does not pass in the same way for every observer, but it depends on how they are moving through space. A classic example of this is a radioactive particle. We will measure this to have a certain half life when it is at rest (relative to ourselves). However, if it is moving at a relativistic speed (relative to ourselves), we will measure it to have a much longer half-life. It is as though the particle has an internal clock, and there is a 50% chance that it will spontaneously decay with each tick of that clock. However, the clock runs slower when the particle is in motion, just as predicted by the Lorentz transformations.

This geometry provides a background on which physics takes place. It is not all there is to physics. Additionally, we require an additional set of abstract objects which represent the matter (or perhaps our knowledge of material states). This second abstraction builds upon the representation of space time, since we say that there is a certain likelihood (or amplitude) that a particular particle exists at a particular place, and the representation of matter thus depends on the coordinate system being used. When we perform a coordinate transformation, we must also at the same time transform the details of how we represent a material object. So the representation of matter is linked to the representation of space time, but contains additional information.

We can also introduce natural coordinate systems for each observer, i.e. where the observer is at the origin of their coordinate system (and the different observers use a common standard to define distances and durations in time; for example via the frequency of light from a particular atom and by fixing the speed of light). So two observers A and B each have their own coordinate system with themselves at the origin.

The claim is that when we specify "B is at location L1", we must go through this process of mapping to geometry and then a coordinate system. Often this is done unconsciously, and we are not precise about all the details, but it is still done. Even those notions of space and time which don't quantify things precisely ("It's over there somewhere") still bring in an observer-dependent layer of abstraction, but just express it in terms of words or something else other than numbers. The problem with these notions is that they are less precise; to make them more definite we need to use the full numerical representation.

Every statement of location or time needs to specify which coordinate system it is referring to; otherwise the statement is meaningless, it is just a collection of numbers with no context. So we ought to say "L1 and L2 are identical in B's coordinate system and not identical in A's coordinate system," and there is clearly no contradiction. The error is failing to recognise that every statement we make about objects in space and time has this coordinate system dependence and therefore we need to state the coordinate system alongside the description of where and when it is.

The statement of relativity is that coordinate systems must be relative to each other, and that there is no absolute coordinate system. This I take to be so deeply enshrined in physics that I don't think that anybody can rationally challenge it. It also makes philosophical sense: there is no right way in which to construct the abstraction. We have to pick one, but ultimately we are free to choose. "Absolute space" is interpreted to mean that there is some preferred frame of reference, and with regards to describing the locations and times of physical objects this is not the case.

Symmetry

Those of you who have read my work before will be amazed that I have made it this far on a post about physics without mentioning the word symmetry. It is time that I rectified that. First of all, I need to ask what is a symmetry, and for that I need to first clarify the nature of a mapping.

First of all we need the concept of a set, which is a collection of things. These things could either be abstract concepts (such as numbers), or physical objects (such as cows in the field). A mapping relates the members of two different sets (for example, we could give each cow a number). Mappings can exist between real objects and abstract objects, or one abstract space and another (I think to relate one set of physical objects with another, we need to go through an abstraction as an intermediary). One can also map from a set of one particular things to another representation of the same set. For example a mathematical function might map from the set of all real numbers to another set of all real numbers. Mappings might be many to one (surjective), one to many (injective) or one to one (bijective). In physics, we are primarily interested in bijective mappings, since we want to be able to extract data from the real world, perform calculations, and then apply the results of those calculations back to the real world. This requires a one to one equivalence between the real world object and our representation of it. However, many to one mappings also play a role. For example, we can have a fundamental theory (such as of quarks and gluons) and a higher level theory (such as of protons and neutrons). Protons and neutrons have less degrees of freedom that in the fundamental theory, so we transform the coordinates and average over those we don't need. We go from a theory with N degrees of freedom to one with n<N degrees of freedom, i.e. a many to one mapping.

So let us suppose that we have a mapping between the members of set A and the members of set B. Let us also suppose that there are some relationships between the members of set A, for example member A₁ is next to A₂ in a sequence. Then, if the mapping to set B is successful, we can identify an analogue to that relationship between the members of set B. This is why getting the geometry of space time right is so important. Professor Feser himself adopts structural realism, where the reason that the abstract representation used by physics is successful is that it maintains the same relationships between the representations of the objects as exists in reality. I quite agree with him that this is crucial. Modern physics starts with a mapping between the points of physical space (and perhaps time) to an abstract geometrical space. There are various relationships between points in physical space, and we want the points in the geometrical space to have the same relationships. But a geometrical space has its own internal rules and relationships between its points, which differ from one geometry to another. Physics will only accurately represent reality if the geometry we choose has the same internal relationships between locations and times as exist in real physical space.

There is a second mapping in physics, which is between the actual beings and substances that occupy our physical universe, and an abstract representation. This second mapping is only partial, in that not everything about the physical substances can be captured in the abstract representation. But physical substances interact with each other, and are associated with particular locations in space and time, and we want these interactions to be mirrored in some way in the abstract representation. and the correspondence to a point in geometrical space retained. In this case, we represent the potentia of substances with the eigenstates of various operators defined in a particular Hilbert space. The movement between one potentia to another (the process of actualisation) in reality is mirrored by a movement in the abstraction between one (abstract) state and another. We need to construct the Hilbert space so that the same transitions are possible as are allowed in reality.

Once we have this picture of a space and various objects existing within it, we can think of a transformation. A transformation is a mapping of a system onto itself. Moving around the various objects represents a transformation. We need one further ingredient to have a symmetry, and that is to say that certain configurations of these objects are equivalent. A symmetry is then a transformation that moves the objects from one configuration to an equivalent one.

For example, consider the following collection of points:

We can transform these points in various ways. For example, we can move the point at the top right a little bit to the left. This will change the picture. But there are also symmetries. For example, we can swap two points, or rotate the whole picture through 90 degrees. And it is obvious that we can rotate things in reality, as well as in the representation. When we have a mapping, it is essential that the abstract structure preserves the symmetries of the thing they represent. For example, the mapping between the blue points and the red points below preserves the symmetries.

But it is also possible to have partial mappings where only some of the points are carried over into the representation. Consider the below, where the left hand side represents reality and the right hand side our physical representation.

In this case, reality has more symmetry than the representation. There are more ways we can transform the points which leave the whole picture the same. When we apply this to physics, it is actually OK. Physics would only provide a partial representation of reality, but we can still use it to understand the interactions which respect the symmetries which are preserved. We can use mean field methods to incorporate the effects of the missing interactions between the things which are mapped to the representation and those which aren't. The representation won't tell us everything about reality, but what it does tell us will still accurately inform us about some of it. Remember that, in physics, we map from reality to a representation, perform calculations in the representation, and then map back to reality so we can learn something about the real world, and how it will evolve. We can lose symmetries when we decrease the number of degrees of freedom.

But what about the reverse situation? What if there are symmetries in the physical representation which are absent in reality? This is clearly troubling. There are two different ways in which this can happen. The first (and not relevant here) is that some configurations are seen as equivalent in the representation but not in reality: some transformations will then be symmetries of the physics, but not symmetries in reality. This would suggest, however, that the physical representation was inaccurate, and we would have to find a better one. The second possibility is analogous to what we have above, where there are more degrees of freedom in the representation than there are in reality. This is also troubling, because it would imply that there are additional relationships in the representation than exist in reality (I am supposing that the representation has the smallest degrees of freedom possible that allows it to accurately predict the results of experiment). These objects would influence the things in the physical representation which correspond to the real objects, but there is nothing in reality corresponding to them. It is like having causes which exist in the physical representation, but not in the real world. That is, I hope we all agree, unacceptable. Thirdly, we can have transformations between different representations of reality. Here we are not adding new data, but expressing the same data in a different (but physically equivalent way). This requires that each representation is in a one to one mapping with that subset of reality it captures, and most importantly that all the different representations we are transforming between capture exactly the same aspects of reality (otherwise they could not be physically equivalent).

So we can reduce the degree of symmetry when moving from reality to an abstract representation, or from one abstraction to the next level of abstraction, but we can't increase it.

Now identifying a symmetry relies on two principles, the transformation or relationship between different points (or potentia in things), and the statement that two physical configurations are equivalent. As a structural realist, Professor Feser would agree that there must be something corresponding to the transformation in physical reality if the physical description is accurate. Equally, if two configurations are equivalent in reality, if it is to be in any way accurate, they must also be so regarded in the abstracted representation (and if they are not equivalent, they must be distinct in the representation). Thus if we need to construct the representation with a particular symmetry in order to make it line up with reality, then that symmetry must also be present in reality. So if you give the idea of structural realism to a contemporary physicist, part of what he will do to break it down into something he can work with is to demand that both reality and the physical abstraction should respect the same symmetries.

What we are doing, as physicists, is constructing a physical representation which can be compared through experiment against reality. The hope is that one day we will have a theory that correctly predicts the probabilities for various different outcomes, as well as the various properties of matter. And we are, I believe, getting close. We can work out what the symmetries of the theory are. That's easy, because it is an abstraction, and we can pick it apart in detail. We can then conclude that reality has the same symmetries. It might have additional symmetries as well, but it must have those identified by the theory.

That's the principle, at least. The next question is how does it work out in practice?

A symmetry is a transformation of the degrees of freedom of a system that leads to an equivalent configuration of those degrees of freedom. How do we determine which degrees of freedom are equivalent? The central object in physics is what is called the action. In classical physics, the action is a mathematical function that determines the equations of motion, which describe how the system evolves in time. In quantum physics, the action doesn't determine the evolution of the system (since things aren't deterministic), but allows us to compute the amplitude (or likelihood) for each possible outcome, and from the amplitude we can calculate probabilities, and compare these against measured frequencies. When we discuss symmetries in physics, we mean that the action is unchanged after a particular transformation.

This is where the first alarm bell might start ringing for the philosopher. The action is a mathematical function. It only exists in the representation, but not in reality. Are symmetries of the action really what we ought to be worrying about when we think about symmetries of reality?

In quantum physics, the action is constructed from the integral of the Lagrangian density over all space and time. In practice, the integral over time is a little complicated: the operators are time ordered. We divide the calculation into slices of infinitesimal duration. We first of all fix to a particular coordinate system (which one we choose doesn't matter; we can always transform to any of the others). We start with an initial state (which represents a set giving the amplitude or likelihood that the system is in each possible state or potentia). Then we apply the correct operator to calculate what the amplitude will be at the next moment of time, and then the next moment, and so on, until we reach the time of the final state. Once we have the amplitude of the final state, we can calculate the various probabilities that the system is in any given configuration. Throughout this calculation, we need to ensure that all the operators remain in temporal sequence. With normal mathematical functions, we wouldn't need to take care of these details, but due to the more complicated commutation relations of quantum operators -- the order in which we do things makes a difference -- we have to be more careful.

The Lagrangian density is constructed from what we as physicists call creation and annihilation operators (and which I keep getting told that I ought to call generation and corruption operators when discussing the philosophy of physics). These represent transitions between different states, and the way they are put together in the Lagrangian density thus encodes which transitions are possible and could happen in reality. In other words, the Lagrangian density, and consequently the action, encodes the final causes of nature. The symmetries play a crucial role in the construction of quantum field theory in that they constrain the possible interactions between different types of particle. If the action was not invariant under a symmetry, then this could imply one of two things. Either additional operators are added to the Lagrangian when we apply the transformation. Each operator implies a different possible interaction or decay. So if the symmetry were violated, we might observe particle A decaying into particle B or particle C if we were stationary with respect to it, while someone else driving past us in a car watching the same particle might see it decay into B, C or D. The second thing that might change are coupling constants, or the relative amplitudes for each possible decay. I might observe that an ensemble of particle A decays to B more than it does C, while my friend in the car, watching the same particles undergoing the same decay events, would see more decays to C. Both of these scenarios are clearly absurd, but they are precisely what what would be implied if the action was not invariant under the relevant symmetry related to the transformation that describes the difference between my motion and that of my friend (in this example, Lorentz symmetry). So when a physicist constructs the action so it satisfies various symmetries, all he is really doing is demanding that reality makes sense. Thus although the symmetries and the action might appear to be merely unphysical mathematical marks on a page, violating the mathematical symmetries would have real world effects. Once again, that the action obeys a symmetry is in a one to one correspondence with something seen in the real world. Symmetries of the action are the things we ought to be looking at when thinking about symmetries of reality.

One of the symmetries of the physical action is Lorentz symmetry, the symmetry that underlies special relativity. The symmetry is slightly modified in general relativity, since the metric is more complicated, but there is still a symmetry that plays the same role and leads to the same conclusions. General relativity also demands that the action is unchanged when subject to more symmetries than are present in special relativity, so it makes the conclusions even stronger. The Lorentz symmetry is one which leaves the metric of a hyperbolic SO(3,1) geometry invariant. The metric is what is used to define length. If ds represents a small unit of length, and dx, dy, and dz small displacements in each of the spatial dimensions, then we are all familiar with Pythagoras' theorem, which, in three dimensions, reads,

ds² = dx² + dy² + dz²

This relation, in part, defines an Euclidean geometry. A hyperbolic geometry adds in the time coordinate as well, so the "distance" is defined by

ds² = dx² + dy² + dz² - (cdt)²,

where c represents the speed of light. The Lorentz symmetry group is that which preserves this equation. So if we use a transformation in the group to map the coordinates so dx becomes dx' and so on, we find

ds² = dx² + dy² + dz² - (cdt)² = (dx')² + (dy')² + (dz')² - (cdt')²,

The Lorentz symmetry group reduces to six individual transformations. The first three are the rotations around each of the three spatial axes (which, as we all know, preserve Euclidean length, but leave time untouched). The second three are the Lorentz transformations proper. They are the analogues of rotations in a hyperbolic geometry. These mix the space and time coordinates. Physically, they correspond to changing from moving with one constant velocity to moving with another constant velocity. The Lorentz transformations change the coordinates so that x' depends on both x and t. Just as the rotations allow us to map between points in one dimension and another, so the Lorentz transformations allow us to map between moments in time and points in space. The future in one coordinate system is the present in another. Physics is constructed to ensure that the amplitudes for each possible interactions between particles (the final causes) are correctly represented in each of these coordinate systems.

The metric is expressed in terms of infinitesimal durations. Would the symmetry then just map points next to each other in time, and not require the whole four dimensional space? But if we say that the symmetry only maps the neighbouring points of time means that only those neighbouring points exist, then we ought to conclude the same thing about space. Nobody denies the existence of points far from me when considering a space-space rotation. We should conclude the same thing when considering a rotation between space and time.

I should now be fair and say that there are two different constructions of quantum field theory. The first is the Lagrangian approach, which I have been discussing so far, which is unashamedly four dimensional. Many physicists (including myself) prefer the Lagrangian approach because it makes the underlying symmetries more obvious. The second approach is the Hamiltonian one, which constructs its operators in three dimensions. This might seem to offer the presentist a way out of the problem I am working towards. There is a three dimensional construction of modern physics. Why don't we just use that, and forget about Lagrangians and actions? The problem is that the commutation relations between the operators in the Hamiltonian construction are carefully constructed to satisfy the Lorentz symmetry. We don't need to do this to maintain mathematical consistency. It is far easier, in fact, to construct operators which violate Lorentz symmetry. We have to ask why we have to construct the Hamiltonian with this restriction; and the obvious answer is that Lorentz symmetry corresponds to something in the real world. So even the three dimensional formulation can't escape the symmetries of four dimensional space time.

Equally, one can question how much of the symmetries I have been discussing depends on the use of coordinates. As I described it, the symmetries relate two different coordinate systems. They tell us which coordinate systems are equivalent to each other in a particular geometry. But introducing a coordinate system is a further abstraction that takes us away from reality. So is the symmetry just an artefact of our introduction of the coordinates? No, because the symmetries don't depend on the introduction of coordinates, but on the underlying geometry. An E³⊗ E¹ geometry has different symmetries to SO(3,1). Different quantities are conserved, and different choices of coordinate are equivalent. For example, in special relativity, the quantity I labelled ds² is the same for all observers (who are not accelerating with respect to each other). This quantity defines the path of a light beam in a vacuum across both space and time, and it is measures the square of the distance travelled minus the square of the time taken. Both distance and durations of time (perhaps infinitesimal durations) are features of reality. The statement that this quantity is the same for all non-accelerating observers is thus a statement that has its roots in reality. The symmetry transformations also correspond directly to things we can do in reality: we can rotate things around, or move at different speeds. The combination of this quality being invariant under these transformations implies the symmetry. So it is not just an artefact of our introduction of the coordinate system.

Is time simply another dimension of space?

Just because we are treating space and time as contained within one four dimensional block does not mean that time is just the same sort of thing as space. I want to highlight three differences. Two I have already mentioned. The first is that minus sign in the metric; that is of significant importance. The second is the need for time ordering in the calculation of the amplitude: this is an admission that we cannot ignore the causal sequence. The third is related to energy and momentum. In quantum physics, momentum is a measure of how likely a particle is to move in space, or, on average, how fast it moves. Energy is the corresponding quantity relating to time; it tells us how quickly the particle's internal clock moves (particle states are defined by both an eigenvector representing the state and a complex phase -- which can be represented by a circle. This phase is constantly changing. Energy tells us how many revolutions the particle does in a given duration of time). But while momentum can be either positive or negative -- particles can move forwards and backwards in space -- energy is always positive. This means that particles can only move forwards in time.

The first of these differences is well acknowledged by the philosophers who favour the tenseless theory of time, but the second two, arising from relativistic quantum field theory rather than special relativity alone, are perhaps less well known. Together, they mean that a) there is a direction in time, from past to future, and b) moments of time follow one after another. These differences apply to time but not space. Note that the Lorentz symmetry does not allow us to change the ordering of events in time between one inertial frame and another. Now, one can argue that we don't need physics to tell us this. Of course we don't. We know them from basic experience. One would hope, of course, that the correct physical theory would lead us to the same conclusion (and if it doesn't, that's a sign that the theory is still incomplete), and it does. My point is this. The abstraction used in contemporary physics is both a) overtly four dimensional, and b) acknowledges the direction and succession of temporal events. Much of the discussion that I have read in philosophy seems to assume that these two attributes are contradictory: we have to choose between the A theory which denies a) and accepts b), and the B theory which denies b) and accepts a). But the two properties exist side by side in the physical abstraction without contradiction, which means that they could exist side by side in a philosophical model as well.

I should finally make one further comment on the physics, and this is to bring in modern cosmology. A lot of the discussion of the philosophy of time revolves around whether or not there is a preferred inertial frame, or coordinate system, which is implied by the A theory of time, and which special relativity denies. The standard model of cosmology relies on the big bang, an initial singularity which marks the beginning of space and time. The big bang created particles with a huge amount of energy, and these moved around freely. As they did so, they emitted and absorbed a lot of radiation. It was a perfect black box; there was no restriction on the frequencies of the radiation emitted and absorbed. As the universe expanded and cooled, there was a phase transition into the situation we have today, where the quarks, gluons and electrons are bound together into protons, neutrons and atoms. This means that they are no longer free to absorb every possible frequency of radiation. What happened to all that radiation left over from the initial expansion? It's still out there. We call it the cosmic microwave background, and it is a residue of the big bang. It is argued that the microwave background and initial singularity provide us with a preferred reference frame, which can be used to define one particular coordinate system, and thus objectively refer to the same time everywhere in the universe, as required by the A theory.

There is a sense in which the microwave background does break the symmetries of special relativity. But it is different from the sense to which we apply those symmetries. The physical representation describes the relationships between the different possible states of the system, and how they could interact with each other. As soon as we introduce matter, we break the symmetries: some of the states are actualised or occupied, and others aren't. The microwave background is just a particular example of this phenomena. But I was not discussing actual configurations of matter, but the action, which describes the possible interactions. The action is not affected by the actual configuration of matter; it can be applies to any of them, and each time spits out the right result describing how that configuration will evolve. So the existence of the microwave background, or any other configuration of matter, doesn't undermine the importance of Lorentz symmetry, both in the physical representation and in reality.

So that's the physical picture. Let's think about what it means for the philosophy.

The A and B theories of time in light of the physics

I'll start with presentism. It should be immediately obvious that presentism violates Lorentz symmetry. It requires that only one moment of time, the same moment throughout the universe, is real, and there is a distinction between the past and future on one hand and the present on the other. The geometrical and coordinate representations contain points for all of past present and future. This means that we can map between the present in reality and one time slice in the representation, but there isn't anything to map to the locations in the representation which represent past and future events. This means that the representation would have Lorentz symmetries, mixing time and space, but reality can't. The presentism theory explicitly breaks the Lorentz symmetries. It therefore forbids transformations to coordinate systems which have different notions of time. But those transformations are possible and used in the abstract representation of physics (as stated, they correspond to changes in velocity). Thus, if presentism were true, the representation would have a greater degree of symmetry than reality. As stated above, this is impossible (if the predictions of the abstract representation are to accurately match the real world, and the real world is complete and doesn't have any missing causes).

The growing block picture is only slightly less bad. One could, perhaps, argue that since only things in the past light cone can causally effect present objects, one can get away with only applying Lorentz symmetries between past times and the present. This might be enough to do the necessary work in the action. One excludes all future points from both reality and the representation. However, the growing block model still requires an unambiguous definition of what the "present" means in different locations. In special relativity, What is in the past for one observer is present or future for another. Any plane we draw across space-time is entirely arbitrary, and will cut out some points from reality which play a role in the physical representation. Even if we can use the microwave background to define a preferred frame, this still represents a breaking of the symmetries of SO(3,1) geometry, and thus the growing block can be ruled out.

I have more sympathy for the moving spotlight theory. Here both past and future are real. This means that the prime objection to the other two A-theories is no longer valid. We do have something in reality which can be mapped to those points in space time. We still, of course, have the problem that the spotlight is extended across space but only an instant of time. This is coordinate system dependent, and breaks Lorentz symmetry. However, we could say that the spotlight is a part of reality which is not captured by the physical representation. This is allowed because everything needed by the representation is real in reality; the spotlight doesn't take anything away from reality (as presentism and the growing block theories do) which is present in the representation, but adds something else to it. This still begs the question as to which frame the spotlight is in, but maybe we can use the microwave background to define it. I am still, however, not wholly enthused by the spotlight approach, because it all seems a bit arbitrary and add-hoc to me. It adds additional complexity to reality which I don't think is truly necessary.

I further note that the A theory as a whole regards time as objective and independent of space. This leads at best to something like the E³⊗E¹ geometry required by Newtonian mechanics (the spatial geometry need not be Euclidean, but it must be seperated from time), and at worst (in the case of presentism) to the correct geometry of the universe being just E³, as Galileo and his predecessors supposed. This is discredited; physics built from this geometry of space time doesn't correspond to reality.

So now we turn to the B-theory, as defined by Edward Feser, and again I have reservations. My first reservation is in the definition itself. Let's remind ourselves of that definition.

The B-theory (or tenseless theory) of time. The B- theory takes an eternalist view according to which all events - whether past, present, or future - are all equally real, as different parts of a single unchanging Parmedian "block". Temporal passage and the now are, on this view, illusionary. An event is now or present only relative to our consciousness of it, and not as a matter of objective fact.

As can be surmised from the above, I regard that all events at all times must be equally real. But there are other aspects to this definition, and what is implied about it later, that I disagree with. I particularly dislike use of the word "unchanging" in this context. To discuss change or an absence of change, we need some measure to parametrise that process of change. In our normal lives, that parameter is time, but if Professor Feser is looking at the whole block from the outside, he can't be using time, since time is one of the dimensions of the block. And, of course, this would just be the fallacy of evicovation if we compare the meaning of "unchanging" in this context with the word "unchanging" as used in the context of presentism. For when presentism talks about things changing, it discusses things which change in time. But the four dimensional view also allows for things to change in time. So if we use the same meaning of "change" as change in time in both presentism and the four dimensional view, the four-dimensional view is not unchanging.

Nor is the four-dimensional view Parmedian. A Parmedian philosophy denies the actualisation of potentia, but rather states that the same state remains actual forever. However, the four dimensional view allows for actualisation. At one time, state A is actual and state B exists potentially. At a slightly later time, state B is actual and C is potential. This is the same process as described in presentism; the only change we need to make is instead of saying that something is actual means that it presently exists, we say that things are actual and exist at specified times. Removing presentism need not severely affect the rest of Aristotlean philosophy of science.

Next, the B theory Feser criticises treats time as though it were just like another dimension of space, only slightly different. The philosophers seem to be aware of the difference in the metric, but not the other differences I discuss above.

Finally, the B theory is described as tenseless. I would only partially agree with this. There are two different viewpoints we can study a B theory universe from. The first is the objective, timeless, view from the perspective of eternity, where we view the montage of history not in succession but together as a whole. Here there is no past, present or future, but something eternal equally belongs to all times. This perspective is indeed tenseless.

But we can also think of what the four dimensional view looks like from the perspective of a being in time. The being would be located at a particular location and moment in space time. Thus the concepts of here and now would both be very relevant for such a being. As I described, since the eigenvalues of the time evolution operator are always positive, such a being would only ever see time flowing forwards. That means that the notions of "past" and "future" would be relevant for such a being, and different from the distinction (for example) between left and right. If the being moved right, then he could move back left again, to return to where he was. But he could not revisit the past. Thus the common sense position is fully in line with the four dimensional view of the universe.

There is, however, no objective moving spotlight. The notions of now, past, present and future are not objective, but observer dependent. And again, this is in accord with our common sense. The word present was not the same thing as I started writing this sentence as it will be when I enscribe the final word of it. The notion is thus subjective, not only changing between one individual and another (present for me as I write this is different from present for you as you read it), but even for the same individual at different moments of their life. Again, we find that the four dimensional view implies a perspective in accordance with our common experience.

Thus we are left with a view of time which is neither tensed nor tenseless, but a third option. Like the B theory, it regards all of past, present, and future to be equally real, and that there is no one time which we can objectively call the present. But, like the A theory and unlike the B theory, it does view time as having a direction and moments of time following in succession, and for beings in time (such as ourselves), there is a subjective but perfectly valid notion of past, present, and future. Even though they are subjective, these tensed features are still a real feature of the universe. This position I take to be both fully in accord with both our common experience, and with what modern physics tells us.

Professor Feser's criticisms of the B-theory

So let us look at some of Professor Feser's arguments for the A theory and against the B theory. I will address the arguments detailed in three sections of Aristotle's revenge, sections 4.3.2, 4.3.3, 4.3.4. Being my usual rebellious self, I will address these sections in reverse order.

Arguments against the spatialization of time.

As stated above, I agree with Professor Feser that time shouldn't just be treated as being another spatial dimension. Where I disagree with him is in his belief that this is necessitated by the four-dimensional universe. So I will agree with some of his arguments in this section, but not all of all of them.

1. Different regions of space exist all at once, while different moments of time exist successively. There are two parts of this a) time proceeds in succession, and space does not; b) at each moment of time, one can specify a region of space. The first point I agree with, since my preferred model acknowledges the succession of time. The second point seems to me to be irrelevant, since (in a four dimensional view), after fixing to a given inertial frame (which is necessary to identify regions of space at the same time), one can just as easily specify a point in the x direction, and identify a three dimensional region in y, z and t and study what happens in that region. In a four dimensional view, we can construct volumes in two spatial dimensions and time just as easily as in three spatial dimensions.
2. The spatial dimensions differ profoundly from time. a) You can rotate in space to convert width to height, but not convert a spatial dimension to a temporal one. b) You use a ruler to measure length and a clock to measure time, but not vice versa.

   Both of these statements are false. As mentioned, a Lorentz transformation is the exact analogue of a rotation, mixing time and space coordinates: the differences in the mathematical expression arise solely from that minus sign in the metric. Of course, both Lorentz transformations and rotations leave the metric unchanged, so you will always have a direction you can identify as being temporal (i.e. with a minus sign in the metric), but that will be a mixture of the old space and time coordinates. The differences between rotations and Lorentz are acknowledged by those who advocate the four dimensional view, and don't argue against it.

   Secondly, you can use clocks to measure distance. One can, for example, fire a pulse of laser light, have it bounce off a mirror at the end of the target, time how long it takes to make the round trip, and calculate from that and the known speed of light the length of the object. Coupled with an interferometer, this is, in fact, the most precise and accurate way of computing distances. Indeed, there is a sense in which all distances these days are measured using a clock. We used to define a meter length using a rod of metal stored in Paris as a guide. But that was not precise enough: the length of the rod varies with temperature, and it can degrade in time. These fluctuations are barely noticeable, but for some applications they matter. We define the second (the unit of time) in terms of the frequency of one of the spectral lines of Caesium, and we define the speed of light in the vacuum to be a fixed number of meters per second. The meter is then read off from the distance that light travels in a second. All contemporary rulers are calibrated against this definition of the meter, so all distances are ultimately measured by using a clock.

3. Time has a direction and flow that space lacks. Agreed, but this doesn't deny all four dimensional models of space time, but only some of them.
4. A region of space can be occupied by one thing, then vacated by it, and occupied by another thing. Time is not like that. Again, this objection is just an artefact of three dimensional thinking. One can take two thin metal plates, a meter square, and place them parallel to each other but a short distance apart. Leave them there for about one three hundred millionth of a second before removing them. Let us define the x direction as along the perpendicular to the plates. Each plate occupies a three dimensional region of space and time, with length of a meter along each of the y and z directions, and a length (also measured in meters) of the speed of light times a three hundred millionth of a second in the t direction (which also comes to about a meter). Looking along the x direction, we get a succession: not plate, first plate, not plate, second plate. The only difference between this and the case where we put a meter cube on the table, remove it, and then put a second cube there is that there is a distinct ordering in the example where the objects are seperated in time, which is absent when they are seperated in space. That rejects some B theory models, but not all of them.
5. The geometrical treatment of space time cannot be the wole story. Physical objects occupy space time, while lines, planes and points don't. Physicists have never claimed that the four dimensional representation of space and time is mapped to things. It is a representation of space time itself. Things are represented by a different abstraction, which pictures fundamental particles as excitations of a quantum field, and composite objects of excitations of an effective field theory generated by mapping the fundamental fields to a different basis, and averaging over redundant degrees of freedom. These representations of fields occupy points or regions in the geometrical space time just as the real object occupies regions in the real space time. This objection confuses between the two representations.
6. The treatment of space and time is abstract in physics. We start with a mapping to geometry, i.e. mapping space to lines and planes. Secondly, physicists deal with the abstraction, rather than space and time themselves. There is far more to geometry than just lines and planes. A geometrical surface, like physical space and time, is a collection of points with certain rules about how those points relate to each other. (Of course, Euclid started defining his geometry from a set of axioms involving lines and circles, but we have moved on since then, and now think of geometry in different ways.) Now it is certainly true that physicists make an abstraction of space and time, first to a geometrical space and then a coordinate space. And we make further abstractions into a mathematical representation when incorporating matter. But if this abstraction is to be used to make successful predictions, it needs to mirror certain aspects of reality. This includes maintaining the structure of the relationships between different points in space time. We know what the abstraction of space-time looks like. We know the way that space and time are related in it. Knowledge of the real space and time is a bit more opaque, but we know that space and time need to have the same sort of relations as are present in the abstraction. Real space and time have to be the sort of things that could map to the physicist's abstraction. It is not true that the abstraction tells us nothing about real space time. As I have argued, I believe that the abstraction satisfies Lorentz symmetry implies that there has to be an analogue to this symmetry in real space time, which means that real space time has to be four dimensional. There are many ways in which real space time differ from the abstraction, and some aspects of real space time can't be deduced from the abstraction, but that doesn't mean that a three dimensional reality is not inconsistent with it.
7. The four dimensional view of space time collapses the distinction between time and eternity. On the contrary, it maintains it. Time passes for things within the universe, eternity is for those outside it. Temporal beings have a subjective notion of the present, as from inside a B theory universe. The eternal being sees all of reality together, as one would expect from something outside a B theory universe.

Professor Feser on Relativity

In section 4.3.3.2 of his book, Edward Feser discusses about whether the A theory of time can be made consistent with the special theory of relativity. He presents four approaches to reconcile presentism with relativity, and then mentions that the growing block and moving spotlight theory might offer others. [Note that I have more than four bullet points below; I have split some of his approaches into different variations]. However, since he subsequently rejects both the growing block and spotlight theory, he cannot consistently use them to overthrow the objections to relativity if his preferred model of presentism fails, and then later reject these ideas in favour of presentism -- even though presentism would have been rejected as inconsistent with relativity. So I will concentrate on his arguments that presentism can be reconciled with relativity.

1. The anti-realist approach. The physics of relavity doesn't tell us anything about reality. But neither Professor Feser nor I consider anti-realism feasible. If relativity was disconnected from reality, it wouldn't be necessary to make accurate predictions.
2. Relativity only offers epistimilogical signifance, and tells us that we cannot know whether events are simultaneous, while there might be an objective sense where they are. There is, however, as discussed, more implied by relativity than just the impossibility to identify to events at different locations as being simultaneous.
3. Relativity is only inconsistent with presentism if space and time exist as a substance, but this isn't true. Of course, space and time aren't substances. Physicists use quantum fields to represent substances, and place them in the representation of space and time. But the Lorentz transformations act on the fields; in other words on the representation of substances. So this doesn't answer the issue.
4. Einstein and Minkowski got the physics wrong. For example, Lorentz's own construction is empirically identical identical to special relativity, but is formulated in a three dimensional universe. I will admit that it is possible to extract time from space in physics, and proceed considering one time slice at a time. This is done by the Hamiltonian construction of quantum field theory. But, the point is, there is an infinite number of ways in which we could construct such a 3+1 dimensional quantum field theory. The question then becomes why is the one that is correct in practice the only one which allows the universe to have the symmetries of a four dimensional hyperbolic geometry? The obvious answer is that in fact the real space time is the equivalent of a four dimensional hyperbolic geometry. Any alternative explanation would lack any sort of explanatory power. Plus, of course, the four dimensional Lagrangian formulation of quantum field theory also exists, and provides a more realistic starting point to understanding quantum gravity. In the four dimensional approach, Lorentz symmetry is essential to constraining the form of the action. Without it (for example, if there was only a rotational symmetry), we would expect there to be various interactions between particles which we don't observe. The strength of those interactions would be controlled by a coupling constant. The presentist has to assume that those coupling constants, which could in principle be any real number, are zero for no reason. The four dimensionalist can explain why they are zero: they are eliminated by a symmetry requirement. So while the Lorentz approach might work in the context of straight-forward special relativity, the demands of quantum field theory are more stringent, and this makes this approach implausible.
5. Special relativity is not actually correct, but only an approximation to General Relativity, and that an approximation to a quantum theory of gravity. So it might be that the true theory of physics is more favourable to presentism. Special relativity asserts that the action describing the true fundamental theory of physics satisfies a global symmetry linking space and time coordinates (i.e the symmetry is a linear function of space and time). General relativity asserts that it also satisfies the equivalent local symmetry (i.e. the transformations can vary from one location to another). General relativity is not a replacement of special relativity, but an addition to it. It makes things worse for the presentist; it adds more transformations linking space and time coordinates. I personally expect that the quantum theory of gravity will emerge from adding that local symmetry to the action, in the same way that special relativity is incorporated by demanding that the action satisfies Lorentz symmetry. That view is a minority. Of the other possibilities, loop quantum gravity and causal dimensional triangulation are standard four dimensional approaches; the Hartle-Hawking approach is four dimensional but removes the remaining distinctions between space and time, while superstring theory makes things even worse for the presentist by adding seven more dimensions to the block universe. All models of quantum gravity make things worse, not better, for the presentist. Since both classical general relativity and the standard model of particle physics imply Lorentz symmetry, it would be very surprising if the theory of quantum gravity, which combines them, violates it.
6. We can affirm the reality of only part of the manifold, i.e a point and its past light cone. The problem is that I sitting here just outside London have my own past light cone, while Professor Feser on the Pacific Coast of the US has his own past light cone, as indeed does Marty the Martian, and the little furry creature from Alpha Centauri. Which of these past light cones is the one that exists in reality? The answer is all of them, in which case we no longer have presentism, but the growing block theory.
7. A certain space time point is all that exists. Finally, we have an option which is consistent with Lorentz symmetry. But it is hardly palatable, since it would entail saying that the point where I currently am exists, while my friend down the street is forced to live at a non-existent point in space time. As indeed does my friend who lives two years in my current future.
8. Physics does not describe an exhaustive description of physical reality. The arguments for presentism tell us that there must be a privileged slicing of Minkowski space time, even if physics cannot tell us what it is. The Minkowski manifold merely represents the set of locations where events could happen. The presentist accepts the existence of space time points at different times, but denies that they are occupied. But the Lorentz symmetries don't just apply to space and time, but the transformations act on the operators representing substances, and transform them into different states. The fields are spread out over all of space time, and the transformations link those fields at what are different times in different coordinate systems.
9. Special relativity isn't using the word time in the same sense that it is in a presentist metaphysics. To some extent this is correct. The presentist and the four dimensionalist have different interpretations of time, and this leaks into the definitions. However, we have a common experience of time, and both the fourth dimension in special relativity and the presentist vision of time are intended to be representations of that experience.
10. The Einstein-Podolski-Rosen experiment shows that there is a contradiction between special relativity and quantum physics, since it implies non-local action at a distance. When we perform a measurement on one of a pair of entangled particles, the wavefunction of the second one collapses. The EPR paradox does not necessarily imply non-local action. It says that either property realism, where all the properties of a particle are simultaneously defined, or locality (and consequently special relativity) must be violated. It therefore does not rule out locality in itself. Indeed, it is not inconsistent with other types of realism, such as those in which the particle might be in a defined state which has certain properties undefined. The idea that states rather than potentia are more fundamental is very much Aristotlian. I discuss my own interpretation of the EPR experiment here.

So, none of these models hit the mark. None of them address the main argument for four-dimensionalism, or show how the underlying symmetries of the standard model emerge in the context of these models. They only consider classical special relativity, but not quantum special relativity. While some might be effective against some forms of the four-dimensional view of time, they don't address all forms of it.

Feser's arguments for the A-theory

So what of Feser's arguments for the A theory of time?

1. Socrates died is a true statement, and the past tense in that phrase is a key aspect of that truth. It is true that Socrates died; it is not true that Socrates is dying or will die. Tense is thus a real part of the world. This only makes sense that there is a present is a real fact of the world, which only makes sense on some form of A-theory of time.

   Professor Feser offers two counter arguments to this argument. The first is that a statement that Socrates drank hemlock can be translated to a statement that Socrates is drinking hemlock in 399BC, or he is drinking hemlock earlier than this sentence. Feser finds this argument unconvincing because the translation doesn't capture all the meaning of the originals.

   The second argument is that even if the whole meaning of the statement is captured by the translation, the truth statements within it are. Feser responds that this still entails contradictions; for example "Nobody is now uttering a sentence" is inconsistent if the word now only refers to something relative to the utterance of a sentence.

   This argument, however, only responds to those four dimensional theories which deny tense altogether. What I am advocating is something different. I accept that tense is a real part of the four dimensional universe, but argue that it is subjective rather than objective. Thus for everything within the universe (which includes us), there is indeed a present, past and future. But it is still not an objective feature; which moment is the present depends on when you are in time. The statement that Socrates died is thus a conditional truth; it is true in all times after the death of Socrates, and was false at all times before it. This is no different to saying "this man is in front of me" or "he is behind me" is conditional on where you are standing in the queue. Many statements are conditional on one's inertial frame or point of reference; they can be rephrased as objective truths if we specify the point of reference alongside the statement. But without this specification, they are neither true nor false absolutely, but incomplete. This is true for spatial relationships, it is true for different inertial frames (for example the perceived colour of something depends on whether the light is blue or red shifted). Why can it not also be true for temporal statements?

   Nor is it necessary to tie the condition that the statement is true to the utterance of any sentence. The conditions under which the statement is true or false depend on a point in space time. For every point in space time in the future light cone of Socrates' death, it is true that he died. It doesn't matter whether somebody is uttering or thinking about it: he is still dead. For every point in the past light cone, it is true that he will die. For every point neither in the future or past light cone, then it is additionally dependent on the coordinate system which is used.

   Note that this is not just desperately explaining something away. Conditional tensed statements are precisely what one expects to see in the form of the B theory which I am advocating.

2. Conscious experience taken at face value also points to there being something special about the present and the passing of time. The present appears to have a brief duration rather than being an instant. It is an experience of change. It is continuous and ongoing. It is directed towards the future. Professor Feser refutes several attempts to deny these four points, which I will pass over since I would agree with them, albeit with some reservations about his discussion of whether the present has a duration or is a succession of durationless instants. This is similar to questions of continuity in mathematics. We would start by dividing time into small segments of non-zero duration; perform whatever calculations we need, and then take the limit that the duration tends to zero (and the number of these slices becomes infinite). This would not really be a durationless treatment of time (since we start by dividing it into slices of a fixed duration), but nor is it really the picture where we experience the present as a duration of time, since we finish by taking the limit that the duration of the present tends towards zero.

   I would also agree that this experience refutes those forms of the B theory which deny that time has a direction or that moments pass in succession. It does not rule out those formulations of the B theory which state that there is a direction and succession in time, but all moments in time are equally real, and there is no objective notion of the present. I cannot see how our experience of there being a present can distinguish between metaphysical approaches where the experience of the present is an objective feature of reality, or only a subjective experience of beings in time. From both of these viewpoints, one would deduce that we would experience the passing of time as Professor Feser describes it.

   Professor Feser's response to arguments similar to this is to state that the B-theory denies that there is any dynamic change or temporal passage anywhere, which would deny beings in time the possibility to experience it. One cannot relegate the experience of change to merely an illusion of the mind, since the mind itself exhibits the passage of time, even while performing the chains of reasoning by which people try to deny change. Nor can we easily confine the experience of the passing of time to the mind; we don't merely experience change ourselves, but observe it outside ourselves, if nothing else as our perspective shifts. We can relate previous experiences to present ones, and predict future experiences, while noting the differences in the experiences. I think that Professor Feser's argument is compelling to show that the passage of time is a real feature of the world. So this argument works well against those formulations of the B theory which he criticises. But I have been argueing is that there is an alternative four dimensional viewpoint which does accept change and temporal passage. The argument from experience says nothing against this.

Professor Feser's arguments for the A theory of time are thus arguments for the passage and direction of time. They simply do not address the four dimensional approach which I advocate.

Conclusion

I have argued that the symmetries underlying contemporary physics are inconsistent with presentism and the growing block theory. The moving spotlight theory also has problems, though to a lesser extent. However, modern physics also confirms the time ordering of causes, and, through the positivity of energy (energy is the eigenvalue of the operator describing changes in time), that the flow of time (unlike the flow of space) has a direction. This conclusion is confirmed by our own experience. Thus the version of the B theory discussed by Professor Feser, which denies the reality of change or the direction of time, is also ruled out. But this leaves (at least) one more option: a four dimensional approach which acknowledges the passage of time, but denies that there is an objective notion of one time as the present. Rather, the experience of the present is merely a subjective aspect of the world, experienced by those beings within time. The denial of an objective notion of the present makes this a B theory, but not the sort of B theory discussed by the philosophers (or at least those who Professor Feser addressed).